Multivariable Operator Theory: The Jörg Eschmeier Memorial Volume 3031505344, 9783031505348

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Ernst Albrecht Raúl Curto Michael Hartz Mihai Putinar Editors

Multivariable Operator Theory The Jörg Eschmeier Memorial Volume

Ernst Albrecht • Raúl Curto Michael Hartz • Mihai Putinar Editors

Multivariable Operator Theory The Jörg Eschmeier Memorial Volume

Previously published in Complex Analysis and Operator Theory “Special Issue: Multivariable Operator Theory” Volume 16, Issues 18, 2022 and Volume 17, Issues 18, 2023

Editors Ernst Albrecht Fachrichtung Mathematik Universität des Saarlandes Saarbrücken, Germany

Raúl Curto Department of Mathematics University of Iowa Iowa City, IA, USA

Michael Hartz Fachrichtung Mathematik Universität des Saarlandes Saarbrücken, Germany

Mihai Putinar Department of Mathematics University of California, Santa Barbara Santa Barbara, CA, USA

Spin-off from the journal Complex Analysis and Operator Theory. Vol. 16, Issues 18, 2022 and Vol. 17, Issues 18, 2023. ISBN 978-3-031-50534-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Contents

Jörg Eschmeier’s Mathematical Work .................................................................. 1 E. Albrecht, R. E. Curto, M. Hartz and M. Putinar: Complex Analysis and Operator Theory 2023, 2022: 17:3 (22 November 2022) https://doi.org/10.1007/s11785-022-01290-z Rochberg’s Abstract Coboundary Theorem Revisited ...................................... 19 C. Badea and O. Devys: Complex Analysis and Operator Theory 2022, 2022: 16:115 (3 November 2022) https://doi.org/10.1007/s11785-022-01293-w Dilation Theory and Functional Models for Tetrablock Contractions ............. 37 J. A. Ball and H. Sau: Complex Analysis and Operator Theory 2023, 2023: 17:25 (11 January 2023) https://doi.org/10.1007/s11785-022-01282-z Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results ............................................................................................. 77 W. Bauer and M. A. Rodriguez Rodriguez: Complex Analysis and Operator Theory 2022, 2022: 16:77 (30 June 2022) https://doi.org/10.1007/s11785-022-01248-1 A Question About Invariant Subspaces and Factorization .............................. 115 H. Bercovici and W. S. Li: Complex Analysis and Operator Theory 2022, 2022: 16:33 (5 March 2022) https://doi.org/10.1007/s11785-021-01183-7 Dilations and Operator Models of W-Hypercontractions ................................ 119 M. Bhattacharjee, B. Krishna Das, R. Debnath and S. Panja: Complex Analysis and Operator Theory 2023, 2023: 17:22 (6 January 2023) https://doi.org/10.1007/s11785-022-01314-8 The Joint Spectrum for a Commuting Pair of Isometries in Certain Cases ................................................................................................... 155 T. Bhattacharyya, S. Rastogi and U. Vijaya Kumar: Complex Analysis and Operator Theory 2022, 2022: 16:83 (18 July 2022) https://doi.org/10.1007/s11785-022-01257-0 Geometric Invariants for a Class of Submodules of Analytic Hilbert Modules Via the Sheaf Model................................................................ 195 S. Biswas, G. Misra and S. Sen: Complex Analysis and Operator Theory 2023, 2022: 17:2 (19 November 2022) https://doi.org/10.1007/s11785-022-01300-0

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Approximation in the Mean on Rational Curves .............................................. 219 S. Biswas and M. Putinar: Complex Analysis and Operator Theory 2022, 2022: 16:73 (24 June 2022) https://doi.org/10.1007/s11785-022-01254-3 Composition Operators on Function Spaces on the Halfplane: Spectra and Semigroups ..................................................................................... 229 I. Chalendar and J. R. Partington: Complex Analysis and Operator Theory 2023, 2023: 17:44 (8 April 2023) https://doi.org/10.1007/s11785-023-01348-6 Weighted Join Operators on Directed Trees ..................................................... 243 S. Chavan, R. Gupta and K. B. Sinha: Complex Analysis and Operator Theory 2023, 2023: 17:36 (2 March 2023) https://doi.org/10.1007/s11785-023-01334-y A Note on Joint Spectrum in Function Spaces .................................................. 345 P. Cui and R. Yang: Complex Analysis and Operator Theory 2023, 2023: 17:81 (16 July 2023) https://doi.org/10.1007/s11785-023-01383-3 Left-Invertibility of Rank-One Perturbations................................................... 361 S. Das and J. Sarkar: Complex Analysis and Operator Theory 2022, 2022: 16:109 (28 October 2022) https://doi.org/10.1007/s11785-022-01295-8 A-Isometries and Hilbert-A-Modules Over Product Domains......................... 383 M. Didas: Complex Analysis and Operator Theory 2022, 2022: 16:71 (18 June 2022) https://doi.org/10.1007/s11785-022-01243-6 Conjugations on L2( T N ) and Invariant Subspaces ........................................... 407 P. Dymek, A. Płaneta and M. Ptak: Complex Analysis and Operator Theory 2022, 2022: 16:104 (2 October 2022) https://doi.org/10.1007/s11785-022-01251-6 Best Approximations in a Class of Lorentz Ideals ............................................ 419 Q. Fang and J. Xia: Complex Analysis and Operator Theory 2022, 2022: 16:51 (2 April 2022) https://doi.org/10.1007/s11785-022-01220-z Szegö’s Theorem on Hardy Spaces Induced by Rotation-Invariant Borel Measures .................................................................................................... 445 K. Guo and Q. Zhou: Complex Analysis and Operator Theory 2022, 2022: 16:45 (25 March 2022) https://doi.org/10.1007/s11785-022-01218-7 On the Structure of Conditionally Positive Definite Algebraic Operators ............................................................................................ 469 Z. J. Jabłoński, I. B. Jung and J. Stochel: Complex Analysis and Operator Theory 2022, 2022: 16:90 (5 August 2022) https://doi.org/10.1007/s11785-022-01265-0 Truncations of Random Unitary Matrices Drawn from Hua-Pickrell Distribution .......................................................................... 491 Z Lin, Y. Qiu and K. Wang: Complex Analysis and Operator Theory 2023, 2022: 17:6 (5 December 2022) https://doi.org/10.1007/s11785-022-01306-8 Five Hilbert Space Problems in Operator Algebras ......................................... 515 L. W. Marcoux: Complex Analysis and Operator Theory 2022, 2022: 16:116 (3 November 2022) https://doi.org/10.1007/s11785-022-01296-7

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Schanuel’s Lemma for Exact Categories ........................................................... 531 M. Mathieu and M. Rosbotham: Complex Analysis and Operator Theory 2022, 2022: 16:76 (28 June 2022) https://doi.org/10.1007/s11785-022-01250-7 Riesz–Kolmogorov Type Compactness Criteria in Function Spaces with Applications..................................................................................... 543 M. Mitkovski, C. B. Stockdale, N. A. Wagner and B. D. Wick: Complex Analysis and Operator Theory 2023, 2023: 17:40 (25 March 2023) https://doi.org/10.1007/s11785-023-01346-8 The Cauchy Transform of the Square Root Function on the Circle ............... 575 R. Mortini and R. Rupp: Complex Analysis and Operator Theory 2022, 2022: 16:38 (19 March 2022) https://doi.org/10.1007/s11785-022-01224-9 Function Theory from Tensor Algebras ............................................................ 587 P. S. Muhly and B. Solel: Complex Analysis and Operator Theory 2022, 2022: 16:92 (18 August 2022) https://doi.org/10.1007/s11785-022-01262-3 Operator Theory on Noncommutative Polydomains, I .................................... 621 G. Popescu: Complex Analysis and Operator Theory 2022, 2022: 16:50 (2 April 2022) https://doi.org/10.1007/s11785-022-01225-8 Radial-like Toeplitz Operators on Cartan Domains of Type I ........................ 723 R. Quiroga-Barranco: Complex Analysis and Operator Theory 2022, 2022: 16:75 (27 June 2022) https://doi.org/10.1007/s11785-022-01258-z A Short Proof for the Twisted Multiplicativity Property of the Operator-Valued S-transform ................................................................. 761 R. Speicher: Complex Analysis and Operator Theory 2022, 2022: 16:81 (11 July 2022) https://doi.org/10.1007/s11785-022-01259-y Spectral Analysis Near Regular Point of Reducibility and Representations of Coxeter Groups ............................................................ 769 M. I. Stessin: Complex Analysis and Operator Theory 2022, 2022: 16:70 (17 June 2022) https://doi.org/10.1007/s11785-022-01244-5 Stratified Hilbert Modules on Bounded Symmetric Domains ......................... 799 H. Upmeier: Complex Analysis and Operator Theory 2023, 2023: 17:74 (5 July 2023) https://doi.org/10.1007/s11785-023-01377-1 Functions and Operators in Real, Quaternionic, and Cliffordian Contexts..................................................................................... 833 F. -H. Vasilescu: Complex Analysis and Operator Theory 2022, 2022: 16:117 (5 November 2022) https://doi.org/10.1007/s11785-022-01292-x Yet Another Approach to Poly-Bergman Spaces .............................................. 861 N. Vasilevski: Complex Analysis and Operator Theory 2022, 2022: 16:74 (27 June 2022) https://doi.org/10.1007/s11785-022-01252-5 Quantum Permutation Matrices ........................................................................ 875 M. Weber: Complex Analysis and Operator Theory 2023, 2023: 17:37 (15 March 2023) https://doi.org/10.1007/s11785-023-01335-x

Complex Analysis and Operator Theory (2023) 17:3 https://doi.org/10.1007/s11785-022-01290-z

Complex Analysis and Operator Theory

Jörg Eschmeier’s Mathematical Work Ernst Albrecht1 · Raúl E. Curto2 · Michael Hartz1 · Mihai Putinar3,4 Received: 22 July 2022 / Accepted: 12 October 2022 / Published online: 22 November 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract An outline of Jörg Eschmeier’s main mathematical contributions is organized both on a historical perspective, as well as on a few distinct topics. The reader can grasp from our essay the dynamics of spectral theory of commutative tuples of linear operators during the last half century. Some clear directions of future research are also underlined.

Introduction Jörg Eschmeier grew up mathematically in the vibrant atmosphere of late XX-th Century German Modern Analysis. His doctoral advisor, Heinz Günther Tillmann was a scientific grandson of Otto Toeplitz. He instilled with high competence in Jörg a lifelong fascination with distribution theory and general duality theory in locally convex spaces. This happened around the late 1970-ies at the University of Münster, where also George Maltese offered Jörg the challenge of an active research group. At the same time and in the same place, function theory of several complex variables was blooming, turning Münster into one of the leading world centers on the subject. Jörg was exposed early on during his studies to analytic techniques, homological algebra methods and geometric interpretations of the intricate nature of holomorphic functions of several complex variables. Throughout his brilliant career, he masterfully combined these two main streams of his student years. We collect below a few pointers to Jörg

M.H. was partially supported by a GIF grant and by the Emmy Noether Program of the German Research Foundation (DFG Grant 466012782). M.P. was partially supported by a Simons Foundation collaboration grant. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial" edited by Ernst Albrecht, Raúl Curto, Michael Hartz and Mihai Putinar. Communicated by Daniel Alpay.

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Mihai Putinar [email protected]; [email protected]

Extended author information available on the last page of the article

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Eschmeier’s highly original discoveries. Our text merely touches upon his deep impact on contemporary operator theory, conveying a fraction of his superb scientific orientation and elegant style of pursuing research.

1 Duality and Spectral Decompositions The reader of this essay has unquestionably been fascinated by some sort of spectral decompositions. It is said that John von Neumann was asked by reporters on his death bed what the highest scientific discovery of his outstanding career was. To the stupefaction of all he mentioned the spectral theorem for unbounded self-adjoint operators, an esoteric concept for the layman, leaving aside his contributions to mathematical economics, game theory, nuclear weapons, electronic computers, fluid mechanics and much more.

Jörg in 2018 The ubiquitous eigenvector of a finite, self-adjoint matrix has to be replaced when dealing with infinite self-adjoint matrices carrying a continuous spectrum by a subspace of vectors representing a localized window in the spectrum. This necessary step in mathematical spectral analysis stirred countless discussions among quantum physicists. Generalized eigenfunctions were proposed by them as a natural substitute, obtained however with the price of stepping outside the original Hilbert space. Dirac’s generalized function δ stands aside in this respect. On a totally perpendicular direction and in full resonance with the Bourbaki tendencies of the day, groups of mathematicians explored in the second part of 20th Century several comprehensive axiomatic approaches to spectral decomposition. The third volume of Dunford and Schwartz monumental monograph [112] contains ample references on this forgotten chapter of operator theory. We owe to the genius of Erret

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Bishop [99] the leap forward and to the unmatched insight of Foia¸s [119] the foundational concept of decomposable operator. Let X be a Banach space over the complex numbers and let T ∈ L(X ) be a linear bounded operator acting on X . The operator T is called decomposable if, for every finite open cover of its spectrum, σ (T ) = σ (T , X ) ⊂ U1 ∪ U2 ∪ . . . ∪ Un there are T -invariant subspaces X 1 , X 2 , . . . , X n with localized spectra of the respective restrictions of T : σ (T , X j ) ⊂ U j , 1 ≤ j ≤ n, and spanning the whole space: X = X 1 + X 2 + · · · + X n . Normal operators on Hilbert space, compact operators, and representations of functions algebras carrying a partition of unity are all decomposable.

1988 Timisoara Operator Theory Conference. From left to right: Sasha Helemskii, Mihai Putinar, Jörg Eschmeier, Henry Helson At the beginning of a theory, variants of the definitions and permanence problems usually have to be clarified. In the original definition of a decomposable operator Foia¸s in [119] required the subspaces X j to be spectral maximal and in the last chapter of their monograph [104] Colojoar˘a and Foia¸s ask if the restrictions of decomposable operators are again decomposable. As Jörg told one of us (E.A.), he first came in touch with the theory of decomposable operators in a student seminar of Tillman in 1977/78, which was announced by his assistant Erich Marschall as follows: “Here is a Lecture Notes volume [114] with a positive answer to that question and a manuscript [90] with a negative answer. Let us find out which one is correct.” In [73] Jörg published a further class of examples which are much more natural and less technical than the example in [90]. After this seminar he wrote his (unpublished) Diploma Thesis [60] on local decomposability (in the sense of Vasilescu [133, 134]) and functional calculi for closed linear operators on Banach spaces. In particular (using methods from [91]) he

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already obtained most of the characterization of such operators described in Theorem IV.4.26 of [135]. With the proper definition of joint spectrum σ (τ, X ) of a tuple of commuting linear bounded operator τ proposed by Taylor [131], the notion of decomposability carries over, but the challenging questions and complications multiply. Jörg Eschmeier’s dissertation of 1981 [61], summarized in his Inventiones article [63], is devoted to the study of spectral localization within the novel, at that time, Taylor analytic functional calculus. His lucid insight builds on solid ground well cultivated in Germany at that time, notably an earlier contribution of one of us (E.A.) [89], who is incidentally a mathematical nephew of Jörg via the filiation Tillmann-Gramsch. Jörg navigates there with ease and efficiency through the formidable Cauchy-Weil integral representation formulas invoked by Taylor. He would then continue for two good decades to explore the subject. But progress was not possible without Bishop’s ideas. In a nutshell, Bishop claimed that it is not generalized eigenvectors à la Dirac’s distribution that illuminates general spectral decomposition behavior, but analytic functionals. Fortunately, the necessary passage from Schwartz distributions to analytic functionals or hyperfunctions was much explored by mathematical analysts of the 1980-s decade. To be more specific, in order to better understand the spectral characteristics of the linear bounded operator T ∈ L(X ), Bishop points to the space of X -valued analytic functions defined on an open set U , O(U ) X , and the linear pencil: z I − T : O(U ) X −→ O(U ) X . The resolvent of T , restricted to values of z outside the spectrum, is lurking around. The operator T is said to have the single valued extension property if the map z I − T above is injective for every open set U . In that case a localized spectrum of T with respect to a vector makes sense. The operator T satisfies Bishop’s condition (β) if the map z I − T is injective with closed range for every open set U . A timely 1983 observation of one of us (M.P.) [125] interprets these notions in terms of sheaf theory: FT (U ) = coker(z I − T : O(U ) X −→ O(U ) X ) = O(U ) X )/(z I − T )O(U ) X , is an analytic sheaf if T has the single valued extension property and it is an analytic sheaf of Fréchet spaces if T satisfies property (β). Note that the canonical identification FT (C) = X , T = Mz , takes place, with the operator T represented by multiplication by the complex variable. Moreover, the spectrum of T is in this case equal to the support of the sheaf model FT , the support of a section (a.k.a. vector) x ∈ X is the local spectrum, and so on. The crucial insight came from the proof that T is decomposable if and only if the canonical sheaf model (and as a matter of fact any sheaf model) is soft [125]. Now, analytic sheaves were at home both at Münster and Bucharest. A fruitful collaboration between Jörg and Mihai was built on this rich ground, leading to a dozen joint publications.

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The rest is technique, sheaf cohomology back and forth, in the treacherous topological homological setting. Returning to Bishop [99], his clear message was to consider in parallel the nonpointwise behavior of the linear pencils z I −T and z I −T  , where T  is the topological dual of T . This is for the simple reason that elements of the dual of coker(z I −  T ) on O(U ) X are analytic functionals φ ∈ O(U )X which fulfill the generalized eigenvector equation (z I − T  )φ = 0. Playing this distribution theory game at the abstract multidimensional level bares fruit: (1) A commutative tuple τ of linear bounded operators satisfies Bishop’s property (β) if and only if admits a quasi-coherent analytic, Fréchet sheaf model; (2) A commutative tuple τ of linear bounded operators is decomposable if and only if both τ and τ  satisfy property (β); (3) A commutative tuple τ of linear bounded operators is the restriction of a decomposable tuple to a joint invariant subspace if and only if τ satisfies property (β); (4) Two quasi-similar tuples of operators subject to property (β) have equal joint spectra and equal essential joint spectra; (5) Division of distributions by complex analytic functions follows from a Bishop’s type property with respect to smooth functions; (6) The functoriality of the sheaf model of a commutative tuple of linear bounded operators implies Riemann-Roch Theorem on singular analytic spaces. Some of the definitive results enumerated above were obtained in several increments, each a source of joy and bewilderment [2, 30, 32, 79, 127]. The monograph [37] collects such advances obtained until early 1990-ies and unified by the concept of analytic sheaf model. A decade later, the monograph by Laursen and Neumann [123] offered a complementary, cohomology free account of local spectral theory (in the case of single operators).

Jörg in 2019

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2 Invariant Subspaces The famous results and in particular the methods of proof in Scott Brown’s work [100, 101] on invariant subspaces for subnormal operators and hyponormal operators with thick spectrum on Hilbert spaces had a great impact for the further development in the research on the invariant subspace problem. In [101], Scott Brown used the fact (due to [126]) that a hyponormal operator T is subscalar (and hence subdecomposable) to obtain nontrivial invariant subspaces for T when the spectrum σ (T ) is thick in the sense that for some non empty set G ⊂ C the set σ (T ) ∩ G is dominating in G. He also noticed that subdecomposability is sufficient for his result. In [75] Jörg obtained a first result for general Banach spaces: If T ∈ L(X ) is subscalar with int(σ (T )) = ∅, then T has nontrivial invariant subspaces. By [2], subdecomposability is equivalent to Bishop’s property (β), which is a local property and has a nice duality theory. Using these results Jörg removed restrictions on the operators or on the underlying Banach spaces to obtain some first localized invariant subspace results. Finally, in joint work with Bebe Prunaru [34], he proved that every continuous linear operator T on an arbitrary Banach space X = {0} which, for some compact set S ⊂ C, satisfies (β) or the dual property (δ) on C\S and for which there is a bounded open set V ⊂ C with S ∩ V = ∅ or S ⊂ V , then the following holds: (1) If σ (T ) is dominating in V than T has a nontrivial invariant subspace. (2) If the essential spectrum of T is dominating in V then the lattice Lat(T ) of invariant subspaces is rich. Even more generally, the authors showed in [27] a corresponding result for operators T for which the localizable spectrum of T or of T  is thick (in the above mentioned sense), has a non-trivial invariant subspace. Notice, that some thickness condition on the spectrum is necessary, as there exists an example due to Charles Read [128] of a quasinilpotent and hence even decomposable operator on some Banach space without any non-trivial invariant subspaces. Following the success of the Scott Brown technique for single operators, Jörg set out to extend these ideas to tuples of commuting operators. The natural goal is to prove that any contractive tuple of commuting operators with sufficiently rich spectrum has non-trivial invariant subspaces. In the multivariate setting, several difficulties arise. As mentioned before, spectral theory becomes substantially more difficult. Moreover, just like there is more than one generalization of the unit disc to higher variables, there are several reasonable notions of contractive tuples of operators. Finally, the Sz.-Nagy– Foia¸s H ∞ -functional calculus, which is a crucial ingredient in the classical Scott Brown technique, is generally not available in the multivariable setting. In [82], Jörg proved that every commuting row contraction that possesses a spherical dilation and whose Harte spectrum is dominating in the ball admits non-trivial invariant subspaces. Along the way, he established an H ∞ (Bd )-functional calculus, which itself inspired further research approximately 20 years later [98, 103]. He also established a version of this theorem on the polydisc [44]. In the one variable setting, the Scott Brown technique was refined by Olin and Thomson [124] to show that a subnormal operator T not only admits nontrivial invari-

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ant subspace, but is even reflexive. This means that the weak operator topology closed algebra generated by T can be completely recovered from its invariant subspace lattice. Jörg successfully established variants of this result in the multivariable setting [9, 43, 46, 48, 84]. The difficulty of this subject can perhaps be appreciated from the fact that the question of whether every subnormal operator tuple is reflexive remains open to this day.

Jörg in 2009, at the occasion of EA’s 65th birthday

3 Multivariable Operator Theory The interplay between functional analysis and function theory of several complex variables entered around 1970-ies into a new era thanks to Joseph Taylor’s innovative insights, in particular his novel joint spectrum and the topological-homological approach to functional calculi [131, 132]. Jörg was a central figure and inspired contributor to this new chapter of modern analysis. His many notable contributions would fill an entire volume, way beyond the size of the present biographical note. We outline a couple of snapshots. In duality with the fibre product of complex spaces, the tensor product of analytic modules is inviting at analyzing the blend of spectral behaviors of its factors. Two traditional constructions of operator theory stand aside in this respect. More specifically, consider a pair of Banach spaces X , Y and commutative tuples of linear bounded operators acting on them S ∈ L(X )m and T ∈ L(Y )n . With a fixed crossnorm, the completed tensor product X ⊗ Y carries the commutative (n + m)-tuple τ = (S ⊗ I , I ⊗ T ). One of Jörg’s early works [74] (see also the third chapter in his Habilitation Thesis [72]) offers a complete evaluation of the joint spectrum of τ , its joint essential spectrum, and the values of the Fredholm index. A second natural path of melting tuples of linear transforms into new ones has to do with the so-called elementary operators. Namely, let J be a bilateral ideal in the space of linear bounded operators L(K , H ) acting between two Hilbert spaces K , H . Let S ∈ L(H )n and

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T ∈ L(K )n be tuples of commuting elements. The operator n  S j AT j , R : J −→ J , R(A) = j=1

is an elementary operator. A complete spectral picture of R has quite surprising implications, for instance to solving non-commutative operator equations relevant to control and stability theory of systems. The spectrum, essential spectrum, and Fredholm index of R were first computed by one of us (R.E.C.) together with Lawrence A. Fialkow [105, 106], and further refined under the assumption of an intricate geometric condition, known as the “finite fibre property" [118]. The note [36] shows, employing cohomology and analytic localization techniques, that the finite fibre property is always satisfied. The investigations of tensor products of analytic modules and separately of elementary operators continue to flourish. As defined, Taylor’s joint spectrum of a commutative tuple of linear operators is quite elusive, requiring a grasp of higher torsion spaces of a pair of analytic modules. Cohomologically trivial cases are in this sense a treat, as much as, keeping the proportion, pseudoconvex domains are among all domains in Cd . For this reason, a collection of simple examples is both precious and inspiring. Take for instance a bounded pseudoconvex domain  in Cd and a finite system of bounded analytic functions defined on : f = ( f 1 , f 2 , . . . , f n ). The note [35] contains a description of Taylor’s joint spectrum of f , as acting on Bergman space L a2 (): σ ( f , L a2 ()) = f (). In the same setting, assuming the boundary of  is smooth, strictly pseudoconvex, the joint essential spectrum is σe ( f , L a2 ()) =



f (U ∩ ),

U

where U runs over all open neighborhoods of ∂. These results can be interpreted as solutions to division problem with L 2 -bounds, in the spirit of the celebrated Corona Problem. Bounded analytic interpolation in several complex variables is significantly more challenging than the classical Nevanlinna-Pick or Carathéodory-Fejér 1-dimensional problems. A major breakthrough was recorded in the 1990-ies with the isolation of the Schur-Agler class of analytic multipliers on a Hilbert space of analytic functions with a reproducing kernel. It was Jim Agler who recognized at that time that this new algebra provides the correct analog for stating and proving multivariate analogs of the quasi-totality of known bounded analytic interpolation results in one dimension [88]. Jörg entered into the first line of avant-garde researchers charting this new territory, as for instance, the articles [28, 29, 39] amply illustrate. A commuting tuple T = (T1 , . . . , Tn ) of operators is said to be Fredholm if the cohomology groups H p (T ) of the Koszul complex are all finite dimensional. The dimensions of H p (T ) and related objects naturally carry operator theoretic meaning. It turns out that they can also be related to certain analytical quantities. In his paper

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[49], Jörg showed that the limits limk→∞ dim H p (T k )/k n exist and in fact agree with the so-called Samuel multiplicities of stalks of cohomology sheaves; moreover, they agree with the generic dimension of H p (z − T ) near z = 0. This result employs a beautiful blend of operator theory, commutative algebra and analytic geometry. Jörg continued his investigations in [47, 51]. The model theory of Sz.-Nagy and Foia¸s yields a complete unitary invariant for (completely non-unitary) contractions, namely the characteristic function. This is closely related to the theory of the Hardy space on the disc and classical inner functions. Jörg, together with several coauthors, made advances into a corresponding multivariable theory [6] and into setting of more general reproducing kernel Hilbert spaces [4, 58, 85]. Another topic in which Jörg was active is the characterization of the essential commutant of the analytic Toeplitz operators. A theorem of Davidson [107] shows that, on the Hardy space, an operator T commutes modulo the compacts with every analytic Toeplitz operator if and only if T is a compact perturbation of a Toeplitz operator with symbol in H ∞ + C. Very general extensions to the multivariable setting were given by Eschmeier and coauthors in [16, 17, 20].

Jörg in 2019

4 Arveson-Douglas Conjecture Classically, there is an intimate relationship between the study of contraction operators on Hilbert space and the Hardy space H 2 on the unit disc. In the theory of tuples of commuting operators on a Hilbert space, the appropriate generalization of the Hardy space is the Drury–Arveson space Hd2 , see [93]. This is the reproducing kernel Hilberts space of analytic functions on the Euclidean unit ball Bd in Cd with reproducing kernel K (z, w) =

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Given Jörg’s experience in multivariable operator theory and in several complex variables, it is no surprise that he made important contributions to this subject early on [28]. Since the coordinate functions z 1 , . . . , z d are multipliers of Hd2 , the space Hd2 becomes a module over the polynomial ring. To each (closed) submodule M ⊂ Hd2 , one associates the operators S j : M ⊥ → M ⊥,

f → PM ⊥ (z j f ),

which are commuting linear operators on M ⊥ . A very influential conjecture, made by Arveson [92] and refined by Douglas [110], asserts that if M is the closure of a homogeneous ideal I of polynomials, then the cross commutators [S j , Sk∗ ] belong to the Schatten class S p for all p > dim Z (I ), where Z (I ) denotes the zero set of I . This conjecture has attracted a large amount of attention. The initial motivation for the Arveson–Douglas conjecture came from Arveson’s work on the curvature invariant of certain operator tuples [92], but the conjecture turned out to be interesting for other reasons as well. For instance, if I is a homogeneous ideal of infinite co-dimension so that the Arveson–Douglas conjecture holds for M = I , then the quotient of the Toeplitz C ∗ -algebra associated with M modulo the compacts is commutative. In fact, the quotient is isomorphic to C(Z (I ) ∩ ∂Bd ). As Douglas observed, this gives rise to a K -homology element of the space Z (I ) ∩ ∂Bd [110]. After the Arveson–Douglas conjecture was formulated, it was verified in a number of special cases. Jörg reduced the conjecture to a certain operator inequality [53]. This led to a unified proof of all cases in which the conjecture was known to hold at the time. A few years later, Jörg Eschmeier and Miroslav Engliš achieved a spectacular breakthrough by showing that the conjecture holds if the homogeneous ideal I is the vanishing ideal of a homogeneous variety that is smooth away from the origin [19]. A similar, related result was independently obtained by Douglas, Tang and Yu around the same time [113]. These works have inspired a lot of research, and the area remains very active to this day.

5 Teaching and Mentoring Jörg Eschmeier was a very highly regarded teacher and mentor. His lectures were widely known for their clarity and precision. They were frequently attended by a large number of students, and even his more advanced courses in functional analysis and complex analysis drew in students from outside of mathematics. At Saarland University, he won the award for the best lecture in mathematics five times, more often than any of his colleagues in the department. In addition, Jörg was an extremely dedicated mentor for bachelor’s, master’s and doctoral theses. He was very generous with his time and invested considerable effort into advising students. A particular gem among Jörg’s lectures were those about several complex variables. His lecture notes formed the basis for his book [57]. The book stands out in that it provides a self-contained treatment of important theorems in several complex vari-

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ables, assuming only basic undergraduate analysis as a prerequisite. The book starts with basic properties of holomorphic functions in several variables, treats analytic sets and domains of holomorphy, proves Oka’s theorems and explains the solution of the Levi problem. The last chapter contains beautiful applications to functional analysis, namely the Arens-Calderón functional calculus, Shilov’s idempotent theorem and the Arens-Royden theorem. Through his lectures and his book, Jörg made this fascinating subject accessible to many students. Another very nice example of informative and well structured writing are his lectures on invariant subspaces contained in Part III of [7]. Already during his period as an Alexander von Humboldt Fellow, Jörg was involved in the mentoring of Roland Wolff, a doctoral student of George Maltese, whom he introduced to the field of Bergman and Hardy spaces in several variables. Roland Wolff finished his thesis Spectral theory on Hardy spaces in several complex variables [137] in Saarbrücken, where he became the first assistent of Jörg. Part of this theses is also published in [138]. The following is a list of all completed doctoral theses written under the mentorship of Jörg at Saarbrücken:

(1) Michael Didas, On the structure of von Neumann n-tuples over strictly pseudoconvex sets [108]. Michael Didas considers operator tuples T = (T1 , . . . , Tn ) on a Hilbert space H admitting a contractive functional calculus T : A(D) → L(H ), where D is a strictly pseudoconvex subset of a Stein submanifold of Cn which admit a ∂ D-unitary dilation. Using dual algebra methods he obtains reflexivity results and invariant subspace results for large classes commuting operator tuples T = (T1 , . . . , Tn ) on a Hilbert space with dominating Harte- or Taylor spectra in D. The main parts of this thesis have been published in [109]. Though Michael Didas left the university he always stayed in contact with Jörg and they published seven joint articles. (2) Eric Réolon, Zur Spektraltheorie vertauschender Operatortupel: Fredholmtheorie und subnormale Operatortupel [129]. In his thesis Eric Réolon obtains a Banach space variant of a Fredholm index formula for essentially normal tuples given in [37], Theorem 10.3.15. He also shows that an operator tuple on a Hilbert space is essentially subnormal if and only if it has an essentially normal extension and if and only if it has an extension to a compact perturbation of a normal tuple. For single operators this had been shown by N.S. Feldman in [117]. (3) Christoph Barbian, Beurling-Type Representation of Invariant Subspaces in Reproducing Kernel Hilbert Spaces [94]. Christoph Barbian studies invariant subspaces of reproducing kernel Hilbert spaces, including the difficult case of the Bergman space on the unit disc. He introduces the notion of Beurling decomposability, and obtains criteria for this property to hold. Among other things, this has implications for multivariable spectra of multiplication tuples. This thesis as well as the following ones is available for download. For further developments see also [95–97]. (4) Dominik Faas, Zur Darstellungs- und Spektraltheorie für nichtvertauschende Operatortupel [116]. This thesis is related to Jörg’s work on Samuel multiplici-

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(5)

(6)

(7)

(8)

(9)

ties. A local closed range theorem for semi-Fredholm valued functions was later improved to a global version by Dominik Faas and Jörg in [21]. Kevin Everard, A Toeplitz projection for multivariable isometries [115]. For compact sets K ⊂ Cn and closed subalgebras A of C(K ) Jörg introduced in [46] the notion of an A-isometry. This class of commuting n-tuples of operators includes spherical isometries and n-tuples of commuting isometries. This allowed Jörg together with Michael Didas and Kevin Everard to introduce associated analytic Toeplitz operators [16]. The thesis of Kevin Everard completes that approach (see also [20]). Michael Wernet, On semi-Fredholm theory and essential normality [136]. Michael Wernet contributes to four areas of Jörg’s interests: He extends Jörg’s results of [50], he shows that a number of positive results on the Arveson-Douglas conjecture can be extended to arbitrary graded Hilbert modules and that the validity of the conjecture is equivalent for a large class of analytic functional Hilbert spaces, he generalizes an essential von Neumann inequality of Matthew Kennedy and Orr Shalit ( [121], Theorem 6.1), and using a result of [21] answers a question by Ronald Douglas ( [111], Question 1) and (extending some results of Jörg and Johannes Schmitt [40]) gives a partial answer to another question of Ronald Douglas in [111], Question 3. Dominik Schillo, K-contractions, and perturbations of Toeplitz operators [130]. For many analytic Hilbert function spaces with reproducing kernels K Dominik Schillo obtains model theorems for K -contractions. In a second part he studies Toeplitz operators associated with regular A-isometries and uses methods from [16] to characterize finite rank and Schatten-p-class perturbations of analytic Toeplitz operators. Part of these results have also been published in [17]. Sebastian Langendörfer, On unitarily invariant spaces and Cowen-Douglas theory [122]. A rather general version of a characterization of Toeplitz operators with pluriharmonic symbols on unitarily invariant with appropriate reproducing kernel and an extension of results by Chang, Chen and Fang [102] to the several variable case are given. See also the joint works [22–24] of Sebastian Langendörfer with Jörg. Daniel Kraemer, Toeplitz operators on Hardy spaces [120]. A several variable Toeplitz operator theory is developed on Hardy type H p (G) spaces which is applicable for bounded symmetric domains and bounded strictly pseudoconvex domains. In particular, in the situation of strictly pseudoconvex domains he obtains a generalization of Jörg’s spectral mapping theorem from [55].

The high quality of these theses reflects the outstandig quality of Jörg Eschmeier as an academic teacher. Acknowledgements We are grateful to Steliana Eschmeier for her support in the preparation of this article. Author Contributions All authors contributed equally to this article. They all reviewed the manuscript before submission.

Declarations Conflict of interest The authors declare no competing interests.

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References Complete List of Jörg Eschmeier’s Works 1. Ambrozie, C., Eschmeier, J.: “A commutant lifting theorem on analytic polyhedra”. In: Topological algebras, their applications, and related topics. Vol. 67. Banach Center Publ. Polish Acad. Sci. Inst. Math., Warsaw, (2005), pp. 83–108. https://doi.org/10.4064/bc67-0-7 2. Albrecht, E., Eschmeier, J.: Analytic functional models and local spectral theory. Proc. Lond. Math. Soc. 75(3), 323–348 (1997). https://doi.org/10.1112/S0024611597000373 3. Albrecht, E., Eschmeier, J., Neumann, M.M.: “Some topics in the theory of decomposable operators”. In: Advances in invariant subspaces and other results of operator theory (Timi¸soara and Herculane, 1984). Vol. 17. Oper. Theory Adv. Appl. Birkhäuser, Basel, (1986), pp. 15–34 4. Bhattacharjee, M., Eschmeier, J., Keshari, D.K., Sarkar, J.: Dilations, wandering subspaces, and inner functions. Linear Algebra Appl. 523, 263–280 (2017). https://doi.org/10.1016/j.laa.2017.02.032 5. Bhattacharyya, T., Eschmeier, J., Sarkar, J.: Characteristic function of a pure commuting contractive tuple. Integral Equ. Oper. Theory 53(1), 23–32 (2005). https://doi.org/10.1007/s00020-004-1309-5 6. Bhattacharyya, T., Eschmeier, J., Sarkar, J.: On CNC commuting contractive tuples. Proc. Indian Acad. Sci. Math. Sci. 116(3), 299–316 (2006). https://doi.org/10.1007/BF02829747 7. Dales, H.G., Aiena, P., Eschmeier, J., Laursen, K., Willis, G.A.: Introduction to Banach algebras, operators, and harmonic analysis. Vol. 57. Cambridge: Cambridge University Press, 2003, pp. xi+324 8. Davidson, K.R., Douglas, R.G., Eschmeier, J., Upmeier, H. (Eds.): “Hilbert modules and complex geometry. Abstracts from the workshop held April 20-26, 2014”. English. In: Davidson, K.R., Douglas, R.G., Eschmeier, J., and Upmeier, H. (eds), Oberwolfach Rep. 11.2 (2014). pp. 1139–1219 9. Didas, M., Eschmeier, J.: Subnormal tuples on strictly pseudoconvex and bounded symmetric domains. Acta Sci. Math. (Szeged) 71(3–4), 691–731 (2005) 10. Didas, M., Eschmeier, J.: Unitary extensions of Hilbert A(D)modules split. J. Funct. Anal. 238(2), 565–577 (2006). https://doi.org/10.1016/j.jfa.2006.02.002 11. Didas, M., Eschmeier, J.: Inner functions and spherical isometries. Proc. Am. Math. Soc. 139(8), 2877–2889 (2011). https://doi.org/10.1090/S0002-9939-2011-11034-7 12. Douglas, R.G., Eschmeier, J.: “Spectral inclusion theorems”. In: Mathematical methods in systems, optimization, and control. Vol. 222. Oper. Theory Adv. Appl. Birkhäuser/Springer Basel AG, Basel, (2012), pp. 113–128. https://doi.org/10.1007/978-3-0348-0411-0_10 13. Didas, M., Eschmeier, J.: Derivations on Toeplitz algebras. Canad. Math. Bull. 57(2), 270–276 (2014). https://doi.org/10.4153/CMB-2013-001-9 14. Didas, M., Eschmeier, J.: Dual Toeplitz operators on the sphere via spherical isometries. Integral Equ. Oper. Theory 83(2), 291–300 (2015). https://doi.org/10.1007/s00020-015-2232-7 15. Douglas, R.G., Eschmeier, J., Upmeier, H.: (Eds.) “Hilbert modules and complex geometry. Abstracts from the workshop held April 5th–April 11th, (2009)”. English. In: Douglas, R.G., Eschmeier, J., and Upmeier, H. (eds), Oberwolfach Rep. 6.2 (2009). pp. 1047–1100 16. Didas, M., Eschmeier, J., Everard, K.: On the essential commutant of analytic Toeplitz operators associated with spherical isometries. J. Funct. Anal. 261(5), 1361–1383 (2011). https://doi.org/10. 1016/j.jfa.2011.05.005 17. Didas, M., Eschmeier, J., Schillo, D.: On Schattenclass perturbations of Toeplitz operators. J. Funct. Anal. 272(6), 2442–2462 (2017). https://doi.org/10.1016/j.jfa.2016.11.007 18. Engliš, M., Eschmeier, J.: “Corrigendum to “Geometric Arveson- Douglas conjecture” [Adv. Math. 274 (2015) 606-630] [ 3318162]”. In: Adv. Math. 278 (2015), p. 254. https://doi.org/10.1016/j.aim. 2015.04.004 19. Engliš, M., Eschmeier, J.: Geometric Arveson-Douglas conjecture. Adv. Math. 274, 606–630 (2015). https://doi.org/10.1016/j.aim.2014.11.026 20. Eschmeier, J., Everard, K.: Toeplitz projections and essential commutants. J. Funct. Anal. 269(4), 1115–1135 (2015). https://doi.org/10.1016/j.jfa.2015.03.017 21. Eschmeier, J., Faas, D.: Closed range property for holomorphic semi-Fredholm functions. Integral Equ. Oper. Theory 67(3), 365–375 (2010). https://doi.org/10.1007/s00020-010-1786-7 22. Eschmeier, J., Langendörfer, S.: Cowen-Douglas tuples and fiber dimensions. J. Operator Theory 78(1), 21–43 (2017). https://doi.org/10.7900/jot.2016may04.2134 23. Eschmeier, J., Langendörfer, S.: Multivariable Bergman shifts and Wold decompositions. Integral Equ. Oper. Theory 90(5), 56 (2018). https://doi.org/10.1007/s00020-018-2481-3

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E. Albrecht et al. 24. Eschmeier, J., Langendörfer, S.: Toeplitz operators with pluriharmonic symbol on the unit ball. Bull. Sci. Math. 151, 34–50 (2019). https://doi.org/10.1016/j.bulsci.2019.01.006 25. Eschmeier, J., Laursen, K.B., Neumann, M.M.: Multipliers with natural local spectra on commutative Banach algebras. J. Funct. Anal. 138(2), 273–294 (1996). https://doi.org/10.1006/jfan.1996.0065 26. Eschmeier, J., Putinar, M.: “Some remarks on spherical isometries”. In: Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000). Vol. 129. Oper. Theory Adv. Appl. Birkhäuser, Basel, 2001, pp. 271–291 27. Eschmeier, J., Prunaru, B.: Invariant subspaces and localizable spectrum. Integral Equ. Oper. Theory 42(4), 461–471 (2002). https://doi.org/10.1007/BF01270923 28. Eschmeier, J., Putinar, M.: Spherical contractions and interpolation problems on the unit ball. J. Reine Angew. Math. 542, 219–236 (2002). https://doi.org/10.1515/crll.2002.007 29. Eschmeier, J., Putinar, M.: “On bounded analytic extension in Cn ”. In: Spectral analysis and its applications. Vol. 2. Theta Ser. Adv. Math. Theta, Bucharest, (2003), pp. 87–94 30. Eschmeier, J., Putinar, M.: Spectral theory and sheaf theory. III. J. Reine Angew. Math. 354, 150–163 (1984) 31. Eschmeier, J., Putinar, M.: The sheaf ε is not topologically at over O. An. Univ. Craiova Mat. Fiz.Chim. 15, 1–4 (1987) 32. Eschmeier, J., Putinar, M.: Bishop’s condition (β ) and rich extensions of linear operators. Indiana Univ. Math. J. 37(2), 325–348 (1988). https://doi.org/10.1512/iumj.1988.37.37016 33. Eschmeier, J., Putinar, M.: On quotients and restrictions of generalized scalar operators. J. Funct. Anal. 84(1), 115–134 (1989). https://doi.org/10.1016/0022-1236(89)90113-4 34. Eschmeier, J., Prunaru, B.: Invariant subspaces for operators with Bishop’s property (β) and thick spectrum. J. Funct. Anal. 94(1), 196–222 (1990). https://doi.org/10.1016/0022-1236(90)90034-I 35. Eschmeier, J., Putinar, M.: Spectra of analytic Toeplitz tuples on Bergman spaces. Acta Sci. Math. 57(1–4), 83–100 (1993) 36. Eschmeier, J., Putinar, M.: The finite fibre problem and an index formula for elementary operators. Proc. Am. Math. Soc. 123(3), 743–746 (1995). https://doi.org/10.2307/2160795 37. Eschmeier, J., Putinar, M.: Spectral decompositions and analytic sheaves. English. Vol. 10. Oxford: Oxford Univ. Press, 1996, pp. x+362 38. Eschmeier, J., Putinar, M.: Spectral decompositions and analytic sheaves. Vol. 10. London Mathematical Society Monographs. New Series. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996, pp. x+362 39. Eschmeier, J., Patton, L., Putinar, M.: Carathéodory-Fejér interpolation on polydisks. Math. Res. Lett. 7(1), 25–34 (2000). https://doi.org/10.4310/MRL.2000.v7.n1.a3 40. Eschmeier, J., Schmitt, J.: Cowen-Douglas operators and dominating sets. J. Oper. Theory 72(1), 277–290 (2014). https://doi.org/10.7900/jot.2013jan21.1976 41. Eschmeier, J., Fialkow, L., Harte, R., Magajna, B., Mathieu, M., Smith, R.R.: “Open problems”. In: Elementary operators and applications (Blaubeuren, 1991). World Sci. Publ., River Edge, NJ, 1992, pp. 249–251 42. Eschmeier, J.: On the essential spectrum of Banach-space operators. Proc. Edinburgh Math. Soc. 43(3), 511–528 (2000). https://doi.org/10.1017/S0013091500021167 43. Eschmeier, J.: “Algebras of subnormal operators on the unit polydisc”. In: Recent progress in functional analysis (Valencia, 2000). Vol. 189. North-Holland Math. Stud. North-Holland, Amsterdam, 2001, pp. 159–171. https://doi.org/10.1016/S0304-0208(01)80043-8 44. Eschmeier, J.: Invariant subspaces for commuting contractions. J. Oper. Theory 45(2), 413–443 (2001) 45. Eschmeier, J.: “On the structure of spherical contractions”. In: Recent advances in operator theory and related topics (Szeged, 1999). Vol. 127. Oper. Theory Adv. Appl. Birkhäuser, Basel, 2001, pp. 211–242 46. Eschmeier, J.: On the reflexivity of multivariable isometries. Proc. Am. Math. Soc. 134(6), 1783–1789 (2006). https://doi.org/10.1090/S0002-9939-05-08139-6 47. Eschmeier, J.: On the Hilbert-Samuel multiplicity of Fredholm tuples. Indiana Univ. Math. J. 56(3), 1463–1477 (2007). https://doi.org/10.1512/iumj.2007.56.2930 48. Eschmeier, J.: Reflexivity for subnormal systems with dominating spectrum in product domains. Integral Equ. Oper. Theory 59(2), 165–172 (2007). https://doi.org/10.1007/s00020-007-1521-1 49. Eschmeier, J.: Samuel multiplicity and Fredholm theory. Math. Ann. 339(1), 21–35 (2007). https:// doi.org/10.1007/s00208-007-0103-5

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Jörg Eschmeier’s Mathematical Work 50. Eschmeier, J.: Fredholm spectrum and growth of cohomology groups. Studia Math. 186(3), 237–249 (2008). https://doi.org/10.4064/sm186-3-3 51. Eschmeier, J.: Samuel multiplicity for several commuting operators. J. Oper. Theory 60(2), 399–414 (2008) 52. Eschmeier, J.: Grothendieck’s comparison theorem and multivariable Fredholm theory. Arch. Math. (Basel) 92(5), 461–475 (2009). https://doi.org/10.1007/s00013-009-3173-7 53. Eschmeier, J.: Essential normality of homogeneous submodules. Integral Equ. Oper. Theory 69(2), 171–182 (2011). https://doi.org/10.1007/s00020-010-1809-4 54. Eschmeier, J.: Quasicomplexes and Lefschetz numbers. Acta Sci. Math. (Szeged) 79(3–4), 611–621 (2013) 55. Eschmeier, J.: The essential spectrum of Toeplitz tuples with symbols in H ∞ + C. Studia Math. 219(3), 237–246 (2013). https://doi.org/10.4064/sm219-3-4 56. Eschmeier, J.: “On the maximal ideal space of a Sarason-type algebra on the unit ball”. In: The corona problem. Vol. 72. Fields Inst. Commun. Springer, New York, (2014), pp. 69–83. https://doi.org/10. 1007/978-1-4939-1255-1_4 57. Eschmeier, J.: Funktionentheorie mehrerer Veränderlicher. German. Berlin: Springer Spektrum, (2017), pp. xii + 209. https://doi.org/10.1007/978-3-662-55542-2 58. Eschmeier, J.: Bergman inner functions and m-hypercontractions. J. Funct. Anal. 275(1), 73–102 (2018). https://doi.org/10.1016/j.jfa.2017.10.018 59. Eschmeier, J.: Wandering subspace property for homogeneous invariant subspaces. Banach J. Math. Anal. 13(2), 486–505 (2019). https://doi.org/10.1215/17358787-2018-0052 60. Eschmeier, J.: “Lokale Zerlegbarkeit und Funktionalkalküle abgeschlossener linearer Operatoren auf Banachräumen”. Diplomarbeit. Westfälische Wilhelms-Universität Münster, 1979, vi+71 p 61. Eschmeier, J.: Spectral decompositions and functional calculus for commuting tuples of continuous and closed operators in Banach space. German. Math. Inst. Univ. Münster, 2. Ser. 20, 104 S. (1981). 1981 62. Eschmeier, J.: “Spektralzerlegungen und Funktionalkalküle für vertauschende Tupel stetiger und abgeschlossener Operatoren in Banachräumen”. In: Schr. Math. Inst. Univ. Müunster (2) 20 (1981), pp. vii+104 63. Eschmeier, J.: Local properties of Taylor’s analytic functional calculus. Invent. Math. 68, 103–116 (1982). https://doi.org/10.1007/BF01394269 64. Eschmeier, J.: Operator decomposability and weakly continuous representations of locally compact abelian groups. J. Oper. Theory 7(2), 201–208 (1982) 65. Eschmeier, J.: Equivalence of decomposability and 2-decomposability for several commuting operators. Math. Ann. 262(3), 305–312 (1983). https://doi.org/10.1007/BF01456012 66. Eschmeier, J.: On two notions of the local spectrum for several commuting operators. Michigan Math. J. 30(2), 245–248 (1983). https://doi.org/10.1307/mmj/1029002855 67. Eschmeier, J.: Are commuting systems of decomposable operators decomposable? In. J. Oper. Theory 12(2), 213–219 (1984) 68. Eschmeier, J.: “Some remarks concerning the duality problem for decomposable systems of commuting operators”. In: Spectral theory of linear operators and related topics (Timi¸soara/Herculane, 1983). Vol. 14. Oper. Theory Adv. Appl. Birkhäuser, Basel, 1984, pp. 115–123 69. Eschmeier, J.: Spectral decompositions and decomposable multipliers. Manuscripta Math. 51(1–3), 201–224 (1985). https://doi.org/10.1007/BF01168353 70. Eschmeier, J.: Analytic index formulas for tensor product systems. Proc. Roy. Irish Acad. Sect. A 87(2), 121–135 (1987) 71. Eschmeier, J.: “Analytic spectral mapping theorems for joint spectra”. In: Operators in indefinite metric spaces, scattering theory and other topics (Bucharest, 1985). Vol. 24. Oper. Theory Adv. Appl. Birkhäuser, Basel, 1987, pp. 167–181 72. Eschmeier, J.: Analytische Dualität und Tensorprodukte in der mehrdimensionalen Spektraltheorie. (Analytic duality and tensor products in multidimensional spectral theory). German. Schriftenr. Math. Inst. Univ. Münster, 2. Ser. 42, 1–132 (1987). 1987 73. Eschmeier, J.: A decomposable Hilbert space operator which is not strongly decomposable. Integral Equ. Oper. Theory 11(2), 161–172 (1988). https://doi.org/10.1007/BF01272116 74. Eschmeier, J.: Tensor products and elementary operators. J. Reine Angew. Math. 390, 47–66 (1988). https://doi.org/10.1515/crll.1988.390.47

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E. Albrecht et al. 75. Eschmeier, J.: Invariant subspaces for subscalar operators. Arch. Math. 52(6), 562–570 (1989). https:// doi.org/10.1007/BF01237569 76. Eschmeier, J.: Operators with rich invariant subspace lattices. J. Reine Angew. Math. 396, 41–69 (1989). https://doi.org/10.1515/crll.1989.396.41 77. Eschmeier, J.: “Multiplication operators on Bergman spaces are reflexive”. In: Linear operators in function spaces (Timi¸soara, 1988). Vol. 43. Oper. Theory Adv. Appl. Birkhäuser, Basel, 1990, pp. 165–184 78. Eschmeier, J.: “Analytically parametrized complexes of Banach spaces”. In: Elementary operators and applications (Blaubeuren, 1991). World Sci. Publ., River Edge, NJ, 1992, pp. 163–177 79. Eschmeier, J.: Bishop’s condition (β) and joint invariant subspaces. J. Reine Angew. Math. 426, 1–22 (1992). https://doi.org/10.1515/crll.1992.426.1 80. Eschmeier, J.: Representations of Alg Lat(T). Proc. Am. Math. Soc. 117(4), 1013–1021 (1993). https://doi.org/10.2307/2159528 81. Eschmeier, J.: Representations of H ∞ (G) and invariant subspaces. Math. Ann. 298(1), 167–186 (1994). https://doi.org/10.1007/BF01459732 82. Eschmeier, J.: Invariant subspaces for spherical contractions. Proc. Lond. Math. Soc. 75(1), 157–176 (1997). https://doi.org/10.1112/S0024611597000300 83. Eschmeier, J.: “C00 -representations of H ∞ (G) with dominating Harte spectrum”. In: Banach algebras ’97 (Blaubeuren). de Gruyter, Berlin, 1998, pp. 135–151 84. Eschmeier, J.: Algebras of subnormal operators on the unit ball. J. Oper. Theory 42(1), 37–76 (1999) 85. Eschmeier, J., Toth, S.: “K-inner functions and K-contractions”. In: Operator theory, functional analysis and applications. Vol. 282. Oper. Theory Adv. Appl. Birkhaüser/Springer, Cham, [2021] © 2021, pp. 157–181 86. Eschmeier, J., Vasilescu, F.-H.: On jointly essentially selfadjoint tuples of operators. Acta Sci. Math. (Szeged) 67(1–2), 373–386 (2001) 87. Eschmeier, J., Wolff, R.: Composition of inner mappings on the ball. Proc. Am. Math. Soc. 130(1), 95–102 (2002). https://doi.org/10.1090/S0002-9939-01-06302-X

Other References 88. Agler, J.: “On the representation of certain holomorphic functions defined on a polydisc”. In: Topics in operator theory: Ernst D. Hellinger memorial volume. Vol. 48. Oper. Theory Adv. Appl. Basel: Birkhäuser, 1990, pp. 47–66 89. Albrecht, E.: Funktionalkalküle in mehreren Veränderlichen für stetige lineare Operatoren auf Banachräumen. Manuscripta Math. 14, 1–40 (1974). https://doi.org/10.1007/BF01637620 90. Albrecht, E.: On two questions of I. Colojoar˘a and C. Foia¸s. Manuscr. Math. 25, 1–15 (1978). https:// doi.org/10.1007/BF01170354 91. Albrecht, E.: On decomposable operators. Integral Equ. Oper. Theory 2, 1–10 (1979). https://doi.org/ 10.1007/BF01729357 92. Arveson, W.: The Dirac operator of a commuting d-tuple. J. Funct. Anal. 189(1), 53–79 (2002). https://doi.org/10.1006/jfan.2001.3827 93. Arveson, W.: Subalgebras of C∗ -algebras. III. Multivariable operator theory. Acta Math. 181(2), 159–228 (1998). https://doi.org/10.1007/BF02392585 94. Barbian, C.: “Beurling-Type Representation of Invariant Subspaces in Reproducing Kernel Hilbert Spaces”. English. Dissertation. Universität des Saarlandes, (2007), p. 156. https://doi.org/10.22028/ D291-26053 95. Barbian, C.: Beurling-type representation of invariant subspaces in reproducing kernel Hilbert spaces. Integral Equ. Oper. Theory 61(3), 299–323 (2008). https://doi.org/10.1007/s00020-008-1590-9 96. Barbian, C.: Approximation properties for multiplier algebras of reproducing kernel Hilbert spaces. Acta Sci. Math. 75(3–4), 655–663 (2009) 97. Barbian, C.: A characterization of multiplication operators on reproducing kernel Hilbert spaces. J. Oper. Theory 65(2), 235–240 (2011) 98. Bickel, K., Hartz, M., McCarthy, J.E.: A multiplier algebra functional calculus. Trans. Am. Math. Soc. 370(12), 8467–8482 (2018) 99. Bishop, E.: A duality theorem for an arbitrary operator. Pac. J. Math. 9, 379–397 (1959)

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Authors and Affiliations Ernst Albrecht1 · Raúl E. Curto2 · Michael Hartz1 · Mihai Putinar3,4 Ernst Albrecht [email protected] Raúl E. Curto [email protected] Michael Hartz [email protected] 1

Fachrichtung Mathematik, Universität des Saarlandes, 66123 Saarbrücken, Germany

2

Department of Mathematics, University of Iowa, Iowa City, IA, USA

3

University of California, Santa Barbara, CA, USA

4

Newcastle University, Newcastle upon Tyne, UK

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Complex Analysis and Operator Theory (2022) 16:115 https://doi.org/10.1007/s11785-022-01293-w

Complex Analysis and Operator Theory

Rochberg’s Abstract Coboundary Theorem Revisited Catalin Badea1 · Oscar Devys1 Received: 29 April 2022 / Accepted: 14 October 2022 / Published online: 3 November 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract Rochberg’s coboundary theorem provides conditions under which the equation (I − T )y = x is solvable in y. Here T is a unilateral shift on Hilbert space, I is the identity operator and x is a given vector. The conditions are expressed in terms of Wold-type decomposition determined by T and growth of iterates of T at x. We revisit Rochberg’s theorem and prove the following result. be an isometry acting on a Hilbert space ∞ Let T ∗k kT H and let x ∈ H. Suppose that k=0   √ x < ∞. Then x is in the range of (I − T ) if (and only if)  nk=0 T k x  = o( n). When T is merely a contraction, x is a coboundary under an additional assumption. Some applications to L 2 -solutions of the functional equation f (x) − f (2x) = F(x), considered by Fortet and Kac, are given. Keywords Coboundary theorems · Unilateral shifts · Wold decomposition · Functional equations Mathematics Subject Classification 47A05 · 47A35 · 39B05

To the memory of Jörg Eschmeier. Communicated by Mihai Putinar. This work was supported in part by the project FRONT of the French National Research Agency (grant ANR-17-CE40-0021), by the Labex CEMPI (ANR-11-LABX-0007-01) and by the Max Planck Institute of Mathematics (Bonn). This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

B 1

Catalin Badea [email protected] CNRS UMR 8524 - Laboratoire Paul Painlevé, University of Lille, F-59000 Lille, France

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1 Introduction 1.1 Coboundaries Let T be a bounded linear operator acting on a complex Banach space X . An element x of X is called a coboundary for T if there is y ∈ X such that x = y − T y. Coboundaries are related to the behavior of the ergodic sums Sn (T )x := x + T x + · · · + T n−1 x, n ≥ 1. A variant of the mean ergodic theorem for power bounded operators on reflexive Banach spaces has been proved by von Neumann for Hilbert spaces and by Lorch in the general case; see for instance [14]. Recall that T is said to be power bounded if supn≥1 T n  < ∞. We have 

 1 X = x ∈ X : lim Sn (T )x exists = {y ∈ X : T y = y} ⊕ (I − T )X . n→∞ n In particular, as a consequence of this ergodic decomposition, we have x ∈ (I − T )X

⇔

lim

n→∞

1 Sn (T )x = 0. n

One can say more about the rate of convergence of (1/n)Sn (T )x to zero when x is a coboundary. Indeed, when there exists a solution y of the equation y − T y = x, the ergodic sums satisfy Sn (T )x = y − T n y. It follows that (Sn (T )x)n∈N is bounded. Therefore      1  Sn (T )x  = O 1 . (1.1)  n n This rate of convergence to zero, namely O(1/n), characterizes coboundaries of power bounded operators on reflexive spaces. Indeed, the converse result (whenever T is power bounded and X is reflexive, an element x satisfying (1.1) is a coboundary for T ) has been proved by Browder [1] and rediscovered by Butzer and Westphal [2]. We also note (see for instance [3, 4, 11] and the references therein) that if (I − T )X is not closed, then for every sequence (an )n≥1 of positive real numbers converging to zero, there exists x ∈ (I − T )X \(I − T )X such that   1   Sn (T )x  ≥ an , ∀n ≥ 1. n  In particular, there is no general rate of convergence in the mean ergodic theorem outside coboundaries.

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1.2 Rochberg’s Theorem Browder’s theorem has been extended to the case that T is a dual operator on a dual Banach space by Lin [16]; see also Lin and Sine [17]. We refer the reader to the introduction of [6], and the references cited therein, for the history of Browder’s theorem and for other extensions and generalizations. We mention here only two references, namely [19] and [13], dealing with the Hilbert space situation. Any of these Hilbert or Banach space abstract characterizations is not strong enough to obtain as consequences classical results of Fortet and Kac [9, 12] who dealt with the case X = L 2 (0, 1) and S f (x) = f (2x). This operator S is the Koopman operator associated with the doubling map on the torus; see the last section of this manuscript for more information about coboundaries of S. This √ situation has been remedied by Rochberg [20], who showed that a condition of o( n) growth of ergodic sums at x is sufficient to ensure that x is a coboundary for a unilateral shift on Hilbert space. Notice that the Koopman operator S acts as a unilateral shift on the subspace of L 2 (0, 1) of functions whose zeroth Fourier coefficient vanishes. We need the following classical definition in order to state Rochberg’s abstract coboundary theorem. Definition 1.1 Let T be an isometry acting on Hilbert space H. A closed subspace K of H is called wandering for T whenever T p K ⊥ T q K for p, q ∈ N, p = q. The isometry T is called a (unilateral) shift if H possess a closed subspace K, wandering for T and such that ∞ 

T n K = H.

n=0

Theorem 1.2 ([20]) Let S be a shift and let f be an element of H. Using the notation of the preceding definition, we denote by f j the projection of f onto the closed subspace S j K. Suppose that there exists β > 0 such that  f j  = O(2−β j ). Then there exists g in H such that (I − S)g = f if and only if 1 lim n→∞ n

 n   Sk   k=0

2   f  = 0. 

Remark 1.3 The condition  f j  = O(2−β j )

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C. Badea and O. Devys

is of course dependent of the decomposition of H associated with the unilateral shift S. It implies S ∗ j f  = O(2−β j ). 1.3 Statement of the Main Results In the next theorem the unilateral shift S is replaced by an arbitrary isometry T and of the norm of the projection f j by the convergence of the series ∞the growth ∗ j f . The statement of the result does not depend on the Wold decomjT j=0 position, at least not in an explicit way. For the convenience of the reader, the Wold decomposition theorem is recalled below. Theorem 1.4 implies Rochberg’s theorem and it allows to recover Kac’s results about the coboundaries of the Koopman operator of the doubling map. Theorem 1.4 Let T be an isometry acting on a Hilbert space H and let x ∈ H. Suppose that ∞ kT ∗k x < ∞. (1.2) k=0

Then there exists y ∈ H such that x = (I − T )y if and only if 1 lim n→∞ n

 n 2     T k x  = 0.    k=0

Note however that the condition (1.2) implies that x is necessarily an element of the shift part of the isometry T . Considering coboundaries of adjoints of isometries, we notice that the identity I −T = (T ∗ −I )T shows that every coboundary of the isometry T is also a coboundary for its adjoint T ∗ . It follows from [7, Proposition 4.3] that when the isometry T is not invertible (i.e., not a unitary operator), there are coboundaries for T ∗ which are not coboundaries for T . The following result, more general than Theorem 1.4, is about coboundaries of contractions (operators of norm no greater than one). Theorem 1.5 Let T be a linear operator acting on a Hilbert space H with T  ≤ 1. Let x ∈ H and denote Sn (T )x := x + T x + · · · + T n−1 x. Suppose that (1.2) holds, as well as √ (1.3) Sn (T )x = o( n), n → ∞ and

n



 Sk (T )x2 − T Sk (T )x2 = o(n), n → ∞.

(1.4)

k=1

Then there exists y ∈ H such that x = (I − T )y. In addition, y can be chosen such that T y = y. We obtain the following consequence.

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Corollary 1.6 Let T be a linear operator acting on a Hilbert space H with T  ≤ 1. Let x ∈ H and denote Sn (T )x := x + T x + · · · + T n−1 x. Suppose that (1.2) and (1.3) hold, as well as

∞ Sk (T )x2 − T Sk (T )x2 < ∞. k

(1.5)

k=1

Then there exists y ∈ H such that x = (I − T )y and T y = y. Some remarks are in order. Theorem 1.5 and its consequence Corollary 1.6 show that the coboundary equation can be solved within the maximal isometric subspace M = {x ∈ H : T n x = x for every n ≥ 0}. We refer to [18] and [15] for the canonical decomposition of a contraction into the maximal isometric subspace and its orthogonal. Conditions (1.4) and (1.5) are easily verified when T is an isometry. The conditions (1.2) and (1.3) are always satisfied when T  < 1; however (1.5) is not, unless x = 0. In fact, T y = y and T  < 1 imply that y = 0 and thus x = 0. Of course, as (I − T ) is invertible when T  < 1 by Carl Neumann’s lemma, the coboundary equation x = (I − T )y is always solvable in this case. 1.4 Outline of the Paper A proof of Theorem 1.4 is given in the next section. The more general Theorem 1.5 and its consequence Corollary 1.6 are proved in Sect. 3. Some applications to the functional equation g(x) − g(2x) = f (x) are presented in the next section. The last section collects the acknowledgments, and (imposed) conflict of interest and data availability statements.

2 Proof of Theorem 1.4 We first recall Wold’s decomposition Theorem (see [18, Chapter 1]). Theorem 2.1 (Wold decomposition) Let T be an isometry on a Hilbert H. Then H decomposes as an orthogonal sum H = H0 ⊕ H1 such that H0 and H1 are reducing for T , the restriction of T to H0 is a unitary operator and the restriction of T to H1 is a unilateral shift (one of the subspaces can eventually reduce to {0}). This decomposition is unique; in particular, we have H0 =

∞ 

T n H and H1 =

n=0

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∞  n=0

23

T n K , where K = H  T H.

C. Badea and O. Devys

 Proof 1.4 If x = (I − T )y, then nk=0 T k x = x − T n+1 x. Therefore n of Theorem k k=0 T x is bounded since the isometry T is clearly power-bounded. In particular, 1 lim n→∞ n

 n 2    k  T x  = 0.   

1 lim n→∞ n

 n 2    k  T x  = 0.   

k=0

Suppose now that

k=0

We want to show the existence of a solution y of the equation (I − T )y = x. Let H = H0 ⊕ H1 be the Wold’s decomposition associated with T . We notice that x ∈ H1 . Indeed, if x = x0 + x1 according to Wold’s decomposition of H, then lim T ∗k x1  = 0 and T ∗n x0  = x0 , ∀n ∈ N.

k→∞

Therefore lim T ∗n x = x0 .

n→∞

On the other hand, it follows from (1.2) that lim T ∗n x = 0.

n→∞

We obtain that x ∈ H1 . In particular, if H1 is reduced to {0}, then x = 0 = (I − T )0. Therefore, without loss of any generality, we can assume that T is a shift. For each n ∈ N, we denote Pn the projection onto the subspace T n K. For u ∈ H, by n n we set u n := Pn (u), u := j=0 u j and Rn := u − u n . Suppose that y is solution of the equation (I − T )y = x. We first obtain, by projecting to T k K for each k ∈ N, the following system of equations : ⎧ x0 = y0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪x1 = y1 − T y0 ⎨ .. . ⎪ ⎪ ⎪ xk = yk − T yk−1 ⎪ ⎪ ⎪ ⎪ ⎩.. .

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We then obtain ⎧ ⎪ ⎪ y0 = x0 ⎪ ⎪ ⎪ y1 = x1 + T y0 = x1 + T x0 ⎪ ⎪ ⎨. .. ⎪ ⎪ ⎪ yk = xk + T yk−1 = xk + T xk−1 + · · · + T k−1 x1 + T k x0 ⎪ ⎪ ⎪ ⎪ . ⎩. . Consider now, for each r ∈ N, the element yr =

r

T k xr −k ∈ T r K.

k=0

  We will prove that r∞=0 yr 2 is convergent, thus showing that y = r∞=0 yr is well defined in H. In that case, for every r ∈ N, we have

Pr (I − T )y = yr − T yr −1 =

r

T j xr − j −

r −1

j=0

T j+1 xr −1− j

j=0

= xr . This shows that (I − T )y = x. To prove that r∞=0 yr 2 is finite, we need two more results. Lemma 2.2 Let u ∈ H be such that 1 lim n→∞ n



j≥0 T

∗ j u

< +∞. Then

 n 2 ∞    k  T u  = u2 + 2Re u; T k u.    k=0

k=1

 k Proof We first notice that the sum ∞ k=1 u; T u is absolutely convergent since ∗ j ∗ (T u) j≥0 is summable. For each n ∈ N , we have 1 n

⎛ ⎞⎞ ⎛  n 2 n   1   T k u = ⎝ T i u2 + 2Re ⎝ T i u; T j u⎠⎠    n k=0 i=0 0≤i< j≤n ⎛ ⎞⎞ ⎛ n 1 ⎝ = u2 + 2Re ⎝ u; T j−i u⎠⎠ n i=0

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0≤i< j≤n

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C. Badea and O. Devys

 n  n+1 2 2 r u + Re = (n − r + 1)u; T u n n r =1  n  n n+1 1 = u2 + 2Re u; T r u − (r − 1)u; T r u . n n r =1

r =1

On the other hand, we have  n  n 1  1  r  (r − 1)u; T u ≤ u (r − 1)T ∗r u.  n  n r =1

r =1

Using again the summability of the sequence (T ∗ j u) j≥0 and the Kronecker’s lemma (see for instance [21, Lemma IV.3.2]), we get n 1 (r − 1)u; T r u −→ 0. n→∞ n r =1

As the series



k≥1 u; T

1 lim n→∞ n

k u

is convergent, we obtain, as n tends to infinity,

 n 2 ∞     T k u  = u2 + 2Re u; T k u.    k=0

k=1

  Lemma 2.3 Let u ∈ H. For every r ∈ N we have  2  r   j  1   T u r − j  = lim  n→∞ n  j=0 

2    n  j r   .  T u     j=0

Proof Let n ≥ r . For k ∈ N we have ⎧k j ⎛ ⎞ ⎪ j=0 T u k− j ⎪ ⎪  n ⎨ r T j u k− j T j u r ⎠ = rj=0 Pk ⎝ ⎪ T j u k− j ⎪ j=0 ⎪ ⎩ j=k−n 0 Using the decomposition of H as H = 1 n

∞

n=0

if if if if

0 ≤ k < r, r ≤ k ≤ n, n < k ≤ n + r, k > n + r.

T n K, we obtain

2  2  2      r −1  k n  r       n 1 1 j r j j     T u  = T u k− j  + T u k− j      n n     j=0 k=0  j=0 k=r  j=0

26

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 2  n+r  r  1  j  + T u k− j    . n  k=n+1  j=k−n We have ⎛  2 ⎞  r −1  k   ⎟ 1⎜ j   T u ⎝ k− j  ⎠ −→ 0  n→∞ n  k=0  j=0 and ⎛  2  2 ⎞     r r r    ⎟ 1 1 ⎜  j j    −→ 0, T u k− j  = ⎝ T u k− j     ⎠ n→∞ n n   k=n+1  j=k−n k=1  j=k n+r

as well as  2 2  2       n  r r r       1 n −r +1  j j j      T u = T u −→ T u k− j  r− j  r− j  .    n→∞  n n      k=r j=0 j=0 j=0 We thus obtain 2  2   r    n     1 j r j   lim  T u = T u r− j  .    n→∞ n   j=0   j=0   We finally show that



2 r ≥0 yr 

< ∞. Using Lemma 2.3, we have for each r ∈ N,

2  r   1   yr 2 =  T i xr −i  = lim n→∞ n   i=0

 n 2     T i xr  .    i=0

Using the parallelogram identity for the vectors x r + Rr = x, we get 2 n

 n 2   1  i r T x  =    n i=0

 n  n 2 2    1   i r  i  T x +  T (x − Rr )     n i=0 i=0  n 2  2  i  −  T Rr  .  n i=0

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C. Badea and O. Devys

Make now n tends to infinity. Using Lemma 2.2 for Rr and x r − Rr , and the hypothesis 2  1  n T k x  −→ 0, we obtain n

k=0

n→∞

 n 2     T i xr     i=0  n 2  1 1  i r  = lim  T (x − Rr ) − 2 lim n→∞ n  n→∞ n 

2 2yr 2 = lim n→∞ n

i=0 2

 n 2     i T Rr     i=0

= x − Rr  − 2Rr  ∞

 x r − Rr ; T k (x r − Rr ) − 2Rr ; T k Rr  + 2Re r

2

k=1 ∞

x r ; T k x r  − x r ; T k Rr  = x  − Rr  + 2Re r 2

2

k=1

 − Rr ; T x  − Rr ; T Rr  k r

k

= x r 2 − Rr 2 + 2Re

∞ ∞ x r ; T k x r  − 2Re Rr ; T k x. k=1

k=1

Using now Lemma 2.2 applied to x r and Lemma 2.3, we get ∞ 2 ∞   1   x r ; T k x r  = lim  T i xr  x r 2 + 2Re n→∞ n   k=1

i=0

= yr 2 . We can infer that 2yr 2 = yr 2 − Rr 2 − 2Re

∞ Rr ; T k x, k=1

so yr 2 = −Rr 2 − 2Re

∞

Rr ; T k x.

k=1

For each fixed r we have Rr = T r +1 T ∗(r +1) x. Thus Rr  = T ∗(r +1) x. As +∞

jT ∗ j x < +∞,

j=1

28

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Rochberg’s Abstract Coboundary Theorem...

we obtain that (Rr 2 )r is summable. It suffices to show that ∞  ∞    k   Rr ; T x < ∞.   r =0 k=1

We have ∞   r  ∞         Rr ; T k x  Rr ; T k x =  Rr ; T k x +     k=1 k=1 k=r +1  r  ∞    k k  =  Rr ; T x + Rk ; T x   k=r +1 ∞

k=1

≤

r k=1

Rr T k x +

k=r +1 ∞



≤ x r Rr  +



Rk 

k=r +1

 ≤ x r T

Rk T k x

∗(r +1)

x +

∞

 T

∗(k+1)

x .

k=r +1

Using again the summability of (r T ∗r x)r , we get ∞ ∞

T ∗k x =

r =0 k=r +1

Therefore

∞

2 r =0 yr 

∞

kT ∗k x < ∞.

k=0

< ∞.

 

3 The Case of Contractions We now prove Theorem 1.5 and its consequence Corollary 1.6. Proof of Theorem 1.5 Let D denote the defect operator D = (I − T ∗ T )1/2 , which is well defined since T is a contraction. As T x2 + Dx2 = T ∗ T x, x + (I − T ∗ T )x, x = x2 , the operator R : 2 (H) → 2 (H) given by R(x0 , x1 , x2 , . . .) = (T x0 , Dx0 , x1 , x2 , . . .)

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C. Badea and O. Devys

and with matrix representation ⎤

⎡

T ⎢D ⎢ ⎢ R=⎢ I ⎢ I ⎣

..

⎥ ⎥ ⎥ ⎥, ⎥ ⎦

(3.1)

.

is an isometry. We can thus apply Theorem 1.4 to R. The iterates of R are given by R k (x0 , x1 , x2 , . . .) = (T k x0 , DT k−1 x0 , DT k−2 x0 , . . . , DT x0 , Dx0 , x1 , x2 , . . .) while their adjoints are given by R ∗k (x0 , x1 , x2 , · · · ) = (T ∗k x0 + T ∗(k−1) Dx1 + · · · + T ∗ Dxk−1 , Dxk , xk+1 , xk+2 , · · · ). Denote x˜ = (x, 0, 0, · · · ) ∈ 2 (H) and y˜ = (y, y1 , y2 , · · · ) ∈ 2 (H). The equation x˜ = (I − R) y˜ reduces to the system of equations x = (I − T )y, y1 = Dy, y2 = y1 , y3 = y2 , etc. As y˜ ∈ 2 (H), we obtain y1 = y2 = · · · = 0. Therefore the equation x˜ = (I − R) y˜ in 2 (H) is equivalent to x = (I − T )y and Dy = 0. Every positive (i.e. positive semi-definite) operator has the same kernel as its positive square-root; thus (I − T ∗ T )y = 0. Therefore T y = y.  ∗k ˜ < ∞ An easy computation shows that the summability condition ∞ k=0 kR x ∞ ∗k is equivalent to k=0 kT x < ∞. Notice now that R k x˜ = R k (x, 0, 0, . . .) = (T k x, DT k−1 x, . . . , Dx, 0, 0, . . .). Therefore n k=0

R k x˜ =

 n k=0

T k x, D

n−1

 Tkx , D

k=0

n−2





T k x , . . . , Dx, 0, 0, . . . .

k=0

√ Hence, using the notation Sn (T )x = x + T x + · · · + T n−1 x, the o( n) condition  n    √  k  R x˜  = o( n)    k=0

30

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Rochberg’s Abstract Coboundary Theorem...

is equivalent to   n n   √  k  = o( T x n) and D(Sk (T )x)2 = o(n).     k=0

k=0

The proof is now complete using the identity Du2 = u2 − T u2 .

 

Corollary 1.6 follows from Theorem 1.5 and Kronecker’s lemma, already used in the proof of Theorem 1.4.

4 Coboundaries of the Doubling Map Let val2 (n) be the 2-valuation of n, that is val2 (n) = k if n = m2k with m ∈ / 2Z. For n ∈ Z, we denote by fˆ(n) = f ∈ L 2 (0, 1).

#1 0

f (t)e−int dt the n-th Fourier coefficient of

Corollary 4.1 Suppose f is a periodic function of period 1 such that f ∈ L 2 (0, 1), $

1

f (t) dt = 0

(4.1)

0

and there exists ε > 0 such that ∞

2    val2 (n)4+ε  fˆ(n) < ∞.

(4.2)

n=−∞

Then there is a function g in L 2 (0, 1) of period one such that f (t) = g(t) − g(2t) a.e. if and only if 1 lim n→∞ n

$ 0

   

n 1  i=0

2   f (2 t) dt = 0  i

Proof We use Theorem 1.4 applied to the isometry T : L 2 (0, 1) −→ L 2 (0, 1) defined by T f (t) = f (2t), t ∈ (0, 1) mod 1.

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C. Badea and O. Devys

We first remark that condition (4.1) is justified by the fact that T acts as a shift operator on the subspace of L 2 (0, 1) of functions whose zeroth Fourier coefficient vanishes. The condition 2 $  n  1 1  i  f (2 t) dt = 0 lim  n→∞ n 0   i=0

is exactly the condition 1 lim n→∞ n

 n   Tk   k=0

2   f  = 0, 

which appears in Theorem 1.4. We want to show that ∞

kT ∗k f  < ∞.

k=0

Recall that T acts as a shift operator on the subspace of L 2 (0, 1) of functions whose zeroth Fourier coefficient vanishes. Let (an ) = ( fˆ(n)) be the sequence of Fourier coefficients of f . We have a0 = 0. The iterates of the adjoint of T at f can be computed as ∞

T ∗k f (t) =

a j2k e2iπ jt , k ∈ N.

j=−∞

For ε > 0, using the change n = j2k in the order of summation, we get ∞

kT ∗k f  =

k=0

∞

⎛ k⎝

k=0

=

∞

∞

|a j2k |2 ⎠

j=−∞

1/2 ⎛ ⎝

k −(1+ε)

∞

k=0 ∞

j=−∞

|an |2

n=−∞

k=0

≤2

⎞1/2

∞

⎞1/2 1/2 ⎛ ∞ ∞ ⎝ k −(1+ε) k 3+ε |a j2k |2 ⎠

k=0

=

⎛

k −(1+ε)/2 ⎝k 3+ε

∞ 

|a j2k |2 ⎠

j=−∞

k=0

≤

⎞1/2

∞

1/2  k

−(1+ε)

∞

val 2 (n)

⎞1/2 k 3+ε ⎠

k=0

val2 (n)

1/2

4+ε

|an |

2

.

n=−∞

k=0

32

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Rochberg’s Abstract Coboundary Theorem...

Thus, under our hypothesis about the Fourier coefficients, we have ∞

kT ∗k f  < ∞.

k=0

  Corollary 4.2 [20] Let f be a periodic function of period 1 such that f ∈ $

1

L 2 (0, 1),

f (t) dt = 0

0

and there exists α > 0 such that ∞

| fˆ((2k + 1))2i |2 = O(2−αi ).

(4.3)

k=−∞

Then there is a function g in L 2 (0, 1) of period one such that f (t) = g(t) − g(2t) a.e. if and only if 1 lim n→∞ n

$ 0

   

n 1  i=0

2   f (2 t) dt = 0  i

Proof The result follows from Corollary 4.1 with ε = 1, say. Indeed, using the condition (4.3), one can estimate ∞

∞ ∞ 2    val2 (n)5  fˆ(n) = i 5 | fˆ((2k + 1))2i |2

n=−∞

i=1 k=−∞ ∞ 5



i=1

i < ∞. 2αi  

Remark 4.3 Condition (4.3) is condition (a) from Theorem 4 in [20]. It has been proved in [20] that each of other three conditions of Hölder type, called there (b), (c) and (d), implies the condition (4.3). Mark Kac has already considered in [12] the case when f is in the Hölder class C 0,α for some α > 1/2. We refer to [5, 9, 10] for other contributions concerning the functional equation f (t) = g(t) − g(2t). Remark 4.4 All the remarks at the end of the paper [20] apply also in our situation. In particular, the generalization to the functional equation f (t) = g(t) − g(nt) (for a fixed integer n) is immediate.

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33

C. Badea and O. Devys Acknowledgements Some of the results presented here are part of the 2012 PhD thesis of the second-named author [8] written under the supervision of the first-named author. We wish to thank several persons who encouraged us to present these results to a larger audience and/or to revisit them. As Amor Towles said: “For as it turns out, one can revisit the past quite pleasantly, as long as one does so expecting nearly every aspect of it to have changed”. Special thanks are due to Michael Lin for several interesting comments and remarks. We would like to thank the anonymous referee for a careful reading of the manuscript and very useful suggestions. The first-named author would like to thank the Max Planck Institute for Mathematics in Bonn for providing excellent working conditions and support. Data Availability No datasets were generated or analysed during the current study.

Declarations Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.

References 1. Browder, F.: On the iteration of transformations in noncompact minimal dynamical systems. Proc. Amer. Math. Soc. 9, 773–780 (1958) 2. Butzer, P.L., Westphal, U.: The mean ergodic theorem and saturation. Indiana Univ. Math. J. 20, 1163–1174 (1970/1971) 3. Badea, C., Müller, V.: On weak orbits of operators. Topology Appl. 156, 1381–1385 (2009) 4. Badea, C., Grivaux, S., Müller, V.: The rate of convergence in the method of alternating projections. Algebra i Analiz 23 (2011), no. 3, 1–30; reprinted in St. Petersburg Math. J. 23 (2012), no. 3, 413–434 5. Ciesielski, Z.: On the functional equation f (t) = g(t) − g(2t). Proc. Amer. Math. Soc. 13, 388–392 (1962) 6. Cohen, G., Lin, M.: Double coboundaries for commuting contractions. Pure Appl. Funct. Anal. 2, 11–36 (2017) 7. Derriennic, Y., Lin, M.: Fractional Poisson equations and ergodic theorems for fractional coboundaries. Israel J. Math. 123, 93–130 (2001) 8. Devys, O.: Localisation spectrale à l’aide de polynômes de Faber et équation de cobord, Ph.D. thesis, Université de Lille, (2012) 9. Fortet, R.: Sur une suite également répartie. Studia Math. 9, 54–69 (1940) 10. Fukuyama, K.: On a gap series of Mark Kac. Colloq. Math. 81(2), 157–160 (1999) 11. Gomilko, A., Haase, M., Tomilov, Y.: On rates in mean ergodic theorems. Math. Res. Lett. 18(2), 201–213 (2011)  12. Kac, M.: On the distribution of values of sums of the type f (2k t). Ann. of Math. (2) 47, 33–49 (1946) 13. Kozma, G., Lev, N.: Exponential Riesz bases, discrepancy of irrational rotations and BMO. J. Fourier Anal. Appl. 17(5), 879–898 (2011) 14. Krengel, U.: Ergodic theorems. With a supplement by Antoine Brunel. de Gruyter Studies in Mathematics, 6. Walter de Gruyter and Co., Berlin. viii+357 pp. ISBN: 3-11-008478-3 (1985) 15. Levan, N.: Canonical decompositions of completely nonunitary contractions. J. Math. Anal. Appl. 101, 514–526 (1984) 16. Lin, M.: On quasi-compact Markov operators. Ann. Probab. 2, 464–475 (1974) 17. Lin, M., Sine, R.: Ergodic theory and the functional equation (I − T )x = y. J. Oper. Theory 10, 153–166 (1983) 18. Sz.-Nagy, B., Foia¸s, C., Bercovici, H., Kérchy, L.: Harmonic analysis of operators on Hilbert space. Second edition. Revised and enlarged edition. Universitext. Springer, New York, 2010 (First edition published in 1967) 19. Robinson, E.: Sums of stationary random variables. Proc. Amer. Math. Soc. 11, 77–79 (1960) 20. Rochberg, R.: The equation (I − S)g = f for shift operators in Hilbert space. Proc. Amer. Math. Soc. 19, 123–129 (1968)

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Rochberg’s Abstract Coboundary Theorem... 21. Shiryaev, A.N.: Probability. 2. Third edition. Translated from the 2007 fourth Russian edition by R. P. Boas and D. M. Chibisov. Graduate Texts in Mathematics, 95. Springer, New York, (2019) 22. Zygmund, A.: Trigonometric series. Vol. I, II. Third edition. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, (2002). xii; Vol. I: xiv+383 pp.; Vol. II: viii+364 pp. ISBN: 0-521-89053-5 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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35

Complex Analysis and Operator Theory (2023) 17:25 https://doi.org/10.1007/s11785-022-01282-z

Complex Analysis and Operator Theory

Dilation Theory and Functional Models for Tetrablock Contractions Joseph A. Ball1 · Haripada Sau2 Received: 1 July 2022 / Accepted: 18 September 2022 / Published online: 11 January 2023 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023

Abstract A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator T can be dilated to a unitary U, i.e., T n = PH U n |H for all n = 0, 1, 2, . . .. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain  contained in Cd , (ii) the contraction operator T is replaced by an contraction, i.e., a commutative operator d-tuple T = (T1 , . . . , Td ) on a Hilbert space H such that r (T1 , . . . , Td )L(H) ≤ supλ∈ |r (λ)| for all rational functions with no singularities in  and the unitary operator U is replaced by an -unitary operator tuple, i.e., a commutative operator d-tuple U = (U1 , . . . , Ud ) of commuting normal operators with joint spectrum contained in the distinguished boundary b of . For a given domain  ⊂ Cd , the rational dilation question asks: given an -contraction T on H, is it always possible to find an -unitary U on a larger Hilbert space K ⊃ H so that, for any d-variable rational function without singularities in , one can recover r (T ) as r (T ) = PH r (U)|H . We focus here on the case where  = E, a domain in C3 called the tetrablock. (i) We identify a complete set of unitary invariants for a

Dedicated to the memory of Jörg Eschmeier: a fine mentor, researcher, and colleague Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht. The research of the second named author was supported by DST-INSPIRE Faculty Fellowship DST/INSPIRE/04/2018/002458.

B

Joseph A. Ball [email protected] Haripada Sau [email protected]; [email protected]

1

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA

2

Department of Mathematics, Indian Institute of Science Education and Research Pune, Pune, Maharashtra 411008, India

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37

J. A. Ball, H. Sau

E-contraction (A, B, T ) which can then be used to write down a functional model for (A, B, T ), thereby extending earlier results only done for a special case, (ii) we identify the class of pseudo-commutative E-isometries (a priori slightly larger than the class of E-isometries) to which any E-contraction can be lifted, and (iii) we use our functional model to recover an earlier result on the existence and uniqueness of a E-isometric lift (V1 , V2 , V3 ) of a special type for a E-contraction (A, B, T ). Keywords Commutative contractive operator-tuples · Functional model · Unitary dilation · Isometric lift · Spectral set · Pseudo-commutative contractive lift Mathematics Subject Classification Primary 47A13; Secondary 47A20 · 47A25 · 47A56 · 47A68 · 30H10

1 Introduction Suppose that we are given a commutative tuple T = (T1 , . . . , Td ) of operators on a Hilbert space H together with a bounded domain  contained in d-dimensional Euclidean space Cd . We now recall the notion of  being a spectral set and  being a complete spectral set for the commutative d-tuple T and refer to the original paper of Arveson [7] for additional details on this and the related matters which follow. We say that  is a spectral set for T if it is the case that r (T)B(H) ≤ sup |r (λ)| for r ∈ Rat() λ∈

(1.1)

where we set Rat() equal to the space of all d-variable scalar-valued rational functions r having no singularities in  and B(H) equal to the Banach algebra of all bounded linear operators on H. Here r (T) can be defined via the functional calculus given by r (T) = p(T1 , . . . , Td )q(T1 , . . . , Td )−1 where ( p, q) is a coprime pair of d-variable polynomials such that r = p/q. In analogy with what happens for the case  equal to the unit disk D (see the discussion below), we say simply that T is an -contraction if it is the case that  is a spectral set for T. We say that  is a complete spectral set for T if (1.1) continues to hold when one substitutes matrix rational functions R(λ) = [ri j (λ)]i, j=1,...,n = [ pi j (λ)qi j (λ)−1 ] having no singularities in : R(T)B(Hn ) ≤ sup R(λ)Cn×n . λ∈

The seminal result of Arveson (see [7]) is that  is a complete spectral set for T if and only if there is a commutative d-tuple of normal operators N = (N1 , . . . , Nd )  ⊃ H with joint spectrum contained in the distinguished on a larger Hilbert space K boundary b of  (in which case we say that N is a -unitary for short) so that, for

38

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Dilation Theory and Functional Models for Tetrablock…

any rational function r with no singularities in  as above, it is the case that r (T) on H can be represented as the compression of r (N) to H, i.e., r (T) = PH r (N)|H where PH is the orthogonal projection of K onto H. It is easy to see that a necessary condition for  to be a complete spectral set for a given operator d-tuple T is that  be a spectral set for T. The rational dilation problem for a given domain  is to determine if the converse holds: given , is it always the case that an operator tuple T having  as a spectral set in fact has  as a complete spectral set (and hence then any T having  as a spectral set has a d-normal dilation N)? Let us mention that it is often convenient to reformulate the problem of existence of an -unitary dilation instead as the problem of existence of a -isometric lift (see e.g. the introduction of [9]). Here we say that the operator tuple V = (V1 , . . . , Vd ) on a Hilbert space K is a -isometry if V extends to a -unitary operator tuple  ⊃ K. We say that V = (V1 , . . . , Vd ) on K U = (U1 , . . . , Ud ) on a Hilbert spaces K is a lift of T = (T1 , . . . , Td ) on H if H ⊂ K and r (V)∗ |H = r (T)∗ for r ∈ Rat(), or equivalently, V is a coextension of T in the sense that PH r (V)|H = r (T) and r (V)H⊥ ⊂ H⊥ for r ∈ Rat(). It suffices to consider only minimal -unitary dilations and minimal -isometric lifts.  It is always the case that the restriction of a -unitary dilation to the subspace r ∈Rat() r (U)H gives rise to a minimal -isometric lift, and conversely, the minimal -unitary extension of a minimal -isometric lift gives rise to a minimal -unitary dilation for T. Finally we point out that it is often convenient to be more flexible in the definition of an -isometric lift and of an -unitary dilation by not insisting that H is  but rather allow an isometric identification map  : H → K and a subspace of K or K    : H → K. Thus we say that the pair (, V) is an an -isometric lift for T on H if  : H → K is an isometric embedding, V is -isometric on K and r (V)∗  = r (T)∗  is an isometric  , U) is a -unitary dilation of T if  : H → K for r ∈ Rat(), while ( ∗  embedding, U is -unitary on K, and  r (U) = r (T) for r ∈ Rat(). The motivating classical example for this setup is the case where  is the unit disk D ⊂ C. In this case, the distinguished boundary bD of D is the same as the boundary ∂D which is the unit circle T and a bD-normal operator is just a unitary operator. Since D is polynomially convex, it suffices to work with polynomials rather than rational functions with no poles in D. By choosing the polynomial p to be p = χ and χ (λ) = λ, we see that T  ≤ 1 (i.e., that T be a contraction) is necessary for D to be a spectral set for T . The fact that this condition is also sufficient, i.e., that the inequality  p(T ) ≤ sup | p(λ)| λ∈D

holds for any contraction operator T and polynomial p, is a classical inequality known as von Neumann’s inequality going back to [34]. to show that D is a complete spectral

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39

J. A. Ball, H. Sau

set for any contraction operator T , we may use the easier side of Arveson’s theorem and show instead that any contraction operator T has a D-unitary dilation. But for the case  = D, according to our conventions, a D-unitary operator is just a unitary operator U (i.e., U ∗ U = UU ∗ = IK  ). But any contraction operator T on H dilating  ⊃ H is exactly the content of the Sz.-Nagy dilation to a unitary operator U on K theorem (see [32, Chapter II]). Over the ensuing decades there have been sporadic attempts to find other domains (both contained in C or more generally contained in Cd ) for which one can settle the rational dilation question one way or the other (i.e., positively or negatively). Among single-variable domains (as observed in the introduction of [9] where precise references are given), it is known that rational dilation holds if  ⊂ C is a simply connected domain (simply use a conformal map to reduce to the disk case) or is doublyconnected, but fails if  has two or more holes (see [2, 21]). As for multivariable domains, perhaps the first class to be understood are the polydisks Dd with d ≥ 2: for d = 2 rational dilation holds due to the Andô dilation theorem [5] while for d ≥ 3 rational dilation fails (see [29, 35]). More recently the rational dilation problem has been investigated for other concrete multivariable domains originally discovered due to connections with the μ-synthesis problem in Robust Control Theory (see the original Doyle-Packard paper [20] as well as the book [24] for a more expository treatment). We mention in particular the symmetrized bidisk  = {(s, p) ∈ C2 : s = (λ1 + λ2 ), p = λ1 λ2 for some (λ1 , λ2 ) ∈ D2 }

(1.2)

and a domain in C3 called the tetrablock and denoted by E:     a a E := (a, b, detX ) : X =  with X  < 1 . b b

(1.3)

As might be expected, the domain  behaves like D2 with respect to the rational dilation problem as both domains are contained in C2 : specifically, rational dilation holds for the domain  (see [3, 4, 13]) and there is a functional model analogous to the Sz.-Nagy–Foias model for the disk case (see [4, 14]), at least for the pure case. The situation of the rational dilation problem for the tetrablock E is less clear: there is a sufficient and a necessary condition for the existence of a E-isometric lift of a certain form [9, 12] but a definitive resolution of the problem in full generality remains elusive (see [9, 26]). However it is shown in [30] that, at least in the pure case, it is still possible to construct a functional representation of a pure -contraction as the compression to H of a certain lift triple (A , B , T ) which formally looks like an tetrablock isometry but is not guaranteed to satisfy all of the required commutativity conditions. A similar phenomenon holds for the case where  = Dd with d ≥ 2 (see [11]): for this case, as pointed out above, there are indeed counterexamples to show that rational dilation fails, but there is nevertheless a weaker type of lift (called pseudo-commutative Dd -isometric lift)) which generates a functional model for the given Dd -contractive d-tuple T = (T1 , . . . , Td ) even when rational dilation fails.

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In this paper we focus on the case  = E. As was the case in [12], the most definitive results are for the case of what we shall call a special tetrablock contraction, i.e., a tetrablock contraction (A, B, T ) which has a tetrablock isometric lift (V1 , V2 , V3 ) such that V3 = V is a Sz.-Nagy–Foias minimal isometric lift for the single contraction operator T . As in [12], we identify the additional commutativity conditions (2.8) which must be imposed on the Fundamental Operator pair (G 1 , G 2 ) of (A∗ , B ∗ , T ∗ ) which characterizes when (A, B, T ) is special. There results a Douglas-type functional model (as in [19] for the single contraction operator setting) for the tetrablock contraction which also exhibits the tetrablock isometric lift (V1 , V2 , V3 ), all in a functional-model form rather than via block-matrix constructions as in [12]. This Douglas-type model can in turn be converted to a Sz.-Nagy–Foias-type model; the Sz.-Nagy–Foias characteristic function T for the contraction operator T , together with the the fundamental operators (G 1 , G 2 ) for the adjoint tetrablock contraction (A∗ , B ∗ , T ∗ ), along with some additional information needed to handle the case where T is not a pure contraction, form what we call a characteristic tetrablock data set for (A, B, T ) in terms of which one can write down the functional model. Conversely, we identify a collection of objects which we call a special tetrablock data set: specifically, (i) a pure contractive operator function (D, D∗ , ), (ii) a pair of operators (G 1 , G 2 ) on the coefficient space D∗ , (iii) a tetrablock unitary (R, S, W ) acting on D · L 2 (D), such that (iv) all these together satisfy a natural invariant-subspace compatibility condition. From such a characteristic tetrablock data set we construct a functional model such that the embedded functional-model operator triple is the most general special tetrablock contraction up to unitary equivalence, with its special tetrablock isometric lift also embedded in the functional model. We also are careful to push the theory as far as we can without the assumption that the original tetrablock contraction is special. In this case we identify a class of operator triples (V1 , V2 , V3 ) with V3 equal to a minimal isometric lift for T to which (A, B, T ) can be lifted: here V! and V2 commute with V3 but not necessarily with each other and it appears that V1 , V2 need not be contractions. In this case there is no converse direction: there is no guarantee that the compression of a general pseudo-commutative tetrablock isometry (V1 , V2 , V3 ) on K back to H will yield a tetrablock contraction. Let us note that the recent paper of Bisai and Pal [16] contains closely related results. These authors basically compute the Z -transform of the Schäffer-type construction of the unique special tetrablock isometric lift (V1 , V2 , V3 ) (where V3 is equal to the minimal Sz.-Nagy isometric lift of T ) to arrive at a functional model for this lift. Our approach on the other hand uses the Douglas lifting approach to construct the functional model directly with the existence and uniqueness of the special tetrablock isometric lift falling out as part of the construction. When the tetrablock contraction is not special and no such lift is possible, the same construction still leads to a functional model but (V1 , V2 , V3 ) is only a pseudo-commutative tetrablock isometry and there is no tetrablock isometric lift constructed in this way. The results for the special case arise as a special case (the case where the Fundamental Operator pair (G 1 , G 2 ) for the tetrablock contraction (A∗ , B ∗ , T ∗ ) satisfy the additional commutativity conditions (2.8)) of the general functional-model construction. The paper [16] also obtains a noncommutative functional model for a non-special case, based on the work of Durszt

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[25] (a variation of the approach of Douglas for the construction of the minimal isometric lift for the case of a single contraction operator T ), but with the additional hypothesis that A and B commute not only with T but also with T ∗ . It is clear that the complete unitary invariant for a pure tetrablock contraction (A, B, T ) consists of the characteristic function T of T together with the Fundamental Operator pair (G 1 , G 2 ) of (A∗ , B ∗ , T ∗ ); for the non-pure case (where T is no longer inner) we add a certain tetrablock unitary (R, S, W ) acting on T H 2 (DT ) which is part of our model (see Theorem 4.5 below), while Bisai-Pal add the Fundamental Operator pair (F1 , F2 ) for (A, B, T ) and argue that ( T , (F1 , F2 ), G 1 , G 2 )) is a complete unitary invariant. It remains to be seen which is the more relevant and useful in the future. It is now becoming clear that the domains Dd (polydisk),  (symmetrized bidisk), E (tetrablock) as well as Dds (symmetrized polydisk) all have common features with respect to the associated operator theory and applications to the rational dilation problem for each of these domains. The paper [11] shows how a program completely parallel to that done here for the tetrablock case can be worked out equally well for the polydisk case  = Dd (where rational dilation is known to fail when d ≥ 3). In all these settings, there appear a pair of unitary invariants called Fundamental Operators which play a key role as part of a set (including the Sz.-Nagy–Foias characteristic function of an appropriate contraction operator determined by the operator tuple) of unitary invariants for the operator tuple of whatever class. The notion of Fundamental Operators as a fundamental object of interest seems to have appeared first in connection with the symmetrized bidisk  [14], then in connection with the un-symmetrized polydisk [11, 31], and now also in connection with the symmetrized polydisk (see [28]). Often the proper notion of Fundamental Operators for one setting is found by making a correspondence of the less understood setting with some other better understood setting, and then adapting definitions for the first to become definitions for the second. In particular, many of the results for the tetrablock case were originally found by adapting from results for the symmetrized bidisk case (see e.g. [12]), and it has been shown how one can deduce the bidisk functional model from the tetrablock functional model (see [31]). In this spirit in a future publication we plan to show how the results from [11] for the polydisk case (most of which are just statements parallel to what is done here for the tetrablock case) can alternatively be derived as a corollary of the corresponding results for the tetrablock case via the simple observation: if T = (T1 , . . . , Td ) is a commutative, contractive operator d-tuple, then for 1 ≤ i ≤ d, if we set T(i) = 1≤ j≤d : j =i T j , then for each i = 1, . . . , d the d-tuple (Ai , Bi , P) = (Ti , T(i) , 1≤ j≤d T j ) is a tetrablock contraction; the d = 2 case can be found in [31, Section 3, Version 3]. Finally, let us point out that it is possible to reformulate the rational dilation problem for a given domain  as a problem about unital representations of a unital function algebra: given a contractive representation π : f ∈ A → π( f ) ∈ B(H) which is contractive (π( f )B(H) ≤  f B(H) where the unital representation property is that π(1A ) = IH and π( f 1 · f 2 ) = π( f 1 )π( f 2 ), is it automatically the case that the representation is completely contractive, i.e., still contractive after tensoring with Cn×n for any n ∈ N? To recover the original formulation as a special case, one can take A = Rat() where the closure is in the C ∗ -algebra C(b) (continuous functions on

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the distinguished boundary b). However with this more general formulation one can consider function algebras which go beyond Rat(), e.g., the constrained subalgebra C · 1 + z 2 A(D) of the disk algebra A(D) = Rat(D). Alternatively, it is often possible to represent the algebra A as conformally equivalent to the algebra of all functions analytic on some algebraic curve  = C embedded in some higher-dimensional closed complex manifold (the Neil parabola intersected with the bidisk for the case of C · 1 + z 2 A(D)). For the state of knowledge (up to 2018) on this direction of dilation theory including much discussion and references on earlier work, we refer to the paper of Dritschel and Undrakh [23]. We shall not pursue this direction here. The paper is organized as follows. After the present Introduction, in Sect. 2 we collect assorted definitions and illustrative results concerning tetrablock contractions, tetrablock isometries, and tetrablock unitaries, including a direct proof of the existence of the Fundamental Operator pair for a given tetrablock contraction, which will be needed in the sequel. Here we also show how to associate a tetrablock unitary (R, S, W ) with a tetrablock contraction (A, B, T ) in a canonical way; this is the key ingredient needed to eliminate the purity assumption on the contraction operator T required in earlier work on this problem (see [30]). Section 3 shows how a lifting framework for the tetrablock-contraction setting can be constructed as an embellishment of the Douglas-model lifting framework [19] originally formulated as an approach to the Sz.-Nagy dilation theorem for a single contraction operator T , with the pseudocommutative tetrablock-isometric lift (V1 , V2 , V3 ) having V3 = T and V1 and V2 constructed by making use of the Fundamental Operator pair for the tetrablock contraction (A∗ , B ∗ , T ∗ ). The final Sect. 4 identifies the invariants required to write down a functional model equipped with a model operator triple (A, B, T ) which is concrete functional-model version of a general tetrablock contraction.

2 The Fundamentals of Tetrablock Contractions This section gives a brief introduction to the operator theory associated with the tetrablock. 2.1 Tetrablock Contractions The tetrablock, denoted by E, is the non-convex but polynomially convex domain in C3 given by (1.3). From this formula for E is is easy to read off the following symmetry properties. Proposition 2.1 The tetrablock E has the following symmetry properties: (1) E is invariant under complex conjugation: (a, b, t) ∈ E ⇔ (a, b, t) ∈ E. (2) E is invariant under interchange of the first two coordinates: (a, b, t) ∈ E ⇔ (b, a, t) ∈ E.

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The distinguished boundary of E, i.e., the Šilov boundary with respect to the algebra of functions that are analytic in E and continuous on E, is given by     a a bE := (a, b, detX ) : X =  is a unitary b b (see [1, Theorem 7.1]). From this characterization it is easy to see that bE is also invariant under the two involutions (a, b, t) → (a, b, t) and (a, b, t) → (b, a, t). Several tractable characterizations of the tetrablock can be found in [1, Theorem 2.2]; we pick two of these that will be used in what follows. Theorem 2.2 For a point (a, b, t) ∈ C3 , the following are equivalent: (i) (a, b, t) ∈ E; (ii) with the rational function : D × C3 → C defined as (z, (a, b, t)) =

a − zt , 1 − zb

(2.1)

supz∈D | (z, (a, b, t))| < 1; and if ab = t then, in addition, |b| < 1; (iii) with as in (2.1), supz∈D | (z, (b, a, t))| < 1; and if ab = t then, in addition, |a| < 1. Moreover, when item (i) is replaced by (a, b, t) ∈ E, then all the strict inequalities in items (ii) and (iii) are replaced by non-strict inequalities. Remark 2.3 Note that in Theorem 2.2, the equivalence of (i) ⇔ (iii) is an immediate consequence of the equivalence of (i) ⇔ (ii) in view of the invariance of E under the involution (a, b, t) → (b, a, t). Recall the notions of tetrablock unitary, tetrablock isometry and tetrablock contraction given in the Introduction. Several algebraic characterizations of tetrablock isometries and tetrablock unitaries are known; see Theorems 5.4 and 5.7 in [12]. We recall the ones that are useful for our purposes here. Here we use the notation r (X ) for the spectral radius of a Hilbert-space operator X . It is then not difficult to see that the E-symmetries noted in Proposition 2.1 imply the same symmetries on the respective operator classes (with respect to the class of E-isometries which requires a little extra care), as noted in the next result. We leave the easy verification as an exercise for the reader. Proposition 2.4 Suppose that (A, B, T ) is a triple of bounded operators on a Hilbert space H. Then: (1) (A, B, T ) is a E-contraction ⇔ (A∗ , B ∗ , T ∗ ) is a E-contraction ⇔ (B, A, T ) is a E-contraction. (2) (A, B, T ) is a E-isometry ⇔ (B, A, T ) is a E-isometry. (3) (A, B, T ) is a E-unitary ⇔ (A∗ , B ∗ , T ∗ ) is a E-unitary ⇔ (B, A, T ) is a Eunitary.

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Theorem 2.5 Let (A, B, T ) be a commutative triple of bounded Hilbert space operators. Then the following are equivalent: (i) (ii) (iii) (iv) (v)

(A, B, T ) is a tetrablock isometry (respectively unitary); (A, B, T ) is a tetrablock contraction and T is an isometry (respectively unitary); A = B ∗ T , B is a contraction and T is an isometry (respectively unitary); and B = A∗ T , A is a contraction and T is an isometry (respectively unitary). B = A∗ T , r (A) ≤ 1 and r (B) ≤ 1, and T is an isometry (respectively unitary).

2.2 Pseudo-commutative Tetrablock Isometries and Unitaries We propose to introduce the notions of pseudo-commutative tetrablock unitary and pseudo-commutative tetrablock isometry for an operator triple (A, B, T ) by using criterion (iii) or equivalently (iv) in Theorem 2.5 but with the weakening the commutativity hypothesis imposed on the whole triple (A, B, T ) to just the condition that A and B commute with T (but not necessarily with each other). As we are also dropping the condition that A or B be a contraction, a more proper term would be noncontractive pseudo-commutative tetrablock isometry, but, as this term will be consistent throughout, we settle on the shorter term for brevity. The resulting definition is as follows. We leave it to the reader to verify that the two formulations are equivalent. Definition 2.6 Let (A, B, T ) be a triple of bounded Hilbert-space operators. We say that the triple (A, B, T ) is a pseudo-commutative tetrablock isometry (respectively, unitary) if any of the following equivalent conditions holds: (1) T is an isometry (respectively, unitary) and AT = T A,

BT = T B,

A = B∗T .

(2.2)

B = A∗ T .

(2.3)

(2) T is an isometry (respectively, unitary), and AT = T A,

BT = T B,

Remark 2.7 From Definition 2.6 and Theorem 2.5, we see that any tetrablock isometry/unitary is also a pseudo-commutative tetrablock isometry/unitary but not conversely. If we wish to emphasize that we are referring to the logically more special tetrablock isometry/ unitary rather than the more general pseudo-commutative tetrablock isometry/unitary, we often will say strict tetrablock isometry/unitary for emphasis. Remark 2.8 We now present a couple of elementary observations on pseudocommutative versus strict tetrablock unitaries which we hope give the reader some additional insight. (1) We remark that if (A, B, T ) is a pseudo-commutative tetrablock unitary, then A and B are not necessarily normal operators as would happen in the strict case. For example, pick a non-normal contraction G 1 acting on a Hilbert space E and consider the triple (MG ∗1 , Mζ G 1 , Mζ ) on L 2 (E). It is easy to see that this triple Reprinted from the journal

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is a pseudo-commutative tetrablock unitary. However, neither A nor B is normal unless G 1 is so. Also note that if (A, B, T ) is a pseudo-commutative tetrablock unitary, then so is the adjoint triple (A∗ , B ∗ , T ∗ ). This can be seen by observing that the adjoint of the identities in (2.2) with T unitary can be converted to the identities (2.3) for (A∗ , B ∗ , T ∗ ) with T ∗ still unitary. Note next that if (A, B, T ) is a pseudo-commutative tetrablock unitary, then A∗ A = T ∗ B B ∗ T = BT ∗ T B ∗ = B B ∗ , B ∗ B = T ∗ A A∗ T = AT ∗ T A∗ = A A∗ . Thus we always have A∗ A = B B ∗ ,

A A∗ = B ∗ B

(2.4)

for a pseudo-commutative tetrablock unitary (A, B, T ). As a first consequence of (2.4), we see that if A is normal, then B ∗ B = A A∗ = A∗ A = B B ∗ and B is also normal. Similarly if B is normal, then A is also normal. In conclusion, if (A, B, T ) is a pseudo-commutative unitary such that one of A or B is normal, then so is the other. (2) We note as a consequence of (iii) ⇒ (i) in Theorem 2.5 that in particular if (A, B, T ) is a strict tetrablock unitary (so we also have AB = B A), then the operators A and B are normal. One can see this directly from the considerations here as follows. As a strict tetrablock unitary in particular meets all the requirements for membership in the pseudo-commutative tetrablock unitary class, we know that (2.4) holds. Combining this with the commutativity relation AB = B A then gives us A∗ A = A∗ B ∗ T = B ∗ A∗ T = B ∗ B = A A∗ showing that A is normal. The same computation with the roles of A and B interchanged then shows that B is also normal. The full strength of (iii) ⇒ (i) in Theorem 2.5 is that in addition the commutative normal triple (A, B, T ) has joint spectrum in the boundary of the tetrablock E; for this somewhat deeper fact we refer to [12]. The next result gives a feel for how close pseudo-commutative tetrablock isometries come to being strict tetrablock isometries. Theorem 2.9 Let (A, B, T ) be a pseudo-commutative tetrablock isometry on a Hilbert space H. (1) Then the spectral radius r (AB) of the product operator AB is given by r (AB) = max{A2 , B2 .

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(2.5)

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(2) Suppose in addition that AB = B A and r (A) ≤ 1, r (B) ≤ 1. Then both A and B are contraction operators (max{A, B} ≤ 1) and (A, B, T ) is a strict tetrablock isometry. Proof The proof follows the ideas of Bhattacharyya  [12,  pp.1619-1620]. We first consider statement (1). Form two operators X 1 = B0 A0 and X 2 = T0 T0 on H H . From the two relations B ∗ T = A and A∗ T = B we deduce that X 1 = X 1∗ X 2 where  ∗ X 2 X 2∗ = T 0T T 0T ∗  0I 0I since T is an isometry. Hence X 1 X 1∗ = X 1∗ X 2 X 2∗ X 1  X 1∗ X 1 , i.e., X 1 is a hyponormal operator. By a theorem of Stampfli (see [17, Proposition 4.6], it follows that r (X 1 ) = X 1 . We compute the operator norm of X 1 as follows:

    ∗ 

AA 0

0 A 0 B∗



= max{A2 , B 2 }.

= X 1  =

B 0 A∗ 0 0 B B ∗

2

∗ hence X 1  = max{A, B}. To compute r (X 1 ), note first that X 1 X 1 = and AB 0 0 B A and hence

X 12n =

  (AB)n 0 . 0 (B A)n

Consequently, 1

1

1

1

r (X 1 ) = lim max (AB)n  2n , (B A)n  2n } = max{r (AB) 2 , r (B A) 2 }. n→∞

However a general fact is that the nonzero spectrum of AB is the same as the nonzero spectrum of B A, and hence r (AB) = r (B A). Thus r (X 1 ) = X 1  gives us (2.5) and the proof of statement (1) is complete. As for statement (2), a known fact is that if A and B commute, then the spectrum of the product operator AB is given by σ (AB) = {λ · μ : λ ∈ σ (A), μ ∈ σ (B)}. Hence the hypothesis that r (A) ≤ 1 and r (B) ≤ 1 implies that r (AB) ≤ 1 as well. But then from the conclusion of statement (1) already proved, we conclude that both A and B are contraction operators, and the proof of statement (2) is now complete. (Note that this also proves (v) ⇒ (iii) or (iv) in Theorem 2.5.)   Example 2.10 (1) A pseudo-commutative/strict tetrablock isometry. Let E be a coefficient Hilbert space and H 2 (E) = H 2 ⊗ E be the associated Hardy space of E-valued functions. Let G 1 and G 2 be operators on E and set A = MG ∗1 +zG 2 , Reprinted from the journal

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Then it is immediate that T is an isometry and that A and B commute with T . The special coupled form of the pencils defining A and B enables us to show that A = B∗T : B ∗ T = (I H 2 ⊗ G ∗2 + Mz ⊗ G 1 )∗ · (Mz ⊗ IG )

= (I H 2 ⊗ G 2 + Mz∗ ⊗ G ∗1 ) · (Mz ⊗ IG ) = (I H 2 ⊗ G ∗1 ) + (Mz ⊗ G 2 ) = MG ∗1 +zG 2 = A

and similarly B = A∗ T . For (A, B, T ) to be a strict tetrablock isometry, we need in addition that AB = B A and that A ≤ 1 (in which case also B = A∗ T  ≤ 1 as well). To ensure that A ≤ 1 requires that G 1 and G 2 are not too large in the precise sense that sup G ∗1 + zG 2  ≤ 1.

(2.7)

z∈T

To check the condition AB = B A, we compute AB = MG ∗1 +zG 2 MG ∗2 +zG 1

= (I H 2 ⊗ G ∗1 + Mz ⊗ G 2 ) · (I H 2 ⊗ G ∗2 + Mz ⊗ G 1 ) = I H 2 ⊗ G ∗1 G ∗2 + Mz ⊗ (G ∗1 G 1 + G 2 G ∗2 ) + Mz2 ⊗ G 1 G 2

while a similar computation gives us B A = I H 2 ⊗ G ∗2 G ∗1 + Mz ⊗ (G 1 G ∗1 + G ∗2 G 2 ) + Mz2 ⊗ G 2 G 1 . We conclude that in this example, (A, B, T ) is a strict tetrablock isometry exactly when (2.7) together with the following commutativity conditions hold: G 1 G 2 = G 2 G 1 , G ∗1 G 1 + G 2 G ∗2 = G 1 G ∗1 + G ∗2 G 2 ,

(2.8)

sometimes also written more compactly in terms of commutators as [G 1 , G 2 ] = 0, [G ∗1 , G 1 ] = [G ∗2 , G 2 ] where in general [X , Y ] is the commutator: [X , Y ] = X Y − Y X . (2) A pseudo-commutative/strict tetrablock unitary. It is easy to use the spectral theory for unitary operators (a particular case of the spectral theory of normal operators) to write down a model for the general pseudo-commutative/strict tetrablock unitary (R, S, W ). as follows. By Definition 2.6 we see in particular that W is unitary. By the spectral theory for general normal operators (see e,g, any

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of [6, 18] or [8, Chapter 2]), after a unitary change of coordinates, we can represent W as the operator Mζ of multiplication  by the coordinate function (Mζ : h(ζ ) → ζ h(ζ )) on a direct-integral space T Hζ ν(dζ ) determined by a scalar spectral measure ν supported on T and a measurable multiplicity function W = Mζ , it follows ζ → dim Hζ .. Since the operators R and S commute with  that R and S are represented as decomposable operators on T Hζ d˚(ı), i.e., R = Mφ : h(ζ ) → φ(ζ )h(ζ ) and S = Mψ : h(ζ ) → ψ(ζ )h(ζ ) for measurable functions such that φ(ζ ) ∈ B(Hζ ), ψ(ζ ) ∈ B(Hζ ) for a.e. ζ . The fact that in addition R = S ∗ W then forces φ(ζ ) = ψ(ζ )∗ · ζ for a.e. ζ , Thus any pseudocommutative tetrablock unitary has the form (R, S, W ) = (Mψ ∗ ·ζ , Mψ , Mζ ) acting on

 T

Hζ ν(dı)

(2.9)

If (R, S, W ) is a strict tetrablock unitary, then in addition we must have that ψ(ζ ) is a contractive normal operator on Hζ for a.e. ζ in order to guarantee in addition that RS = S R and that R ≤ 1, S ≤ 1. By this analysis we conclude that (2.9) (with ψ(ζ ) constrained to be contractive normal for a.e. ζ for the strict case) is the general form for a pseudo-commutative/strict tetrablock unitary. In a lessfunctional form, to write a pseudo-contractive tetrablock triple (R, S, W ), the free parameters are: (i) a unitary operator W , and (ii) an operator S commuting with W ; then the associated pseudo-commutative tetrablock contraction is (W ∗ S, S, W ); for this to be strict, one must require in addition that the operator S in the commutant of W be a normal contraction. To deduce the von Neumann-Wold decomposition for a tetrablock isometry, the next lemma is useful. Only the special case where the operator S in the statement is a shift will be needed for our application, in which case the result is well-known (see e.g. [33, page 22]). For completeness we present here a proof of the general result. Lemma 2.11 Let W be a unitary operator on H2 , S an operator on H1 such that S ∗n → 0 in the strong operator topology as n → ∞. If X is a bounded operator from H2 to H1 such that X W = S X , then X = 0. Proof From X W = S X we get by iteration that X W n = S n X for n = 1, 2, . . .. Taking adjoints gives then W ∗n X ∗ = X ∗ S ∗n . Apply this identity to an arbitrary fixed vector x ∈ H2 to get W ∗n X ∗ x = X ∗ S ∗n x for all n ≥ 1. Apply W n to both sides of this equation to get W n W ∗n X ∗ x = W n X ∗ S ∗n x for n = 1, 2, . . .. As W is unitary, this becomes X ∗ x = W n X ∗ S ∗n x. Taking norms then gives X ∗ x = W n X ∗ S ∗n x = X ∗ S ∗n x ≤ X ∗ S ∗n x → 0 as n → ∞ by the assumed strong convergence of powers of S ∗ to zero, forcing X ∗ (and hence also X ) to be the zero operator.   The von Neumann-Wold decomposition (see [32, 34, 36]) ensures that if T is an isometry acting on a Hilbert space H, then T can be represented as an operator as the

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external direct sum Mz ⊕ U of a shift operator Mz acting on a Hardy space H 2 (E) and a unitary operator W on F for some coefficient Hilbert spaces E and F. The following result not only gives a model for an arbitrary pseudo-commutative/strict tetrablock isometry (A, B, T ), but also can be seen as a pseudo-commutative/strict tetrablock-isometry analogue of the classical von Neumann–Wold decomposition for a single isometric Hilbert space operator T . Theorem 2.12 Let (A, B, T ) be an operator-triple on the Hilbert space H. (1) Then (A, B, T ) is a pseudo-commutative tetrablock isometry on H if and only if there exist Hilbert spaces E, F, operators G 1 , G 2 acting on E subject to (2.7), along with a pseudo-commutative   tetrablock unitary (R, S, W ) acting on F, such 2 (E ) H that H is isomorphic to and under the same isomorphism (A, B, T ) is F unitarily equivalent to 

     MG ∗2 +zG 1 0 MG ∗1 +zG 2 0 Mz 0 , , . 0 W 0 R 0 S

(2.10)

(2) Then (A, B, T ) is a strict tetrablock isometry on H if and only if (A, B, T ) is unitarily equivalent to the operator triple as in (2.10) (with G 1 , G 2 subject to  2 (E ) H (2.7)) acting on a space , where in addition the operator-pencil coefficients F (G 1 , G 2 ) satisfy the system of operator identities (2.8), and the triple (R, S, W ) is a strict tetrablock unitary (i.e., we also have the relation RS = S R with R and S contraction operators).  2  Remark 2.13 We shall think of a triple of operators on H (E ) as in (2.10) as a F functional model for a pseudo-commutative/strict tetrablock isometry/unitary. The H 2 (E)-component clearly has a functional form while the second component can be brought to a measure-theoretic functional form as in item (2) in Example 2.10. Proof The sufficiency (for both the pseudo-commutative and the strict case) follows from Example 2.10. We now suppose that (A, B, T ) is a strict tetrablock isometry. Let us apply the Wold decomposition  2  to the isometry T : there exist Hilbert spaces E, F, and a unitary τ : H → H (E ) such that F

  2   2  Mz 0 H (E) H (E) : → τTτ = 0 W F F ∗



for some unitary W on F. Next assume that τ (A, B)τ ∗ =



    2   2  B B A11 A12 H (E) H (E) , 11 12 : → . A21 R B21 S F F

Now use these matrix representations and equate the (12)-entries of the relation AT = T A to get A12 W = Mz A12 . Therefore A12 = 0 by Lemma 2.11. Similarly, from

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the relation BT = T B we have B12 = 0. Compare the (21)-entries of the relation A = B ∗ T to get A21 = 0. The same treatment for the relation B = A∗ T gives B21 = 0. Therefore we are left with the following relations ∗ Mz (and B11 = A∗11 Mz ); A11 Mz = Mz A11 , B11 Mz = Mz B11 , A11 = B11

RW = W R, SW = W S, R = S ∗ W (and S = W R ∗ ).

The second set of the above relations together with the fact that W is a unitary implies that (R, S, W ) is a pseudo-commutative tetrablock unitary. The first two intertwining relations in the first set implies that there exist bounded analytic functions , : D → B(E) such that A11 = M , B11 = M . The remaining relations in the first set then give us ∗ Mz , M = M

∗ M = M Mz .

There now only remains a tedious computation with the power series expansions of  and to see that the remaining relations in the second set forces  and to have the coupled linear forms (z) = G ∗1 + zG 2 and (z) = G ∗2 + zG 1 for some operators G 1 , G 2 ∈ B(E). Again the relation (2.7) is equivalent to M being a contraction operator. From Definition 2.6 it follows that (M , M , Mz ) is a pseudo-commutative tetrablock isometry. The completes the proof for the pseudocommutative setting. Suppose now that (A, B, T ) is a tetrablock isometry. Then in particular (A, B, T ) satisfies all the requirements to be a pseudo-commutative tetrablock isometry, so all the preceding analysis applies. We then see that (A, B, T ) is unitarily equivalent to the triple in (2.10). As (A, B, T ) now is actually a tetrablock isometry, we have that AB = B A. The unitary equivalence then forces 

MG ∗1 +zG 2 0 0 R



    MG ∗2 +zG 1 0 MG ∗2 +zG 1 0 MG ∗1 +zG 2 0 = 0 S 0 S 0 R

which can be split up as two commutativity conditions MG ∗1 +zG 2 MG ∗2 +zG 1 = MG ∗2 +zG 1 MG ∗1 +zG 2

(2.11)

RS = S R.

(2.12)

By reversing the computations done in item (1) of Example 2.10, we see that the intertwining (2.11) forces the set of conditions (2.8) Moreover, the condition (2.12) is exactly the missing ingredient needed to promote (R, S, W ) from a pseudocommutative tetrablock unitary to a strict tetrablock unitary. This completes the proof.  

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Remark 2.14 It will be useful to have a terminology for an intermediate class of operator triples (A, B, T ) which sits between strict tetrablock isometries and general pseudo-commutative tetrablock isometries. Let us say that the triple (A, B, T ) is a semi-strict tetrablock isometry if (A, B, T ) is a pseudo-commutative isometry with Wold decomposition as in (2.10) is such that the pseudo-commutative tetrablock unitary component (R, S, W ) is actually a strict tetrablock unitary, i.e., R and S are contractions which commute with each other as well as with W which is unitary. We next present an analogue of the single-variable operator theory fact that any isometry can always be extended to a unitary. Corollary 2.15 Pseudo-commutative/strict tetrablock isometries can be extended to pseudo-commutative/strict tetrablock unitaries. More precisely: (1) A triple (A, B, T ) is a pseudo-commutative tetrablock isometry if and only if it extends to a pseudo-commutative tetrablock unitary. Moreover there exists an extension that acts on the space of minimal unitary extension of the isometry T . (2) A triple (A, B, T ) is a strict tetrablock isometry if and only if it extends to a strict tetrablock unitary acting on the space of the minimal unitary extension of the isometry T . Proof If a triple extends to a pseudo-commutative tetrablock unitary, then from Definition 2.6 we can read off that it is also a pseudo-commutative tetrablock isometry. Similarly, if a triple extends to a strict tetrablock unitary, we can read off from criterion (iii) or (iv) in Theorem 2.5 that the triple itself must be a strict tetrablock isometry. We now address the converse. In view of Theorem 2.12, we can assume without loss of generality that a pseudo-commutative tetrablock isometry (A, B, T ) is given in the form:      2    2   MG ∗1 +zG 2 0 MG ∗2 +zG 1 0 Mz 0 H (E) H (E) , , → : 0 W F F 0 R 0 S for some operators G 1 , G 2 on E and for some pseudo-commutative tetrablock unitary (R, S, W ) acting on F. Consider H 2 (E)⊕F as a subspace of L 2 (E)⊕F in the natural way. Then the triple 

     2    2  MG ∗2 +ζ G 1 0 MG ∗1 +ζ G 2 0 Mζ 0 L (E) L (E) , , → : 0 W F F 0 R 0 S

is an extension of (A, B, T ). The unitary Mζ ⊕W is clearly a minimal unitary extension of the isometry Mz ⊕ W . And since MG ∗1 +ζ G 2 = MG∗ ∗ +ζ G 1 Mζ and (R, S, W ) is 2 a pseudo-commutative tetrablock unitary, the above triple is a pseudo-commutative tetrablock unitary by Definition 2.6.   MG ∗ +zG 0 2 1 If we start with a strict tetrablock isometry, then we shall also have that 0 R    2  MG ∗ +zG 0 1 2 commutes with on H (E ) , or equivalently, RS = S R and the Toeplitz F 0 S operator symbols equal to the pencils G ∗1 + zG 2 and G ∗2 + zG 1 commute: (G ∗1 + zG 2 )(G ∗2 + zG 1 ) = (G ∗2 + zG 1 )(G ∗1 + zG 2 ).

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But then it is straightforward to see that this implies the commutativity of the associated Laurent operators acting on L 2 (E): MG ∗1 +ζ G 2 MG ∗2 +ζ G 1 = MG ∗2 +ζ G 1 MG ∗1 +ζ G 2 . and hence also 

 on

L 2 (E )

F

MG ∗1 +ζ G 2 0 0 R



    MG ∗2 +ζ G 1 0 MG ∗2 +ζ G 1 0 MG ∗1 +ζ G 2 0 = . 0 S 0 S 0 R

 . Moreover the extension of MG ∗1 +zG 2 on H 2 (E) to MG ∗1 +ζ G 2 on

L 2 (E) is norm-preserving, and hence the latter is contractive whenever the former is contractive, and similarly for MG ∗2 +zG 1 and MG ∗2 +ζ G 1 . We now have enough observations to conclude by criterion (iii) or (iv) in Theorem 2.5 that   MG ∗ +ζ G 0   MG ∗ +ζ G 0   Mζ 0    L 2 (E )  1 2 2 1 , , 0 W on is a tetrablock unitary as required. F 0 S 0 R   Another one-variable fact is the result due to Sz.-Nagy–Foias (see [32, Theorem I.3.2]): any contraction operator T on a Hilbert space H can be decomposed as 0 T = Tcnu where T cnu is a completely nonunitary (c.n.u.) contraction operator 0 U (meaning there is no reducing subspace of H such that T |H is unitary) and where U is unitary. There is a analogous result for the setting of tetrablock unitaries and tetrablock contractions. We say that the tetrablock contraction (A, B, T ) on H is a c.n.u. tetrablock contraction if there is no nontrivial jointly reducing subspace Hu ⊂ H for (A, B, T ) such that (A, B, T )|Hu is a tetrablock unitary. The following result appears in [27] Theorem 2.16 Let (A, B, T ) be a tetrablock contraction on H. Then H has an internal orthogonal direct-sum decomposition H = Hc.n.u. ⊕ Hu with Hc.n.u. and Hu jointly reducing for (A, B, T )|Hc.n.u. equal to a c.n.u. tetrablock contraction and (A, B, T )|Hu equal to a tetrablock unitary. The at first surprising fact is that the same decomposition H = Hc.n.u. ⊕ Hu inducing the canonical decomposition of the contraction operator T into is c.n.u. part Tc.n.u. and its unitary part Tu turns out to also be jointly reducing for the whole operator triple (A, B, T ) and induces the canonical tetrablock decomposition for the tetrablock contraction (A, B, T ), i.e., (A, B, T )|Hc.n.u. is a c.n.u. tetrablock contraction, and (A, B, T )|Hu is a tetrablock unitary. Remark 2.17 The import of Theorem 2.16 for model theory is the same as is the case for the classical case: since the model theory for a (strict) tetrablock unitary is already well understood (see item (2) in Remark 2.8, it follows that it is perfectly satisfactory to focus on the case where (A, B, T ) is a c.n.u. tetrablock contraction for the purposes of model theory.

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2.3 A Canonical Construction of a Tetrablock Unitary from a Tetrablock Contraction In this section we start with a tetrablock contraction (A, B, T ) and construct a tetrablock unitary which is uniquely associated with (A, B, T ) in a sense that will be made precise later in this subsection. This will be used later to construct a concrete functional model for a pseudo-commutative tetrablock-isometric lift for (A, B, T ) which can be viewed as a functional model for (A, B, T ) itself. We start by using the fact that the last entry T in our tetrablock contraction (A, B, T ) is a contraction operator. Hence there exist a positive semidefinite operator Q T ∗ such that Q 2T ∗ := SOT- lim T n T ∗n .

(2.13)

Define the operator X T∗ ∗ : Ran Q T ∗ → Ran Q T ∗ densely by X T∗ ∗ Q T ∗ h = Q T ∗ T ∗ h.

(2.14)

This is an isometry because for all h ∈ H, X T∗ ∗ Q T ∗ h2 = Q 2T ∗ T ∗ h, T ∗ h = lim T n T ∗n T ∗ h, T ∗ h n→∞

= lim T n→∞

n+1

T

∗(n+1)

h, h = Q 2T ∗ h, h = Q T ∗ h2 .

(2.15)

Since A is a contraction, we have for all h ∈ H AQ 2T ∗ A∗ h, h = lim T n A A∗ T ∗n h, h ≤ limT n T ∗n h, h = Q 2T ∗ h, h. n

n

The same computation for the contraction B will yield the same inequality involving B in place of A. Consequently, the operators A T ∗ .BT ∗ : Ran Q T ∗ → Ran Q T ∗ defined densely by A∗T ∗ Q T ∗ h = Q T ∗ A∗ h and BT∗ ∗ Q T ∗ h = Q T ∗ B ∗ h

(2.16)

are contractions, and extend contractively to all of Ran Q T ∗ by a limiting process. Furthermore, from the definitions it is easy to see that (A T ∗ , BT ∗ , X T ∗ ) is a commutative triple since by assumption we know that (A∗ , B ∗ , T ∗ ) is commutative. This and (2.14) imply that if f is a three-variable polynomial, then for all h ∈ H,  f (A∗T ∗ , BT∗ ∗ , X T∗ ∗ )Q T ∗ h = Q T ∗ f (A∗ , B ∗ , T ∗ )h

≤  f (A∗ , B ∗ , T ∗ )h|| ≤  f (A∗ , B ∗ , T ∗ )h   ≤ sup | f | h E

where in the last inequality we used the fact that (A, B, T ) is a tetrablock contraction. This inequality together with the fact that X T∗ ∗ is an isometry implies that 54

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(A∗T ∗ , BT∗ ∗ , X T∗ ∗ ) is a tetrablock isometry. By Corollary 2.15, (A∗T ∗ , BT∗ ∗ , X T∗ ∗ ) has ∗ , W ∗ ) acting on a space which we shall call a tetrablock unitary extension (R ∗D , S D D ∗ QT ∗ ⊇ Ran Q T ∗ , where W D acting on QT ∗ is the minimal unitary extension of X T∗ ∗ . Definition 2.18 Let (A, B, T ) be a tetrablock contraction and let (R D , S D , W D ) be the tetrablock unitary constructed from (A, B, T ) as above. We say that (R D , S D , W D ) is the canonical tetrablock unitary associated with the tetrablock contraction (A, B, T ). The next result assures us that canonical tetrablock unitaries associated with the same tetrablock contraction (A, B, T ) are the same up to unitary equivalence. Theorem 2.19 Let (A, B, T ) on H and (A , B  , T  ) on H be two tetrablock con , W  ) equal to the respective canonical tractions with (R D , S D , W D ) and (R D , S D D  tetrablock unitaries. If (A, B, T ) and (A , B  , T  ) are unitarily equivalent via τ , then  , W  ) are unitarily equivalent via ω : Q ∗ → Q  (R D , S D , W D ) and (R D , S D τ T D T ∗ 

ωτ : W Dn Q T ∗ h → W Dn Q T ∗ τ h

(2.17)

for all n ≥ 0 and h ∈ H. Proof Let the spaces QT ∗ , QT  ∗ and the operators {A T ∗ , BT ∗ , Q T ∗ }, {A T ∗ , BT ∗ , Q T ∗ } be obtained as above from (A, B, T ) and (A , B  , T  ), respectively. Since τ is a unitary intertwining T and T  , it intertwines T ∗ and T ∗ and thus τ Q T ∗ = Q T ∗ τ . Therefore by definition (2.16) it follows that τ (A T ∗ , BT ∗ , Q T ∗ ) = (A T ∗ , BT ∗ , Q T ∗ )τ . By definition of ωτ it is clear that ωτ W D = W D ωτ . Therefore for every h ∈ H and n ≥ 0, ωτ R D W Dn Q T ∗ h = ωτ W Dn+1 W D∗ R D Q T ∗ h ∗ = ωτ W Dn+1 S D Q T ∗ h [using Theorem 2.5, part (iv)] (n+1)

= WD

(n+1)

τ BT∗ ∗ Q T ∗ h = W D

(n+1) ∗ S D Q T ∗ τ h ∗ (n+1) Q T ∗ [since SD W D  n R D W D Q T ∗ τ h [using R D ωτ W Dn Q T ∗ h.

BT∗ ∗ Q T ∗ τ h

= WD = = =

 ω . A similar computation shows that ωτ S D = S D τ

∗ W  = W  S ∗ ] SD D D D

Theorem 2.5, part (iii)]

 

2.4 The Fundamental Operators Much of the theory of tetrablock contractions is heavily based on a pair of operators that is uniquely associated with a tetrablock contraction. These are called the fundamental operators, the existence of which was proved in [12] with appeal to connections between tetrablock contractions and symmetrized-bidisk contractions. We state the result and sketch a more self-contained proof with the appeal to symmetrized-bidisk

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theory eliminated. In the sequel we shall use the notation ν(X ) to denote the numerical radius of the operator X on the Hilbert space H: ν(X ) :=

sup

x∈H : x=1

|X x, xH |.

Theorem 2.20 Let (A, B, T ) be a tetrablock contraction on a Hilbert space H. (i) (See [12, Theorem 3.4]) There exist two unique operators F1 and F2 acting on DT with the numerical radii at most one such that A − B ∗ T = DT F1 DT and B − A∗ T = DT F2 DT .

(2.18)

Moreover, the operators F1 , F2 are such that ν(F1 + z F2 ) ≤ 1 for all z ∈ D. (ii) (See [12, Corollary 4.2]) The operators F1 , F2 are alternatively characterized as the unique bounded operators on DT such that (X 1 , X 2 ) = (F1 , F2 ) satisfies the system of operator equations DT A = X 1 DT + X 2∗ DT T and DT B = X 2 DT + X 1∗ DT T .

(2.19)

Proof Let (A, B, T ) be a tetrablock contraction on H. Since for every z ∈ D, (z, ·) as in item (ii) of Theorem 2.2 is analytic in an open set containing E, and E is polynomially convex, a limiting argument implies that (ζ, (A, B, T )) is a contraction for every ζ ∈ T, or equivalently, on simplifying I − (ζ, (A, B, T ))∗ (ζ, (A, B, T ))  0 we get (I − T ∗ T ) + (B ∗ B − A∗ A) − ζ (B − A∗ T ) − ζ (B − A∗ T )∗  0. Similarly, applying item (iii) of Theorem 2.2 we have for every α ∈ T, (I − T ∗ T ) + (A∗ A − B ∗ B) − α(A − B ∗ T ) − α(A − B ∗ T )∗  0. On adding the above two positive operators and then simplifying we get  DT2  Re α (A − B ∗ T ) + β(B − A∗ T )

(2.20)

for every α, β ∈ T. We now make use the following lemma of independent interest: Lemma 2.21 (See [13, Lemma 4.1]) Let  and D be two operators such that D D ∗  Re α for all α ∈ T. Then there exists an operator F acting on Ran D ∗ with numerical radius at most one such that  = D F D ∗ .

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Sketch of proof Apply the Fejer-Riesz factorization theorem of Dritschel-Rovnyak [22, Theorem 2.1] to the Laurent operator-valued polynomial P(eiθ ) = 2D D ∗ − eiθ  − e−iθ  ∗ . Along the way one makes use of the standard Douglas lemma (∃ X ∈ B(H) with AX = B ⇔ B B ∗  A A∗ ) and a criterion for a Hilbert space operator to have numerical radius at most 1: X ∈ B(H) has ν(X ) ≤ 1 ⇔ Re (β X )  IH for all β ∈ T. Note that Lemma 2.21 can itself be viewed as a quadratic, numerical-radius version of the Douglas lemma.   We apply Lemma 2.21 to the case (2.20) for each β to get a numerical contraction F(β) such that (A − B ∗ T ) + β(B − A∗ T ) = DT F(β)DT .

(2.21)

On adding equations (2.21) for the cases β = 1 and −1, we get A − B ∗ T = DT F1 DT where F1 :=

F(1) + F(−1) . 2

(2.22)

Thus putting β = 1 in (2.21) and combining with (2.21) gives us B − A∗ T = DT (F(1) − F1 )DT = DT F2 DT where F(1) − F(−1) . F2 := 2

(2.23)

We conclude that F1 and F2 so constructed satisfy equations (2.18). It is easy to see that in general X ∈ B(DT ) with DT X DT = 0 ⇒ X = 0.

(2.24)

Applying this observation to the homogeneous version of equations (2.18) implies that the solutions (F1 , F2 ) of (2.18) must be unique whenever they exist.  On the other hand, if we combine (2.22) with (2.23) we see that F(β) := F1 + β F2 gives us a second solution of (2.21). Hence   − F(β))DT = 0 where F(β) − F(β) ∈ B(DT ) for all β ∈ T. DT ( F(β)  Again by (2.24), we see that F(β) = F(β) for all β ∈ T. But we saw above (as a consequence of Lemma 2.21) that F(β) is a numerical contraction for all β ∈ T. As   we now know that F(β) = F(β), we conclude that the pencil F(β) = F1 + β F2 is a numerical contraction for all β ∈ T. By applying the Maximum Modulus Theorem to the holomorphic function (F1 + β F2 )h, h for each fixed h ∈ H, we see that F1 + z F2 is a numerical contraction for all z ∈ D. This completes the proof of item (i) in Theorem 2.20. To see that F1 and F2 satisfy equations (2.19), simply multiply DT on the left of each equation and use the identities (2.18) to simplify. To show that F1 , F2 are the

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unique operators on DT satisfying these two equations, it is enough to show that X = 0 and Y = 0 are the only operators in B(DT ) satisfying X DT + Y ∗ DT T = 0, Y DT + X ∗ DT T = 0. To show that X = 0, compute DT X DT = −DT Y ∗ DT T = T ∗ DT X DT T = −T ∗ DT Y ∗ DT T 2 = T ∗2 DT X DT T 2 . Thus by iteration of the above process DT X DT = T ∗n DT X DT T n . This shows that X = 0 because for every h ∈ H lim DT T n h2 = lim T n h2 − lim T n+1 h2 = 0. n

n

n

A similar argument gives Y = 0. This is the idea of the proof due by Bhattacharyya [12].   The unique operators F1 , F2 in item (i) of Theorem 2.20 will be referred to as the fundamental operators for the tetrablock contraction (A, B, T ), as in [12]. As we have seen in Proposition 2.1, if (A, B, T ) is a E-contraction, so also is (A∗ , B ∗ , T ∗ ). For the construction of the functional model for a tetrablock contraction (A, B, T ), it turns out to be more convenient to work with the fundamental operators for the adjoint tetrablock contraction (A∗ , B ∗ , T ∗ ) which we denote as (G 1 , G 2 ). Thus there exists exactly one solution (X 1 , X 2 ) = (G 1 , G 2 ) of the system of equations A∗ − BT ∗ = DT ∗ X 1 DT ∗ ,

B ∗ − AT ∗ = DT ∗ X 2 DT ∗

(2.25)

with equivalent characterization as the unique solution (X 1 , X 2 ) = (G 1 , G 2 ) of the second system of equations DT ∗ A∗ = X 1 DT ∗ + X 2∗ DT ∗ T ∗ ,

DT ∗ B ∗ = X 2 DT ∗ + X 1∗ DT ∗ T ∗

(2.26)

Then from Example 2.10 we see immediately that the operator pair (MG ∗1 +zG 2 , MG ∗2 +zG 1 , Mz ) acting on the Hardy space H 2 (DT ∗ ) is of the correct form to be a pseudo-commutative tetrablock isometry. We would like to establish conditions under which this a priori only pseudo-commutative E-isometry is actually a strict E-isometry. This follows from the following result. Theorem 2.22 (See [12]) Suppose that (G 1 , G 2 ) is the Fundamental Operator pair for the E-contraction (A∗ , B ∗ , T ∗ ). Let (V1 , V2 , V3 ) be the operator triple (V1 , V2 , V2 ) = (MG ∗1 +zG 2 , MG ∗2 +zG 1 , Mz ) on H 2 (DT ∗ ).

(2.27)

Then:

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(1) (V1 , V2 , V3 ) is a pseudo-commutative tetrablock isometry having the additional property that r (V1 ) ≤ 1, r (V2 ) ≤ 1. (2) Suppose in addition that the Fundamental Operator pair (G 1 , G 2 ) satisfy the commutativity conditions. Then (V1 , V2 , V3 ) is a strict tetrablock isometry. Corollary 2.23 Let (G 1 , G 2 ) be the Fundamental Operator pair for the tetrablock contraction (A∗ , B ∗ , T ∗ ) and set (V1 , V2 , V3 ) as in (2.27). Then (V1 , V2 , V3 ) is a (strict) tetrablock isometry if and only if the commutativity conditions (2.8) hold. Proof of Corollary 2.23 If the commutativity conditions (2.8) are satisfied, then statement (2) of Theorem 2.22 says that (V1 , V2 , V3 ) is a tetrablock isometry. Conversely, if (V1 , V2 , V3 ) is a tetrablock isometry, in particular (V1 , V2 , V3 ) is commutative so the commutativity conditions (2.8) are satisfied.   Proof of Theorem 2.22 We first consider statement (1). That (V1 , V2 , V3 ) is a pseudocommutative tetrablock isometry follows from the fact that it has the required form (2.6) as presented in Example 2.10. It remains to use the fact that (G 1 , G 2 ) is a Fundamental Operator pair for the tetrablock contraction (A∗ , B ∗ , T ∗ ) (to see that we also have r (V1 ) ≤ 1 and r (V2 ) ≤ 1. By Theorem 2.20 (applied to (A∗ , B ∗ , T ∗ ) in place of (A, B, T )), we know that ν(G 1 + zG 2 ) ≤ 1 for all z ∈ D. By the Andô criterion for the numerical radius of an operator to be no more than 1, this means that β(G 1 + αG 2 ) + β(G ∗1 + αG ∗2 )  2IDT ∗ ∀α, β ∈ D. Rearrange this inequality as (βG ∗1 + βαG 2 ) + (βG 1 + βαG ∗2 ) = β(G ∗1 + β 2 αG 2 ) + β(G 1 + β αG ∗2 )  2IDT ∗ . 2

By the Andô criterion applied in the reverse direction, this tells us that ν(G ∗1 + zG 2 ) ≤ 1 for all z ∈ D. But in general the numerical radius dominates the spectral radius; thus r (G ∗1 + zG 2 ) ≤ ν(G ∗1 + zG 2 ) ≤ 1 for all z ∈ D. If we choose λ ∈ C with |λ| > 1, then λIDT ∗ − (G ∗2 + zG 1 ) is invertible, and in fact the operator-valued function z → (λIDT ∗ − (G ∗2 + zG 1 ))−1 is in H ∞ (B(DT ∗ ). We thus conclude that in fact r (MG ∗2 +zG 1 ) ≤ 1. Reprinted from the journal

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All the above analysis applies to the pair (G 2 , G 1 ) in place of (G 1 , G 2 ), as (G 2 , G 1 ) is the Fundamental Operator pair for the E-contraction (B ∗ , A∗ , T ∗ ); hence we also have also r (MG ∗2 +zG 1 ) ≤ 1. This completes the proof of statement (1). We now consider statement (2). As we are now assuming that (G 1 , G 2 ) satisfy the commutativity conditions (2.8), it follows that MG ∗1 +zG 2 commutes with MG ∗2 +zG 1 . We are now in a position to apply statement (2) in Theorem 2.9 to conclude that the a priori only pseudo-commutative E-contraction (MG ∗1 +zG 2 , MG ∗2 +zG 1 , Mz ) is in fact a strict E-contraction.  

3 Functional Models for Tetrablock Contractions In this section we produce two functional models for tetrablock contractions, the first inspired by model theory of Douglas [19], and the second by the model theory of Sz.-Nagy and Foias [32]. We note that so far only a functional model is known for the special case when the last entry is a pure contraction; see [30, Theorem 4.2]. 3.1 A Douglas-Type Functional Model Let T be any contraction on H. Define the operators O DT ∗ ,T ∗ : H → H 2 (DT ∗ ) as O DT ∗ ,T ∗ (z)h =

∞ 

z n DT ∗ T ∗n h, for every h ∈ H,

(3.1)

n=0

and  D : H →



H 2 (D T ∗ )

QT ∗

 by 

D h =

O DT ∗ ,T ∗ (z)h QT ∗ h

 for all h ∈ H,

(3.2)

where Q T ∗ is as in (2.13). Then the computation  D h2 =O DT ∗ ,T ∗ (z)h2H 2 (D =

∞ 

T∗)

+ Q T ∗ h2

DT ∗ T ∗n h2 + lim T ∗n h2 n→∞

n=0

= (h2 − T ∗ h2 ) + (T ∗ h2 − T ∗2 h2 ) + · · · ) + lim T ∗n h2 n→∞

= h

2

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shows that  D is an isometry. Note the following intertwining property of O DT ∗ ,T ∗ : O DT ∗ ,T ∗ (z)T ∗ h =

∞ 

z n DT ∗ T ∗n+1 h = Mz∗

n=0

∞ 

z n DT ∗ T ∗n h

n=0

= Mz∗ O DT ∗ ,T ∗ (z)h.

(3.3)

This together with intertwining (2.14) of Q T ∗ implies 

Mz 0 D T = 0 WD ∗

∗

D .

(3.4)

This shows that the the pair  VD :=

  2   2  Mz 0 H (DT ∗ ) H (DT ∗ ) : → 0 WD QT ∗ QT ∗

is an isometric lift of T . This construction is by Douglas, where he also showed that this lift is minimal (see [19]). Now let (A, B, T ) be a tetrablock contraction acting on H and G 1 , G 2 be the fundamental operators of (A∗ , B ∗ , T ∗ ). Let (R D , S D , W D ) acting on QT ∗ be the canonical tetrablock unitary associated with (A, B, T ). Consider the operators 

      2  MG ∗1 +zG 2 0 MG ∗2 +zG 1 0 Mz 0 H (DT ∗ ) , , on . 0 WD QT ∗ 0 RD 0 SD

(3.5)

We claim that  D (A∗ , B ∗ , T ∗ ) ∗  ∗  ∗   MG ∗2 +zG 1 0 MG ∗1 +zG 2 0 Mz 0 D , , , = 0 WD 0 RD 0 SD where  D : H →



H 2 (D T ∗ )

(3.6)



is the isometry as in (3.2).   2  O DT ∗ ,T ∗ H (D T ∗ ) Recalling the definition  D = : H → , we see that the threeQ ∗ QT ∗ T fold intertwining condition (3.6) splits into two three-fold intertwining conditions QT ∗

O DT ∗ ,T ∗ (A∗ , B ∗ , T ∗ ) = (MG∗ ∗ +zG 2 , MG∗ ∗ +zG 1 , Mz∗ )O DT ∗ ,T ∗ ,

(3.7)

∗ Q T ∗ (A∗ , B ∗ , T ∗ ) = (R ∗D , S D , W D∗ )Q T ∗ .

(3.8)

1

2

The last equation in (3.7) combined with the last equation in (3.8) we have already  seen as the condition that  D is the isometric identification map implementing M0z W0D as the Douglas minimal isometric lift of T (see (3.4)). We shall next check only the

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first equation in (3.7) and the first equation in (3.8) as the verification of the respective second equations is completely analogous. Thus it remains to check O DT ∗ ,T ∗ A∗ = MG∗ ∗ +zG 2 O DT ∗ ,T ∗ , ∗

QT ∗ A =

(3.9)

1

R ∗D Q T ∗ .

(3.10)

Note that (3.10) is part of the construction of the canonical tetrablock unitary associated with the original tetrablock contraction (A, B, T ) (see (2.16)). To check (3.9), let us rewrite the condition in function form: DT ∗ (I − zT ∗ )−1 A∗ = G 1 DT ∗ (I − zT ∗ )−1 + G ∗2 DT ∗ (I − zT ∗ )−1 T ∗ As A commutes with T , we can rewrite this as DT ∗ A∗ (I − zT ∗ )−1 = (G 1 DT ∗ + G ∗2 DT ∗ T ∗ )(I − zT ∗ )−1 . We may now cancel off the resolvent term (I − zT ∗ )−1 to get a pure operator equation DT ∗ A∗ = G 1 DT ∗ + G ∗2 DT ∗ T ∗ .

(3.11)

Let us now recall that the operators (G 1 , G 2 ) on DT ∗ were chosen to be the Fundamental Operators for the tetrablock contraction (A∗ , B ∗ , T ∗ ). Thus by our earlier discussion of Fundamental Operators for tetrablock contractions (see Theorem 2.20), we know that (X 1 , X 2 ) = (G 1 , G 2 ) satisfies the identities (2.26), the first of which gives the same condition on (G 1 , G 2 ) as (3.11). Thus this choice of (G 1 , G 2 ) indeed leads to a solution of (3.9), and the proof of (3.6) is complete. (Of course the second equation in (2.26) amounts to the verification of the second equation in (3.7).) This then means that the operator triple with isometric embedding operator  D         MG ∗2 +zG 1 0 MG ∗1 +zG 2 0 Mz 0 , , (3.12) (V1 , V2 , V3 ) :=  D , 0 WD 0 RD 0 SD is a lift of the tetrablock contraction (A, B, T ). As (R D , S D , W D ) is E-unitary as part of the canonical construction in Sect. 2.3, we see from the form of the top components of (V1 , V2 , V3 ) in (3.12) that (V1 , V2 , V3 ) is a semi-strict E-isometry and is a strict Eisometry exactly when the top-component triple (MG ∗1 +zG 2 , MG ∗2 +zG 1 , Mz ) is a strict E-isometry. By Theorem 2.22, this in turn happens exactly when the Fundamental Operator pair (G 1 , G 2 ) satisfies the commutativity conditions (2.8). In summary we have verified most of the following result. We note that item (2) recovers a result of Bhattacharyya-Sau [15] via functional-model methods rather than by block-matrixconstruction methods. Theorem 3.1 (1) Let (A, B, T ) be a tetrablock contraction on H and let V3 on K ⊃ H be the (essentially unique) minimal isometric lift for the contraction operator T . Then there is a unique choice of operators (V1 , V2 ) on K so that the triple V = (V1 , V2 , V3 ) is a semi-strict tetrablock isometric lift for (A, B, T ).

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(2) A necessary and sufficient condition that there be a strict tetrablock isometric lift V = (V1 , V2 , V3 ) for (A, B, T ) with the isometry V3 equal to a minimal isometric lift for T is that the fundamental operators (G 1 , G 2 ) for the adjoint tetrablock contraction (A∗ , B ∗ , T ∗ ) satisfy the system of operator equations (2.8). In this case the operator pair (V1 , V2 ) on K is uniquely determined once one fixes a choice (essentially unique) for a minimal isometric lift V3 for T . (3) The lift (V1 , V2 , V3 ) can be given in functional form in the coordinates of the Douglas model as follows. For (A, B, T ) equal to a tetrablock contraction on a Hilbert space H let (G 1 , G 2 ) be the fundamental operators of (A∗ , B ∗ , T ∗ ), let canonically associated with (A, B, T ) as (R D , S D , W D ) be the tetrablock unitary   O D ∗ ,T ∗ T in Definition 2.18, and let  D = be the Douglas isometric embedding QT ∗  2  map from H into H Q(D∗T ∗ ) . Then T

        MG ∗2 +zG 1 0 MG ∗1 +zG 2 0 Mz 0 D , , , 0 WD 0 RD 0 SD

(3.13)

is a semi-strict (strict exactly when (G 1 , G 2 ) satisfies (2.8)) tetrablock isometric lift for (A, B, T ). In particular (A, B, T ) is jointly unitarily equivalent to  PH D

       MG ∗1 +zG 2 0 MG ∗2 +zG 1 0 Mz 0  , , , 0 W D H 0 RD 0 SD D

(3.14)

where H D is the functional model space given by  H D := Ran  D ⊂

 H 2 (DT ∗ ) . QT ∗

and any other semi-strict E-isometric lift (V1 , V2 , V3 ) with V3 =  2  H (D T ∗ ) is equal to (3.13). Q ∗

(3.15) 

Mz 0 0 WD

 on

T

Proof The discussion preceding the statement of the theorem amounts to a proof of statement (3) in Theorem 3.1. Statements (1) and (2) apart from the uniqueness assertion amounts to a coordinate-free (abstract, model-free) interpretation of the results of statement (3). It remains only to discuss the uniqueness assertion in statements (1) and (2). This can also be formulated in terms of the  model as follows: Given a  O D ∗ ,T ∗ T be the Douglas embedding tetrablock contraction (A, B, T ), let  D = QT ∗   2 map and let K D = H Q(D∗T ∗ ) be the Douglas minimal isometric lift space for T with T   Mz 0 VD = 0 W D on K D equal to the Douglas minimal isometric lift for T . Suppose that        B12 B11   = A11 A12 ,  B = A 21 A 22  A B21  B22 Reprinted from the journal

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   2    B, M0z are two operators on K D = H Q(D∗T ∗ such that A, T commutative tetrablock isometric lift for T . Then necessarily

0 WD



is a pseudo-

    ∗ MG ∗2 +zG 1 0  = MG 1 +zG 2 0 ,  B= . A 0 RD 0 SD where the pair (G 1 , G 2 ) is equal to the pair of Fundamental Operators for the tetrablock contraction (A∗ , B ∗ , T ∗ ), and where (R D , S D , W D ) is the tetrablock unitary canonically associated with the tetrablock contraction (A, B, T ) as in Definition 2.18. To prove this model-theoretic reformulation of the uniqueness problem, we proceed as follows. We are given first of all that the triple 

  12 11 A A  A=   , A21 A22

     B12 B11  Mz 0  B=   , 0 WD B21 B22

(3.16)

is a pseudo-commutative tetrablock isometry. According to Definition 2.6 we have the following:         Mz 0 Mz 0  Mz 0 = Mz 0 A  and   • (i) A B = 0 WD 0 WD 0 W D B;    0 WD  ∗ Mz 0 , and  ∗ Mz 0 ; and • (ii)  B=A B=A 0 WD 0 WD  ≤ 1. • (iii)  A  and  As in the proof of Theorem 2.12, conditions (i) ad (ii) force A B to have the block-diagonal form   ( A, B) =



   MG MG ∗ +z G ∗ +z G 2 0 1 0 1 2 , . 22  0 A 0 B22

22 ,  for some pseudo-commutative tetrablock unitary ( A B22 , W D ), and operators ∗    ∗ + z G 2 are contraction G 1 , G 2 ∈ B(DT ∗ ) so that the linear pencils G 1 + z G 2 and G 1 valued for all z ∈ D. We now use the fact that the triple (3.16) is a pseudo-commutative   lift of (A, B, T ), i.e., the operators A, B satisfy the conditions      ∗ MG O DT ∗ ,T ∗ O DT ∗ ,T ∗ 2 0 ∗ +z G 1 = A∗ ∗ QT ∗ QT ∗ 0 A 22 and 

∗ MG ∗ +z G  2

0

1

0  B∗

22

    O DT ∗ ,T ∗ O DT ∗ ,T ∗ = B∗. QT ∗ QT ∗

Equivalently, ∗ ∗ ∗ ∗ MG  O DT ∗ ,T ∗ = O DT ∗ ,T ∗ A , MG  O DT ∗ ,T ∗ = O DT ∗ ,T ∗ A ∗ +z G ∗ +z G

(3.17)

∗ ∗22 ,  and ( A B22 , W D∗ )Q T ∗ = Q T ∗ (A∗ , B ∗ , T ∗ ).

(3.18)

1

2

1

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22 ,  22 commutes with W D We first show that ( A B22 ) = (R D , S D ). Note that since A ∗  and W D is a unitary, A22 commutes with W D as well, and so we use (3.18) to compute ∗22 (W Dn Q T ∗ h) = W Dn A ∗22 Q T ∗ h = W Dn Q T ∗ A∗ h A = W Dn R ∗D Q T ∗ h = R ∗D (W Dn Q T ∗ h). 22 = R D . Since {W Dn Q T ∗ h : n ≥ 0 and h ∈ H} is dense in QT ∗ , we have A 1 , G 2 ) are the fundamental operators Similarly,  B22 = S D . Next we show that (G ∗ ). To this end, we can use (3.17) and the power series expansion of of (A∗ , B ∗ , T  O D ∗ ,T ∗ (z) = n≥0 DT ∗ T ∗n h to arrive at the equations 1 DT ∗ + G ∗2 DT ∗ T ∗ = DT ∗ A∗ and G 2 DT ∗ + G ∗1 DT ∗ T ∗ = DT ∗ B ∗ . G By part (ii) of Theorem 2.20 applied to the tetrablock contraction (A∗ , B ∗ , T ∗ ), 2 ) must be equal to the Fundamental Operator pair for (A∗ , B ∗ , T ∗ ). 1 , G   (G 3.2 A Sz.-Nagy–Foias Type Functional Model Sz.-Nagy and Foias gave a function-space realization of QT ∗ and thereby produced a concrete functional model for a contraction T . In their analysis a crucial role is played by what they called the characteristic function associated with T : T (z) := −T + zO DT ∗ ,T ∗ T |DT : DT → DT ∗ .

(3.19)

The name suggests the fact the characteristic function T enables one to write down an explicit functional model on which there is a compressed multiplication operator T which recovers the original operator T up to unitary equivalence in case T is a c.n.u. contraction (see Chapter VI of [32]). Let T (ζ ) be the radial limit of the characteristic function defined almost everywhere on T. Consider

T (ζ ) := (I − T (ζ )∗ T (ζ ))1/2 . Sz.-Nagy and Foias showed in [32] that 

VNF

0 Mz := 0 M | ζ L 2 (D T

  2   2  H (DT ∗ ) H (DT ∗ ) : →

T L 2 (DT )

T L 2 (DT ) T)

is a minimal isometric lift of T via some isometric embedding  H 2 (DT ∗ ) =: KNF

T L 2 (DT )

 NF : H →

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such that    H 2 (DT ∗ ) T · H 2 (DT ).  =

T

T L 2 (DT ) 

HNF := Ran NF

(3.21)

Any two minimal isometric lifts of a given contraction T are unitarily equivalent; see Chapter I of [32]. In [10] an explicit unitary u min : QT ∗ → T L 2 (DT ) is found that intertwines W D and Mζ | L 2 (D ) and T

T



NF

 I H 2 (D T ∗ ) 0 = D . 0 u min

(3.22)

It is possible to give a concrete Sz.-Nagy–Foias type functional model using the transition map u min : QT ∗ → T L 2 (DT ) as appeared (see (3.22)) in the case of a single contractive operator above. We must replace the canonical tetrablock unitary (R D , S D , W D ) by its avatar on the function space T L 2 (DT ): (RNF , SNF , WNF ) = u ∗min (R D , S D , W D )u min .

(3.23)

Then the following functional model is a straightforward consequence of Theorem 3.1 and (3.22). Theorem 3.2 Let (A, B, T ) be a tetrablock contraction on a Hilbert space H such that T is c.n.u., and (G 1 , G 2 ) be the fundamental operators of (A∗ , B ∗ , T ∗ ). Then (A, B, T ) is jointly unitarily equivalent to  PHNF

       MG ∗1 +zG 2 0 MG ∗2 +zG 1 0 Mz 0  , , ,  0 W 0 RNF 0 SNF NF HNF

(3.24)

where HNF is the functional model space given by (see (3.21))    H 2 (DT ∗ ) T · H 2 (DT ). 

T

T L 2 (DT )

 HNF = Ran NF =

Note that in the special case when T ∗n → 0 strongly as n → ∞, the space QT ∗ = 0 and hence also T L 2 (DT ) = 0, i.e., T is inner. Therefore in this special case the models above simply boil down to the following which was obtained in [30]. Theorem 3.3 (See [30, Theorem 4.2]) Let (A, B, T ) be a pure tetrablock contraction on a Hilbert space H and (G 1 , G 2 ) be the fundamental operators of (A∗ , B ∗ , T ∗ ). Then (A, B, T ) is jointly unitarily equivalent to PRan O D

T ∗ ,T

∗

(MG ∗1 +zG 2 , MG ∗2 +zG 1 , Mz ))|Ran O D

66

T ∗ ,T

∗

.

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4 Tetrablock Data Sets: Characteristic and Special In this section we provide some preliminary results towards a Sz.-Nagy–Foias-type model theory for tetrablock-contraction operator tuples (A, B, T ). Note that if T is unitary, then the characteristic function T , as in (3.19), is trivial (i.e., equal to the zero operator between the zero spaces). As the most general contraction operator T is the direct sum of a unitary Tu with a completely nonunitary (c.n.u.) part Tcnu and the model theory for unitary operators is easily handled by spectral theory, it is natural for model theory purposes to restrict to the case where T is c.n.u. One then associates a functional model spaces  KT =

H 2 (DT ∗

T L 2 (DT )



 T H 2 (DT ) ⊂ KT

T

 HT = KT 

together with functional-model operators TT and VT by     Mz 0  Mz 0 on KT . TT = PHT on H , V = T T 0 M ζ H 0 Mζ T

Then it is immediate that: (NF1) VT is an isometry on KT .  T ⊥ 2 (NF2) HT = T H (DT ) is invariant for VT , and hence VT is an isometric lift for TT . Less immediately obvious are other features of the model: (NF3) (See [32, Theorem VI.2.3]) If T is c.n.u., then TT is unitarily equivalent to T . (NF4) (See [32, Theorem VI.3.4].) If T on H and T  on H are two c.n.u. contraction operators, then T is unitarily equivalent to T  if and only if T coincides with T  in the following sense: there exist unitary change-of-coordinate maps φ : DT → DT  and φ∗ : DT ∗ → DT ∗ so that φ∗ T (λ) = T  (λ)φ for all λ ∈ D. Often in the literature this property is described simply as: the characteristic function T is a complete unitary invariant for c.n.u. contraction operator T . The Sz.-Nagy–Foias theory goes still further by identifying the coincidence-envelop of the characteristic functions (3.19), i.e., the set of all contractive operator functions (D, D∗ , ) coinciding with the characteristic function T for some c.n.u. contraction operator T , as simply any contractive operator function (D, D∗ , ) which is pure in the sense that  (0)u < u for any u ∈ D such that u = 0. Then we can start with any pure COF (D, D∗ , ), form K( ) and H( ) according to     H 2 (D∗ ) H 2 (D) ⊂ K( ) , H( ) = K( )  K( ) = D D · L 2 (D) Reprinted from the journal

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where D is the -defect operator function D (ζ ) = (ID − (ζ )∗ (ζ )) 2 . Then we can form the model operators     Mz 0 on K( ), T( ) = PH( ) V( ) V( ) = . 0 Mζ H( ) Then we have the additional results: (NF5) (See [32, Theorem VI.3.1].) Given any pure COF , T( ) is a c.n.u. contraction operator on H( ) with characteristic operator function T( ) coinciding with . In this way the loop is closed: the study of c.n.u. contraction operators is the same as the study of pure COFs. To explain generalizations to the setting of tetrablock contractions, we first introduce some useful terminology. For this discussion, just as in the classical Sz.-Nagy–Foias settings, it makes sense to restrict to c.n.u. tetrablock contractions (see Theorem 2.16 and Remark 2.17). Definition 4.1 Let us say that any collection of objects  = ( , (G 1 , G 2 ), ψ} consisting of (i) a pure COF function (D, D∗ , ), (ii) a pair of operators (G 1 , G 2 ) on the coefficient space D∗ , and (iii) a measurable function ψ on T such that, for a.e. ζ ∈ T, ψ(ζ ) is a contractive normal operator on D (ζ ) = Ran D (ζ ). is a tetrablock data set. Remark 4.2 Canonically associated with any such ψ as in item (iii) in Definition 4.1 | is the tetrablock unitary triple (R, S, W ) on the direct integral space ⊕ T D (ζ ) |dζ 2π given by R = Mψ ∗ ·ζ , S = Mψ , W = Mζ and (as one sweeps over all possible such ψ), this is the general tetrablock unitary | operator triple (R, S, W ) on the space ⊕ T D (ζ ) |dζ 2π with the unitary operator W equal to W = Mζ (multiplication by the coordinate function) (see Example 2.10 (2)). Thus item (iii) in the definition of tetrablock data set can be equivalently rephrased as: (iii’) a tetrablock unitary operator-triple (R, S, W ) on the direct-integral space |dζ | T D (ζ ) 2π such that the last unitary component W is equal to multiplication by the coordinate function W = Mζ . However, for convenience of notation, we shall continue to use the notation (R, S, W ) for the third component of a tetrablock data set  = ( , (G 1 , G 2 ), (R, S, W )) with the convention (iii’) also part of the definition.

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Definition 4.3 Given a c.n.u. tetrablock contraction (A, B, T ) we say that  A,B,T = ( , (G 1 , G 2 ), ψ) is the characteristic tetrablock data set for (A, B, T ) if (i) (D, D∗ , ) is equal to the Sz.-Nagy–Foias characteristic function (DT , DT ∗ , T ) for the c.n.u. contraction operator T , (ii) (G 1 , G 2 ) is equal to the Fundamental Operator pair for the adjoint tetrablock contraction (A∗ , B ∗ , T ∗ ), and (iii) (R, S, D) is given by ((R, S, W )) = (RNF , SNF , WNF ) := u ∗min (R D , S D , W D )u min where (R D , S D , W D ) is the tetrablock unitary on QT ∗ determined by tetrablock contraction (A, B, T ) according to Definition 2.18, and where u min : T L 2 (DT ) → QT ∗ is the unitary identification map identifying the Sz.-Nagy–Foias lifting residual space T L 2 (DT ) with the Douglas lifting residual space QT ∗ . Then it is clear that the characteristic tetrablock data set  A,B,T for a c.n.u. tetrablock contraction (A, B, T ) is a tetrablock data set. The natural notion of equivalence for tetrablock-data sets is the following. Definition 4.4 Let (D, D∗ , ), (D , D∗ ,  ) be two purely contractive analytic functions. Let G 1 , G 2 ∈ B(D∗ ), G 1 , G 2 ∈ B(D∗ ), and (R, S, W ) on L 2 (D) and (R  , S  , W  ) on  L 2 (D ) be two tetrablock unitaries (with W and W  equal to Mζ on their respective spaces). We say that the two triples ( , (G 1 , G 2 ), (R, S, W )) and (  , (G 1 , G 2 ), (R  , S  , W  )) coincide if: (i) (D, D∗ , ) and (D , D∗ ,  ) coincide, (ii) the unitary operators φ, φ∗ involved in the coincidence of (D, D∗ , ) and (D , D∗ ,  ) satisfy the additional intertwining conditions: φ∗ (G 1 , G 2 ) = (G 1 , G 2 )φ∗ and ωφ (R, S, W ) = (R  , S  , W  )ωφ , where ωφ : L 2 (D) →  L 2 (D ) is the unitary map induced by φ according to the formula ωφ := (I L 2 ⊗ φ)|

L

2 (D )

.

(4.1)

Given a characteristic tetrablock data set (A,B,T ) = ( , (G 1 , G 2 ), (R, S, W )) for a tetrablock contraction (A, B, T ) we can write down a functional model:  K() =

   H 2 (D∗ ) H 2 (D) , H() = K  D D L 2 (D)

with functional-model operators  V() =

     MG ∗2 +zG 1 0 MG ∗1 +zG 2 0 Mz 0 , , on K(), 0 Mζ 0 R 0 S

T() = PH() V()|H() .

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The tetrablock analogue of items (NF1)-(NF2) in our discussion of the Sz.-Nagy–Foias model above fails without the additional assumptions: namely, it is not the case that V is a tetrablock isometry as well as that V is a lift for T unless we also impose the condition (2.8) on (G 1 , G 2 ). Nevertheless, the analogue of (NF3) does hold: given that  is the characteristic tetrablock triple for (A, B, T ), it is the case that (A, B, T ) is unitarily equivalent to T(): this is the content of the last part of item (3) in Theorem 3.1 (after a conversion to Sz.-Nagy–Foias rather than Douglas coordinates): see (3.14) and (3.15). The next theorem amounts to the analogue of (NF4) in our list of features for the Sz.-Nagy–Foias model above. Theorem 4.5 Let (A, B, T ) and (A , B  , T  ) be two tetrablock contractions acting on H and H , respectively. Let    ( T , (G 1 , G 2 ), (RNF , SNF , WNF ), ( T  , (G 1 , G 2 ), (RNF , SNF , WNF ))

be the characteristic tetrablock data sets for (A, B, T ) and (A , B  , T  ), respectively. (1) If (A, B, T ) and (A , B  , T  ) are unitarily equivalent, then their characteristic tetrablock data sets coincide. (2) Conversely, if T and T  are c.n.u. contractions and the characteristic tetrablock data sets of (A, B, T ) and (A , B  , T  ) coincide, then (A, B, T ) and (A , B  , T  ) are unitarily equivalent. Proof First suppose that (A, B, T ) and (A , B  , T  ) are unitarily equivalent via a unitary τ : H → H . The fact that T and T  coincide is a part of the Sz.-Nagy–Foias theory [32]. Indeed, note that τ (I − T ∗ T ) = (I − T ∗ T  )τ and τ (I − T T ∗ ) = (I − T  T ∗ )τ and therefore by the functional calculus for positive operators, τ DT = DT  τ and τ DT ∗ = DT ∗ τ

(4.2)

and thereby inducing two unitary operators φ := τ |DT : DT → DT  and φ∗ := τ |DT ∗ : DT ∗ → DT ∗ .

(4.3)

Consequently φ∗ T = T  φ. Next, since the fundamental operators are the unique operators that satisfy the equations for (X 1 , X 2 ) = (G 1 , G 2 ): A∗ − BT ∗ = DT ∗ X 1 DT ∗ and B ∗ − AT ∗ = DT ∗ X 2 DT ∗ , one can easily obtain using (4.2) that φ∗ (G 1 , G 2 ) = (G 1 , G 2 )φ∗ .

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(4.4)

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Finally the proof of the forward direction will be complete if we establish that    (RNF , SNF , WNF ) = ωφ∗ (RNF , SNF , WNF )ωφ

(4.5)

where ωφ = (I L 2 ⊗ φ)| L 2 (D ) : T L 2 (DT ) → T  L 2 (DT  ). For this we first note T T that      I 2 ⊗ φ∗∗ 0 O DT ∗ ,T ∗ I 0 = NF = H NF τ 0 ωφ∗ QT ∗ 0 u min     I 2 ⊗ φ∗∗ 0 O DT ∗ ,T ∗ I 0 τ, = H 0 ωφ∗ 0 u min Q T ∗ from which we read off that u min Q T ∗ = ωφ∗ u min Q T ∗ τ.

(4.6)

Now since QT ∗ = span{W Dn Q T ∗ τ h : h ∈ H, n ≥ 0} and u min has the intertwining property u min W D = Mζ u min , we use (4.6) to compute ωφ · u min · ωτ∗ (W Dn Q T ∗ τ h) = ωφ · u min (W Dn Q T ∗ h) = ωφ Mζn u min Q T ∗ h = Mζn ωφ · u min Q T ∗ h

= Mζn u min Q T ∗ τ h = u min (W Dn Q T ∗ τ h). Consequently ωφ · u min · ωτ∗ = u min .

(4.7)

Using this identity and the intertwining properties of the unitaries involved, it is now easy to establish (4.5). Conversely, suppose T and T  are c.n.u. contractions, φ : DT → DT  and φ∗ : DT ∗ → DT ∗ be the unitary operators involved in the coincidence of the characteristic tetrablock data sets    ((G 1 , G 2 ), (RNF , SNF , WNF ), T ), ((G 1 , G 2 ), (RNF , SNF , WNF ), T  ).

By Definition 4.4, it follows that the unitary   2   2  H (DT ∗ ) H (DT ∗ ) I H 2 ⊗ φ∗ 0 : U= → 0 ωφ

T L 2 (DT )

T  L 2 (DT  ) 

 and intertwines the model operators as in identifies the model spaces HNF and HNF (3.24) associated with (A, B, T ) and (A , B  , T  ), respectively. This completes the proof of Theorem 4.5.  

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The question remains as to what additional coupling conditions must be imposed on a tetrablock data set  to assure that  coincides with the characteristic tetrablock data set for a c.n.u. tetrablock contraction (A, B, T ). In the Sz.-Nagy–Foias theory, the data set (or invariant) consists of a single COF, and the only additional requirement is that it must be pure. From the results of Sect. 2.4 we see that any characteristic tetrablock data  A,B,T = ( , (G 1 , G 2 ), (R, S, W )) for a c.n.u. tetrablock contraction operator-triple (A, B, T ) satisfies the additional conditions (expressed directly in terms of the components of  A,B,T rather than in terms of (A, B, T )): (i) is a pure COF (see [32, Theorem VI.3.1]), (ii) the numerical radius conditions ν(G ∗1 + zG 2 ) ≤ 1, ν(G ∗2 + zG 1 ) ≤ 1 for all z ∈ D hold, implying that also the spectral radius conditions r (MG ∗1 +zG 2 ) ≤ 1, r (MG ∗2 +zG 1 ) ≤ 1 (see Theorems 2.22 (1)). (iii) It is almost the case that the spectral radius and norm agree for MG ∗1 +zG 2 and MG ∗2 +zG 1 in the following sense (see Theorem 2.9 (1)): r (MG ∗1 +zG 2 · MG ∗2 +zG 1 ) = max{MG ∗1 +zG 2 2 , MG ∗2 +zG 1 2 }. However we do not expect that just imposing these conditions is sufficient to guarantee that such a tetrablock data set  will coincide with the characteristic tetrablock data set for some tetrablock contraction, so we do not expect to have an analogue of (NF5) at this level of generality. Let us now specialize our class of tetrablock contractions to what we call special tetrablock contractions, i.e., any tetrablock contraction (A, B, T ) with the special property that the Fundamental Operator pair (G 1 , G 2 ) for (A∗ , B ∗ , T ∗ ) satisfies the additional pair of operator equations (2.8). By Theorem 3.1, this is equivalent to (A, B, T ) having a minimal tetrablock isometric lift (V1 , V2 , V3 ) acting on a minimal Sz.-Nagy isometric-lift space for T with V3 equal to a minimal isometric lift for the single contraction operator T . This suggests that we define a special tetrablock data set as follows. For convenience in later discussion we shall now write the last component (R, S, W ) simply as ψ for a measurable contractive-normal operator-valued function ζ → ψ(ζ ) ∈ B(D (ζ ) ) according to the convention explained in Remark 4.2. Definition 4.6 We say that the tetrablock data set  = ((D, D∗ , ), (G 1 , G 2 ), ψ)

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(4.8)

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is a special tetrablock data set if the following conditions hold: (i) The operators G 1 , G 2 ∈ B(D∗ ) satisfy the commutativity conditions (2.8), i.e., [G 1 , G 2 ] = 0, [G ∗1 , G 1 ] = [G ∗2 , G 2 ] as well as the pencil-contractivity condition G ∗1 + zG 2  ≤ 1 for all z ∈ D and then also G ∗2 + zG 1  ≤ 1 for all z ∈ D. (ii) the space



D





 f : f ∈ H 2 (D) is jointly invariant under the operator triple

     MG ∗1 +zG 2 0 MG ∗2 +zG 1 0 Mz 0 , , . 0 Mζ 0 Mψ(ζ )∗ ·ζ 0 Mψ(ζ )

(4.9)

Given a special tetrablock data set ( , (G 1 , G 2 ), ψ), we say that the space  H=

   H 2 (D∗ ) H 2 (D)  2 D D L (D)

(4.10)

is the functional model space and the (commutative) operator triple (A, B, T) given by  PH

       MG ∗1 +zG 2 0 MG ∗2 +zG 1 0 Mz 0  , ,  ∗ 0 M 0 Mψ(ζ ) ·ζ 0 Mψ(ζ ) ζ H

(4.11)

the functional-model operator triple associated with the data set. The following theorem is our one analogue of item (NF5) in our list of features of the Sz.-Nagy– Foias model. Theorem 4.7 If  = ( , (G 1 , G 2 ), ψ) is a special tetrablock data set, then the associated model operator triple (A, B, T) as in (4.11), is a tetrablock contraction that lifts to the tetrablock isometry as in (4.9). Moreover, the tetrablock data set  coincides with the characteristic triple of (A, B, T). Proof By part (i) of Definition 4.6, the triple as in (4.9) is a strict tetrablock isometry, and by part (ii), it lifts the model triple (A, B, T) as in (4.11). Thus in particular, (A, B, T) is a tetrablock contraction and the first part of the theorem follows. For the second part, we use the Sz.-Nagy–Foias model theory for single contractions and Theorem 3.1 as follows. Apply Theorem VI.3.1 in [32] to the purely contractive analytic function to conclude that the characteristic function T of T coincides with , i.e., there exists unitary operators u : D → DT and u ∗ : D∗ → DT∗ such that u ∗ · (z) = T (z) · u for all z ∈ D. Let us set (G 1 , G 2 ) := u ∗ (G 1 , G 2 )u ∗∗ and Reprinted from the journal

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(R  , S  , W  ) := ωu (Mψ(ζ )∗ ·ζ , Mψ(ζ ) , Mζ )ωu∗ . Since G 1 , G 2 satisfy the commutativity conditions, G 1 , G 2 also satisfy the same conditions, and consequently the triple 

     MG ∗2 +zG 1 MG ∗1 +zG 2 0 0 Mz 0 , , 0 Mζ 0 ωu Mψ(ζ )∗ ·ζ ωu∗ 0 ωu Mψ ωu∗

(4.12)

is a strict tetrablock isometry. Note that since       T u 0 u∗ 0 = ,

T 0 ω u 0 ωu the unitary operator τ :=

   2    H (D∗ ) H 2 (DT∗ ) u∗ 0 : → o ωu

L 2 (D)

T L 2 (DT )

takes the functional model space H as in (4.10) onto 

   H 2 (DT∗ ) T H 2 (DT ). 

T

T L 2 (DT )

Therefore by part (ii) of Definition 4.6, the tetrablock isometry as in (4.12) is a lift of (A, B, T) via the embedding ι · τ |H , where  ι:

     H 2 (DT ) H 2 (DT∗ ) T → 

T

T L 2 (DT )

T L 2 (DT )

is the inclusion map. Since 

     H 2 (DT ) H 2 (DT ) Mz 0 : → 0 Mζ

T L 2 (DT )

T L 2 (DT )

is a minimal isometric lift of T, by part (2) of Theorem 3.1, there is a unique such tetrablock isometric lift with the last entry of the lift fixed. If G1 , G2 are the Fundamental Operators of (A∗ , B∗ , T∗ ), then by Theorem 3.2, 

     MG1∗ +zG2 0 MG2∗ +zG1 0 Mz 0 , , 0 WNF 0 RNF 0 SNF

is another tetrablock isometric lift of (A, B, T), where (RNF , SNF , WNF ) is the canonical tetrablock unitary associated with (A, B, T). Therefore we must have (G1 , G2 ) = (G 1 , G 2 ) = u ∗ (G 1 , G 2 )u ∗∗ and (RNF , SNF , WNF ) = ωu (Mψ(ζ )∗ ·ζ , Mψ(ζ ) , Mζ )ωu∗ .  

This is what was needed to be shown.

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Epilogue. It is easy to write down tetrablock data sets as in (4.8). Given such a tetrablock data set, it may not be very tractable to determine if in addition it satisfies conditions (i) and (ii) in Definition 4.6. However it is not so difficult to cook up viable examples. For example, we note that the commutativity conditions in (i) are automatic if we choose G 1 and G 2 to be scalar operators on D∗ . We can arrange the pencil contractivity condition to hold just by choosing G 1 and G 2 to be sufficiently small. If we choose the operator ψ(ζ ) to be a scalar for each ζ , then we are forced to choose ψ(ζ ) = G ∗2 ζ + G 1 . Since we have already chosen G 2 and G 1 so that the pencil contractivity condition holds, then ψ(ζ ) is contractive and of course a scalar operator is normal. Then all conditions are satisfied. In this way we get a whole class of tractable examples of special tetrablock contractions for any pure COF . If D∗ and D are both at most one-dimensional, all the examples are of this form. Data availability Data sharing is not applicable to this article as no datasets were generated or analysed during this currenct study.

Declarations Conflict of interest The authors state that there are no conflicts of interest.

References 1. Abouhajar, A.A., White, M.C., Young, N.J.: A Schwarz lemma for a domain related to μ-synthesis. J. Geom. Anal. 17, 717–750 (2007) 2. Agler, J., Harland, J., Raphael, B.: Classical function theory, operator dilation theory, and machine computation on multiply-connected domains. Mem. Am. Math. Soc. 191(892), viii+159 (2008) 3. Agler, J., Young, N.J.: Operators having the symmetrized bidisc as a spectral set. Proc. Edinb. Math. Soc. (2) 43, 195–210 (2000) 4. Agler, J., Young, N. J.: A model theory for -contractions. J. Oper. Theory 49, 45–60 (2003) 5. Andô, T.: On a Pair of Commuting Contractions. Acta Sci. Math. (Szeged) 24, 88–90 (1963) 6. Arveson, W.B.: An Invitation to C ∗ -Algebras, Graduate Texts in Mathematics 39. Springer, Heidelberg (1976) 7. Arveson, W.B.: Subalgebras of C ∗ -algebra. II. Acta Math. 128, 271–308 (1972) 8. Arveson, W.B.: A Short Course in Spectral Theory, Graduate Texts in Mathematics, vol. 209. Springer, New York (2002) 9. Ball, J.A., Sau, H.: Rational dilation of tetrablock contractions revisited. J. Funct. Anal. 278, 108275 (2020) 10. Ball, J.A., Sau, H.: Sz.-Nagy–Foias functional models for pairs of commuting Hilbert-space contraction operators, in preparation 11. Ball, J.A., Sau, H.: Functional models for commuting Hilbert-space contractions. In: Operator Theory, Operator Algebras and Their Interactions with Geometry and Topology: Ronald G. Douglas memorial volume, 11–54, Oper. Theory Adv. Appl., 278. Birkhäuser/Springer, Cham (2020) 12. Bhattacharyya, T.: The tetrablock as a spectral set. Indiana Univ. Math. J. 63, 1601–1629 (2014) 13. Bhattacharyya, T., Pal, S., Shyam Roy, S.: Dilations of -contractions by solving operator equations. Adv. Math. 230, 577–606 (2012) 14. Bhattacharyya, T., Pal, S.: A functional model for pure -contractions. J. Oper. Theory 71, 327–339 (2014) 15. Bhattacharyya, T., Sau, H.: Explicit and unique construction of tetrablock unitary dilation in a certain case. Complex Anal. Oper. Theory 10, 749–768 (2016)

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J. A. Ball, H. Sau 16. Bisai, B., Pal, S.: A model theory for operators associated with a domain related to μ-synthesis. Collect. Math. (2021). https://doi.org/10.1007/s13348-021-00341-6 17. Conway, J.B.: The Theory of Subnormal Operators, Mathematical Surveys and Monographs, vol. 36. American Mathematical Society, Providence (1991) 18. Dixmier, J.: von Neumann Algebras (with a preface by E.C. Lance), translated from the second French edition by F. Jellett, North-Holland Math. Library, vol. 27. North Holland Publishing Co., Amsterdam (1981) 19. Douglas, R.G.: Structure theory for operators. I. J. Reine Angew. Math. 232, 180–193 (1968) 20. Doyle, J.C., Packard, A.: The complex structured singular value. Automatica 29(1), 71–109 (1993) 21. Dritschel, M.A., McCullough, S.: Failure of rational dilation on a triply connected domain. J. Am. Math. Soc. 18, 873–918 (2005) 22. Dritschel, M. A., Rovnyak, J.: The operator Fejér-Riesz theorem. In: A Glimpse at Hilbert Space Operators: Paul R. Halmos in Memoriam. Oper. Theory Adv. Appl., vol. 207, pp. 223–254. Birkhäuser Verlag, Basel (2010) 23. Dritschel, M.A., Undrakh, B.: Rational dilation problems associated with constrained algebras. J. Math. Anal. Appl. 467, 95–131 (2018) 24. Dullerud, G.E., Paganini, F.: A Course in Robust Control Theory: A Convex Approach, Texts in Applied Mathematics 36. Springer, New York (2000) 25. Durszt, E.: Contractions as shifts. Acta Sci. Math. (Szeged) 48, 129–134 (1985) 26. Pal, S.: The failure of rational dilation on the tetrablock. J. Funct. Anal. 269, 1903–1924 (2015) 27. Pal, S.: Canonical decomposition of a tetrablock contraction and operator model. J. Math. Anal. Appl. 438, 274–284 (2016) 28. Pal, S.: Dilation, functional model and a complete unitary invariant for C·0 n -contractions. arXiv:1708.06015v2 [math.FA] (2021) 29. Parrott, S.: Unitary dilations for commuting contractions. Pacific J. Math. 34, 481–490 (1970) 30. Sau, H.: A note on tetrablock contractions. N. Y. J. Math. 21, 1347–1369 (2015) 31. Sau, H.: Andô dilations for a pair of commuting contractions: two explicit constructions and functional models. arXiv:1710.11368 [math.FA] 32. Sz.-Nagy, B., Foias, C., Bercovici, H., Kerchy, L.: Harmonic Analysis of Operators on Hilbert Space, 2nd edn. Revised and enlarged edition, Universitext, Springer, New York (2010) 33. Sz.-Nagy, B., Foias, C.: On the structure of intertwining operators. Acta Sci. Math. (Szeged) 35, 225–254 (1973) 34. von Neumann, J.: Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math. Ann. 102, 49–131 (1930) 35. Varopoulos, NTh.: On an inequality of von Neumann and an application of the metric theory of tensor products to operator theory. J. Funct. Anal. 16, 83–100 (1974) 36. Wold, H.: A Study in the Analysis of Stationary Time Series, 2nd edn. Stockholm (1954) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Complex Analysis and Operator Theory (2022) 16: 77 https://doi.org/10.1007/s11785-022-01248-1

Complex Analysis and Operator Theory

Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results Wolfram Bauer1

· Miguel Angel Rodriguez Rodriguez1

Received: 30 April 2022 / Accepted: 23 May 2022 / Published online: 30 June 2022 © The Author(s) 2022

Abstract We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and Hardy spaces over the unit ball in Cn . As an application we discuss semi-simplicity and the spectral invariance of these algebras. The different function Hilbert spaces are dealt with in parallel in successive chapters so that a direct comparison of the results is possible. As a new aspect of the theory we define commutative Toeplitz algebras over spaces of functions in infinitely many variables and present some structural results. The paper concludes with a short list of open problems in this area of research. Keywords Bergman and Hardy space · Gaussian measure in infinite dimensions · Fock space of functions in infinitely many variables · Commutative Banach algebras Mathematics Subject Classification Primary 47B35; Seconday 47L80 · 32A36

1 Introduction During the last years there has been an intensive study of commutative Banach and C ∗ algebras generated by Toeplitz operators acting on different function Hilbert spaces In memory of Jörg Eschmeier. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

A previous version of this chapter was published Open Access under a Creative Commons Attribution 4.0 International License at http://link.springer.com/10.1007/s11785-022-01248-1.

B

Wolfram Bauer [email protected] Miguel Angel Rodriguez Rodriguez [email protected]

1

Institut für Analysis, Leibniz Universität, Welfengarten 1, 30167 Hannover, Germany

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such as Bergman, Hardy or Fock spaces, see [3–7, 12, 13, 15, 20, 24–28, 30–34]. We call them Toeplitz algebras and remark that interesting examples already appear if the underlying domain is complex one dimensional. As is well known, Toeplitz operators acting on the Hardy space over the unit circle only commute in rare cases (see [10] for a precise statement). However, when passing to the standard weighted Bergman space over the unit disc D in C one discovers a variety of commutative C ∗ Toeplitz algebras. When assuming some “richness” of the symbol class and commutativity of the generating Toeplitz operators simultaneously in the weight parameter one even obtains a complete classification of such algebras based on the structure of geodesic pencils in the Poincaré hyperbolic disc, see [33]. One source of commutativity of a C ∗ Toeplitz algebra over D is the invariance of the symbols of the generating operators under the action of a maximal commutative subgroup of the automorphism group Aut(D). In [27] the authors have extended these results to complex domains of dimension n > 1 and studied commutative C ∗ algebras generated by Toeplitz operators over the unit ball Bn in Cn . Again, families of such algebras subordinate to the maximal commutative subgroups of the automorphism group of the domain could be constructed and a spectral decomposition of Toeplitz operators in each of these algebras was derived in a rather explicit form, see [27, 32, 34]. However, a classification result as was mentioned in dimension n = 1 is still missing. In the higher dimensional setting n > 1 the dependence of the symbol functions on different (groups of) coordinates is another source of commutativity and provides additional flexibility in the construction of commutative Toeplitz C ∗ algebras. In [12] the authors extend the results in [27, 33] replacing the unit ball by a general bounded symmetric domain of higher rank. Based on tools from representation theory this paper characterizes subgroups of the automorphism group that induce commutative C ∗ Toeplitz algebras generated by operators having symbols that are constant along the orbits. Another line of research is concerned with the structure of commutative Toeplitz Banach algebras which are not C ∗ . More precisely, symbol classes subordinate to a given abelian subgroup of Aut(Bn ) having the property that the corresponding Toeplitz operators generate a commutative Banach algebra while the generated C ∗ algebra is non-commutative have been considered [3–7, 32, 34]. Typical problems concern the structural properties of these algebras, such as a description of their maximal ideal spaces and Gelfand transform, semi-simplicity or spectral invariance inside the full algebra of bounded operators. It is natural to study the existence and structure of the above commutative Toeplitz algebras for different function Hilbert spaces such as the Hardy space over the unit sphere [24, 30] or the Fock space of Gaussian square integrable entire functions [4, 13]. In fact, there is an interesting interplay between these cases and the analysis of one of these spaces may be useful in the study of another. The aim of this paper is twofold. First, we present a survey of the recent research on commutative Toeplitz Banach and C ∗ algebras for the weighted Bergman spaces and the Hardy space over the unit ball and the unit sphere in Cn , respectively. By adding some new results we will complement this survey. However, due to the constantly growing literature on the subject, not all aspects of the theory can be dealt with in detail. Secondly, we introduce a new set up in the construction of operator algebras by considering Toeplitz operators over the Fock space in infinitely many variables.

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Such operators have been introduced in [21, 22] and generalize the notion of Toeplitz operators on the Fock-Segal-Bargmann space H 2 (Cn , μ) of Gaussian square integrable entire functions in Cn . New effects in the analysis of Toeplitz operators can be observed and in parts are a consequence of the infinite dimensional measure theory of the underlying Hilbert space H = 2 (N). Another new feature from a topological point of view is the non-nuclearity of the compact open topology on the space of entire functions over H . In the classical setting of the space H 2 (Cn , μ) commutative C ∗ and Banach algebras generated by Toeplitz operators were considered in [4, 13, 15]. The main Sects. 3–5 of the paper discuss commutative Toeplitz algebras for the above mentioned function Hilbert spaces and all sections have the same structure. This allows to easily compare the results and methods in the different cases. The structural analysis of commutative Toeplitz algebras is a current topic of interest in Operator Theory and Complex Analysis. A number of open problems intends to stimulate further research in the area. Therefore, in the last section we have collected a (certainly incomplete) list of questions which arose from the analysis presented in this work. We now describe the structure of the paper. In Sect. 2 we explain some notation that are standard in the literature and will be used throughout. Commutative algebras generated by Toeplitz operators acting on the (weighted) Bergman space over the unit ball have been studied most intensively (see [3–7, 20, 26, 27, 31–34] and the literature cited therein) and will be treated in Sect. 3. We introduce various symbol classes which either are invariant or have a homogeneity property under a torus action on Bn . The induced Toeplitz operators generate commutative Banach algebras. The main result concerns the Gelfand theory of such algebras. As an application we discuss semi-simplicity, description of the radical and spectral invariance in the algebra of all bounded operators on the Bergman space. Section 4 carries out the corresponding analysis in case of the classical Hardy space H 2 (S 2n−1 ) over the unit sphere S 2n−1 in Cn , see [1, 10, 24, 30]. An useful ingredient to the proofs is a decomposition of Toeplitz operators (with certain symbols) as an infinite sum of Bergman space Toeplitz operators over Bn−1 with integer weights. This provides a link to the results in Sect. 3. We present some original results, which extend and generalize for general dimensions the results regarding the case n = 3, which was presented in [24]. Section 5 starts with a reminder on the construction of Gaussian measures on an infinite dimensional separable Hilbert space H [9, 14, 18]. Without restriction we assume that H = 2 (N) is the space of square summable sequences over the natural numbers N. We introduce the corresponding Fock space, Toeplitz operators and consider commutative Toeplitz algebras in this set up, see [21, 22]. The structural results on these algebras are new and therefore we have added some proofs. Various open questions remain unsolved. In the concluding Sect. 6 we compare some of the results on operator algebras over different function Hilbert spaces. Moreover, a list of open problems will be mentioned.

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2 Notations We will often divide n-tuples of numbers in groups as follows. Let m ∈ N and k = (k1 , . . . , km ) ∈ Nm such that n = k1 + · · · + km . Given a tuple of complex numbers u = (u 1 , . . . , u n ), we define u ( j) , j = 1, . . . , m, by u (1) = (u 1 , . . . , u k1 ) ··· u ( j) = (u k1 +···+k j−1 +1 , . . . , u k1 +···+k j ), j > 1 ··· u (m) = (u k1 +···+km−1 +1 , . . . , u n ). Hence we have u = (u (1) , . . . , u (m) ). We will write u ( j) = (u j,1 , . . . , u j,k j ) for the entries of u ( j) . Throughout the paper we will denote by Z+ the set N ∪ {0} and, for any integer m > 1, we will repeatedly use the usual multi-index notations for the elements of Zm +: α! = α1 ! · · · αm !, |α| = α1 + · · · + αm ,

αm z α = z 1α1 · · · z m , z ∈ Cm .

We denote by Bn the open unit ball of Cn , by S 2n−1 = ∂Bn the unit sphere in and by T = S 1 the unit circle. Moreover, τ (Bn ) will denote the basis of Bn as a Reinhardt domain:

Cn

τ (Bn ) = {(r1 , . . . , rn ) ∈ Rn+ : r12 + · · · + rn2 < 1}. As usual, we write L(X ) for the space of bounded operators acting on a Banach space X. 2.1 On the Spectrum of an Operator Since we will constantly deal with different notions of spectrum, we set up the following notations. Given an operator S ∈ L(X ), we denote by sppt (S) the point-spectrum, understood as the set of all eigenvalues of S. If S belongs to an algebra A, then σA (S) means the spectrum of S as an element of A and sp(S) denotes the spectrum of S as an operator with respect to the algebra L(X ). Furthermore, ess-sp(S) denotes the essential spectrum of S. If U is a bounded subset of Cn , the polynomially convex hull of U is defined as the  of all z ∈ Cn such that for every (analytic) polynomial p ∈ C[z 1 , . . . , z n ] we set U have | p(z)| ≤ sup | p(w)|. w∈U

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. Recall that a set U is called polynomially convex if U = U As is well-known, a set is polynomially convex if and only if it is the maximal ideal space of some finitely generated commutative unital Banach algebra. As a consequence, if A is the commutative unital Banach algebra generated by an operator S, then σA (S) =  sp S. For more details consult [16, 23].

3 Bergman Space Over the Unit Ball 3.1 The Bergman Space For λ > −1 define the (probability) measure dvλ on Bn by dvλ (z) = cλ (1 − |z|2 )λ dv(z), where dv denotes the usual Lebesgue measure on Cn ∼ = R2n and cλ is a normalizing constant given by cλ =

(n + λ + 1) . π n (λ + 1)

We denote by A2λ (Bn ) the weighted Bergman space with weight parameter λ, being the space of all holomorphic functions from L 2 (Bn , dvλ ). As is well known, the Bergman space A2λ (Bn ) is a reproducing kernel Hilbert space, with a standard orthonormal basis (eα )α∈Zn+ consisting of normalized monomials:  eα (z) =

(n + |α| + λ + 1) α z , α ∈ Zn+ . α!(n + λ + 1)

(3.1)

We denote by P the orthogonal projection from L 2 (Bn , dvλ ) onto A2λ (Bn ). Given m and k as in the previous section, we can decompose the Bergman space with respect to these data into an orthogonal sum. So, for any κ ∈ Zm + consider the finite dimensional subspace Hκ = span{eα : |α( j) | = κ j ,

j = 1, . . . , m}.

(3.2)

Then we have A2λ (Bn ) =



Hκ .

(3.3)

κ∈Zm +

Given a bounded measurable function ϕ ∈ L ∞ (Bn ) we define the Toeplitz operator with symbol ϕ as Tϕ : A2λ (Bn ) −→ A2λ (Bn ) : Tϕ ( f ) = P(ϕ f ). Reprinted from the journal

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Our main task is to study certain operator algebras generated by Toeplitz operators with special symbols. In particular, as it was indicated in the introduction, some symbol classes subordinate to the maximal abelian subgroups of automorphisms of Bn lead to Banach algebras that are commutative for each weight parameter λ > −1, but such that the C ∗ algebra generated by them is no longer commutative. In this work, we present operator algebras that appear when applying such ideas in case of different function spaces. Since the best understood cases are algebras associated with the so-called quasi-radial symbols, we begin our study with this setting and use it as a model case to introduce some of the main ideas. 3.2 Quasi-radial Symbols A bounded measurable function ϕ = ϕ(z) defined on Bn is called k-quasi-radial if it only depends on the groupal radii |z ( j) |, j = 1, . . . , m. That is, if there is a bounded m 2 2 function  ϕ defined on τ (Rm + ) = {(r1 , . . . , rm ) ∈ R+ : r1 + . . . + rm < 1} such that ϕ(z) =  ϕ (|z (1) |, . . . , |z (m) |), ∀z ∈ Bn . Alternatively, a bounded measurable function ϕ is k-quasi-radial if and only if it is invariant under the action of the Cartesian product of unitary groups U(1) × · · · ×U(m) , where U( j) = U(Ck j ) denotes the group of unitary k j × k j -matrices and each U( j) acts on Ck j . This notion interpolates between some well known special cases: ϕ is radial when m = 1 and separately radial when m = n. n Let L ∞ k−qr (B ) denote the space of all k-quasi-radial functions and Tk-qr the corre∗ sponding C algebra generated by Toeplitz operators with symbols in this set. Lemma 3.1 [32, Lemma 3.1] Given a k-quasi-radial function a = a(r1 , . . . , rm ), where r j := |z ( j) |, the Toeplitz operator Ta is diagonal with respect to the standard orthonormal basis (3.1). More precisely, we have Ta z α = γa,k,λ (α)z α , α ∈ Zn+ , where γa,k,λ (α) = γ a,k,λ (|α(1) |, . . . , |α(m) |)  2m (n + |α| + λ + 1) m = a(r1 , . . . , rm )(1 − |r |2 )λ (λ + 1) j=1 (k j − 1 + |α( j) |)! τ (Bm ) ×

m 

2|α( j) |+2k j −1

rj

dr j .

(3.4)

j=1

Since the above formula depends only on the quantities |α( j) |, the functions γ a,k,λ generate an algebra of bounded functions on the set Zm + . Moreover, as was shown in ∗ [32], this algebra separates points of Zm + and thus we can identify Tk-qr with a C algebra of continuous functions on a suitable compactification M(Tk-qr ) of Zm . + 82

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As was observed in [5], the algebra Tk-qr contains the C ∗ algebra of all functions on m Z+ having limits at infinity. In particular, it contains all orthogonal projections Pκ from A2λ (Bn ) onto Hκ , κ ∈ Zm + . Furthermore, as a non-trivial fact, it turns out to contain ( j) also all orthogonal projections Q d mapping A2λ (Bn ) onto the infinite dimensional spaces

( j)

Hd

= span{eα : |α( j) | = d},

(d ∈ Z+ , j ∈ {1, . . . , m})

(3.5)

that is, the projections 

( j)

Qd =

Pκ .

κ∈Zm + ,κ j =d

Note that for each fixed j ∈ {1, . . . , m} one obtains an orthogonal decomposition of the Bergman space: 

A2λ (Bn ) =

( j)

Hd .

(3.6)

d∈Z+

Finally, we recall a useful fibration of the compact set M(Tk-qr ) of maximal ideals which was presented in [3]. Consider the set = {0, 1}m and for each θ ∈ define Jθ = { j : θ j = 1}. We set Zθ+ :=



Z+ ( j) and κθ := {(κ j1 , . . . , κ j|θ| ) : j p ∈ Jθ },

j∈Jθ

where Z+ ( j) denotes a copy of Z+ . Given θ ∈ we set Mθ := μ ∈ M(Tk-qr ) :



( j) μ(Q d )

0 for all d ∈ Z+ , if θ j = 0 = . 1 for some d ∈ Z+ , if θ j = 1

Thus the set Mθ consists of all points of the maximal ideal space that are reached by nets (κα ) in Zm + such that the coordinate (κα ) j tends to infinity if and only if θ j = 0. Since the other entries (κα ) j are essentially constant, we can further decompose the set Mθ as 

Mθ =

Mθ (κθ ),

κθ ∈Zθ+ ( j)

where Mθ (κθ ) := {μ ∈ Mθ : μ(Q κ j ) = 1 for all j ∈ Jθ }. In particular, we have m M1 = Zm + and M1 (κ) = {κ} for any κ ∈ Z+ . Here and subsequently we use the notation 1 := (1, . . . , 1) ∈ {0, 1}m . By construction we obtain the following result.

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Lemma 3.2 [3, Lemma 3.13] The compact space of maximal ideals of Tk-qr admits the following decomposition into mutually disjoints sets M(Tk-qr ) =



Mθ =

θ∈

 

Mθ (κθ ).

(3.7)

θ∈ κθ ∈Zθ

+

The elements of Tk-qr , being diagonal with respect to the canonical basis, will customary be written as Dγ , where γ denotes the corresponding eigenvalue function defined on Zm + . Furthermore, given an element μ ∈ M(Tk-qr ), we will usually write γ (μ) for μ(Dγ ), i.e. the evaluation of the multiplicative functional μ in Dγ . 3.3 Quasi- and Pseudo-homogeneous Symbols The next ingredient to the construction of the algebras we are studying is a family of Toeplitz operators whose symbols are subordinated to the aforementioned group U(1) × · · · × U(m) (see [25]). To introduce these symbols we need some notations. Let z ∈ Bn and recall the decomposition z = (z (1) , . . . , z (m) ). We write r j = |z ( j) |, for each j = 1, . . . , m, and express z ( j) as: z ( j) = r j ξ( j) , ξ( j) ∈ S 2k j −1 := ∂Bk j , r j ∈ R+ . Furthermore, we decompose the tuples ξ( j) into polar coordinates as ξ( j) = (ξ j,1 , . . . , ξ j,k j ) = (s j,1 t j,1 , . . . , s j,k j t j,k j ), where k −1

s( j) = (s j,1 , . . . , s j,k j ) ∈ S+j

k

:= S k j −1 ∩ R+j

and t( j) = (t j,1 , . . . , t j,k j ) ∈ Tk j . The first version of these symbols was introduced in [32] and developed in [3– 5]. It leads to Toeplitz operators with so-called quasi-homogeneous symbols. These functions are defined by p

q

φ j (z) = ξ( j)( j) ξ( j)( j) ,

(3.8)

k

where j ∈ {1, . . . , m} and p( j) , q( j) ∈ Z+j are such that p( j) · q( j) = 0 and | p( j) | = |q( j) |.

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This class of functions was generalized afterwards to the family of pseudohomogeneous symbols (see [17, 34]), which are defined as p

φ j (z) = b(s( j) )t( j)( j) , k −1

(3.9)

k

where b ∈ L ∞ (S+j ) and p( j) ∈ Z+j is such that | p( j) | = 0. We remark that a Toeplitz operator Tφ with symbol φ of the form (3.8) or (3.9) leaves Hκ invariant and that the restriction Tφ | Hκ is nilpotent for every κ ∈ Zm . In particular, sp(Tφ | Hκ ) = {0}. Both classes of functions are invariant under the action of T on Ck j given by (t, z ( j) ) ∈ T × Ck j −→ (t z j,1 , . . . , t z j,k j ).

(3.10)

It turns out that this condition suffices for much of the previous results to hold and so this was the approach in [30], where the so-called generalized pseudo-homogeneous symbols were introduced. These symbols are defined by φ j (z) = c(s( j) , t( j) ),

(3.11)

k −1

where c ∈ L ∞ (S+j × Tk j ) is invariant under the aforementioned action of T (restricted to Tk j ) on its second component. Finally, as was shown in [25], the most general form of these kind of symbols is given by adjoining a quasi-radial dependence to (3.11). That is, symbols of the form φ j (z) = g(|z (1) |, . . . , |z (m) |, s( j) , t( j) ).

(3.12)

Whatever the case may be, it can be shown that the corresponding Toeplitz operator Tφ j also leaves all subspaces Hκ invariant and so it can be decomposed as Tφ j =



Tφ j | Hκ .

(3.13)

κ∈Zm +

We note that the symbols of the form (3.11) (and its above particular cases) can be regarded as functions defined on Bn , Bk j or even Cn and Ck j . To avoid lengthy notation we will use the same symbol to represent any of these functions. As an immediate consequence of the decomposition (3.13), the operator Tφ j commutes with all operators from the C ∗ algebra Tk-qr . Moreover, summarizing the results from [25], the action of Tφ j is given by the following proposition. Proposition 3.3 [25, Corollary 7.7] Let j ∈ {1, . . . , m}, φ j as in (3.12) and α, β ∈ Zn+ . If κ j := |α( j) | = |β( j) | or α(l) = β(l) for some l = j, then Tφ j eα , eβ  = 0. Otherwise Reprinted from the journal

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Tφ j eα , eβ  =

2m−1 (n + λ + |α| + 1)  α( j) !β( j) ! l = j (kl + κl − 1)!

π k j (λ + 1) 

×

k j −1 k τ (Bm )×S+ ×T j

× (1 − |r |2 )λ

m 

α( j) +β( j) +1k j α( j) −β( j) t( j)

g(r , s( j) , t( j) )s( j)

rl2kl +2κl −1 drl ds( j) dt( j) ,

(3.14)

l=1 k

where 1k j := (1, . . . , 1) ∈ Z+j . It can be shown that the restriction of such an operator Tφ j to the spaces Hκ is naturally unitarily equivalent to a tensor product of operators where all factors except the one at the j th position equal the identity operator. So, for different indices j1 , j2 ∈ {1, . . . , m} the operators Tφ j1 and Tφ j2 , acting on different “positions” with respect to the tensor product, commute. (See Section 7 from [25] for more details). As is known, the Bergman space A2λ (Bn ) does not carry a natural tensor product structure such as the Fock space, and so in general the phenomenon described above has to be studied locally on Hκ instead of on the whole space. Nevertheless, when we restrict ourselves to symbols of the form (3.11), we obtain a useful representation for these operators. Consider the tensor product of weightless Bergman spaces A20 (Bk1 ) ⊗ · · · ⊗ 2 A0 (Bkm ). This is a Hilbert space with a canonical orthonormal basis given by eα(1) ⊗ · · · ⊗ eα(m) , α ∈ Zn+ , (1) (m) ( j)

where eα( j) is the canonical basic monomial of the space A20 (Bk j ), see (3.1). We note that, by identifying the corresponding orthonormal basis, the space A2λ (Bn ) is isomorphic to this tensor product of Bergman spaces. More specifically, we denote by U : A2λ (Bn ) → A20 (Bk1 ) ⊗ · · · ⊗ A20 (Bkm ) the unique unitary operator such that U (eα (z)) = eα(1) ⊗ · · · ⊗ eα(m) , α ∈ Zn+ . (1) (m) Lemma 3.4 Let j ∈ {1, . . . , m} and let φ j be a symbol of the form (3.11). We have ( j)

Tφ j = U ∗ (I ⊗ · · · ⊗ Tφ j ⊗ · · · ⊗ I )U ,

(3.15)

( j)

where Tφ j is the Toeplitz operator with symbol φ j acting on the Bergman space A20 (Bk j ).

Although the proof of this lemma is straightforward, one way of understanding why this representation holds consists of analyzing the action of the corresponding Toeplitz operators on the Fock space with symbols given by (3.11) (changing trivially its domain from Bk j to Ck j ). Indeed, since those symbols φ j do not depend on the quantities |z ( j) |, the formulas appearing on the Fock space turn out to be the same

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as the corresponding ones on the Bergman space, so that one can naturally identify these operators. Then using the inherent tensor product structure of the Fock space one easily shows that the corresponding Toeplitz operators can be viewed as some tensor products of operators as in the above lemma. This approach was followed, for example, in [4] and [33]. We introduce some notations suggested by the tensor product structure mentioned ( j) be the finite-dimensional subspace of above. For j ∈ {1, . . . , m} and d ∈ Z+ let H d A20 (Bk j ) defined by ( j) = span{eα( j) : |α( j) | = d, α( j) ∈ Zk+j }. H d ( j) Note that U ∗ maps ( j) ⊗ · · · ⊗ A20 (Bkm ), A20 (Bk1 ) ⊗ · · · ⊗ H d ( j) on the j th position, onto the space H ( j) given by (3.5). being H d d 3.4 Commutative Banach Algebras We will present the most general setting and explain the particular cases. However, it is worth mentioning that, although the quasi-homogeneous case is the simplest one regarding the symbols, it has the interesting property that for each portion j we can take several generators, which leads to some non-trivial properties of the algebra. Example 3.5 Suppose n > 1 and consider the case m = 1. Let h ∈ {1, . . . , n − 1} and let P be the subset of all tuples ( p, q) from Zn+ × Zn+ such that ph+1 = · · · = pk1 = q1 = · · · = qh = 0 and | p| = |q|. For each ( p, q) ∈ P, let ψ( p,q) denote the quasi-homogeneous function given by p

q

ψ( p,q) (z) = ξ(1) ξ(1) ,

z = 0.

Then {Tψ( p,q) : ( p, q) ∈ P} is a commuting set of operators and so the Banach algebra generated by them together with Tk-qr is a commutative unital Banach algebra. Furthermore, it can be shown that the infinite set of generators Tψ( p,q) can be reduced to a finite set. Although the maximal ideal space of the unital Banach algebra generated by the operators from {Tψ( p,q) : ( p, q) ∈ P} is known, it is still an open question to find an explicit description of this set, being the polynomially convex hull of a subset of Ch(n−h) . As a consequence of this, not much can be said about the spectral invariance of the algebras studied in this context (compare with Proposition 3.12 below). See [3, 5, 32] for more details. The action of a Toeplitz operator with arbitrary pseudo-homogeneous symbol on Hκ could be more complicated, as (3.14) indicates. In general, it is no longer possible to take more than one generator for each portion j without losing commutativity.

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Since much of the results of this section still hold for quasi-homogeneous symbols, we proceed to describe the Gelfand theory of the corresponding algebras when the symbols are of the more general form (3.11). Thus fix for each j = 1, . . . , m a generalized pseudo-homogeneous symbol φ j of the form (3.11) and let Tph be the (commutative) unital Banach algebra generated by the Toeplitz operators Tφ j , j = 1, . . . , m. 3.5 Gelfand Theory We recall that for j ∈ {1, . . . , m} one has a decomposition ( j)

Tφ j =



( j)

Tφ j | H( j) , d

d∈Z+ ( j)

( j)

where Tφ j is the operator from (3.15). Since each Tφ j | H( j) is an operator on a finited dimensional space one has the following result. Lemma 3.6 (See [30, Section 3.3]). Let j ∈ {1, . . . , m} and φ j be as in (3.11), ( j) considered as a function on Bk j , and ζ j ∈ sp(Tφ j ). Then there is a sequence of ( j) ( j) ( j) for some sequence unimodular functions ( f n )n such that, for all n, f n ∈ H d(n) (d(n))n , and ( j)

( j)

lim (Tφ j − ζ j ) f n  = 0.

(3.16)

n→∞ ( j)

Moreover, if ζ j ∈ sppt (Tφ j ) then one can choose d(n) = d for any fixed d such that ( j)

( j)

ζ j ∈ sppt (Tφ j | H( j) ) and if ζ j ∈ ess-sp(Tφ j ), then one can suppose that d(n) → ∞. d

( j)

Using a suitable sequence of tensor products of the above functions f n one obtains the following characterization for the maximal ideal space of the algebra Tph . Proposition 3.7 [29, Proposition 3.5]. The compact set M(Tph ) of maximal ideals of Tph can be identified with the set   (1) (m) M(Tph ) = sp(Tφ1 ) × · · · × sp(Tφm ). We introduce now our main object of study. Let Tk-qr,ph be the Banach algebra generated by both algebras Tk-qr and Tph . By our previous discussion, Tk-qr,ph is a commutative Banach algebra for each weight parameter λ > −1 and, in general, it is no longer commutative when it is extended to a C ∗ algebra. Denote by T k-qr,ph the non-closed dense subalgebra of Tk-qr,ph generated by all finite sums of finite products of the generators. Now we proceed to develop the Gelfand theory of this algebra, presenting the main results and ideas. Since some of the proofs are non-trivial and lenghty, we refer to [29] for a detailed discussion.

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First, we determine the maximal ideal space of Tk-qr,ph . By restricting multiplicative functionals to the generating subalgebras and applying Proposition 3.7 we obtain the following inclusion:   M(Tk-qr,ph ) ⊂ M(Tk-qr ) × M(Tph ) = M(Tk-qr ) × sp(T φ1 ) × · · · × sp(T φm ). Thus one needs to determine all points from the Cartesian product that lie in the maximal ideal space of Tk-qr,ph . For the following remarks and results, we recall decomposition (3.7) and the notations introduced before it. Furthermore, we will constantly write ζ for the tuple ζ = (ζ1 , . . . , ζm ) ∈ Cm . Roughly speaking (and under some additional conditions) it turns out that the compact set of maximal ideals M(Tk-qr,ph ) consists of (the polynomially convex hull of) those ordered pairs (μ, ζ ) from the above Cartesian product such that μ ∈ Mθ (κθ ) for θ ∈ {0, 1}m and κ ∈ Zθ+ , and

ζj ∈

⎧ ⎨sp(T ( j) | ( j) ), if j ∈ Jθ φj H κj

⎩ess-sp(T ( j) ), φj

otherwise.

Although the first condition holds in general, we have to assume special properties of the symbols φ j for the second one to hold. As is well known (see for example [11]), the C ∗ algebra T0 (C(Bk j )) generated by Toeplitz operators with continuous symbols on Bk j contains the ideal of compact operators K(A20 (Bk j )) and the quotient

T0 (C(Bk j ))/K(A20 (Bk j )) is isometrically isomorphic to the C ∗ algebra C(∂Bk j ) via the isomorphism generated by the mapping Ta + K(A20 (Bk j )) −→ a|∂Bk j .

In particular, for a Toeplitz operator Ta ∈ T0 (C(Bk j )), we have ess-sp(Ta ) = a(∂Bk j ). The compactness of the semi-commutators T|φ j |2 − Tφ j Tφ∗j will be an essential ingredient to our analysis. Therefore, for the rest of the section, we will assume that the symbols φ j , considered as functions on Bk j , extend continuously to the boundary of Bk j . This condition was assumed for symbols of the form (3.9) in [17]. Proposition 3.8 [29, Theorem 4.4] The maximal ideal space of Tk-qr,ph can be identified with the set  

Mθ (κθ ) × Mθ,κθ ,1 × · · · × Mθ,κθ ,m ,

θ∈ κθ ∈Zθ

+

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where with the previous notation κθ = (κ j1 , . . . , κ j|θ| ), and jl ∈ Jθ Mθ,κθ , j =

⎧ ⎨sp(T ( j) | ( j) ), if j ∈ Jθ φj H κj

( j) ⎩e ss-sp(T ), φj

otherwise.

Furthermore, the Gelfand transform is generated by the following map on the generators of the algebra: 

Dγρ T ρ −→

ρ∈F



γρ (μ)ζ ρ , (μ, ζ ) ∈ Mθ (κθ ) × Mθ,κθ ,1 × · · · × Mθ,κθ ,m ,

ρ∈F

(3.18) ρ

ρ

ρ := T 1 · · · T m and the operators D where F ⊂ Zm γρ are + is a finite subset, T φ1 φm diagonal operators in Tk-qr with eigenvalue sequence γρ .

It is not difficult to recover the known particular cases. If n = 2 (and then necessarily m = 1), we get M(Tk-qr ) = M(1) ∪ M(0) = Z+ ∪ M∞ , where M∞ represents the points at infinity. Since for the quasi-homogeneous ( j) and pseudo-homogeneous cases, all matrices Tφ1 | H(1) are nilpotent and thus d

sp(Tφ(1) | (1) ) = {0}, we have 1 H d

  M(Tk-qr,ph ) = (Z+ × {0}) ∪ M∞ × e ss-sp(Tφ(1) ) , 1 which coincides with the previous results from [5, 17]. 3.6 Applications We make use of the above results to obtain structural information about the algebra Tk-qr,ph . A non-trivial problem concerns the characterization of its radical. Since the first works on this topic, one usually starts the study of the radical by analyzing the non-closed subalgebra Rad(Tk-qr,ph ) ∩ T k-qr,ph . After a short examination one can detect some typical elements of this algebra. These operators have the form Dγ



( j) ( j)

Q d pd (Tφ j ),

(3.19)

d∈FL ( j)

according to the orthogonal decomposition (3.6). Here, pd is the polynomial ( j)

pd (z) = (z − ζ1 ) · · · (z − ζ M ), 90

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where sp(Tφ j | H( j) ) = {ζ1 , . . . , ζ M }, and Dγ ∈ Tk-qr is such that γ (μ) = 0 for any d

( j)

μ ∈ Mθ with θ j = 0. Moreover, FL = {d ∈ Z+ : | sp(Tφ j | H( j) )| ≤ L}, L ∈ Z+ . d Indeed, provided such an element belongs to the algebra Tk-qr,ph , one can easily see that it is mapped to zero by all multiplicative functionals in Propositon 3.8. Nevertheless it is still not clear whether such an operator always belong to Tk-qr,ph . A complete answer to this (and a generalization of Proposition 3.10 below) could probably be obtained by analyzing the (asymptotical) behaviour of the cardinality of ( j) the finite sets sp(Tφ j | H( j) ) in the parameter d. In particular, it would be useful to know d

( j)

under which conditions on the symbol φ j the quantity | sp(Tφ j | H( j) )| tends to infinity d as d → ∞. ( j) We remark that if φ j is of the form (3.8) or (3.9), then all matrices Tφ j | H( j) are d

( j)

nilpotent and thus pd (z) = z for every d ∈ Z+ . Hence we have 

( j) ( j)

Q d pd (Tφ j ) = Tφ j ,

d∈FL

so that (3.19) can be written as Dγ Tφ j , recovering the corresponding operators described in [4, 5]. ( j) As we can see, the polynomials pd above turn out to play an essential role. One can furthermore characterize semi-simplicity by means of these polynomials: Proposition 3.9 [29, Proposition 3.9]. The Banach algebra Tk-qr,ph is semi-simple, ( j) i.e. Rad(Tk-qr,ph ) = {0}, if and only if all matrices Tφ j | H( j) satisfy the condition ( j)

d

( j)

pd (Tφ j | H( j) ) = 0, that is, if and only if all such matrices are diagonalizable in the d sense that their Jordan canonical form is diagonal. Although a complete characterization of Rad(Tk-qr,ph ) ∩ Tk−qr ,qh is still missing for the most general case, in all examples we know, the operators of the form (3.19) generate this algebra. We present some important cases: Proposition 3.10 [29, Proposition 7.4] Assume that one of the following conditions holds: (1) m = 1, ( j) (2) All matrices Tφ j | H( j) are nilpotent, d

( j)

(3) For each j ∈ {1, . . . , m}, the number of distinct eigenvalues of the matrix Tφ j | H( j) d tends to infinity as d → ∞.  := Rad(Tk-qr,ph ) ∩ Tk−qr ,qh is the non-closed algebra generated by all operThen R ators of the form (3.19).  coincides with the whole As a non-trivial fact, it turns out that the closure of R radical:

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Proposition 3.11 [29, Corollary 6.9]. Suppose that φ j is continuous up to the boundary of Bk j for every j ∈ {1, . . . , m}. Then  Rad(Tk-qr,ph ) = clos Rad(Tk-qr,ph ) ∩ T k-qr,ph ).

(3.20)

This result was partially proved for the case n = 2 in [5] and for the case m = 1 (and any positive integer n) in [4]. We remark that it is still valid for the quasi-homogeneous case taking more than one generator in each division. (See Section 6 from [29]). Finally, we state an interesting result regarding the spectral invariance, which can be completely characterized in this setting. We observe a different behaviour from what was obtained in the quasi-homogeneous case. Recall that an algebra A ⊂ L(X ) of bounded operators acting on some Banach space X , is called spectral invariant or closed under inversion if the inverse of every invertible operator A ∈ A (invertible with respect to algebra of all bounded operators L(X )) belongs to A. We may shortly express this statement in the form A ∩ L(X )−1 = A−1 , where A−1 denotes the group of invertible elements in A. For a commutative unital Banach algebra A, spectral invariance is not automatic and can be studied in terms of its multiplicative functionals. Indeed, given an operator S ∈ A, invertible with respect to the algebra of bounded operators, its inverse S −1 will also belong to A if and only if the Gelfand transform of S is nowhere zero. Proposition 3.12 [29, Proposition 5.1] The algebra Tk-qr,qh is spectral invariant if and ( j) only if the sets ess-sp(Tφ j ) are polynomially convex for j = 1, . . . , m.

4 Hardy Space Over the Unit Ball In this section we construct and analyze families of commutative Banach algebras generated by Toeplitz operators on the Hardy space over the unit sphere. One of the first approaches to the study of such algebras is [1]. Partially by adapting methods in [27] the authors described commutative Toeplitz C ∗ algebras by studying symbols invariant under the action of maximal abelian groups of automorphisms. However, there is a different approach based on the known Bergman space theory. In [24], the Hardy space is decomposed into a direct sum of Bergman spaces (with different integer weight parameters). Then results obtained in the Bergman space setting can be applied. This approach permits also the study of commutative Toeplitz Banach algebras as in the preceding section. We recall the ideas and further extend the analysis in [24]. 4.1 The Hardy Space For the study of the Hardy space we follow the notations and the constructions from [24].

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Let S 2n−1 = ∂Bn denote the unit sphere in Cn and by dσ denote the normalized 2n−1 as z = (z  , z ), where z  ∈ surface measure of S 2n−1 . Denote n the points z of S n−1  2 and z n ∈ C with |z n | = 1 − |z | . B As usual we define the Hardy space H 2 (S 2n−1 ) as the (closed) subspace of the Hilbert space L 2 (S 2n−1 , dσ ) consisting of functions f satisfying the tangential Cauchy-Riemann equations:   ∂ ∂ f = 0, 1 ≤ k < j ≤ n. − zj L k, j f = z k ∂z j ∂z k The orthogonal projection from L 2 (S 2n−1 , dσ ) onto H 2 (S 2n−1 ) is called the Szegö projection and will be denoted by P. There is an interesting relation between the Hardy space and Bergman spaces over Bn−1 with integer weights: Proposition 4.1 [24, Theorem 2.1, Corollary 2.2] There is a unitary operator U from L 2 (S 2n−1 , dσ ) onto  L 2 (Bn−1 , dv p ) p∈Z+

under which H 2 (S 2n−1 ) is mapped onto

 p∈Z+

A2p (Bn−1 ).

Given a bounded measurable function ϕ defined on S 2n−1 we define the Toeplitz operator T ϕ acting on the Hardy space H 2 (S 2n−1 ) as Tϕ f = P(ϕ f ),

f ∈ H 2 (S 2n−1 ). p

To avoid confusion we will denote by Tϕ the Toeplitz operator with symbol ϕ ∈ acting on the Bergman space A2p (Bn−1 ). Now let ϕ ∈ L ∞ (S 2n−1 ) be a function of the form (4.1) ϕ(z) = ϕ(z  , |z n |) = ϕ(z  , 1 − |z  |2 ). L ∞ (Bn−1 )

Note that to ϕ we can associate a unique function ϕ ∈ L ∞ (Bn−1 ) such that ϕ(z 1 , . . . , z n−1 ) = ϕ(z 1 , . . . , z n−1 , 1 − |z  |2 ).

(4.2)

Toeplitz operators with that kind of symbols behave well with respect to the decomposition from Theorem 4.1 as the following result shows. Proposition 4.2 [24, Theorem 3.1] Let ϕ(z  , |z n |) be a bounded measurable symbol T ϕ , acting defined on S 2n−1 . Under the above isomorphism U , the Toeplitz operator p on the Hardy soace H 2 (S 2n−1 ) is unitarily equivalent to the operator p∈Z+ Tϕ , acting on 

A2p (Bn−1 ),

p∈Z+

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where ϕ = ϕ(z  ) is of the form (4.2). By means of Propositions 4.2 and 4.1, one can easily apply our construction in the setting of the Bergman space to the Hardy space. p Given a weight parameter p ∈ Z+ and κ ∈ Zm + , let Hκ denote the subspace Hκ 2 n−1 of the Bergman space A p (B ) as defined in (3.2). Then by Proposition 4.1, we can further decompose the Hardy space as the direct sum 

H 2 (S 2n−1 ) =

Hκp .

p∈Z+ ,κ∈Zm +

Consider a positive integer m and a tuple of positive integers k = (k1 , . . . , km ) such that k1 + · · · + km = n − 1. Taking into account our representation of points in S 2n−1 as z = (z  , z n ), with z  ∈ Bn−1 , we divide the tuple z  into m groups as before (see Sect. 2). Thus we will write z  = (z (1) , . . . , z (m) ). 4.2 Quasi-radial Symbols 2n−1 ) be the set of all symbols a of the form (4.1) such that the associated Let L ∞ k−qr (S n−1 ). By Proposition 4.2 and Lemma 3.1, a Toeplitz symbol a belongs to L ∞ k−qr (B ∞ 2n−1 operator Ta with a ∈ L k−qr (S ) acts as a constant multiple of the identity on all p subspaces Hκ ,

Ta =



γa,k ( p, κ)I .

p∈Z+ ,κ∈Zm +

Here γa,k is the function given by γa,k ( p, κ) := γa,k, p (κ), where γa,k, p was defined in (3.4). Thus Ta is a diagonal operator whose eigenvalue set depends on κ as well as on the weight parameter p ∈ Z+ . Again, we denote by Tk-qr the C ∗ algebra generated 2n−1 ). by Toeplitz operators with symbols from L ∞ k−qr (S First, we present some facts in the particular case m = 1 and k = (k1 ) = (n − 1). That is, we consider the space H 2 (S 2k1 +1 ). In this case, for a k-quasi-radial function a ∈ L ∞ (0, 1), the eigenvalues γa,k of the corresponding Toeplitz operator have the form  2m (n + κ + p)! 1 γa,k (κ, p) = a(r )(1 − r 2 ) p r 2κ+2k1 −1 dr , (κ, p) ∈ Z2+ . p!(k1 − 1 + κ)! 0 So, the algebra Tk-qr is isomorphic to an algebra of bounded functions on Z2+ . We remark that γa,k (κ, p) = γa,(2) (κ + k1 − 2, p), where γa,(2) represents the corresponding function for the particular case n = 3 (corresponding to operators acting on H 2 (S 5 )). This case was studied in [24, Section 6] and, due to the above relation, one can imitate the analysis done there.

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In particular, we conclude that the algebra Tk-qr separates the points of Z2+ and,   2 ), where Z 2 = Z2 ∪ S is the compactfurthermore, it contains the C ∗ algebra C(Z ∞ + + + 2 iθ : θ ∈ [0, π/2]}. Here eiθ ∼ ification of Z+ by the infinitely far quarter-circle S∞ = {e∞ ∞ is the point reached by sequences ((k , p )) with tan θ = lim

→∞

k . p

Finally, we consider the symbol g ∈ L ∞ (0, 1) (see Section 6 in [24]) given by g(r ) =

1 (ln(1 − r −2 ))−i . (1 + i)

(4.3)

As was shown in the aforementioned work, given (κ, p) ∈ Z2+ , we have   γg,(2) (κ, p) = ( p + 1)−i dκ+1 + O

1 p+1

 ,

i where dκ+1 = dκ (1 + κ+1 ) and d1 = 1 + i. By some careful analysis one can show 1 that indeed O( p+1 ) is a function of (κ, p) that converges uniformly to zero whenever κ + p → ∞ with κp ≤ 1. Moreover, we have limκ→∞ dκ = (1 + i)−1 . As a consequence, we can separate some of the points of M(Tk-qr )\Z2+ for different integer coordinates κ: If (κα , pα ) is a net convergent to an element μ in M(Tk-qr ) and κα pα ≤ 1 for all α, then μ ∈ dκ0 +1 T if and only if, for some subnet ((καβ , pαβ ))β , we have καβ = κ0 . Now we proceed to examine the general case m ≥ 1 by means of the above remarks. Given j ∈ {1, . . . , m} and a function a ∈ L ∞ (0, 1), we denote by a( j) the function defined on S 2n−1 by

a( j) (z) = a(|z ( j) |).

(4.4)

Lemma 4.3 Let j ∈ {1, . . . , m} and a ∈ L ∞ (0, 1). Then γa( j) ,k (κ, p) = γa,(k j ) (κ j , p + |τ j (κ)|), κ ∈ Zm +, where γa,(k j ) is the eigenvalue sequence corresponding to the Toeplitz operator with radial symbol a acting on the space H 2 (S 2k j +1 ). Proof It follows directly from Fubini’s Theorem applied to the representation (3.4) of eigenvalues and well-known properties of the Beta function.   Given (κ, p) ∈ Zm + × Z+ consider the following subspaces of the Hardy space Hκp = span{eαp : |α( j) | = κ j , Reprinted from the journal

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j = 1, . . . , m},

W. Bauer, M. A. Rodriguez Rodriguez p

where eα denote the normalized monomials from the Bergman space A2p (Bn−1 ) given p by (3.1). We denote by P(κ, p) the orthogonal projection from H 2 (S 2n−1 ) onto Hκ . m+1 Corollary 4.4 The algebra Tk-qr contains all projections P(κ, p) , (κ, p) ∈ Z+ .

 2 ). Proof This is clear for the case k = (k1 ), since all such projections belong to C(Z + The general case then follows from Lemma 4.3 by multiplying adequate projections   for each coordinate κ j . According to our previous remarks about g in (4.3), the eigenvalue function γ g ( j) ,k , m+1 defined on Z+ and corresponding to an Toeplitz operator on H 2 (S 2n−1 ), has the m+1 property that for every net ((κ α , p α ))α in Z+ with

in M(Tk-qr ) to μ ∈

m+1 M(Tk-qr )\Z+ ,

κ αj

p α +|τ

j (κ

α )|

≤ 1 and convergent

we have

γ g ( j) ,k (μ) ∈ dρ+k j −1 T

if and only if

lim κ αj = ρ. α

m+1 with different “finite” coorWe can use this to separate the points of M(Tk-qr )\Z+ dinates:

Corollary 4.5 Let j ∈ {1, . . . , m} and ρ ∈ Z+ . Then there is an operator Dh j,ρ ∈ Tk-qr m+1 with h j,ρ (κ, p) ∈ [0, 1] for all (κ, p) ∈ Z+ and such that, for any net ((κ α , p α ))α m+1 m+1 in Z+ convergent to μ ∈ M(Tk-qr )\Z+ , h j,ρ (μ) =

1 if limα κ αj = ρ 0 otherwise.

Proof Let f 1 : C → [0, 1] be a bump function equal to 1 on the set dρ+k j −1 T and  2 → [0, 1] equal to 0 outside a sufficiently small neighborhood of this set. Let f : Z 2

+

0·i ) = 1 and equal to 0 outside a sufficiently small be a bump function with f 2 (e∞ neighborhood of this point. Then the function f 1 (Dγ g( j) ,k )D f2 ◦ι ∈ Tk-qr has the required properties, where

 2 ι : M(Tk-qr ) → Z +  2)⊂T is the inclusion obtained from the inclusion of C ∗ algebras C(Z k-qr . +

 

For the sake of simplicity, in the following analysis we will denote the weight parameter p as κm+1 , so that the point (κ, p) ∈ Zm + × Z+ will be also written as m+1 . As in the previous section, let = {0, 1}m+1 (κ, p) = (κ1 , . . . , κm , κm+1 ) ∈ Z+  and for every θ ∈ consider the sets Jθ = { j : θ j = 1} and Zθ+ = j∈Jθ Z+ ( j). We denote by Mθ the set of all points μ from M(Tk-qr ) that are limits of a net α )) in Zm+1 such that κ α → ∞ if and only if θ = 0. Furthermore, ((κ1α , . . . , κm+1 α j + j 96

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for each κθ = (κ j1 , . . . , κ j|θ| ) ∈ Zθ+ we define the sets Mθ (κθ ) by   α Mθ (κθ ) = μ ∈ Mθ : μ = lim(κ1α , . . . , κm+1 ) with κ αj = κ j , for all j ∈ Jθ . α

Corollary 4.6 The compact set of maximal ideals of Tk-qr admits the following decomposition into disjoint sets M(Tk-qr ) =

 

Mθ (κθ ).

θ∈ κθ ∈Zθ

+

Proof It follows from separating the points by means of the operators from Corollary m+1 , similarly as in the Bergman space 4.5 and the projections P(κ, p) , (κ, p) ∈ Z+ setting.   4.3 Commutative Banach Algebras Note that we can reproduce the construction from Sect. 3.3 for the z  -component of the tuples (z  , z n ) ∈ S 2n−1 . Thus we can naturally define symbols φ j ∈ L ∞ (S 2n−1 ) such that the corresponding functions φ j ∈ L ∞ (Bn−1 ) are of the form (3.12). Moreover, such symbols do not depend on z n , and with respect to the decomposition in Proposition 4.2 the corresponding Toeplitz operators T φ j decompose as Tφj =



p

Tφ j .

p∈Z+ p

Here Tφ j is the Toeplitz operator with symbol φ j acting on the Bergman space

A2p (Bn−1 ), already studied in Sect. 3. From the integral expression (3.14) one observes that when the functions φ j are p of the form (3.11), the action of Tφ j does not depend on p so that the operator is essentially the same for different weight parameters. This observation simplifies some of the calculations. We will consider only this case, leaving the study of symbols of the form (3.12) for a future work.

4.4 Gelfand Theory and Applications Fix for each j ∈ {1, . . . , m} a symbol φ j of the form (3.11) and let Tph be the Banach algebra generated by the operators T φ 1 , . . . , T φ m . By the analysis of the previous section, we note that this is a commutative Banach algebra, which in general is noncommutative, when extended to a C ∗ algebra. ( j) To avoid notation, we will use the Toeplitz operators Tφ j acting on the Bergman spaces A20 (Bk j ) from Sect. 3.3. Since the action of the operators T φ j does not depend on the weight parameter p, one can see that the maximal ideal space remains the same as in the Bergman space case:

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Proposition 4.7 The compact space of maximal ideals of the algebra Tph coincides with the set   (1) (m) M(Tph ) = sp(Tφ1 ) × · · · × sp(Tφm ). Proof A similar argument as in the Bergman space case.

 

Let Tk-qr,ph be the Banach algebra generated by the algebra Tk-qr and the Toeplitz operators T φ 1 , . . . , T φ m . As usual, we denote by M(Tk-qr,ph ) its maximal ideal space. Reasoning as before, we can identify M(Tk-qr,ph ) with a subset of the Cartesian product of the maximal ideal spaces of the generating algebras:   (1) (m) M(Tk-qr,ph ) ⊂ M(Tk-qr ) × M(Tph ) = M(Tk-qr ) × sp(Tφ1 ) × · · · × sp(Tφm ). Using the same arguments as in the Bergman space case, one can detect almost all points belonging to the maximal ideal space. There is, however, an interesting difference compared to the Bergman space setting given by the following proposition. Proposition 4.8 If (μ, ζ ) ∈ M(Tk-qr,ph ), θ ∈ and μ ∈ Mθ (κθ ) (see (3.7)), then ( j) ζ j ∈ sp(Tφ j | H( j) ) for every j ∈ {1, . . . , m} with θ j = 1. κj

( j)

Proof Let q be the characteristic polynomial of the matrix (Tφ j | H( j) ). We assume that κj

θ = 0 and distinguish two cases:

(1) If θ = 1, then P(κ, p) q(T φ j ) = 0 and thus, after evaluating the functional μ, we get q(ζ j ) = 0. (2) Otherwise, let j ∈ {1, . . . , m} with θ j = 1 and write ρ := κ j . Let I be the closed ideal in Tk-qr,ph generated by the projections P(κ, p) , (κ, p) ∈ m+1 Z+ , and let π : Tk-qr,ph −→ Tk-qr,ph /I be the canonical projection. One easily sees that the multiplicative functional (μ, ζ ) =: ψ(μ,ζ ) can be factorized as (μ,ζ ) ◦ π, ψ(μ,ζ ) = ψ (μ,ζ ) is a multiplicative functional defined on Tk-qr,ph /I. where ψ Consider now the operator S = Dh j,ρ q(T φ j ), where Dh j,ρ is given by Corollary 4.5. By construction, one can check that S + I = 0 and thus (μ,ζ ) (S + I) = 0. 1 · q(ζ j ) = h j,ρ (μ)q(ζ j ) = ψ(μ,ζ ) (S) = ψ ( j)

In both cases (1) and (2) we have q(ζ j ) = 0 and thus ζ j ∈ sp(Tφ j | H( j) ). κj

 

As a consequence, we see that for a multiplicative functional (μ, ζ ) ∈ M(Tk-qr,ph ) ( j) we have ζ j ∈ ess-sp(Tφ j ) only when μ ∈ Mθ with θ j = 0, that is, roughly speaking,

when μ has an "infinite j th coordinate". We note that this fact is independent of the behaviour of the coordinate κm+1 = p, which plays the role of the weight parameter in the space decomposition in Proposition 4.1.

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Proposition 4.9 The maximal ideal space of Tk-qr,ph coincides with the set  

Mθ (κθ ) × Mθ,κθ ,1 × · · · × Mθ,κθ ,m .

θ∈ κθ ∈Zθ

+

where Mθ,κθ , j =

⎧ ⎨sp(T ( j) | ( j) ), if j ∈ Jθ \{m + 1} φj H κj

( j) ⎩e ss-sp(T ), φj

otherwise.

Furthermore, the Gelfand transform is generated by the following map on the generators of the algebra Tk-qr,ph :  ρ∈F

Dγρ T ρ −→



γρ (μ)ζ ρ , (μ, ζ ) ∈ Mθ (κθ ) × Mθ,κθ ,1 × · · · × Mθ,κθ ,m ,

ρ∈F

(4.5) ρ

ρ

m 1 ρ where F ⊂ Zm + is a finite subset and T := T φ1 · · · T φm .

Proof It follows from Proposition 4.8 and similar arguments as in the Bergman space setting.   Compare with [24] for the case n = 3. Although the maximal ideal space has almost the same form as in the Bergman space setting, it is worth mentioning that, according to our remarks before the proposition, we have the above set Jθ \{m + 1}, instead of just Jθ , as in the Bergman space. This will probably change for symbols of the form (3.12). We can apply this result to obtain some structural information on the algebra Tk-qr,ph as in the previous section. However, there is still much work to do in this direction. As a matter of example, we can search for typical elements of the radical of Tk-qr,ph . However, such operators seem to be more complicated compared to those found in the Bergman space case. Indeed, one would consider elements similar to the operator ( j) (3.19), replacing the projections Q d by operators appearing in Corollary 4.5. Nevertheless, one can still characterize semi-simplicity, having the same phenomenon as in the Bergman space. (We use for simplicity once again the notations introduced in that case). Proposition 4.10 The algebra Tk-qr,ph is semi-simple, i.e. Rad(Tk-qr,ph ) = {0}, if and ( j) ( j) ( j) only if all matrices Tφ j | H( j) satisfy the condition pd (Tφ j | H( j) ) = 0, that is, if and d d only if all such matrices are diagonalizable in the sense that their Jordan canonical form is diagonal. Proof This follows from the same arguments as in the Bergman space setting.

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5 Fock Space Over Infinite Dimensional Hilbert Spaces In this section we consider Toeplitz operators acting on the Segal-Bargmann space over an infinite dimensional Hilbert space (see [21, 22]) and commutative Banach algebras generated by such operators. We start by recalling the notion of Gaussian measures on a real separable Hilbert space (H , ·, ·) (see [18] for further details and proofs). Let (·, ·) be a continuous scalar product (positive definite bilinear form) in H and by H  ∼ = H we denote the dual space consisting of all continuous linear functionals. Then there exists a bounded self-adjoint, injective and positive operator B on H such (x, y) = Bx, y

for all

x, y ∈ H .

(5.1)

Let F ⊂ H be a subspace of finite dimension n ∈ N. Bochner’s Theorem implies that on F there is a unique Radon measure ν F with Fourier transform χν F (y) satisfying:  (y,y) ei(x,y) dν F (x) = e− 4 for all y ∈ F. χν F (y) := F

We call ν F the Gaussian measure on F associated with the inner product (·, ·). Define the annihilator space   F ◦ = ϕ ∈ H  : ϕ(x) = 0 for x ∈ F and consider the finite dimensional quotient H  /F ◦ . Let π:H∼ = H  → H  /F ◦ denote the canonical projection. Given any Borel subset X ⊂ H  /F ◦ we can consider the preimage N X := π −1 (X ) ⊂ H . We call N X a cylinder set with base X and we refer to H  /F ◦ as its generating space. When F runs through the collection of all finite dimensional subspaces of H then the corresponding cylinder sets form an algebra denoted by C(H ). Moreover, the σ -algebra generated by C(H ) coincides with the Borel σ -algebra B(H ) of H . Consider F ⊂ H equipped with the restriction of (·, ·) as a Hilbert space. There is a canonical isomorphism IF : F ∼ = F  −→ H  /F ◦ : v → (·, v) + F ◦ , which can be applied to define a Radon measure μ F on B(H  /F ◦ ) by   μ F (X ) := ν F I F−1 (X ) . It can be verified that the family of measures (μ F ) F , where F runs through the finite dimensional subspaces of H fulfills the following compatibility condition: let G ⊂ H be finite dimensional with F ⊂ G and consider the canonical map p : H  /G ◦ → H  /F ◦ : ϕ + G ◦ → ϕ + F ◦ .

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Then, for each Borel set X ∈ H  /F ◦ we have:   μ F (X ) = μG p −1 (X ) .

(5.2)

Condition (5.2) implies that there is a well-defined real valued set function μ on C(H ) such that μ(N X ) = μ F (X ) where X ∈ B(H  /F ◦ ). Clearly 0 ≤ μ(N ) ≤ 1 for all cylinder sets N ∈ C(H ) and μ(H ) = 1. We call μ a Gaussian cylinder measure. It is important to note that a cylinder measure in general is not σ -additive. It only is defined on the algebra of cylinder sets and may not extend to a measure on the full Borel σ -algebra B(H ). A characterization of the σ -additivity of cylinder measures in terms of the embedding (H , ·, ·) ⊂ (H , (·, ·))

(5.3)

is given by the following result: Proposition 5.1 [18] The Gaussian cylinder measure μ on C(H ) above extends to an σ -additive Borel measure on B(H ) if and only if the operator B in (5.1) is trace class. In this case the embedding (5.3) is of Hilbert-Schmidt type. In this section we consider complex separable Hilbert space H . Since H can as well be seen as a real Hilbert space the above construction of Gaussian measures applies. The operator B appearing there will be assumed to be complex linear. To simplify the setting we consider the model case H = 2 (N) of all squaresummable complex valued sequences equipped with the standard inner product. Let [ε j = (δ j ) ∈ H : j ∈ N] denote the canonical orthonormal basis of H and assume that B is a positive diagonal nuclear (trace class) operator, i.e. Bε j = λ j ε j

where

λ j > 0 and



λ j = tr(B) < ∞.

j∈N

We write Z0 ⊂ ZN + for the set of all sequences α = (αn )n with non-negative integer entries such that αn = 0 for all but finitely many n ∈ N. Let z = (z 1 , z 2 , . . .) be the coordinates of H . With α ∈ Z0 we use the usual notation: z α = z 1α1 z 2α2 . . .

α! = α1 !α2 !α3 ! . . .

λα = λα1 1 λα2 2 λα3 3 . . . .

According to Proposition 5.1, the induced Gaussian cylinder measure is a σ -additive measure on B(H ) and will be denoted by μ B . Note that the space L 2B := L 2 (H , dμ B ) contains all complex polynomials in H .

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Definition 5.2 We define the Fock space over H induced from B by (see [21, 22]):

A2B (H )





= span eα (z) =

 1 α z ⊂ L 2B , : α ∈ Z 0 λα α!

where the closure is taken in L 2B . It can be checked that the monomials [eα : α ∈ Z0 ] form an orthonormal basis of A2B (H ). In the following we call it the standard orthonormal basis. Remark 5.3 In the classical case where we deal with a finite number of variables (i.e. H = Cn for some n ∈ N) it is well-known that A2B (Cn ) can be interpreted as a Hilbert space of (pointwisely defined) entire functions with reproducing kernel. In fact, well-known apriori estimates (see [35]) imply that L 2 -convergence in A2B (Cn ) implies uniform convergence on compact subsets of Cn . In the setting of functions in infinitely many variables, i.e. H = 2 (N), we only have the following weaker 1 property: let V ⊂ (H , ·, ·) be an open set and assume that z 0 ∈ V ∩ B 2 H . We equip 1 1 the range H 1 := B 2 H with the Hilbert space norm  ·  1 := B − 2 · . Then it can 2 2 be shown that there is an open neighbourhood Wz 0 ⊂ (H 1 ,  ·  1 ) of z 0 (open in the 2 2 topology of H 1 ) and a constant C z 0 > 0 such that for any holomorphic function f in 2 V:   sup | f (z)| : z ∈ Wz 0 ≤ C z 0



1 | f (w)| dμ B (w) 2

2

.

V

Hence L 2B -convergence in A2B (H ) implies uniform compact convergence in the subspace (H 1 ,  ·  1 ) only. In conclusion we may consider elements in the Fock space 2

2

A2B (H ) as holomorphic functions on H 1 ⊂ H . In general they do not extend to holo2 morphic functions on H . Also note that H 1 is dense in H of measure μ B (H 1 ) = 0. 2

2

Let P : L 2B → A2B (H ) denote the orthogonal projection. Given any bounded measurable function ϕ ∈ L ∞ (H , μ B ) we define the Toeplitz operator Tϕ with symbol ϕ in the usual way: Tϕ : A2B (H ) → A2B (H ),

Tϕ ( f ) = P(ϕ f ),

f ∈ A2B (H ).

Clearly Tϕ  ≤ ϕ∞ such that Tϕ defines a bounded operator on the Fock space. Now we pass to operator symbols with additional structure such that the corresponding Toeplitz operators commute. We are interested in the generated commutative Banach algebras and, in particular, in extensions of the results in [3–5, 13, 32, 34] we consider the infinite dimensional setting H = 2 (N).

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5.1 Quasi-radial Symbols Let k = (k j ) j∈N be a fixed integer sequence. We use k to subdivide any z = (z 1 , z 2 , . . .) ∈ H into groups as follows: z = (z (1) , z (2) . . .) where z ( j) = (z k1 +...+k j−1 +1 , . . . , z k1 +...+k j ) ∈ Ck j , j ∈ N. Furthermore, we assume that the eigenvalue sequence λ = (λn )n of the operator B in the above construction is compatible with k. By this we mean that there is a sequence (u j ) j∈N of positive real numbers such that k

λ( j) = (u j , u j , . . . , u j ) ∈ R+j ,

( j ∈ N).

Clearly, in the case where k = (1, 1, 1, . . .) there is no additional assumption on the sequence λ ∈ 1 (N). Let a : 2 (N) → C be a k-quasi-radial measurable and bounded symbol, i.e. a = a(z) only depends on the infinite vector (|z ( j) |) j∈N . Standard arguments from representation theory show that the Toeplitz operator Ta acts on the standard orthonormal basis E B := [eα : α ∈ Z0 ] as a diagonal operator with eigenvalues γa . More precisely: Ta eα = γa (κ)eα

where

κ = κ(α) = (|α( j) |) j ∈ Z0 .

(5.4)

In the following we denote by Tk-qr the C ∗ algebra in L(A2B (H )) generated by all Toeplitz operators with bounded k-quasi-radial symbols. According to (5.4) we can identify Tk-qr with a C ∗ subalgebra of the bounded complex-valued functions Cb (Z0 ) on the discrete set Z0 . Since the system E B forms an orthonormal basis of A2B (H ) we obtain:      γa (κ) = Ta eα , eα = aeα , eα = a|eα |2 dμ B . H

If, in addition, a is cylindrical, k-quasi-radial, i.e. a is of the form   a(z) = a˜ |z (1) |, . . . , |z (L) |

(5.5)

L ), then we can evaluate γ (κ) more explicitly in with some L ∈ N and a˜ ∈ L ∞ (R+ a form of a finite dimensional integral:

γa (κ) =

L  j=1

1 (k j − 1 + κ j )!

 L R+

a˜

√

u 1 r1 , . . . ,

√

 u L r L r k−1+κ˜ e−|r | dr1 . . . dr L , (5.6)

L , κ˜ = (κ , . . . , κ ) and |r | = r + . . . + r . where 1 = (1, . . . , 1) ∈ Z+ 1 L 1 L

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Remark 5.4 If a ≥ 0, then the eigenvalue sequence γa (κ) is a moment sequence. Note that the corresponding moment problem which asks for a characterization of moment sequences in infinitely many variables via representing measures has been studied in the literature (cf. [2, 19]). In particular, there is an infinite dimensional generalization of Haviland’s Theorem. The final sections of the algebra Tk-qr are well understood (see [13]). More precisely, consider the natural inclusions Zn+ ⊂ Z0 with n ∈ N defined via extension by zero, i.e. ιn : (z 1 , . . . , z n ) → (z 1 , . . . , z n , 0, 0 . . .). Let n = k1 + . . . + k L where L ∈ N and consider the algebra:  (L) Tk-qr := C ∗ γa (κ) : a as in (5.5) and κ = ι L (κ) ∈ Z0 ).

(5.7)

Clearly, for all n ∈ N as above: (L)

Tk-qr ⊂ Tk-qr ⊂ Cb (Z0 ). L. Note that the eigenvalues γa (κ) in (5.7) only depend on κ˜ = (κ1 , . . . , κ L ) ∈ Z+ (L) L ). Hence, we can identity Tk-qr with a subalgebra of Cb (Z+ L and consider the C ∗ In [13] the authors define the square root metric ρ L on Z+ L L L that are unisubalgebra Cb,u (Z+ , ρ L ) in Cb (Z+ ) of all bounded functions on Z+ formly continuous with respect to ρ L . It is an interesting observation in [13] that the L , ρ ) in general is strictly larger than the L-fold tensor product of algebra Cb,u (Z+ L Cb,u (Z+ , ρ1 ). The main result of [13] (which generalizes [15] to the case L > 1) immediately implies: (L)

L , ρ ). Theorem 5.5 [13, 15] The algebra Tk-qr is isometrically isomorphic to Cb,u (Z+ L

For each L ∈ N we can consider the C ∗ algebra   L L ˜ 0, 0 . . .) : γ ∈ Tk-qr , κ˜ = (κ1 , . . . , κ L ) ∈ Z+ ). ⊂ Cb (Z+ T L := γ (κ, (L)

Then, clearly, Tk-qr ⊂ T L . However, one can show more: (L)

Proposition 5.6 Both C ∗ algebras above coincide, i.e. Tk-qr = T L . Proof Let γ ∈ Tk-qr . Then κ → γ (κ) uniformly can be approximated by a sequence ( ) , where  is a finite sum of finite products of eigenvalues of the form: γa1 (κ)γa2 (κ) . . . γak (κ),

(k ∈ N).

Each a j for j = 1, . . . , k is a bounded k-quasi-radial function on H = 2 (N). Let L . Choose α ∈ Z such that κ = L ∈ N be fixed and pick κ˜ = (κ1 , . . . , κ L ) ∈ Z+ 0 (κ, ˜ 0, 0, . . .) = κ(α). Now, we replace a j by a cylindrical function without changing the eigenvalue γa j (κ). Consider the orthogonal projection   n : H → span ε j : 1 ≤ j ≤ n ⊂ H ,

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where n = k1 + . . . + k L . By Fubini’s Theorem we obtain:  γa j (κ) =

a j |eα |2 dμ B H 

=  =

n H n H

(I −n )H

a j (x + y)|eα (x + y)|2 dμ(I −n )B (y)dμn B (x)

a˜ j (x)|eα (x)|2 dμn B (x) = γa˜ j (κ).

In the last equality we used the definition:  a˜ j (x) :=

(I −n )H

  a j n x + y dμ(I −n )B (y).

Note that a˜ j is a bounded cylindrical k-quasi-radial function. Hence, for all κ = (L) (κ, ˜ 0, . . .) of the above form we can replace  by an element in Tk-qr which proves the assertion.   Consider now the C ∗ algebra:   (L) L A := γ ∈ Cb (Z0 ) : γ (κ, ˜ 0, 0, . . .) ∈ Tk-qr for all κ˜ ∈ Z+ for all L ∈ N . (5.8) From Proposition 5.6 it follows that Tk-qr ⊂ A.

(5.9)

Furthermore, consider the C ∗ subalgebra Tc-k-qr ⊂ Tk-qr generated by Toeplitz operators having cylindrical bounded k-quasi radial symbols, i.e. symbols of the type (5.5). With L ∈ N define the projections: π L : Z0 → Z0 : πn (κ) := (κ1 , . . . , κ L , 0, 0 . . .). Lemma 5.7 We have the equality Tc-k-qr = {γ ∈ A : lim γ ◦ π L = γ L→∞

 uniformly on Z0 .

Proof Let γ ∈ Tc-k-qr . From Tc-k-qr ⊂ Tk-qr ⊂ A we conclude that γ ∈ A. Let γ˜ be a finite sum of finite products of eigenvalues of Toeplitz operators with symbols of the form (5.5). From the expression (5.6) of the eigenvalues we conclude that γ˜ ◦ π L = γ˜ for L ∈ N sufficiently large. Hence lim γ˜ ◦ π L = γ˜

L→∞

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uniformly on Z0 . Since such sequences are uniformly dense in Tc-k-qr we conclude that γ fulfills (5.10) as well. Conversely, let γ ∈ A such that lim L→∞ γ ◦π L = γ uniformly on Z0 . By definition (L) of A we have γ ◦ π L ∈ Tk-qr ⊂ Tc-k-qr . Hence γ ∈ Tc-k-qr .   Example 5.8 We show that the algebra Tc-k-qr contains Toeplitz operators with noncylindrical symbols, i.e. functions that depend on infinitely many variables. For simplicity we only consider the case k = (1, 1, . . .). Let ρ ∈ 1 (N) with 1 > ρ j > 0 for all j ∈ N. Consider a sequence ( f  )∈N of non-constant continuous functions f  : R+ → R+ such that 1 − ρ ≤ f  (y) ≤ 1 + ρ for all y ∈ C,  ∈ N. Define a k-quasi-radial (non-cylindrical) function F : 2 (N) → R+ as the infinite product: F(z) = F(z 1 , z 2 , . . .) =

∞ 

f  (|z  |).

=1

For each L ∈ N put FL = F ◦  L =

L

=0 f  (|z  |).

Then

∞       F(z) − FL (z)| = |FL (z)|1 − f  (|z  |)

≤ exp

∞ 

=L+1

∞        log(1 + ρ ) 1 − exp log f  |z  | .

=1

=L+1

Note that for all z ∈ 2 (N): ∞  =L+1

∞  ρ ≤ log(1 − ρ ) ρ − 1 =L+1

≤ R L (z) :=

∞ 

∞ ∞     log f  |z  | ≤ log(1 + ρ ) ≤ ρ .

=L+1

=L+1

=L+1

Since ρ ∈ 1 (N) we conclude that lim L→∞ R L (z) = 0 uniformly in 2 (N). Therefore lim L→∞ FL = F uniformly. It follows that γ F = lim L→∞ γ FL ∈ Tc-k-qr . Example 5.9 Let k = (1, 1, . . .) and γ ∈ Cb (Z0 ). We define the generating function G γ of γ by: G γ (x) =



γ (κ)x κ

for

x ∈  ⊂ H = 2 (N),

(5.11)

κ∈Z0

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where  denotes the domain of convergence of G γ . Since γ is bounded on Z0 it is easy to see that 

x = (x1 , x2 , . . .) ∈ 1 (N) : |x j | < 1,

 ∀ j ∈ N ⊂ .

Let γ ∈ Cb (Z0 ) be defined by γ (κ) :=

1 if κ = ε j for some j ∈ N, 0 else.

Note that actually γ ∈ A. Then G γ (x) =

∞ 

xj

where x = (x1 , x2 , . . .) ∈  = 1 (N) ⊂ 2 (N).

j=0

In particular, let η ∈ C with |η| < 1 and consider xη := −(η, η2 , η3 , . . .) ∈ . Then G γ (xη ) = −

∞ 

ηj =

j=1

η η−1

such that the map D := {y ∈ C : |y| < 1}  η → G γ (xη ) extends to a meromorphic function with a simple pole at η = 1. We now consider the generating function of an eigenvalue sequences. Let a ∈ L ∞ (H , μ B ) be a k-quasi-radial function. Let δ ∈ (0, 1) and consider   δ := x ∈ 1 (Z+ ) : |x j | < δλ j . From exp{δz2 } ∈ L 1 (H , μ B ) and Lebesgue’s Theorem on dominated convergence we obtain for all x ∈ δ : G γa (x) =

 κ∈Z0

a|eκ |2 x κ dμ B

H

⎛ ⎞  ∞  |z κ |2 x κ  |z j |2 x j ⎠ dμ B (z). dμ B (z) = = a(z) a(z) exp ⎝ λκ κ! λj H H 

κ∈Z0

j=1

Note that the integral on the right hand side extends to a holomorphic function on {x = (x1 , x2 , . . .) ∈ 2 (N) : Re(x j ) ≤ 0, j ∈ N}. Assume that there is δ > 0 such that 

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is an open zero-neighborhood in C. Note that this is the case if, e.g., we choose λ j := θ j where θ j ∈ (0, 1). Again we consider the map ⎛

 D  η → G γa (xη ) =

a(z) exp ⎝− H

∞  |z j |2 η j+1 j=0

λj

⎞ ⎠ dμ B (z).

The assignment η → G γa (xη ) extends to a real analytic function in R+ . In particular, it has no singularity in η = 1. We conclude that there is no bounded k-quasi-radial function with γa = γ . To end this subsection, we briefly analyze the maximal ideal space of the algebra Tk-qr . We introduce a corresponding version of the subspaces used in the previous sections. Given κ ∈ Z0 denote by Hκ the finite dimensional subspace     Hκ = span eα : α( j)  = κ j , ∀ j ∈ N . Moreover, Pκ will denote the orthogonal projection from A2B (H ) onto Hκ . On the ( j) other hand, for d ∈ Z+ and j ∈ N we define the infinite dimensional space Hd by ( j)

Hd

    = span eα : α( j)  = d ,

( j)

( j)

and we denote by Q d the orthogonal projection from A2B (H ) onto Hd . ( j) By (5.4) we see that the operators from Tk-qr leave all subspaces Hd and Hκ ( j) invariant. Furthermore, by Theorem 5.5, we conclude that all projections Q d belong to Tk-qr . Nonetheless, it is interesting to note that the orthogonal projections Pκ , being ( j) an infinite product of projections of the form Q d , may not belong to this algebra. A proof of this however is missing.   conjecture ideals of Tk-qr . As in the Let M Tk-qr denote the compact space of maximal  by means of these operators. Let Bergman space case, we can decompose M T k-qr  

be the set of all sequences θ j j∈N with θ j ∈ {0, 1} for all j ∈ N. Given θ ∈   we define the set Jθ = j : θ j = 1 and, in this case, we let Zθ+ denote the set of all sequences of integers κθ such that (κθ ) j = 0 for all j ∈ / Jθ . Set & % %   if θ j = 0 0 for all d ∈ Z0 , ( j) , Mθ = μ ∈ M Tk-qr : μ(Q d ) = 1 for some d ∈ Z0 , if θ j = 1 and ( j)

Mθ (κθ ) = {μ ∈ Mθ : μ(Q (κθ ) j ) = 1 for all j ∈ Jθ }.   Then, as before, we can write M Tk-qr as a disjoint union:      M Tk-qr = Mθ = Mθ (κθ ). θ∈

θ∈ κθ ∈

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5.2 Commutative Banach Algebras and Gelfand Theory Finally, we present the corresponding commutative Banach algebras and their Gelfand theory. Following our previous constructions we consider a sequence of generalized   pseudo-homogeneous symbols φ j j∈N , where each φ j (formally defined on Bk j ) is a function of the form (3.11). That is, our symbols φ j are formally cylindrical functions defined on H depending only on the coordinates z ( j) , such that the associated function on Ck j coincides with the extension of a function (3.11) defined on Bk j . For the sake of simplicity, we will denote by φ j all of these functions. Thus, as one easily sees, we have an infinite dimensional version of the tensor product introduced in Sect. 3.3. That is, for each of the operators Tφ j there is a uniquely ( j)

associated Toeplitz operator Tφ j acting on the Bergman space A20 (Bk j ) such that

Tφ j eα , eβ A2 (H ) = B

0, ( j) ( j) ( j)

Tφ j eα( j) , eβ( j) A2 (Bk j ) ,

if α( j  ) = β( j  ) , for some j  ∈ N, otherwise.

0

( j)

Here eα( j) denotes the canonical basic monomial of A20 (Bk j ). Denote by Tph the unital Banach algebra generated by the operators (Tφ j ) j∈N .   Corollary 5.10 The sequence of operators Tφ j j∈N is a commutative family of operators. In particular, the Banach algebra Tph is commutative. Proposition 5.11 The compact set of maximal ideals of Tph can be (topologically) identified with the following set       sp Tφ j . M Tph = j∈N

Proof This follows from standard arguments as in the Bergman space case.

 

Denote by Tk-qr,ph the Banach algebra generated by the algebras Tk-qr and Tph . To conclude, we characterize the maximal ideal space of the algebra Tk-qr,ph . As in the previous sections, we have the following natural inclusion: M(Tk-qr,ph ) ⊂ M(Tk-qr ) × M(Tph ) = M(Tk-qr ) ×



  sp Tφ j .

j∈N

Although some special care has to be taken in the corresponding calculations, it is not hard to find the points which belong to the maximal ideal space following the same methods as before. Proposition 5.12 The maximal ideal space of Tk-qr,ph coincides with the set  

Mθ (κθ ) ×

θ∈ κθ ∈Zθ

j∈N

+

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where Mθ,κθ , j =

⎧ ⎨sp(Tφ( j) | ( j) ), if j ∈ Jθ j H (κθ ) j

( j) ⎩e ss-sp(T ), φj

otherwise.

Furthermore, the Gelfand transform is generated by the following map on the generators of the algebra:  ρ∈F

Dγρ T ρ −→



γρ (μ)ζ ρ , (μ, ζ ) ∈ Mθ (κθ ) ×

ρ∈F



Mθ,κθ , j ,

(5.12)

j∈N ρ

ρ

where F ⊂ Z0 is a finite subset and T ρ := Tφ11 · · · Tφmm . Proof This follows from the same arguments as in the Bergman space setting, approximating operators by finite sums of finite products of the generators.   It seems reasonable to expect that Tk-qr,ph shares similar properties with the corresponding algebra on the Bergman space (e.g. being not simple but spectral invariant). Yet we leave this for future works, as it requires some detailed analysis.

6 Conclusion and Open Problems To conclude the present work we compare our results for different function Hilbert spaces. In each section we introduced a commutative Banach algebra Tk-qr,ph generated by a commutative C ∗ algebra Tk-qr and a commutative (not C ∗ ) Banach algebra Tph . The model cases are the corresponding algebras defined on the Bergman space A2λ (Bn ). The other cases are somehow infinite extensions of this: an infinite direct sum for the Hardy space and an infinite tensor product for the Fock space. The C ∗ algebras Tk-qr have already been an interesting object of study in the recent literature. The C ∗ algebra generated by radial Toeplitz operators on the Bergman space A2λ (Bn ) was characterized in [8], whereas the corresponding result for C ∗ algebras generated by Toeplitz operators with quasi-radial symbols on the Fock space F 2 (Cn ) on finitely many variables can be found in [13, 15]. In particular, in both cases it turns out that the closed linear span of the generators coincides with the corresponding C ∗ algebra. However, a complete characterization of these algebras for all other function Hilbert spaces and non-radial quasi-radial symbols, presented in this work, remains an open problem (see Problem 1). As for the algebras Tph , we have seen that their structure is essentially the same in all three cases, so that the work done on the Bergman space can be reproduced on the Hardy and Fock spaces without much effort. This will probably change when studying symbols of the more general form (3.12), where the corresponding weight parameters have to be carefully analyzed. In all three cases, after analyzing individually the generating algebras, we introduced the corresponding algebra Tk-qr,ph and developed its Gelfand theory. In this

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context, much of the work done on the Bergman space can be reproduced on the other spaces. However, in the case of the Hardy space some careful examination of the set M(Tk-qr ) has to be done to show that the points at infinity are separated. We recall that this was not necessary in the Bergman and Fock spaces due to the existence of some particular orthogonal projections inside the algebra. Finally, in the Bergman space setting, we showed how these results can give structural information (description of the radical and spectral invariance) of the algebra Tk-qr,ph . Such an analysis can probably be performed also on the Hardy and Fock spaces along the same lines and we leave this for future works. To close this survey, we list some open problems collected along the paper.

(1) Find a characterization of the algebras Tk-qr (e.g. as uniformly bounded continuous function with respect to some metric) for the remaining cases: (not radial) quasiradial symbols on the Bergman space, quasi-radial symbols on the Hardy space and quasi-radial symbols on the Fock space in infinitely many variables. Does Tk-qr coincide with the (linear) closure of Toeplitz operators with k-quasi-radial symbols? Do we have equality in (5.9)? (See Sects. 3.2, 4.2 and 5.1). (2) Is it possible to determine an asymptotic behaviour for the eigenvalues of the ( j) ( j) matrices Tφ j | H( j) ? Under which conditions do we have | sp(Tφ j | H( j) )| → ∞ as d d d → ∞? (See Sect. 3.6). (3) Find a description of the algebra Rad(Tk-qr,ph ) ∩ T k-qr,ph for general symbols of the form (3.11). (See Sect. 3.6). (4) Develop the Gelfand theory for the corresponding algebras taking more general symbols φ j such as symbols of the form (3.12) or symbols without the continuity assumptions. (See Sect. 3.3).

Funding Open Access funding enabled and organized by Projekt DEAL. The second author was partially supported by Consejo Nacional de Ciencia y Tecnología (Conacyt), Mexico. Availability of data and material Not applicable.

Declarations Conflict of interest The authors have no relevant financial or non-financial interest to disclose. Code availability Not applicable. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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A Question About Invariant Subspaces and Factorization Hari Bercovici1

· Wing Suet Li2

Received: 3 August 2021 / Accepted: 3 November 2021 / Published online: 5 March 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract We ask whether a connection between isometric functional calculus and factorization of linear functionals, known to hold for the case of a single contraction operator, persists in the case of commuting pairs—or, more generally, n-tuples—of contractions. A positive answer has consequences concerning the jointly invariant subspaces of the commuting operators. We recall first that an algebra A of bounded linear operators on a complex Hilbert space H is said to have property (A1 ) if every weak*-continuous functional ϕ : A → C can be represented as ϕ(T ) = T x, y, T ∈ H, for some vectors x, y ∈ H. When one proves that a given algebra has property (A1 ), one usually obtains an estimate of the form xy ≤ Cϕ, for some constant C, independent of ϕ. Suppose now that G ⊂ Cd is a bounded open set for some d ∈ N, and denote by ∞ H (G) the Banach algebra consisting of all bounded holomorphic functions defined in G. The algebra H ∞ (G) has a natural weak* topology such that a sequence in

Dedicated to the memory of Jörg Eschmeier. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz and Mihai Putinar. W. S. Li was supported in part by a grant from the Simons Foundation.

B

Hari Bercovici [email protected] Wing Suet Li [email protected]

1

Mathematics Department, Indiana University, Bloomington, IN 47405, USA

2

Mathematics Department, Georgia Institute of Technology, Atlanta, GA 30332, USA

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H ∞ (G) converges if and only it converges pointwise and is uniformly bounded. We are interested in unital representations  from H ∞ (G) to the algebra B(H) of bounded linear operators on H. Recall that B(H) also has a natural weak* topology, arising from its duality with the trace class. Problem Suppose that  : H ∞ (G) → B(H) is a unital algebra representation such that 1. (u) = u∞ for every u ∈ H ∞ (G), and 2.  is weak*-to-weak* continuous. Does it follow that the algebra {(u) : u ∈ H ∞ (G)} has property (A1 )? The answer is known to be in the affirmative when G ⊂ C is a disk [4, 5]. Versions of this problem, usually with the stronger hypothesis of the existence of a dominating spectrum, were proved by several authors. We only mention here Eschmeier [6] for a rather general setting and Ambrozie–Müller [1] for a Banch space version in case G is a polydisk. A particular case arises from considering a pair (T1 , T2 ) of commuting contractions on H. When these contractions are completely nonunitary, it was shown in [3] that there is a version of the Sz.-Nagy–Foias functional calculus that yields a representation of H ∞ (D2 ), where D ⊂ C is the unit disk. A different argument, along with a dilation of this representation, is given in [2] in the special case in which T1 and T2 are of class C00 in the sense of [8]. Our purpose in [2] was to give an affirmative answer to the above problem in this special case. Eschmeier [7] pointed out a subtle error in our argument, and therefore the problem must be considered to be open even in this particular case. For the record, we comment briefly on the nature of this error. The argument of [2] relies on the fact that one can consider that there is some measure ν on the distinguished boundary T2 of D2 such that H can be viewed as a subspace of L 2 (ν) ⊗ 2 in such a way that the operators T1 and T2 are compressions to H of the operators of multiplication by the two coordinates on T2 . Lemma 4.2 of [2] shows that, given ε > 0 and a Borel set σ ⊂ T2 with ν(σ ) > 0, there exists a function u ∈ H ∞ (D2 ) such that ν({ζ ∈ T2 : |u(ζ ) − χσ (ζ )| > ε} < ε. In other words, |u| is close to 1 on most of σ and close to 0 on most of T2 \σ . If f ∈ H is such that (u) f  is very close to u∞  f , it follows that f is concentrated mostly on the set on which |u| is close to 1. Then [2,Proposition 4.3] asserts, incorrectly, that f must be concentrated mostly on σ itself. Indeed, it may well be that much of f lives on a set ω ⊂ T2 \σ where |u| is close to 1. Finding an argument along these lines would require either finding u such that |u| < ε almost everywhere on T2 \σ , or an improvement in the basic factorization [2,Theorem 2.3]. The first alternative seems unlikely to succeed because σ may be, for instance, a one dimensional arc. Data Availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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References 1. Ambrozie, C., Müller, V.: Dominant Taylor spectrum and invariant subspaces. J. Oper. Theory 61(1), 63–73 (2009) 2. Bercovici, H., Li, W.S.: Isometric functional calculus on the bidisk and invariant subspaces. Bull. Lond. Math. Soc. 25, 582–590 (1993) 3. Briem, E., Davie, A.M., Øksendal, B.K.: A functional calculus for pairs of commuting contractions. J. Lond. Math. Soc. (2) 7, 709–718 (1974) 4. Brown, S.W., Chevreau, B., Pearcy, C.: On the structure of contraction operators. II. J. Funct. Anal. 76, 30–55 (1988) 5. Chevreau, B.: Sur les contractions à calcul fonctionnel isométrique. II. J. Oper. Theory 20, 269–293 (1988) 6. Eschmeier, J.: C00 -Representations of H ∞ (G) with Dominating Harte Spectrum, Banach Algebras ’97 (Blaubeuren), pp. 135–151. de Gruyter, Berlin (1998) 7. Eschmeier, J.: Private communication 8. Sz.-Nagy, B., Foias, C., Bercovici, H., Kérchy, L..: Harmonic Analysis of Operators on Hilbert Space, 2nd edn. Universitext, Springer, New York (2010) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Complex Analysis and Operator Theory (2023) 17:22 https://doi.org/10.1007/s11785-022-01314-8

Complex Analysis and Operator Theory

Dilations and Operator Models of W -Hypercontractions Monojit Bhattacharjee1 · B. Krishna Das2 · Ramlal Debnath2 · Samir Panja2 Received: 3 December 2022 / Accepted: 12 December 2022 / Published online: 6 January 2023 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023

Abstract We revisit the study of ω-hypercontractions corresponding to a single weight sequence ω = {ωk }k≥0 introduced by Olofsson ([22] J Oper Theory 74:249–280, 2015) and find an analogue of Nagy-Foias characteristic function in this setting. Explicit construction of characteristic functions is obtained and it is shown to be a complete unitary invariant. By considering a multi-weight sequence W and W-hypercontractions we extend Olofsson’s work ([22] J Oper Theory 74:249–280, 2015) in the multi-variable setting. Model for W-hypercontractions is obtained by finding their dilations on certain weighted Bergman spaces over the polydisc corresponding to the multi-weight sequence W. This recovers and provides a different proof of the earlier work of Curto and Vasilescu ([13] Indiana Univ Math J 42:791–810, 1993 and [14] Indiana Univ 12 Math J 44:727–746, 1995) for y-contractive multi-operators through a particular choice of multi-weight sequence.

To the memory of Professor Jörg Eschmeier Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

B

Monojit Bhattacharjee [email protected]; [email protected] B. Krishna Das [email protected]; [email protected] Ramlal Debnath [email protected]; [email protected] Samir Panja [email protected]; [email protected]

1

Department of Mathematics, Birla Institute of Technology and Science - Pilani, K. K. Birla Goa Campus, South Goa 403726, India

2

Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

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Keywords Hypercontraction · Weighted shift · Characteristic function · Bergman space over the polydisc · Commuting contractions · Bounded analytic functions Mathematics Subject Classification 47A13 · 47A20 · 47A45 · 47A56 · 46E22 · 47B32 · 32A36 · 47B20 List of Symbols N Set of all natural numbers. Set of all positive integers including 0. Z+ Zn+ {α = (α1 , . . . , αn ) | αi ∈ Z+ , i = 1, . . . , n}. e (1, . . . , 1) ∈ Zn+ . Set of all positive real numbers including 0. R+ {γ = (γ1 , . . . , γn ) | γi ∈ R+ , i = 1, . . . , n}. Rn+ Complex n-space. Cn z (z 1 , . . . , z n ) ∈ Cn . z 1α1 · · · z nαn for all α ∈ Zn+ . zα T n-tuple of commuting operators (T1 , . . . , Tn ). T1α1 · · · Tnαn for all α ∈ Zn+ . Tα n D Open unit polydisc {z | |z i | < 1, i = 1, . . . , n}. B(H) Set of all bounded linear operators on a Hilbert space H.

1 Introduction Dilations of operators on Hilbert spaces is a mathematical tool which is used to understand operators in terms of simple and well-understood operators. The basic idea of dilation of an operator T on a Hilbert space H is to find a well-understood operator V on K such that T is a part of V , that is, there exists a V ∗ -invariant subspace Q ⊆ K such that T ∼ = PQ V |Q . In a similar vain, by fixing a well-understood operator (more generally a class of operators), one can ask for a characterization of operators which are part of the fixed operator (or the class of operators). First significant result in this direction is due to Sz.-Nagy and Foias in which states that a contraction T on a Hilbert space H is pure (that is, T ∗n → 0 as n → ∞ in the strong operator topology) if and only if T is a part of the shift operator Mz on a vector valued Hardy space HE2 (D). Here for a Hilbert space E, ⎧ ⎫ ⎨ ⎬   HE2 (D) = f : D → E | f (z) = an z n , an 2 < ∞, z ∈ D, an ∈ E ⎩ ⎭ n≥0

n≥0

is the E-valued Hardy space over D and Mz : HE2 (D) → HE2 (D) is the shift operator, defined by, (Mz f )(w) = w f (w) for all w ∈ D. In other words, T is a pure contraction

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if and only if there exist a Hilbert space E and an Mz∗ -invariant subspace Q ⊆ HE2 (D) such that T ∼ = PQ Mz |Q . Another remarkable consequence of Sz.-Nagy and Foias dilation result is that there exist a Hilbert space E∗ and a B(E∗ , E)-valued inner multiplier θT (that is, θT : D → B(E∗ , E) is a contractive analytic function such that θT is isometry-valued a.e. on the unit circle), known as the characteristic function of T (cf. [21]), such that Q = HE2 (D) θT HE2∗ (D). The above dilation result is also extended for general contractions and is the stepping stone of Sz.-Nagy and Foias theory for contractions. Subsequently, by considering Bergman shift on vector-valued weighted Bergman space, Agler in his seminal paper [1] extended the above result. He showed that a contraction T is a part of the Bergman shift Mz acting on some E-valued weighted Bergman space A2m (E) with kernel K m (z, w) = (1 − z w) ¯ −m IE ,

(z, w ∈ D)

if and only if T is a pure m-hypercontraction, that is, T is a pure contraction and K m−1 (T , T ∗ )



m  k m T k T ∗k ≥ 0. := (−1) k k=0

Recently, Olofsson in [22] extended it further for shifts on weighted Bergman spaces corresponding to a certain class of weight sequences. He showed that if ω is a weight sequence then T is a part of the Bergman shift on some E-valued weighted Bergman space A2ω (E) if and only if T is a pure ω-hypercontraction. The reader is referred to Sect. 2 and Theorem 2.3 below for terminologies and a detailed description of the result. There are also several works in the multi-variable setting and an incomplete list of references is [4–6, 8–10, 13, 14, 18, 20, 26]. . The purpose of the present article is twofold. Firstly, we revisit the study of hypercontractions corresponding to a class of weight sequences as considered in [22]; these hypercontractions are known as ω-hypercontractions where ω is a weight sequence (see Sect. 2 for the definition). Using dilations of such hypercontractions, we find an analogue of Sz.-Nagy and Foias characteristic functions in this setting. Explicit construction of such characteristic functions is given in terms of triples which we call as characteristic triple. As expected, it is also shown that the characteristic function is a complete unitary invariant. This generalizes the work of [11, 16] for the case n = 1. The main ideas behind this consideration comes from a recent article [11]. For recent developments on this topic in different context, the reader is referred to [7, 16, 17, 19, 23, 24]. Section 2 is devoted to discuss these. Secondly, we extend Olofsson’s result [22] in the polydisc setting. We say that (i) W = (ω1 , . . . , ωn ) is a multi-weight sequence if for all i = 1, . . . , n, ωi = {ωm }m≥0

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is a weight sequence (see Definition 2.1 below for the definition of a weight sequence). We introduced the notion of W-hypercontractions corresponding to a multi-weight sequence W and obtained their models by finding their dilations on some weighted Bergman space over the polydisc. Our method of multi-variable dilation is driven by the idea of using one variable dilation result, obtained by Olofsson, at a time and it is well supported by a commutant lifting type result obtained in this setting. For a particular choice of multi-weight sequence W, we recover the dilation result of Curto and Vasilescu [13] with a different proof. It is worth mentioning here that in the setting of unit ball in Cn , Schilo (see [25, Theorem 3.21]) extended Olofsson’s results. To describe our result succinctly we need to develop some notations and terminology. Let T = (T1 , . . . , Tn ) ∈ B(H)n be an n-tuple of commuting contractions. For β = (β1 , . . . , βn ) ∈ Zn+ with β ≥ e = (1, . . . , 1), there is a natural  choice of 1 multi-weight sequence denoted by Wβ = (ωβ1 , . . . , ωβn ), where ωβi = βi +l−1 ( l ) l≥0 

i +l−1)! = (β and βi +l−1 l (βi −1)!l! , for all i = 1, . . . , n. We mention here that, Wβ for β = n (β1 , . . . , βn ) ∈ R+ with β ≥ e = (1, . . . , 1) also defines a multi-weight sequence. For a Hilbert space E and β ∈ Zn+ with β ≥ e = (1, . . . , 1), A2Wβ (E) is the E-valued weighted Bergman space over Dn with kernel K Wβ IE , where

K Wβ (z, w) =

n  i=1

1 (1 − z i w¯ i )βi

(z = (z 1 , . . . , z n ) ∈ Dn , w = (w1 , . . . , wn ) ∈ Dn ).

Set −1 KW (T , T ∗ ) = β



(−1)|α|

0≤α≤β

β! T α T ∗α . α!(β − α)!

n Here for α = (α1 , . . . , αn ), β = (β1 , . . . , βn ) ∈ Zn+ , |α| = i=1 αi , α! = α1 ! · · · αn !, and α ≤ β if and only if αi ≤ βi for all i = 1, . . . , n. We say that an n-tuple of commuting contraction is a part of the multi-shift (Mz 1 , . . . , Mz n ) on A2Wβ (E) if there exists a joint (Mz∗1 , . . . , Mz∗n )-invariant subspace Q of A2Wβ (E) such that Ti ∼ = PQ Mzi |Q for all i = 1, . . . , n. In this set up, Curto and Vasilescu [14] proved that for β ∈ Zn+ with β ≥ e, a commuting tuple of contractions T = (T1 , . . . , Tn ) is a part of the multi-shift (Mz 1 , . . . , Mz n ) on A2Wβ (E) if and only if T is pure and satisfies −1 KW (T , T ∗ ) ≥ 0. β

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In this article, we show that this result is true for a large class of kernels corresponding to multi-weight sequences. For a multi-weight sequence W = (ω1 , . . . , ωn ), we set S(W) := {(ωλ 1 , . . . , ωλ n ) : ωλ i ∈ {ωλi , 1}, i = 1, . . . , n}, where we denote by 1 the constant weight sequence 1. For a coefficient Hilbert space E, we denote by A2W (E) the reproducing kernel Hilbert space corresponding to the kernel K W on the polydisc Dn defined by K W (z, w) =



1

α∈Zn+

(1) (n) ωα1 · · · ωαn

¯ α IE . (z w)

By one of the assumptions in the definition of multi-weight sequence, the analytic function kW (z) =



1

α∈Zn+

(n) ωα(1) 1 · · · ωαn

zα

on Dn associated to K W does not vanish on Dn . Suppose that  1 cα z α (z ∈ Dn ) = kW (z) n

(1.1)

α∈Z+

is the Taylor expansion of 1/kW . For an n-tuple of commuting contractions T = (T1 , . . . , Tn ) ∈ B(H)n , and r ∈ (0, 1)n , using the hereditary functional calculus introduced by Agler in [2], we define DW ,T (r) :=



cα rα T α T ∗α ,

(1.2)

α∈Zn+

where cα as in (1.1) and if r = (r1 , . . . , rn ), rα = r1α1 · · · rnαn . Definition 1.1 An n-tuple of commuting contractions T = (T1 , . . . , Tn ) ∈ B(H)n is said to be an W-hypercontraction corresponding to a multi-weight sequence W = (ω1 , . . . , ωn ) if DW ,T (r) ≥ 0 for all W ∈ S(W) and r ∈ (0, 1)n . In addition, if each Ti is a pure contraction, then we say that T is a pure W-hypercontraction. With these terminologies, one of the main theorems of this article is the following. Theorem 1.2 Let W be a multi-weight sequence. An n-tuple of commuting contractions T = (T1 , . . . , Tn ) on H is a part of the multi-shift (Mz 1 , . . . , Mz n ) on A2W (E) for some Hilbert space E if and only if T is a pure W-hypercontraction. This theorem is proved in Sect. 4 as Theorem 4.4 by finding dilation of such an n-tuple of commuting contraction T . We also consider the case when T is an Whypercontraction but not necessarily pure and find their dilations on direct sums of

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some weighted Bergman spaces. The structure of the dilation map and the dilating operators resemble that of Curto and Vasilescu model for general multi-operators. See Theorem 4.6 for more details. In the case when W = Wβ for some β ∈ Zn+ with β ≥ e, we show that T is a −1 (T , T ∗ ) ≥ 0. Therefore, pure Wβ -hypercontractions if and only if T is pure and K W β for the particular choice of multi-weight sequence W = Wβ , Theorem 1.2 recovers the classical result of Curto and Vasilescu [13, 14] with a different proof. For the choice of multi-weight sequence Wβ when β ∈ Rn+ with β ≥ e, it also provides a natural generalization. Moreover, the class of multi-weight sequences is wide enough to include tensor product of reproducing kernel Hilbert spaces corresponding to certain Nevanlinna-Pick kernels over D (see Example 3.2 below). Section 3 is devoted to study multi-weight sequences and W-hypercontractions. In Sect. 4, we find dilations of pure W-hypercontractions and more generally for W-hypercontractions.

2 Characteristic Functions for !-Hypercontractions We construct characteristic functions of ω-hypercontractions in this section. We recall the notion of ω-hypercontractions and their dilations first. Let ω = {ωk }k≥0 be a pos1

itive decreasing sequence such that ω0 = 1 and lim inf k→∞ ωkk = 1. Corresponding to the sequence ω and a Hilbert space E, we denote by A2ω (E) the E-valued weighted Bergman space; the Hilbert function space consists of f ∈ O(D, E), the space of E-valued analytic functions f on the open unit disc (D), such that f (z) =



ak z k and f 2ω :=

k≥0



ak 2 ωk < ∞ (ak ∈ E, z ∈ D).

k≥0

It is also a reproducing kernel Hilbert space with the kernel K ω : D × D → B(E) defined by K ω (ζ, η) =

 1 (ζ η) ¯ k IE ωk

(ζ, η ∈ D).

k≥0

If the co-efficient Hilbert space is C, we simply write A2ω to denote A2ω (C). The multiplication operator Mz , known as shift operator, on A2ω (E) is defined by (Mz f )(η) = η f (η) for all η ∈ D. A straight forward computation shows that Mz 2 = supk

ωk+1 . ωk

Thus an equivalent condition for the shift operator to be a contraction is that the sequence ω has to be a decreasing sequence. The kernel function K ω has an associated

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analytic function kω on D defined by kω (z) =

 1 zk . ωk k≥0

Properties of this associated analytic function has been very crucial in Olofsson’s consideration. In fact, the class of weight sequences ω that he considered in [22] are so that the associated analytic function kω on D possess some additional properties. The first natural property is that kω is non-vanishing. Then k1ω is also analytic on D  and suppose that k1ω (z) = k≥0 ck z k . Other properties that kω needs to satisfy are as follows: ω has non negative Taylor coefficients for 0 < r < 1, where (P1) The function kkω,r kω,r (z) = kω (r z). k have uniformly bounded Taylor coefficients for 0 < r < 1. (P2) The quotients kω,r ω (P3) The Taylor coefficients of the reciprocal function k1ω is absolutely summable and

the absolute sum of Taylor coefficients of bounded family.

kω,r kω

for 0 < r < 1 form a uniformly

The first two properties are essential to obtain dilations of ω-hypercontractions and we briefly indicate below the role played by these properties. The above discussion also prompt us to make the following definition. Definition 2.1 A weight sequence is a positive decreasing sequence ω = {ωn }n≥0 1

such that ω0 = 1, lim inf n→∞ ωnn = 1 and the corresponding analytic function kω is non-vanishing on D and satisfies (P1) and (P2) as above. Natural examples of weight sequences are the constant sequence ωn = 1 for all n ≥ 0 1 and for a fixed m ∈ N, ωn = n+m−1 for all n ≥ 0. The constant sequence case ( n ) corresponds to the Hardy space where as the later corresponds to the Bergman space defined above with kernel K m (z, w) = (1−z1w) ¯ m (z, w ∈ D). For the rest of this section we fix a weight sequence ω. We suppose that the reciprocal of the associated analytic function kω has the following power series expansion:  1 (z) = cn z n (z ∈ D). kω

(2.1)

n≥0

Now we recall the notion of ω-hypercontraction introduced by Olofsson in [22]. Definition 2.2 A bounded linear operator T ∈ B(H) is said to be an ωhypercontraction if T is a contraction and satisfies Dω,T (r ) :=



r n cn T n T ∗n ≥ 0

n≥0

for all r ∈ (0, 1), where cn ’s are as in (2.1).

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It can be shown that for the choice ω = {ωn }n≥0 where ωn =

1 for all n ≥ 0, ω(n+m−1 n ) hypercontractivity for a contraction T is same as m-hypercontractivity in the sense of Agler [1] (see [22, Theorem 4.5] for a proof). Thus the notion of ω-hypercontractions is a natural generalization of m-hypercontractions. Moreover, it has been shown in [22] that every ω-hypercontraction is part of the shift operator Mz on the weighted Bergman space A2ω (E) for some suitable Hilbert space E; we briefly recall this dilation result next. Let T ∈ B(H) be an ω-hypercontraction. Then using the property (P1), it can be shown that the SOT limit of the operator Dω,T (r ) exists as r → 1. We denote

Dω,T (1) := SOT − lim Dω,T (r ), r →1

and we define the defect operator and the defect space of T as

1/2 Dω,T := Dω,T (1) and Dω,T := ran(Dω,T ), respectively. On the other hand, the property (P2) helps one to establish the identity h 2 =

 1 Dω,T T ∗k h 2 + lim T ∗k h 2 k→∞ ωk

(h ∈ H).

(2.2)

k≥0

Then it is evident from the above identity that the map πω,T : H → A2ω (Dω,T ) defined by πω,T h(z) =

 1 (Dω,T T ∗k h)z k ωk

(h ∈ H, z ∈ D)

(2.3)

k≥0

is a contraction and πω,T T ∗ = Mz∗ πω,T . Moreover, setting Q 2T := SOT − limk→∞ T k T ∗k and QT := ran Q T , we have an isometry ω,T : H → A2ω (Dω,T ) ⊕ QT defined by   (ω,T h)(z) = (πω,T h)(z), Q T h (h ∈ H, z ∈ D) and ω,T T ∗ = (Mz ⊕U )∗ ω,T , where U is a co-isometry on QT such that U ∗ Q T h = Q T T ∗ h for all h ∈ H. Summarizing the above discussion we have the following dilation result. Theorem 2.3 (c.f. [22]) Let ω = {ωk }k≥0 be a weight sequence. If T is an ωhypercontraction on H, then there exist an isometry ω,T : H → A2ω (Dω,T ) ⊕ QT and a co-isometry U on QT such that ω,T T ∗ = (Mz ⊕ U )∗ ω,T .

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In addition, if T is pure then there exists an isometry πω,T : H → A2ω (Dω,T ) such that πω,T T ∗ = Mz∗ πω,T . In the above theorem, the co-isometry U can be made to a unitary by taking a coextension. We do not include it in the statement as we shall use the present form of the theorem in later section. The observant reader might have noticed that we have not used (P3) to obtain the above dilation. But (P3) provides a necessary and sufficient condition for a contraction T ∈ B(H) to be an ω-hypercontraction. To be more precise, a contraction T is ω-hypercontraction if and only if Dω,T (1) ≥ 0. For more details we refer the reader to [22, Theorem 6.2]. For Hilbert spaces E1 and E2 , an operator-valued analytic map θ : D → B(E1 , E2 ) is a multiplier from HE21 (D) to A2ω (E2 ) if θ f ∈ A2ω (E2 ) for all f ∈ HE21 (D). We denote by M(HE21 (D), A2ω (E2 )), the space of all multipliers from HE21 (D) to A2ω (E2 ). We also use Mθ , for θ ∈ M(HE21 (D), A2ω (E2 )), to denote the associated multiplication operator by θ , that is, Mθ f = θ f

( f ∈ HE21 (D)).

A multiplier θ ∈ M(HE21 (D), A2ω (E2 )) is said to be partially isometric if Mθ is a partially isometric operator from HE21 (D) to A2ω (E2 ). Such a partially isometric multiplier naturally occur in the Beurling-Lax-Halmos type characterization of invariant subspaces of vector-valued weighted Bergman spaces (see [24]). The characterization relevant for us is the following. Theorem 2.4 (c.f. [24]) If S is an Mz -invariant subspace of A2ω (E∗ ), then there exist a Hilbert space E and a partially isometric multiplier θ ∈ M(HE2 (D), A2ω (E∗ )) such that S = θ HE2 (D). Now combining Theorem 2.3 and Theorem 2.4, we can associate a partially isometric multiplier to a pure ω-hypercontraction as follows. Corollary 2.5 Let T ∈ B(H) be a pure ω-hypercontraction. Then there exist a Hilbert space E and a partially isometric multiplier θ ∈ M(HE2 (D), A2ω (Dω,T )) such that ∼ PQ Mz |Q , T = θ θ where Qθ = ran πω,T , πω,T is the dilation map corresponding to T as in Theorem 2.3 2 and Q⊥ θ = θ HE (D). In the rest of this section, we compute such a partially isometric multiplier explicitly corresponding to each pure ω-hypercontraction. The construction is based on a recently developed technique found in [11] in the context of m-hypercontractions. Let T ∈

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B(H) be a pure ω-hypercontraction. Since T is a contraction, we define DT := (I − T T ∗ )1/2 . It follows from Lemma 3.6 in [22] that for h ∈ H, DT h 2 = Dω,T h 2 +

 1 1 Dω,T T ∗k h 2 . − ωk ωk−1

(2.4)

k≥1

Also, since T is pure it follows from (2.2) that h 2 =

 1 Dω,T T ∗n h 2 . ωn

(2.5)

n≥0

Consider the map Cω,T : H → l 2 (Z+ , Dω,T ) defined by √ Cω,T (h) = { ρn Dω,T T ∗n h}n≥0 , (h ∈ H) where ρ0 = 1 and ρn = h ∈ H, Cω,T h 2 =

1 ωn

−



1 ωn−1

≥ 0 for all n ≥ 1. Then by the identity (2.4), for

ρn Dω,T T ∗n h 2

n≥0

 1 1 Dω,T T ∗n h 2 = Dω,T h + − ωn ωn−1 2

n≥1 2

= DT h ≤ h . 2

Thus Cω,T is a contraction. Now by identity (2.5), ∗ πω,T = IH = πω,T

 1 2 T n Dω,T T ∗n , ωn n≥0

and consequently,  1  2 2 T n Dω,T T ∗n − ρn T n Dω,T T ∗n ωn n≥0 n≥0

 1 2 = − ρn T n Dω,T T ∗n ωn

∗ IH − Cω,T Cω,T =

n≥0

=



1

n≥1

ωn−1

2 T n Dω,T T ∗n

 1 2 = T n+1 Dω,T T ∗n+1 ωn n≥0

= T T ∗.

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 T∗ : H → H ⊕ l 2 (Z+ , Dω,T ) defined by Cω,T

 This shows that the map X T =

X T (h) = (T ∗ h, Cω,T h) (h ∈ H) is isometry. By adding a Hilbert space E, if necessary, we get a unitary operator   U := X T YT : H ⊕ E → H ⊕ l 2 (Z+ , Dω,T ),   B , D where B = PH YT ∈ B(E, H) and D = Pl 2 (Z+ ,Dω,T ) YT ∈ B(E, l 2 (Z+ , Dω,T )), we have the following result which will lead us to construct the characteristic function.

where YT = U |E : E → H ⊕ l 2 (Z+ , Dω,T ) is a contraction. By setting YT =

Theorem 2.6 Let T ∈ B(H) be a pure ω-hypercontraction. Then the map Cω,T : H → l 2 (Z+ , Dω,T ) defined by √ Cω,T (h) = { ρn Dω,T T ∗n h}n≥0 , where ρ0 = 1 and 1 1 − for all n ≥ 1, ρn = ωn ωn−1 is a contraction and there exist a Hilbert space E and bounded operators B ∈ B(E, H) 2 and D = {Dn }∞ n=0 ∈ B(E, l (Z+ , Dω,T )) where each Dn ∈ B(E, Dω,T ) such that 

 T∗ B : H ⊕ E → H ⊕ l 2 (Z+ , Dω,T ) Cω,T D

is unitary. The fact that each triple (E, B, D) – which appears in the above theorem – gives rise to a characteristic function of T , motivates us to make the following definition. Definition 2.7 A triple (E, B, D) consisting of a Hilbert space E and bounded linear operators B ∈ B(E, H) and D ∈ B(E, l 2 (Z+ , Dω,T )) is said to be a characteristic triple of a pure ω-hypercontraction T on B(H) if 

 T∗ B : H ⊕ E → H ⊕ l 2 (Z+ , Dω,T ) Cω,T D

is unitary. It turns out that characteristic triple is unique in the following sense. Theorem 2.8 If (E1 , B1 , D1 ) and (E2 , B2 , D2 ) are two characteristic triple of a pure ω-hypercontraction T ∈ B(H), then there exists a unitary U : E2 → E1 such that (E2 , B2 , D2 ) = (U ∗ E1 , B1 U , D1 U ).

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 Proof The proof follows from the observation that their range is same.

   B B1 and 2 are isometries and D1 D2  

We are now in a position to state the main result of this section which provides an explicit method to construct characteristic functions. The proof of the theorem is similar to Theorem 3.1 in [11] and we only include a sketch of the proof here. Theorem 2.9 Let T be a pure ω-hypercontraction on H, and let (E, B, D) be a characteristic triple of T . Then θT (z) =

√

ρn Dn z n + z Dω,T

n≥0

 1 z n T ∗n B ωn n≥0

is a partially isometric multiplier in M(HE2 (D), A2ω (Dω,T )) such that 2 ∼ Q⊥ T = θT HE (D) and T = PQT Mz |QT ,

where ρ0 = 1, ρn =

1 ωn

−

1 ωn−1

for all n ≥ 1.

 Sketch of the proof For a contraction A and z ∈ D, we set K ω (z, A) := n≥0 where the series converges as z A is a strict contraction. Now, note that (1 − zT ∗ )K ω (z, T ∗ ) =

1 n n ωn z A ,

 1  1 1 z n T ∗n = + − ρn z n T ∗n . ω0 ωn ωn−1 n≥1

n≥0



 T∗ B A direct calculation using the above identity and the unitary property of , Cω,T D it can be shown that K ω (η, ζ )IDω,T −

θT (η)θT (ζ )∗ = Dω,T K ω (η, T ∗ )K ω (ζ¯ , T )Dω,T (ζ, η ∈ D). (2.6) 1 − ηζ¯

Then using some standard arguments in the theory of reproducing kernel Hilbert spaces, we conclude that θT ∈ M(HE2 (D), A2ω (Dω,T )) and Mθ∗T K ω (., ζ )h = K 1 (., ζ )θT (ζ )∗ h (ζ ∈ D, h ∈ Dω,T ), where K 1 (η, ζ ) = (1 − ηζ¯ )−1 . Consequently, (I − MθT Mθ∗T )K ω (., ζ )h = (K ω (., ζ )IDω,T − K 1 (., ζ )θT (.)θT (ζ )∗ )h, 130

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and therefore by (2.6), (I − MθT Mθ∗T )K ω (., ζ )h = Dω,T K ω (., T ∗ )K ω (ζ¯ , T )Dω,T h for all ζ ∈ D and h ∈ Dω,T . On the other hand, the adjoint of the dilation map for the ∗ pure ω-hyeprercontraction πω,T : A2ω (Dω,T ) → H is given by ∗ K ω (., ζ )h = K ω (ζ¯ , T )Dω,T h πω,T

(ζ ∈ D, h ∈ Dω,T ).

Then it is easy to see that ∗ πω,T πω,T K ω (., ζ )h = Dω,T K ω (., T ∗ )K ω (ζ¯ , T )Dω,T h,

for all ζ ∈ D and h ∈ Dω,T . Combining all these we have the required identity ∗ πω,T πω,T + MθT Mθ∗T = I A2ω (Dω,T ) . 2 That is, θT is a partially isometric multiplier such that Q⊥ T = θT HE (D).

 

We call the partially isometric multiplier θT obtained in the above theorem corresponding to a characteristic triple (E, B, D) as characteristic function of T . Let T1 and T2 be two pure ω-hypercontractions on H1 and H2 , respectively. Also, let θT1 and θT2 be characteristic functions corresponding to the characteristic triples (E1 , B1 , D1 ) and (E2 , B2 , D2 ) of T1 and T2 , respectively. Then the characteristic functions θT1 and θT2 is said to be coincide if there exists two unitaries τ : E2 → E1 and τ∗ : Dω,T1∗ → Dω,T2∗ such that θT2 (z) = τ∗ θT1 (z)τ

(z ∈ D).

We end the section with the observation that characteristic function is completely unitary invariant for pure ω-hypercontractions. We omit the proof as it is exactly same as Theorem 3.2 in [11]. Theorem 2.10 Let T1 and T2 be two pure ω-hypercontractions on H1 and H2 , respectively. Then T1 and T2 are unitary equivalent if and only if θT1 and θT2 coincide.

3 W -Hypercontractions (i)

Let, for each i = 1, . . . , n, ωi = {ωm }m≥0 be a weight sequence. Then the ntuple of weight sequences W = (ω1 , . . . , ωn ) is called a multi-weight sequence. Corresponding to such a multi-weight sequence W and a Hilbert space E, consider the E-valued weighted Bergman space A2W (E) over Dn consists of f ∈ O(Dn , E) such that   f (z) = aα z α and f 2 := aα 2 ωα(1) · · · ωα(n) < ∞ (aα ∈ E, z ∈ Dn ). n 1 α∈Zn+

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The Hilbert space A2W (E) is unitarily equivalent to A2ω1 ⊗· · ·⊗ A2ωn ⊗E via the natural unitary which sends the monomials z α e to z 1α1 ⊗ · · · ⊗ z nαn ⊗ e for all α ∈ Zn+ and e ∈ E; it is also a reproducing kernel Hilbert space with kernel K W (z, w) = K ω1 (z 1 , w1 ) · · · K ωn (z n , wn )IE  1 ¯ α IE (z, w ∈ Dn ). = (z w) (1) (n) α∈Zn ωα1 · · · ωαn +

The analytic function associated to K W is crucial in what follows and has the form kW (z) = kω1 (z 1 ) · · · kωn (z n ) = where kωi (z) =



1

α∈Zn+

(n) ωα(1) 1 · · · ωαn

z α (z ∈ Dn ),



1 m m≥0 ω(i) z is the associated analytic function corresponding to the m kernel function K ωi and the weight sequence ωi . Since each ωi is a weight sequence, kωi does not vanish on D and consequently kW does not vanish on Dn and

 1 = cα(1) · · · cα(n) zα , n 1 kW (z) n

(3.1)

α∈Z+

where the coefficients satisfy  1 (i) m = cm z (z ∈ D, 1 ≤ i ≤ n). kωi (z) m≥0

The several variable analogue of properties (P1), (P2) and (P3) that the analytic function kW should satisfy are as follows: W (P1 ) The function kkW,r has non negative Taylor coefficients for all r ∈ (0, 1)n , where for r = (r1 , . . . , rn ) and z ∈ Dn ,

kW ,r (z) = kω1 (r1 z 1 ) · · · kωn (rn z n ). have uniformly bounded Taylor coefficients for all r ∈ (P2 ) The quotients kW,r W (0, 1)n . 1 is absolutely summable (P3 ) The Taylor coefficients of the reciprocal function kW k

and the absolute sum of Taylor coefficients of uniformly bounded family.

kW,r kW

for all r ∈ (0, 1)n form a

We observe that the analytic function kW automatically inherits properties (P1 ) and (P2 ). Proposition 3.1 Let W = (ω1 , . . . , ωn ) be a multi-weight sequence and kW be the associated analytic function on Dn as above. Then kW satisfies (P1 ) and (P2 ). In addition if, for all i = 1, . . . , n, kωi satisfies (P3) then kW also satisfies (P3 ).

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Proof Let r, s ∈ (0, 1]n . Then for each i and |z i | < min(1/ri , 1/si ), we have ⎞⎛ ⎞ ⎛ kωi (ri z i ) ⎝ 1 n n ⎠ ⎝  (i) m m ⎠  (i) r z cm si z i am (ri , si )z im , = = (i) i i kωi (si z i ) ωn n≥0

m≥0

m≥0

where (i) am (ri , si ) =

 0≤k≤m

rik

(i) c s m−k . (i) m−k i

ωk

(3.2)

Then  kW ,r (z) kω (r1 z 1 ) kω (rn z n ) = 1 ··· n = aα (r, s)z α , kW ,s (z) kω1 (s1 z 1 ) kωn (sn z n ) n α∈Z+

(1)

(n)

where aα (r, s) = aα1 (r1 , s1 ) · · · aαn (rn , sn ) for all α ∈ Zn+ . This in particular shows that kW inherits (P1 ) and (P2 ) from that of kωi . Now if each kωi satisfies (P3), then it follows from (3.1) and the identity above   that kW satisfies (P3 ). This completes the proof. Although there are abundant examples of analytic functions on Dn satisfying (P1 ), (P2 ) and (P3 ). We provide a few examples corresponding to a certain type of natural multi-weight sequence. Example 3.2 (i) For any β = (β1 , . . . , βn ) ∈ Rn+ with β ≥ e, we consider the multi

1 = and βi +l−1 weight sequence Wβ = (ωβ1 , . . . , ωβn ), where ωβi = βi +l−1 l ( l ) l≥0 (βi +l) n (βi )l! , for i = 1, . . . , n. Then the analytic function on D corresponding to Wβ is 1 1 kWβ (z) = kωβ1 (z 1 ) · · · kωβn (z n ) = ··· (1 − z 1 )β1 (1 − z n )βn   β + α − 1    β + α − 1  1 1 n n z 1α1 · · · z nαn . = α1 αn α1 ≥0

αn ≥0

By Corollary 5.5 in [22], the analytic function kωβi (z i ) = (1−z1 )βi , for each i = i 1, . . . , n, has the properties (P1), (P2), and (P3). Therefore, using Proposition 3.1, the analytic function kWβ (z) on Dn corresponding to the multi-weight sequence Wβ also satisfies the properties (P1 ), (P2 ) and (P3 ). Moreover, the reproducing kernel Hilbert space corresponding to the multi-weight sequence Wβ is the weighted Bergman space over Dn with kernel K Wβ (z, w) =

n  i=1

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(z, w ∈ Dn ).

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In particular, it is easy to see that for β = (1, . . . , 1), each of the weight sequence becomes the constant sequence 1, which we denote by 1, and the corresponding reproducing kernel Hilbert space is the Hardy space over Dn . (ii) For each i = 1, . . . , n, we consider the sequence ωi = {ωk(i) }k≥0 such that it 1

(i)

(i) k

is a positive decreasing sequence satisfying ω0 = 1, lim inf n→∞ ωk = 1, and the  associated analytic function kωi (z) = l≥0 1(i) z l (z ∈ D) is a finite product of analytic ωl

functions corresponding to some Nevanlinna-Pick kernels [3]. Then by Proposition (i) 5.4 in [22], each ωi = {ωk }k≥0 is a weight sequence and the associated analytic function kωi satisfies (P3). Therefore, using Proposition 3.1, W = (ω1 , . . . , ωn ) is a multi-weight sequence and the associated analytic function kW satisfies (P3 ). Recall that an n-tuple of commuting contractions T = (T1 , . . . , Tn ) ∈ B(H)n is an W-hypercontraction corresponding to a multi-weight sequence W = (ω1 , . . . , ωn ) if DW ,T (r) ≥ 0 for all W ∈ S(W) and r ∈ (0, 1)n , where DW ,T (r) :=



cα rα T α T ∗α ,

(3.3)

α∈Zn+

cα as in (3.1) and S(W) := {(ωλ 1 , . . . , ωλ n ) : ωλ i ∈ {ωλi , 1}, i = 1, . . . , n}. For a non-empty subset  = {λ1 , . . . , λm } of I = {1, . . . , n}, we set T := (Tλ1 , . . . , Tλm ), and W := (ωλ1 , . . . , ωλm ). One of the important properties is that W-hypercontractivity is preserve for subtuples of an W-hypercontraction. Proposition 3.3 Let T be an W-hypercontraction for some multi-weight sequence W. Then for any non-empty subset  ⊆ I , T is a W -hypercontraction. Proof We only consider  = {1, . . . , n − 1} and show that T is an W hypercontraction, as the argument for general  ⊆ I is similar. Let W = ) ∈ S(W ) and set W := (W , 1) = (ω1 , . . . , ωn−1 , 1) ∈ S(W). (ω1 , . . . , ωn−1 Since T = (T1 , . . . , Tn ) is an W-hypercontraction and W ∈ S(W), DW ,T (r) ≥ 0 for all r ∈ (0, 1)n . Then using Agler’s hereditary calculus to the identity kW (r1 z 1 , . . . , rn z n ) = (1 − rn z n )−1 kW (r1 z 1 , . . . , rn−1 z n−1 ), we get DW ,T (r) = DW ,T (r ) − rn Tn DW ,T (r )Tn∗ , 134

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where r = (r1 , . . . , rn−1 ). Now using telescoping sum we have DW ,T (r ) =

l 

rnk Tnk DW ,T (r)Tn∗k + rnl+1 Tnl+1 DW ,T (r )Tn∗(l+1) .

k=0

Since rn ∈ (0, 1), taking limit as l → ∞ we conclude that D

W ,T



(r ) =

∞ 

rnk Tnk DW ,T (r)Tn∗k ≥ 0.

k=0

 

This completes the proof.

The following proposition is the key to define defect operator and defect space corresponding to an W-hypercontraction. Proposition 3.4 Let W be a multi-weight sequence. Let T be an n-tuple of commuting contractions such that DW ,T (r) ≥ 0 for all r ∈ (0, 1)n . Then SOT − lim DW ,T (r) r→e is a positive operator, where e = (1, . . . , 1). Moreover, if the associated analytic function kW satisfies (P3 ) then lim DW ,T (r) =

r→e



cα T α T ∗α ,

α∈Zn+

where the sum converges in the operator norm in B(H). Proof For the existence of the strong operator limit, it is enough to show that if r, s ∈ (0, 1)n with r ≤ s then DW ,T (s) ≤ DW ,T (r). Let r ≤ s. Without any loss of generality, we assume that ri = si for all i = 1, . . . , m and ri < si for all i = m + 1, . . . , n. Then using the identity kW (sm+1 z m+1 ) · · · kW (sn z n ) kW (s1 z 1 ) · · · kW (sn z n ) = kW (r1 z 1 ) · · · kW (rn z n ) kW (rm+1 z m+1 ) · · · kW (rn z n )  = aα (sm+1 , . . . , sn , rm+1 , . . . rn )z α , α∈Zn−m +

(sm+1 , rm+1 ) · · · aα(n) where aα (sm+1 , . . . , sn , rm+1 , . . . rn ) = aα(m+1) n−m (sn , r n ) for all 1 n−m α ∈ Z+ , and Agler’s hereditary functional calculus we have that 

DW ,T (r) =

aα (sm+1 , . . . sn , rm+1 , · · · , rn )T α DW ,T (s)T ∗α .

α∈Zn−m +

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Moreover, since (ωm+1 , . . . , ωn ) is a multi-weight sequence, by (P1 ), aα (sm+1 , . . . sn , rm+1 , · · · , rn ) ≥ 0 for all (α ∈ Zn−m + ) and a(0,...,0) (sm+1 , . . . sn , rm+1 , · · · , rn ) = 1. This shows that DW ,T (s) ≤ DW ,T (r). Finally, if kW satisfies (P3 ), by a simple  use of Lebesgue’s dominated convergence theorem we also have DW ,T (r) → α∈Zn+ cα T α T ∗α as r → e in B(H). This completes the proof.   Remark 3.5 The proof of the above proposition also suggests that if r ∈ (0, 1)m and s ∈ (0, 1)n−m then SOT −

lim

r→(1,...,1)

DW ,T (r, s) ≥ 0.

We denote the positive operator whose existence is shown in the above proposition by DW ,T (e) := SOT − lim DW ,T (r), r→e

and the defect operator and the defect space of T by DW ,T := DW ,T (e)1/2 and DW ,T := ranDW ,T ,

(3.5)

respectively. It could be difficult in general to determine when an n-tuple of commuting contractions T = (T1 , . . . , Tn ) is an W-hypercontraction as it asks to verify infinitely many inequalities. However, for certain multi-weight sequences it becomes easier to verify. For instance, consider the multi-weight sequence Wγ for γ = (γ1 , . . . , γn ) ∈ Zn+ with γ ≥ e as in 3.2. In this case, we show that the notion of γ -contractive multioperator in the sense of [14] is same as Wγ -hypercontraction. Recall that an n-tuple of commuting contractions T = (T1 , . . . , Tn ) is a γ -contractive multi-operator if for all 0 ≤ β ≤ γ , −1 KW (T , T ∗ ) = β

 0≤α≤β

β! T α T ∗α ≥ 0. α!(β − α)!

−1 Observe that the operator K W (T , T ∗ ) can also be represented as β −1 (T , T ∗ ) = (I − C T1 )β1 · · · (I − C Tn )βn (I ), KW β

where for an operator A ∈ B(H), the completely positive map C A : B(H) → B(H) is defined by C A (X ) = AX A∗ (X ∈ B(H)). We begin with the following lemma.

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Lemma 3.6 Let γ = (γ1 , . . . , γn ) ∈ Rn+ and r ∈ (0, 1)n . If T = (T1 , . . . , Tn ) ∈ B(H)n satisfies DWγ ,T (r) ≥ 0 then DWβ ,T (r) ≥ 0 for all β ∈ Rn+ such that 0 ≤ β ≤ γ. Proof Let 0 ≤ β ≤ γ . For (z 1 , . . . , z n ) ∈ Dn and r ∈ (0, 1)n , applying Agler’s hereditary functional calculus to the identity (1 − r z 1 )β1 . . . (1 − r z n )βn 1 = (1 − r z 1 )γ1 . . . (1 − r z n )γn , (1 − r z 1 )γ1 −β1 . . . (1 − r z n )γn −βn we have DWβ ,T (r) =



rδ

δ∈Zn+



γ −β +δ−e δ T DWγ ,T (r)T ∗δ ≥ 0, δ

where for β = (β1 , . . . , βn ) and α = (α1 , . . . , αn ), the proof.

β  α

=

β1  α1

···

βn  αn . This completes  

To simplify computations we borrow the following notation from [14]. For β ∈ Zn+ , β we denote by T the map on B(H) defined by β

T := (I − C T1 )β1 · · · (I − C Tn )βn and for r = (r1 , . . . , rn ) ∈ (0, 1)n , β

rT := (I − r1 C T1 )β1 · · · (I − rn C Tn )βn . Theorem 3.7 Let T = (T1 , . . . , Tn ) ∈ B(H)n be an n-tuple of commuting contractions and γ ∈ Zn+ such that γ ≥ e. Then the following are equivalent. (a) T is an γ -contractive multi-operator. (b) T is an Wγ -hypercontraction. Proof If T is an Wγ -hypercontraction then by Lemma 3.6, it follows that DWβ ,T (r) ≥ 0 for all 0 ≤ β ≤ γ and r ∈ (0, 1)n . Then by taking limit as r → e and using −1 (T , T ∗ ) ≥ 0 for all 0 ≤ β ≤ γ . Proposition 3.4, we conclude that DWβ ,T (e) = K W β This proves (b) ⇒ (a). To prove (a) ⇒ (b), let 0 ≤ β ≤ γ . We work on one component at a time and β only show that (T ,r Tn ) (I ) ≥ 0 for all r ∈ (0, 1), where T = (T1 , . . . , Tn−1 ), as β

repetition of the same argument will then establish rT (I ) ≥ 0 for all r ∈ (0, 1)n . To β β this end, we show that (T ,r Tn ) (I ) ≥ T (I ) ≥ 0. If β = (β1 , . . . , βn ) and βn = 0, then there is nothing to prove. Assume that βn > 0 and set β = (β1 , . . . , βn−1 ). First note that β

β

β

(T ,r Tn ) (I ) − T (I ) = T [(I − rC Tn )βn (I ) − (I − C Tn )βn (I )] Reprinted from the journal

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= (1 − r )

β n −1

β

T C Tn (I − rC Tn )(βn −1−l) (I − C Tn )l (I ).

l=0 β

It is now enough to show that T (I − rC Tn )k (I − C Tn )(βn −1−k) (I ) ≥ 0 for all β

(β ,βn −1)

k = 0, . . . , βn − 1. For k = 0, T (I − C Tn )(βn −1) (I ) = T hypothesis. For k = 1,

(I ) ≥ 0 by the

β

T (I − rC Tn )(I − C Tn )(βn −2) (I ) β

= T [(I − C Tn )(βn −2) (I ) − rC Tn (I − C Tn )(βn −2) (I )] (β ,βn −2)

= T ≥ =

(β ,βn −2)

(I ) − rC Tn T

(I )

(β ,β −2) (β ,β −2) T n (I ) − C Tn T n (I ) (β ,β −1) T n (I ).

(β ,βn −2)

(as T

β

(I ) ≥ 0 and 0 < r < 1)

(β ,βn −1)

This shows that T (I − rC Tn )(I − C Tn )(βn −2) (I ) ≥ T calculation as above one can show that β

(I ) ≥ 0. By a similar

β

T (I − rC Tn )2 (I − C Tn )(βn −3) (I ) ≥ T (I − rC Tn )(I − C Tn )(βn −2) (I ) (β ,βn −1)

≥ T

(I ) ≥ 0.

Repeating this k-times we have the following chain of inequalities: β

β

T (I − rC Tn )k (I − C Tn )(βn −1−k) (I ) ≥ T (I − rC Tn )k−1 (I − C Tn )(βn −k) (I ) β

(β ,βn −1)

≥ · · · ≥ T (I − rC Tn )(I − C Tn )(βn −2) (I ) ≥ T

(I ) ≥ 0.  

This completes the proof. Even for γ ∈ Rn+ , we have a similar result to the above Theorem.

Theorem 3.8 Let T = (T1 , . . . , Tn ) ∈ B(H)n be an n-tuple of commuting contractions and γ ∈ Rn+ such that γ ≥ e. Then the following are equivalent. −1 (a) K W (T , T ∗ ) := (I − C T1 )β1 · · · (I − C Tn )βn (I ) ≥ 0 for all β ∈ Rn+ such that β 0 ≤ β ≤ γ. (b) T is an Wγ -hypercontraction.

Proof If T is an Wγ -hypercontraction then by Lemma 3.6, it follows that DWβ ,T (r) ≥ 0 for all β ∈ Rn+ such that 0 ≤ β ≤ γ . Then by taking limit as r → e and using −1 (T , T ∗ ) ≥ 0 for all 0 ≤ β ≤ γ . Proposition 3.4, we conclude that DWβ ,T (e) = K W β This proves (b) ⇒ (a). To prove (a) ⇒ (b), it is enough to prove that for 0 ≤ β ≤ γ and r ∈ (0, 1), (β ,β ) (T ,r nTn ) (I ) ≥ 0, where β = (β1 , . . . , βn ) and β = (β1 , . . . , βn−1 ). Let r ∈ (0, 1) 138

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β

be fixed. If βn = 0, then (T ,r nTn ) (I ) = T (I ) ≥ 0. Assume that βn > 0. First we β

consider the case when βn ∈ Z+ . If 1 ≤ βn , then using the fact T (I ) ≥ 0, we have (β ,1)

β

β

(β ,1)

(T ,r Tn ) (I ) = T (I − rC Tn )(I ) ≥ T (I − C Tn )(I ) = T

(I ) ≥ 0.

(β ,1)

Similarly if 2 ≤ βn , then using (T ,r Tn ) (I ) ≥ 0, (β ,2)

(β ,1)

(β ,1)

(T ,r Tn ) (I ) = (T ,r Tn ) (I − rC Tn )(I ) ≥ (T ,r Tn ) (I − C Tn )(I ) (β ,1)

= T

(β ,2)

(1 − rC Tn )(I ) ≥ T

(I ) ≥ 0. (β ,β )

By repeating this method sufficient number of times we conclude that (T ,r nTn ) (I ) ≥ 0 / Z+ . Let [βn ] be the largest integer if βn is an integer. Next we suppose that βn ∈ which is less than or equal to βn . We set δ = βn − [βn ]. Since 0 < δ < 1, observe that  (β ,[βn ]) k (1 − x)δ = 1 − ∞ k=1 bk x , where bk ≥ 0 for all k ≥ 1. Now using (T ,r Tn ) (I ) ≥ 0, (β ,β )

(T ,r nTn ) (I ) β

= T (I − rC Tn )βn (I ) β

= T (I − rC Tn )[βn ] (I − rC Tn )δ (I ) (β ,[β ])

∞ 

(β ,[β ])

∞ 

= (T ,r Tnn ) (I ) −

(β ,[β ])

bk r k C Tkn (T ,r Tnn ) (I )

k=1

≥ (T ,r Tnn ) (I ) −

(β ,[β ])

bk C Tkn (T ,r Tnn ) (I )

k=1

=

(β ,[β ]) (T ,r Tnn ) (I

− C Tn )δ (I ).

To complete the proof of the theorem, we apply the similar strategy as in the proof of the above theorem to get a chain of inequalities. That is, β

β

T (I − C Tn )δ (I − rC Tn )[βn ] (I ) ≥ T (I − C Tn )δ+1 (I − rC Tn )[βn ]−1 (I ) β

β

β

≥ · · · ≥ T (I − C Tn )δ+[βn ]−1 (I − rC Tn )(I ) ≥ T (I − C Tn )δ+[βn ] (I ) = T (I ) ≥ 0.

 

This completes the proof. We end this section with a few examples of W-hypercontractions.

Examples 3.9 (i) We say T = (T1 , . . . , Tn ) is a Szegö tuple on H if T is a commuting ∗ tuple of contractions on H such that S−1 n (T , T ) ≥ 0 where Sn is the Szegö kernel n of the Hardy space over D . Moreover, we say T is a Brehmer tuple ( [12]) if T is a Szegö tuple for any non-empty subset  of I . If T is a Brehmer tuple then, by Theorem 3.7, T is an W-hypercontraction for the multi-weight sequence W = (1, . . . , 1).

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(ii) Let T = (T1 , . . . , Tn ) be an n-tuple of commuting co-isometries on H and W = (ω1 , . . . , ωn ) be a multi-weight sequence. For any W ∈ S(W), we have 

DW ,T (r) =

cα rα T α T ∗α =

α∈Zn+

1 IH kW (r)

(r ∈ (0, 1)n ).

Since, kW (r) > 0 for any W ∈ S(W), so T is an W-hypercontraction for any multi-weight sequence W. (iii) Let R = (R1 , . . . , Rn ) be an n-tuple of commuting contractions on H such that R1 is an co-isometry and R = (R2 , . . . , Rn ) is an W -hypercontraction. Then for any weight sequence ω1 , consider the multi-weight sequence W = (ω1 , W ). ˜ ∈ S(W). Then clearly W ˜ ∈ S(W ) and observe that for Let W = (ω1 , W) n r ∈ (0, 1) , DW ,R (r) =



cα(1) · · · cα(n) rα R1α1 R2α2 · · · Rnαn R1∗α1 R2∗α2 · · · Rn∗αn n 1

α∈Zn+

=



cα(1) · · · cα(n) rα R2α2 · · · Rnαn R1α1 R1∗α1 R2∗α2 · · · Rn∗n n 1

α∈Zn+

=

 α1 ≥0

=

 cα(1) r α1 DW˜ ,R (r ) 1 1

1 DW˜ ,R (r ) ≥ 0, kω1 (r1 )



where r = (r2 , . . . , rn ). Thus R = (R1 , . . . , Rn ) is an W-hypercontraction on H. (iv) Let M z = (Mz 1 , . . . , Mz n ) be the n-tuple of multi-shifts on A2W (E) for some multi-weight sequence W, that is Mzi f (w) = wi f (w) (i = 1, . . . , n, and w ∈ Dn ). Note that for f (z) =



α∈Zn+

M z∗β f (w) =

aα z α and β = (β1 , . . . , βn ) ∈ Zn+ ,  ωα(1)+β · · · ωα(n)+β n n 1 1

α∈Zn+

(1) (n) ωα1 · · · ωαn

aα+β wα .

Then  ωα(1)+β · · · ωα(n)+β n n 1 1 2

M z∗β f 2 =

α∈Zn+

2

(n) ωα(1) 1 · · · ωαn

140

aα+β 2 .

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Now, for r = (r1 , . . . , rn ) ∈ (0, 1)n , DW ,M z (r) f , f  =



cβ r β M z∗β f 2

β∈Zn+

=



⎛ cβ r β ⎝

β∈Zn+

=



 ωα(1)+β · · · ωα(n)+β n n 1 1 2

(n) ωα(1) 1 · · · ωαn

α∈Zn+

⎛

ωα1 · · · ωαn ⎝ (1)2

(n)2



α 

aα+β 2 ⎠ ⎞

cβ r β (1)

(n)

β≥0 ωα1 −β1 · · · ωαn −βn

α∈Zn+

=

⎞

2

⎠ aα 2

ωα(1) · · · ωα(n) aα (e, r) aα 2 , n 1 2

2

α∈Zn+

where for the last equality we use identities as in 3.2. By the property (P1 ), aα (e, r) ≥ 0 for all r ∈ (0, 1)n and therefore, DW ,M z (r) ≥ 0 for all r ∈ (0, 1)n . Thus the n-tuple of multi-shifts M z on A2W (E) is an W-hypercontraction. In fact, Mz is a pure W-hypercontraction. To see this first observe that the space span{z α h : α ∈ Zn+ , h ∈ E} is dense in A2W (E). For fixed α ∈ Zn+ and h ∈ E, if we consider k ∈ Z+ such that k > |α|, then Mz∗ki (z α h) = 0. Finally, since {Mz∗ki }k≥1 is uniformly bounded we get that Mz∗ki → 0 in the strong operator topology. This shows that Mzi is a pure contraction for all i = 1, . . . , n.

4 Model For W -Hypercontractions The main purpose of this section is to find dilations of W-hypercontractions. This multi-variate dilation is obtained using one-variable dilation at a time. The key to use one variable dilation theory is a commutant lifting result which we describe first. Let T = (T1 , . . . , Tn ) be an W-hypercontraction on H corresponding to a multi-weight sequence W = (ω1 , . . . , ωn ). Then considering the subset  = {i} of I and using Proposition 3.3 we have that each Ti is an ωi -hypercontraction for all i = 1, . . . , n. Thus Dωi ,Ti (1) := SOT − limr →1 Dωi ,Ti (r ) ≥ 0 for all i = 1, . . . , n, and recall that the corresponding defect operator and defect space are Dωi ,Ti = (Dωi ,Ti (1))1/2 and Dωi ,Ti = ran Dωi ,Ti , respectively. In what follows, we denote by Wˆ i (1 ≤ i ≤ n) the multi-weight sequence obtained from W = (ω1 , . . . , ωn ) by deleting ωi , that is, Wˆ i := (ω1 , . . . , ωi−1 , ωi+1 , . . . , ωn ). Proposition 4.1 Let W be a multi-weight sequence and let T = (T1 , . . . , Tn ) be an Whypercontraction on H. Suppose (Mz ⊕UT1 ) on A2ω1 (Dω1 ,T1 )⊕QT1 is the dilation of T1 with the dilation map ω1 ,T1 : H → A2ω1 (Dω1 ,T1 ) ⊕ QT1 as obtained in Theorem 2.3. Reprinted from the journal

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Then there exists an Wˆ 1 -hypercontraction V = (V2 , . . . , Vn ) on A2ω1 (Dω1 ,T1 ) ⊕ QT1 such that ω1 ,T1 Ti∗ = Vi∗ ω1 ,T1 and Vi (Mz ⊕ UT1 ) = (Mz ⊕ UT1 )Vi ,

(i = 2, . . . , n)

where Vi = (I A2ω ⊗ Ai ) ⊕ X i (i = 2, . . . , n) for some commuting operator tuples 1 (A2 , . . . , An ) on Dω1 ,T1 and (X 2 , . . . , X n ) on QT1 . Proof Let 2 ≤ i ≤ n and consider the subset  = {1, i} of I . Since T is a W hypercontraction, then for the multi-weight sequence W = (ω1 , 1) ∈ S(W ), we have DW ,T (1, 1) ≥ 0, that is, Dω1 ,T1 (1) − Ti Dω1 ,T1 (1)Ti∗ ≥ 0. Applying Douglas factorization lemma ( [15]) to the above inequality we have a contraction Ai on Dω1 ,T1 such that Dω1 ,T1 Ti∗ = Ai∗ Dω1 ,T1 .

(4.1)

Thus, we get an (n−1)-tuple of commuting contractions A = (A2 , . . . , An ) on Dω1 ,T1 . We now show that A is an Wˆ 1 -hypercontraction. To this end, we only consider the multi-weight sequence Wˆ 1 and show that DWˆ ,A (r) ≥ 0 for all r ∈ (0, 1)n−1 . As the 1 required positivity corresponding to other multi-weight sequences in S(Wˆ 1 ) can be shown similarly. For any r ∈ (0, 1)n−1 and h ∈ H, we have DWˆ

1 ,A



(r)Dω1 ,T1 h, Dω1 ,T1 h =

rα cα(2) · · · cα(n) Dω1 ,T1 Aα A∗α Dω1 ,T1 h, h n 2

(n−1) α=(α2 ,...,αn )∈Z+



=

rα cα(2) · · · cα(n) T α Dω1 ,T1 (1)T ∗α h, h n 2 (n−1)

α=(α2 ,...,αn )∈Z+



=

rα cα(2) · · · cα(n) lim T α Dω1 ,T1 (s)T ∗α h, h n 2

(n−1) α=(α2 ,...,αn )∈Z+

= lim

s→1



s→1

(r, s)α cα T α T ∗α h, h

α∈Zn+

= lim DW ,T (r, s)h, h ≥ 0. s→1

Here the positivity in the last equality follows from Remark 3.5. This proves that A is an Wˆ 1 -hypercontraction. On the other hand, recall from the construction of dilation in Theorem 2.3 that Q 2T1 = SOT − limn→∞ T1n T1∗n , QT1 = ran Q T1 and the co-isometry UT1 on QT1 is defined by the identity UT∗1 Q T1 h = Q T1 T1∗ h for all h ∈ H. Now for any 2 ≤ i ≤ n, since Ti is a contraction we have Ti Q 2T1 Ti∗ ≤ Q 2T1 . Again applying Douglas factorization lemma to the inequality we get a contraction X i on QT1 such that X i∗ Q T1 = Q T1 Ti∗ (i = 2, . . . , n).

(4.2)

It is now easy to see that (UT1 , X 2 , . . . , X n ) is an n-tuple of commuting contractions on QT1 . As before, we show that X = (X 2 , . . . , X n ) is an Wˆ 1 -hypercontraction and

142

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for this we only show that DWˆ ,X (r) ≥ 0 for all r ∈ (0, 1)n−1 as the argument for 1 other multi-weight sequences is similar. Now for h ∈ H and r ∈ (0, 1)n−1 , DWˆ

1 ,X



(r)Q T1 h, Q T1 h =

rα cα(2) · · · cα(n) T α ( lim T1k T1∗k )T ∗α h, h n 2 k→∞

(n−1)

α=(α2 ,...,αn )∈Z+



= lim

k→∞

rα cα(2) · · · cα(n) T α T ∗α T1∗k h, T1∗k h n 2

(n−1) α=(α2 ,...,αn )∈Z+

= lim DWˆ ,T (r)T1∗k h, T1∗k h ≥ 0. 1 k→∞

Thus X is an Wˆ 1 -hypercontraction. We set 

Vi := (I A2ω ⊗ Ai ) ⊕ X i ∈ B A2ω1 (Dω1 ,T1 ) ⊕ QT1 (i = 2, . . . , n). 1

Then it is evident that V = (V2 , . . . , Vn ) is an Wˆ 1 -hypercontraction. It remains to verify that V satisfies the required commuting and intertwining relations. For any h ∈ H and i = 2, . . . , n, ω1 ,T1 Ti∗ h =



zk

 1 Dω1 ,T1 T1∗k Ti∗ h, Q T1 Ti∗ h ωk

zk

 1 ∗ Ai Dω1 ,T1 T1∗k h, X i∗ Q T1 h ωk

k≥0

=

 k≥0

 1

 = (I A2ω ⊗ Ai∗ ) z k Dω1 ,T1 T1∗k h, X i∗ Q T1 h 1 ωk k≥0

=

Vi∗ ω1 ,T1 h.

Thus ω1 ,T1 Ti∗ = Vi∗ ω1 ,T1 for all i = 2, . . . , n. Finally, as each X i commutes with  UT1 it follows that Mz ⊕ UT1 commutes with each Vi . This completes the proof.  Remark 4.2 If T1 is a pure contraction in the above proposition, then Q T1 = 0 and therefore X i = 0 for all i = 2, . . . , n. Thus, in this case, the Wˆ 1 -hypercontraction V = (V2 , . . . , Vn ) will be of the form Vi = I A2ω ⊗ Ai for all i = 2, . . . , n. 1

The following lemma is needed to prove the general dilation result below. Lemma 4.3 Let T , X , A be as in Proposition 4.1 and let  ⊆ {2, . . . , n}. Then (i) Dω1 ,T1 DW ,A (1, . . . , 1)Dω1 ,T1 = DW∪{1} ,T∪{1} (1, . . . , 1). (ii) Q T1 DW ,X  (1, . . . , 1)Q T1 = SOT − limk→∞ T1k DW ,T (1, . . . , 1)T1∗k . Proof Let  = (λ1 , . . . , λm ) ⊆ {2, . . . , n}. Then Dω1 ,T1 DW ,A (1, . . . , 1)Dω1 ,T1 

 = Dω1 ,T1 SOT − lim rα cα(λ11 ) · · · cα(λmm ) Aα A∗α  Dω1 ,T1 r→(1,...,1)

Reprinted from the journal

α∈Zm +

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= SOT − = SOT −



lim

r→(1,...,1)

lim

α∈Zm +



r→(1,...,1)

(λ )

rα c(α11 ) · · · cα(λmm ) Tα Dω2 1 ,T1 T∗α (by (4.1)) ∗α m) T α rα cα(1) c(λ1 ) · · · cα(λm+1 ∪{1} T∪{1} 1 α2

α∈Zm+1 +

= DW∪{1} ,T∪{1} (1, . . . , 1). For the second part of the lemma we again do a similar computation. Q T1 DW ,X  (1, . . . , 1)Q T1 

 α ∗α = Q T1 SOT − lim rα cα(λ11 ) · · · cα(λmm ) X  X  Q T1 r→(1,...,1)

= SOT − = SOT −

lim

r→(1,...,1)

lim

r→(1,...,1)



α∈Zm +

rα cα(λ11 ) · · · cα(λmm ) Tα Q 2T1 T∗α (by (4.2))

α∈Zm +



rα cα(λ11 ) · · · cα(λmm ) Tα (SOT − lim T1k T1∗k )T∗α k→∞

α∈Zm +

= SOT − lim T1k DW ,T (1, . . . , 1)T1∗k . k→∞

In the second last equality, one can interchange of limit and sum using Lebesgue’s dominated convergence theorem and the interchange of limits in the last equality can be justified by showing that the double limit exists (see the appendix below for more details). This completes the proof.   Dilations of pure W-hypercontractions are very concrete and less complicated to describe compared to that of general W-hypercontractions. We consider this simpler case first which also helps facilitate the understanding of our dilation method. Recall that for a multi-weight sequence W = (ω1 , . . . , ωn ) and a non-empty subset  = {λ1 , . . . , λm } of I , we denote by W the multi-weight sequence (ωλ1 , . . . , ωλm ). When  = {1, . . . , i} ⊆ I then we simply use Wi] and W[i+1 to denote W and Wc , respectively, with the convention that W[n+1 = ∅. Let T = (T1 , . . . , Tn ) be a pure W-hypercontraction on H. Let Mz on A2ω1 (Dω1 ,T1 ) be the dilation of T1 with the canonical dilation map πω1 ,T1 : H → A2ω1 (Dω1 ,T1 ) as in Theorem 2.3. Then by Proposition 4.1 and Remark 4.2, we get an W[2 (2) (2) hypercontraction A(2) = (A2 , . . . , An ) on Dω1 ,T1 such that (2)

πω1 ,T1 Ti∗ = (I A2ω ⊗ Ai )∗ πω1 ,T1 1

(i = 2, . . . , n).

Since each Ti is pure then by the intertwining relation 4.1 A(2) is also a pure W[2 hypercontraction. We now apply Proposition 4.1 to A(2) as follows. Let πω ,A(2) : 2

144

2

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Dω1 ,T1 → A2ω2 (Dω

(2) 2 ,A2

(2)

) be the dilation map of A2 . Then we get a pure W[3 -

(3) hypercontraction A(3) = (A(3) 3 , . . . , An ) such that

πω

(2) 2 ,A2

Ai(2)∗ = (I A2ω ⊗ Ai(3)∗ )πω

(i = 3, . . . , n).

(2) 2 ,A2

2

Set 1 := πω1 ,T1 : H → A2ω1 (Dω1 ,T1 ) and 2 := I A2ω ⊗ πω

(2) 2 ,A2

1

: A2ω1 (Dω1 ,T1 ) → A2ω1 ⊗ A2ω2 (Dω

(2) 2 ,A2

) = A2W2] (Dω

(2) 2 ,A2

).

Then a moments thought reveals that the map 2 ◦ 1 satisfies (2 ◦ 1 )T1∗ = Mz∗1 (2 ◦ 1 ), (2 ◦ 1 )T2∗ = Mz∗2 (2 ◦ 1 ), and for all i = 3, . . . , n, (2 ◦ 1 )Ti∗ = (I A2

W2] (D

2)

(3)

⊗ Ai )∗ (2 ◦ 1 ).

Repeating the above procedure j-times we get a pure W[ j+1 -hypercontraction ( j+1) ( j+1) A( j+1) = (A j+1 , . . . , An ) and an isometry  j := I A2

W j−1]

⊗ πω

( j) j ,A j

: A2W j−1] (Dω ,A( j−1) ) → A2W j] (Dω ,A( j) ), j−1 j j−1 j

such that for all i = 1, . . . , j, ( j ◦ · · · ◦ 2 ◦ 1 )Ti∗ = Mz∗i ( j ◦ · · · ◦ 2 ◦ 1 ), and for all i = j + 1, . . . , n, ( j ◦ · · · ◦ 2 ◦ 1 )Ti∗ = (I A2

( j+1)

W j]

⊗ Ai

)( j ◦ · · · ◦ 2 ◦ 1 ).

Thus after n-th step we will have the following chain of isometries 1

2

3

n

0 → H −→ A2ω1 (Dω1 ,T1 ) −→ A2W2] (Dω ,A(2) ) −→ · · · −→ A2W (Dω ,A(n) ) n n 2 2 such that the isometry T := n ◦ · · · ◦ 2 ◦ 1 : H → A2W (Dω ,A(n) ) satisfies n n T Ti∗ = Mz∗i T (i = 1, . . . , n). Thus T dilates to the weighted Bergman shift (Mz 1 , . . . , Mz n ) on A2W (Dω ,A(n) ) via n n the dilation map T . We summarize this in the next result.

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Theorem 4.4 Let W be a multi-weight sequence and let T = (T1 , . . . , Tn ) be a pure W-hypercontraction on H. Then there exist a Hilbert space E and a joint (Mz∗1 , . . . , Mz∗n )-invariant subspace Q of A2W (E) such that (T1 , . . . , Tn ) ∼ = PQ (Mz 1 , . . . , Mz n )|Q . We now consider dilations of general W-hypercontractions and find their explicit dilation. In order to make the proof of the dilation result shorter, we take out some part of the proof and prove it as a separate lemma. The lemma is a special case of one operator being a co-isometry. Lemma 4.5 Let (V , Y1 , . . . , Yn−1 ) ∈ B(H)n be an n-tuple of commuting contractions such that V is a co-isometry. Suppose that the (n − 1)-tuple Y = (Y1 , . . . , Yn−1 ) is an W-hypercontractive tuple and dilates to an W-hypercontractive tuple R =  (R1 , . . . , Rn−1 ) on K through the dilation map  : H → K := ⊆I \{n} A2W (E ), where ⎞ ⎛    and Ri = ⎝ Ri ⎠ , = ⊆I \{n}

⊆I \{n}

such that for each  = {λ1 , . . . , λm },  : H → A2W (E ) defined by  h(z) =

 

1 ωα(λ11 ) · · · ωα(λmm )

α∈Zm +

  Y∗α h z α ,

Rλj = Mz j on A2W (E ) ( j = 1, . . . , m) and for i ∈ / , Ri = I A2 ⊗ Vi for some W co-isometry Vi on E = ran , with  ∈ B(H), satisfying  Yi∗ = (Vi )∗  for all i ∈ / . If the co-isometry V satisfies the relations V ∗  V ∗ = ∗  for each  ⊆ {1, . . . , n − 1}, then V lifts to a co-isometry W on K such that the tuple (V , Y1 , . . . , Yn−1 ) on H dilates to R = (W , R1 , . . . , Rn−1 ) on K. Proof By the hypothesis, for each  ⊆ {1, . . . , n − 1}, V ∗  V ∗ = ∗  . Therefore for each  ⊆ {1, . . . , n − 1}, by the Douglas factorization lemma, there exists a co-isometry W on E such that W∗  =  V ∗ . We define a co-isometry W on K = ⎛ W =⎝





⊆I \{n}

A2W (E ) as ⎞

(I A2

⊆I \{n}

W

146

⊗ W  )⎠ .

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Now, we observe that for any  ⊆ {1, . . . , n − 1}, h ∈ E and z ∈ D|| ,  V ∗ h(z) =

  α∈Zm +

= =

1 (λ ) (λ ) ωα11 · · · ωαmm

 

1 (λ )

(λ

 ∗ ∗α  V Y h z λα11 · · · z λαmm   )

α∈Zm +

ωα11 · · · ωαmm

α∈Zm +

(λ ) (λ ) ωα11 · · · ωαmm

 

  Y∗α V ∗ h z λα11 · · · z λαmm

1

 W∗  Y∗α h z λα11 · · · z λαmm

= W∗  h(z). Thus V ∗ = W ∗ . For the commutativity of the tuple (W , R1 , . . . , Rn−1 ), we fix i ∈ {1, . . . , n − 1} and show that W Ri = Ri W . To this end, it is enough to show that for each  ⊆ {1, . . . , n − 1}, (I A2 ⊗ W )Ri = Ri (I A2 ⊗ W ). If i = λ j ∈ , W

W

then Ri = Mz j and there is nothing to prove. If i ∈ / , then Ri = I A2

W

the commutativity can be read from the following:

⊗ Vi and

W∗ Vi∗  = W∗  Yi∗ =  V ∗ Yi∗ =  Yi∗ V ∗

= Vi∗ W∗  .  

This completes the proof.

We are now ready to prove the model for general W-hypercontractive tuples. It is an exact generalization of the model given in Theorem 2.8 in [14]. Theorem 4.6 Let T = (T1 , . . . , Tn ) be an W-hypercontraction on H for some multiweight sequence W. Then there exist  ∈ B(H) corresponding to each subset  of I with E = ran , an isometry  : H → K :=

 ⊆I

A2W (E ),

and an n-tuple of commuting contractions R = (R1 , . . . , Rn ) on K such that Ti∗ = Ri∗  (i = 1, . . . , n), where with respect to the above decomposition of K =



⎛  and Ri = ⎝

⊆I

Reprinted from the journal

 ⊆I

147

⎞ Ri ⎠

(4.3)

M. Bhattacharjee et al.

such that if  = {λ1 , . . . , λm } then  : H → A2W (E ) is defined by  h(z) =

 

1 (λ )

(λ

ωα11 · · · ωαmm

α∈Zm +

 ∗α  T h zα   )

and  Ri

=

I A2

W

⊗ Vi if i ∈ / Mz j if i = λ j ∈ 

for some co-isometry Vi on E . Moreover, for every subset  = {λ1 , . . . , λm } of I , β

∗β

∗  = SOT − lim Tc DW ,T (1, . . . , 1)Tc β→∞

and for all i ∈ / ,  Ti∗ = (Vi )∗  . Proof We prove the theorem by induction on n. For n = 1, if T is a ω-hypercontraction {1} on H then by Theorem 2.3 the result holds with E{1} = Dω,T , E∅ = QT , R1 = Mz on A2ω (Dω,T ), R1∅ = U , {1} = Dω,T , ∅ = Q T and  = {1} ⊕ {∅} with {1} h(z) =

  1 {1} T ∗k h z k and {∅} h = Q T h (h ∈ H, z ∈ D). ωk k≥0

Now we assume that the result is true for n − 1. Let T = (T1 , . . . , Tn ) be an Whypercontraction. Suppose (Mz ⊕U ) on A2ω1 (Dω1 ,T1 )⊕QT1 is the dilation of T1 with the canonical dilation map ω1 ,T1 : H → A2ω1 (Dω1 ,T1 )⊕QT1 as obtained in Theorem 2.3. Then applying Proposition 4.1 we get W[2 -hypercontractions A = (A2 , . . . , An ) on Dω1 ,T1 and X = (X 2 , . . . , X n ) on QT1 such that ω1 ,T1 Ti∗ = Vi∗ ω1 ,T1 (i = 2, . . . , n), where Vi = (I A2ω ⊗ Ai ) ⊕ X i for all i = 2, . . . , n. We now apply the induction 1 hypothesis to both A and X . Since A is an W[2 -hypercontraction, then by the hypothesis we get Hilbert spaces E = ran for all subset  of {2, . . . , n}, an isometry  A : Dω1 ,T1 → K :=





⊆{2,...,n}

A2(W[2 ) (E ) =





⊆{2,...,n}

A2W (E ),

A and for  = {λ , . . . , λ },  A : D 2 where  A = ⊕⊆{2,...,n}  2 m ω1 ,T1 → AW (E )  is defined by A  h(z) =

 

1 (λ )

α∈Zm−1 +

(λ

ωα22 · · · ωαmm

148

 ∗α  A h zα )  

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β



∗β

|c |

and ∗  = SOT − limβ→∞ Ac DW ,A (1, . . . , 1)Ac (β ∈ Z+ ). We also have an (n − 1)-tuple of commuting contractions R = (R2 , . . . , Rn ) on K having the structure as in (4.3) such that  A Ai∗ = Ri ∗  A . Once again applying the hypothesis to the W[2 -hypercontraction X on QT1 we also get Hilbert spaces E = ran for all subset  of {2, . . . , n}, an isometry 



 X : QT1 → K :=





⊆{2,...,n}

A2(W[2 ) (E ) =

⊆{2,...,n}



A2W (E ),

X and for  = {λ , . . . , λ },  X : Q → A2 (E ) is where  X = ⊕⊆{2,...,n}  2 m T1  W  defined by X h(z) = 

  (λ )

(λ

ωα22 · · · ωαmm

α∈Zm−1 +

1

 ∗α  X h zα )  

β



∗β

|c |

and ∗  = SOT − limβ→∞ X c DW ,X  (1, . . . , 1)X c (β ∈ Z+ ). We also have an (n − 1)-tuple of commuting contractions R = (R2 , . . . , Rn ) on K having structure as in (4.3) such that  X X i∗ = Ri ∗  X . Since U is a co-isometry on QT1 which commutes with X , then by Lemma 4.5 we get a co-isometry W on K such that W commutes with R and  X U ∗ = W ∗  X . We now have all the ingredient to construct the required dilation of T . For a subset  of I , we set  E :=

 if 1 ∈  E\{1} and K := A2W (E ). E otherwise ⊆I

Then note that A2ω1 ⊗ K = =





⊆{2,...,n}



⊆I ,1∈

A2ω1 ⊗ A2W (E )

A2W (E ) and K = (A2ω1 ⊗ K ) ⊕ K .

We consider the dilation map for T as  = ((I A2ω ⊗  A ) ⊕  X ) ◦ ω1 ,T1 : H 1   A2ω1 ⊗ A2W (E ) A2W (E ) = K. → ⊆{2,...,n}

⊆{2,...,n}

More explicitly, the dilation map has the following decomposition

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=

 ⊆{2,...,n}

=



 A (I A2ω ⊗  ) ◦ πω1 ,T1 1

 ⊆{2,...,n}

X   ◦ Q T1

 ,

⊆I

where 

A (I A2ω ⊗ \{1} ) ◦ πω1 ,T1 if 1 ∈ 

 :=

1

X ◦Q  T1

otherwise.

Moreover, for  ⊆ I with 1 ∈  and h ∈ H, 

1

α∈Zm +

ωα(1) 1

 h(z) =

···

 1 \{1} Dω1 ,T1 T ∗α h z α ,

ωα(m) m

(4.4)

and for 1 ∈ / , 

 h(z) =

α∈Zm−1 +

1 ωα(2) 1

···

1 ωα(m) (m−1)



 Q T1 T ∗α h z α .

(4.5)

Here for the above description of the dilation map we have used the intertwining relations Ai∗ Dω1 ,T1 = Dω1 ,T1 Ti∗ and X i∗ QT1 = QT1 Ti∗ for all i = 2, . . . , n, which can be read from Eqs. (4.1) and (4.2), respectively. The dilating tuple of commuting contractions R = (R1 , . . . , Rn ) on K is defined with respect to the decomposition K = (A2ω1 ⊗ K ) ⊕ K as

R1 = (Mz ⊗ IK ) ⊕ W and Ri = (I A2ω ⊗ Ri ) ⊕ Ri 1

(i = 2, . . . , n).

We now do a routine calculation to see that R is a dilation of T . First note that T1∗ = ((I A2ω ⊗  A ) ⊕  X )(Mz ⊕ U )∗ ω1 ,T1 1

= ((I A2ω ⊗  A )Mz∗ ⊕  X U ∗ )ω1 ,T1 1

= ((Mz∗ ⊗ IK ) ⊕ W ∗ )((I A2ω ⊗  A ) ⊕  X )ω1 ,T1 =

1

R1∗ ,

and similarly for all i = 2, . . . , n, Ti∗ = ((I A2ω ⊗  A ) ⊕  X )(I A2ω ⊗ Ai ⊕ X i )∗ ω1 ,T1 1

= ((I A2ω ⊗ 1

=

1

Ri ) ⊕



∗

Ri ) ((I A2ω ⊗  A ) ⊕  X )ω1 ,T1 1

Ri∗ .

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We left it to the reader to check that each of the operator Ri has the form as in (4.3). To complete the proof, we now need to construct  ∈ B(H) corresponding to each subset  of I which satisfies the moreover part of the theorem. To this end, we define   :=



\{1} Dω1 ,T1 if 1 ∈  otherwise.  Q T1

Rest of the proof is divided into two cases. Case I: 1 ∈ . In this case, setting  :=  \ {1}, we have



∗  = Dω1 ,T1 ∗  Dω1 ,T1 β

∗β

|c |

= SOT − lim Dω1 ,T1 Ac DW ,A (1, . . . , 1)Ac Dω1 ,T1 (β ∈ Z+ ) β→∞

β

∗β

= SOT − lim Tc Dω1 ,T1 DW ,A (1, . . . , 1)Dω1 ,T1 Tc β→∞

β

∗β

= SOT − lim Tc DW ,T (1, . . . , 1)Tc β→∞

( by part (i) of Lemma 4.3).





Also if i ∈ /  then by construction   Ai∗ = (Vi  )∗   , where Vi  is the co-isometry corresponding to Ri  , that is, Ri  = I A2 ⊗ Vi  . Also since Ri = I A2ω ⊗ Ri  = I A2

W



W

1



⊗ Vi  , the co-isometries corresponding to Ri and Ri  are the same, that is,

Vi = Vi  . Consequently,

 Ti∗ =   Dω1 ,T1 Ti∗ =   Ai∗ Dω1 ,T1 = (Vi  )∗   Dω1 ,T1 = (Vi )∗  . Case II: 1 ∈ / . In this case, setting  =  ∪ {1}, we have



∗  = Q T1 ∗  Q T1 β

∗β

| c |

= SOT − lim Q T1 X  c DW ,X  (1, . . . , 1)X  c Q T1 (β ∈ Z+ ) β→∞

β

∗β

= SOT − lim T c Q T1 DW ,X  (1, . . . , 1)Q T1 T c β→∞

β

∗β

= SOT − lim Tc DW ,T (1, . . . , 1)Tc β→∞

(by part (ii) of Lemma 4.3).









If i ∈ /  and i ≥ 2 then by the hypothesis  X i∗ = (Vi  )∗  , where Vi  is the co-isometry corresponding to Ri  . In this case, by construction the co-isometry corresponding to Ri and Ri  are the same, that is, Vi = Vi  . Consequently,





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 Q T1 = Vi  .

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Finally, for i = 1 the co-isometry corresponding to R1 is W1 and it follows from Lemma 4.5 that  U ∗ = W1∗   . Thus





 T1∗ =  Q T1 T1∗ =  U ∗ Q T1 = W1∗  Q T1 = W1∗  .  

This completes the proof.

5 Appendix Let T be an W-hypercontraction corresponding to a multi-weight sequence W. Then for a subset  = {λ1 , . . . , λm } of I , consider the map f : (0, 1)m × Zn−m → B(H), + defined by β

∗β

f (r , β) = Tc DW ,T (r )Tc , where r = (r1 , . . . , rm ) ∈ (0, 1)m and β = (β1 , . . . , βn−m ) ∈ Zn−m + . We claim that SOT − lim(r ,β)→(e ,∞) f (r , β) exists, where e = (1, . . . , 1). Indeed, it is enough to show that for r ≤ s and α ≤ β, f (s , β) ≤ f (r , α). Let j ∈ c and without any loss of generality assume that λm < i. Then the multiweight sequence W = (W , 1) ∈ S(W ), where  = {λ1 , . . . , λm , i}. Since T is a W -hypercontraction, then for any r ∈ (0, 1) DW ,T (r , r ) = DW ,T (r ) − r Ti DW ,T (r )Ti∗ ≥ 0 (r ∈ (0, 1)).

(5.1)

By Remark 3.5, taking limit as r → 1 in 5.1 we get Ti DW ,T (r )Ti∗ ≤ DW ,T (r ) for all i ∈ c . Consequently, for any β = (β1 , . . . , βn−m ) ∈ Zn−m + , β

∗β

Tc DW ,T (r )Tc ≤ DW ,T (r ). Now, for α ≤ β, β

∗β

Tαc DW ,T (r )T∗αc − Tc DW ,T (r )Tc   ∗α = Tαc DW ,T (r )−T β−α D ∗β−α Tc ≥ 0. (r )T 



W ,T

c

c

That is, β

∗β

Tc DW ,T (r )Tc ≤ Tαc DW ,T (r )T∗αc

(5.2)

for all α ≤ β.

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Finally, using Proposition 3.4 and the inequality 5.2, for r ≤ s and α ≤ β, f (r , α) = Tαc DW ,T (r )T∗αc

β

∗β

≥ Tαc DW ,T (s )T∗αc ≥ Tc DW ,T (s )Tc = f (s , β). This proves the claim. Acknowledgements The authors are grateful to Prof. Jaydeb Sarkar for some insightful discussions at the beginning of this project. The first named author acknowledges IIT Bombay for warm hospitality, and his research is supported by the DST-INSPIRE Faculty Fellowship No. DST/INSPIRE/04/2020/001250. The second author is supported by the Mathematical Research Impact Centric Support (MATRICS) grant, File No: MTR/2021/000560, by the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India. The third named author acknowledges IIT Bombay for its warm hospitality. The research of the third named author is supported by the institute post-doctoral fellowship of IIT Bombay.

References 1. Agler, J.: Hypercontraction and Subnormality. J. Oper. Theory 13, 203–217 (1985) 2. Agler, J.: The Arveson extension theorem and coanalytic models. Int. Eq. Oper. Theory 5(5), 608–631 (1982) 3. Agler, J., McCarthy, J.E.: Pick interpolation and hilbert function spaces, Graduate Stud. in Math., vol. 44, Amer. Math. Soc., Providence, RI (2002) 4. Ambrozie, C., Engliš, M., Müller, V.: Operator tuples and analytic models over general domains in Cn . J. Oper. Theory 47, 287–302 (2002) 5. Arazy, J., Engliš, M.: Analytic models for commuting operator tuples on bounded symmetric domains. Trans. Amer. Math. Soc. 355, 837–864 (2003) 6. Ball, J.A., Bolotnikov, V.: Weighted Bergman spaces: shift-invariant subspaces and input/state/output linear systems. Int. Eq. Oper. Theory 76(3), 301–356 (2013) 7. Ball, J.A., Bolotnikov, V.: Contractive multipliers from Hardy space to weighted Hardy space. Proc. Amer. Math. Soc. 145, 2411–2425 (2017) 8. Barik, S., Das, B.K.: Isometric dilations of commuting contractions and Brehmer positivity, Complex Analysis and Operator Theory 16, paper No. 69, 25 pp (2022) 9. Barik, S., Das, B.K., Haria, K.J., Sarkar, J.: Isometric dilations and von Neumann inequality for a class of tuples in the polydisc. Trans. Amer. Math. Soc. 372(2), 1429–1450 (2019) 10. Barik, S., Das, B.K., Sarkar, J.: Isometric dilations and von Neumann inequality for finite rank commuting contractions. Bull. Sci. Math. 165, 102915 (2020) 11. Bhattacharjee, M., Das, B.K., Sarkar, J.: Hypercontactions and factorizations of multipliers in one and several variables, J. Operator Theory (to appear). arXiv:1812.08143 12. Brehmer, S.: Über vertauschbare Kontraktionen des Hilbertschen Raumes. Acta Sci. Math. (Szeged) 22, 106–111 (1961) 13. Curto, R.E., Vasilescu, F.-H.: Standard operator models in the polydisc. Indiana Univ. Math. J. 42, 791–810 (1993) 14. Curto, R.E., Vasilescu, F.-H.: Standard operator models in the polydisc, II. Indiana Univ. Math. J. 44, 727–746 (1995) 15. Douglas, R.G.: On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc. 17, 413–415 (1966) 16. Eschmeier, J.: Bergman inner functions and m-hypercontractions. J. Func. Anal. 275, 73–102 (2018) 17. Popescu, G.: Noncommutative hyperballs, wandering subspaces, and inner functions. J. Funct. Anal. 276, 3406–3440 (2019) 18. McCullough, S., Richter, S.: Bergman-type reproducing kernels, contractive divisors, and dilations. J. Funct. Anal. 190(2), 447–480 (2002) 19. McCullough, S., Trent, T.: Invariant subspaces and Nevanlinna-Pick kernels. J. Funct. Anal. 178(1), 226–249 (2000)

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M. Bhattacharjee et al. 20. Müller, V., Vasilescu, F.-H.: Standard models for some commuting multioperators. Proc. Amer. Math. Soc. 117, 979–989 (1993) 21. Sz.-Nagy, B., Foias, C.: Harmonic analysis of operators on Hilbert space, North-Holland, AmsterdamLondon (1970) 22. Olofsson, A.: Parts of adjoint weighted shifts. J. Oper. Theory 74, 249–280 (2015) 23. Sarkar, J.: An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces - II. Complex Anal. Oper. Theory 10, 769–782 (2016) 24. Sarkar, J.: An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces - I. J. Oper. Theory 73, 433–441 (2015) 25. Schillo, D.: K-contractions, and perturbations of Toeplitz operators, PhD Thesis, Saarland University (2018) https://www.math.uni-sb.de/ag/eschmeier/lehre/examen/SchilloDr.pdf 26. Shimorin, S.: Wold-type decompositions and wandering subspaces for operators close to isometries. J. Reine Angew. Math. 531, 147–189 (2001) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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Complex Analysis and Operator Theory (2022) 16: 83 https://doi.org/10.1007/s11785-022-01257-0

Complex Analysis and Operator Theory

The Joint Spectrum for a Commuting Pair of Isometries in Certain Cases Tirthankar Bhattacharyya1

· Shubham Rastogi1

· U. Vijaya Kumar1

Received: 12 November 2021 / Accepted: 17 June 2022 / Published online: 18 July 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract We show that the joint spectrum of two commuting isometries can vary widely depending on various factors. It can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case and relate it to the modified bi-shift. Mathematics Subject Classification Primary 47A13 · 47A45 · 47A65

1 Introduction An isometry V is called pur e if V ∗n converges to 0 strongly as n → ∞. This is equivalent to saying that V is the unilateral shift of multiplicity equal to the dimension of the range of the defect operator I − V V ∗ . The famous Wold decomposition [17, 23] tells us that given an isometry V on a Hilbert space H, the space H breaks uniquely into a direct sum H = H0 ⊕ H0⊥ of reducing subspaces such that V |H0 is a unitary and V |H⊥ is a pure isometry. This 0 immediately implies that for a non-unitary isometry V (i.e., when the defect operator is

In fond memory of Jörg Eschmeier. Communicated by Michael Hartz. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial" edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

B

Tirthankar Bhattacharyya [email protected] Shubham Rastogi [email protected] U. Vijaya Kumar [email protected]

1

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Published online: 18 July 2022

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positive and not zero), the spectrum σ (V ) is the closed unit disc D = {z ∈ C : |z| ≤ 1}. The situation for a pair of commuting isometries is vastly different. The topic of commuting isometries has been vigorously pursued in the last two decades, see [1, 2, 4–6, 11, 15, 16, 18, 20, 21] and the references therein. In [6] and [5], the novel idea of using graphs has led to a clear understanding of structures. The defect operator C(V1 , V2 ) is introduced in [12] and [13], as C(V1 , V2 ) = I − V1 V1∗ − V2 V2∗ + V1 V2 V2∗ V1∗ . In [13] and [16], the authors provide the characterization of (V1 , V2 ), when the defect is positive, negative or zero. It is well known (see [11, 13]) that a pair has positive defect if and only if it is doubly commuting, and it has negative defect if and only if it is dual doubly commuting. In all the three cases, the defect is either a projection or negative of a projection. In general the defect is the difference of two projections; see [16]. In this paper we study the pairs of commuting isometries, whose defect is the difference of two mutually orthogonal projections. We characterize such pairs in Theorem 2.1 and we classify them in Table 1. We also provide the characterization for a few cases in Table 1; see Lemma 6.3 and Lemma 6.9. We rephrase the structure of (V1 , V2 ) in each case, which appears in Table 1, in a unified approach using the Berger-CoburnLebow (BCL) Theorem. The joint spectrum is studied in detail for all the cases except the last one appearing in Table 1. There is the related concept of the fringe operators: F1 : ker V1∗ → ker V1∗

and

F2 : ker V2∗ → ker V2∗

F1 (x) = Pker V1∗ V2 (x)

and

F2 (x) = Pker V2∗ V1 (x).

defined by, (1.1)

In various characterizations of Table 1, we shall point out the criteria in terms of the fringe operators for possible use in examples. 1.1 The Joint Spectrum If (T1 , T2 ) is a pair of commuting bounded operators on H, then for defining (see [14, 22]) the Taylor joint spectrum σ (T1 , T2 ), one considers the Koszul complex K (T1 , T2 ): δ0

δ1

δ2

δ3

0→H→H⊕H→H→0

(1.2)

where δ1 (h) = (T1 h, T2 h) for h ∈ H and δ2 (h 1 , h 2 ) = T1 h 2 − T2 h 1 for h 1 , h 2 ∈ H. From the way the complex is constructed, ran δn−1 ⊆ ker δn . When ran δn−1 = ker δn for all n = 1, 2, 3 we say that the Koszul complex K (T1 , T2 ) is exact or the pair (T1 , T2 ) is non-singular. A pair (λ1 , λ2 ) ∈ C2 is said to be in the joint spectrum σ (T1 , T2 ) if

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the pair (T1 − λ1 I , T2 − λ2 I ) is singular. In the case of a singular pair, we say that the non-singularity breaks at the stage n if ran δn−1 = ker δn . Observe that the non-singularity breaks at stage 1 if and only if (λ1 , λ2 ) is a joint eigenvalue for (T1 , T2 ) and the non-singularity breaks at stage 3 if and only if the joint range of (T1 −λ1 I , T2 −λ2 I ) is not the whole space H. If (λ1 , λ2 ) is a joint eigenvalue of (T1∗ , T2∗ ), then by the fact that ran T1 + ran T2 = H implies ker T1∗ ∩ ker T2∗ = {0}, the non-singularity of the Koszul complex K (T1 − λ1 I , T2 − λ2 I ) breaks at stage 3. There are a few elementary results which we record as a lemma so that we can refer to it later. Lemma 1.1 Let H and K be two non-zero Hilbert spaces. Let (T1 , T2 ) be a pair of commuting bounded operators on H. (1) σ (T1 , T2 ) ⊆ σ (T1 ) × σ (T2 ). (2) If there is a non-trivial joint reducing subspace H0 for (T1 , T2 ), i.e., if H = H0 ⊕ H0⊥ and  Ti =

H0 H0⊥  Ti0 0 H0 0 Ti1 H0⊥

then σ (T1 , T2 ) = σ (T10 , T20 ) ∪ σ (T11 , T21 ). (3) (z 1 , z 2 ) ∈ σ (T1 , T2 ) if and only if (z 1 , z 2 ) ∈ σ (T1∗ , T2∗ ). (4) σ (IK ⊗ T1 , IK ⊗ T2 ) = σ (T1 , T2 ) = σ (T1 ⊗ IK , T2 ⊗ IK ). (5) Let (S1 , S2 ) be a pair of commuting bounded operators on a Hilbert space K. If (T1 , T2 ) is jointly unitarily equivalent to (S1 , S2 ), then σ (T1 , T2 ) = σ (S1 , S2 ). (6) For any T in B(H) and S in B(K), the joint spectrum σ (T ⊗ IK , IH ⊗ S) is the Cartesian product σ (T ) × σ (S). Thus, for commuting isometries V1 and V2 , we have σ (V1 , V2 ) ⊆ D2 . The joint spectrum of a pair of commuting unitary operators is contained in the torus T2 , where T = {z ∈ C : |z| = 1} is the unit circle in the complex plane. This note uses the fundamental pairs of isometries consisting of multiplication operators to describe the structure of (V1 , V2 ). These are fundamental in the sense that in each case the sign of the defect operator is dictated by the fundamental pair alone. When the defect operator C(V1 , V2 ) of two commuting isometries is positive or negative, but not zero, then the whole space H breaks into a direct sum of reducing subspaces H = H0 ⊕ H0⊥ in the style of Wold where the restriction of (V1 , V2 ) on the H0 part is the fundamental pair and the restriction of (V1 , V2 ) on the H0⊥ part has defect zero. What the fundamental pair is depends on whether C(V1 , V2 ) is positive or negative. These are the contents of Theorem 4.11 and Theorem 5.10. The fundamental pairs are such that in both cases (of C(V1 , V2 ) positive or negative), the joint spectrum of (V1 , V2 ) is the whole closed bidisc D2 . These are done in Theorem

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4.11 and Theorem 5.11. If the defect operator C(V1 , V2 ) is zero, the joint spectrum of (V1 , V2 ) is contained in the topological boundary of the bidisc. The structure theorem in the case ran V1 = ran V2 , shows that (V1 , V2 ) is the direct sum of a prototypical pair (see Sect. 6.1.1) and a pair of commuting unitaries. The joint spectrum is computed. The joint spectrum of the prototypical pair of this case, is neither the closed bidisc nor contained inside the topological boundary of the bidisc. In the case ran V2  ran V1 , the joint spectrum σ (V1 , V2 ) ⊆ {(z 1 , z 1 z 2 ) : z 1 , z 2 ∈ D}. The above inclusion is sharp; see Example 6.12, and it can be a strict inclusion; see Example 6.13. Note that {(z 1 , z 1 z 2 ) : z 1 , z 2 ∈ D} has measure non-zero and it is not equal to the closed bidisc. In each case above except the case ran V2  ran V1 , we point out the stage of the Koszul complex where non-singularity is broken. 1.2 The Berger-Coburn-Lebow Theorem For a Hilbert space E, the Hardy space of E-valued functions on the unit disc in the complex plane is  HD2 (E)

=

f : D → E | f is analytic and f (z) =

∞ 

n

an z with

n=0

∞ 

 an E < ∞ . 2

n=0

Here the an are from E. This is a Hilbert space with the inner product ∞  n=0

an z , n

∞ 

 bn z

n

∞  an , bn E =

n=0

n=0

and is identifiable with HD2 ⊗ E where HD2 stands for the Hardy space of scalar-valued functions on D. We shall use this identification throughout the paper, often without any further mention, and Mz denotes the multiplication operator by the coordinate function z on HD2 . For λ ∈ D, let kλ be the function in HD2 given by kλ (z) =

∞ 

n

zn λ =

n=0

1 1 − zλ

.

It is well-known that the span of {kλ : λ ∈ D} is dense in HD2 . The space of B(E)-valued bounded analytic functions on D will be denoted by HD∞ (B(E)). Naturally, if ϕ ∈ HD∞ (B(E)), then it induces a multiplication operator Mϕ on HD2 (E). One of the main tools for us is the Berger-Coburn-Lebow (BCL) theorem [3]. Theorem 1.2 Let (V1 , V2 ) be a commuting pair of isometries acting on H. Then, up to unitary equivalence, the Hilbert space H breaks into a direct sum of reducing subspaces H = H p ⊕ Hu such that

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(1) There is a unique (up to unitary equivalence) triple (E, P, U ) where E is a Hilbert space, P is a projection on E and U is a unitary on E such that H p = HD2 (E), the functions ϕ1 and ϕ2 defined on D by ϕ1 (z) = U ∗ (P ⊥ + z P) and ϕ2 (z) = (P + z P ⊥ )U ,

(1.3)

are commuting multipliers in HD∞ (B(E)) and (V1 |H p , V2 |H p ) is equal to (Mϕ1 , Mϕ2 ). (2) V1 |Hu and V2 |Hu are commuting unitary operators. The result of the theorem above will be called the BCL representation of (V1 , V2 ). Using Theorem 1.2 we can compute the defect operator (see [13]), because Mϕ1 = I H 2 ⊗ U ∗ P ⊥ + Mz ⊗ U ∗ P and Mϕ2 = I H 2 ⊗ PU + Mz ⊗ P ⊥ U . Hence D

D

C(Mϕ1 , Mϕ2 ) = (I − Mz Mz∗ ) ⊗ (U ∗ PU − P) = E 0 ⊗ (U ∗ PU − P) where E 0 is the one dimensional projection onto the space of constant functions in HD2 . Together with the fact that the defect operator of a pair of commuting unitary operators is zero, this means, in the decomposition H = H p ⊕ Hu with H p = HD2 (E), C(V1 , V2 ) = (E 0 ⊗ (U ∗ PU − P)) ⊕ 0.

(1.4)

Definition 1.3 (1) A BCL triple (E, P, U ) is a Hilbert space E, along with a projection P and a unitary U . It is said to be the BCL triple for the pair of commuting isometries (V1 , V2 ) if E, P and U are as in Theorem 1.2, part (1). (2) Given a BCL triple (E, P, U ), the functions ϕ1 , ϕ2 : D → B(E) will always be as defined in (1.3). (3) A pair of commuting isometries (V1 , V2 ) is called pure if Hu = {0} in its BCL representation. In fact, Hu is the unitary part of the product V = V1 V2 in its Wold decomposition and E = ker V ∗ . Thus Hu = {0} if and only if V is pure. The Berger-Coburn-Lebow theorem has an interesting consequence in the case when the part H p is non-zero. We have ϕ1 (z)ϕ2 (z) = z for every z ∈ D and hence Mϕ1 Mϕ2 = Mz ⊗ IE . Hence, by the spectral mapping theorem for joint spectra (see for example [8]), {z 1 z 2 : (z 1 , z 2 ) ∈ σ (Mϕ1 , Mϕ2 )} = σ (Mz ⊗ IE ) = D. Thus, if H p = {0}, then σ (Mϕ1 , Mϕ2 ) cannot be contained in the torus T2 . Hence by Lemma 1.1 part (2), σ (V1 , V2 ) cannot be contained in the torus T2 . Let (E, P, U ) be a BCL triple. It is easy to see that if the Koszul complex K (ϕ1 (z)− λ1 I , ϕ2 (z)−λ2 I ) breaks at stage 3 for some z ∈ D, then the Koszul complex K (Mϕ1 −

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λ1 I , Mϕ2 −λ2 I ) breaks at stage 3. More generally, we show that, in all the cases under consideration in this note except the case ran V2  ran V1 , we have ∪z∈D σ (ϕ1 (z), ϕ2 (z)) = σ (Mϕ1 , Mϕ2 ).

(1.5)

See Theorem 3.8, Theorem 4.15, Theorem 5.13 and Theorem 6.8. In the following, when we consider the pair (V1 , V2 ) of commuting isometries, V denotes the product V1 V2 , and {en : n ∈ Z} denotes the standard orthonormal basis of l 2 (Z). We end this section with the comment that one of the strong points of the BCL theorem is that it models the Vi in terms of functions of one variable whereas Vi could, a priori, be dependent on two variables (multipliers on the Hardy space of the bidisc, for example). This strength will be greatly exploited in this note.

2 The Defect Operator Recall that ([12] and [13]) the defect operator of a pair of commuting isometries (V1 , V2 ) is defined as C(V1 , V2 ) = I − V1 V1∗ − V2 V2∗ + V1 V2 V2∗ V1∗ .

(2.1)

It is easy to see that (see [16]) the defect C(V1 , V2 ) = Pker V1∗ − PV2 (ker V1∗ ) = Pker V2∗ − PV1 (ker V2∗ ) ,

(2.2)

ker V1∗ ⊕ V1 (ker V2∗ ) = ker V2∗ ⊕ V2 (ker V1∗ ) = ker V1∗ V2∗ .

(2.3)

and

Note that the defect operator lives on ker V ∗ in the sense that the defect operator is zero on the orthogonal component of ker V ∗ . Equation (2.2) shows that the defect is always a difference of two projections. Let P1 = Pker V1∗

and

P2 = PV2 (ker V1∗ ) .

(2.4)

Define H1 := ran P1 ∩ ker P2 = ker V1∗ ∩ ker V2∗ ,

H2 := ran P2 ∩ ker P1 = V1 (ker V2∗ ) ∩ V2 (ker V1∗ ), H3 := ran P1 ∩ ran P2 = ker V1∗ ∩ V2 (ker V1∗ ), H4 := ker P1 ∩ ker P2 = V1 (ker V2∗ ) ∩ ker V2∗ .

Notice that Hi ⊥ H j if i = j, and the Hi are reducing for P1 and P2 and H1 ⊕ H2 ⊕ H3 ⊕ H4 ⊆ ker V ∗ .

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Theorem 2.1 The following are equivalent: (a) (b) (c) (d) (e)

The defect C(V1 , V2 ) is the difference of two mutually orthogonal projections. ker V ∗ = H1 ⊕ H2 ⊕ H3 ⊕ H4 . ker V1∗ = H1 ⊕ H3 . V1 (ker V2∗ ) = H2 ⊕ H4 . If (E, P, U ) is the BCL triple for (V1 , V2 ), U ∗ (ran P) = (U ∗ (ran P) ∩ ran P) ⊕ (U ∗ (ran P) ∩ ran P ⊥ ).

(2.6)

Proof Suppose C(V1 , V2 ) = Q 1 − Q 2

with

ran Q 1 ⊥ ran Q 2

(2.7)

for some projections Q 1 , Q 2 in ker V ∗ . Notice that the pair (Q 1 , Q 2 ) satisfying (2.7) is unique if it exists. Then for such a pair (V1 , V2 ) we have ⎛

⎞ Iran Q 1 0 0 C(V1 , V2 ) = ⎝ 0 −Iran Q 2 0⎠ = P1 − P2 . 0 0 0

(2.8)

For any two projections P and Q in H and x ∈ H, P x − Qx = x implies that P x = x and Qx = 0. Using this fact and (2.8) we see that: H1 = ran Q 1

H2 = ran Q 2 .

and



 IK 0 Note that P1 = P2 = on K ⊕ L for some K, L that satisfy ker V ∗ = 0 0L H1 ⊕ H2 ⊕ K ⊕ L. Hence, by (2.8) ran P1 = H1 ⊕ K and ran P2 = H2 ⊕ K. Thus K = ran P1 ∩ ran P2 = H3 . Similarly L = ker P1 ∩ ker P2 = H4 . Therefore ker V ∗ = H1 ⊕ H2 ⊕ H3 ⊕ H4 , and in this decomposition ⎛

I ⎜0 ⎜ P1 = ⎝ 0 0

0 0 0 0

0 0 I 0

⎞ 0 0⎟ ⎟, 0⎠ 0

⎛ 0 ⎜0 ⎜ P2 = ⎝ 0 0

0 I 0 0

0 0 I 0

⎞ 0 0⎟ ⎟. 0⎠ 0

(2.9)

This proves (a) ⇒ (b), and (b) ⇒ (a) is trivial. Note the fact that, if E1 , E2 , E3 and E4 are any subspaces of E satisfying E1 ⊕ E2 = E = E3 ⊕ E4 , then E1 = (E1 ∩ E3 ) ⊕ (E1 ∩ E4 ) if and only if E2 = (E2 ∩ E3 ) ⊕ (E2 ∩ E4 ). Using the above fact the equivalence (c) ⇐⇒ (d) follows from (2.3). The implication (b) ⇒ (c) is clear. The implication (c) ⇒ (b) follows from the equivalence of (c) and (d) and (2.3).

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The status of (V1 , V2 )

BCL characterization

Hi = 0

C(V1 , V2 ) = 0 and both V1 and V2 are unitaries

E =0

Hi = 0 for i = 1, 2, 4, H3 = 0

C(V1 , V2 ) = 0 and V2 is a unitary, V1 is not a unitary

P=I

Hi = 0 for i = 1, 2, 3, H4 = 0

C(V1 , V2 ) = 0 and V1 is a unitary, V2 is not a unitary

P=0

Hi = 0 for i = 1, 2, Hi = 0 for i = 3, 4

C(V1 , V2 ) = 0 and both V1 and V2 are not unitaries

P is non-trivial and U reduces ran P

for

i = 1, 2, 3, 4

H1 = 0, Hi = 0 for i = 2, 3, 4

C(V1 , V2 ) ≤ 0 and C(V1 , V2 ) = 0

U (ran P ⊥ )  ran P ⊥

H2 = 0, Hi = 0 for i = 1, 3, 4

C(V1 , V2 ) ≥ 0 and C(V1 , V2 ) = 0

U (ran P)  ran P

Hi = 0 for i = 3, 4, Hi = 0 for i = 1, 2

ran V1 = ran V2 and Vi is not a unitary

P is non-trivial and U (ran P) = ran P ⊥

H4 = 0, Hi = 0 for i = 1, 2, 3

ran V1  ran V2 and V2 is not unitary

P = I and U (ran P ⊥ )  ran P

H3 = 0, Hi = 0 for i = 1, 2, 4

ran V2  ran V1 and V1 is not unitary

P = 0 and U (ran P)  ran P ⊥

Hi = 0 for i = 1, 2, 3, 4

Unknown

—

Note that ker Mϕ∗1 = ran(I − Mϕ1 Mϕ∗1 ) = 1 ⊗ ran(U ∗ PU ) = 1 ⊗ U ∗ (ran P), ker Mϕ∗2 = 1⊗ran P ⊥ and Mϕ2 (ker Mϕ∗1 ) = 1⊗ran P. Now the proof of (a) ⇐⇒ (e) follows from the equivalence (a) ⇐⇒ (c).   Using the (e) part of the above theorem we easily get an example of a pair (V1 , V2 ) whose defect is not a difference of two mutually orthogonal projections.     1 1 10 and P = . The BCL triple Example 2.2 Let E = C2 , U = √1 2 −1 1 00 (E, P, U ) does not satisfy (2.6), hence the defect of the pair (V1 , V2 ) corresponding to this BCL triple, is not the difference of two mutually orthogonal projections. The Table 1 gives a neat classification. We leave the proofs to the reader for the first two columns. The contents of the third column will unfold as we progress. One can notice that certain cases are not mentioned in the table. That is because those cases cannot occur.

3 The Zero Defect Case 3.1 Structure This subsection is mainly a rephrasing of known results. If one of the Vi ’s is a unitary, then it is trivial to check that the defect C(V1 , V2 ) is zero. The following example is

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the prototypical example of a pure pair of commuting isometries with defect zero; this example serves as a building block in the general structure, see Theorem 3.4. Example 3.1 Let L be a non-zero Hilbert space and W be a unitary on L. Consider the commuting pair of isometries (Mz ⊗ IL , I H 2 ⊗ W ) on HD2 ⊗ L. As I ⊗ W is a unitary, D

the defect C(Mz ⊗ IL , I H 2 ⊗ W ) = 0. Also, σ (Mz ⊗ IL , I H 2 ⊗ W ) = D × σ (W ), D D by Lemma 1.1 part (6). Also see [7]. Now consider the unitary  : HD2 ⊗ L → HD2 ⊗ L given by  

∞  m=0

 am z

m

=

∞ 

W m (am )z m .

(3.1)

m=0

Then, (Mz ⊗ W ∗ )∗ = Mz ⊗ I and (I ⊗ W )∗ = I ⊗ W . This says that (Mz ⊗ W ∗ , I ⊗ W ) and (Mz ⊗ I , I ⊗ W ) are jointly unitarily equivalent. In particular, we have C(Mz ⊗ W ∗ , I ⊗ W ) = 0 and σ (Mz ⊗ W ∗ , I ⊗ W ) = D × σ (W )

(3.2)

for any unitary W . We proceed towards the structure of an arbitrary pair (V1 , V2 ) with C(V1 , V2 ) = 0. A pair of commuting isometries (V1 , V2 ) is called doubly commuting if V1 commutes with V2∗ . The following lemma is proved in [13] and [16], it is relating positivity of the defect operator C(V1 , V2 ) with double commutativity of the pair (V1 , V2 ). We give a short proof here. Lemma 3.2 Let (V1 , V2 ) be a pair of commuting isometries on a Hilbert space H. Then the following are equivalent: (a) C(V1 , V2 ) ≥ 0. (b) (V1 , V2 ) is doubly commuting. (c) If (E, P, U ) is the BCL triple for (V1 , V2 ), then U (ran P) ⊆ ran P. Proof Since commuting unitaries are always doubly commuting, it is enough to prove the equivalences when (V1 , V2 ) = (Mϕ1 , Mϕ2 ). By virtue of (1.4), we have C(V1 , V2 ) ≥ 0 if and only if U ∗ PU ≥ P which happens if and only if ran P is invariant under U . Now, Mϕ1 Mϕ∗2 = Mϕ∗2 Mϕ1 if and only if (I − Mz Mz∗ ) ⊗ (U ∗ PU ∗ P ⊥ ) = 0 if and only if P ⊥ U P = 0 if and only if ran P is invariant under U .  

This completes the proof. We recall some characterization results from [13] and add a few new ones.

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Lemma 3.3 Let (V1 , V2 ) be a pair of commuting isometries on a Hilbert space H. Then the following are equivalent: (a) C(V1 , V2 ) = 0. (b) ker V2∗ is a reducing subspace for V1 and V1 |ker V2∗ is a unitary. (c) ker V1∗ is a reducing subspace for V2 and V2 |ker V1∗ is a unitary. (d) The fringe operators F1 and F2 are unitaries. (e) ker V1∗ and ker V2∗ are orthogonal and their direct sum is ker V ∗ . (f) (ran V1  ran V ) ⊕ (ran V2  ran V ) ⊕ ran V = H. (g) If (E, P, U ) is the BCL triple for (V1 , V2 ), then ran P reduces U . Proof The equivalences of (a), (b), (c), (d) and (e) follows easily from (2.2) and (2.3). (e) ⇒ ( f ): Suppose (e) is true. We shall show that ker V1∗ = (ran V2  ran V ). Suppose x ∈ ker V1∗ , which implies x ∈ ran V2 and x ∈ (ran V )⊥ . So x ∈ (ran V2  ran V ). If x ∈ (ran V2  ran V ), then x ∈ ran V2 and x ∈ ker V ∗ . So x ∈ ker V1∗ . Similarly, ker V2∗ = ran V1  ran V . Hence ( f ) is true. ( f ) ⇒ (e): Suppose ( f ) is true. Since ran V ⊆ ran V1 , we have ran V1 = (ran V1  ran V ) ⊕ ran V . Therefore ker V1∗ = (ran V1 )⊥ = (ran V2  ran V ). Similarly, ker V2∗ = (ran V1  ran V ). Hence ker V1∗ ⊕ ker V2∗ = ker V ∗ . (a) ⇔ (g): We use the formula C(V1 , V2 ) = (E 0 ⊗ (U ∗ PU − P)) ⊕ 0 from (1.4). This gives C(V1 , V2 ) = 0 if and only if U ∗ PU − P = 0 if and only if ran P reduces U .  

Thus, completes the proof.

We now write the structure theorem given in Popovici [18, Sec. 4], which highlights the importance of Example 3.1. We give a proof for the completeness. Theorem 3.4 Let (V1 , V2 ) be a pair of commuting isometries on H with defect zero. Let (E, P, U ) be the BCL triple for (V1 , V2 ). Let E1 = ran P and E2 = ran P ⊥ . Then E1 , E2 are reducing subspaces for U , i.e.,     I E1 0 U1 0 in B(E1 ⊕ E2 ) and P = E = E1 ⊕ E2 , U = 0 U2 0 0 for some unitaries U1 and U2 on E1 and E2 respectively.  ran(V1 V2 )n and in this Also, H = (HD2 ⊗ E1 ) ⊕ (HD2 ⊗ E2 ) ⊕ K where K = n≥0

decomposition, ⎛

⎛ ⎞ ⎞ M z ⊗ I E1 I H 2 ⊗ U1 0 0 0 0 D ∗ 0 I H 2 ⊗ U2 0 ⎠ , V2 = ⎝ V1 = ⎝ 0 M z ⊗ I E2 0 ⎠ , D 0 0 W1 0 0 W2 up to unitarily equivalence, for some unitary Ui on Ei , i = 1, 2 and commuting unitaries W1 , W2 on K.

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Proof Let (E, P, U ) be the BCL triple for (V1 , V2 ). By Lemma 3.3, we have E1 reduces U and in the decomposition E = E1 ⊕ E2 , we have     U1 0 I 0 . U= and P = E1 0 U2 0 0  In this case, ϕ1 (z) =  M ϕ1 =

   zU1∗ 0 U1 0 and ϕ , for z ∈ D. Therefore, (z) = 2 0 U2∗ 0 zU2

   0 0 Mz ⊗ U1∗ I H 2 ⊗ U1 D . 0 I H 2 ⊗ U2∗ , Mϕ2 = 0 M z ⊗ U2 D

Note that (Mz ⊗U1∗ , I H 2 ⊗U1 ) and (Mz ⊗IE1 , I H 2 ⊗U1 ) are jointly unitarily equivalent D D and (I H 2 ⊗ U2∗ , Mz ⊗ U2 ) and (I H 2 ⊗ U2∗ , Mz ⊗ IE2 ) are jointly unitarily equivalent; D D see Example 3.1. This completes the proof, by Theorem 1.2.   Remark 3.5 In the structure theorem (Theorem 3.4) one can write the U1 , U2 and E1 , E2 explicitly in terms of V1 and V2 as follows: E1 = ker V1∗ , E2 = ker V2∗ and U1 = V2 |ker V1∗ , U2 = V1∗ |ker V2∗ . This is because of Lemma 3.3. A comment is in order. Over several decades Marek Słoci´nski, first by himself and then with his collaborators, has developed a complete structure theorem on commuting pairs of isometries; see [5] and the references therein. One of his early results is the following. Theorem 3.6 (M. Słoci´nski [21]) Let (V1 , V2 ) be a pair of doubly commuting isometries on a Hilbert space H. Then there exists a unique decomposition H = Hss ⊕ Hsu ⊕ Hus ⊕ Huu , where the subspace Hi j reduces both V1 and V2 for all i, j ∈ {s, u}. Moreover, V1 on Hi j is a shift if i = s and unitary if i = u and V2 is a shift if j = s and unitary if j = u. By Theorem 3.4, and the fact that if one of the Vi ’s is a unitary then the defect is zero, we have C(V1 , V2 ) = 0 if and only if (V1 , V2 ) is doubly commuting and Hss in Theorem 3.6 is {0}. 3.2 Joint Spectrum If (V1 , V2 ) is pure and defect C(V1 , V2 ) = 0, then by Theorem 3.4 and Remark 3.5, ⎧ ⎪ if V2 is unitary, ⎨D × σ (U1 ) ∗ σ (V1 , V2 ) = σ (U2 ) × D (3.3) if V1 is unitary, ⎪ ⎩ ∗ D × σ (U1 ) ∪ σ (U2 ) × D if neither V1 nor V2 is a unitary, where U1 = V2 |ker V1∗ , U2 = V1∗ |ker V2∗ . Reprinted from the journal

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Lemma 3.7 Let (V1 , V2 ) be a pair of commuting isometries with defect zero and ker V ∗ = {0}. Let (E, P, U ) be the BCL triple for (V1 , V2 ). Let U1 = U |ran P and U2 = U |ran P ⊥ . Then ⎧ ⎪ if V2 is a unitary, ⎨{(zλ, λ) : λ ∈ σ (U1 )} σ (ϕ1 (z), ϕ2 (z)) = {(μ, zμ) : μ ∈ σ (U2 )} if V1 is a unitary, ⎪ ⎩ {(zλ, λ), (μ, zμ) : λ ∈ σ (U1 ), μ ∈ σ (U2 )} otherwise.

(3.4) Here, for every point in the joint spectrum, the non-singularity breaks at stage 3. Proof The proof in the case when neither V1 nor V2 is a unitary is done below in detail. The other cases follow similarly. Letting E1 = ran P and E2 = ran P ⊥ , it is an easy check that both E1 and E2 are non-trivial. By Lemma 3.3, E1 reduces U . Hence in the decomposition E = E1 ⊕ E2 , we have  ϕ1 (z) =

zU1∗ 0 0 U2∗



  U1 0 , 0 zU2

and

ϕ2 (z) =

and

σ (ϕ2 (z)) = σ (U1 ) ∪ σ (zU2 )

for z ∈ D. Then, σ (ϕ1 (z)) = σ (zU1∗ ) ∪ σ (U2∗ ) for all z ∈ D. For z ∈ D, we have ϕ1 (z)ϕ2 (z) = z IE1 ⊕E2 = ϕ2 (z)ϕ1 (z). Therefore, for all z = 0, by Lemma 1.1 part (1) and polynomial spectral mapping theorem, we get σ (ϕ1 (z), ϕ2 (z)) ⊆ {(zλ, λ), (μ, zμ) : λ ∈ σ (U1 ), μ ∈ σ (U2 )}. For z = 0, it is easy to see that σ (ϕ1 (0), ϕ2 (0)) ⊆ {(0, λ), (μ, 0) : λ ∈ σ (U1 ), μ ∈ σ (U2 )}. To prove the other containments, λ ∈ σ (U1 ). Then U1 − λI is not onto, because let    k1 h1 ,k = ∈ E1 ⊕ E2 . Then, U1 is normal. Let h = h2 k2  (ϕ1 (z) − zλI )k + (ϕ2 (z) − λI )h =

 z(U1∗ − λI )k1 + (U1 − λI )h 1 . (U2∗ − zλI )k2 + (zU2 − λI )h 2

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Since U1 is a normal operator, ran(U1 − λI ) = ran(U1∗ − λI ) and hence the first component of the above spans only ran(U1 − λI ). Hence we have ran(ϕ1 (z) − zλI ) + ran(ϕ2 (z) − λI ) = E1 ⊕ E2 . Therefore (zλ, λ) ∈ σ (ϕ1 (z), ϕ2 (z)). Similarly, if μ ∈ σ (U2 ), we have ran(ϕ1 (z) − μI ) + ran(ϕ2 (z) − zμI ) = E1 ⊕ E2 . Therefore (μ, zμ) ∈ σ (ϕ1 (z), ϕ2 (z)). So, σ (ϕ1 (z), ϕ2 (z)) = {(zλ, λ), (μ, zμ) : λ ∈ σ (U1 ), μ ∈ σ (U2 )} for z ∈ D.   Theorem 3.8 With the hypothesis of Lemma 3.7, we also have ⎧ ⎪ if V2 is a unitary, ⎨D × σ (U1 ) σ (Mϕ1 , Mϕ2 ) = ∪z∈D σ (ϕ1 (z), ϕ2 (z)) = σ (U2∗ ) × D if V1 is a unitary, ⎪ ⎩ D × σ (U1 ) ∪ σ (U2∗ ) × D otherwise.

⎧ ⎪ if V2 is a unitary, ⎨D × σ (U1 ) ∗ and, for every point (z 1 , z 2 ) ∈ σ (U2 ) × D if V1 is a unitary, the ⎪ ⎩ D × σ (U1 ) ∪ σ (U2∗ ) × D otherwise, non-singularity of K (Mϕ1 − z 1 I , Mϕ2 − z 2 I ) breaks at stage 3. Proof As in Lemma 3.7, we prove only the case when neither V1 nor V2 is a unitary, other cases follow similarly. We saw in the proof of Lemma 3.7 that for any point z ∈ D, a pair of points (z 1 , z 2 ) ∈ σ (ϕ1 (z), ϕ2 (z)) if and only if ran(ϕ1 (z) − z 1 I ) + ran(ϕ2 (z) − z 2 I ) = E1 ⊕ E2 . Hence, if (z 1 , z 2 ) ∈ σ (ϕ1 (z), ϕ2 (z)), ran(Mϕ1 − z 1 I ) + ran(Mϕ2 − z 2 I ) = H 2 (E1 ⊕ E2 ). Therefore, (z 1 , z 2 ) ∈ σ (Mϕ1 , Mϕ2 ), which implies that ∪z∈D σ (ϕ1 (z), ϕ2 (z)) ⊆ σ (Mϕ1 , Mϕ2 ). Note that ∪z∈D σ (ϕ1 (z), ϕ2 (z)) = ∪z∈D {(zλ, λ), (μ, zμ) : λ ∈ σ (U1 ), μ ∈ σ (U2 )}

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(3.5)

T. Bhattacharyya et al.

= D × σ (U1 ) ∪ σ (U2∗ ) × D.

(3.6)

Since  M ϕ1 =

0 Mz ⊗ U1∗ 0 I H 2 ⊗ U2∗



 and

M ϕ2 =

D

 0 I H 2 ⊗ U1 D , 0 M z ⊗ U2

we have σ (Mϕ1 , Mϕ2 ) = σ (Mz ⊗ U1∗ , I H 2 ⊗ U1 ) ∪ σ (I H 2 ⊗ U2∗ , Mz ⊗ U2 ). D

D

By (3.2), we have σ (Mϕ1 , Mϕ2 ) = D × σ (U1 ) ∪ σ (U2∗ ) × D.

(3.7)

Therefore from (3.6) and (3.7) we have σ (Mϕ1 , Mϕ2 ) = D × σ (U1 ) ∪ σ (U2∗ ) × D = ∪z∈D σ (ϕ1 (z), ϕ2 (z)). The final thing to note is that for every point (z 1 , z 2 ) in the set D×σ (U1 )∪σ (U2∗ )×D, the non-singularity breaks at stage 3 and this is a direct consequence of (3.5).   To conclude the section, we note that by Theorem 1.2, σ (V1 , V2 ) = σ (Mϕ1 , Mϕ2 )∪ σ (V1 |Hu , V2 |Hu ). Hence Theorem 3.8 tells that ∪z∈D σ (ϕ1 (z), ϕ2 (z)) ⊆ σ (V1 , V2 ).

(3.8)

The equality in (3.8) holds if and only if σ (V1 |Hu , V2 |Hu ) ⊆ σ (Mϕ1 , Mϕ2 ).

4 The Negative Defect Case 4.1 The Prototypical Example When the defect operator is negative, a fundamental example plays an important role in much the same way the unilateral shift plays its role in the Wold decomposition of a single isometry. We shall first describe this example and then show how it is a part of every pair of commuting isometries with negative defect. The Hardy space of E-valued functions on the bidisc D2 is HD2 2 (E) =

⎧ ⎨ ⎩

f : D2 → E | f is analytic and f (z 1 , z 2 ) =

⎫ ∞ ⎬  with am,n 2E < ∞ . ⎭

∞ 

am,n z 1m z 2n

m,n=0

m,n=0

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This is a Hilbert space with the inner product 

∞ 

am,n z 1m z 2n ,

m,n=0

∞ 

 bm,n z 1m z 2n

m,n=0

=

∞ 

am,n , bm,n E

m,n=0

and is identifiable with HD2 2 ⊗E where HD2 2 stands for the Hardy space of scalar-valued functions on D2 . Let U : HD2 2 → HD2 2 be the unitary defined by ⎧ m +2 m 1 ⎪ z2 2 if m 1 ≥ m 2 , ⎨z 1 m1 m2 m 1 +1 m 2 −1 U (z 1 z 2 ) = z 1 z2 if m 1 + 1 = m 2 , ⎪ ⎩ m 1 m 2 −2 if m 1 + 2 ≤ m 2 . z1 z2

(4.1)

on the orthonormal basis {z 1m 1 z 2m 2 }m 1 ,m 2 ≥0 . The pair of multipliers by the coordinate functions (Mz 1 , Mz 2 ) forms a pair of doubly commuting isometries on HD2 2 . There is a natural isomorphism between the Hilbert spaces HD2 2 and HD2 ⊗ HD2 wherein z 1m 1 z 2m 2 is identified with z 1m 1 ⊗ z 2m 2 . In this identification, the pair of coordinate multipliers (Mz 1 , Mz 2 ) is identified with (Mz ⊗ I , I ⊗ Mz ). Definition 4.1 The pair of bounded operators τ1 := U ∗ Mz 1 and τ2 := Mz 2 U on the Hardy space of the bidisc HD2 2 will be called the fundamental isometric pair with negative defect. The following lemma justifies the name except the word f undamental which will be clear from Theorem 4.11. Lemma 4.2 The pair (τ1 , τ2 ) is a pair of commuting isometries with defect negative and non-zero. Proof It is simple to check that the unitary U defined in (4.1) commutes with Mz 1 z 2 , the operator of multiplication by the function z 1 z 2 . That proves commutativity of τ1 and τ2 . They are isometries because each is a product of an isometry and a unitary. Now, let W1 = ker(τ1∗ ) = span{z 22 , z 23 , z 24 , . . . }

and

W2 = ker(τ2∗ ) = span{1, z 1 , z 12 , . . . }.

Then, τ2 (W1 ) = span{z 2 , z 22 , z 23 , . . . }. Thus, we have C(τ1 , τ2 ) = PW1 − Pτ2 (W1 ) = −Pspan{z 2 } ≤ 0.  

That completes the proof. We shall prove the following lemma to compute the joint spectrum of (τ1 , τ2 ).

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Lemma 4.3 For any λ ∈ D, we have   (ran(τ2 − λI ))⊥ = (I − λτ2 )−1 x : x ∈ ker Mz∗2 .

(4.2)

 ¯ −1 = ¯n n Proof Using the Neumann series (I − λA) n≥0 λ A for λ ∈ D and any contraction A, it is straightforward that the equality (ran(A − λI ))⊥ = {(I − λ¯ A)−1 x : x ∈ ker A∗ } is satisfied when A is an isometry. The proof is complete by noting that ker τ2∗ =   ker U ∗ Mz∗2 = ker Mz∗2 . Recall the Koszul complex for a pair of commuting bounded operators (T1 , T2 ) from (1.2): δ0

δ1

δ2

δ3

0 → H → H ⊕ H → H → 0.

(4.3)

It is well-known that the most difficult stage to treat for the purpose of showing lack of exactness is the stage 2. Proposition 4.4 The fundamental isometric pair with negative defect has the full closed bidisc D2 as its joint spectrum. Moreover, for every point in the open bidisc D2 , the non-singularity breaks at stage 2. Proof Let λ1 , λ2 ∈ D. We shall find a non-zero function h 2 ∈ (ran(τ2 − λ2 I ))⊥ such that (τ1 − λ1 I )h 2 ∈ ran(τ2 − λ2 I ).

(4.4)

This would imply that there exists h 1 ∈ H = HD2 2 such that (τ1 − λ1 I )h 2 = (τ2 − λ2 I )h 1 producing a pair (h 1 , h 2 ) in ker δ2 which would not be in ran δ1 . ))⊥ obtained in Lemma To that end, we shall use the description of (ran(τ2 − λI m ∗ ∗ 4.3. Since any element from ker τ2 = ker Mz 2 is of the form ∞ m=0 am z 1 for a square summable sequence {am }m≥0 , it follows from Lemma 4.3 that 

∞ 

(ran(τ2 − λ2 I ))⊥ = (I − λ2 τ2 )−1  =

m=0 ∞ 



(λ2 Mz 2 U )

n

n=0

=

am z 1m :

⎧ ∞ ⎨  ⎩

∞ 



am z 1m

λ2 am z 1m+2n z 2n :

m,n=0

:

∞ 

 |am | < ∞ 2

m=0

∞  m=0

170

|am |2 < ∞

m=0

m=0 n



∞ 

⎫ ⎬ |am |2 < ∞ . ⎭

(4.5)

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Our candidate for h 2 to satisfy (4.4) is h2 =

1 (1 − λ2 z 12 z 2 )(1 − λ1 z 1 )

∞ 

=

n

m+2n n λ2 λm z2 . 1 z1

m,n=0

By (4.5), this function is in (ran(τ2 − λ2 I ))⊥ . We shall verify below that (τ1 − λ1 I )h 2 is in the closure of ran(τ2 − λ2 I ). Since |λ2 | < 1 and τ2 is an isometry, (τ2 − λ2 I ) is bounded below and hence its range is closed. That will complete the proof. First note that U

∞ 

∞ 

n

λ2 am z 1m+2n z 2n =

m,n=0

n

λ2 am z 1m+2n+2 z 2n = Mz21

m,n=0

∞ 

n

λ2 am z 1m+2n z 2n

m,n=0

(4.6) and (Mz∗1 − λ1 I )h 2 = Mz∗1 (h 2 ) − λ1 h 2 ∞ 

=

n

m+2n−1 n λ2 λm z2 − 1 z1

m,n=0 (m,n) =(0,0) ∞ 

= =

n

m+2n−1 n λ2 λm z2 + 1 z1

=

∞ 

n

λ2 z 12n−1 z 2n −

n=1 n

λ2 λm+1 z 1m+2n z 2n + 1

m,n=0 ∞ 

n

λ2 λm+1 z 1m+2n z 2n 1

m,n=0

m=1,n=0 ∞ 

∞ 

∞ 

∞ 

n

λ2 λm+1 z 1m+2n z 2n 1

m,n=0 n

λ2 z 12n−1 z 2n −

n=1

∞ 

n

λ2 λm+1 z 1m+2n z 2n 1

m,n=0

n λ2 z 12n−1 z 2n .

(4.7)

n=1

We now compute the inner product between a typical element of (ran(τ2 − λ2 I ))⊥ and (τ1 − λ1 I )h 2 by using the two equations above. 

∞ 

(τ1 − λ1 I )h 2 ,

 n

λ2 am z 1m+2n z 2n

m,n=0



∞ 

∗

= (U Mz 1 − λ1 I )h 2 ,  = Mz 1 h 2 , U =

 n

λ2 am z 1m+2n z 2n

m,n=0 ∞ 



n

λ2 am z 1m+2n z 2n



∞ 

 n

λ2 am z 1m+2n z 2n

− λ1 h 2 , m,n=0 m,n=0     ∞ ∞   n n m+2n n m+2n n 2 Mz 1 h 2 , Mz 1 λ2 a m z 1 z 2 − λ1 h 2 , λ2 a m z 1 z2 m,n=0 m,n=0

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 (Mz∗1

=  =

∞  n=1

− λ1 I )h 2 ,

n λ2 z 12n−1 z 2n ,



∞ 

n

λ2 am z 1m+2n z 2n

m,n=0 ∞ 

 n

λ2 am z 1m+2n z 2n

= 0.

m,n=0

This shows that (τ1 − λ1 I )h 2 ∈ ran(τ2 − λ2 I ) = ran(τ2 − λ2 I ) and hence completes the proof.   Note 4.5 In Remark 5.4, we shall see that there is a joint invariant subspace M for (τ1 , τ2 ) such that the defect operator of (τ1 |M , τ2 |M ) is positive (and not zero). 4.2 General Theory for the Negative Defect Case Here we shall show that the fundamental example above is a typical example. This helps us to compute the joint spectrum of any commuting pair of isometries with negative defect. In [11], Gaspar and Gaspar introduced the dual doubly commuting pairs. If (V¯1 , V¯2 ) is the minimal unitary extension to H¯ of (V1 , V2 ) acting on H, then the pair of com∗ ∗ muting isometries (V¯1 |H¯ H , V¯2 |H¯ H ) is called the dual of (V1 , V2 ). If the dual is doubly commuting, then (V1 , V2 ) is called a dual doubly commuting pair. A pair (V1 , V2 ) of commuting isometries is called a bi-shift (see [19]) if there is p p q q a wandering subspace R (i.e., V1 1 V2 2 (R) ⊥ V1 1 V2 2 (R) if ( p1 , p2 ), (q1 , q2 ) ∈ Z2+ and ( p1 , p2 ) = (q1 , q2 )) such that H=



V1n 1 V2n 2 (R).

(n 1 ,n 2 )∈Z2+

(V1 , V2 ) is called a modified bi-shift if it is pure and its dual is a bi-shift. Popovici used the concepts above greatly in his papers [19] and [20]. We are thankful to him for sending us his papers. First we shall give a characterizing lemma for this case; see also [13]. Lemma 4.6 Let (V1 , V2 ) be a pair of commuting isometries on a Hilbert space H. Then the following are equivalent: (a) (b) (c) (d) (e) (f) (g) (h)

C(V1 , V2 ) ≤ 0 and C(V1 , V2 ) = 0. V2 (ker V1∗ )  ker V1∗ . V1 (ker V2∗ )  ker V2∗ . The adjoint of the fringe operators are isometries and not unitaries. (V1 , V2 ) is dual doubly commuting and C(V1 , V2 ) = 0. ker V1∗ is orthogonal to ker V2∗ and ker V1∗ ⊕ ker V2∗ = ker V ∗ . C(V1 , V2 ) is the negative of a non-zero projection. If (E, P, U ) is the BCL triple for (V1 , V2 ), then U (ran P ⊥ )  ran P ⊥ .

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Proof The equivalence (e) ⇔ (h) is proved in [11]. All other proof are along the same lines as the proofs of various parts of Lemma 3.3.   The geometrical structure and a model for dual doubly commuting isometries is known due to [11, 18]. Here we observe that the multiplication operators Mϕi , i = 1, 2 associated to (V1 , V2 ) as in Theorem 1.2, have some special forms in this case. This also helps us getting a model in the Hardy space of the bidisc. Some steps of the proof are used to obtain Theorem 4.15. The wandering space arguments used in the proof of the following theorem, appears in [18, Thm 4.3]. Let ψ1 , ψ2 : D → B(l 2 (Z)) be the multipliers associated with the BCL triple 2 (l (Z), p− , ω) where p− is the projection onto span{en : n < 0} and ω is the bilateral shift on l 2 (Z). Theorem 4.7 Let (V1 , V2 ) be a pair of commuting isometries such that C(V1 , V2 ) is a non-zero negative operator. Let (E, P, U ) be the BCL triple for (V1 , V2 ). Then, up to unitary equivalence E = (l 2 (Z) ⊗ L) ⊕ E2 for some non-trivial closed subspace L and a closed subspace E2 of E. Moreover,

Mϕi =

2 2 2  HD (l (Z)) ⊗ L HD (E2 )  Mψi ⊗ IL 0 HD2 (l 2 (Z)) ⊗ L 0 Mϕi | H 2 (E2 ) HD2 (E2 )

(4.8)

D

with C(Mϕ1 | H 2 (E2 ) , Mϕ2 | H 2 (E2 ) ) = 0. In particular, D

D

σ (Mψ1 , Mψ2 ) = σ (Mψ1 ⊗ IL , Mψ2 ⊗ IL ) ⊆ σ (Mϕ1 , Mϕ2 ) ⊆ σ (V1 , V2 ). (4.9) Proof Since C(V1 , V2 ) ≤ 0 and C(V1 , V2 ) = 0, by Lemma 4.6, U (ran P ⊥ )  ran P ⊥ . Consider L := (ran P ⊥  U (ran P ⊥ )) = {0}. Let m, n ∈ Z and m > n. Then for x, y ∈ L we have U m x, U n y = U m−n x, y = 0, because U m−n x ∈ U (ran P ⊥ ). Therefore U m (L) ⊥ U n (L) if m, n ∈ Z and m = n. Set E1 := ⊕n∈Z U n (L). Clearly U reduces E1 . Now U (ran P ⊥ )  ran P ⊥ implies: ⊕n≥0 U n (L) ⊆ ran P ⊥ .

(4.10)

For all x ∈ L, y ∈ ran P ⊥ and n < 0, U n x, y = x, U −n y = 0 implies that ⊕n in Hλ,μ (D2 ). On the zero set {(0, w2 ) : w2 ∈ D}, the frame is given by p−1

p

F1 (z, (0, w2 )) =

(λ) p z 1 , p!(1 − z 2 w¯ 2 )μ

and hence

F1 (z, (0, w2 )) 2 =

(λ) p  (μ)n |w2 |2n , p! n! n≥0

where (λ) p = λ(λ + 1) . . . (λ + p − 1) is the Pochhammer symbol. The curvature at 0 relative to this frame is ∂ ∂¯ log F1 (z, (0, w2 )) 2{w2 =0} =

μ(λ) p . p!

(5.4)

k+ p

We note that Ik =< z 1 > is contained in I for each k ∈ N. If the completion [I] of the ideal I in Hλ,μ (D2 ) is equivalent to the completion of [I] of the ideal [I]

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taken in Hλ ,μ (D2 ), then the completion [Ik ] of Ik in Hλ,μ (D2 ) is equivalent to the   completion [Ik ] of Ik in Hλ ,μ (D2 ). This follows from the fact that the kernel K [I ] p p p p and K [I ] factor: K [I ] (z, w) = z 1 χ (z, w)w¯ 1 and K [I ] (z, w) = z 1 χ  (z, w)w¯ 1 . Thus  if [I] and [I] are equivalent, then there exists a non-zero holomorphic map ϕ on D2 that induces the unitary module map between I] and [I] . Following the same idea as in the the proof of [4, Lemma 4.1], [Ik ] and [Ik ] are equivalent via the module map induced by ϕ. By considering the case k = 1, from Theorem 5.1 and the curvature computation in Eq. (5.4), it follows that μ(λ) p = μ (λ ) p and μ(λ) p+1 = μ (λ ) p+1 . Thus, we have λ = λ and μ = μ . im > Consider the submodule [I] that is the completion of the ideal I =< z 1i1 , . . . , z m λ n λ (D ) for some m < n and λ = (λ1 , . . . , λn ). The reproducing kernel of H (Dn ) in H " n (1 − z i w¯ i )λi . On the zero set {(0, , . . . , 0, wm+1 , . . . , wn ) ∈ Cn : wi ∈ is 1/ i=1 D, m + 1 ≤ i ≤ n}, the frame is given by Fk (z, (0, , . . . , 0, wm+1 , . . . , wn )) =

ik !

(λk )ik z 1ik , ¯ i )λi i=m+1 (1 − z i w

"n

and hence

Fk (z, (0, , . . . , 0, wm+1 , . . . , wn )) 2 =

ik !

(λk )ik , 2 λi i=m+1 (1 − |wi | )

"n

for 1 ≤ k ≤ m. Since Fi , F j  = 0 for i = j, the metric is a m × m diagonal matrix with diagonals Fk 2 as above. Thus, we just need to look at ∂2 log F1 (z, (0, , . . . , 0, wm+1 , . . . , wn )) 2 ∂wi ∂ w¯ i for 1 ≤ k ≤ m and m +1 ≤ i ≤ n as the mixed derivatives are zero at the origin. So the curvature computation at the origin and calculation similar to the case of the principal ideal, we have the completion [I] of the ideal I in Hλ (Dn ) and the completion [I]  in H λ (Dn ) are equivalent if and only if λ = λ .

6 On the Relationship of Curvature with the Generators Two different sets of generators for an ideal I give rise to two distinct holomorphic frames for the holomorphic hermitian vector bundle associated to the completion [I] in the Hilbert module H. The curvature computed relative to these frames are equivalenrt via an invertible map which is explicitly computed below in the case when  is a bounded domain containing the zero vector in Cm and { p1 , . . . , pt }, {q1 , . . . , qt } are two sets of generators of the polynomial ideal I consisting of homogeneous polynomials of the same degree. Let H be an analytic Hilbert module in O() and M be the closure of I in H with the property that codimV (M) = t. Then, by [3, Lemma

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4.7] we obtain that  ¯ (·, w)|w=0 , . . . , pt ( D)K ¯ (·, w)|w=0 and p1 ( D)K   ¯ (·, w)|w=0 , . . . , qt ( D)K ¯ (·, w)|w=0 q1 ( D)K



are two bases of ker D M ∗ . As a result, from [4, Lemma 2.1] and [4, Proposition 2.10] t it follows that there exists a constant invertible matrix A = (ai j )i, j=1 such that, qj =

t 

ai j pi , 1 ≤ j ≤ t.

(6.1)

i=1

Applying Theorem 4.3 separately with the choice of { p1 , . . . , pt } and {q1 , . . . , qt }, we p p q q obtain the sets {F1 , . . . , Ft }, {F1 , . . . , Ft }, respectively which consist of the antiholomorphic maps from V (M) to M satisfying conditions 1) to 4) of the theorem. Now, we claim the following.   q   p p q Lemma 6.1 For each w ∈ V (M), F1 (w) . . . Ft (w) = F1 (w) . . . Ft (w) A∗ . Proof Fix an arbitrary point w0 ∈ V (M). Then from condition 1 of Theorem 4.3 there exist a neighbourhood w0 of w0 in , anti-holomorphic maps Fk w , p , Fk w ,q : 0 0 V (M) → M, k = 1, . . . , t such that p q (a) Fk w , p (v) = Fk (v), Fk w ,q (v) = Fk (v), for all v ∈ V (M) ∩ w0 , k = 1, . . . , t 0 0 and t  j (b) K (·, u) = i=1 pi (u)Fi w , p (u), K (·, u) = tj=1 q j (u)Fw ,q (u), for all u ∈ 0 0 w0 . Now, applying Eq. (6.1) to the second equality of b) we obtain that K (·, u) =

t 

pi (u)

t 

i=1

j

a¯ i j Fw

0

j=1

 (u) , ,q

 q for all u ∈ w0 . Finally, observe that, for each i ∈ {1, . . . , t}, v → tj=1 a¯ i j F j (v) is an anti-holomorphic map from V (M) → M satisfying conditions 1.a) and 1.b) of Theorem 4.3 with respect to the set { p1 , . . . , pt }. So, from condition 2 of Theorem 4.3, it follows that, for each i = 1, . . . , t, v ∈ V (M), p

Fi (v) =

t 

q

a¯ i j F j (v),

j=1

 

proving the lemma. p {F1 , . . . ,

p

q {F1 , . . . ,

q Ft }

Note that each of the sets Ft } and canonically induces an p q anti-holomorphic frame of E M on V (M). If we denote the frames as FM and FM respectively, then we have K E M (FM ) = K E M (FM A∗ ) = (A∗ )−1 · K E M (FM ) · A∗ , p

q

q

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q

where K E M (FM ), K E M (FM ) are the curvature matrices of the bundle E M with p q respect to the frames FM , FM , respectively. Thus, we have proved the following. Proposition 6.2 Let  be a bounded domain in Cm and { p1 , . . . , pt }, {q1 , . . . , qt } be two generators of the polynomial ideal I consisting of homogeneous polynomials of same degree. Also, let H ⊆ O() be an analytic Hilbert module and M be the closure p q of I in H with the property that codimV (M) = t. Furthermore, assume that FM , FM are the global frames of E M obtained by applying Theorem 4.3 on M with respect to the generators mentioned above. Then there exists a constant invertible matrix A such that K E M (FM ) = (A∗ )−1 · K E M (FM ) · A∗ . p

q

Corollary 6.3 Let , I be as above, H, H ⊆ O() be two analytic Hilbert modules and M, M be the closure of I in H, H respectively with codimV (M) = q q p p p q codimV (M ) = t. Suppose that FM := {F1 , . . . , Ft }, FM := {F1 , . . . , Ft } are the global frames of E M , E M corresponding to the generators { p1 , . . . , pt }, {q1 , . . . , qt }, respectively. If the modules M and M are "unitarily" equivalent, then there exists a constant invertible matrix A such that K E M (FM ) = (A∗ )−1 · K E M (FM ) · A∗ , p

p

q

q

where K E M (FM ), K E M (FM ) are the curvature matrices of E M , E M with respect p q to the frames FM , FM , respectively. Proof If we apply Theorem 4.3 on M with respect to the generator {q1 , . . . , qt }, we q q will obtain a collection of anti-holomorphic maps {F1 , . . . , Ft } from V (M) to M. q This set canonically induces a global frame of E M . Let us denote the frame by FM . Then, by Proposition 6.2 it follows that there exists a constant invertible matrix A such that K E M (FM ) = (A∗ )−1 · K E M (FM ) · A∗ . p

q

Finally, from Theorem 5.1 we obtain that q

q

K E M (FM ) = K E M (FM )  

which proves the corollary.

Data availability Data sharing is not applicable to this article as no data sets were generated or analysed during the current study.

References 1. Agrawal, O.P., Salinas, N.: Sharp kernels and canonical subspaces. Am. J. Math. 110(1), 23–47 (1988)

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S. Biswas et al. 2. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68(3), 337–404 (1950) 3. Biswas, S.: Geometric invariants for a class of semi-Fredholm Hilbert Modules, Ph.D. thesis, Indian Statistical Institute of Science (2010) 4. Biswas, S., Misra, G.: Resolution of singularities for a class of Hilbert modules. Indiana Univ. Math. J. 61, 1019–1050 (2012) 5. Biswas, S., Misra, G., Putinar, M.: Unitary invariants for Hilbert modules of finite rank. Journal fr die reine und angewandte Mathematik (Crelles Journal) 662, 165–204 (2012) 6. Chen, X.M., Douglas, R.G.: Localization of Hilbert modules. Mich. Math. J. 39(3), 443–454 (1992) 7. Chen, X.M., Guo, K.: Analytic Hilbert Modules. Chapman and Hall/CRC, London (2003) 8. Chirka, E.M.: Complex Analytic Sets. Kluwer, Dordrecht (1989) 9. Cowen, M.J., Douglas, R.G.: Complex geometry and operator theory. Acta Math. 141(3–4), 187–261 (1978) 10. Curto, R.E., Salinas, N.: Generalized Bergman kernels and the Cowen–Douglas theory. Am. J. Math. 106, 447–488 (1984) 11. Douglas, R.G., Misra, G.: On quasi-free Hilbert modules. N. Y. J. Math. 11, 547–561 (2005) 12. Douglas, R.G., Misra, G., Varughese, C.: Some geometric invariants from resolutions of Hilbert modules. In: Borichev, A.A., Nikolski, N.K. (eds.) Systems, Approximation, Singular Integral Operators, and Related Topics Operator Theory: Advances and Applications, vol. 129, pp. 241–270. Birkhäuser, Basel (2001) 13. Douglas, R.G., Paulsen, V.I.: Hilbert Modules Over Function Algebras. Longman Sc & Tech, New York (1989) 14. Duan, Y., Guo, K.: Dimension formula for localization of Hilbert modules. J. Oper. Theory 62, 439–452 (2009) 15. Ghara, S., Misra, G.: Decomposition of the tensor product of two Hilbert modules. In: Operator Theory, Operator Algebras and Their Interactions with Geometry and Topology-Ronald G. Douglas Memorial, Operator Theory: Advances and Applications, vol. 278, pp. 221–265 16. Lojasiewicz, S.: Introduction to Complex Analytic Geometry. Birkhäuser, Basel (2013) 17. Paulsen, V.I., Raghupathi, M.: An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Cambridge Studies in Advanced Mathematics, vol. 152. Cambridge University Press, Cambridge (2016) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Authors and Affiliations Shibananda Biswas1 · Gadadhar Misra2,3 · Samrat Sen4 Shibananda Biswas [email protected] Samrat Sen [email protected] 1

Department of Mathematics and Statistics, Indian Institute of Science Education and Research Kolkata, Nadia, Mohanpur, West Bengal 741246, India

2

Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore 560059, India

3

Department of Mathematics, Indian Institute of Technology, Gandhinagar 382055, India

4

4 L B.G. Bye Lane, Naktala, Kolkata 700047, India

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Complex Analysis and Operator Theory (2022) 16:73 https://doi.org/10.1007/s11785-022-01254-3

Complex Analysis and Operator Theory

Approximation in the Mean on Rational Curves Shibananda Biswas1

· Mihai Putinar2,3

Received: 7 January 2022 / Accepted: 1 June 2022 / Published online: 24 June 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue L 2 -space to the existence of analytic bounded point evaluations. Analogues to the complex plane results of Thomson and Brennan are obtained on rational curves. Keywords Bounded point evaluation · Rational approximation · Subnormal operator · Rational curve Mathematics Subject Classification 41A10 · 41A20 · 47B20 · 14H45

1 Introduction Let n be a positive integer and C[z] denote the algebra of polynomials in n complex variables z = (z 1 , z 2 , . . . , z n ). Let V ⊂ Cn be a complex affine curve, that is, an algebraic variety (common zero set of finitely many polynomials) of dimension 1. When necessary, we consider V as an algebraic variety endowed with its reduced

Jörg Eschmeier, in memoriam. Communicated by Ernst Joachim Albrecht. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

B

Shibananda Biswas [email protected] Mihai Putinar [email protected]; [email protected]

1

Department of Mathematics and Statistics, Indian Institute of Science Education and Research Kolkata, Mohanpur, 741246 Nadia, West Bengal, India

2

University of California at Santa Barbara, Santa Barbara, CA, USA

3

Newcastle University, Newcastle upon Tyne, UK

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structural sheaf. Given a positive Borel measure μ supported by a compact subset K of V we consider the closure P 2 (μ) of C[z] in L 2 (μ). A central question of function theory is the relationship between the density of polynomials in Lebesgue space, that is P 2 (μ) = L 2 (μ) versus the existence of bounded point evaluations λ for P 2 (μ): | p(λ)| ≤ C p2,μ ,

p ∈ C[z].

Note that, if such a point exists, then it belongs to V. If the above estimate holds, with the same constant, for an open set, we say that the measure μ admits analytic bounded point evaluations. By enlarging the notion of (analytic) bounded point evaluation beyond polynomials, such as to rational functions with prescribed pole location, one has to specify the algebra of analytic functions for which the above bound holds. In the case of measures defined on the complex plane, this density problem is classical, naturally related to Szegö’s Limit Theorem (on the circle), determinateness of the moment problem on the line, the structure of cyclic subnormal operators [1, 5, 10]. We owe to Jim Thomson [13] the definitive answer, encrypted in one definitive statement which culminates more than half a century of partial results: For a positive Borel measure μ, compactly supported on the complex plane and without point masses, P 2 (μ) = L 2 (μ) if and only if there exist analytic bounded point evaluations. A different proof of Thomson’s theorem appears in [3], together with authoritative historical comments. The cloud of such point evaluations gives essential information about the building blocks of subnormal operators [5]. Several generalizations of Thomson’s Theorem to the case of rational functions have been proposed, see the recent survey [6] for a detailed account, or [14] for some recent advances. In the present note we rely on Brennan’s rational approximation theorem, reported with an original proof in [4]. Our aim is to start investigating conditions assuring the validity of Thomson’s Theorem on algebraic curves.

2 Algebraic Curves with Polynomial Parametrization To fix ideas we start with the simplest framework. In this section we assume that the algebraic curve V admits a polynomial parametrization. This is a well studied subclass of rational curves, not neglected by its relevance to numerical algebraic geometry [12]. To simplify terminology, we say that V is a polynomial curve. A key algebraic observation is that one can change the parametrization of V to a proper polynomial parametrization [12, Theorem 6.11], which in turn is normal [12, Corollary 6.21]. To be more specific, given a polynomial curve V, there exists a map P = ( p1 , . . . , pn ) : C → V, pi ∈ C[ζ ], 1 ≤ i ≤ n, with the property that P is bijective away from a finite set of points and avoids at most finitely many points of V. The complex coordinate on the parameter space is ζ ∈ C.

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That is the fibre P −1 P(ζ ) has cardinality one, with the exception of finitely many points ζ ∈ C, where it is finite. We denote by O the sheaf of complex analytic functions. Since the map P is finite, Grauert’s Theorem states that the direct image sheaf P∗ OC , defined by the sheaf associated to the presheaf P∗ OC (U ) = O(P −1 (U )) for U open in Cn , is coherent as a OCn -module. We refer to [8, Chapter I, Section 3] for the proof and terminology. Moreover, P∗ OC (Cn ) = O(C).

(2.1)

Consider the natural pull-back morphism  : OCn −→ P∗ OC defined by h → P ∗ h = h ◦ P. Lemma 2.1 Let P = ( p1 , . . . , pn ) : C → Cn , pi ∈ C[z], 1 ≤ i ≤ n, be an injective map away from a finite set of points. Then dim O(C)/P ∗ O(Cn ) < ∞. Proof The kernel I of  is the ideal sheaf defining the curve V = P(C). The short exact sequence 0

I

OCn

ψ

Im

0

implies the coherence of the image sheaf Im. In view of the subsection I.1.5 of [8, page 48], the cokernel S of  is zero at every point y ∈ Cn with card(P −1 (y)) = 1. Hence S is a coherent analytic sheaf, supported by finitely many points of Cn , that is dim S (Cn ) < ∞. The long exact sequence of cohomology induced from the short exact sequence 0

Im

P∗ OC

q

S

0,

where q is the quotient morphism, implies H 1 (Cn , Im) = 0. Hence, from equation 2.1 the desired finiteness follows.

Returning to the polynomial density in Lebesgue space, we carry the data from V to the parameter space, and there we invoke Thomson’s Theorem. More precisely, let μ be a positive Borel measure supported by a compact subset K of V. Since we are seeking non-trivial bounded point evaluations, we assume that μ does not have point

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masses, that is μ({y}) = 0 for all y ∈ K . The pre-image set L = P −1 (K ) is compact since the map P is finite. Let A ⊂ K denote the finite subset of points y ∈ K with the cardinality of P −1 ({y}) bigger than one. The singular points of V are included in A. Let B = P −1 (A), also a finite subset of L. Denote K  = K \ A and L  = L \ B. Then the restriction map P  = P| L  : L  −→ K  is bijective. Finally, let μ = χ K  μ denote the restriction of the measure μ to K  , and take the push-forward measure ν = (P −1 )∗ μ . The positive measure ν is supported by V and does not possess atoms. Since the measure μ does not carry point masses, L 2 (μ) = L 2 (μ ) isometrically, and consequently P 2 (μ) = P 2 (μ ). Let φ be a continuous function on Cn . Then     φdμ = φdμ = φ ◦ Pdν. K

K

In other terms the pull-back map P ∗ : L 2 (μ) −→ L 2 (ν) is isometric. Due to the local structure of an algebraic curve, continuous functions in the ambient space separate, modulo finitely many singular points, different branches of the curve, see for instance Section 2.5 in [12]. Hence, by passing to Borel functions, we find that the isometric map P ∗ is onto, hence a unitary operator. Proposition 2.2 Under the assumptions above, the space P 2 (μ) admits analytic bounded point evaluations if and only if P 2 (ν) admits analytic bounded point evaluations. Proof Assume that α ∈ L is an analytic bounded point evaluation with respect to P 2 (ν). Let P(α) = β. For every p ∈ C[z] one finds | p(β)| = | p ◦ P(α)| ≤ C p ◦ P2,ν = C p2,μ where C > 0 is a universal constant. Therefore P 2 (μ) admits analytic point evaluations filling a neighborhood of β. Conversely, assume that P 2 (μ) admits bounded point evaluations (with the same bound C) at every point of an open subset U of V. While the map P : C −→ V may not be surjective, it avoids only finitely many points of V in its range, cf. Theorem 2.2.43 in [12]. Choose a point β ∈ U which possesses a pre-image, that is α ∈ C such that P(α) = β. We claim that at α, is a bounded point evaluation for P 2 (ν). The algebra of entire functions O(Cn ) is dense in P 2 (μ), and we have deduced from Grauert’s finiteness theorem that P ∗ O(Cn ) is a finite codimensional subspace of O(C). Hence there exists a finite dimensional space of entire functions W ⊂ O(C) with the property that P ∗ O(Cn ) + W is a dense subspace of P 2 (ν). Since α is a bounded point evaluation for both P ∗ O(Cn ) and W , and dim W < ∞, we infer that α is a

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bounded point evaluation for the entire space P 2 (ν), with a locally bounded constant C(α) = sup p(α)

p2,ν . Theorem 2.3 Let V ⊂ Cn be a polynomial curve and let μ be a positive Borel measure supported by a compact subset of V. Assume that μ does not have point masses and P 2 (μ) = L 2 (μ). Then, and only then, there exist analytic bounded point evaluations for P 2 (μ). Proof Choose, as before in this section, a normal, polynomial parametrization P : C −→ Cn of the curve V. Let ν denote the pull-back measure of μ on C. We claim P 2 (ν) = L 2 (ν). Suppose by contradiction P 2 (ν) = L 2 (ν). Lemma 2.1 implies that P ∗ P 2 (μ) is a finite codimenion subspace of L 2 (ν). But the pull-back map P ∗ is unitary at the level of L 2 spaces. Hence P 2 (μ) is a finite codimension subspace of L 2 (μ). The multiplication operator Mi on P 2 (μ) by the coordinate function z i is subnormal, for every i, 1 ≤ i ≤ n. The corresponding normal extension Ni , is represented by the multiplication by z i on L 2 (μ). With respect to the decomposition L 2 (μ) = P 2 (μ) ⊕ P 2 (μ)⊥ , we can write Ni in 2 × 2 blocks:  Ni =

Mi Si 0 Ti



with both Si and Ti finite rank operators, 1 ≤ i ≤ n. The normality block operator equation of these extensions yields: [Mi∗ , Mi ] = Si Si∗ and Si∗ Si = [Ti , Ti∗ ]. Since Si is a finite rank operator and trace of Si∗ Si vanishes, we find Si = 0 for all i, 1 ≤ i ≤ n. This in turn shows that both Mzi and Ti ’s are normal operators. By assumption P 2 (μ) = L 2 (μ), that is the block carrying the commuting normal matrices Ti is non-trivial. Then a common eigenvector of the Ti ’s exists, implying the existence of a point mass for the measure μ. A contradiction. According to Thomson’s theorem [13], the space P 2 (ν) admits a non-empty open set of bounded point evaluations. Hence by Proposition, 2.2, P 2 (μ) has analytic bounded point evaluations. For the only then part, we note that if P 2 (μ) admits bounded point evaluations, then P 2 (μ) = L 2 (μ) as otherwise the unitarity of the pull back map implies P 2 (ν) = L 2 (ν) which is a contradiction to Thomson’s theorem via Proposition, 2.2. This completes the proof.



3 Rational Curves The general case of rational curves is not much different. This time we change the base via a rational map, and invoke a generalization of Thomson’s theorem proved by Brennan [4]. We state the main result and indicate the very similar deduction molded on the proof detailed in the previous section.

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Theorem 3.1 Let V be a rational curve in Cn and let μ be a positive Borel measure without point masses, supported by a compact subset of V. Then P 2 (μ) = L 2 (μ) if and only if there are analytic P 2 (μ)-bounded point evaluations. Proof Let R = (r1 , r2 , . . . , rn ) be an n-tuple of rational functions which properly parametrizes the affine curve V. That is, denoting by S ⊂ C the poles of R, the holomorphic map R : C \ S −→ V is one to one, except finitely many points, and it covers V except finitely many points. We refer to Section 4.4.2 in [12] for terminology and basic results. Let ρ denote a sufficiently large radius, so that the support of the measure μ is contained in the ball B(0, ρ). The pull-back U = R −1 B(0, ρ) is an open subset of C, of finite connectivity, with piece-wise smooth boundary. In particular we can assume that every connected component of the complement of U has positive diameter. The restricted analytic map R : U −→ B(0, ρ) has finite fibres, hence it is proper. Grauert’s finiteness theorem implies that the direct image sheaf R∗ OU is coherent and R∗ OU (B(0, ρ)) = O(U ). See again Theorem I.1.5 in [8]. As in the previous section, the coherence of R∗ OU and the injectivity of R modulo a finite set imply that dim O(U )/R ∗ O(B(0, ρ)) < ∞. Next we define the pull-back measure ν on U as in the previous proof: 

 φ dμ =

φ ◦ R dν,

for every continuous function φ : B(0, ρ) −→ C. Let R 2 (U , ν) denote the closure in L 2 (ν), of rational functions with poles on the complement of U . Runge’s approximation theorem implies that R 2 (U , ν) is also the closure of the algebra O(U ) in L 2 (ν). The counterpart of Proposition 2.2 has the same proof: There exist analytic bounded point evaluations with respect to P 2 (μ) if and only if there exists analytic bounded point evaluations with respect to R 2 (U , ν).

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Theorem 1 in [4] asserts, under the positive diameter assumption of the connected components of U and the lack of point masses, that R 2 (U , ν) = L 2 (ν) if and only if

there exist analytic bounded point evaluations with respect to R 2 (U , ν). Corollary 3.2 A commuting subnormal tuple with Taylor’s joint spectrum contained in a rational curve admits joint invariant subspaces. Proof For a commuting subnormal tuple with Taylor’s joint spectrum contained in a rational curve, it is enough to show the same for cyclic subnormal tuple of operators, in fact, by [7, Remark 2.2], it suffices to show the same for the tuple of multiplication operator Mz = (Mz 1 , . . . , Mz n ) by the coordinate functions on P 2 (μ). From Theorem 3.1, it follows that if β = (β1 , . . . , βn ) is a bounded point  evaluation point, then ∩ ker(Mzi − βi )∗ is non-trivial and hence the closure of (z i − βi )P 2 (μ) is a

non-trivial invariant subspace for Mz .

4 Concluding Remarks 4.1 Analytic Coordinate Charts The above proof carries verbatim on an algebraic curve V, in case the measure μ is compactly supported in a coordinate chart V ⊂ V. More specifically, assuming the existence of a bi-holomorphic map f : U −→ V where U ⊂ C is an open set with complement C \ U consisting of finitely many connected sets of positive diameter. 4.2 Resolution of Singularities Uniform approximation by analytic functions defined on an open Riemann surface is much better understood, with definitive results generalizing Runge’s Theorem or even Mergelyan’s Theorem, see for instance [2, 11]. Let V be an affine algebraic curve. The well known desingularization procedure provides a birational transform p : X −→ V, where X is an open Riemann surface, see for instance [9]. Our main proof carries without major adaptation to the map p, raising the following intriguing question. Open Problem. Let X be an open Riemann surface of finite genus, and let μ be a positive Borel measure on X without point masses. Let U ⊂ X be an open, relatively compact subset of X , with finitely many components of X \U , none reduced to a point. Assume the closed support of the measure μ is contained in U . Then analytic functions O(U ) are dense in L 2 (μ) if and only if there are no corresponding bounded analytic point evaluations. The precautions in stating the above question with specific topological constraints on the open set U are resonant to Brennan’s main result proved in the complex plane

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X = C, [4]. A simple application of the much stronger, uniform approximation results, can be formulated as follows. Remark 4.1 Let X be an open Riemann surface. Assume that the measure μ is supported by a piecewise smooth curve ⊂ X with the property that the complement X \ is connected. Then P 2 (μ) = L 2 (μ). Simply because O(X ) is dense in the space of continuous functions C( ), hence in L 2 (μ), according to Scheinberg’s master theorem [11]. The situation of an elliptic curve X , with as one of the generating cycles of homology of the projective completion of X , is notable in this respect. 4.3 Generic Linear Projections Let V ⊂ Cn be an affine algebraic curve and let μ be a positive Borel measure supported by V. If there are L 2 (μ)-bounded point evaluations for polynomials, then any generic linear map L : Cn −→ C will detect them. Indeed, Thomson’s Theorem applied to the measure L ∗ μ on C and the fact that L is an open map, except the exceptional case when a component of V is contained in the fibre of L, imply | p(L(λ)| ≤ C p ◦ L2,μ = C p2,L ∗ μ . A simple example shows that the converse does not hold. Indeed, let V be the curve in C2 given by the equation z 1 z 2 = 1, and let μ be the positive measure 

 p(z 1 , z 2 )dμ =

π

−π

p(eit , e−it )dt, p ∈ C[z 1 , z 2 ].

If (λ, λ1 ) were an analytic bounded point for this measure, by taking polynomial in z 1 , or z 2 , and say |λ| > 1, one would obtain the impossible estimate  |q(λ)| ≤ C 2

π −π

|q(eit )|2 dt, q ∈ C[z].

As a matter of fact, one finds easily the isometric identification P 2 (μ) ≡ L 2 (T, dθ ), where T is the unit circle. The pull-back map φ ∗ : f → f ◦ φ from L 2 (μ) to L 2 (T, dθ ), where φ : z → (z, 1z ) from T into V, gives the isometric identification, since μ = φ∗ (dθ ) and φ ∗ C[z 1 , z 2 ] = C[z] + C[ 1z ] is dense in L 2 (T, dθ ). Hence, the space P 2 (μ) does not carry bounded point evaluations. On the other hand, considering the linear map L(z 1 , z 2 ) = z 1 + az 2 , one finds that for all complex numbers a of modulus different than 1, the measure L ∗ μ is integration

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along an ellipse, against a positive weight times arc length. Hence the space P 2 (L ∗ μ) does admit bounded analytic point evaluations. In particular P 2 (L ∗ μ) = H 2 (D) for a = 0. Data availability Data sharing is not applicable to this article as no data sets were generated or analysed during the current study.

References 1. Akhiezer, N.I.: Theory of Approximation Frederick Ungar Publishing Co., New York (1956) 2. Bishop, E.: Subalgebras of functions on a Riemann surface. Pacific J. Math. 8, 29–50 (1958) 3. Brennan, J. E.: Thomson’s theorem on mean-square polynomial approximation. (Russian) ; translated from Algebra i Analiz 17(2), 1–32 (2005); St. Petersburg Math. J. 17(2), 217–238 (2006) 4. Brennan, J.E.: The structure of certain spaces of analytic functions. Comput. Meth. Funct. Theo. 8(1–2), 625–640 (2008) 5. Conway, J.B.: The theory of subnormal operators, Mathematical Surveys and Monographs, 36, American Mathematical Society, (1991) 6. Conway, J.B., Yang, L.: Approximation in the mean by rational functions, arXiv: 1904.06446v6 7. Curto, R.E., Salinas, N.: Spectral properties of cyclic subnormal m-tuples. Amer. J. Math. 107(1), 113–138 (1985) 8. Grauert, H., Remmert, R.: Theory of Stein spaces, Translated by Alan Huckleberry. Classics in Mathematics. Springer-Verlag, Berlin (2004) 9. Kollár, J.: Lectures on resolution of singularities. Ann. Math. Stud., 166. Princeton University Press, Princeton, NJ, vi+208 pp (2007) 10. Krein, M.G.: On a generalization of some investigations of G. Szegö, V. Smirnoff and A. Kolmogoroff, C. R. (Doklady) Acad. Sci. URSS (N.S.) 46, 91-94 (1945) 11. Scheinberg, S.: Uniform approximation by functions analytic on a Riemann surface. Ann. of Math. (2). 108(2), 257–298 (1978) 12. Sendra, J.R., Winkler, F., Pérez-Díaz, S.: Rational algebraic curves. A computer algebra approach, Algorithms and Computation in Mathematics, 22. Springer, Berlin, (2008) 13. Thomson, J.E.: Approximation in the mean by polynomials. Ann. of Math. (2) 133(3), 477–507 (1991) 14. Yang, L.: Reproducing kernel of the space R t (K , μ). Operator theory, operator algebras and their interactions with geometry and topology, 521–534, Oper. Theory Adv. Appl., 278, Birkhüser/Springer, Cham, (2020) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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Complex Analysis and Operator Theory (2023) 17:44 https://doi.org/10.1007/s11785-023-01348-6

Complex Analysis and Operator Theory

Composition Operators on Function Spaces on the Halfplane: Spectra and Semigroups I. Chalendar1 · J. R. Partington2 Received: 9 December 2022 / Accepted: 11 March 2023 / Published online: 8 April 2023 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023

Abstract This paper considers composition operators on Zen spaces (a class of weighted Bergman spaces of the right half-plane related to weighted function spaces on the positive half-line by means of the Laplace transform). Generalizations are given to work of Kucik on norms and essential norms, to work of Schroderus on (essential) spectra, and to work by Arvanitidis and the authors on semigroups of composition operators. The results are illustrated by consideration of the Hardy–Bergman space; that is, the intersection of the Hardy and Bergman Hilbert spaces on the half-plane. Keywords Composition operator · Hardy space · Bergman space · Spectrum · Essential spectrum · Operator semigroup Mathematics Subject Classification 30H10 · 30H20 · 47B33 · 47D03

1 Introduction and Background Material The subject of this paper involves properties of composition operators on holomorphic function spaces on the right half-plane C+ , both as individual operators and as elements of one-parameter semigroups. One difficulty, even in the case of the Hardy space

Communicated by Aurelian Gheondea. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

B

I. Chalendar [email protected] J. R. Partington [email protected]

1

LAMA, (UMR 8050), UPEM, UPEC, CNRS, Université Gustave Eiffel, 77454 Marne-la-Vallée, France

2

School of Mathematics, University of Leeds, Leeds LS2 9JT, UK

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H 2 (C+ ), is that not all composition operators on the spaces are bounded, although in many cases the bounded operators have been characterised, as we explain below. The literature on semigroups of composition operators is not as extensive as it is for the disc, although it has been an object of study since the work of Berkson and Porta [5]. In this note we shall concentrate on the so-called Zen spaces (weighted Hardy– Bergman spaces), which form a large class of spaces with applications in systems and control theory [11, 12]. For these information on the bounded composition operators is also available, thanks to Kucik [13]. The remainder of this section presents necessary background material. In Sect. 2 we provide new results on the norm and spectral radius of composition operators. Then, in Sect. 3, we consider the spectral theory of composition operators on the half-plane with linear fractional symbols, extending several results of Schroderus [15]. Finally, in Sect. 4, we consider semigroups of composition operators on Zen spaces, providing generalizations of results of Arvanitidis [3]. 1.1 Zen Spaces Kucik [13] considered composition operators on Zen spaces A2ν , which are isometrically Laplace transforms of weighted Hardy spaces L 2w (0, ∞) = L 2 (0, ∞, w(t)dt). For general background see [6], but we provide the basic facts now. Let ν be a positive regular Borel measure on [0, ∞) satisfying the doubling condition R := sup t>0

ν[0, 2t) < ∞. ν[0, t)

The Zen space A2ν is defined to consist of all analytic functions F on C+ such that the norm, given by  F = sup 2

>0 C+

|F(s + )|2 dν(x) dy

is finite, where we write s = x + i y for x ≥ 0 and y ∈ R. The best-known examples here are: (1) For ν = δ0 , a Dirac mass at 0, we obtain the Hardy space H 2 (C+ ); (2) For ν equal to Lebesgue measure we obtain the Bergman space A2 (C+ ). Often we shall have ν({0}) = 0, in which case F2 can be written simply as  C+

|F(s)|2 dν(x) dy.

Theorem 1.1 [11] Suppose that w is given as a weighted Laplace transform 

∞

w(t) = 2π

e−2r t dν(r ),

(t > 0).

0

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Then the Laplace transform provides an isometric map L : L 2 (0, ∞, w(t)dt) → A2ν . The result also holds for Hilbert-space valued functions [1]. The following result generalizes the Elliott–Jury theorem in [8] and the Elliott–Wynn theorem in [9]. Theorem 1.2 [13] The composition operator Cφ is bounded on A2ν if and only if φ has a finite nonzero angular derivative L = ∠ lim z→∞ z/φ(z) at ∞. In that case L inf

t>0

w(t) w(t) ≤ Cφ 2 ≤ L sup . w(Lt) t>0 w(Lt)

This gives correctly Cφ  = case.

√

(1)

L in the Hardy case and Cφ  = L in the Bergman

1.2 Semigroups of Composition Operators Berkson and Porta [5] give the following criterion for an analytic function G to generate a one-parameter semigroup of analytic mappings of the right half-plane C+ to itself; that is, solutions to ∂φt (z) = G(φt (z)), ∂t

φ0 (z) = z,

(2)

on C+ ,

(3)

in terms of the following condition: x

∂(ReG) ≤ ReG ∂x

where x = Rez. The associated composition operators Cφt are bounded on H 2 (C+ ) if and only if the non-tangential limit L t := ∠ lim z/φt (z) z→∞

1/2

exists and is non-zero. It will be positive, and then Cφt  = L t (see [8, 14]). Arvanitidis [3] showed that a necessary and sufficient condition for boundedness of the composition operators is that δ := ∠ lim z→∞ G(z)/z exists. In this case Cφt  = e−δt/2 and the semigroup is quasicontractive. The following result is taken from [4]. Theorem 1.3 For an operator A given by A f = G f on D(A) ⊆ H 2 (C+ ), the following are equivalent: (i) A generates a quasi-contractive C0 -semigroup of bounded composition operators on H 2 (C+ ); (ii) Condition (3) holds and ∠ lim z→∞ G(z)/z exists.

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2 The Essential Norm and Spectral Radius We know from [13, Lem. 4] that in every A2ν space the normalized reproducing kernels kn /kn  tend weakly to 0 as n → ∞. Now we can adapt the proof that there is no compact composition operator as follows. Using the isometry between L 2 (0, ∞; w(t) dt) and A2ν we may pull back the reproducing kernel of A2ν at a point λ ∈ C+ by the inverse Laplace transform to obtain kλ (t) := e−λt /w(t) and its norm is kλ , kλ 1/2 or 

∞

0

e−2Reλt dt w(t)

1/2 .

(4)

Theorem 2.1 Let Cφ be a bounded composition operator on the Zen space A2ν corresponding to a weight w on (0, ∞). Let L denote the finite nonzero angular derivative L = ∠ lim z→∞ z/φ(z) at ∞. Then w(t) w(t) ≤ Cφ 2e ≤ Cφ 2 ≤ L sup . t>0 w(Lt) t>0 w(Lt)

L inf

(5)

Proof For every δ > 0 there is a compact operator Q on A2ν such that Cφ e + δ ≥ Cφ − Q ≥ lim sup (Cφ − Q)∗ kn /kn  n→∞

= lim sup kφ(n) /kn  n→∞

since Q ∗ kn /kn  → 0 in norm. Since this is true for all δ > 0 we have  Cφ 2e

∞

≥ lim sup n→∞

0

   ∞ e−2nt e−2Reφ(n)t dt / dt w(t) w(t) 0

by (4). Hence, for every 0 < M < L we have  Cφ 2e

∞

≥ lim sup n→∞

0



∞

= lim sup n→∞

0

  ∞ −2nt  e−2nt/M e dt / dt w(t) w(t) 0    ∞ e−2nt e−2un M du / dt w(Mu) w(t) 0

w(u) ≥ M inf . u>0 w(Mu) Now, using Theorem 1.2, we have the estimate for the essential norm given in (5).  

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For the case of the Hardy and weighted Bergman spaces we therefore recover the formula Cφ  = Cφ e from [8, 9]. Note that the left and right hand sides of (1) and (5) may certainly differ, and as an example we shall consider the Hardy–Bergman space H 2 (C+ ) ∩ A2 (C+ ). On the disc the Hardy space H 2 (D) is contained in the Bergman space A2 (D) but no such inclusion either way exists on the half-plane, as we shall now explain. We may give H 2 (C+ ) ∩ A2 (C+ ) a Hilbert space structure with the norm defined by  f 2H 2 (C

+ )∩A

2 (C

+)

=  f 2H 2 (C

+)

+  f 2A2 (C ) . +

Since the Hardy space is isomorphic by means of the Laplace transform to L 2 (0, ∞) and the Bergman space to L 2 (0, ∞, dt/t) we see that neither space is contained in the other and that the Hardy–Bergman space H 2 (C+ ) ∩ A2 (C+ ) corresponds to L 2 (0, ∞, w(t) dt) with the weight w(t) = 1 + 1/t. t +1 , so that the sup and inf are 1 and 1/L in some In this case w(t)/w(Lt) = t+L order. For the spectral radius ρ(Cφ ) we may apply (1) to get an estimate which is once more tight in the case of the Hardy and weighted Bergman spaces. Corollary 2.2 For a bounded composition Cφ on a space A2ν corresponding to a weight w, the spectral radius ρ(Cφ ) satisfies 

w(t) L lim sup inf n t) t>0 w(L n→∞

1/n

  w(t) 1/n ≤ ρ(Cφ ) ≤ L lim inf sup , n n→∞ t>0 w(L t) 2

where L = ∠ lim z→∞ z/φ(z). Proof For the iterate φ (n) the corresponding angular derivative L n = ∠ lim z→∞ z/φ (n) (z) is simply L n , being the product of n terms each tending to L. Thus L n inf

t>0

w(t) w(t) ≤ Cφn 2 ≤ L n sup , n t) w(L n t) w(L t>0

from which the result follows by the standard spectral radius formula.

 

3 Spectral Theory For the spectral theory of composition operators on the half-plane, little seems to be known, although Schroderus [15] has determined the spectrum and essential spectrum in the case of linear fractional mappings φ for Hardy and weighted Bergman spaces. The associated composition operator Cφ is bounded if and only if φ is a parabolic or hyperbolic mapping fixing ∞; that is, φ(s) = μs + s0 with μ > 0 and Res0 ≥ 0 (we shall discuss this in detail below). The same applies in general Zen spaces, by Theorem 1.2.

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Schroderus restricts herself to Hardy spaces and weighted Bergman spaces on the upper half-plane. Transforming to the right half-plane these are Zen spaces A2ν with measures dν(x) = x α d x for α > −1 (in the following the Hardy space is formally identified with the case α = −1). Her results in this particular context are: Theorem 3.1 [15, Thm. 1.1, Thm 1.2] (1) In the parabolic case φ(s) = s + s0 , where Res0 ≥ 0 and s0 = 0, the spectrum and essential spectrum of Cφ coincide and equal (a) T if s0 ∈ iR; (b) {e−s0 t : t ≥ 0} ∪ {0} if s0 ∈ C+ . (2) In the hyperbolic case φ(s) = μs + s0 , where μ ∈ (0, 1) ∪ (1, ∞) and Res0 ≥ 0, the spectrum and essential spectrum of Cφ coincide and equal (a) {λ ∈ C : |λ| = μ−(α+2)/2 } if s0 ∈ iR; (b) {λ ∈ C : |λ| ≤ μ−(α+2)/2 } if s0 ∈ C+ . 3.1 The Parabolic Case For a general Zen space, part (1) of Theorem 3.1 is easy to prove. Proposition 3.2 Let φ(s) = s + s0 where Res0 ≥ 0 and s0 = 0. Then the spectrum and essential spectrum of Cφ on A2ν coincide and equal (a) T if s0 ∈ iR; (b) {e−s0 t : t ≥ 0} ∪ {0} if s0 ∈ C+ . Proof The composition operator is seen to be unitarily equivalent to the multiplication operator on L 2 (0, ∞, w(t)dt) given by multiplication by the function t → e−s0 t . Thus it is a normal operator: its spectrum and essential spectrum equal the closure of {e−s0 t : t ≥ 0}, as in (a) and (b). 3.2 The Hyperbolic Case In this subsection we take φ(s) = μs + s0 , where μ ∈ (0, 1) ∪ (1, ∞) and Res0 ≥ 0. Schroderus observed that we have the following simplifications: For s0 = i y, with y ∈ R, the operator Cφ is similar to Cψ with ψ(s) = μs. Indeed, if ρ(s) = s + i y/(μ − 1) then ρ −1 ◦ ψ ◦ ρ(s) = μ(s + i y/(μ − 1)) − i y/(μ − 1) = μs + i y. Similarly, for s0 = x + i y with x > 0 and y ∈ R, let ψ(s) = μs + x and take the same ρ. Then ρ −1 ◦ ψ ◦ ρ(s) = μ(s + i y/(μ − 1)) + x − i y/(μ − 1) = μs + x + i y.

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Thus, since Cρ is a unitary map on every A2ν , we need only consider the spectrum of Cφ when φ(s) = μs + x for μ ∈ (0, 1) ∪ (1, ∞) and x ≥ 0. In case (a) we can estimate the spectral radius of Cφ and Cφ−1 , although this tells us only that the spectrum is contained in an annulus (which may be a circle). In case (b) we have that the spectrum is contained in a disc. We showed in [6, Prop. 3.5] that the norm of the weighted composition operator Cψ   w(μt) 1/2 1 sup corresponding to ψ(s) = μs with μ > 0 is exactly . The method μ t>0 w(t) of [6] also shows that for all x ≥ 0 and φ(s) = μs + x the composition operator has norm equal to the norm of the mapping f → g in L 2 (0, ∞; w(t) dt), where g(t) =

1 −xt/μ e f (t/μ) μ

since 

∞

0

 1 ∞ −st −xt/μ e e f (t/μ) dt μ  ∞0 e−μsτ e−xτ f (τ ) dτ =

e−st g(t) dt =

0

and g2 =

1 μ2



∞

e−2xt/μ | f (t/μ)|2 w(t) dt =

0

1 μ



∞

e−2xτ | f (τ )|2 w(μτ ) dτ,

0

whence Cφ 2 =

e−2xt w(μt) 1 sup . μ t>0 w(t)

Now by looking at iterates we can obtain an explicit formula for the spectral radius of μn − 1 Cφ as the nth iterate of φ is given by φn (s) = μn s + xn , where xn = x, and μ−1 this has the same form as φ. That is,  1/2n e−2xn t w(μn t) 1 ρ(Cφ ) = √ lim sup . μ n→∞ t>0 w(t)

(6)

In the time domain (that is, using the inverse Laplace transform), the operator Cφ corresponds to the operator Bφ on L 2 (0, ∞, w(t)dt) with Bφ f (t) = a f (bt)e−ct

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such that we have  ∞  −ct −st a f (bt)e e dt = 0

∞

a f (u)e−cu/b e−su/b du/b

0

= (a/b)(L f )(s/b + c/b) = (L f )(μs + x), and so a = b = 1/μ and c = x/μ. With A f (t) = μ1 f (t/μ)e−xt/μ on L 2 (0, ∞, w(t)dt) we find that  0

∞

1 f (t/μ)e−xt/μ g(t)w(t) dt = μ

so that A∗ g(u) = g(μu)e−xu



∞

f (u)e−xu g(μu)

0

w(μu) w(u) du w(u)

w(μu) . w(u)

3.2.1 The Case x = 0 It is enough to consider the spectrum of Cφ for φ(s) = μs with μ > 1 since the case 0 < μ < 1 may be studied by taking inverses. Theorem 3.3 For the composition operator Cφ on A2ν where φ(s) = μs with μ > 1 and ν determining the weight w on (0, ∞) we have σ (Cφ ) ⊆ {z ∈ C : r ≤ |z| ≤ R}, where   w(μn t) 1/(2n) 1 r = √ lim inf μ n→∞ t>0 w(t) and   w(μn t) 1/(2n) 1 . R = √ lim sup μ n→∞ t>0 w(t) Proof This follows from (6).

 

w(μt) , we have eigenvalues of A∗ given by μα , with w(t) eigenvectors f (t) = t α /w(t), provided that these eigenvectors lie in the space L 2 (0, ∞, w(t)dt). Indeed the eigenvalues then have infinite multiplicity, since f χ E is an eigenvector for any measurable E ⊂ R+ such that μE = E, the notation χ E denoting the characteristic (indicator) function of a set E. Clearly there are infinitely many distinct subsets E ⊂ R such that μE = E. For example, if μ > 1 we may take With A∗ f (t) = f (μt)

E=

∞ 

χ(μn ,μn (1+δ))

n=−∞

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for any δ with 0 < δ < μ − 1. The same is true for 0 < μ < 1 working with 1/μ instead of μ: this will be required later. Thus we have the following lower bound for the essential spectrum σe (Cφ ): Theorem 3.4 For the composition operator Cφ on A2ν where φ(s) = μs with μ > 1 and ν determining the weight w on (0, ∞) it holds that σe (Cφ ) contains all α such that  ∞ |t α |2 /w(t) < ∞. (7) 0

Proof The eigenvectors of f (t) = t α /w(t) of the unitarily equivalent operator A∗ lie   in the space L 2 (0, ∞, w(t)dt) if and only if (7) holds. Note that the α occurring in Theorem 3.4 form an annulus (if the set is non-empty), and in some cases (e.g. the Hardy–Bergman space below) this enables us to find the spectrum exactly. 3.2.2 The Case x > 0 Here the calculations are necessarily more difficult, but we do have the spectral radius formula (6) to give a disc in which the spectrum is contained. In many cases all the points in the interior are eigenvalues, either of A or A∗ . Indeed, they are again eigenvalues of infinite multiplicity, and hence in the essential spectrum, as we see in the following two propositions, Proposition 3.5 For μ > 1, if α ∈ C and β = x/(μ − 1) > 0 are such that the function f : t → t α e−βt lies in L 2 (0, ∞, w(t)dt), then A has the eigenvalue 1/μα+1 and f χ E is an eigenvector for any measurable E ⊂ R+ such that μE = E. Proof This is an easy calculation since A f (t) =

1 μ

f (t/μ)e−xt/μ .

 

Proposition 3.6 For 0 < μ < 1 if α ∈ C and β = x/(1 − μ) > 0 are such that the function f (t) = t α e−βt /w(t) lies in L 2 (0, ∞, w(t)dt), then A∗ has the eigenvalue μα and f χ E is an eigenvector for any measurable E ⊂ R+ such that μE = E. Proof This is similarly straightforward, since A∗ f (t) = f (μt)e−xt w(μt)/w(t).   One technique from [15] is not available in general: for the Hardy and standard weighted Bergman spaces, the weighted composition operators with μ and 1/μ are related by taking adjoints. This is not true unless w(t) is a power of t. However, as we shall see in the next section the information here is enough to allow us to determine the exact spectrum in an important new example (as well as the cases covered by Schroderus).

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3.3 The Hardy–Bergman Space As a significant example of a situation not covered by older results, let us look again at the Hardy–Bergman space H 2 (C+ ) ∩ A2 (C+ ). To determine the spectrum we may take any equivalent norm, so we take w(t) = 1 + 1/t as before. Theorem 3.7 Let X = H 2 (C+ ) ∩ A2 (C+ ) be the Hardy-Bergman space and Cϕ ∈ L(X ). a) If ϕ(s) = μs with μ > 0, then σ (Cϕ ) is the annulus    1 1 1 1 ≤ |z| ≤ max . σ (Cφ ) = z ∈ C : min ,√ ,√ μ μ μ μ b) If ϕ(s) = μs + (x + i y) with μ, x > 0 and y ∈ R, then σ (Cϕ ) is the closed disc σ (Cφ ) = {z ∈ C : |z| ≤ 1/μ}. In all cases we have σe (Cφ ) = σ (Cφ ). The proof of this theorem follows from the two subsections detailed below. 3.3.1 The Case x = 0 For φ(s) = μs with μ > 0 we have Cφ 2 =

1 1 w(μt) 1 + 1/(μt) μt + 1 1 sup = sup = sup . μ t>0 w(t) μ t>0 1 + 1/t μ t>0 μt + μ

This is

1/μ2 if 0 < μ < 1, 1/μ if μ > 1.

Thus for μ > 1 we have Cφn 2 =

1 μn

while Cφ−n 2 = μ2n . That is, the spectrum of Cφ lies in the annulus {z ∈ C : μ1 ≤ |z| ≤ √1μ }, but not in a circle. So we see that Theorem 3.1 (2)(a) does not hold in the general case of a Zen space.

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We now show that the (essential) spectrum equals the whole annulus. To do this we use Theorem 3.4. The function t α /(1 + 1/t) lies in the space L 2 (0, ∞, (1 + 1/t) dt) if and only if 

∞ 0

|t α |2 (1 + 1/t) dt < ∞, (1 + 1/t)2

so that −1 < Reα < −1/2. From this we see that all points in the annulus {z ∈ C : 1 ∗ √1 μ < |z| < μ } are eigenvalues of A so that for μ > 1 we have 1 1 . σ (Cφ ) = z ∈ C : ≤ |z| ≤ √ μ μ 

By considering inverses, we see that for 0 < μ < 1  1 1 σ (Cφ ) = z ∈ C : √ ≤ |z| ≤ . μ μ 3.3.2 The Case x > 0 We begin with the spectral radius formula (6). Elementary calculus shows that the supremum in (6) is at t = 0 except if 2xn < μn − 1. This requires μ > 1. If the supremum is at t = 0 it is 1/μn . Otherwise (with μ > 1) we can estimate the supremum for each n by considering the two cases (recall that xn → ∞ as n → ∞): n log μ , when the expression is clearly at most μ−n (the second factor is 2xn bounded by 1); n log μ , when we may ignore the exponential and obtain an upper bound (2) 0 ≤ t ≤ 2xn (1) t ≥

1 + 1/(μn t) 1 + 2xn /(nμn log μ) 1 + C1 /n ≤ ≤ , 1 + 1/t 2xn /n log μ C2 μn−1 /n where C1 and C2 are independent of n. Taking the maximum of these two estimates and then the 2n-th root gives a limit √ 1/ μ. That is ρ(Cφ ) ≤ √1μ × √1μ = μ1 for all x > 0. Now we use Propositions 3.5 and 3.6 to find eigenvectors. For μ > 1 the range of admissible α is {Reα > 0} and the eigenvalue is 1/μα+1 , so that every point of the disc {z ∈ C : 0 < |z| < 1/μ} is an eigenvector of A and we deduce that σ (Cφ ) = {z ∈ C : |z| ≤ 1/μ}. Similarly, for 0 < μ < 1 the range of admissible α is {Reα > −1} and the eigenvalue is μα , so again we deduce that σ (Cφ ) = {z ∈ C : |z| ≤ 1/μ}, and the same holds for σe (Cφ ).

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4 Semigroups of Composition Operators 4.1 Norm Estimates Our first task is to generalize the result of Arvanitidis [3] showing that a necessary and sufficient condition for boundedness of a semigroup (Cφt ) of composition operators on H 2 (C+ ) with generator A : f → G f is that δ := ∠ lim z→∞ G(z)/z exists. In this case Cφt  H 2 (C+ ) = e−δt/2 . This relies on a theorem in [7] (which is stated for the disc), which gives the angular derivative, and hence the norm, in terms of G. In particular, L t = e−δt . By virtue of Theorem 1.2 (Kucik) we know that the condition for boundedness does not depend on which Zen space we use, and all that changes is the norm of the operator. We may thus state the following easy theorem. Theorem 4.1 For a semigroup of analytic self-maps (φt ) on C+ , the following are equivalent: (1) the non-tangential limit δ := ∠ lim z→∞ G(z)/z exists; (2) the semigroup (Cφt )t≥0 consists of bounded operators on A2ν . In this case, we have L t inf

x>0

w(x) w(x) ≤ Cφt 2 ≤ L t sup , w(L t x) w(L t x) x>0

where L t = e−δt for t ≥ 0. Proof This follows from [3, Thm. 3.4] together with the estimate (1) from [13].

 

For example, with the Hardy space H 2 (C+ ) considered by Arvanitidis, we have Cφt  = e−δt/2 , while for the standard Bergman space A2 (C) we have the apparently new result that Cφt  = e−δt . In [2] there is a result applying to analytic function spaces, i.e., Banach spaces X of holomorphic functions on a domain  for which point evaluations are continuous functionals. The result applies to such spaces satisfying the following supplementary condition: (E) If (z n ) is a sequence in  such that z n → z ∈  ∪ {∞} and limn→∞ f (z n ) exists in C for all f ∈ X then z ∈ . Theorem 4.2 Suppose that (Tt )t≥0 is a C0 semigroup on a function space X with property (E) such that for some G ∈ Hol() the generator A is the operator f → G f for all f ∈ dom(A). Then (Tt ) is a semigroup of composition operators. Similar results are given in [10], for function spaces on the disc D. We may now apply Theorem 4.2 to our situation. Theorem 4.3 Suppose that (Tt )t≥0 is a C0 semigroup on a Zen space A2ν such that for some G ∈ Hol(C+ ) the generator A is the operator f → G f for all f ∈ dom(A). Then (Tt ) is a semigroup of composition operators.

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Proof For the halfplane we need to find sequences (z n ) in C+ tending to ∞ or a point on the imaginary axis for which lim f (z n ) does not exist for some f ∈ A2ν . Now if (z n ) is such that ( f (z n )) converges for all f ∈ A2ν then by the uniform boundedness theorem the reproducing kernels at (z n ) are uniformly bounded. But if Rez n → 0 monotonically then the integrals in (4) with λ = z n tend to ∞ (note that w is a decreasing function). This shows that condition (E) holds for A2ν (take sequences tending to either a point on the imaginary axis or to infinity with real parts tending to 0). The result follows from Theorem 4.2.   Remark 4.4 We may also consider the question of groups of composition operators on A2ν . However, since we have same semigroups for all the spaces under consideration, with different norms, we have the same groups. Proposition 4.4 of [4] asserts that for a C0 -quasicontractive group of bounded composition operators on on H 2 (C+ ) we have G(z) = ps + iq for real p and q, and φt (s) = e pt s + iqp (e pt − 1) if p = 0, or φt (s) = s + iqt if p = 0. The same will apply to any Zen space A2ν . 4.2 Semigroups of Linear Fractional Mappings As in Sect. 3 we may discuss parabolic or hyperbolic mappings fixing ∞. That is, φt (s) = μt s + st , where μt > 0 and st ∈ C+ . The semigroup relation φt+u = φt ◦ φu gives us μt = e pt for some p ∈ R, and (1) st = α(e pt − 1) for some α ∈ C+ if p = 0 (the hyperbolic case), or (2) st = αt for some α ∈ C+ if p = 0 (the parabolic case). Using (2) we see that G(z) = pz + pα in the hyperbolic case and G(z) = α in the parabolic case. An immediate corollary of Theorem 3.7 is the following description of the spectrum. Corollary 4.5 Let X = H 2 (C+ ) ∩ A2 (C+ ) be the Hardy-Bergman space and (Cϕt )t≥0 ⊂ L(X ).. a) If ϕt (s) = e pt s + α(e pt − 1) with p = 0, then σ (Cϕt ) is the annulus    1 1 1 1 σ (Cφt ) = z ∈ C : min pt , pt/2 ≤ |z| ≤ max pt , pt/2 . e e e e b) If ϕt (s) = s + αt, then σ (Cϕt ) is the closed disc σ (Cφt ) = {z ∈ C : |z| ≤ e− pt } In all cases the spectrum and essential spectrum coincide. Author Contributions Isabelle Chalendar and Jonathan Partington wrote the manuscript.

Declarations Conflict of interest The authors declare no competing interests.

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References 1. Alajyan, A.E., Partington, J.R.: Weighted operator-valued function spaces applied to the stability of delay systems. Oper. Matrices 15(4), 1257–1266 (2021) 2. Arendt, W., Chalendar, I.: Generators of semigroups on Banach spaces inducing holomorphic semiflows. Isr. J. Math. 229(1), 165–179 (2019) 3. Arvanitidis, A.G.: Semigroups of composition operators on Hardy spaces of the half-plane. Acta Sci. Math. (Szeged) 81(1–2), 293–308 (2015) 4. Avicou, C., Chalendar, I., Partington, J.R.: Analyticity and compactness of semigroups of composition operators. J. Math. Anal. Appl. 437(1), 545–560 (2016) 5. Berkson, E., Porta, H.: Semigroups of analytic functions and composition operators. Mich. Math. J. 25(1), 101–115 (1978) 6. Chalendar, I., Partington, J.R.: Norm estimates for weighted composition operators on spaces of holomorphic functions. Complex Anal. Oper. Theory 8(5), 1087–1095 (2014) 7. Contreras, M.D., Díaz Madrigal, S., Pommerenke, Ch.: On boundary critical points for semigroups of analytic functions. Math. Scand. 98(1), 125–142 (2006) 8. Elliott, S., Jury, M.T.: Composition operators on Hardy spaces of a half-plane. Bull. Lond. Math. Soc. 44(3), 489–495 (2012) 9. Elliott, S.J., Wynn, A.: Composition operators on weighted Bergman spaces of a half-plane. Proc. Edinb. Math. Soc. 54, 373–379 (2011) 10. Gallardo-Gutiérrez, E.A., Yakubovich, D.V.: On generators of C0 -semigroups of composition operators. Isr. J. Math. 229(1), 487–500 (2019) 11. Jacob, B., Partington, J.R., Pott, S.: On Laplace-Carleson embedding theorems. J. Funct. Anal. 264(3), 783–814 (2013) 12. Jacob, B., Partington, J.R., Pott, S.: Applications of Laplace-Carleson embeddings to admissibility and controllability. SIAM J. Control Optim. 52, 1299–1313 (2014) 13. Kucik, A.S.: Weighted composition operators on spaces of analytic functions on the complex half-plane. Complex Anal. Oper. Theory 12(8), 1817–1833 (2018) 14. Matache, V.: Weighted composition operators on H 2 and applications. Complex Anal. Oper. Theory 2(1), 169–197 (2008) 15. Schroderus, R.: Spectra of linear fractional composition operators on the Hardy and weighted Bergman spaces of the half-plane. J. Math. Anal. Appl. 447(2), 817–833 (2017) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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Complex Analysis and Operator Theory (2023) 17:36 https://doi.org/10.1007/s11785-023-01334-y

Complex Analysis and Operator Theory

Weighted Join Operators on Directed Trees Sameer Chavan1 · Rajeev Gupta2 · Kalyan B. Sinha3,4 Received: 17 July 2021 / Accepted: 20 January 2023 / Published online: 2 March 2023 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023

Abstract A rooted directed tree T = (V , E) with root root can be extended to a directed graph T∞ = (V∞ , E ∞ ) by adding a vertex ∞ to V and declaring each vertex in V as a parent of ∞. One may associate with the extended directed tree T∞ a family of semigroup structures b with extreme ends being induced by the join operation  and the meet operation  from lattice theory (corresponding to b = root and b = ∞ respectively). Each semigroup structure among these leads to a family of densely (b) defined linear operators Wλu acting on 2 (V ), which we refer to as weighted join operators at a given base point b ∈ V∞ with prescribed vertex u ∈ V . The extreme (root) and weighted meet operators ends of this family are weighted join operators Wλu (∞)

Wλu . In this paper, we systematically study the weighted join operators on rooted directed trees. We also present a more involved counterpart of weighted join operators (b) Wλu on rootless directed trees T . In the rooted case, these operators are either finite

Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht. Sameer Chavan and Rajeev Gupta are supported by P. K. Kelkar Fellowship and Inspire Faculty Fellowship, respectively. Kalyan B. Sinha acknowledges the support of SERB-Distinguished Fellowship as well as of INSA Senior Scientist Scheme.

B

Sameer Chavan [email protected] Rajeev Gupta [email protected] Kalyan B. Sinha [email protected]

1

Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India

2

School of Mathematics and Computer Science, IIT Goa, India

3

J. N. Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India

4

Indian Statistical Institute, Kolkata, India

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rank operators, diagonal operators or rank one perturbations of diagonal operators. In the rootless case, these operators are either possibly infinite rank operators, diagonal operators or (possibly unbounded) rank one perturbations of diagonal operators. In both cases, the class of weighted join operators overlaps with the well-studied classes of complex Jordan operators and n-symmetric operators. An important half of this paper is devoted to the study of rank one extensions W f ,g of weighted join operators Wλ(b) on rooted directed trees, where f ∈ 2 (V ) and g : V → C is unspecified. u Unlike weighted join operators, these operators are not necessarily closed. We provide a couple of compatibility conditions involving the weight system λu and g to ensure closedness of W f ,g . These compatibility conditions are intimately related to whether or not an associated discrete Hilbert transform is well-defined. We discuss the role of the Gelfand-triplet in the realization of the Hilbert space adjoint of W f ,g . Further, we describe various spectral parts of W f ,g in terms of the weight system and the tree data. We also provide sufficient conditions for W f ,g to be a sectorial operator (resp. an infinitesimal generator of a quasi-bounded strongly continuous semigroup). In case T is leafless, we characterize rank one extensions W f ,g , which admit compact resolvent. Motivated by the above graph-model, we also take a brief look into the general theory of rank one non-selfadjoint perturbations. Keywords Directed tree · Join · Meet · Rank one perturbation · Discrete Hilbert transform · Commutant · Gelfand-triplet · Sectorial · Complex Jordan · n-symmetric · Form-sum Mathematics Subject Classification Primary 47B37 · 47B15 · 47H06; Secondary 05C20 · 47B20

Contents 1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Directed Trees . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Hilbert Space Operators . . . . . . . . . . . . . . . . . . . 1.3 Rank One Operators . . . . . . . . . . . . . . . . . . . . . Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Semigroup Structures on Extended Directed Trees . . . . . . . 2.1 Join and Meet Operations on Extended Directed Trees . . . 2.2 A Canonical Decomposition of an Extended Directed Tree 3 Weighted Join Operators on Rooted Directed Trees . . . . . . . 3.1 Closedness and Boundedness . . . . . . . . . . . . . . . . 3.2 A Decomposition Theorem . . . . . . . . . . . . . . . . . 3.3 Commutant . . . . . . . . . . . . . . . . . . . . . . . . . 4 Rank One Extensions of Weighted Join Operators . . . . . . . 4.1 Compatibility Conditions and Discrete Hilbert Transforms 4.2 Closedness and Relative Boundedness . . . . . . . . . . . 4.3 Adjoints and Gelfand-Triplets . . . . . . . . . . . . . . . . 4.4 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . 5 Special Classes . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Sectoriality . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Normality . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Symmetricity . . . . . . . . . . . . . . . . . . . . . . . .

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1 Background The present work is yet another illustration of the rich interplay between graph theory and operator theory that includes the recent developments pertaining to the weighted shifts on directed trees [18, 19, 21, 38]. This work exploits the order structure of directed trees to introduce a class of possibly unbounded linear operators to be referred (b) to as weighted join operators Wλu based at the vertex b and with prescribed vertex u ∈ V . We capitalize on the fact that any directed tree has a natural partial ordering induced by the notion of the directed path. This ordering satisfies all the requirements of the so-called spiral-like ordering (SLO) introduced and studied by Pruss for p-regular trees (see [59, Definition 6.1]; see also [17, Definition 3] for modified and extended definition). This allows us to define join and meet operations on a directed tree (refer to [31, Chapter 4] for the basics of lattice theory). These operations, in turn, induce (root) (∞) and weighted meet operators Wλu on the so-called weighted join operators Wλu a directed tree. The present work is devoted to a systematic study of this class. In case (b) the directed tree is rooted, it turns out that weighted join operators Wλu are either (possibly unbounded) finite rank operators, diagonal operators or bounded rank one perturbations of (possibly unbounded) diagonal operators. In case the directed trees are rootless, the situation being more complex allows unbounded rank one perturbations of (b) (possibly unbounded) diagonal operators. In particular, weighted join operators Wλu on rootless directed trees need not be even closable. A substantial part of this paper is devoted to the study of rank one extensions W f ,g of weighted join operators. The so-called compatibility conditions (which control the rank one perturbation f  g with (b) the help of the diagonal operator Dλu ) play an important role in the spectral theory of these operators. We also discuss the problem of determining the Hilbert space adjoint of W f ,g . Our analysis of this problem relies on the idea of the Hilbert rigging (refer to [16]). The notion of the rank one extensions of weighted join operators is partly motivated by the graph-model arising from the semigroup structures on directed trees. Interestingly, the above graph-model plays a decisive role in deriving various spectral properties of these operators. The class of rank one perturbations of diagonal operators has been studied extensively in the context of hyperinvariant subspace problem [26–29, 37, 44, 48, 69] and spectral analysis [10–12, 24, 32, 73]. The reader is referred to [65] for a survey on the classical theory of self-adjoint rank one perturbations of self-adjoint operators (refer also to [49] for its connection with the theory of singular integral operators). The class of rank one perturbations of diagonal operators also arises naturally in a problem of

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domain inclusion in the context of weighted shifts on directed trees (see [38, Theorem 4.2.2]). We find it necessary to comment upon the relationship of the present work to the existing literature. The class of weighted join operators has essentially no intersection with the existing class (RO), as studied in [28], of bounded rank one perturbations of bounded diagonal operators. Unlike the case of operators in (RO), commutants of weighted join operators are not necessarily abelian. It turns out that there are no non-normal hyponormal weighted join operators (refer to [41, 58] for basics of unbounded hyponormal operators). In the context of bounded rank one perturbations of bounded normal operators, similar behaviour has been observed in [44]. On the other hand, the class of weighted join operators and their rank one extensions contains bounded as well as unbounded complex Jordan operators (and n-symmetric operators) in abundance (refer to [1–4, 8, 9, 36, 45, 51] for the basic theory of Jordan operators, n-symmetric operators, and their connections with the classes of n-normal operators and n-isometries). Further, it overlaps with the class of sectorial operators, and also provides a family of examples of non-normal compact operators with large null summand in the sense of Anderson [6]. We would also like to draw attention to the works [7, 13, 53, 55, 56, 72] on the spectral theory of unbounded operator matrices on non-diagonal domains (refer also to the authoritative exposition [71] on this topic). The rank one extensions W f ,g of weighted join operators fit into the class of operator matrices with not necessarily of diagonal domain. Further, under some compatibility conditions, W f ,g is diagonally dominant in the sense of [72] with the exception that exactly one of its entries is not closable. In Sects. 1.1 and 1.2 of this section, we collect preliminaries pertaining to the directed trees and the Hilbert space operators respectively (the reader is referred to [31, 38] for the basics of graph theory, and [64, 66] for that of Hilbert space operators). In particular, we set notations and introduce some natural and known classes of directed trees and unbounded Hilbert space operators, which are relevant to the investigations in this paper. In Sect. 1.3, we collect several simple but basic properties of bounded and unbounded rank one operators. We conclude this section with a prologue including some important aspects and the layout of the paper. 1.1 Directed Trees A pair T = (V , E) is said to be a directed graph if V is a non-empty set and E is a non-empty subset of V × V \{(v, v) : v ∈ V }. An element of V (resp. E) is referred to n of distinct vertices is said as a vertex (resp. an edge) of T . A finite sequence {vi }i=1 to be a circuit in T if n ≥ 2, (vi , vi+1 ) ∈ E for all 1 ≤ i ≤ n − 1 and (vn , v1 ) ∈ E. Given u, v ∈ V , by a directed path from u to v in T , we understand a finite sequence {u 1 , . . . , u k } in V such that u 1 = u, (u j , u j+1 ) ∈ E (1 ≤ j ≤ k − 1) and u k = v. We say that two distinct vertices u and v of T are connected by a path if there exists a finite sequence {u 1 , . . . , u k } of distinct vertices of T such that

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u 1 = u, (u j , u j+1 ) or (u j+1 , u j ) ∈ E (1 ≤ j ≤ k − 1) and u k = v. A directed graph T is said to be connected if any two distinct vertices of T can be connected by a path in T . For a subset W of V , define Chi(W ) :=



{v ∈ V : (u, v) ∈ E}.

u∈W

One may define inductively Chi n (W ) for a non-negative integer n as follows:  n

Chi

(W ) :=

W Chi(Chi n−1 (W ))

if n = 0, if n ≥ 1.

Given v ∈ V and an integer n ≥ 0, we set Chi n (v) := Chi n ({v}). An element of Chi(v) is called a child of v. For a given vertex v ∈ V , consider the set Par(v) := {u ∈ V : (u, v) ∈ E}. If Par(v) is a singleton, then the unique vertex in Par(v) is called the parent of v, which we denote by par(v). Define the subset Root(T ) of V by Root(T ) := {v ∈ V : Par(v) = ∅}. An element of Root(T ) is called a root of T . If Root(T ) is a singleton, then its unique element is denoted by root. We set V ◦ :=V \Root(T ). A directed graph T = (V , E) is called a directed tree if T has no circuits, T is connected and each vertex v ∈ V ◦ has a unique parent. A subgraph of a directed tree T which itself is a directed tree is said to be a subtree of T . A directed tree T is said to be (i) rooted if it has a unique root. (ii) locally finite if card(Chi(u)) is finite for all u ∈ V , where card(X ) stands for the cardinality of the set X . (iii) leafless if every vertex has at least one child. (iv) narrow if there exists a positive integer m such that card(Chi n (root)) ≤ m, n ∈ N.

(1.1.1)

The smallest positive integer m satisfying (1.1.1) will be referred to as the width of T . Remark 1.1 Note that any narrow directed tree is necessarily locally finite. However, the converse is not true. Consider, for instance, the binary tree (see [38, Example 4.3.1]). It is worth noting that there exist narrow directed trees with card(V≺ ) = ℵ0 , where V≺ denotes the set of branching vertices of T defined by V≺ := {u ∈ V : card(Chi(u)) ≥ 2}

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0

2

4

6

...

1

3

5

7

...

Fig. 1 A narrow tree T = (V , E) with width m = 2 and V≺ = 2N

(see Fig. 1). This is not possible if the directed tree is leafless. Let T = (V , E) be a rooted directed tree with root root. For each u ∈ V , the depth of u is the unique non-negative integer du such that u ∈ Chi du (root) (see [38, Corollary 2.1.5]). We discuss here the convergence of nets associated with rooted directed trees induced by the depth. Define the relation ≤ on V as follows: v ≤ w if dv ≤ dw ,

(1.1.2)

where dv denotes the depth of v in T . Note that V is a partially ordered set with partial order relation ≤, that is, ≤ is reflexive and transitive. Further, if V is infinite, then given two vertices v, w ∈ V , there exists u ∈ V such that v ≤ u and w ≤ u (the reader is referred to the discussion prior to [21, Remark 3.4.1] for details). In this text, we will frequently be interested in the nets {μv }v∈V of complex numbers induced by the above partial order (the reader is referred to [66, Chapter 2] for the definition and elementary facts pertaining to nets). Let T = (V , E) be a directed tree. For a vertex u ∈ V , we set par 0 (u) = u. Note that the correspondence par(·) : u → par(u) is a partial function in V . For a positive integer n, by the partial function par n (·), we understand par(·) composed with itself n-times. The descendant of a vertex u ∈ V is defined by ∞  Des(u) := · Chi n (u) (disjoint sum) n=0

(see the discussion prior to [38, Eqn (2.1.10)]). Note that Tu = (Des(u), E u ) is a rooted subtree of T with root u, where E u := {(v, w) ∈ E : v, w ∈ Des(u)}.

(1.1.3)

The ascendant or ancestor of a vertex u ∈ V is defined by Asc(u) := {par n (u) : n ≥ 1}. In particular, a vertex is its descendant, while it is not its ascendant. Although this is not standard practice, we find it convenient. Note that a directed path from u to v in T , denoted by [u, v], is unique whenever it exists. Indeed, since there exists a path from u to v in T , v ∈ Des(u) and dv ≥ du . In this case, it is easy to see that [u, v] = {par n (v) : n = dv − du , dv − du − 1, . . . , 0}.

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Further, for u, v ∈ V , we set  (u, v] :=

[u, v]\{u} if v ∈ Des(u), ∅ otherwise.

We also need the following subsets of V : For u ∈ V and v ∈ Des(u), Desv [u] := Des(u)\(u, v] Desv (u) := Des(u)\[u, v].

 (1.1.4)

Note that Desu [u] = Des(u) and Desu (u) = Des(u)\{u}. 1.2 Hilbert Space Operators For a subset  of the complex plane C, let int(),  and C\ denote the interior, the closure and the complement of  in C respectively. We use R to denote the real line, and z and z denote the real and imaginary parts of a complex number z respectively. The conjugate of the complex number z will be denoted by z¯ , while arg(z) stands for the argument of a non-zero complex number z. We reserve the notation N for the set of non-negative integers, while Z (resp. Z+ ) stands for the set of all integers (resp. all positive integers). Unless stated otherwise, all the Hilbert spaces occurring below are complex, infinite-dimensional and separable. Let H be a complex, separable Hilbert space with the inner product ·, · H and the corresponding norm ·H . Whenever there is no ambiguity, we will suppress the suffix and simply write x, y

 and x in place of x, y H and xH respectively. By span{x ∈ H : x ∈ W } (resp. {x ∈ H : x ∈ W }), we mean the smallest linear subspace (resp. smallest closed linear subspace) generated by the subset W of H. In case W = {x}, we use the simpler notation [x] for the linear span of W . The orthogonal complement of a closed subspace W of H is denoted by H  W . Sometimes H  W is denoted by W ⊥ . Let S be a densely defined linear operator in H with domain D(S). The symbols ker S and ran S will stand for the kernel of S and the range of S respectively. We use σ p (S), σap (S), σ (S) to denote the point spectrum, the approximate-point spectrum, and the spectrum of S respectively. It may be recalled that σ p (S) is the set of eigenvalues of S, that σap (S) is the set of those λ in C for which S − λ is not bounded below, and that σ (S) is the complement of the set of those λ in C for which (S − λ)−1 exists as a bounded linear operator on H. Here, by S − λ, we understand the linear operator S − λI with I denoting the identity operator on H. We reserve the symbol B(H) for the unital C ∗ -algebra of bounded linear operators on H. The resolvent set ρ(S) of S is defined as the complement of σ (S) in C. The resolvent function R S : ρ(S) → B(H) is given by R S (λ) := (S − λ)−1 , λ ∈ ρ(S). The regularity domain π(S) of S is defined as the complement of σap (S) in C. For μ ∈ π(S), we refer to the linear subspace ran(S − μ)⊥ of H the deficiency subspace

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of S at μ and its dimension d S (μ) := dim ran(S − μ)⊥ ,

(1.2.1)

the defect number of S at μ. By the multiplicity function of S, we understand the function m S : σ p (S) → Z+ ∪ {ℵ0 } assigning with each eigenvalue λ of S, the dimension of the eigenspace E S (λ) of S corresponding to λ. We extend m S to the entire complex plane by setting m S (λ) = 0, λ ∈ C\σ p (S). We say that a densely defined linear operator S in H is Fredholm if the range of S is closed, dim ker S and dim ker S ∗ are finite. The essential spectrum σe (S) of S is the complement of the set of those λ ∈ C for which S − λ is Fredholm. The Fredholm index ind S : C\σe (S) → Z is given by ¯ λ ∈ C\σe (S) ind S (λ) := m S (λ) − m S ∗ (λ), (the reader is referred to [46, 52, 64] for elementary properties of various spectra of unbounded linear operators). Let T be a densely defined linear operator in H with domain D(T ). The closure (resp. adjoint) of T is denoted by T (resp. T ∗ ), whenever it exists. A subspace D of H is said to be a core of a closable linear operator T if D ⊆ D(T ), D = H, and T |D = T . If S is a linear operator in H such that D(S) ⊆ D(T ) and Sh = T h for every h ∈ D(S), then we say that T extends S (denoted by S ⊆ T ). Note that two operators S and T are same if and only if S ⊆ T and T ⊆ S. A closed linear subspace M of H is said to be invariant for T if T (M ∩ D(T )) ⊆ M. In this case, the restriction of T to M is denoted by T |M . Note that if T has invariant domain, that is, T D(T ) ⊆ D(T ), then T admits polynomial calculus in the sense that p(T ) is a well-defined linear operator with domain D(T ) for every complex polynomial p in one variable. A closed linear subspace M of H is reducing for T if there exist linear operators T0 in M and T1 in M⊥ such that T = T0 ⊕ T1 . The commutant of a linear operator T is given by {T } := {A ∈ B(H) : AT ⊆ T A}. In case T ∈ B(H), {T } = {A ∈ B(H) : AT = T A}. If PM is an orthogonal projection of H onto a closed subspace M of H, then PM ∈ {T } if and only if M is a reducing subspace for T (see [64, Proposition 1.15]). We recall definitions of some well-studied classes of unbounded linear operators, which are relevant to the present investigations (refer to [52, 57, 62, 64]). A densely defined linear operator T in a complex Hilbert space H is said to be (i) self-adjoint if D(T ) = D(T ∗ ) and T ∗ x = T x for all x ∈ D(T ).

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(ii) normal if D(T ) = D(T ∗ ) and T ∗ x = T x for all x ∈ D(T ). (iii) nilpotent if T has invariant domain such that T n = 0 for some positive integer n. The smallest positive integer with this property is referred to as the nilpotency index of T . (iv) complex Jordan if there exist a normal operator M and a nilpotent linear operator N such that any one of the following holds: (a) M ∈ B(H), T = M + N and M ∈ {N } , (b) N ∈ B(H), T = M + N and N ∈ {M} . 1.3 Rank One Operators We will see that the rank one (possibly unbounded) operators form building blocks in the orthogonal decomposition of weighted join operators (see Theorem 3.13). Hence we find it necessary to collect below several elementary properties of rank one operators. Let H be a complex Hilbert space. For x, y ∈ H, the injective tensor product x ⊗ y of x and y is defined by x ⊗ y(h) = h, y x, h ∈ H. Clearly, x ⊗ y is a rank one bounded linear operator. In fact, x ⊗ y = xy. Conversely, every rank one bounded linear operator arises in this fashion. Indeed, if T ∈ B(H) is a rank one operator with range spanned by a unit vector y ∈ H, then for every x ∈ H, T x = αx y for some scalar αx ∈ C, and hence T x = T x, y y = x, T ∗ y y = (y ⊗ T ∗ y)(x), x ∈ H (cf. [67, Proposition 2.1.1]). It is worth noting that a diagonal operator Dλ on H with respect to an orthonormal basis {e j } j∈J and diagonal entries λ:={λ j } j∈J ⊆ C (counted with multiplicities) can be rewritten uniquely (up to permutation) in terms of injective tensor products as follows: Dλ =



λj ej ⊗ ej,

j∈J

where J is a directed set. Note that Dλ is a bounded linear operator on H if and only if λ forms a bounded subset of C. The analysis of weighted join operators relies on a thorough study of bounded and unbounded rank one operators. As we could not locate an appropriate reference for a number of facts essential in our investigations, we include their statements and elementary verifications. Lemma 1.2 Let H be a complex Hilbert space of dimension bigger than 1. For unit vectors x, y, z, w ∈ H, we have the following statements:

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(i) x ⊗ y is an algebraic operator: p(x ⊗ y) = 0 with p(λ) = λ(λ − x, y ), λ ∈ C.

(1.3.1)

(ii) σ (x ⊗ y) = {0, x, y } = σ p (x ⊗ y). Moreover, the eigenspace Ex⊗y (μ) corresponding to the eigenvalue μ of x ⊗ y is given by Ex⊗y (0) = [y]⊥ , Ex⊗y ( x, y ) = [x]. Thus the multiplicity mx⊗y (μ) of the eigenvalue μ is given by mx⊗y (0) = ℵ0 if dim H = ℵ0 , mx⊗y ( x, y ) = 1 if x, y = 0. (iii) The resolvent function Rx⊗y : ρ(x ⊗ y) → B(H) of x ⊗ y at μ is given by Rx⊗y (μ) = −



 1  (μ − x, y )P[x]⊥ + x ⊗ P[x]⊥ y + μ¯ x , p(μ)

where p is as given in (1.3.1). (iv) The commutant {x ⊗ y} of x ⊗ y is given by

{x ⊗ y} = A ∈ B(H) : Ax = Ax, x x, A∗ y = Ax, x y . (v) x ⊗ y is normal if and only if there exists a unimodular scalar α ∈ C such that x = α y. (vi) x ⊗ y is self-adjoint if and only if x = ± y. Proof It is easy to see the following: x ⊗ α y = α¯ x ⊗ y, α ∈ C, (x ⊗ y)∗ = y ⊗ x, (x ⊗ y)(z ⊗ w) = z, y x ⊗ w.

(1.3.2)

To see (i), note that by (1.3.2), x ⊗ y satisfies (x ⊗ y)2 = x, y x ⊗ y. To see (ii), note that (x ⊗ y)(z) = 0 if z ∈ [y]⊥ , (x ⊗ y − x, y )x = 0. Further, by (i) and the spectral mapping property for polynomials [67], p(σ (x ⊗ y)) = σ ( p(x ⊗ y)) = {0}, where p(z) = z(z − x, y ). The desired conclusions in (ii) are now immediate.

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To see the formula for the resolvent function Rx⊗y in (iii), let f , g ∈ H be such that (x ⊗ y − μ) f = g. Writing h = P[x]⊥ h + h, x x for h ∈ H, and comparing coefficients, we obtain P[x]⊥ f = −

1 P ⊥ g, g, x = f , y − μ¯ x . μ [x]

It follows that g, x = P[x]⊥ f + f , x x, y − μ¯ x

= P[x]⊥ f , y + f , x ( x, y − μ) 1 = − P[x]⊥ g, y + f , x ( x, y − μ). μ This yields f = P[x]⊥ f + f , x x   1 1 1 g, x + P[x]⊥ g, y x. = − P[x]⊥ g + μ x, y − μ μ It is now easy to see that Rx⊗y (μ) has the desired expression. To see (iv), note that A commutes with x ⊗ y if and only if z, y Ax = Az, y x, z ∈ H.

(1.3.3)

If z ∈ [y]⊥ , then Az, y x = 0, and hence Az ∈ [y]⊥ . This shows that A∗ maps [y] into [y]. Letting z = y in (1.3.3), we obtain the necessity part of (iv). Conversely, if Ax = Ay, y x, then (1.3.3) is equivalent to A(z − z, y y), y = 0, which is equivalent to A∗ y = Ay, y y. To see (v), note that by (1.3.2), (x ⊗ y)∗ = y ⊗ x, and apply (iv) to A = y ⊗ x. The sufficiency part of (vi) follows immediately from (1.3.2). Assume that (x ⊗ y)∗ = x ⊗ y. By (1.3.2), x ⊗ y = y ⊗ x. By (v), x = α y for some α ∈ C. Thus α y ⊗ y = x ⊗ y = y ⊗ x = α¯ y ⊗ y. It follows that α ∈ R. Since x, y are unit vectors, α = ± 1.

 

Recall that a densely defined linear operator T in H admits a compact resolvent if there exists λ ∈ C\σ (T ) such that (T − λ)−1 is compact. It may be concluded from Lemma 1.2(iii) that x ⊗ y has a compact resolvent if and only if dim H is finite. Let us discuss a class of unbounded, densely defined, but not necessarily closed (rank one) operators, which we denote as f  g, f ∈ H and g is unspecified (to be chosen later). Fix an orthonormal basis {e j } j∈J of H for some directed set J , and let g = {g( j)} j∈J .

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Define f  g in H by D( f  g) =

⎧ ⎨

⎫ ⎬



h∈H: h( j)g( j) is convergent , ⎩ ⎭ j∈J ⎛ ⎞ ∞  f  g(h) = ⎝ h( j)g( j)⎠ f , h ∈ D( f  g). j∈J

Note that f  g is densely defined with span{e j : j ∈ J } ⊆ D( f  g). Thus the Hilbert space adjoint ( f  g)∗ of f  g is well-defined. Recall that for p ∈ [1, ∞],  p (J ) is the Banach space of all p-summable complex functions f : J → C endowed with the norm

 f p =

⎧⎛ ⎞1/ p ⎪ ⎪  ⎪ ⎪ ⎨⎝ | f ( j)| p ⎠ if p < ∞, j∈J ⎪ ⎪ ⎪ ⎪ | f ( j)| sup ⎩

if p = ∞.

j∈J

It turns out that ( f  g)∗ is not densely defined unless g ∈ 2 (J ). Indeed, we have the following result: Lemma 1.3 Let J be an infinite directed set. Fix an orthonormal basis {e j } j∈J of H, f ∈ H\{0}, and let g = {g( j)} j∈J . Then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

f  g admits a bounded linear extension to H. f  g is closed. f  g is closable. g ∈ 2 (J ). D( f  g) = H.

In case g ∈ / 2 (J ), ( f  g)∗ is not densely defined, σ ( f  g) = C, and

σ p ( f  g) =

⎧   ⎪ ⎨ 0, f ( j)g( j) if f ∈ D( f  g), ⎪ ⎩{0}

j∈J

otherwise.

Proof To see the equivalence of (i)–(v), it suffices to check that (iii) ⇒ (iv) and (v) ⇒ (i). Suppose that f  g is a closable operator with closed extension A. Since A is a densely defined closed operator, by [52, Theorem. 1.5.15], D(A∗ ) is dense in H (see also [64, Theorem 1.8]). Assume that g ∈ / 2 (J ). Then h ∈ D(A∗ ) if and only if there exists a positive real number c such that | Ax, h | ≤ cx, x ∈ D(A).

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However, for x =

 j∈F

g( j)e j with F ⊆ J and card(F) < ∞, Ax, h =



|g( j)|2 f , h .

j∈F

It follows that ⎛ ⎝



⎞1/2 |g( j)|2 ⎠

| f , h | ≤ c.

j∈F

Since g ∈ / 2 (J ), we must have f , h = 0. This shows that D(A∗ ) ⊆ H  [ f ]. Since f is non-zero, this contradicts the fact that the Hilbert space adjoint A∗ of A is densely defined. This completes the verification of the implication (iii) ⇒ (iv). The implication (v) ⇒ (i) may be derived from the uniform boundedness principle [23] applied to the family { f ⊗ gn }n≥0 of bounded linear operators, where {gn }n≥0 is any finitely supported sequence converging pointwise to g. To see the remaining part, assume that g ∈ / 2 (J ). Arguing as in the preceding paragraph (with A replaced by f  g), we obtain D(( f  g)∗ ) ⊆ H  [ f ]. The assertion that σ ( f  g) = C follows from the fact that any densely defined operator with a non-empty resolvent set is closed  (see [22, Lemma 1.17]). Note that any h ∈ D( f  g) such that j∈J h( j)g( j) = 0 (there are infinitely many such vectors h) is an eigenvector for f  g corresponding to the eigenvalue 0. Suppose f ∈ D( f  g). Then, as in the proof of Lemma 1.2(iii), it can be seen that j∈J f ( j)g( j) is an eigenvalue of f  g corresponding to the eigenvector f . Since any eigenvector of f  g corresponding to a non-zero eigenvalue belongs to [ f ], we conclude in this case that ⎧ ⎫ ⎨  ⎬ f ( j)g( j) . σ p ( f  g) = 0, ⎩ ⎭ j∈J

This also shows that if f ∈ / D( f  g), then f  g can not have a non-zero eigenvalue.   In case f  g is closable, the bounded extension of f  g, as ensured by Lemma 1.3, is precisely f ⊗ g. Otherwise, it may be concluded from (1.3.4) that D(( f  g)∗ ) = H  [ f ]. We conclude this section with a brief discussion on an interesting family of unbounded rank one operators. Example 1.4 Let J be an infinite directed set. Let f ∈ 2 (J )\{0} and let g ∈  p (J ), 1 ≤ p ≤ ∞. Then f  g defines a densely defined rank one operator in 2 (J ) with domain D( f  g) = q (J ) ∩ 2 (J ), where 1 ≤ q ≤ ∞ is such that 1p + q1 = 1. Indeed, since ( f  g)(h)2 ≤ hq g p  f 2 , h ∈ q (J ),

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f  g is bounded when considered as a linear transformation from q (J ) into 2 (J ). Moreover, since  p (J ) ⊆ 2 (J ) for 1 ≤ p ≤ 2, by Lemma 1.3, f  g belongs to B(2 (J )) if and only if 1 ≤ p ≤ 2. Thus for g ∈  p (J )\2 (J ) for some 2 < p ≤ ∞, f  g is a densely defined unbounded rank one operator in 2 (J ) with domain  q (J ) ∩ 2 (J ).

Prologue In the following discussion, we attempt to explain some important aspects of the present work with the aid of a family of rank one perturbations of weighted join operators. In the following exposition, we have tried to minimize the graph-theoretic prerequisites. In particular, we avoided the rather involved graph-theoretic definition : u, b ∈ V } of the so-called weighted join operators. of the class {Wλ(b) u Let T = (V , E) be a rooted directed tree with root root and let b, u ∈ V . Consider (b) (b) the closed subspace 2 (Uu ) of 2 (V ), where the subset Uu of V is given by Uu(b)

⎧ ⎪ if b = u, ⎨V \{u} = Asc(u) ∪ Desb (u) if b ∈ Desu (u), ⎪ ⎩ otherwise Desu (u)

(1.3.5)

(b)

(see (1.1.4)). Consider the standard orthonormal basis {ev }v∈U (b) of 2 (Uu ) and let (b)

(b)

u

Du denote the diagonal operator in 2 (Uu ) defined as Du(b) ev = λuv ev , v ∈ Uu(b) , where the diagonal entries of Du(b) are given by λuv :=dv − du , v ∈ Uu(b)

(1.3.6) (b)

with dv denoting the depth  of v ∈ V in T . Consider the rank one operator Nu eu ⊗ e Au , where e Au := v∈Au (dv − du )ev and the subset Au of V is given by  Au =

=

[u, b] if b ∈ Des(u), Asc(u) ∪ {b, u} otherwise. (b)

We also need the (possibly unbounded) rank one operator ew gx , where w ∈ V \Uu , x ∈ R and gx : Uu(b) → C is given by gx (v) = dvx , v ∈ Uu(b) .

(1.3.7)

From the viewpoint of the spectral theory, we will be interested in the the family (b) W :={Ww,x : w ∈ V \Uu , x ∈ R} of rank one extensions of weighted join operators

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defined as follows:

D(Ww,x ) = (h, k) : h ∈ D(Du(b) ) ∩ D(ew  gx ), k ∈ 2 (V \Uu(b) )  Ww,x =

 (b) Du 0 (b) . ew  gx Nu

Clearly, the linear operator Ww,x is densely defined in 2 (V ). Further, it is not difficult to see that Ww,x is unbounded unless Du(b) belongs to B(2 (Uu(b) )) and gx ∈ 2 (Uu(b) ). Many conclusions can be drawn about the family W of rank one extensions of weighted join operators. The analysis of Sects. 4 and 5 of this paper leads us to the following. Theorem 1.5 Let T = (V , E) be a rooted directed tree with root root and let b, u ∈ V . Assume that (Des(u), E u ) is a countably infinite narrow subtree of T (see (1.1.3)) and (b) (b) let Uu be as defined in (1.3.5). Then, for any w ∈ V \Uu and x ∈ R, the spectrum of Ww,x is a proper subset of C if and only if x < 1/2. In case x ∈ (−∞, 1/2), we have the following statements: (i) Ww,x is a closed operator with the domain being the orthogonal direct sum of D(Du(b) ) and 2 (V \Uu(b) ). (b) (ii) σ (Ww,x ) = {dv − du : v ∈ Uu ∪ {u}} = σ p (Ww,x ).

(b) (iii) σe (Ww,x )\{0} = dv − du : v ∈ Uu ∪ {u}, card(Chi dv (root)) = ∞ . Moreover, ind Ww,x = 0 on C\σe (Ww,x ). (iv) Ww,x is a sectorial operator, which generates a strongly continuous quasibounded semigroup. (v) Ww,x is never normal. (vi) If, in addition, T is leafless, then Ww,x admits a compact resolvent if and only if the set V≺ of branching vertices of T is disjoint from Asc(u). Remark 1.6 If x ≥ 1/2, then σ (Ww,x ) = C. Assume that x < 1/2. Then, by (1.3.5), / Des(u). Otherwise, σ (Ww,x ) = N if b ∈

k∈ N:k∈ / {1, . . . , db − du } ∪{−1, . . . , −du } ⊆ σ (Ww,x ) ⊆ N ∪ {−1, . . . , −du }. The exact description of σ (Ww,x ), x < 1/2 depends on the set V≺ ∩[u, b] and the leafstructure of branches emanating from this part (see Fig. 2 for a pictorial representation of the spectra of Ww,x , x ∈ R, where x varies over the X -axis while the spectra σ (Ww,x ) are plotted in the YZ -plane). Note that the spectral behaviour of the family {Ww,x }x∈R changes at x = 1/2 (see Fig. 2). A proof of Theorem 1.5 will be presented towards the end of Sect. 4. We remark that the conclusion of Theorem 1.5 may not hold in case Des(u) is not a narrow subtree of T (see Remark 5.9). It is tempting to arrive at the conclusion that almost everything about Ww,x can be determined. However, this is not the case. Although, the Hilbert

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1/2 •

•

σ(Ww,x ) : discrete x < 1/2

•

X-axis σ(Ww,x ): plane x  1/2

Y -axis

Fig. 2 Spectra of the family Ww,x , x ∈ R

space adjoint of Ww,x is densely defined in case −1 ≤ x < 1/2, neither we know its ∗ . Further, in case x ≥ 1/2, we do domain nor do we have a neat expression for Ww,x not know whether or not Ww,x is closable. Plan of the paper. We conclude this section with the layout of the paper. Section 2 provides the graph-theoretic framework essential in the study of weighted join operators. In particular, we introduce the notion of the extended directed tree and exploit the order structure on a rooted directed tree T = (V , E) to introduce a family of semigroup structures on the extended directed trees to be referred to as join operations at a base point (see Proposition 2.11). The join operation based at root (resp. ∞) is precisely the join (resp. meet) operation. We exhibit a pictorial illustration of the decomposition of the set V of vertices into descendant, ascendant, and the rest with respect to a fixed vertex (see (2.2.1) and Fig. 4). This decomposition of V helps to understand the action of join and meet operations. We conclude this section with (b) a description of the set Mu (w) of vertices which join to a given vertex u at another given vertex w (see Proposition 2.14). In Sect. 3, we see that the semigroup structures on the extended rooted directed tree T = (V , E), as introduced in Sect. 2, induce a family of operators referred . We show that for b = ∞, these operators are to as weighted join operators Wλ(b) u closed and that the linear span of the standard orthonormal basis of 2 (V ) forms a core for Wλ(b) (see Proposition 3.5). We further discuss the boundedness of weighted u join operators (see Theorem 3.7). One of the main results in this section decomposes a weighted join operator with a base point in V as the sum of a diagonal operator and a bounded operator of rank at most one (see Theorem 3.13). In case the base point is turns out to be possibly an unbounded finite rank ∞, the weighted meet operator Wλ(b) u (b)

operator. Among various properties of weighted join operators, it is shown that Wλu has large null summand except for at most finitely many choices of u (see Corollary 3.15). It is also shown that the weighted join operator is either a complex Jordan operator of index 2 or it has a bounded Borel functional calculus (see Corollary 3.16). We also characterize compact weighted join operators and discuss an application to

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the theory of commutators of compact operators (see Proposition 3.17 and Corollary 3.18). We further provide a description of the commutant of a bounded weighted join operator (see Theorem 3.19). We exhibit a concrete family of weighted join operators to conclude that the commutant of a weighted join operator is not necessarily abelian. In Sect. 4, we capitalize on the graph-theoretic framework developed in earlier sections to introduce and study the class of rank one extensions of weighted join operators. In particular, we discuss the closedness, the structure of the Hilbert space adjoint and the spectral theory of rank one extensions of weighted join operators. We introduce two compatibility conditions which control the unbounded rank one component in the matrix representation of rank one extensions of weighted join operators and discuss their connection with certain discrete Hilbert transforms (see Proposition 4.7). Moreover, we characterize these conditions under some sparsity conditions on the weight systems of weighted join operators (see Proposition 4.10). We provide a complete spectral picture (including a description of spectra, point-spectra and essential spectra) for rank one extensions of weighted join operators (see Theorem 4.15). It turns out that the spectra of rank one extensions of weighted join operators satisfying the so-called compatibility condition I can be recovered from its point spectra. In case the compatibility condition I does not hold, either rank one extensions of weighted join operators are not closed or their domain of regularity is empty (see Corollary 4.18). We give an example of a rank one extension of a weighted join operator with spectrum properly containing the topological closure of its point spectrum (see Example 4.19). Further, under the assumption of compatibility condition I, we characterize rank one extensions of weighted join operators on leafless directed trees which admit compact resolvent (see Corollary 4.20). Given an unbounded closed subset σ of the complex plane, we construct a non-trivial rank one extension of a weighted join operator with the spectrum same as σ (see Corollary 4.22). Finally, we specialize Theorem 4.15 to weighted join operators and conclude that these operators are not complete unless they are complex Jordan (see Theorem 4.23 and Corollary 4.25). In Sect. 5, we investigate various special classes of rank one extensions of weighted join operators. We exhibit a family of rank one extensions of weighted join operators, which are sectorial (see Proposition 5.1). A similar result is obtained for the generators of quasi-bounded strongly continuous semigroups (see Proposition 5.2). Further, we characterize the classes of hyponormal, cohyponormal, n-symmetric weighted join operators (see Propositions 5.3 and 5.5). It turns out that there are no non-normal hyponormal weighted join operators. On the other hand, if a weighted join operator is n-symmetric, then either n = 1 or n ≥ 3. We also discuss the normality and the symmetricity of rank one extensions of weighted join operators (see Propositions 5.4 and 5.7). Towards the end of this section we present a proof of Theorem 1.5. In Sect. 6, we discuss the counterpart of the theory of weighted join operators for rootless directed trees. The definition of join operation at a given base point becomes less obvious in view of the absence of depth of a vertex in a rootless directed tree. One of the main results in this section is a decomposition theorem analogous to Theorem 3.13 (b) (see Theorem 6.6). It turns out that a weighted join operator Wλu on a rootless directed tree could be a possibly unbounded rank one perturbation of (unbounded) diagonal (b) operator. Unlike the case of a rooted tree, the weighted join operator Wλu , b = ∞,

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need not be even closable (see Corollary 6.8). We also obtain a counterpart of Corollary 3.16 for weighted join operators on rootless directed trees (see Corollary 6.10). Finally, we briefly discuss some difficulties in the study of the rank one extensions of these operators. In Sect. 7, we glimpse at the general theory of unbounded non-self-adjoint rank one perturbations of diagonal operators or the forms associated with diagonal operators. In particular, we discuss the sectoriality of rank one perturbations of diagonal operators and some of its immediate applications to the spectral theory. We also discuss the role of some compatibility conditions in the sectoriality of the form-sum of the form associated with a sectorial diagonal operator and a form associated with not necessarily square-summable functions f and g. We conclude the paper with an epilogue including several remarks and possible lines of investigations.

2 Semigroup Structures on Extended Directed Trees In this section, we provide the graph-theoretic framework essential to introduce and study the so-called weighted join operators and their rank one extensions on rooted directed trees (see Sects. 3 and 4). In particular, we formally introduce the notion of the extended directed tree and exhibit a family of semigroup structures on it. We also present a canonical decomposition of a rooted directed tree suitable for understanding the actions of weighted join operators. 2.1 Join and Meet Operations on Extended Directed Trees In what follows, we always assume that card(V ) = ℵ0 . The following notion of the extended directed tree plays an important role in unifying theories of weighted join operators and weighted meet operators. Definition 2.1 (Extended directed tree) Let T = (V , E) be a directed tree. The extended directed tree T∞ associated with T is the directed graph (V∞ , E ∞ ) given by V∞ = V · {∞},

E ∞ = E · {(u, ∞) : u ∈ V }.

Remark 2.2 The element ∞ ∈ V∞ is descendant of every vertex in V . Indeed, ∞ ∈ Chi(u) for every u ∈ V . In view of Friedman’s notion of the graph with boundary (see [30, Pg 490]), it’s worth pointing out that ∞ is a boundary point when T∞ is considered as the graph with boundary. A pictorial representation of an extended directed tree T∞ = (V∞ , E ∞ ) with the vertex set V = {root, v1 , v2 , . . .} is given in Fig. 3. Definition 2.3 (Join operation) Let T = (V , E) denote a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T .

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v2

...

v4

∞

root

v1

...

v3

Fig. 3 An extended directed tree T∞

Given u, v ∈ V∞ , we say that u joins v if either u ∈ Des(v) or v ∈ Des(u). Further, we set ⎧ ⎪ ⎨u if u ∈ Des(v), u  v = v if v ∈ Des(u), ⎪ ⎩ ∞ otherwise. By Remark 2.2, ∞  u = ∞ = u  ∞ for any u ∈ V∞ . Further, the join operation  satisfies the following: ⎫ • (Commutativity) u  v = v  u for all u, v ∈ V∞ , ⎪ ⎪ ⎬ • (Associativity) (u  v)  w = u  (v  w) for all u, v, w ∈ V∞ , • (Neutral element) u  root = u = root  u for all u ∈ V∞ , ⎪ ⎪ ⎭ • (Absorbing element) u  ∞ = ∞ = ∞  u for all u ∈ V∞ .

(2.1.1)

We summarize (2.1.1) in the following lemma. Lemma 2.4 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Then the pair (V∞ , ) is a commutative semigroup admitting root as a neutral element and ∞ as an absorbing element. Before we define the meet operation, let us introduce the following useful notation. For u, v ∈ V , par(u, v):={w ∈ V : par n (u) = w = par m (v) for some m, n ∈ N}. Definition 2.5 (Meet operation) Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Let u, v ∈ V . We say that u meets v if there exists a unique vertex ω ∈ V such that sup

w∈par(u,v)

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dw = dω .

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In this case, we set u  v = ω. In case u ∈ V∞ , we set ∞  u = u = u  ∞. Remark 2.6 Note that u ∈ par(v, v) ⇒ u  v = u = v  u.

(2.1.2)

In fact, if u = par l (v) for some l ∈ N, then par(u, v) = {w ∈ V : par n (u) = w = par m (v) for some m, n ∈ N} = {w ∈ V : par m (u) = w for some m ∈ N}. The conclusion in (2.1.2) is now immediate. Lemma 2.7 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Any two vertices u, v ∈ V always meet. Further, they meet in a unique vertex ω belonging to [root, u] ∩ [root, v], so that maxw∈par(u,v) dw = dω . Proof Note that the set par(u, v) is non-empty. Indeed, root ∈ par(u, v), since par du (u) = root = par dv (v). Further, for any w ∈ par(u, v), there exist m, n ∈ N such that dw = du − n = dv − m ≤ min{du , dv } < ∞. This shows that sup

w∈par(u,v)

dw < ∞.

(2.1.3)

We claim that par(u, v) is finite. To see this, in view of (2.1.3), it suffices to check that for any two distinct vertices x, y ∈ par(u, v), dx = d y . Note that for some integers n 1 , n 2 ∈ N, par n 1 (u) = x, par n 2 (u) = y.

(2.1.4)

dx = du − n 1 , d y = du − n 2 ⇒ dx = d y + n 2 − n 1 .

(2.1.5)

It follows that

Since x = y, by (2.1.4), n 1 = n 2 . Hence, by (2.1.5), dx = d y . This also shows that sup par(u, v) is attained at a unique vertex in par(u, v). The remaining part also follows from this.  

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Let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with the rooted directed tree T . Then the meet operation  satisfies the following: ⎫ • (Commutativity) u  v = v  u for all u, v ∈ V∞ , ⎪ ⎪ ⎬ • (Associativity) (u  v)  w = u  (v  w) for all u, v, w ∈ V∞ , • (Neutral element) u  ∞ = u = ∞  u for all u ∈ V∞ , ⎪ ⎪ ⎭ • (Absorbing element) u  root = root = root  u for all u ∈ V∞ .

(2.1.6)

We summarize (2.1.6) in the following lemma. Lemma 2.8 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Then the pair (V∞ , ) is a commutative semigroup admitting ∞ as a neutral element and root as an absorbing element. The operations meet and join can be unified in the following manner. Definition 2.9 (Join operation at a base point) Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Fix b ∈ V∞ and let u, v ∈ V∞ . Define the binary operation b on V∞ by ⎧ ⎪ ⎪u  v ⎪ ⎨u u b v = ⎪v ⎪ ⎪ ⎩ uv

if u, v ∈ Asc(b), if v = b, if b = u, otherwise.

Remark 2.10 Clearly, root = . Further, by Remark 2.2, ∞ = . Thus we have a family of countably many operations, the first of which (corresponding to root) is the join operation, while the farthest operation (corresponding to ∞) is the meet operation. Proposition 2.11 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Then, for every b ∈ V , the pair (V∞ , b ) is a commutative semigroup admitting b as a neutral element and ∞ as an absorbing element. Proof Let b ∈ V . The fact that (V∞ , b ) is commutative and associative may be deduced from Lemmata 2.4 and 2.8. To complete the proof, note that b is a neutral   element, while ∞ is an absorbing element for (V∞ , b ). 2.2 A Canonical Decomposition of an Extended Directed Tree For a directed tree T = (V , E) and u ∈ V , the set V of vertices can be decomposed into three disjoint subsets: V = Des(u) · Asc(u) · Vu ,

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where Vu is the complement of Des(u)  Asc(u) in V (see Fig. 4 for a pictorial representation of this decomposition with u:=v4 ). Note that if T∞ = (V∞ , E ∞ ) is the extended directed tree associated with T , then V∞ decomposes as follows: V∞ = Des(u) · Asc(u) · Vu ,

(2.2.2)

where ∞ ∈ Des(u) by the very definition of the extended directed tree. C AUTION Whenever we consider the decomposition in (2.2.1), it is understood that the directed tree under consideration is T (and not the extended directed graph T∞ ), so that ∞ ∈ / Des(u). It turns out that the cardinality of Vu being infinite is intimately related to the large null summand property of weighted join operators (see Corollary 3.15). We record the following general fact for ready reference. Proposition 2.12 Let T = (V , E) be a directed tree and let u ∈ V . If Vu is as given in (2.2.1), then the following statements hold: (i) If T is rooted, then card(Vu ) = ℵ0 if and only if card(V \Des(u)) = ℵ0 . (ii) If T is leafless, then card(Vu ) = ℵ0 if and only if there exists a branching vertex w ∈ Asc(u). (iii) If T is leafless, then either card(Vu ) = 0 or card(Vu ) = ℵ0 . Proof Note that (i) follows from (2.2.1) and the fact that card(Asc(u)) < ∞ for any u ∈ V provided T is rooted. To see (ii), suppose T is leafless and assume that there exists w ∈ Asc(u) such that card(Chi(w)) ≥ 2. Thus there exists a vertex v ∈ Chi(w) such that v ∈ / Asc(u) ∪Des(u). Further, Des(v) is contained in Vu . Since T is leafless, card(Des(v)) = ℵ0 . This proves the sufficiency part of (ii). On the other hand, if all vertices in Asc(u) are non-branching, then V = Des(u) · Asc(u), which, by (2.2.1),   implies that Vu = ∅. This also yields (iii). To see the role of canonical decompositions (2.2.1) and (2.2.2) in determining the join and meet of two vertices, let us see an example. Example 2.13 Consider the rooted directed tree T = (V , E) as shown in Fig. 4. To get an essential idea about the operations of meet and join, let us compute v4  v and v4  v for v ∈ V . Note that ⎧ ⎪ ⎨v if v ∈ {v4 , v7 , . . . , }, v4  v = v4 if v ∈ {v2 , v0 , root}, ⎪ ⎩ ∞ if v ∈ {v1 , v3 , . . .} ∪ {v5 , v8 , . . .}. Similarly, one can see that ⎧ v4 ⎪ ⎪ ⎪ ⎨ v v4  v = ⎪ v0 ⎪ ⎪ ⎩ v2

if v if v if v if v

∈ {v4 , v7 , . . .}, ∈ {v2 , v0 , root}, ∈ {v1 , v3 , . . .}, ∈ {v5 , v8 , . . .}.

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v5

v8

...

v4

v7

...

v2

v0

root

... v1

v3

v6

...

Fig. 4 A rooted directed tree T = (V , E), where V is the disjoint union of Des(v4 ) = {v4 , v7 , . . .}, Asc(v4 ) = {v2 , v0 , root}, and Vv4 = {v1 , v3 , . . .} ∪ {v5 , v8 , . . .}

Note that two vertices in V can join at the vertex ∞, while two vertices in V always meet in V .  Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Fix u ∈ V . Then, for any b ∈ V∞ \{u} and v ∈ V , the binary operation b on V∞ satisfies the following: ⎧ u ⎪ ⎪ ⎪ ⎪ ⎪ v ⎪ ⎪ ⎪ ⎪ ⎪ ⎨v u b v = u ⎪ ⎪ ⎪ v ⎪ ⎪ ⎪ ⎪ ⎪u ⎪ ⎪ ⎩ v

if b ∈ Asc(u) ∪ Vu , v ∈ Asc(u), if b ∈ Des(u), v ∈ Asc(u), if b ∈ Asc(u), v ∈ Des(u), if b ∈ Des(u), v ∈ [u, b], if b ∈ Des(u)\{∞}, v ∈ Desb (u), if b = ∞, v ∈ Desb (u), if b ∈ Vu , v ∈ Des(u),

where [u, v] denotes the directed path from u to v in a directed tree T . The above discussion is summarized in the following table: We conclude this section with a useful result describing the set of vertices, which join to a given vertex (with respect to a base point) at another given vertex. Proposition 2.14 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V and w ∈ V∞ , define Mu(b) (w) := {v ∈ V : u b v = w}. Then the following statements hold:

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(2.2.3)

S. Chavan et al. Table 1 Join operation u b v at the base point b

b→ v↓

Asc(u)

Des(u)\{u, ∞}

u

Vu

{∞}

Asc(u)

u

v

v

u

v

Desb (u)

v

v

v

v

u

[u, b]

−

u

u

−

u

Vu \{b}

∞

∞

v

∞

uv

{b}

u

u

b

u

u

(i) If b = ∞, then ⎧ j

j−1 (u)) if w = par j (u), j = 1, . . . , d , ⎪ u ⎨Des(par (u))\Des(par (b) Mu (w) = Des(u) if w = u, ⎪ ⎩ ∅ otherwise.

(ii) If b ∈ V and u ∈ Asc(b), then ⎧ {w} ⎪ ⎪ ⎪ ⎨[u, b] Mu(b) (w) = ⎪ Vu ⎪ ⎪ ⎩ ∅

if w ∈ Asc(u)  Desb (u), if w = u, if w = ∞, otherwise.

(iii) If b ∈ V and u ∈ / Asc(b), then ⎧ {w} ⎪ ⎪ ⎪ ⎨Asc(u) ∪ {u, b} Mu(b) (w) = ⎪ Vu \{b} ⎪ ⎪ ⎩ ∅

if b = u or w ∈ Desu (u), if u = b and w = u, if w = ∞, otherwise.

(iv) If b ∈ V \{u}, then V∞ \Mu(b) (∞) = Asc(u) ∪ Des(u) ∪ {b}. (b)

Proof By the definition of join operation b , Mu (w) equals ⎧ {v ∈ Asc(b) : u  v = w} ∪ {v ∈ V \Asc(b) : u  v = w} ⎪ ⎪ ⎪ ⎨ {v ∈ Asc(b) : v ∈ Des(w)} ∪ ({b} ∩ V ) ⎪ {w} ⎪ ⎪ ⎩ {v ∈ V : u  v = w}

if u ∈ Asc(b), w = u, if u ∈ Asc(b), w = u, if b = u, if u ∈ / Asc(b), u = b.

The desired conclusions in (i)–(iii) can be easily deduced from the facts that Asc(∞) = V , u  v ∈ par(u, v), u  v ∈ Des(u) ∩ Des(v), u, v ∈ V .

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The parts (i)–(iii) may also be deduced from Table 1. To see (iv), let b ∈ V \{u}. As seen above, Mu(b) (∞)

 {v ∈ Asc(b) : u  v = ∞} ∪ Vu if u ∈ Asc(b), = {v ∈ V \{b} : u  v = ∞} otherwise,

where Vu is as given in (2.2.1). Thus, in any case, Mu(b) (∞) = Vu \{b}, and hence Asc(u) ∪ Des(u) ∪ {b} ⊆ V∞ \Mu(b) (∞). (b)

To see the reverse inclusion, let v ∈ V∞ \Mu (∞). Since ∞ ∈ Des(u), we may assume that v = ∞. Then u b v ∈ V , and hence we may further assume that v = b. If u, v ∈ Asc(b), then v ∈ Asc(u) ∪ Des(u). Otherwise, u  v = u b v ∈ V , and hence u ∈ Des(v) or v ∈ Des(u). In this case also, v ∈ Asc(u) ∪ Des(u). This yields the desired equality in (iv).   The last result turns out to be crucial in decomposing the so-called weighted join operator as a direct sum of a diagonal operator and a finite rank operator.

3 Weighted Join Operators on Rooted Directed Trees Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . In what follows, 2 (V ) stands for the Hilbert space of square-summable complex functions on V equipped with the standard inner product. Note that the set {eu }u∈V is an orthonormal basis of 2 (V ), where eu : V → C is the indicator function of {u}. The convention e∞ = 0 will be used throughout this text. Note that 2 (V ) is a reproducing kernel Hilbert space. Indeed, for every v ∈ V , the evaluation map f → f (v) is a bounded linear functional from 2 (V ) into C. For a non-empty subset W of V , 2 (W ) may be considered as a subspace of 2 (V ). Indeed, if one extends f : W → C by setting F := f on W and 0 on V \W , then the mapping U : 2 (W ) → 2 (V ) given by U f = F is an isometric homomorphism. Sometimes, the orthogonal projection P2 (W ) of 2 (V ) onto 2 (W ) 2 will be denoted by PW . We  say that a closed subspace M of  (V ) is supported on a subset W of V if M = {ev : v ∈ W }. In this case, we refer to W as the support of M and we write supp M := W . Definition 3.1 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V and b ∈ V∞ , by the the weight system λu = {λuv }v∈V∞ , we understand the subset {λuv }v∈V∞ of complex numbers such that λu∞ = 0. (i) The diagonal operator Dλu on T is given by    | f (v)|2 |λuv |2 < ∞ D(Dλu ) := f ∈ 2 (V ) : v∈V

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Dλu f :=

f (v)λuv ev ,

f ∈ D(Dλu ).

v∈V (b)

(ii) The weighted join operator Wλu on T is given by

(b) D(Wλu ) := f ∈ 2 (V ) : Λu(b) f ∈ 2 (V ) , (b)

(b)

Wλu f := Λu(b) f ,

f ∈ D(Wλu ),

(b)

where Λu is the mapping defined on complex functions f on V by 

(Λu(b) f )(w) :=

λuv f (v), w ∈ V

(3.0.1)

(b) v∈Mu (w)

(b)

with the set Mu (w) given by (2.2.3). The operator Wλ(∞) is referred to as the weighted meet operator. u Remark 3.2 Several remarks are in order. (i) It is well-known that Dλu is a densely defined closed linear operator in 2 (V ). Its adjoint Dλ∗u is the diagonal operator with diagonal entries {λuv }v∈V . Furthermore, Dλu is normal and DV is a core for Dλu , where DU = span {ev : v ∈ U }, U ⊆ V ,

(3.0.2)

(see [38, Lemma 2.2.1]). (b) (b) (ii) Note that D(Wλu ) forms a subspace of 2 (V ). Indeed, if f , g ∈ D(Wλu ), then (b)

(b)

for every w ∈ V , the series (Λu f )(w) and (Λu g)(w) converge, and hence (b) so does (Λu ( f + g))(w). In particular, by Proposition 2.14, these series are (b) (b) finite sums in case b = ∞. Also, Λu ( f + g) ∈ 2 (V ) if Λu ( f ) ∈ 2 (V ) and (b) Λu (g) ∈ 2 (V ). Indeed,

Λu(b) ( f + g)2 ≤ 2 Λu(b) ( f )2 + Λu(b) (g)2 . (b)

(iii) For every v ∈ V , ev ∈ D(Wλu ) and (b)

(Wλu ev )(w) =



λuη ev (η) = λuv eub v (w), w ∈ V .

(3.0.3)

(b) η∈Mu (w)

In particular, (b)

(b)

DV := span {ev : v ∈ V } ⊆ D(Wλu ), Wλu DV ⊆ DV . 268

(3.0.4)

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Thus all positive integral powers of Wλu are densely defined and the Hilbert (b) ∗ Wλu

(b) ∗

(b)

space adjoint of Wλu is well-defined. To see the action of Wλu , let 2 f ∈  (V ) and w ∈ V . Note that (b)

Wλu f , ew =



f (v)λuv eub v (w) =

v∈V



f (v)λuv = f , gw ,

(b) v∈Mu (w)

 where Mu(b) (w) is as given in (2.2.3) and gw = λ¯ e . Thus we (b) v∈Mu (w) uv v obtain     (b) ∗ (b) g(w)λ¯ uv ev , g ∈ D((Wλu )∗ ). Wλu g = w∈V v∈M (b) (w) u

(iv) Finally, note that if b = u, then Wλ(b) is the diagonal operator Dλu . u Let us see two simple yet instructive examples of rooted directed trees in which the associated weighted join operators take a concrete form. Both these examples of rooted directed trees have been discussed in [38, Eqn (6.2.10)] in the context of weighted shifts on directed trees. Example 3.3 (With no branching vertex) Consider the directed tree T1 with the set of vertices V = N and root = 0. We further require that Chi(n) = {n + 1} for all n ∈ N. (b) Let m ∈ V and n, b ∈ V∞ . By (3.0.3), the weighted join operator Wλm on T is given by ⎧ λmn emin{m,n} ⎪ ⎪ ⎪ ⎨ λmn em (b) Wλm en = ⎪ λ mn en ⎪ ⎪ ⎩ λmn emax{m,n}

if m < b and n < b, if n = b, if m = b and n ∈ N, otherwise,

where we used the assumption that λm∞ = 0 and the convention that max{m, ∞} = ∞. In particular,  (b) Wλm en

= (0)

λmn emax{m,n} if b = 0, λmn emin{m,n} if b = ∞. (∞)

The matrix representations of Wλm and Wλm with respect to the ordered orthonormal basis {en }n∈N are given by

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⎛

Wλ(0) m

0 ⎜ .. ⎜. ⎜ ⎜0 ⎜ ⎜ λm0 ⎜ =⎜ ⎜0 ⎜ .. ⎜. ⎜ ⎜ ⎜ ⎝ ⎛

Wλ(∞) m

λm0 ⎜0 ⎜ ⎜ .. ⎜. ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

··· ··· · · · λmm 0 ··· 0 λmm+1 .. . 0 .. . 0 ··· λm1 0 · · · .. .0 0 .. . λmm−1 0 .. .

··· 0

···

λmm+2 λmm+3

0 .. .

. ⎞

··· 0 ··· λmm λmm+1 0 .. .

..

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 .. .

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ···⎟ ⎟ ⎟ ···⎟ ⎠

(∞) Thus Wλ(0) is an at most rank one perturbation of a diagonal operator, while Wλm is m a finite rank operator with a range contained in the linear span of {ek : k = 0, . . . , m}. 

Example 3.4 (With one branching vertex) Consider the directed tree T2 with a set of vertices V = {0} ∪ {2 j − 1, 2 j : j ≥ 1} and root = 0. We further require that Chi(0) = {1, 2}, Chi(2 j − 1) = {2 j + 1} and Chi(2 j) = {2 j + 2}, j ≥ 1. Let m ∈ V and n ∈ V∞ . By (3.0.3), the weighted join (b) operator Wλm on T is given by ⎧ ⎪ ⎪λmn emn ⎪ ⎨λ e mn m (b) Wλm en = ⎪λmn en ⎪ ⎪ ⎩ λmn emn

if m, n ∈ Asc(b), if n = b, if m = b and n ∈ N, otherwise.

In particular, if m and n are positive integers, then  (0) Wλm en

=

λmn emax{m,n} if m, n are odd or m, n are even, 0 otherwise,

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2

4

...

m

...

1

3

...

...

...

0

Fig. 5 A pictorial representation of T2 with prescribed vertex m

 (∞) Wλm en

=

λmn emin{m,n} if m, n are odd or m, n are even, otherwise. λmn e0 (0)

(∞)

The matrix representations of Wλm and Wλm with respect to the ordered orthonormal basis {e2n }n∈N ∪ {e2n+1 }n∈N are given by ⎛

(0)

Wλm

0 ⎜ .. ⎜ . ⎜ ⎜ 0 ⎜ ⎜λm0 ⎜ =⎜ ⎜ 0 ⎜ .. ⎜ . ⎜ ⎜ ⎜ ⎝

⎞

··· ··· λm2 · · · λmm 0 ··· ··· 0 · · · 0 λmm+2 0 .. .. . . 0 λmm+4 0 .. . 0 λmm+6 .. .. . .

⎛

Wλ(∞) m

λm0 0 · · · ⎜ 0 λm2 0 · · · ⎜ ⎜ .. . ⎜ . 0 .. 0 ⎜ ⎜ .. . λmm−2 =⎜ ⎜ ⎜ 0 ⎜ ⎜ .. ⎜ . ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⊕ 0, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

λm1 λm3 · · · ··· 0 ··· λmm λmm+2 · · · 0 .. .

0 .. .

···

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(0)

(see Fig. 5). It turns out that Wλm is at most a rank one perturbation of a diagonal (∞)

operator, while Wλm is a finite rank operator with range contained in the linear span  of {em , em−2 , . . . , e0 }. The fact, as illustrated in the preceding examples, that the weighted join operator is either diagonal, a rank one perturbation of a diagonal operator or a finite rank operator holds in general (see Theorem 3.10).

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3.1 Closedness and Boundedness In this section, we discuss the closedness and boundedness of weighted join operators on rooted directed trees. Unless stated otherwise, b ∈ V∞ denotes the base point of (b) the weighted join operator Wλu . Proposition 3.5 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Let b, u ∈ V and let λu = {λuv }v∈V∞ be a weight system of complex numbers. Then the weighted (b) join operator Wλu on T defines a densely defined closed linear operator. Moreover, . DV := span{ev : v ∈ V } forms a core for Wλ(b) u (b)

Proof We have already noted that Wλu is densely defined (see Remark 3.2). Let (b)

{ f n }n∈N be a sequence converging to f in 2 (V ). Suppose that {Wλu f n }n∈N converges to some g ∈ 2 (V ). Since 2 (V ) is a reproducing kernel Hilbert space, for every w ∈ V,  λuv f n (v) = g(w), lim f n (w) = f (w), lim n→∞

n→∞ (b) v∈Mu (w)

(b)

(b)

where Mu (w) is given by (2.2.3). However, since b = ∞, card(Mu (w)) < ∞ for (b) each w ∈ V (see (ii) and (iii) of Proposition 2.14). It follows that f ∈ D(Wλu ) and (b)

(b)

Wλu f = g. Thus Wλu is a closed linear operator.

, note that by the preceding discussion, Wλ(b) | To see that DV is a core for Wλ(b) u u DV (b)

(b)

is a closable operator such that Wλu |DV ⊆ Wλu . To see the reverse inclusion, let  (b) f = v∈V f (v)ev ∈ D(Wλu ) and let f n :=



f (v)ev , n ∈ N.

v∈V dv ≤n

Then { f n }n∈N ⊆ DV ,  f n − f 2 (V ) → 0 as n → ∞ and (b)



(b)

Wλu f n − Wλu f 22 (V ) = =

f (v)λuv eub v

v∈V dv >n

2 2 (V )

!2  !!  ! f (v)λuv ! , ! w∈V v∈V dw >n ub v=w

f ∈ 2 (V ). It follows that f ∈ which converges to 0 as n → ∞, since Wλ(b) u

D(Wλ(b) | ) and Wλ(b) f = Wλ(b) | f . This yields Wλ(b) | = Wλ(b) . u DV u u DV u DV u

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Remark 3.6 This result no more holds true for the weighted meet operator Wλu . (∞)

Indeed, it may be concluded from Theorem 3.7 below and Lemma 1.3 that Wλu not be even closable.

may

(b)

We discuss next the boundedness of weighted join operators Wλu . Theorem 3.7 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Let u ∈ V , b ∈ V∞ and let λu = {λuv }v∈V∞ be a weight system of complex numbers. Then the weighted join (b) operator Wλu on T is bounded if and only if ⎧ 2 ⎪ if b = ∞, ⎨ (V ) λu belongs to ∞ (V ) if b = u, ⎪ ⎩ ∞  (Des(u)) otherwise.

(3.1.1)

 (b) Proof Let f = v∈V f (v)ev ∈ 2 (V ) be of norm 1 and let Λu be as defined in (b) (b) (3.0.1). Recall that f ∈ D(Wu ) if and only if Λu f ∈ 2 (V ). By (3.0.3), Λu(b) f =



f (v)λuv eub v .

v∈V (b)

It follows that f ∈ D(Wu ) if and only if Λu(b) f 2 =

 !! ! w∈V

!2 ! λuv f (v)!



(3.1.2)

(b) v∈Mu (w)

is finite, where Mu(b) (w) is as given in (2.2.3). We divide the proof into the following three cases: Case I. b = ∞: Let u j := par j (u), j = 0, . . . , du . By Proposition 2.14(i) and (3.1.2), we obtain Λu(∞) f 2 =

 !! ! w∈V

=

(∞)

v∈Mu

du ! 

! !

!2 ! λuv f (v)!

 (w)

!2 ! λuv f (v)! ,



(3.1.3)

j=0 v∈Des(u j )\Des(u j−1 ) (∞)

where we used the convention that Des(par −1 (u)) = ∅. We claim that Wλu belongs to B(2 (V )) if and only if λu ∈ 2 (Des(u j )\Des(u j−1 )),

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j = 0, . . . , du .

(3.1.4)

S. Chavan et al. u2 = root

u1 u0

Fig. 6 The decomposition (3.1.5) of V∞ with u = u 0 of depth 2

If (3.1.4) holds, then by (3.1.3) and the Cauchy–Schwarz inequality, Λu(∞) f 2 ≤  f 2

du 



|λuv |2 ,

j=0 v∈Des(u j )\Des(u j−1 ) (∞)

which shows that Wλu (3.1.3), ! ! sup !

(∞)

∈ B(2 (V )). Conversely, if Wλu 

 f =1 v∈Des(u )\Des(u j j−1 )

! ! λuv f (v)! < ∞,

∈ B(2 (V )), then by

j = 0, . . . , du .

By the standard polar representation, the series above is indeed absolutely convergent, and hence by Riesz representation theorem [23], λu ∈ 2 (Des(u j )\Des(u j−1 )), j = 0, . . . , du . Thus the claim stands verified. To complete the proof, it now suffices to check that V∞

du    = · Des(u j )\Des(u j−1 )

(3.1.5)

j=0

(see Fig. 6). To see that, let v ∈ V∞ . Clearly, Des(u) = Des(u 0 )\Des(u −1 ). Thus we may assume that v ∈ V∞ \Des(u). In view of (2.2.2), we must have v ∈ Asc(u) · Vu . If v ∈ Asc(u), then there exists j ∈ {1, . . . , du } such that v = par j (u) = u j , and hence v ∈ Des(u j )\Des(u j−1 ). Hence we may further assume that v ∈ Vu . By Lemma 2.7, u  v ∈ [root, u] = {u du , . . . , u 0 }. Thus u  v = u j for some j = 0, . . . , du , and therefore v ∈ Des(u j ). However, by the uniqueness of the meet operation, v ∈ / Des(u j−1 ). This completes the verification of (3.1.5).

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Case II. b ∈ V & b ∈ Des(u): = Dλu , and hence Wλ(b) ∈ B(2 (V )) if and only If b = u, then by Remark 3.2, Wλ(b) u u ∞ if λu ∈  (V ). Assume that u = b. Then, by Proposition 2.14(ii),  !! ! w∈V



!2 ! λuv f (v)! =

(b)

v∈Mu (w)





! ! !λuw f (w)!2 +

w∈Asc(u)

!  !2 ! ! +! λuv f (v)! .

! ! !λuw f (w)!2

w∈Desb (u)

v∈[u,b]

Since Asc(u), [u, b] are finite sets, Λu(b) f ∈ 2 (V )

⇐⇒



! ! !λuw f (w)!2 < ∞.

(3.1.6)

w∈Des(u)

It is now clear that Λu(b) f ∈ 2 (V ) for every f ∈ 2 (V ) if and only if λu ∈ ∞ (Des(u)). This shows that (3.1.1) is a necessary condition. Conversely, if (3.1.1) holds then Λu(b) f ∈ 2 (V ) for every f ∈ 2 (V ), and hence by the closed graph theorem together with Proposition 3.5, Wλ(b) defines a bounded linear operator on u 2 (V ). Case III. b ∈ V & b ∈ / Des(u): By Proposition 2.14(iii),  !! ! w∈V



!2 ! λuv f (v)! =

(b)

v∈Mu (w)



! ! ! !λuw f (w)!2 + !!



!2 ! λuv f (v)! .

v∈Asc(u)∪{u,b}

w∈Desu (u)

(b)

Once again, since Asc(u) is a finite set, we must have (3.1.6). It follows that Λu f ∈ 2 (V ) for every f ∈ 2 (V ) if and only if λu ∈ ∞ (Des(u)). The verification of the remaining part in this case is now similar to that of Case II.   Remark 3.8 Note that the weighted join operator Wλ(root) on T is bounded if and only u (∞)

if λu ∈ ∞ (Des(u)). Further, the weighted meet operator Wλu and only if λu ∈ 2 (V ).

on T is bounded if

An examination of Cases II and III of the proof of Theorem 3.7 yields a neat (b) expression for the domain of weighted join operator Wλu , b = ∞: Corollary 3.9 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Let u ∈ V , b ∈ V and let λu = {λuv }v∈V∞ be a weight system of complex numbers. For u ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let Dλu be the diagonal operator with diagonal entries λu . Then, for any b ∈ V , the domain of the weighted on T is given by D(Wλ(b) ) = D(PDes(u) Dλu ). join operator Wλ(b) u u Proof This follows from (3.1.6), which holds for any b ∈ V .

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3.2 A Decomposition Theorem (b)

One of the main results of this section shows that the weighted join operator Wλu on a rooted directed tree can be one of the following three types, viz. a diagonal operator, a rank one perturbation of a diagonal operator or a finite rank operator. Further, we obtain an orthogonal decomposition of Wλ(b) into a diagonal operator and u a rank one operator provided b = u. Among various applications, we exhibit a family of weighted join operators with large null summand. It turns out that either a weighted join operator is complex Jordan or it has bounded Borel functional calculus. Before we state the first main result of this section, we introduce the function eμ,A , which appears in the decomposition of weighted join operators. For a subset A ⊆ V and μ := {μv : v ∈ A} ⊆ C, consider the function eμ,A : V → C given by eμ,A :=



μ¯ v ev .

(3.2.1)

v∈A

Note that eμ,A ∈ 2 (V ) if and only if μ ∈ 2 (A). Theorem 3.10 Let T = (V , E) denote a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let Dλu be the diagonal operator with diagonal entries λu . Then, for any b ∈ V \{u}, the weighted (b) join operator Wλu on T is given by (b)

D(Wλu ) = D(PDes(u) Dλu ),  PAsc(u)∪Desb [u] Dλu + eu ⊗ eλu ,(u,b] if b ∈ Des(u), (b) Wλu = PDes(u) Dλu + eu ⊗ eλu ,Asc(u)∪{b} otherwise. (b)

Remark 3.11 In case b = u, by Remark 3.2(iv), Wλu is the diagonal operator Dλu . In case b = ∞, by (3.0.3), (∞)

Wλu ev = λuv euv , v ∈ V . (∞)

It now follows from Lemma 2.7 that Wλu

(3.2.2)

is a finite rank operator. Let us find an

(∞) explicit expression for Wλu . By (3.1.5), 2 (V ) admits the orthogonal decomposition

2 (V ) =

du "

2 (Des(u j )\Des(u j−1 )),

j=0

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where Des(u −1 ) = ∅ and u j := par j (u) for j = 0, . . . , du . By (3.2.2), with respect (∞) to the above decomposition, Wλu decomposes as (∞) D(Wλu )

=

du " j=0

(∞) Wλu

=

du "

⎫

⎪ ⎪ D eu j  eλu ,Des(u j )\Des(u j−1 ) , ⎪ ⎪ ⎪ ⎬ eu j  eλu ,Des(u j )\Des(u j−1 ) .

j=0

⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(3.2.3)

Thus Wλ(∞) is an orthogonal direct sum of rank one operators u eu j  eλu ,Des(u j )\Des(u j−1 ) ,

j = 0, . . . , du . (b)

Proof Let b ∈ V \{u}. By Corollary 3.9, D(Wλu ) = D(PDes(u) Dλu ). To see the (b) W λu ,

decomposition of Proposition 2.14(iv),

consider the subset

(b) Mu (∞)

of V as given in (2.2.3). By

V \Mu(b) (∞) = Asc(u) ∪ Des(u) ∪ {b} = (V \Vu ) ∪ {b},

(3.2.4)

where Vu is as given in (2.2.1). Note that (3.2.4) induces the orthogonal decomposition  (b) if b ∈ V \Vu , 2 (Asc(u)) ⊕ 2 (Des(u)) ⊕ 2 (Mu (∞))  (V ) = 2 (b) 2 2 2  (Asc(u)) ⊕  (Des(u)) ⊕  (Mu (∞)) ⊕  ({b}) otherwise. 2

(3.2.5)

It may be concluded from Table 1 that (b)

Wλu (2 (Asc(u))) ⊆ 2 (V )  2 (Mu(b) (∞)), (b)

Wλu (2 (Des(u))) ⊆ 2 (Des(u)), (2 (Mu(b) (∞))) = {0}, Wλ(b) u (b)

Wλu (2 ({b})) ⊆ 2 ({u}). With respect to the orthogonal decomposition (3.2.5) of 2 (V ), the weighted join (b) operator Wλu decomposes as follows: ⎡

Wλ(b) u ⎡

Wλ(b) u

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⎤ (b) W11 0 0 (b) (b) ⎦ = ⎣ W21 W22 0 0 0 0 (b)

0 W11 ⎢ W (b) W (b) 21 22 =⎢ ⎣ 0 0 0 0

if b ∈ V \Vu ,

⎤ 0 0 0 λub eu ⊗ eb ⎥ ⎥ if b ∈ Vu . ⎦ 0 0 0 0

277

(3.2.6)

(3.2.7)

S. Chavan et al.

Note that for any v ∈ Asc(u), by Table 1, (b)

(b)

W11 ev = PAsc(u) Wλu ev = λuv PAsc(u) eub v  0 if b ∈ Asc(u) or b ∈ Vu , = Dλu ev if b ∈ Des(u).

(3.2.8)

A similar argument using Table 1 shows that for any v ∈ Asc(u), (b)

W21 ev = λuv PDes(u) eub v ⎞ ⎧⎛ ⎪  ⎪ ⎨⎝ λuw eu ⊗ ew ⎠ ev if b ∈ Asc(u) or b ∈ Vu , = (3.2.9) w∈Asc(u) ⎪ ⎪ ⎩ 0 if b ∈ Des(u). Further, for any v ∈ Des(u), by Table 1, (b)

W22 ev = λuv eub v ⎧ Dλu ev if b ∈ Asc(u) or b ∈ Vu , ⎪ ⎪⎛ ⎞ ⎪ ⎪ ⎨  = ⎝ λuw eu ⊗ew ⎠ ev if b ∈ Des(u) and v ∈ [u, b], (3.2.10) ⎪ ⎪ ⎪ w∈[u,b] ⎪ ⎩ if b ∈ Des(u) and v ∈ / [u, b]. Dλu ev It is now easy to see that

(b)

W22

⎧ ⎪ ⎨ Dλu |2 (Des(u)) = Dλ | + ⎪ ⎩ u 2 (Desb (u))



if b ∈ Asc(u) or b ∈ Vu , λuw eu ⊗ ew if b ∈ Des(u).

w∈[u,b]

In view of (3.2.8), (3.2.9), (3.2.10), one may now deduce the desired decomposition from (3.2.6) and (3.2.7).   Theorem 3.10 together with Remark 3.11 yields the following: Corollary 3.12 (Dichotomy) Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For b, u ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers. Then the (b) weighted join operator Wλu on T is at most rank one perturbation of a diagonal (∞)

operator, while the weighted meet operator Wλu 278

on T is a finite rank operator.

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By Theorem 3.10, any Wλu ∈ B(2 (V )) can be rewritten as C + M, where C is a diagonal operator and M is a nilpotent operator of nilpotency index 2 given by  C=  M=

PAsc(u)∪Desb [u] Dλu if b ∈ Des(u), otherwise, PDes(u) Dλu eu ⊗ eλu ,(u,b] if b ∈ Des(u), eu ⊗ eλu ,Asc(u)∪{b} otherwise.

(b)

However, Wλu is not a complex Jordan operator unless λuu = 0. Indeed, C M − MC = λuu M. It is worth noting that (C M − MC)2 = 0. Unfortunately, the above decomposition is (b) not orthogonal. Here is a way to get such a decomposition of Wλu . Theorem 3.13 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V , b ∈ V \{u} and the weight system λu = {λuv }v∈V∞ of complex numbers, let Dλu be the diagonal (b) operator on T and let Wλu be a weighted join operator on T . Consider the closed (b)

subspace Hu

of 2 (V ), given by  Hu(b) =

2 (Asc(u) ∪ Desb (u)) if b ∈ Des(u), otherwise. 2 (Desu (u))

(3.2.11)

Then the following statements hold: (b)

(i) The weighted join operator Wλu admits the decomposition

(b) (b) (b) Wλu = Dλu ⊕ Nλu on 2 (V ) = Hu(b) ⊕ 2 (V )  Hu(b) , (b)

(b)

where Dλu is a densely defined diagonal operator in Hu

linear rank one operator on 2 (V )  Hu(b) . (b) (b) (ii) Dλu and Nλu are given by (b)

Dλu = Dλu |

(b) Hu

(3.2.12)

(b)

and Nλu is a bounded

(b) , D(Dλu ) = f ∈ Hu(b) : Dλu ( f ⊕ 0) ∈ Hu(b) ,(3.2.13)

(b)

Nλu = eu ⊗ eλu ,Au ,

(3.2.14)

where eλu ,Au is as given in (3.2.1) and the subset Au of V is given by  Au =

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[u, b] if b ∈ Des(u), Asc(u) ∪ {b, u} otherwise.

279

(3.2.15)

S. Chavan et al.

Proof Since the orthogonal projection PH (b) commutes with Dλu , one may appeal to u

[64, Proposition 1.15] to conclude that Hu(b) (identified with a subspace of 2 (V )) is a reducing subspace for Dλu . The desired conclusions in (i) and (ii) now follow from Theorem 3.10.   We find it convenient to denote the orthogonal decomposition (3.2.12) of Wλ(b) , u (b)

(b)

(b)

b = u, as ensured by Theorem 3.13, by the triple (Dλu , Nλu , Hu (b)

(b)

), where Hu ,

(b)

Dλu and Nλu are given by (3.2.11), (3.2.13) and (3.2.14) respectively. For the sake of convenience, we set Hu(u) := 2 (V \{u}).

(3.2.16)

Note that Wλ(u) admits the decomposition (3.2.12) with Nλ(u) = λuu eu ⊗ eu . u u

(b)

In what follows, we will be interested in only those vertices u ∈ V for which Hu is of infinite dimension. In the remaining part of this section, we present some immediate consequences of Theorem 3.13. Corollary 3.14 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V , b ∈ V \{u} and denote the weighted the weight system λu = {λuv }v∈V∞ of complex numbers, let Wλ(b) u (b)

(b)

(b)

join operator on T and let (Dλu , Nλu , Hu

) denote the orthogonal decomposition (b) (b) (b) (b) of Wλu . If λu ∈ ∞ (Des(u)), then Wλu  = max Dλu , Nλu  . Further, ⎛ (b) Dλu 

=

sup (b) v∈supp Hu

|λuv |,

(b) Nλu 

=⎝



⎞1/2 |λuv |2 ⎠

,

v∈Au

where Au is given by (3.2.15). The following two corollaries give more insight into the structure of weighted join operators on rooted directed trees. The first of which is motivated by the work [6]. We say that a densely defined linear operator T in H admits a large null summand if it has an infinite dimensional reducing subspace contained in its kernel. Corollary 3.15 Let T = (V , E) be a leafless, rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V , b ∈ V \{u} and the weight system λu = {λuv }v∈V∞ of complex numbers, let Wλ(b) u denote the weighted join operator on T . If there exists a branching vertex w ∈ Asc(u), (b) then Wλu has a large null summand. Proof Suppose there exists a branching vertex w ∈ Asc(u). By Theorem 3.13, (b) ∗ (b) (2 (V )  Hu(b) )  2 (Au ) ⊆ ker Wλu ∩ ker Wλu , 280

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where Au is a finite set given by (3.2.15). In view of (2.2.1) and (3.2.11), it suffices to check that card(Vu ) = ℵ0 . However, since T is leafless, this follows from Proposition 2.12(ii).   Complex Jordan weighted join operators exist in abundance. Corollary 3.16 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V , (b) b ∈ V \{u} and the weight system λu = {λuv }v∈V∞ of complex numbers, let Wλu denote the weighted join operator on T . Then the following holds: (b)

(i) If λuu = 0, then Wλu is a complex Jordan operator of index 2.

(b)

(ii) If λuu = 0, then there exists a bounded homomorphism : B∞ (σ (Wλu )) → B(2 (V )) given by ), ( f ) = f (Wλ(b) u

f ∈ B∞ (σ (Wλ(b) )), u

where B∞ () denotes the algebra of bounded Borel functions from a closed subset  of C into C. In this case, extends the polynomial functional calculus. (b)

(b)

(b)

(b)

Proof Let (Dλu , Nλu , Hu ) denote the orthogonal decomposition of Wλu . To verify (i), assume that λuu = 0. By Theorem 3.10, (b)

Wλu

(b)

= PH (b) Dλu + Nλu . u

By a routine inductive argument, we obtain (b) k

N λu

= λk−1 uu eu ⊗ eλu ,Au , k ≥ 1.

(3.2.17)

(b)

It follows that Nλu is nilpotent of nilpotency index 2. Also, it is easily seen that (b)

(b)

(b)

(b)

(b)

(b)

Nλu (Wλu − Nλu ) ⊆ (Wλu − Nλu )Nλu = 0, which completes the verification of (i). Assume next that λuu = 0. In view of (3.2.12) and [61, Theorem 13.24], it suffices (b) to check that Nλu given by (3.2.14) admits a Borel functional calculus. By (3.2.17),

) → C, f (Nλ(b) ) is a well-defined for any Borel measurable function f : σ (Wλ(b) u u bounded linear operator given by )= f (Nλ(b) u

f (λuu ) eu ⊗ eλu ,Au . λuu (b)

Further, for any bounded Borel measurable function f on σ (Wλu ), (b)

 f (Nλu ) ≤

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This completes the verification of (ii).

The following is a consequence of Theorem 3.10 and the well-known characterization of diagonal compact operators [23], once it is observed that the class of bounded finite rank operators is a subset of compact operators, dense in the operator norm (in fact, it is also dense in Schatten p-class in its norm for every p ≥ 1) (cf. [38, Corollary 3.4.5]). The reader is referred to [67] for the basics of operators in Schatten classes. Proposition 3.17 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V , (b) b ∈ V \{u} and the weight system λu = {λuv }v∈V∞ of complex numbers, let Wλu denote the weighted join operator on T . Then, for any p ∈ [1, ∞), the following hold: (b)

λuv = 0.  is Schatten p-class if and only if |λuv | p < ∞.

(i) Wλu is compact if and only if (b)

(ii) Wλu

lim

v∈Des(u)

v∈Des(u)

Here the limit and sum are understood in a generalized sense (see (1.1.2)). We conclude this section with an application to the theory of commutators of compact operators. Corollary 3.18 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V , (b) b ∈ V \{u}, and the weight system λu = {λuv }v∈V∞ of complex numbers, let Wλu denote the weighted join operator on T . Then the following statements hold: (i) If

lim

v∈Des(u) (b) that Wλu 

(ii) If

λuv = 0, then there exist compact operators K , L ∈ B(2 (V )) such = K L − LK. |λuv | p < ∞ for some p ∈ [1, ∞), then there exist Schatten class

v∈Des(u)

(b)

operators K , L ∈ B(2 (V )) such that Wλu = K L − L K . Proof This follows from Proposition 3.17, Corollary 3.15 and Anderson’s Theorems [6, Theorems 1 and 3].   3.3 Commutant The main result of this section describes commutants of weighted join operators (cf. (b) [37, Proposition 5.4], [28, Theorem 1.8]). In general, the weighted join operators Wλu do not belong to the class (RO) as introduced in [28]. Indeed, in contrast with [28, (b) Theorem 1.8], the commutant of Wλu need not be abelian (see Corollary 3.21). Theorem 3.19 (Commutant) Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V and b ∈ V \{u}, consider the weight system λu = {λuv }v∈V∞ of complex numbers 282

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and assume that the weighted join operator Wλu on T belongs to B(2 (V )). Let

, Nλ(b) , Hu(b) ) denote the orthogonal decomposition of Wλ(b) , Au be as given in (Dλ(b) u u u (3.2.15), and let Wu be given by

Wu := v ∈ supp Hu(b) : λuv = λuu . (b)

(3.3.1)

(b)

If ker Dλu = {0} and Nλu = 0, then the following statements are equivalent: (b)

(b)

(i) X ∈ B(2 (V )) belongs to the commutant {Wλu } of Wλu . (ii) X ∈ B(2 (V )) admits the orthogonal decomposition )

P f 0 ⊗ eλu ,Au X= eu ⊗ eμu ,Wu S

*



on 2 (V ) = Hu(b) ⊕ 2 (V )  Hu(b) , (b)

(b)

where P is a block diagonal operator in {Dλu } , f 0 ∈ ker(Dλu − λuu ),

μu := {μuv }v∈supp H (b) belongs to 2 (supp Hu(b) ), and S is any operator in (b)

u

B(2 (V )  Hu ) such that Seu = Seu , eu eu and S ∗ eλu ,Au = Seu , eu eλu ,Au . (b)

(b)

Proof Assume that ker Dλu = {0} and Nλu = 0. Let X ∈ B(2 (V )) be such that (b)

(b)

X Wλu = Wλu X . We decompose X as follows: ) X=

P Q R S

*

on 2 (V ) = Hu(b) ⊕ 2 (V )  Hu(b) .

A simple calculation shows that X Wλ(b) = Wλ(b) X is equivalent to u u (b)

(b)

P Dλu = Dλu P,

(b)

(b)

Nλu S = S Nλu ,

(b)

(b)

Nλu R = R Dλu ,

Dλ(b) Q = Q Nλ(b) . u u

(3.3.2)

We contend that R is a finite rank operator with a range contained in [eu ]. To see that, (b) let f ∈ Hu . By (3.2.13) and (3.2.14), R f = R f , eλu ,Au eu , Nλ(b) u

R Dλ(b) f = u



λuv f (v)Rev .

(b) v∈supp Hu

Thus, by the third equation of (3.3.2), we obtain R f , eλu ,Au eu =

 (b) v∈supp Hu

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λuv f (v)Rev .

(3.3.3)

S. Chavan et al. (b)

(b)

Since (3.3.3) holds for arbitrary f ∈ Hu and λuv = 0 for every v ∈ supp Hu , there exists a system μu := {μuv }supp H (b) ⊆ C such that u

Rev = μuv eu , v ∈ supp Hu(b) .

(3.3.4)

This immediately yields 

R f , eλu ,Au eu =

λuu f (v)μuv eu .

(b) v∈supp Hu

Combining this with (3.3.3), we obtain 



λuu f (v)μuv eu =

(b) v∈supp Hu

λuv f (v)μuv eu .

(b) v∈supp Hu

(b)

Since f ∈ Hu is arbitrary, this yields that μuv (λuu − λuv ) = 0 for every v ∈ supp Hu(b) . If v ∈ supp Hu(b) \Wu , then λuu = λuv (see (3.3.1)), and hence μuv = 0. It may be now concluded from (3.3.4) that R = eu ⊗ eμu ,Wu . Thus the claim stands verified. Q = Q Nλ(b) (see (3.3.2)). Note that by (3.2.14) Next we consider the equation Dλ(b) u u Dλ(b) Qeu = Q Nλ(b) eu = λuu Qeu , u u which simplifies to (Dλ(b) − λuu )Qeu = 0. Since Dλ(b) − λuu = 0 on 2 (Wu ) and u u (b)

injective on Hu

 2 (Wu ), Qeu ∈ 2 (Wu ). Moreover, Q = Q(eu ⊗ eλu ,Au ) = (Qeu ) ⊗ eλu ,Au . Dλ(b) u

If Qeu = 0, then so is Dλ(b) Q, and hence Q = 0 (since, by assumption, Dλ(b) is u u (b)

(b)

injective). Suppose Qeu = 0. Then Q : 2 (V )  Hu → Hu is of rank one, since so is Dλ(b) Q and Dλ(b) is injective. Thus Q = f 0 ⊗ g0 for some f 0 ∈ Hu(b) and u u (b)

(b)

(b)

g0 ∈ 2 (V )  Hu . It follows from Dλu Q = Q Nλu and (1.3.2) that (b)

(b)

(Dλu f 0 ) ⊗ g0 = f 0 ⊗ g0 Nλu = ( f 0 ⊗ g0 )(eu ⊗ eλu ,Au ) = g0 (u) f 0 ⊗ eλu ,Au . (3.3.5) Thus for any h ∈ 2 (V )  Hu(b) , (b)

h, g0 Dλu f 0 = g0 (u) h, eλu ,Au f 0 . 284

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Letting h = g0 −

g0 , eλ

(b)

eλ

u ,Au 2

u ,Au



(b)

(b)

eλu ,Au ∈ 2 (V )  Hu , we get h, g0 Dλu f 0 = 0.

However, since Dλu is assumed to be injective and f 0 = 0 (since Q = 0), h, g0 = 0. It follows that | g0 , eλu ,Au | = g0 eλu ,Au . By the Cauchy–Schwarz inequality, we must have g0 = α¯ eλu ,Au for some α ∈ C. Further, α = 0, since g0 = 0 (otherwise Q = 0). This also shows that Q = α f 0 ⊗ eλu ,Au . One may now infer from (3.3.5) (b) (b) (evaluated at g0 = α¯ eλu ,Au ) that Dλu f 0 = λuu f 0 . Further, by (3.3.2), P ∈ {Dλu } (b)

and S ∈ {Nλu } . The fact that P is a block diagonal operator is a routine verification (see [23, Proposition 6.1, Chapter IX]). The remaining part now follows from Lemma 1.2(vi). This completes the proof of (i) ⇒ (ii). The reverse implication is a routine verification using (3.3.2).   (b)

Remark 3.20 The injectivity of Dλu can be relaxed by replacing the basis {ev }v∈V by {eα(v) }v∈V for some permutation α : V → V . We leave the details to the reader. The following result is applicable to the case when the weight system λu : V → C (b) of the weighted join operator Wλu is injective. Corollary 3.21 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V and b ∈ V \{u}, consider the weight system λu = {λuv }v∈V∞ of complex numbers and assume that (b) (b) (b) (b) the weighted join operator Wλu on T belongs to B(2 (V )). Let (Dλu , Nλu , Hu ) . Assume that ker Dλ(b) = {0} and Nλ(b) = denote the orthogonal decomposition of Wλ(b) u u u (b)

/ σ p (Dλu ), then 0. If λuu ∈ (b) (b) (b) {Wλu } = P ⊕ S : P ∈ {Dλu } , S ∈ {Nλu } . (b)

(b)

/ σ p (Dλu ) and let X ∈ {Wλu } . Thus X admits the decomProof Assume that λuu ∈ position as given in (ii) of Theorem 3.19. However, by (3.3.1), Wu = ∅, and hence (b) eu ⊗ eμu ,Wu = 0. Also, since f 0 ∈ ker(Dλu − λuu ), by our assumption, f 0 = 0. This completes the proof.   Even under the assumptions of the preceding corollary, the commutant of a weighted meet operator can be non-abelian. Example 3.22 Let T = (V , E) be a leafless, rooted directed tree and let u ∈ V be such that Vu = ∅ (for example, take the directed tree T as shown in Fig. 4 and let u = v0 ), where Vu is as given in (2.2.1). For b ∈ Des(u), consider the weight system λu = {λuv }v∈V∞ of distinct positive numbers and assume that the weighted join operator Wλ(b) on T belongs to B(2 (V )). Let (Dλ(b) , Nλ(b) , Hu(b) ) denote the u u u (b)

orthogonal decomposition of Wλu . Since Vu = ∅, by (3.2.11) and (3.2.15), we have 2 (V )  Hu(b) = 2 (Au ), where Au = [u, b]. Reprinted from the journal

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(3.3.6)

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(b) (b)

We claim that {Wλu } is non-abelian if and only if dim 2 (V )  Hu ≥ 3. By

} is non-abelian if and only if the preceding corollary, it suffices to check that {Nλ(b) u

2 (b) ≥ 3. We consider the following cases: dim  (V )  Hu 2 (b)

Case 1. dim  (V )  Hu = 1: (b) By (3.3.6), b = u, and hence Wλu is the diagonal operator Dλu with distinct diagonal (b)

entries. In this case, {Wλu } is indeed maximal abelian [23].

Case 2. dim 2 (V )  Hu(b) = 2: Consider the basis {eu , eλu ,Au } of 2 ([u, b]) and let T ∈ {Nλ(b) } . By Lemma 1.2(iv), u T eu = T eu , eu eu , T eλu ,Au = T eu , eu eλu ,Au .

(3.3.7) (3.3.8)

∗

Let αT and βT be scalars such that T eλu ,Au = αT eu + βT eλu ,Au . Then T eλu ,Au , eλu ,Au

=

eλu ,Au , T ∗ eλu ,Au

(3.3.8)

=

αT λuu + βT eλu ,Au 2 , T eu , eu eλu ,Au 2 .

It follows that αT =

eλu ,Au 2 ( T eu , eu − βT ). λuu

(3.3.9)

(b)

Hence, for any S ∈ {Nλu } , we have ST eλu ,Au

= (3.3.7)

=

αT Seu + βT Seλu ,Au (αT Seu , eu + βT α S )eu + βT β S eλu ,Au .

By symmetry, ST eλu ,Au = T Seλu ,Au if and only if αT ( Seu , eu − β S ) = α S ( T eu , eu − βT ). } , by The latter equality follows from (3.3.9). On the other hand, for any S ∈ {Nλ(b) u (b)

(3.3.7), ST eu = T Seu always holds. This shows that {Nλu } is abelian. (b)

= 3: Case 3. dim 2 (V )  Hu Let f ∈ 2 (Au ) be orthogonal to {eu , eλu ,Au }. Consider a bounded linear operator T (b) on 2 (V )  Hu governed by T eu = αeu , T eλu ,Au = βeu + γ f , T f = 0, 286

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(b)

(b)

where α, β, γ are complex numbers. Clearly, T Nλu eu = Nλu T eu , T Nλu f = T f . Moreover, T Nλ(b) eλu ,Au = Nλ(b) T eλu ,Au if and only if Nλ(b) u u u β = (eλu ,Au 2 T eu , eu )/λuu . (b)

Consider another bounded linear operator S on 2 (V )  Hu

governed by

Seu = aeu , Seλu ,Au = beu , S f = ceu + deλu ,Au , where a, b, c, d are complex numbers. A routine calculation shows that S ∈ {Nλ(b) } u if and only if bλuu − aeλu ,Au 2 = 0, cλuu + deλu ,Au 2 = 0. On the other hand, ST f = T S f implies that dγ = 0, and hence for non-zero choices (b) of γ and d, S and T do not commute. This shows that {Nλu } is not abelian. 2 (b)

(b)  ≥ 4, {Nλu } contains a copy of B(C2 ), and Finally, in case dim  (V )  Hu hence it is not abelian. 

4 Rank One Extensions of Weighted Join Operators In this section, we introduce and study the class of rank one extensions of weighted join operators. We introduce the so-called compatibility conditions and discuss their roles in the closedness of these operators. We also discuss the problem of determining the Hilbert space adjoint of these operators. Further, we provide a complete spectral picture for members in this class and discuss some of its applications. Definition 4.1 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For b, u ∈ V and the (b) weight system λu = {λuv }v∈V∞ of complex numbers, let Wλu be a weighted join (b)

(b)

(b)

operator on T . Consider the orthogonal decomposition (Dλu , Nλu , Hu By a rank one extension of in 2 (V ) given by

Wλ(b) u

on T , we understand the linear operator

(b)

) of Wλu .

Wλ(b) [ f , g] u

⎫ (b) (b) (b) ⎪ D(Wλu [ f , g]) = (h, k) : h ∈ D(Dλu ) ∩ D( f  g), k ∈ 2 (V )  Hu ⎪ ⎬   (b) D 0 (b) λu ⎪ Wλu [ f , g] = ⎪ (b) , ⎭ f  g N λu (4.0.1) (b)

(b)

where f ∈ 2 (V )  Hu is non-zero and g : supp Hu → C is unspecified. For the sake of convenience, we use the simpler notation W f ,g in place of Wλ(b) [ f , g]. u

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(b)

Remark 4.2 Since Nλu is bounded and the domains of Dλu and f  g contains the

dense subspace Dsupp H (b) of Hu(b) (see (3.0.2)), W f ,g is densely defined. Since u the sum of a closed operator and a bounded linear operator is closed, the rank one (b) (b) extension W f ,g of Wλu is closed provided g ∈ Hu . This happens in particular (b)

when Des(u) has finite cardinality (see (3.2.11)). In case g ∈ / Hu , W f ,g need not be closed (cf. Corollary 4.18). To see this assertion, consider the situation in which Dλ(b) u (b)

is bounded and g ∈ / Hu . By Lemma 1.3, f  g is not closable, and hence there exists (b) a sequence {h n }n≥0 in Hu such that h n → 0, {( f  g)(h n )}n≥0 is convergent but ( f  g)(h n )  0 as n → ∞. Then (h n , 0) → (0, 0), {W f ,g (h n , 0)}n≥0 is convergent, however, W f ,g (h n , 0)  0, and hence W f ,g is not even closable. Here is a remark about the manner in which W f ,g is defined. Certainly, one could (b) have defined the rank one extension of Wλu with the entry f  g appearing on the extreme upper right corner in (4.0.1). It turns out, however, that the operators defined (b) this way are closed if and only if g ∈ 2 (V )  Hu . From the view point of the (b) spectral theory, these operators are of little importance in case g ∈ / 2 (V )  Hu , and otherwise, these are bounded finite rank perturbations of diagonal operators. Needless to say, the latter class has been studied extensively in the literature (refer, for example, to [26–29, 37, 44, 48, 69]). Also, the way in which W f ,g is defined (cf. [2–4, 60]), it should be referred to as rank one co-extension of Wλ(b) . However, by abuse of u (b)

terminology, we refer to it as a rank one extension of Wλu . In what follows, we will be particularly interested in the following family of rank one extensions of weighted join operators (cf. Proposition 5.7 below). (b)

Example 4.3 Let W f ,g be a rank one extension of the weighted join operator Wλu satisfying the intertwining relation: (b)

(b)

( f  g)Dλu + Nλu ( f  g) = 0,

(4.0.2)

where the linear operator on the left-hand side is defined on the space Dsupp H (b) (see u (3.0.2)). Note that (4.0.2) is equivalent to λuv f + f , eλu ,Au eu = 0, v ∈ supp(g),

(4.0.3)

where Au is given by (3.2.15) and the support supp(h) of the function h : V → C is given by supp(h) := {v ∈ V : h(v) = 0}. Suppose g = 0 and note that supp(g) is non-empty. We make the following observations: (i) If λuv = 0 for some v ∈ supp(g), then by (4.0.3), f , eλu ,Au = 0. Since f = 0, by another application of (4.0.3), λuw = 0 for all w ∈ supp(g).

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(ii) If λuv = 0 for some v ∈ supp(g), then by (4.0.3), f ∈ [eu ]\{0} and λuw = −λuu for every w ∈ supp(g). The above discussion provides the following examples of W f ,g satisfying (4.0.2). (b)

(a) λuv = 0 for v ∈ supp(g), supp Hu 

\supp(g) is infinite and

f (w)λuw = 0.

w∈Au

(b) λuv = −λuu = 0 for every v ∈ supp(g) and f ∈ [eu ]\{0}. For example, if T is leafless and u is a branching vertex, then the condition that (b) supp Hu \supp(g) is infinite in (a) is ensured for any g such that supp(g) = Des(w) provided  w belongs to

Chi(b) if b ∈ Des(u), Chi(u) if b ∈ / Des(u). (b)

In case (b) holds, then the rank one extension of Wλu can be rewritten as the sum of

a diagonal operator and the rank one operator eu  f (u) g + eλu ,Au . In these cases, W f ,g satisfies (b)

(b)

W 2f ,g = (Dλu )2 ⊕ (Nλu )2 on Dsupp H (b) ⊕ (2 (V )  Hu(b) ). u

Thus although W f ,g does not have a diagonal decomposition, the intertwining relation (4.0.2) ensures the same for its square.  The bounded rank one extensions of weighted join operators can be characterized easily. Proposition 4.4 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the (b) rank one extension of the weighted join operator Wλu on T . Then the following are equivalent: (i) W f ,g defines a bounded linear operator on 2 (V ). (b) (b) (b) (ii) g ∈ Hu and Dλu defines a bounded linear operator on Hu .

(iii) g ∈ Hu(b) and

 λu belongs to

∞ (V ) if b = u, ∞  (Des(u)) otherwise.

Proof In view of Theorem 3.7, it suffices to see the equivalence of (i) and (ii). Since (b) (b) Nλu is a bounded linear operator on 2 (V )  Hu , (ii) implies (i). To see the reverse Reprinted from the journal

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implication, assume that W f ,g is bounded linear on 2 (V ). Thus D(Dλu )∩D( f g) = , being a closed operator defined on Hu(b) , Hu(b) . By the closed graph theorem, Dλ(b) u (b)

is a bounded linear operator on Hu

(b)

. Further, for any h ∈ D(Dλu ) ∩ D( f  g), (b)

W f ,g (h, 0)2 = (Dλu h,

 

 h(v)g(v) f )

2

(b)

v∈supp Hu (b)

= Dλu h2 + ( f  g)(h)2 . Since f = 0 and W f ,g is bounded, f  g must be bounded linear, and hence by (b) Lemma 1.3, g ∈ Hu . This completes the proof.   4.1 Compatibility Conditions and Discrete Hilbert Transforms (b)

(b)

It turns out that the inclusion D(Dλu ) ⊆ D( f  g) of domains of Dλu and f  g plays a central role in deciding whether or not a given rank one extension W f ,g of a weighted join operator is closed (cf. Remark 4.2). Indeed, we will see that the socalled compatibility conditions on W f ,g always ensure the above inclusion as well as the closedness of W f ,g . We formally introduce these conditions below (cf. [37, Proposition 2.4(iv)], [48, Proposition 4.1]). Definition 4.5 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the rank one extension of the weighted join operator Wλ(b) on T , where f ∈ 2 (V )  Hu(b) is u (b)

non-zero and g : supp Hu

→ C is given. Set

dist(μ, λu ) = inf |μ − λuv | : v ∈ supp Hu(b) , μ ∈ C,

Γλu = μ ∈ C : dist(μ, λu ) > 0 .

(4.1.1)

(i) We say that W f ,g satisfies compatibility condition I if there exists μ0 ∈ Γλu such that gλu ,μ0 ∈ Hu(b) , where gλu ,μ0 (v) :=

g(v) , v ∈ supp Hu(b) . λuv − μ0

(4.1.2)

(ii) We say that W f ,g satisfies compatibility condition II if the function g satisfies  (b) v∈supp Hu

|g(v)|2 < ∞. |λuv |2 + 1

(4.1.3)

If one of the above conditions holds, then we say that W f ,g satisfies a compatibility condition.

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(b)

Remark 4.6 It turns out that gλu ,μ0 ∈ Hu for some μ0 ∈ Γλu , then gλu ,μ ∈ Hu for every μ ∈ Γλu . This may be derived from |λuv − μ0 |2 ≤ 2(|λuv − μ|2 + |μ0 − μ|2 ), v ∈ supp Hu(b) , μ ∈ Γλu \{μ0 }, and the fact that |μ0 − μ| ≤ c dist(μ, λu ) for some c > 0 (see (4.1.1)).

Here we discuss the relationships between the above compatibility conditions and (b) the domain inclusion D(Dλu ) ⊆ D( f  g). Proposition 4.7 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the rank one extension of the weighted join operator Wλ(b) on T , where f ∈ 2 (V )  Hu(b) is nonu (b)

zero and g : supp Hu → C is given. If W f ,g satisfies a compatibility condition, then (b) we have the domain inclusion D(Dλu ) ⊆ D( f  g). Moreover, if Γλu is non-empty, then the following statements are equivalent: (b)

(i) D(Dλu ) ⊆ D( f  g). (ii) W f ,g satisfies compatibility condition I. (iii) The discrete Hilbert transform Hλu ,g given by Hλu ,g (h) =

 (b)

v∈supp Hu

h(v)g(v) μ − λuv (b)

is well-defined for every μ ∈ Γλu and every h ∈ Hu . (b) (iv) For every μ ∈ Γλu , the linear operator L λu ,μ := ( f  g)(Dλu − μ)−1 defines a (b)

bounded linear transformation from Hu

(b)

into 2 (V )  Hu

.

(b)

Remark 4.8 Note that σ (Dλu ) = C\Γλu (see [64, Example 3.8]). This clarifies the (b)

expression (Dλu − μ)−1 appearing in (iv). It is worth noting that a discrete Hilbert transform appears in [37, Corollary 2.5], which characterizes the set of eigenvalues of a bounded rank one perturbation of a diagonal operator (see also [37, Corollary 2.6]). Also, the operator L λu ,μ , as appearing in Proposition 4.7(iv), is precisely the operator G(μ), as appearing in the Frobenius-Schur-type factorization in [7, Equation (1.6)]. Proof Let h ∈ D(Dλ(b) ). We divide the verification of the domain inclusion D(Dλ(b) )⊆ u u D( f  g) into the following cases: Assume that W f ,g satisfies the compatibility condition I. Then, for some μ0 ∈ Γλu , by the Cauchy–Schwarz inequality, ! ! !



! ! (b) h(v)g(v)! ≤ (Dλu − μ0 )hgλu ,μ0 . (b)

v∈supp Hu

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(b)

where we used (4.1.2) and the fact that D(Dλu − μ0 ) = D(Dλu ). This shows that h ∈ D( f  g). (b) Next assume that W f ,g satisfies the compatibility condition II. Since D(Dλu ) = (b)

(b)

D((Dλu )∗ Dλu + I )1/2 ), by the Cauchy–Schwarz inequality, ! !



!2 ! (b) (b) h(v)g(v)! ≤ (Dλu )∗ Dλu + I )1/2 h2 (b)

v∈supp Hu

 (b)

v∈supp Hu

|g(v)|2 . |λuv |2 + 1

Thus h ∈ D( f  g) in this case, as well The preceding discussion also yields the implication (ii) ⇒ (i). To see the equivalence of (i)–(iv), assume that Γλu is non-empty. (b) (b) (i) ⇒ (ii): Let μ ∈ Γλu . For h ∈ Hu , consider the function kh : supp Hu → C defined by kh (v) =

h(v) , v ∈ supp Hu(b) . λuv − μ

(b)

(b)

(4.1.4) (b)

Clearly, kh belongs to D(Dλu ) for every h ∈ Hu

. By assumption, D(Dλu ) ⊆

D( f  g), and hence the linear functional φg : Hu

→ C given by

(b)

φg (h) =



kh (v)g(v), h ∈ Hu(b)

(4.1.5)

(b) v∈supp Hu



(b) k h (v)g(v) v∈supp Hu (b) is absolutely convergent for every h ∈ Hu . One may now apply the uniform bound g(v) edness principle [66] to the family of linear functionals φ F,g (h) = v∈F h(v) λuv −μ , (b) F is a finite subset of supp Hu to derive the boundedness of φg . By (4.1.4) and (b) (4.1.5), the boundedness of φg in turn is equivalent to gλu ,μ ∈ Hu .

is well-defined. By the standard polar representation, the series

(iii) ⇒ (ii): This may be derived from the uniform boundedness principle (see the verification of (i) ⇒ (ii)). (ii) ⇒ (iii): In view of the Cauchy–Schwarz inequality, it suffices to check that gλu ,μ0 ∈ Hu(b) for some μ0 ∈ Γλu , then gλu ,μ ∈ Hu(b) for every μ ∈ Γλu . This is observed in Remark 4.6. (b) (ii) ⇒ (iv): By the Cauchy–Schwarz inequality, for any h ∈ Hu , (b)

L λu ,μ  = ( f  g)(Dλu − μ)−1 h !  g(v) !! ! =! h(v) ! f  ≤ hgλu ,μ . λuv − μ (b) v∈supp Hu

(b)

Since gλu ,μ ∈ Hu , this shows that L λu ,μ is bounded linear. (iv) ⇒ (iii): This is straightforward.

292

 

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Here is an instance in which the compatibility condition II implies the compatibility condition I. Corollary 4.9 Under the hypotheses of Proposition 4.7 and the assumption that Γλu = ∅, if W f ,g satisfies the compatibility condition II, then it satisfies the compatibility condition I. Proof If W f ,g satisfies the compatibility condition II, then by the first half of Proposition 4.7, we obtain D(Dλ(b) ) ⊆ D( f  g). The desired conclusion now follows from u the implication (i) ⇒ (ii) of Proposition 4.7.   (b)

Following [15], for any g : supp Hu

→ C\{0}, we set

(Γ, g)∗ = {μ ∈ C : gλu ,μ ∈ Hu(b) }.

(4.1.6)

The following has been motivated by the discussion from [15, Pg 2] on discrete Hilbert transforms in a slightly different context. Note that the compatibility condition II is nothing but the existence of admissible weight sequence in the sense of [15]. Proposition 4.10 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the (b) (b) rank one extension of the weighted join operator Wλu on T , where f ∈ 2 (V )Hu (b)

is non-zero and g : supp Hu true:

→ C\{0} be given. Then the following statements are

(i) If {λuv : v ∈ supp Hu(b) } is closed, then W f ,g satisfies the compatibility condition I if and only if Γλu = C\{λuv : v ∈ supp Hu(b) } = (Γ, g)∗ (see (4.1.1) and (4.1.6)). (b) (ii) If {λuv : v ∈ supp Hu } has an accumulation point only at ∞ with each of its entries appearing finitely many times, then W f ,g satisfies compatibility condition II if and only if Γλu = (Γ, g)∗ . (b)

Proof To see (i), assume that {λuv : v ∈ supp Hu } is closed. Clearly, Γλu = C\{λuv : v ∈ supp Hu(b) }.

(4.1.7)

Since g is nowhere vanishing, (Γ, g)∗ ⊆ C\{λuv : v ∈ supp Hu(b) }.

(4.1.8)

Note further that if W f ,g satisfies the compatibility condition I, then gλu ,μ belongs to (b)

Hu

for every μ ∈ Γλu (see Remark 4.6). In this case, Γλu ⊆ (Γ, g)∗ , and hence the

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necessity part in (i) follows from (4.1.7) and (4.1.8). Since C\{λuv : v ∈ supp Hu } (b) is always a non-empty set (by the assumption that card(V ) = ℵ0 , supp Hu is always countable), the sufficiency part of (i) follows from (4.1.6). (b) To see (ii), assume that {λuv : v ∈ supp Hu } has an accumulation point only at ∞ with each of its entries appearing finitely many times. The necessity part follows from (i), (4.1.7) and Corollary 4.9. To see the sufficiency part of (ii), suppose that (Γ, g)∗ = Γλu . By (i), for some μ0 ∈ C, we must have  (b) v∈supp Hu

|g(v)|2 < ∞. |μ0 − λuv |2

(4.1.9)

However, since ∞ is the only accumulation point for {λuv : v ∈ supp Hu(b) }, there exists (sufficiently large) M > 0 such that |μ0 − λuv |2 ≤ M(|λuv |2 + 1) for every (b) v ∈ supp Hu . It now follows from (4.1.9) that W f ,g satisfies the compatibility condition II.   4.2 Closedness and Relative Boundedness In this section, we show that any rank one extension W f ,g of weighted join operator satisfying a compatibility condition is closed (cf. [7, Theorem 1.1], [72, Theorems 2.5 and 2.6]). This is achieved by decomposing W f ,g as A + B, where A is closed and B is A-bounded. We begin recalling some definitions from [46]. Given densely defined linear operators A and B in H, we say that B is A-bounded if D(B) ⊇ D(A) and there exist non-negative real numbers a and b such that Bx2 ≤ aAx2 + bx2 , x ∈ D(A). The infimum of all a ≥ 0 for which there exists a number b ≥ 0 such that the above inequality holds is called the A-bound of B. Note that B is A-bounded if and only if D(B) ⊇ D(A) and there exist non-negative real numbers a and b such that Bx ≤ aAx + bx, x ∈ D(S).

(4.2.1)

For basic facts pertaining to A-bounded operators, the reader is referred to [46, 64, 68]. For the sake of convenience, we recall here the statement of the Kato-Rellich theorem from [46]. Suppose that A is a closed operator in H. Let B be a linear operator such that D(A) ⊆ D(B) and there exist a ∈ (0, 1) and b ∈ (0, ∞) with the property (4.2.1), then the linear operator A + B with the domain D(A) is a closed operator in H (see [46, Theorem 1.1, Chapter IV]). Theorem 4.11 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the rank one 294

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(b)

extension of the weighted join operator Wλu on T , where f ∈ 2 (V )  Hu

is non-

supp Hu(b)

→ C is given. Suppose that W f ,g satisfies a compatibility zero and g : condition. Then W f ,g defines a closed linear operator with the domain given by

(b) D(W f ,g ) = (h, k) : h ∈ D(Dλu ), k ∈ 2 (V )  Hu(b) .

(4.2.2)

Moreover, DV , as given by (3.0.2), forms a core for W f ,g . Proof Suppose that W f ,g satisfies the compatibility condition I for some μ ∈ Γλu . (b) Let a be a positive real number less than 1. Since gλu ,μ ∈ Hu , there exists a finite (b)

subset F of supp Hu

such that  (b) supp Hu \F

|g(v)|2 a ≤ . 2 |λuv − μ| 4  f 2 (b)

Define closed linear operators N F and D F in Hu NF =



λuv ev ⊗ ev ,

v∈F

Further let g F =



(b)

v∈supp Hu

\F

(4.2.3)

by (b)

D F = Dλu − N F .

g(v)ev . We rewrite W f ,g as A + B + C, where

A, B, C (with their natural domains) are densely defined operators in 2 (V ) given by   * * ) ) NF 0 0 0 DF 0 ,C = ,B= A= (b) . 0 0 f  gF 0 f ⊗ (g − g F ) Nλu (b)

Note that D(D F ) = D(Dλu ) and D( f  g F ) = D( f  g). Furthermore, A is a closed linear operator in 2 (V ) and C is a bounded linear operator on 2 (V ). Moreover, D(D F ) ⊆ D( f g F ) (cf. Proposition 4.7). Indeed, by the Cauchy–Schwarz inequality and (4.2.3), for any h ∈ D(D F ), ! ! ! ! √ ! !  a (D F − μ)h ! ! . (4.2.4) h(v)g(v) ! ≤ ! ! ! 2 f !v∈supp Hu(b) \F ! Since C is a bounded operator, we obtain that D(A) ⊆ D(B + C). We claim that for all h ∈ D(A), (B + C)h2 ≤ a Ah2 + b(μ) h2 ,

(4.2.5)

where b(μ) = a|μ|2 +2C2 . To see the claim, let h = (h 1 , h 2 ) ∈ D(A). By repeated applications of |α + β|2 ≤ 2(|α|2 + |β|2 ), α, β ∈ C, we obtain (B + C)h2

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≤

2( f  g F )h 1 2 + 2C2 h2

295

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≤ ≤ ≤

a (D F − μ)h 1 2 + 2C2 h2 2 aD F h 1 2 + a|μ|2 h 1 2 + 2C2 h2

aAh2 + a|μ|2 + 2C2 h2 .

This completes the verification of (4.2.5). Thus B + C is A-bounded with A-bound less than 1. Hence, by the Kato-Rellich Theorem, W f ,g is a closed operator with the domain given by (4.2.2). Next suppose that W f ,g satisfies the compatibility condition II. Let 

G m :=

(b) v∈supp Hu

|g(v)|2 , m ≥ 1. |λuv |2 + m 2

(4.2.6)

Note that 0 ≤ G m ≤ G 1 < ∞, m ≥ 1.

(4.2.7)

We rewrite W f ,g as A+ B +C, where A, B, C (with their natural domains) are densely defined operators in 2 (V ) given by  A=

   ) * (b) 0 0 0 0 Dλu 0 ,B= . ,C = f g 0 0 Nλ(b) 0 0 u

(4.2.8)

Note that A is a closed linear operator in 2 (V ) and C is a bounded linear operator on 2 (V ). Given a positive integer m, consider the inner-product space D(A) endowed with the inner-product x, y A,m = Ax, Ay + m 2 x, y , x, y ∈ D(A). Since A is a closed linear operator, H A,m = (D(A), ·, · A,m ) is a Hilbert space. Further, by Kato’s second representation theorem [46, Theorem 2.23, Chapter VI], D(A) = D((A∗ A + m 2 I )1/2 ), x, y A,m = (A∗ A + m 2 I )1/2 x, (A∗ A + m 2 I )1/2 y , x, y ∈ D(A).

 (4.2.9)

By Proposition 4.7, D(A) ⊆ D(B). Moreover, if h ∈ D(A), then by (4.2.9), we have 

h2A,m =

(|λuv |2 + m 2 )|h(v)|2 < ∞, (b)

v∈supp Hu

and hence by (4.1.3) and (4.2.7), ! ! !



!2 ! h(v)g(v)! ≤ h2A,m G m ≤ G 1 h2A,m

(4.2.10)

(b) v∈supp Hu

296

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(see (4.2.6)). Since C is a bounded linear operator, D(A) ⊆ D(B + C). We claim that for any (arbitrarily small) a > 0, there exists (large enough) b such that (B + C)h2 ≤ aAh2 + bh2 , h ∈ D(A).

(4.2.11)

To see the claim, let h = (h 1 , h 2 ) ∈ D(A). By repeated applications of |α + β|2 ≤ 2(|α|2 + |β|2 ), α, β ∈ C, we obtain (B + C)h2

≤ 2( f  g)h 1 2 + 2C2 h 2 2 (4.2.10) ≤ 2 f 2 h2A,m G m + 2C2 h 2 2 ≤

2 f 2 Ah2 G m + (2m 2  f 2 G m + 2C2 )h2 .

However, an application of Lebesgue dominated convergence theorem together with (4.2.7) shows that G m → 0 as m → ∞. This completes the proof of the claim. Another application of the Kato-Rellich Theorem shows that W f ,g is closed. To prove that DV forms a core for W f ,g , it suffices to check that W f ,g ⊆ W f ,g |DV . To see that, let (h, k) ∈ D(W f ,g ). By (4.2.2), h ∈ D(Dλ(b) ) and k ∈ 2 (V )  Hu(b) . u (b)

Since Dsupp H (b) is a core for Dλu , there exists a sequence {h n }n∈N ⊆ Dsupp H (b) u

(b)

u

(b)

such that h n → h and Dλu h n → Dλu h as n → ∞. Let {kn }n∈N be any sequence in DV \supp H (b) converging to k. Note that by Proposition 4.7, h ∈ D( f  g). Further, u

(b)

(b)

(b)

since gλu ,μ ∈ Hu and (Dλu − μ)h n → (Dλu − μ)h as n → ∞, by the Cauchy– Schwarz inequality, f  g(h n ) → ( f  g)(h) as n → ∞. It is now easy to see that (h n , kn ) → (h, k), W f ,g (h n , kn ) → W f ,g (h, k) as n → ∞. Thus (h, k) ∈ D(W f ,g |DV ) and W f ,g (h, k) = W f ,g |DV (h, k), as desired.

 

The following is a consequence of the proof of Theorem 4.11. Corollary 4.12 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the rank one (b) (b) extension of the weighted join operator Wλu on T , where f ∈ 2 (V )  Hu is non-

zero and g : supp Hu(b) → C is given. Suppose that W f ,g satisfies the compatibility condition II. Then W f ,g decomposes as A + B +C, where A, B, C are densely defined operators given by (4.2.8) such that B + C is A-bounded with A-bound equal to 0. 4.3 Adjoints and Gelfand-Triplets

In this section, we try to unravel the structure of the Hilbert space adjoint of the rank one extension W f ,g of weighted join operators. In particular, we discuss the question of determining the action of the Hilbert space adjoint of W f ,g . For an interesting

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discussion on the relationship between the Hilbert space adjoint and the formal adjoint of an unbounded operator matrix, the reader is referred to [53]. Unfortunately, the situation in our context suggests that there is no obvious way in which one can identify the Hilbert space adjoint of W f ,g with its formal adjoint (that is, the transpose of the (b) matrix formed after taking entry-wise adjoint) unless g ∈ Hu . We conclude this section with a brief discussion on the role of a Gelfand-triplet naturally associated with W f ,g in the realization of W ∗f ,g . We begin with the following proposition which shows that the adjoint of W f ,g coincides with the adjoint of the associated weighted join operator Wλ(b) on a possibly u non-dense subspace of D(W ∗f ,g ). Proposition 4.13 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the (b) (b) rank one extension of the weighted join operator Wλu on T , where f ∈ 2 (V )Hu (b)

is non-zero and g : supp Hu Moreover,

→ C is given. Then W ∗f ,g is a closed operator.

D := {k = (k1 , k2 ) ∈ D(Dλ(b) ) ⊕ (2 (V )  Hu(b) ) : k2 , f = 0} ⊆ D(W ∗f ,g ), u (b)

(b)

W ∗f ,g k = (Dλu )∗ k1 + (Nλu )∗ k2 , k = (k1 , k2 ) ∈ D. Further, if W f ,g satisfies a compatibility condition, then W ∗f ,g is densely defined. Proof Since W f ,g is densely defined in 2 (V ), the Hilbert space adjoint W ∗f ,g is a closed linear operator [64]. Let k = (k1 , k2 ) ∈ D. Then, for any h = (h 1 , h 2 ) ∈ D(W f ,g ), (b)

(b)

W f ,g h, k = Dλu h 1 , k1 + ( f  g)h 1 , k2 + Nλu h 2 , k2

(b)

(b)

= h 1 , (Dλu )∗ k1 + h 2 , (Nλu )∗ k2 , where we used the fact that D((Dλ(b) )∗ ) = D(Dλ(b) ). It follows that k ∈ D(W ∗f ,g ) and u u (b)

(b)

W ∗f ,g k = (Dλu )∗ k1 + (Nλu )∗ k2 . Finally, if W f ,g satisfies a compatibility condition, then by Theorem 4.11, W f ,g is closed, and hence, by [64, Theorem 1.8(i)], W ∗f ,g is densely defined. This completes the proof.   Remark 4.14 Note that for any k ∈ 2 (V )  Hu(b) , (0, k) ∈ / D(W ∗f ,g ). Otherwise, φ(h) = W f ,g h, (0, k)

(b)

= ( f  g)h 1 , k + Nλu h 2 , k , h = (h 1 , h 2 ) ∈ D(W f ,g ) (b)

extends as a bounded linear functional, which is not possible since g ∈ / Hu Lemma 1.3). In particular, (0, f ) ∈ / D(W ∗f ,g ). 298

(see

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Note that the closability of W f ,g is equivalent to the density of the domain D(W ∗f ,g ) of W ∗f ,g (see [64, Theorem 1.8]). In particular, it would be interesting to obtain conditions ensuring the density of D(W ∗f ,g ) (with or without a compatibility condition). In view of the decomposition W f ,g = A + B + C as given in (4.2.8), it is tempting to ask whether W ∗f ,g can be decomposed as A∗ + B ∗ + C ∗ . It may be concluded from [14, Proposition] that if W f ,g is Fredholm such that B is A-compact and B ∗ is A∗ -compact, then W ∗f ,g = A∗ + B ∗ + C ∗ (see [54, Theorem 2.2] for a variant). We will see in the proof of Theorem 4.15(iv) that under the assumption of compatibility condition I, A-compactness of B can be ensured (Recall that B is A-compact if D(A) ⊆ D(B) and B maps {h ∈ D(A) : h + Ah ≤ 1} into a pre-compact set). Another natural problem which arises in finding W ∗f ,g is whether it is possible to have a matrix decomposition of W ∗f ,g similar to the one we have it for W f ,g . One possible candidate for W ∗f ,g is its formal adjoint, that is, the transpose of the operator matrix W × f ,g obtained by taking the Hilbert space adjoint of each entry of W f ,g . A direct application of [53, Theorem 6.1] shows the following: × ∗ (i) W × f ,g is a closable operator such that W f ,g ⊆ W f ,g . (ii) If W × f ,g is densely defined, then W f ,g is closable.

It is evident that there is no natural way to recover W ∗f ,g from W × f ,g . One such way has been shown in [53, Proposition 6.3], which provides a sufficient condition for the equality of the Hilbert space adjoint and the formal adjoint of an unbounded operator matrix. Unfortunately, this result is not applicable to W f ,g unless all its entries are (b) closable operators. Recall that f  g is not even closable in case g ∈ / Hu (see (b) ∗ Lemma 1.3). That’s why to understand the action of W f ,g , we need to replace Hu by a larger Hilbert space. We will see below that the notion of Hilbert rigging turns out to be handy in this context. Consider the inner-product space D(Dλ(b) ) endowed with the inner-product u (b)

(b)

(b)

x, y ◦ := Dλu x, Dλu y + x, y , x, y ∈ D(Dλu ). (b)

Note that H◦ := D(Dλu ) endowed with the inner-product ·, · ◦ is a Hilbert space. (b)

Clearly, the inclusion map i : H◦ → Hu is contractive. Consider further the topological dual H∗◦ of H◦ , which we denote by H◦ . We claim that any element (b) h ∈ Hu can be realized as a bounded conjugate-linear functional in H◦ . To see this, consider the mapping j : Hu(b) → H◦ given by j(h) = φh , where

φh (k) =



k(v)h(v), k ∈ H◦ .

(b) v∈supp Hu

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The contractivity of j follows from the Cauchy–Schwarz inequality and ! ! ! ! ! !  ! ! k(v)h(v)! φh  = sup ! ! ! k◦ =1 ! (b) ! v∈supp Hu ⎛ ⎞1/2 |h(v)|2 ⎟ ⎜  (b) (b) =⎝ = ((Dλu )∗ Dλu + I )−1/2 h, (4.3.1) 2⎠ 1 + |λ | uv (b) v∈supp Hu

and hence the claim stands verified. This also shows that H◦ can be identified with (b) the completion of Hu endowed with the inner-product x, y ◦ = ((Dλ(b) )∗ Dλ(b) + I )−1 x, y , x, y ∈ Hu(b) . u u Thus we have the following chain of Hilbert spaces: H◦  Hu(b)  H◦ , where H◦ is dense in Hu(b) and Hu(b) is dense in H◦ . One may refer to this (b) chain of Hilbert spaces as the Hilbert rigging of Hu by H◦ and H◦ . The triplet (b) (H◦ , Hu , H◦ ) is known as the Gelfand-triplet (refer to [16, Chapter 14] for an (b) abstract theory of rigged spaces). If {(|λuv |2 + 1)−1/2 : v ∈ supp Hu } is squaresummable, then by (4.3.1), the above Hilbert rigging is quasi-nuclear in the sense that (b) the inclusion j : Hu → H◦ is Hilbert-Schmidt (see [16, Pg 121]). Suppose that W f ,g satisfies the compatibility condition II. Define φg : H◦ → C by φg (k) =



k(v)g(v), k ∈ H◦ .

(b) v∈supp Hu

It may be concluded from (4.1.3) that φg ∈ H◦ . This allows us to introduce the (b) bounded linear transformation B : H◦ → 2 (V )  Hu by setting Bk = φg (k) f , k ∈ H◦ , Note that for any l ∈ 2 (V )  Hu(b) and k ∈ H◦ , (B ∗l)(k) = l, Bk = φg (k) l, f = (φg ⊗ f )(l)(k). Thus B ∗ can be identified with g ⊗ f . In particular, the Hilbert space adjoint of W f ,g can be identified with the formal adjoint of W f ,g after replacing the Hilbert space (b) Hu by the larger Hilbert space H◦ .

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4.4 Spectral Analysis We now turn our attention to the spectral properties of rank one extensions of weighted join operators. The main result of this section provides a complete spectral picture for rank one extensions W f ,g of weighted join operators (cf. [70, Theorem 1], [37, Theorem 2.3], [45, Theorem 2.3], [56, Corollary 2.8], [7, Theorem 2.2]). It turns out that W f ,g has non-empty resolvent set if and only if it satisfies the compatibility condition I. Among various applications, we characterize rank one extensions of weighted join operators on leafless directed trees which admit compact resolvent. Theorem 4.15 (Spectral picture) Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let (b) W f ,g be the rank one extension of the weighted join operator Wλu on T , where f ∈ 2 (V )  Hu(b) is non-zero and g : supp Hu(b) → C is given (see (4.0.1)). Then, we have the following statements: (i) The point spectrum σ p (W f ,g ) of W f ,g is given by ⎧ ⎪ if b = u, ⎨{λuv : v ∈ V } σ p (W f ,g ) = {λuv : v ∈ Asc(u) ∪ Desb [u]} ∪ {0} if b ∈ Desu (u), ⎪ ⎩ otherwise. {λuv : v ∈ Des(u)} ∪ {0} (ii) The spectrum σ (W f ,g ) of W f ,g is given by  σ (W f ,g ) =

σ p (W f ,g ) if W f ,g satisfies the compatibility condition I , C otherwise.

If, in addition, W f ,g satisfies the compatibility condition I, then we have the following: (iii) For every μ ∈ C\σ p (W f ,g ), the resolvent of W f ,g at μ is given by  (W f ,g − μ)−1 =

 (b) 0 (Dλu − μ)−1 , −(Nλ(b) − μ)−1 L λu ,μ (Nλ(b) − μ)−1 u u (b)

(4.4.1)

where the linear transformation L λu ,μ := ( f  g)(Dλu −μ)−1 defines a Hilbert(b) Hu

(b)

into 2 (V )  Hu . Schmidt integral operator from (iv) The essential spectrum σe (W f ,g ) of W f ,g is given by 

(b) (b)

< ∞, if dim 2 (V )  Hu σe (Dλu ) σe (W f ,g ) = (b) σe (Dλu ) ∪ {0} otherwise. Moreover, ind W f ,g = 0 on C\σe (W f ,g ).

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Remark 4.16 By [48, Theorem 7.1], for any diagonal operator D with simple point spectrum and perfect spectrum, there exists a bounded rank one perturbation h ⊗ k of an arbitrarily small positive norm such that D + h ⊗ k has no point spectrum. This situation does not appear in the context of rank one extensions of weighted join operators, where the point spectrum is always non-empty. Proof Let μ be a complex number and (h, k) ∈ D(W f ,g ) be a non-zero vector such that W f ,g (h, k) = μ(h, k). By (4.0.1), h ∈ D(Dλ(b) ) ∩ D( f  g), k ∈ 2 (V )  Hu(b) u and ⎛ ⎞ ⎜  ⎟ Dλ(b) h = μ h, ⎝ h(v)g(v)⎠ f + k, eλu ,Au eu = μ k. (4.4.2) u (b)

v∈supp Hu

Case I. h = 0: In this case, k, eλu ,Au eu = μ k. Accordingly, any one of the following possibilities occur: (1) eu (considered as the vector (0, eu )) is an eigenvector of W f ,g corresponding to the eigenvalue μ = λuu . (2) k is an eigenvector of W f ,g corresponding to the eigenvalue μ = 0 provided (b)

b = u, where k ∈ 2 (V )  Hu  [eλu ,Au ]. Here, in the second assertion, we used the facts that dim 2 (Au ) ≥ 2 (since b = u) and

dim 2 (V )  Hu(b) ≥ dim 2 (Au ) (see (3.2.15)). Case II. h = 0: (b) (b) In this case, μ ∈ σ p (Dλu ), and hence μ = λuw for some w ∈ supp Hu and h ∈ E D (b) (μ) = 2 (Ww ), λu

where Ww is as given in (3.3.1). It follows from (4.4.2) that ⎛ ⎝



⎞ h(v)g(v)⎠ f + k, eλu ,Au eu = λuw k.

(4.4.3)

v∈Ww

Taking inner-product with eλu ,Au on both sides, we get ⎛ ⎝



⎞ h(v)g(v)⎠ f , eλu ,Au = (λuw − λuu ) k, eλu ,Au .

(4.4.4)

v∈Ww

Accordingly, any one of the following possibilities occur:

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(1) λuw is a non-zero number equal to λuu : In this case, 

f , eλu ,Au = 0 or

h(v)g(v) = 0.

v∈Ww

Thus k belongs either to [ f ] or to [eu ]. (2) λuw is a non-zero number not equal to λuu : By (4.4.3) and (4.4.4), k takes the form  k=

v∈Wwh(v)g(v)

+

, f , eλu ,Au

f + eu . λuw − λuu

λuw

f,e

(4.4.5)



u ,Au In this case, k belongs to the span of f + λuwλ−λ eu . uu (3) λuw = 0: In this case, any non-zero vector (h, k) with h ∈ 2 (Ww ), k ∈ (b) 2 (V )  Hu satisfying the following identity will be an eigenvector of W f ,g corresponding to the eigenvalue 0:

⎛ ⎝



⎞ h(v)g(v)⎠ f + k, eλu ,Au eu = 0.

v∈Ww

In particular, the cases above show that  σ p (W f ,g ) =

(b)

σ p (Dλu ) ∪ {λuu }

if b = u,

(b)

σ p (Dλu ) ∪ {λuu , 0} otherwise.

(b) The conclusion in (i) is now clear from the fact that σ p (Dλu ) = λuv : v ∈ (b) supp Hu , (3.2.11) and (3.2.16). To see (ii), let μ ∈ C\σ p (W f ,g ), that is, μ is a non-zero number such that μ = λuu and dist(λu , μ) =

inf

(b)

v∈supp Hu

|λuv − μ| > 0.

(4.4.6)

By (4.0.1), (k1 , k2 ) ∈ ran(W f ,g − μ) if and only if there exists (h 1 , h 2 ) ∈ D(W f ,g ) such that ⎛ ⎜ (b) (Dλu − μ)h 1 = k1 , ⎝

⎞ 

⎟ h 1 (v)g(v)⎠ f + h 2 , eλu ,Au eu − μh 2 = k2 . (b)

v∈supp Hu

(4.4.7)

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We claim that (b)

W f ,g − μ is surjective if and only if D(Dλu ) ⊆ D( f  g).

(4.4.8) (b)

To see the claim, suppose that W f ,g − μ is surjective and let h 1 ∈ D(Dλu ). Letting

−μ)h 1 and k2 = 0, by surjectivity of W f ,g −μ, we get (h 1 , h 2 ) ∈ D(W f ,g ) k1 = (Dλ(b) u (b)

such that (4.4.7) holds. However, since Dλu − μ is injective, h 1 = h 1 , and hence (b)

h 1 ∈ D( f  g). To see the reverse implication, assume that D(Dλu ) ⊆ D( f  g),

− μ is invertible, and let k1 ∈ Hu(b) and k2 ∈ 2 (V )  Hu(b) . By (4.4.6), Dλ(b) u (b)

(b)

and hence there exists h 1 ∈ D(Dλu ) such that (Dλu − μ)h 1 = k1 . By assumption, h 1 ∈ D( f  g). Since μ = λuu , the following equation can be uniquely solved for h 2 , eλu ,Au : h 2 , eλu ,Au (λuu − μ) = k2 , eλu ,Au −



 h 1 (v)g(v) f , eλu ,Au . (b)

v∈supp Hu

Since μ = 0, substituting the above value of h 2 , eλu ,Au in (4.4.7) determines h 2 ∈ 2 (V )  Hu(b) uniquely. This completes the verification of (4.4.8). The first part in (ii) now follows from Proposition 4.7. To see the remaining part in (ii), suppose that W f ,g does not satisfy the compatibility condition I. Let Γλu be as given in (4.1.1). Thus there are two possibilities: Case 1. Γλu = ∅: (b) In this case, σ p (Dλu ) is necessarily dense in C, and hence by (i), σ p (W f ,g ) is also dense in C. If possible, then assume that σ (W f ,g ) is not equal to C. Then, by [22, Lemma 1.17], W f ,g is closed. However, the spectrum of a closed operator is always closed (see [64, Proposition 2.6]). This is not possible since σ p (W f ,g ) is a dense and proper subset of C, and hence we must have σ (W f ,g ) = C. Case 2. Γλu = ∅: If possible, then suppose that σ (W f ,g )  C. Thus there exists μ ∈ C\σ (W f ,g ) and a linear operator R(μ) ∈ B(2 (V )) such that (W f ,g − μ)R(μ)h = h, h ∈ 2 (V ).

(4.4.9)

Consider the following decomposition of R(μ): ) R(μ) =

A(μ) B(μ) C(μ) D(μ)

*

on 2 (V ) = Hu(b) ⊕ (2 (V )  Hu(b) ). (b)

Note that for any k ∈ 2 (V )  Hu

,

) * ) * 0 (4.4.9) 0 = (W f ,g − μ)R(μ) k k

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 =

 (b) (Dλu − μ)B(μ)k , (b) f  g(B(μ)k) + (Nλu − μ)(D(μ)k)

(b)

(b)

/ σ (Dλu ), and consequently, which yields (Dλu − μ)B(μ)k = 0. However, μ ∈ (b)

B(μ) = 0. Also, since for any h ∈ Hu , R(μ)

) * h ∈ D(W f ,g − μ) = D(W f ,g ), 0

by the definition of the domain of W f ,g , we must have (b)

A(μ)h ∈ D( f  g), (Dλu − μ)A(μ) = I

(4.4.10)

(b)

(see (4.4.9)). It follows that A(μ) = (Dλu − μ)−1 and hence any arbitrary vector in (b)

(b)

D(Dλu ) is of the form A(μ)h for some h ∈ Hu

. This together with (4.4.10) yields

(b) the inclusion D(Dλu ) ⊆ D( f g). An application of Proposition 4.7, however, shows that W f ,g satisfies the compatibility condition I, which is contrary to our assumption.

This completes the proof of (ii). To see (iii) and (iv), assume that W f ,g satisfies the compatibility condition I. Let μ ∈ C\σ p (W f ,g ). By Proposition 4.7, L λu ,μ is bounded. A routine verification shows that L λu ,μ is an integral operator with kernel K λu ,μ given by K λu ,μ (w, v) :=

g(v) f (w) , w ∈ V \supp Hu(b) , v ∈ supp Hu(b) . λuv − μ (b)

By the compatibility condition I and the assumption that f ∈ 2 (V )  Hu , K λu ,μ (b) (b) belongs to 2 ((V \supp Hu ) × supp Hu ). By [67, Theorem 3.8.5], L λu ,μ is a Hilbert-Schmidt operator. We leave it to the reader to verify that the expression given by (4.4.1) defines the resolvent of W f ,g at μ. To see (iv), let A, B, C be as given by (4.2.8) and note that A+ B +C = W f ,g . Since

A = Dλ(b) ⊕0 on Hu(b) ⊕ 2 (V )Hu(b) , it suffices to check that σe (W f ,g ) = σe (A) u and ind W f ,g = ind A . In view of [46, Theorems 5.26 and 5.35, Chapter IV], it is sufficient to verify that B + C is A-compact (see also the foot note 1 on [46, Pg 244]). Let {h n } be a bounded sequence in D(A) ⊆ D(B) such that {Ah n } is bounded. Since C is a finite rank operator, it suffices to check that {Bh n } has a convergent subsequence. By part (iii), L λu ,μ is a compact operator, and hence so is B(A − μ)−1 . However, {(A − μ)h n } is bounded, and hence {Bh n } admits a convergent subsequence. The remaining part follows from the fact that the index function for a diagonal operator is identically 0.   Remark 4.17 In this remark, we describe eigenspaces of W f ,g (under the hypotheses of Theorem 4.15). To see that, we need some notations. Given a subspace 2 (W ) of

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2 (V ) and a rank one operator h  k from 2 (V ) into 2 (U ), we introduce the linear transformation h  k|2 (W ) from 2 (W ) into 2 (U ) as follows: D(h  k|2 (W ) ) = D(h  k) ∩ 2 (W ), h  k|2 (W ) (l) = h  k(l), l ∈ D(h  k|2 (W ) ). For μ ∈ C, let Wμ be given by Wμ = {w ∈ supp Hu(b) : λuw = μ}.

(4.4.11)

In case μ = λuv for some v ∈ V , we denote Wμ by the simpler notation Wv . Further, we reserve the notation graph(T ) for the graph of a linear operator T in H. If EW f ,g (μ) denotes the eigenspace corresponding to the eigenvalue μ of W f ,g , then we have the following statements: (b)

(a) If λuv = 0 for every v ∈ supp Hu

, then

EW f ,g (0) = {0} ⊕ ker(Nλ(b) ). u (b) If λuv = 0 for some v ∈ supp Hu(b) , then  EW f ,g (0) =

ker( f  g) ˜ if f ∈ [eu ], (b) ker( f  g|2 (W ) ) ⊕ ker(Nλu ) otherwise, v

(b)

→ C is given by where g˜ : Wv ∪ V \supp Hu  g(w) ˜ =

f (u) g(w) if w ∈ Wv , eλu ,Au (w) otherwise.

(c) If μ = λuu is non-zero, then  EW f ,g (μ) =

(b) graph( f˜  g|2 (W ) ) + [eu ] if f ∈ ker Nλu , u

ker( f  g|2 (W ) ) ⊕ [eu ] u

otherwise,

where f˜ = f /λuu . (b) / {0, λuu }, then (d) If μ = λuv for some v ∈ supp Hu and μ ∈

EW f ,g (μ) = graph f˜  g|2 (W ) , u

where f˜ =

1 λuv



f +

f , eλ ,A

u u λuv −λuu eu

 .

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To see the above statements, suppose that (h, k) ∈ EW f ,g (μ). Since b = u, by Case I(2) of the proof of Theorem 4.15, 2

(b)  (V )  Hu(b)  [eλu ,Au ] = ker(Nλu ) ⊆ EW f ,g (0).

(4.4.12)

The desired conclusion in (a) now follows from (4.4.2) and the assumption that λuv = 0 (b) (b) for every v ∈ supp Hu . To see (b), assume that λuv = 0 for some v ∈ supp Hu , and suppose that f = c eu for some non-zero scalar c. By (4.4.3) and (4.4.12), (h, k) ∈ EW f ,g (0) if and only if ⎛ c⎝



⎞ h(v)g(v)⎠ + k, eλu ,Au = 0,

v∈Ww

which is equivalent to (h, k) ∈ ker( ˜ This yields the first part in (b). If f and  f  g). eu are linearly independent, then v∈Wwh(v)g(v) = 0 and k, eλu ,Au = 0. Thus the other part in (b) follows at once from (4.4.12). To see (c), suppose that μ = λuu is non-zero. By Case I(1) and (4.4.3), k = α f + β eu for some α, β ∈ C. Combining this with (4.4.3) yields ⎛ ⎝



⎞ h(v)g(v) − α λuu ⎠ f + α f , eλu ,Au eu = 0

v∈Wu

If f ∈ ker(Nλ(b) ), then f , eλu ,Au = 0, and the above equation determines α uniquely, u whereas β can be chosen arbitrarily to get the conclusion in the first part of (c). If  (b) f ∈ / ker(Nλu ), then by (4.4.4), v∈Wuh(v)g(v) = 0, that is, h ∈ ker( f  g|2 (W ) ). u However, in this case, k ∈ [eu ], which yields the remaining part of (c). To see (d), / {0, λuu }. Once again, by (4.4.5), k = f˜ g(h). This completes assume that μ = λuv ∈ the verification of (d).   The following sheds more light into the spectral picture of rank one extensions of weighted join operators. Corollary 4.18 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the (b) (b) rank one extension of the weighted join operator Wλu on T , where f ∈ 2 (V )Hu (b)

is non-zero and g : supp Hu → C is given. Then σ (W f ,g ) is a proper closed subset of C if and only if W f ,g satisfies the compatibility condition I. Further, we have the following: (a) In case W f ,g satisfies the compatibility condition I, W f ,g defines a closed linear operator such that the following hold: (a1) σ (W f ,g ) = σ p (W f ,g ).

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(a2) π(W f ,g ) = C\σ p (W f ,g ). (b) In case W f ,g does not satisfy the compatibility condition I, the following hold: (b1) σ (W f ,g ) = C. (b2) Either W f ,g is not closed or π(W f ,g ) = ∅. Proof Suppose that σ (W f ,g ) is a proper closed subset of C. Thus there exists μ ∈ C\σ p (W f ,g ) such that W f ,g − μ is surjective. By (4.4.8), we obtain the domain (b) inclusion D(Dλu ) ⊆ D( f  g), and hence by Proposition 4.7, W f ,g satisfies the compatibility condition I. Conversely, if W f ,g satisfies the compatibility condition I, then Γλu is non-empty (see (4.1.1)). It may now be concluded from (i) and (ii) of Theorem 4.15 that σ (W f ,g ) is a proper closed subset of C. To see (a), assume that W f ,g satisfies the compatibility condition I. Since σ (W f ,g ) is a proper subset of C, W f ,g is closed (see [22, Lemma 1.17]). Further, (a1) follows from Theorem 4.15(ii). Since the complement of the regularity domain of a densely defined closed operator is a closed subset of the spectrum that contains the point spectrum (see [64, Proposition 2.1]), the conclusion in (a2) follows. To see (b), assume that W f ,g does not satisfy the compatibility condition I. Clearly, (b1) follows from the first part of this corollary. To see (b2), assume that W f ,g is closed. By [64, Proposition 2.6], (b1)

{μ ∈ π(W f ,g ) : dW f ,g (μ) = 0} = C\σ (W f ,g ) = ∅

(4.4.13)

(see (1.2.1)). Let μ ∈ π(W f ,g ). Since π(W f ,g ) ⊆ C\σ p (W f ,g ), μ ∈ / σ p (W f ,g ). (b) (b) Then, by the proof of Theorem 4.15(i), μ ∈ / σ p (Dλu ) ∪ σ (Nλu ). It follows that for every v ∈ supp Hu(b) , 

−μ 0 Dλ(b) u (b) f  g N λu − μ

 −(λuv

(λuv − μ)−1 ev (b) − μ)−1 g(v)(Nλu − μ)−1 ( f ) (b)

which implies that (W f ,g − μ)D(W f ,g ) is dense in Hu



* ev . = 0 )

. Also,

(W f ,g − μ)(2 (V )  Hu(b) ) = 2 (V )  Hu(b) , which implies that dW f ,g (μ) = 0. This together with (4.4.13) shows that π(W f ,g ) = ∅ completing the proof.   In general, the spectrum of W f ,g may not be the topological closure of its point spectrum. (b)

Example 4.19 Let g : supp Hu

→ C be such that



(b)

v∈supp Hu

|g(v)|2 = ∞ and

card(supp(g)) = ℵ0 = card(supp Hu(b) \supp(g))

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(for instance, one may let u = v2 and supp(g) = Des(v5 ) in the rooted directed tree as given in Fig. 4). Let λu be a weight system such that {λuv : v ∈ supp(g)} ⊆ {z ∈ C : 1 ≤ |z| ≤ 2}, {λuv : v ∈

(b) supp Hu }

= C.

(4.4.14) (4.4.15)

It is easy to see using (4.4.14) that W f ,g does not satisfy the compatibility condition. Hence, by Corollary 4.18, σ (W f ,g ) = C. Further, since σ p (W f ,g ) is not dense in C (see (4.4.15)), we must have σ p (W f ,g )  σ (W f ,g ). Thus the spectral picture of a rank one extension W f ,g of a weighted join operator can be summarized as follows: (i) If g satisfies the compatibility condition I, then σ (W f ,g ) = σ p (W f ,g ) is a proper subset of C. (ii) If g does not satisfy the compatibility condition I, then σ (W f ,g ) = C and σ p (W f ,g ) may be a proper subset of C. In the last case, either W f ,g is not closed or π(W f ,g ) = ∅.



As an application to Theorem 4.15, we characterize those rank one extensions of weighted join operators on leafless directed trees, which admit compact resolvent. Corollary 4.20 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V and b ∈ V \{u}, consider the weight system λu = {λuv }v∈V∞ of complex numbers and (b) let W f ,g be the rank one extension of the weighted join operator Wλu on T , where

f ∈ 2 (V )  Hu(b) is non-zero and g : supp Hu(b) → C is given. Suppose that W f ,g satisfies the compatibility condition I. If T is leafless, then the following are equivalent: (b)

(i) The rank one extension W f ,g of the weighted join operator Wλu on T admits a compact resolvent. (b) (ii) The set {λuv : v ∈ supp Hu } has an accumulation point only at ∞ with each of its entries appearing finitely many times and the set V≺ of branching vertices of T is disjoint from Asc(u). Proof We need a couple of general facts in this proof. (a) The diagonal operator Dλ has compact resolvent if and only if the weight system λ has an accumulation point only at ∞ with each of its entries appearing finitely many times. (b) A finite block matrix with operator entries being bounded linear is compact if and only if all of its entries are compact. To see the equivalence of (i) and (ii), assume that T is leafless. In view of (b), the formula (4.4.1) and Theorem 4.15(iii), W f ,g has compact resolvent if and only if Dλ(b) u (b)

and Nλu have compact resolvents. On the other hand, by Proposition 2.12(ii) and (b)

(3.2.11), 2 (V )  Hu is finite dimensional if and only if V≺ ∩ Asc(u) = ∅. In view (b) of Lemma 1.2(iii), this is equivalent to the assertion that Nλu has compact resolvent. The desired equivalence now follows from (a).  

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Remark 4.21 Assume that T is leafless and b = u. Then, by (3.2.16), 2 (V )  Hu is one-dimensional. One may now argue as the proof of Corollary 4.20 to show that W f ,g has compact resolvent if and only if the set {λuv : v ∈ V } has an accumulation point only at ∞ with each of its entries appearing finitely many times. It is well-known that given any closed subset σ of the complex plane, there exists a diagonal operator Dλ on 2 (N) such that σ (Dλ ) = σ. Here is a variant of this fact for rank one extensions of weighted join operators. Corollary 4.22 Let T = (V , E) be a rooted directed tree and let b ∈ V . Let u ∈ V be such that Tu = (Des(u), E u ) is an infinite directed subtree of T . Then, for any closed, unbounded proper subset σ of the complex plane, there exists a rank one extension (b) W f ,g of a weighted join operator Wλu on T such that the following hold: (b)

(i) g ∈ / Hu , (ii) W f ,g satisfies the compatibility condition I, and (iii) σ (W f ,g ) = σ . Proof Let f = eu and let z 0 ∈ C\σ. Let {μn }n≥1 be a countable dense subset of σ and let {νn }n≥1 be a subset of σ such that |νn − z 0 | ≥ 2n/2 for every integer n ≥ 1

(4.4.16)

(which exists since σ is unbounded). Consider the countable dense subset {λn }n≥1 of σ defined by  λn =

μk if n = 2k, k ≥ 1, νk if n = 2k − 1, k ≥ 1. (b)

By axiom of choice [66, Pg 11], there exists a sequence {vn }n≥1 ⊆ supp Hu that dvn = n for every integer n ≥ 1. Set  λuv = (b)

Define g : supp Hu

such

λn if v = vn , 0 otherwise.

→ C by  g(v) =

λn −z 0 dv

0

if v = vn , otherwise.

Then, by the choice of vertices {vn }n≥1 , the weight system λu and g, ∞

 (b)

v∈supp Hu

 1 |g(v)|2 = < ∞, 2 |λuv − z 0 | n2 n=1

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and hence the rank one extension W f ,g of the weighted join operator Wλu satisfies the compatibility condition I. This combined with Theorem 4.15(ii) yields (iii). On the other hand, by (4.4.16) and the definition of {λn }n≥1 , 

|g(v)|2 ≥ (b)

∞ ∞   |νk − z 0 |2 2k ≥ , |2k − 1|2 |2k − 1|2 k=1

v∈supp Hu

k=1

which shows that g ∈ / Hu(b) . This completes the proof.

 

It is worth noting that the conclusion of Corollary 4.22 does not hold in case σ is a bounded subset of C. Indeed, the boundedness of the spectrum of a rank one (b) extension W f ,g of a weighted join operator Wλu implies that the diagonal operator (b)

Dλu is bounded. Then, by Remark 4.2, W f ,g is not even closable. So, by Theorem 4.11, W f ,g can not satisfy a compatibility condition. A special case of Theorem 4.15 (the case of g = 0) provides a complete spectral picture for weighted join operators. This together with some additional properties is summarized in the next result. We first introduce some notations and definitions. Let T be a densely defined linear operator in H. A complex number μ is said to be a generalized eigenvalue of T if there exists a positive integer k and a non-zero vector f ∈ D(T k ) (to be referred to as generalized eigenvector corresponding to μ) such that (T − μ)k f = 0. The rootspace RT (μ) of T corresponding to the generalized eigenvalue μ is defined as the closed space spanned by the corresponding generalized eigenvectors of T . Any vector in the rootspace of T is referred to as root vector for T . We say that T is complete if it has a complete set of root vectors (refer to [32] for the basics of completeness of root systems and to [11] for completeness of root systems for the class of rank one perturbations of self-adjoint operators). Theorem 4.23 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V and b ∈ V \{u}, be the consider the weight system λu = {λuv }v∈V∞ of complex numbers and let Wλ(b) u weighted join operator on T . Then, we have the following statements: (b)

(b)

(i) The point spectrum σ p (Wλu ) of Wλu is given by ⎧ ⎪ if b = u, ⎨{λuv : v ∈ V } (b) σ p (Wλu ) = {λuv : v ∈ Asc(u) ∪ Desb [u]} ∪ {0} if b ∈ Desu (u), ⎪ ⎩ otherwise. {λuv : v ∈ Des(u)} ∪ {0} (b)

(b)

(b)

(ii) The spectrum σ (Wλu ) of Wλu is the topological closure of σ p (Wλu ). Reprinted from the journal

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(iii) If EW (b) (μ) denotes the eigenspace corresponding to the eigenvalue μ of Wλu , then

λu

⎧ 2 ⎪ if μ = 0, μ = λuu ⎨ (Wμ ) EW (b) (μ) = 2 (Wμ ) ⊕ [eu ] if μ = 0, μ = λuu , ⎪ 2 λu ⎩ 2 (b)

 (W0 ) ⊕  (V )  Hu  [eλu ,Au ] if μ = 0, (b)

where Wμ is given by (4.4.11), Hu is given by (3.2.11) and Au is given by (3.2.15). (b) (iv) The multiplicity function mW (b) : σ p (Wλu ) → Z+ ∪ {ℵ0 } is given by λu

mW (b) (μ) = card {v ∈ {u} ∪ supp Hu(b) : λuv = μ} if μ = 0. λu

In addition, if T is leafless and if there exists a branching vertex w ∈ Asc(u), then mW (b) (0) = ℵ0 . λu

(b)

(v) The rootspace RW (b) (μ) of Wλu corresponding to the generalized eigenvalue μ is given by

λu

⎧ ⎪ E (b) (μ) ⎪ ⎪ ⎨ Wλu RW (b) (μ) = EW (b) (0) λu ⎪ λu ⎪ ⎪ ⎩E (b) (0) ⊕ [e Wλu

λu ,Au

if μ ∈ σ p (Wλ(b) )\{0}, u if μ = 0, λuu = 0, ] if μ = 0, λuu = 0. (b)

Remark 4.24 In case b = u, the spectral picture of Wλu coincides with that of the diagonal operator Dλu (see Remark 3.11). We discuss here the spectral picture of (∞) (∞) Wλu . By (3.2.3), Wλu is an orthogonal direct sum of rank one operators eu j  eλu ,Des(u j )\Des(u j−1 ) ,

j = 0, . . . , du ,

where Des(u −1 ) = ∅ and u j := par j (u) for j = 0, . . . , du . If λu ∈ 2 (V ), then it (∞) follows that Wλu ∈ B(2 (V )), and hence by Lemma 1.2(iii), σ (Wλ(∞) ) = {0} ∪ {λuu j : j = 0, . . . , du } = σ p (Wλ(∞) ). u u (∞)

Assume now that λu ∈ / 2 (V ). By Lemma 1.3, σ p (Wλu ) is given by the same formula (∞)

as above. Further, another application of Lemma 1.3 shows that Wλu and hence

) σ (Wλ(∞) u

is not closed,

= C.

Proof The conclusions in (i) and (ii) follow from (i) and (iii) of Theorem 4.15. Since (b) 2 (Au ) ⊆ 2 (V )  Hu and card(Au ) ≥ 2 (see (3.2.15)), by Lemma 1.2(iii) and (3.2.14), we obtain

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σ p (Nλ(b) ) = {0, λuu } = σ (Nλ(b) ), u u

2 m N (b) (0) = ℵ0 if dim  (V )  Hu(b) = ℵ0 , m N (b) (λuu ) = 1 if λuu = 0, λu λu  [eλu ,Au ]⊥ if μ = 0, λuu = 0 ⇒ E N (b) (μ) = λu if μ = λuu . [eu ] (b)

= ℵ0 if and only if card(Vu ) = ℵ0 , In view of the fact that dim 2 (V )  Hu the conclusions in (iii) and (iv) pertaining to the eigenspaces and multiplicities now follow from Proposition 2.12. To see (v), let k be a positive integer and let μ ∈ C\{0}. By (3.2.17), (b)

(Nλ

u

+ , + , k k   k k l−1 (b) l Nλ = (−μ)k I + λ eu ⊗ eλu ,Au . (−μ)k−l (−μ)k−l u l l uu

− μ)k =

l=0

l=1

(b)

(b)

It is not difficult to see that for any h ∈ 2 (V )  Hu such that (Nλu − μ)k h = 0, we must have h = h(u)eu and μ = λuu . However, eu is an eigenvector, and hence a root vector. This shows that the rootspace of Nλ(b) corresponding to μ is spanned u by eu . Since the generalized eigenvalues and eigenvalues coincide for a diagonal operator, the desired conclusion in (v) follows provided λuu = 0. In case λuu = 0, (b) by Corollary 3.16, Nλu is a nilpotent operator of nilpotency index 2, and hence any (b)

h ∈ 2 (V )  Hu

is a root vector. This completes the verification of (v).

 

A result of Wermer says that the spectral synthesis holds for all normal compact operators [73]. As shown by Hamburger [33], this no longer holds for compact operators. Interestingly, the following result can be used to construct examples of compact non-normal operators which are not even complete. Corollary 4.25 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V and b ∈ V \{u}, consider the weight system λu = {λuv }v∈V∞ of complex numbers and (b) (b) let Wλu be the weighted join operator on T . Then Wλu is complete if and only if λuu = 0 or λuv = 0 for every v ∈ Au \{u}. Proof If λuu = 0, then by (iii) and (v) of Theorem 4.23, the root vectors for Wλ(b) u (b)

forms a complete set. If λuv = 0 for every v ∈ Au \{u}, then by Theorem 3.13, Wλu is a diagonal operator, and hence we get the sufficiency part. To see the necessity part, suppose that λuu = 0 and λuv = 0 for some v ∈ Au \{u}. Thus eλu ,Au is a non-zero vector in 2 (V ), and by (iv) and (v) of Theorem 4.23, μ∈C

RW (b) (μ) ⊆ 2 (V )  [eλu ,Au ]. λu

is not complete. This shows that Wλ(b) u

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5 Special Classes In this section, we discuss some special classes of weighted join operators and their rank one extensions. In particular, we exhibit families of sectorial operators and infinitesimal generators of quasi-bounded strongly continuous semigroups within these classes. Further, we characterize hyponormal operators and n-symmetric operators within the class of weighted join operators on rooted directed trees. We also investigate the classes of hyponormal and n-symmetric rank one extensions W f ,g of weighted join operators. The complete characterizations of these classes seem to be beyond reach at present, particularly, in view of the fact that structures of positive integral powers and the Hilbert space adjoint of W f ,g are complicated. 5.1 Sectoriality A densely defined linear operator T in H is sectorial if there exist a ∈ R, M ∈ (0, ∞) and θ ∈ (0, π2 ) such that λ ∈ ρ(T ) and RT (λ) ≤

M whenever λ ∈ C and | arg(λ − a)| ≥ θ. |λ − a| (5.1.1)

Sometimes we say that T is sectorial with angle θ and vertex a. Note that there is considerable divergence of terminology in the literature, for example, Kato [46] calls it m-sectorial, while we call it just sectorial, the correspondence being a minus sign between the two. For the basic theory of sectorial operators, the reader is referred to [5, 16, 22, 34, 46, 52, 64, 67, 68]). The following result yields a family of sectorial rank one extensions of weighted join operators (cf. [41, Proposition 3]). Proposition 5.1 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the (b) rank one extension of the weighted join operator Wλu on T , where f ∈ 2 (V )  (b)

(b)

Hu is non-zero and g : supp Hu → C is given. Suppose that W f ,g satisfies (b) the compatibility condition II. If {λuv : v ∈ supp Hu } is contained in the sector Sθ,α := {z ∈ C : | arg(z − α)| < θ } for some θ ∈ (0, π/2) and α ∈ R, then W f ,g is a sectorial operator. (b)

Proof Assume that {λuv : v ∈ supp Hu } is contained in Sθ,α for some θ ∈ (0, π/2). After replacing θ by θ +  ∈ (0, π/2) for some  > 0, we may assume without loss (b) of generality that σ (Dλu ) ⊆ Sθ  ,α for some θ  ∈ (0, θ ). It is well-known that the

satisfies the estimate (5.1.1) (see, for instance, [34, Chapdiagonal operator Dλ(b) u ter 2, Section 2.2.1] and [52, Example 4.5.2]). Indeed, for any μ ∈ C\Sθ , (b)

(Dλu − μ)−1  =

sup (b) v∈supp Hu

|λuv − μ|−1

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Fig. 7 The situation of Proposition 5.1

1

=

inf

(b)

v∈supp Hu

|λuv − μ|

1 inf |t − μ|

≤

t∈Sθ  ,α

 ≤

1 |μ−α| 1 1 |μ−α| sin(arg(μ−α)−θ  )

if | arg(μ − α)| ≥ θ + π/2, otherwise

1 (see Fig. 7). Thus (5.1.1) holds with M = sin(θ−θ  ) . By Corollary 4.12, W f ,g = A + B + C, where B + C is A-bounded with A-bound equal to 0 (see (4.2.8) and (b) (4.2.11)). Since A is sectorial (since so is Dλu ), an application of [52, Theorem 4.5.7]   shows that W f ,g is sectorial.

For all relevant definitions and basic theory of strongly continuous quasi-bounded semigroups, the reader is referred to [46, 52, 68]. A result similar to the following has been obtained in [55, Proposition 3.1] for a family of upper triangular operator matrices on non-diagonal domains (cf. [13, Theorem 7.11]). Proposition 5.2 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the rank one extension of the weighted join operator Wλ(b) on T , where f ∈ 2 (V )  u (b)

(b)

Hu is non-zero and g : supp Hu → C is given. Suppose that W f ,g satisfies the compatibility condition II. If {λuv : v ∈ supp Hu(b) } is contained in the right half plane Hα = {z ∈ C : z ≥ α} for some α ∈ R, then W f ,g is the generator of a strongly continuous semigroup {Q(t)}t≥0 satisfying Q(t) ≤ Me−αt for t ≥ 0.

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Proof Assume that {λuv : v ∈ supp Hu } is contained in the right half plane Hα = (b) {z ∈ C : z ≥ α} for some α ∈ R. Thus σ (Dλu ) ⊆ Hα . It is easy to see that (b)

(Dλu − μ)−n  ≤

1 , μ ∈ (−∞, α), n = 1, 2, . . . . |α − μ|n

Hence, by the Hille-Yoshida Theorem (see [52, Theorem 4.3.5], [68, Theorem 2.3.3]), Dλ(b) is the generator of a strongly continuous semigroup {S(t)}t≥0 satisfying the quasiu boundedness condition S(t) ≤ Me−αt , t ≥ 0. By Corollary 4.12, W f ,g = A + B + C, where B + C is A-bounded with A-bound equal to 0 (see (4.2.8) and (4.2.11)). Note that A is the generator of the strongly continuous semigroup {S(t) ⊕ I }t≥0 . The desired conclusion may now be derived from [46, Corollary 2.5, Chapter IX].   If W f ,g is as in the preceding result, one can define fractional powers of W f ,g (refer to [52, Chapter 6]). Further, one may obtain a counterpart of [41, Corollary 2] ensuring H ∞ -functional calculus for rank one extensions of weighted join operators (cf. [20, Proposition 3.4]). We refer the reader to [34] for more details on this topic. 5.2 Normality A densely defined linear operator T in H is said to be hyponormal if D(T ) ⊆ D(T ∗ ) and T ∗ x ≤ T x for all x ∈ D(T ). We say that T is cohyponormal if T is closed and T ∗ is hyponormal. There has been significant literature on the classes of hyponormal and cohyponormal operators (refer to [25, 38–44, 58]). We begin with a rigidity result stating that no weighted join operator can be hyponormal unless it is diagonal. A variant of this fact in the context of bounded operators has been obtained in [44, Theorem 2.3] (cf. [37, Proposition 3.1]). Proposition 5.3 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , be the consider the weight system λu = {λuv }v∈V∞ of complex numbers and let Wλ(b) u weighted join operator on T . Then the following statements are equivalent: (b)

(i) Wλu is normal. (b)

(ii) Wλu is hyponormal.

is cohyponormal. (iii) Wλ(b) u (b)

(iv) Wλu is diagonal with respect to the orthonormal basis {ev }v∈V . (v) b = u or λuv = 0 for every v ∈ Au \{u}, where Au is given by (3.2.15). (b)

Proof By Remark 3.11, Wλu is a diagonal operator if b = u or λuv = 0 for every admits the decomv ∈ Au \{u}, and hence (v) implies (i)–(iv). By Theorem 3.13, Wλ(b) u (b)

(b)

(b)

position (Dλu , Nλu , Hu

(b)

). Thus the Hilbert space adjoint of Wλu is given by (b) ∗

(b)

D(Wλu ) = D(Wλu ), 316

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Wλu

(b) ∗

= Dλu

(b) ∗

⊕ Nλu .

Since Dλ(b) is normal, Wλ(b) is hyponormal (resp. cohyponormal) if and only if so is u u (b) ∗

(b)

Nλu . Note that by (1.3.2), Nλu

= eλu ,Au ⊗ eu . Also, by (1.3.2), for x, y ∈ 2 (V ),

[x ⊗ y, (x ⊗ y)∗ ] = (x ⊗ y)(y ⊗ x) − (y ⊗ x)(x ⊗ y) = y2 x ⊗ x − x2 y ⊗ y. It follows that ⎧ 2 ⎪ ⎨ eλu ,[u,b] ⊗ eλu ,[u,b] − eλu ,[u,b]  eu ⊗ eu if b ∈ Des(u), (b) ∗ (b) [Nλu , Nλu ] = eλu ,Asc(u)∪{u,b} ⊗ eλu ,Asc(u)∪{u,b} ⎪ ⎩ − eλu ,Asc(u)∪{u,b} 2 eu ⊗ eu otherwise. This yields (b) ∗ [Nλu ,

 (b) Nλu ]eu ,

eu =

−eλu ,(u,b] 2 if b ∈ Des(u), 2 −eλu ,Asc(u)∪{b}  otherwise,

which is always negative provided b = u and λuv = 0 for some v ∈ Au \{u}. Further, in this case, ∗

[Nλ(b) , Nλ(b) ]ev , ev = |λuv |2 , v ∈ Au \{u}, u u and hence Nλ(b) is not cohyponormal. This completes the proof. u

 

Let us now investigate the class of normal rank one extensions W f ,g of weighted join operators. Proposition 5.4 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the rank one extension of the weighted join operator Wλ(b) on T , where f ∈ 2 (V )  Hu(b) is nonu (b)

zero and g : supp Hu → C is given. Suppose that W f ,g satisfies the compatibility condition I. Then the following statements are equivalent: (i) W f ,g is normal. (ii) W f ,g is diagonal with respect to the orthonormal basis {ev }v∈V . (iii) g = 0 and either b = u or λuv = 0 for every v ∈ Au \{u}.

(5.2.1)

(b)

Proof Assume that W f ,g is normal. Note that Nλu , being the restriction of W f ,g to (b)

(b)

2 (V )Hu , is hyponormal. By Proposition 5.3, (iii) holds. Thus Nλu = λuu eu ⊗eu . Reprinted from the journal

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Since W f ,g satisfies the compatibility condition I, by Corollary 4.18, σ (W f ,g ) is a proper closed subset of C. Let μ ∈ C\σ (W f ,g ). By Theorem 4.15(iii),  (W f ,g − μ)−1 =

 (b) 0 (Dλu − μ)−1 , (b) (b) −(Nλu − μ)−1 L λu ,μ (Nλu − μ)−1

(5.2.2) (b)

where the linear transformation L λu ,μ is given by L λu ,μ = ( f  g)(Dλu − μ)−1 . On the other hand, by [64, Proposition 3.26(v)], (W f ,g − μ)−1 is normal. Let A = (b) (b) (b) (Dλu − μ)−1 , B = −(Nλu − μ)−1 L λu ,μ and C = (Nλu − μ)−1 . Since A and C are normal, it may be concluded from (5.2.2) that [((W f ,g − μ)

−1 ∗

) , (W f ,g − μ)

−1

* B ∗ C − AB ∗ B∗ B . ]= C ∗ B − B A∗ B B∗ )

Since W f ,g is normal, B = 0. It follows that L λu ,μ = 0, and hence f  g = 0 (b) on D(Dλu ). Since f is non-zero, we must have g = 0. The remaining implications follow from Proposition 5.3.   The methods of proofs of Propositions 5.3 and 5.4 are different. In particular, the unavailability of a formula for the Hilbert space adjoint of W f ,g necessitated us to characterize the normality of W f ,g with the help of the resolvent function. An inspection of the proof of Proposition 5.4 shows that (5.2.1) is a necessary condition for W f ,g to be a hyponormal operator. 5.3 Symmetricity A densely defined linear operator T in H is said to be n-symmetric if n  j=0

(−1)n− j

+ , n T j x, T n− j y = 0, x, y ∈ D(T n ), j

where n is a positive integer. We refer to 1-symmetric operator as symmetric operator. We say that T is strictly n-symmetric if it is n-symmetric, but not (n − 1)-symmetric, where n is a positive integer bigger than 1. For the basic properties of n-symmetric operators and its connection with the theory of differential equations, the reader is referred to [2–4, 8, 9, 35, 36, 62, 63]. The following proposition describes all n-symmetric weighted join operators. It reveals the curious fact that there is no strictly 2-symmetric weighted join operator. On the other hand, strictly 3-symmetric weighted join operators exist in abundance. Proposition 5.5 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For b, u ∈ V and (b) the weight system λu = {λuv }v∈V∞ of complex numbers, let Wλu denote the weighted join operator on T . Then, for any positive integer n, the following are equivalent:

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(i) Wλu is n-symmetric. (ii) The weight system λu satisfies  λuv ∈ R, v ∈

Asc(u) ∪ Desb [u] if b ∈ Des(u), Des(u) otherwise,

and one of the following holds: (a) λuu = 0 and n ≥ 3. (b) λuv = 0 for v ∈ Au \{u}, where Au is as given in (3.2.15). (iii) One of the following holds: (a) λuu = 0 and n ≥ 3. (b) (b) The weighted join operator Wλu is symmetric. (b)

Proof By Theorem 3.13, the weighted join operator Wλu admits the decomposition

, Nλ(b) , Hu(b) ). Thus Wλ(b) is n-symmetric if and only if Dλ(b) and Nλ(b) are n(Dλ(b) u u u u u symmetric. It is easy to see that a diagonal operator is n-symmetric if and only if it is symmetric, which in turn is equivalent to the assertion that its diagonal entries are (b) real. Assume that Nλu is n-symmetric and let v, w ∈ Au . Note that by (3.2.17), + , n  (b) j (b) n− j n− j n Nλu ev , Nλu (−1) ew

j j=0 ⎧ (λuu − λ¯ uu )n if v = u, w = u, ⎪ ⎪ ⎪ + , ⎪ n−1 ⎪  ⎪ n j n− j−1 ⎪(−1)n λ¯ λ¯ n−1 + ⎨ λ¯ uw λuu λ¯ uu (−1)n− j if v = u, w = u, uw uu j = j=1 ⎪ + , ⎪ n−1 ⎪ ⎪ n j−1 n− j−1 ⎪ ⎪ (−1)n− j λuv λ¯ uw λuu λ¯ uu if v = u, w = u. ⎪ ⎩ j j=1

(5.3.1) Thus, by the first identity, λuu is real, and hence, by the second identity, for every w ∈ Au \{u}, ¯ λn−1 uu λuw

+ , n−1  n− j n = 0. (−1) j j=0

Thus λuu = 0 or λuw = 0 for every w ∈ Au \{u}.

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Suppose λuw = 0 for some w ∈ Au \{u} and n ≤ 2. Then, by (5.3.2), λuu = 0, and hence, by the third identity in (5.3.1), + , n−1  n (−1)n− j λuv λ¯ uw λn−2 uu = 0, v ∈ Au \{u}. j j=1

If n = 2, then −2λuv λ¯ uw = 0, which is not possible for v = w (since by assumption λuw = 0). This proves the implication (i) ⇒ (ii). Also, since condition (b) of (ii) is (b) equivalent to the assertion that Nλu is equal to the normal rank one operator λuu eu ⊗eu (see (3.2.14)), we also obtain the equivalence of (ii) and (iii). Finally, the implication (ii) ⇒ (i) may be easily deduced from (5.3.1) and (3.0.4).   (b)

Remark 5.6 The weighted join operator Wλu symmetric. In particular,

Wλ(b) u

is either symmetric or strictly 3-

is never strictly 2-symmetric.

We capitalize on the last proposition to exhibit a family of n-symmetric rank one extensions of weighted join operators. Proposition 5.7 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the rank one (b) (b) extension of the weighted join operator Wλu on T , where f ∈ 2 (V )  Hu is nonzero and g : supp Hu(b) → C is a non-zero function. Assume that f , eλu ,Au = 0. Then the following statements hold:

(i) If n ≥ 2 and supp(g)∩{v ∈ supp Hu(b) : λuv = 0} = ∅, then W f ,g is n-symmetric (b) if and only if {λuv : v ∈ supp Hu } is contained in R and either f (u) = 0 or (5.3.2) holds. (ii) If supp(g)∩{v ∈ supp Hu(b) : λuv ∈ R\{0}} = ∅, then W f ,g is never n-symmetric. Proof By (4.0.1), W nf ,g can be decomposed as  W nf ,g =

(b)

0 (Dλu )n L n (Nλ(b) )n u

 ,

where L n , n ≥ 0 is defined inductively as follows: (b)

(b)

L 0 = 0, L 1 = f  g, L n = L n−1 Dλu + (Nλu )n−1 f  g, n ≥ 2. (b) k

Recall from (3.2.17) that Nλu shows that

)k−1 + L k = f  g(Dλ(b) u

(b)

= λk−1 uu Nλu , k ≥ 1. An inductive argument now k−1 

λuu Nλ(b) ( f  g)(Dλ(b) )k− j−1 , k ≥ 1. (5.3.3) u u j−1

j=1

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Note that W f ,g is n-symmetric if and only if Nλu is n-symmetric and for every (h 1 , h 2 ), (k1 , k2 ) in D(W nf ,g ), + , n  n (b) (b) (Dλu ) j h 1 , (Dλu )n− j k1

(−1)n− j j j=0

+ , n L j h 1 , L n− j k1 = 0, j

(5.3.4)

+ , n−1  (b) n− j n (Nλu ) j h 2 , L n− j k1 = 0. (−1) j

(5.3.5)

+

n−1  j=1

(−1)n− j

j=0

Assume now that n ≥ 2 and f , eλu ,Au = 0. By (5.3.3), L k = f  g(Dλ(b) )k−1 , k ≥ 1. u

(5.3.6)

(b)

It follows that for any v, w ∈ supp Hu , n−1  j=1

(−1)

n− j

+ , + , n−1  n j−1 n− j−1 2 n− j n L j ev , L n− j ew = g(v)g(w) f  λuv λ¯ uw . (−1) j j j=1

(b)

Hence (5.3.4) holds if and only if for any v, w ∈ supp Hu (λuv − λ¯ uv )n + |g(v)|2  f 2

,

+ , n−1  n j−1 n− j−1 λuv λ¯ uv (−1)n− j = 0, j j=1

+ , n−1  j−1 n− j−1 n− j n λuv λ¯ uw g(v)g(w) (−1) = 0, v = w. j

(5.3.7)

j=1

Further, (5.3.5) holds if and only if for every v ∈ V \supp Hu(b) and w ∈ supp Hu(b) , (−1)n λn−1 uw g(w) f (v) + f (u)g(w) ev , eλu ,Au

+ , n−1  n j−1 n− j−1 λuu λ¯ uw (−1)n− j = 0. j j=1

(5.3.8) (b)

(b)

Assume that supp(g)∩{v ∈ supp Hu : λuv = 0} = ∅. In this case, ( f g)Dλu = 0. Hence, by (5.3.6), L k = 0 for k ≥ 2. Let v ∈ supp(g), h 1 = ev = k1 in (5.3.4). Then L 1 ev , L n−1 ev = 0, n ≥ 2,

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which is possible only if n ≥ 3. In this case, (5.3.4) holds if and only if {λuv : v ∈ supp Hu(b) } ⊆ R. Further, (5.3.5) holds if and only if (b)

(Nλu )n−1 ew , L 1 ev = 0, v ∈ supp Hu(b) , w ∈ V \supp Hu(b) , which is possible if and only if either f (u) = 0 or λuu = 0. This completes the verification of (i). (b) Assume next that supp(g) ∩ {v ∈ supp Hu : λuv ∈ R\{0}} = ∅. In this case, there exists η ∈ supp(g) such that λuη ∈ R\{0}. If possible, then assume that W f ,g is n-symmetric. By (5.3.7) with v = η, (λuη − λ¯ uη )n +



|g(η)|2  f 2 (λuη − λ¯ uη )n − (−1)n λnuη − λ¯ nuη = 0. |λuη |2 (b)

However, λuη ∈ R\{0}, and hence n is necessarily an odd integer. Further, since Nλu is also n-symmetric, by (5.3.1) and (5.3.2), λuu ∈ R and λuu = 0 or λuw = 0 for every w ∈ Au \{u}.

(5.3.9)

By (5.3.8) with w = η, for any v ∈ V \(supp Hu(b) ∪ Au ), f (v) = 0. Further, if λuu = 0, then by (5.3.9), eλu ,Au = λuu eu , which by assumption is orthogonal to f . This forces f (u) to be equal to 0. In that case, by (5.3.8) with w = η, f (v) = 0 for all v ∈ Au . This is not possible since f = 0. Thus λuu = 0 and f (u) = 0. Once again, by (5.3.8) with v = u and w = η, (−1)n λn−1 uw

+

n−1 

(−1)

n− j

j=1

+ , n j n− j−1 λuu λuw = 0. j

It follows that (λuu − λuη )n = λnuu . This is not possible, since n is an odd integer and λuu , λuη ∈ R\{0}. Thus we arrive at a contradiction to the assumption that W f ,g is n-symmetric.   Remark 5.8 Let g : supp Hu(b) → C be given. Then W f ,g is symmetric if and only if (b) {λuv : v ∈ supp Hu } is contained in R, g = 0 and λuv = 0 for v ∈ Au \{u}, where Au is as given in (3.2.15). This may be concluded from (5.3.4), (5.3.5) (with n = 1) and Proposition 5.5. We are now in a position to present a proof of Theorem 1.5 (recall the notations (b) Uu , gx , Ww,x as introduced in the Prologue).

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(b)

Proof of Theorem 1.5 Note that 2 (Uu ) is nothing but Hu , while Ww,x is a rank one extension of a weighted join operator in 2 (V ). Clearly, Ww,x is densely defined with {ev : v ∈ V } contained in D(Ww,x ). By (1.3.6) and Theorem 4.15(i), σ p (Ww,x ) = {dv − du : v ∈ Uu(b) ∪ {u}}. Let us find conditions on x ∈ R which ensure that Ww,x satisfies the compatibility condition I. To see that, note first that σ p (Ww,x ) is a closed subset of C and μ0 = / σ p (Ww,x ). By assumption, (Des(u), E u ) is a narrow tree of width m, and −du − 1 ∈ hence by (1.3.6) and (1.3.7), we have the estimate  (b) v∈supp Hu

∞   |gx (v)|2 d2x n 2x v = ≤ m . |λuv − μ0 |2 (dv + 1)2 (n + 1)2 (b)

(5.3.10)

n=0

v∈Uu

Since card(Des(u)) = ℵ0 , we must have  (b)

v∈supp Hu

∞  |gx (v)|2 n 2x ≥ . 2 |λuv − μ0 | (n + 1)2 n=0

It follows from the last two estimates that Ww,x satisfies the compatibility condition I if and only if x < 1/2. It now follows from Corollary 4.18 that σ (Ww,x ) is a proper closed subset of C if and only if x < 1/2. The conclusion in (i), (ii) and (iii) now follow from Theorem 4.11 and Corollary 4.18. Since the spectrum of Ww,x is a subset of {−du + k : k ∈ N}, parts (iv) and (v) may be deduced from Propositions 5.1, 5.2 and 5.4. Finally, part (vi) may be deduced from Corollary 4.20.   Remark 5.9 Note that the conclusion of Theorem 1.5 may not hold in case (Des(u), E u ) is not narrow. For instance, if T is the binary tree, then an examination of (5.3.10) shows that  (b)

v∈supp Hu

|g0 (v)|2 = ∞. |λuv − μ0 |2

In this case, σ (Ww,0 ) = C (see Corollary 4.18). Finally, note that by Proposition (b) 5.5, the weighted join operator Wλu with weight system given by (1.3.6) is strictly 3-symmetric, while by Proposition 5.7(ii), its rank one extension Ww,x , x ∈ R is never n-symmetric.

6 Weighted Join Operators on Rootless Directed Trees In this section, we extend the notion of join operation at a given base point to rootless directed trees and study the associated weighted join operators. This is achieved by introducing a partial order relation on a rootless directed tree.

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6.1 Semigroup Structures on Extended Rootless Directed Trees Let T = (V , E) be a rootless directed tree and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Fix u, v ∈ V∞ . Define u ≤ v if there exists a directed path [u, v] from u to v. Note that ≤ defines a partial order on V∞ . Further, ≤ is anti-symmetric, since T is a directed tree. Moreover, since v ≤ ∞ for every v ∈ V∞ , ∞ can be considered as a maximal element of T∞ , whereas T∞ , being rootless, has no minimal element. One may now define the join operation u  v on T∞ by setting ⎧ ⎪ ⎨u if v ≤ u, u  v = v if u ≤ v, ⎪ ⎩ ∞ otherwise. As in the case of rooted directed trees (see Lemma 2.4), (V∞ , ) is a commutative semigroup admitting ∞ as an absorbing element. Let us now define the meet operation. Let u ∈ V and v ∈ V∞ . Note that by [38, Proposition 2.1.4] and the definition of the extended directed tree, there exists w0 ∈ V such that {u, v} ⊆ Des(w0 ). Thus par n (u) = w0 = par m (v) for some m, n ∈ N. In particular, the set par(u, v) given by par(u, v) := {w ∈ V : par n (u) = w = par m (v) for some m, n ∈ N} is non-empty. Further, w ≤ u for every w ∈ par(u, v). Moreover, it is totally ordered, that is, for any w1 , w2 ∈ par(u, v), w1 ≤ w2 or w2 ≤ w1 . This follows since each vertex in V has a unique parent. One may now define u  v by

u  v = max [w0 , u] ∩ [w0 , v] ,

(6.1.1)

where w0 is any element in par(u, v). Note that u  v ∈ Asc(u) ∩ Asc(v).

(6.1.2)

Since par(u, v) is totally ordered, u  v is independent of the choice of w0 . Further, we set ∞  ∞ = ∞. Once again, (V∞ , ) is a commutative semigroup admitting identity element as ∞ (cf. Lemma 2.8). We leave the verification to the reader. Before we define the join operation at a given base point, we present a decomposition of the extended directed tree. Lemma 6.1 Let T = (V , E) be a rootless directed tree and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Then, for any u ∈ V , we have ∞ 

V∞ = · Des(par j (u))\Des(par j−1 (u)) , j=0

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where we used the convention that Des(par −1 (u)) = ∅. Proof Let v ∈ V . By (6.1.1), there exists a non-negative integer j such that u  v = par j (u). It now follows from (6.1.2) and the uniqueness of u  v that v ∈ Des(par j (u))\Des(par j−1 (u)). Consequently, we get the inclusion V ⊆

∞ 

Des(par j (u))\Des(par j−1 (u)).

j=0

Since ∞ ∈ Des(u), we get the desired equality.

 

Definition 6.2 (Join operation at a base point) Let T = (V , E) be a rootless directed tree and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Fix b ∈ V∞ and let u, v ∈ V∞ . Define the binary operation b on V∞ by ⎧ uv ⎪ ⎪ ⎪ ⎨u u b v = ⎪ v ⎪ ⎪ ⎩ uv

if u, v ∈ Asc(b), if v = b, if b = u, otherwise.

Note that (V∞ , b ) is a commutative semigroup admitting identity element as b. Further, ∞ = . The table for join operation u b v at the base point b for a rootless directed tree is identical with Table 1. Definition 6.3 Let T = (V , E) be a rootless directed tree and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . Fix u, b ∈ V∞ and the weight system λu = {λuv }v∈V∞ of complex numbers, we define the weighted join operator Wλ(b) u (based at b) on T by (b)

D(Wλu ) :=



f ∈ 2 (V ) : Λu(b) f ∈ 2 (V ) ,

(b)

(b)

Wλu f := Λu(b) f ,

f ∈ D(Wλu ),

where Λu(b) is the mapping defined on complex functions f on V by (Λu(b) f )(w) :=



λuv f (v), w ∈ V

(b) v∈Mu (w)

(b)

with Mu (w) given by Mu(b) (w) := {v ∈ V : u b v = w}.

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...

6

3

2

...

m

...

1

4

...

...

0

Fig. 8 The rootless directed tree T3 with prescribed vertex m

Remark 6.4 As in the rooted case, it can be seen that D(Wλ(b) ) forms a subspace of u (b)

(b)

2 (V ). Clearly, ev ∈ D(Wλu ) and (Wλu ev )(w) = λuv eub v (w), w ∈ V . Thus (b)

(b)

DV := span {ev : v ∈ V } ⊆ D(Wλu ), Wλu DV ⊆ DV . (b)

Thus all positive integral powers of Wλu are densely defined and the Hilbert space adjoint

∗ Wλ(b) u

of Wλ(b) is defined. u

To get an idea about the structure of weighted join operators on rootless directed trees, let us discuss one example. Example 6.5 (With one branching vertex) Let T3 denote the directed tree as shown in Fig. 8 (see [38, Eqn (6.2.10)]). Consider the ordered orthonormal basis

{e3n : n ∈ N} ∪ {e3n+2 : n ∈ N} ∪ {e3n+1 : n ∈ N} (m)

of 2 (V ). The matrix representation of the weighted join operator Wλ0 and weighted (∞)

meet operator Wλm on T3 are given by ⎛

(m)

Wλ0

··· 0 .. ⎜ ⎜ . ⎜ ⎜··· 0 ⎜ ⎜ . . . λm3 ⎜ =⎜ ⎜··· 0 ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

··· ··· λm0 λm2 · · · λmm 0 0 0 ··· .. .

0 ··· ··· 0 λmm+3 0 .. . 0 λmm+6 0 .. . 0 λmm+9 .. .. . .

326

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⊕ 0, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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⎛

(∞)

Wλm

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

..

⎞

. λm3

λm0 0 0 λm2 .. . 0 .. .

··· 0 ··· .. .0

···

0 λm1 λm4

···

λmm−3 0 0 λmm .. . 0 .. .

··· λmm+3 · · · ···

0 .. .

⎟ ⎟ ⎟ ···⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

These expressions should be compared with the matrix representations of weighted join operator and weighted meet operator on T2 discussed in Example 3.4.  6.2 A Decomposition Theorem and Spectral Analysis (b)

Note that Wλb = Dλb , the diagonal operator with diagonal entries λb . The structure (b)

of the weighted join operator Wλu , u = b turns out to be quite involved in case of rootless directed trees. We present below a counterpart of Theorem 3.13 for weighted join operators on rootless directed trees.

Theorem 6.6 Let T = (V , E) be a rootless directed tree and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For b ∈ V∞ , u ∈ V \{b} and the weight system λu = {λuv }v∈V∞ of complex numbers, let Dλu be the diagonal operator on T (b) and let Wλu be the weighted join operator on T . Then the following hold: (b)

(i) Assume that b ∈ V . Consider the subspace Hu  Hu(b)

=

of 2 (V ) given by

2 (Asc(u) ∪ Desb (u)) if b ∈ Des(u), otherwise. 2 (Desu (u))

(6.2.1)

admits the decomposition Then the weighted join operator Wλ(b) u

(b) (b) (b) Wλu = Dλu ⊕ Nλu on 2 (V ) = Hu(b) ⊕ 2 (V )  Hu(b) , (b)

(b)

where Dλu is a densely defined diagonal operator in Hu

(6.2.2)

(b)

and Nλu is a rank

one densely defined linear operator on 2 (V )  Hu(b) with invariant domain. (b) (b) Further, Dλu and Nλu are given by (b)

Dλu = Dλu |

(b) Hu

(b)

, D(Dλu ) = { f ∈ Hu(b) : Dλu f ∈ Hu(b) }, (6.2.3) (b)

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where the subset Au of V is given by  Au =

[u, b] if b ∈ Des(u), Asc(u) ∪ {b, u} otherwise.

(ii) Assume that b = ∞. Consider the orthogonal decomposition of 2 (V ) (as ensured by Lemma 6.1) given by  (V ) = 2

∞ "

2 (Des(u j )\Des(u j−1 )),

j=0

where Des(u −1 ) = ∅ and u j := par j (u), j ∈ N. Further, with respect to the (b) above decomposition, Wλu decomposes as (b) D(Wλu ) (b)

=

W λu =

∞ " j=0 ∞ "

D(eu j  eλu ,Des(u j )\Des(u j−1 ) ), eu j  eλu ,Des(u j )\Des(u j−1 ) .

(6.2.5)

j=0

Proof Let Vu be the complement of Des(u) · Asc(u) in V . We divide the proof into three cases. Case I. b ∈ / Des(u): Consider the following decomposition of V :   V = Desu (u) · Au · Vu \{b} . Thus 2 (V ) = 2 (Desu (u)) ⊕ 2 (Au ) ⊕ 2 (Vu \{b}). (b)

Note that 2 (Desu (u)), 2 (Au ) and 2 (Vu \{b}) are invariant subspaces of Wλu . We (b)

claim that the weighted join operator Wλu is given by D(Wλ(b) ) = D(Dλu |2 (Des u

u (u))

) ⊕ D(eu  eλu ,Au ) ⊕ 2 (Vu \{b}),

(b)

Wλu = Dλu |2 (Desu (u)) ⊕ eu  eλu ,Au ⊕ 0. To see the above decomposition, let f ∈ 2 (V ) be of the form f 1 ⊕ f 2 ⊕ f 3 with (b) f 1 ∈ 2 (Desu (u)), f 2 ∈ 2 (Au ), f 3 ∈ 2 (Vu \{b}). Note that f ∈ D(Wλu ) if and (b)

(b)

only if Wλu f ∈ 2 (V ), where Wλu f takes the form 328

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f 1 (v)λuv eub v ⊕

v∈Desu (u)

=



 v∈Au

f 1 (v)λuv ev ⊕



f 2 (v)λuv eub v ⊕

f 3 (v)λuv eub v

v∈Vu \{b}



f 2 (v)λuv eu ⊕ 0.

v∈Au

v∈Desu (u) (b)

It follows that f ∈ D(Wλu ) if and only if f 1 ∈ D(Dλu |2 (Des

u (u))

),

f 2 ∈ D(eu  eλu ,Au ),

f 3 ∈ 2 (Vu \{b}).

This yields the desired orthogonal decomposition of Wλ(b) . u Case II. b ∈ Des(u)\{∞}: Consider the following decomposition of V :   V = Asc(u) ∪ Desb (u) · Au · Vu . Thus 2 (V ) = 2 (Asc(u) ∪ Desb (u)) ⊕ 2 (Au ) ⊕ 2 (Vu ). (b)

Note that 2 (Asc(u) ∪ Desb (u)), 2 (Au ) and 2 (Vu ) are invariant subspaces of Wλu . (b)

As in the previous case, one can verify that the weighted join operator Wλu is given by (b)

D(Wλu ) = D(Dλu |2 (Asc(u)∪Des = Dλu |2 (Asc(u)∪Des Wλ(b) u

b (u))

b (u))

) ⊕ D(eu  eλu ,Au ) ⊕ 2 (Vu ),

⊕ eu  eλu ,Au ⊕ 0.

Case III. b = ∞: The decomposition (6.2.5) follows from (b)

Wλu ev = λuv euv = λuv eu j , v ∈ Des(u j )\Des(u j−1 ).  

This completes the proof. Here are some immediate consequences of Theorem 6.6.

Corollary 6.7 (Dichotomy) Let T = (V , E) be a rootless directed tree and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For b, u ∈ V and the (b) weight system λu = {λuv }v∈V∞ of complex numbers, the weighted join operator Wλu on T is at most rank one (possibly unbounded) perturbation of a diagonal operator, (∞) while the weighted meet operator Wλu on T is an infinite rank operator provided λu ⊆ C\{0}.

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The orthogonal decomposition (6.2.2) of Wλu , as ensured by Theorem 6.6, is given

, Nλ(b) , Hu(b) ), where Hu(b) , Dλ(b) and Nλ(b) are given by (6.2.1), by the triple (Dλ(b) u u u u (6.2.3) and (6.2.4) respectively. Corollary 6.8 Let T = (V , E) be a rootless directed tree and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For b ∈ V∞ , u ∈ V \{b} and the weight system λu = {λuv }v∈V∞ of complex numbers, let Dλu be the diagonal operator (b) on T and let Wλu be a weighted join operator on T . Then the following holds true: (b)

(i) If b ∈ / Des(u), then Wλu is bounded if and only if λu ∈ ∞ (Des(u)) and



|λuv |2 < ∞.

v∈Asc(u)

(ii) If b ∈ Des(u)\{∞}, then Wλ(b) is bounded if and only if λu ∈ ∞ (Asc(u) ∪ u Desb (u)). (b) (iii) If b = ∞, then Wλu is bounded if and only if sup



|λuv |2 < ∞,

j≥0 v∈Des(u )\Des(u j j−1 )

where Des(u −1 ) = ∅ and u j := par j (u) for j ∈ N.  (b) (iv) If b ∈ / Des(u) and v∈Asc(u) |λuv |2 = ∞, then Wλu is not closable. Proof The desired conclusions in (i)–(iii) follow from Theorem 6.2.2, while (iv) follows from (i) and Lemma 1.3.   Remark 6.9 Let us briefly discuss the spectral picture for a weighted join opera(b) tor Wλu on the rootless directed tree T . Consider the orthogonal decomposition (b)

(b)

(b)

(Dλu , Nλu , Hu

(b)

(b)

) of Wλu . In case b ∈ Des(u)\{∞}, the operator Nλu in the decom-

is bounded. Hence the spectral picture of Wλ(b) can be described as position of Wλ(b) u u in the rooted case (see Theorem 4.15). We leave the details to the reader. In case  is bounded and the same remark as b ∈ / Des(u) and v∈Asc(u) |λuv |2 < ∞, Nλ(b) u  above is applicable. Suppose now that b ∈ / Des(u) and v∈Asc(u) |λuv |2 = ∞. Then, is unbounded. Hence, by Lemma 1.3, by Corollary 6.8, Nλ(b) u (b)

(b)

(b)

σ (Wλu ) = C, σ p (Wλu ) = σ p (Dλu ) ∪ {0, λuu }. The verification of the following is similar to that of Corollary 3.16, and hence we skip its verification. Corollary 6.10 Let T = (V , E) be a rootless directed tree and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u ∈ V and b ∈ V \{u}, consider the (b) weight system λu = {λuv }v∈V∞ of complex numbers and let Wλu denote the weighted 330

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(b)

(b)

join operator on T . Consider the orthogonal decomposition (Dλu , Nλu , Hu Wλ(b) u

) of

as ensured by Theorem 6.6. Then the following statements hold: (b)

(b)

(i) If λuu = 0, then Wλu is a complex Jordan operator of index 2 provided Dλu belongs to B(Hu(b) ) or Nλ(b) belongs to B(2 (V )  Hu(b) ). u (b)

(b)

(ii) If λuu = 0 and Nλu ∈ B(2 (V )  Hu functional calculus.

(b)

), then Wλu admits a bounded Borel

As in the case of rooted directed trees (see Definition 4.1), one may introduce the (b) rank one extension W f ,g of the weighted join operator Wλu on a rootless directed

appearing in the decomposition of tree in a similar fashion. In case the operator Nλ(b) u (b)

(b)

Wλu is unbounded, it turns out (due to the fact that Dλu has no “good influence" on (b)

(b)

Nλu ) that W f ,g is not even closable. On the other hand, in case Nλu is bounded, one can obtain counterparts of Theorems 4.11, 4.15 and Propositions 5.1, 5.2 for rank one extensions of weighted join operators on rootless directed trees along similar lines. We leave the details to the reader.

7 Rank One Perturbations The considerations in Sect. 4 around the notion of rank one extensions of weighted join operators were mainly motivated by the graph-model developed in earlier sections. Some of these can be replicated in a general set-up simply by replacing the vertex set of the underlying rooted directed tree by a countably infinite directed set. The results in this section give a few glimpses of this general scenario. In particular, we discuss the role of some compatibility conditions (differing from compatibility conditions I and II as introduced in Sect. 4) in the sectoriality of rank one perturbations of diagonal operators. We also discuss the sectoriality of the form-sum of the form associated with a sectorial diagonal operator and a form associated with not necessarily squaresummable functions f and g. 7.1 Operator-Sum Throughout this section, J denotes a countably infinite directed set and let {e j : j ∈ J } be the standard orthonormal basis of 2 (J ). Let Dλ stand for the diagonal operator in 2 (J ) with diagonal entries λ = {λ j : j ∈ J } given by Dλ e j = λ j e j ,

j ∈ J.

The following main result of this section shows that a compatibility condition ensures the sectoriality of the operator-sum of a sectorial diagonal operator and an unbounded rank one operator.

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Theorem 7.1 Let Dλ be a sectorial operator in 2 (J ) and let f ∈ 2 (J ). Let g : J → C be such that for some z 0 ∈ ρ(Dλ ),  |g( j)|2 < ∞. |λ j − z 0 |2

(7.1.1)

j∈J

Then Dλ + f  g defines a sectorial operator in 2 (J ) with the domain D(Dλ ). Clearly, Theorem 7.1 generalizes Proposition 5.1. In its proof, we need a couple of observations of independent interests. The first of which characterizes the B-boundedness of A in terms of the strict contractivity of B(A − z)−1 for some z ∈ ρ(A), where A is a normal operator satisfying certain growth condition. Proposition 7.2 Let A be a normal operator in H and let B be a linear operator in H with D(A) ⊆ D(B). If there exists z ∈ ρ(A) such that B(A − z)−1  < 1, then Bx ≤ aAx + bx, x ∈ D(A),

(7.1.2)

where a = B(A − z)−1  and b ∈ (0, ∞). Conversely, if there exist a ∈ (0, 1) and b ∈ (0, ∞) such that (7.1.2) holds and if for some θ ∈ R, max{|μ|, n} ≤ |μ − eiθ n|, n ∈ N, μ ∈ σ (A),

(7.1.3)

then B(A − z)−1  < 1 for some z ∈ ρ(A). Remark 7.3 There are two particular instances in which (7.1.3) can be ensured. (i) If A is self-adjoint, then by [64, Corollary 3.14], σ (A) ⊆ R, and hence (7.1.3) holds with θ = ±π/2. (ii) If A is sectorial with vertex at 0, then (7.1.3) holds with θ = π. Proof Note that if z ∈ ρ(A) is such that B(A − z)−1  < 1, then for any x ∈ D(A) Bx = B(A − z)−1 (A − z)x ≤ aAx + bx, where a = B(A − z)−1  and b = |z|B(A − z)−1 . This yields (7.1.2). To see the converse, suppose that (7.1.2) and (7.1.3) hold. For any z ∈ ρ(A), (A−z)−1 is a bounded operator on H with range equal to D(A). Thus (7.1.2) becomes B(A − z)−1 y ≤ aA(A − z)−1 y + b(A − z)−1 y, z ∈ ρ(A), y ∈ H. (7.1.4) Let E(·) denote the spectral measure of A and let n ∈ N. Clearly, by (7.1.3), eiθ n ∈ ρ(A). It follows from the spectral theorem [61, Theorem 13.24] and (7.1.4), that for any y ∈ H,

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B(A − e n) iθ

−1

./ y ≤ a

σ (A)

(7.1.3)

≤

! ! !

!2 μ ! ! E(dμ)(y)2 + b(A − eiθ n)−1 y μ − eiθ n b y. n

ay +

Thus, for sufficiently large integer n, B(A − eiθ n)−1  ≤ a +

b < 1. n  

This completes the proof. We need one more fact in the proof of Theorem 7.1 (cf. Theorem 4.15(iii)).

Proposition 7.4 Let Dλ be a sectorial operator in 2 (J ) and let f ∈ 2 (J ). Let g : J → C be such that for some z 0 ∈ ρ(Dλ ), (7.1.1) holds. Then, for any z ∈ ρ(Dλ ), G z := f  g(Dλ − z)−1 is a Hilbert-Schmidt integral operator with square-summable kernel K z ( j, k) :=

f ( j)g(k) , λk − z

j, k ∈ J .

Moreover, there exists a sequence {z n }n∈N ⊆ ρ(Dλ ) such that lim G z n 2 = 0,

n→∞

where  · 2 denotes the Hilbert-Schmidt norm. Proof Let z ∈ ρ(Dλ ). Then, as in the proof of Theorem 4.15, it can be seen that G z is a Hilbert-Schmidt integral operator with kernel K z ∈ 2 (J × J ). Moreover, G z 22 =  f 2

 |g( j)|2 . |λ j − z|2

(7.1.5)

j∈J

On the other hand, it is easily seen that there exists a sequence {z n }n∈N ⊆ ρ(Dλ ) with the only accumulation point at ∞ such that |λ j − z 0 | ≤ |λ j − z n |, n ∈ N, j ∈ J . Using Lebesgue dominated convergence theorem, we see that  |g( j)|2 → 0 as n → ∞. |λ j − z n |2 j∈J

Hence, by (7.1.5), we obtain the remaining part.

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Proof of Theorem 7.1 As in the proof of Proposition 4.7, it is easily seen that D(Dλ ) ⊆ D( f  g).

(7.1.6)

Also, by Proposition 7.4, for any a ∈ (0, 1), there exists z ∈ ρ(Dλ ) such that the Hilbert-Schmidt norm of f  g(Dλ − z)−1 is less than a. Since the operator norm of any Hilbert-Schmidt operator is less than or equal to its Hilbert-Schmidt norm, it follows from Proposition 7.2 that f  g is Dλ -bounded with Dλ -bound equal to 0. The desired conclusion now follows from [52, Theorem 4.5.7].   The following provides a variant of Corollary 4.12. Since the bounded component Nλ(b) in the rank one extension W f ,g has no effect in the Dλ(b) -boundedness of f  g, u u this variant may be obtained by imitating the proof of Theorem 7.1. Corollary 7.5 Let T = (V , E) be a rooted directed tree with root root and let T∞ = (V∞ , E ∞ ) be the extended directed tree associated with T . For u, b ∈ V , consider the weight system λu = {λuv }v∈V∞ of complex numbers and let W f ,g be the rank one (b) (b) extension of the weighted join operator Wλu on T , where f ∈ 2 (V )  Hu is non(b)

zero and g : supp Hu → C is given. Suppose that W f ,g satisfies the compatibility condition I. Then W f ,g decomposes as A + B + C, where A, B, C are densely defined operators given by (4.2.8) such that B + C is A-bounded with A-bound equal to 0. We conclude this section with a brief discussion on some spectral properties of rank one perturbations of the diagonal operator Dλ . Assume that there exists z 0 ∈ ρ(Dλ ) such that g : J → C satisfies (7.1.1). By (7.1.6), Dλ + f  g is a densely defined operator in 2 (J ) with the domain D(Dλ ). Let μ ∈ C\λ be an eigenvalue of Dλ + f g. Thus there exists a non-zero vector h in 2 (J ) such that for every j ∈ J ,   h(k)g(k) f ( j) = μh( j) λ j h( j) + k∈J

⇒ (μ − λ j )h( j) =



 h(k)g(k) f ( j)

k∈J

⇒ h( j) = a where a =



f ( j) , μ − λj

h(k)g(k) is non-zero. Therefore, we have

k∈J

 f ( j)g( j) = 1. μ − λj

(7.1.7)

j∈J

Notice that expression in (7.1.7) is an analytic function in μ outside the spectrum of Dλ . Also, for μ ∈ C\λ to be an eigenvalue for Dλ + f  g, it has to satisfy (7.1.7). Therefore, the set of all eigenvalues of Dλ + f  g outside the set σ (Dλ ) has to be discrete (cf. [37, Corollary 2.5]). One may argue now as in the proof of Theorem 4.15(iv) using Weyl’s theorem to conclude that σe (Dλ + f  g) = σe (Dλ ).

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7.2 Form-Sum Consider a sectorial diagonal operator Dλ in 2 (J ). By [46, Theorem 3.35, Chapter V] Dλ (and hence Dλ∗ as well) has a unique square root; let us denote it by Rλ and note that Rλ is a sectorial operator with D(Rλ ) = D(Rλ∗ ). Consider the form Q R given by Q R (h, k) := Rλ h, Rλ∗ k , h, k ∈ D(Rλ ).

(7.2.1)

Then having the unbounded form-perturbation of Q R by Q f ,g is to find conditions on f , g, so that the form Q f ,g (h, k) :=



h( j)g( j)

j∈J



f ( j)k( j)

(7.2.2)

j∈J

is well defined for all h, k ∈ D(Rλ ) and to ensure that the perturbation by Q f ,g is small, (so that an application of [46, Theorems 1.33 and 2.1, Chapter VI] can be made through a choice of large enough z in the appropriate sector). In such a case, the formsum Q R + Q f ,g is closed and defines a sectorial operator with domain contained in the domain of Q R (the reader is referred to [46, Chapter VI] for all definitions pertaining to sesquilinear forms in Hilbert spaces). This is made precise in the following theorem. Theorem 7.6 Let Dλ be a sectorial diagonal operator in 2 (J ) with angle θ ∈ (0, π/2) and vertex 0. Let f : J → C and g : J → C be such that for some z 0 ∈ (−∞, 0),  j∈J

 | f ( j)|2 |g( j)|2 0 0 < ∞, < ∞, | λ j − z 0 |2 | λ j − z 0 |2 j∈J

(7.2.3)

where the square root is obtained by the branch cut at the non-positive real axis. Let Q R and Q f ,g be as given by (7.2.1) and (7.2.2). Then Q f ,g (h, k) is defined for all h, k ∈ D(Rλ ). Moreover, the form Q R + Q f ,g is sectorial and there exists a unique sectorial operator T in 2 (J ) with domain contained in the domain of Q R such that Q R (h, k) + Q f ,g (h, k) = T h, k , h ∈ D(T ), k ∈ D(Q R ). Proof Note that z 0 ∈ ρ(Rλ ), and hence by (7.2.3) and the definition of D(Rλ ), Q f ,g (h, k) is well-defined for all h, k ∈ D(Rλ ). To see the remaining part, we make some general observations. Notice first that Q R (h, h) =



λ j |h( j)|2 , Q R (h, h) =

j∈J



λ j |h( j)|2 .

(7.2.4)

j∈J

Further, since Rλ is normal, Rλ h2 ≥ |Q R (h, h)|2 , h ∈ D(Rλ ).

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Furthermore, since Dλ is a sectorial operator with angle θ , |λ j | ≤ tan θ λ j , j ∈ J , |Q R (h, h)| ≤ tan θ Q R (h, h), h ∈ D(Rλ ).

 (7.2.6)

We claim that Q R is a closed form. It suffices to check that Q R is closed (see [46, Pg 336]). Let h ∈ 2 (J ), {h n }n∈N ⊆ D(Q R ) be such that h n → h as n → ∞ and Q R (h n − h m , h n − h m ) → 0 as n and m tend to ∞. It follows that Q R (h n − h m , h n − h m )

(7.2.4)

=



λ j |h n ( j) − h m ( j)|2

j∈J (7.2.6)

≥

√

=

√

=

√



1 1 + tan2 θ 1 1 + tan2 θ 1 1 + tan2 θ

|λ j ||h n ( j) − h m ( j)|2

j∈J

 0 0 | λ j h n ( j) − λ j h m ( j)|2 j∈J

Rλ (h n − h m )2 .

This shows that {Rλ (h n )}n∈N is a Cauchy sequence in 2 (J ). Thus there exists g ∈ 2 (J ) such that Rλ (h n ) → g as n → ∞. Since Rλ is closed, h ∈ D(Rλ ) = D(Q R ) and g = Rλ h. By (7.2.5), Q R (h n − h, h n − h) → 0 as n → ∞. This completes the verification of the claim. We next show that Q f ,g is Q R -bounded with Q R -bound less than 1. To see this, let z ≤ z 0 be a negative real number. Note that |w − z 0 | ≤ |w − z| for any w ∈ C such that | arg w| < θ. It follows from (7.2.3) that ⎫ ⎪ ≤ < ∞, ⎪ ⎪ ⎪ 1/2 1/2 ⎬ 2 2 j∈J |λ j − z| j∈J |λ j − z 0 |  |g( j)|2  |g( j)|2 ⎪ ⎪ ≤ < ∞. ⎪ ⎪ 1/2 1/2 ⎭ 2 2 j∈J |λ j − z| j∈J |λ j − z 0 | 

 | f ( j)|2

| f ( j)|2

(7.2.7)

For any k ∈ D(Rλ ), note that !2 ! ! ! f ( j)k( j)! !

≤

j∈J

≤ (7.2.6)

≤

 | f ( j)|2



1/2 j∈J |λ j

j∈J

2

2

−

z|2

1/2

|λ j

 | f ( j)|2  1/2 j∈J |λ j

− z|2

 | f ( j)|2  1/2

j∈J

|λ j

− z|2

− z|2 |k( j)|2

|λ j ||k( j)|2 +

j∈J

√



|z|2 |k( j)|2



j∈J

1 1 + tan2 θ

 |Q R (k, k)| + |z|2 k2 . (7.2.8)

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Similarly, we can conclude that for any h ∈ D(Rλ ), ! !2  ! ! h( j)g( j)! ≤ 2 ! j∈J

j∈J

|g( j)|2 1/2

|λ j

+

− z|2

√

,

1

|Q R (h, h)| + |z| h 2

1 + tan2 θ

2

. (7.2.9)

Combining (7.2.8) and (7.2.9) together, for any h ∈ D(Rλ ), we obtain 1 ,1 2 2 |Q R (h, h)| | f ( j)|2 2 |g( j)|2 2 2 2 3 |Q f ,g (h, h)| ≤ 2 √ + |z| h 3 . 1/2 1/2 2 2 1 + tan2 θ j∈J |λ j − z| j∈J |λ j − z| +

(7.2.10) Also, note that for any j ∈ J , lim

z→−∞ z≤z 0

f ( j) 1/2 λj

−z

= 0,

lim

z→−∞ z≤z 0

g( j) 1/2 λj

−z

= 0.

We now conclude from (7.2.7) and the Lebesgue dominated convergence theorem that lim

 | f ( j)|2

1/2 z→−∞ z≤z 0 j∈J |λ j

− z|2

= 0,

lim



z→−∞ z≤z 0 j∈J

|g( j)|2 1/2

|λ j

− z|2

= 0.

Let h ∈ D(Rλ ). Then, for any a ∈ (0, 1), there exists z ≤ z 0 such that  | f ( j)|2 1/2

j∈J

|λ j

− z|2


0, |Q f ,g (h, h)| ≤ a|Q R (h, h)| + bh2 .

(7.2.11)

Since Q R is a sectorial form, (7.2.11) together with [46, Theorem 1.33, Chapter VI] implies that Q R + Q f ,g is a sectorial form. The remaining assertion about the existence of T follows from [46, Theorem 2.1, Chapter VI].   We next present a variant of Theorem 7.6, where we discuss the sectoriality of form-sum with Q R replaced by the form Q defined as Q(h, k) := |Dλ |1/2 h, |Dλ |1/2 k , h, k ∈ D(|Dλ |1/2 ),

(7.2.12)

where |A| denotes the modulus of a densely defined closed operator A. Needless to say, we alter compatibility conditions as per the requirement.

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Theorem 7.7 Let Dλ be a sectorial diagonal operator in 2 (J ) with angle θ and vertex 0. Let f : J → C and g : J → C be such that for some β0 > 0,  j∈J

 | f ( j)|2 |g( j)|2 < ∞, < ∞. (|λ j |1/2 + β0 )2 (|λ j |1/2 + β0 )2

(7.2.13)

j∈J

Let Q and Q f ,g be as given by (7.2.12) and (7.2.2). Then Q f ,g (h, k) is defined for all h, k ∈ D(|Dλ |1/2 ). Moreover, the form Q + Q f ,g is sectorial and there exists a unique sectorial operator T in 2 (J ) with domain contained in the domain of Q such that Q(h, k) + Q f ,g (h, k) = T h, k , h ∈ D(T ), k ∈ D(Q). Proof For h, k ∈ D(|Dλ |1/2 ), note that Q f ,g (h, k) =



˜ p)k(q), ˜ K β ( p, q)h(

(7.2.14)

p,q∈J

where, for p, q ∈ J and β ∈ [β0 , ∞), g(q) f ( p) , 1/2 |λ p | + β |λq |1/2 + β ˜ p) := (|λ p |1/2 + β)h( p), k( ˜ p) := (|λ p |1/2 + β)k( p). h( K β ( p, q) :=

Let G β denote the integral operator with kernel K β and notice that G β 22 =

 p∈J

 | f ( p)|2 |g(q)|2 . (|λ p |1/2 + β)2 (|λq |1/2 + β)2 q∈J

From (7.2.13), it is clear that G β is a Hilbert-Schmidt operator for any β ≥ β0 . One may argue as in Proposition 7.4, using Lebesgue dominated convergence theorem, to conclude that G β 2 → 0 as β → ∞. Now one may use (7.2.14) to see that for any h ∈ D(|Dλ |1/2 ), |Q f ,g (h, h)| ≤



|h(q)||g(q)|

q∈J

≤ G β 2





| f ( p)||k( p)|

p∈J

(|λ p |1/2 + β)2 |h( p)|2

p∈J

≤ 2G β 2 (



|λ p ||h( p)|2 + β 2 h2 )

p∈J

= 2G β 2 (Q(h, h) + β 2 h2 ).

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Since limβ→∞ G β 2 = 0, with an arbitrarily small a > 0, we obtain for some b > 0, |Q f ,g (h, h)| ≤ a Q(h, h) + bh2 , h ∈ D(|Dλ |1/2 ). Since Q defines a sectorial form, one may now argue as in the proof of Theorem 7.6 to complete the proof.   In general, the form Q f ,g is far from being closable. In fact, since Q f ,g is associated with the rank one operator f  g, it may be derived from Kato’s first representation theorem [46, Theorem 2.1, Chapter VI] and Lemma 1.3 that Q f ,g is closable if and only if g ∈ 2 (J ). Under suitable compatibility conditions, Theorems 7.6 and 7.7 above ensure that the form-sums Q R + Q f ,g and Q + Q f ,g are indeed closed. Finally, we note that as in the case of operator-sum, under the compatibility condition (7.1.1), it can be seen that the eigenvalues of the sectorial operator T associated with Q + Q f ,g outside σ (Dλ ) is discrete and the essential spectrum of T coincides with that of Dλ .

Epilogue The present paper capitalizes on the order structure of directed trees to introduce and study the classes of weighted join operators and their rank one extensions. In particular, we discuss the issue of the closedness, unravel the structure of the Hilbert space adjoint and identify various spectral parts of members of these classes. Certain discrete Hilbert transforms arise naturally in the spectral theory of rank one extensions of weighted join operators. The assumption that the underlying directed trees are rooted or rootless brings several prominent differences in the structures of these classes. Further, these classes overlap with the well-studied classes of complex Jordan operators, nsymmetric operators and sectorial operators. This work also takes a brief look into the general theory of rank one perturbations. As a natural outgrowth of this work, the study of finite rank extensions of weighted join operators would be desirable. In this regard, we would like to draw attention to the very recent work [50] on finite rank (self-adjoint) perturbations of self-adjoint operators. In the remaining part of this section, we discuss some problems pertaining to the theory of weighted join operators and their rank one extensions, which arise naturally be a weighted join in our efforts to understand these operators. In what follows, let Wλ(b) u operator on a rooted directed tree T = (V , E) and let W f ,g be its rank one extension. Here u ∈ V , b ∈ V , f ∈ 2 (V )  Hu(b) is non-zero and g : supp Hu(b) → C is given. Numerical range and Friedrichs extension (b) (b) (b) (b) Let (Dλu , Nλu , Hu ) denote the orthogonal decomposition of Wλu . Recall that the numerical range (T ) of a densely defined linear operator T is given by (T ) := { T f , f : f ∈ D(T ),  f  = 1}. Since the numerical range of a diagonal operator is contained in the closed convex (b) (b) hull of its diagonal entries, we obtain (Dλu ) ⊆ conv{λuv : v ∈ supp Hu }. where Reprinted from the journal

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conv(A) denotes the closed convex hull of A. Further, for any f ∈ 2 (V )  Hu unit norm, we have  eu ⊗ eλu ,Au f , f = f , eλu ,Au f (u) = f (u) λuv f (v),

of

v∈Au (b)

(b)

where Au is as given in (3.2.15). Thus the numerical range (Nλu ) of Nλu satisfies (b)

(Nλu ) = ⊆

⎧ ⎨ ⎩ ⎧ ⎨ ⎩



f (u)

λuv f (v) :  f 2 (V )

v∈Au

z ∈ C : |z|2 ≤

 v∈Au

|λuv |2

⎫ ⎬ ⎭

⎫ ⎬ =1 ⎭

.

The numerical range of the rank one extension W f ,g of Wλ(b) is given by u 

 (b) (b) Dλ h, h + ( f  g)h, k + Nλ k, k : (h, k) ∈ D(W f ,g ), h2 + k2 = 1 . u

u

Recall that if the numerical range is a proper subset of the complex plane, then the underlying operator is closable (see [46, Theorem 3.4, Chapter V]). It would be interesting otherwise also to find conditions on g (different from the compatibility conditions) so that the numerical range of W f ,g is a proper subset of the complex plane or is contained in a sector. Let us now discuss the so-called Friedrichs extensions of weighted join operators and their rank one extensions [52]. Suppose, for some r ∈ R and M ∈ (0, ∞), we have (b)

(b)

(b)

| Wλu h, h | ≤ M  (Wλu − r )h, h , h ∈ D(Wλu ).

(7.2.15)

By [52, Theorem 2.12.1], there exist a subspace  of 2 (V ), an inner product ·, ·  on  with the corresponding norm  ·  , and a sectorial sesquilinear form F on  such that the following assertions hold: ) is a dense subspace of  (in  ·  ). (a) D(Wλ(b) u (b) (b)

(b) (b) h, k  = (1/2) Wλu h, k + h, Wλu k + (1 − r ) h, k , h, k ∈ D(Wλu ). (b)

(b)

(c) F(h, k) = Wλu h, k for all h, k ∈ D(Wλu ). It turns out that the linear operator A associated with the sectorial sesquilinear form (b) (b) F, referred to as the Friedrichs extension of Wλu , turns out to be Wλu itself. Since (b)

Wλu is a closed linear operator (Proposition 3.5), this fact may be deduced from [52, Lemma 1.6.14] and Theorem 4.15 (see also [46, Theorem 2.9, Chapter VI]). It would be desirable to find conditions on g (similar to compatibility conditions) (b) so that (7.2.15) is ensured for the rank one extension W f ,g of Wλu . In this case, W f ,g 340

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admits a Friedrichs extension. In case W f ,g is closed, then it can be seen once again that this extension coincides with W f ,g itself. The Friedrichs extension would be of interest, particularly, in case either W f ,g is not closed or σ (W f ,g ) is the entire complex plane. One may be keen to know whether or not there exists a rank one extension W f ,g , which is closable but not closed. Hyponormality and n-Symmetricity It has been seen in Proposition 5.3 that the notions of the hyponormality and the normality coincide in the context of rank one extensions of weighted join operators. We do not know whether or not there exists any non-normal hyponormal rank one extension W f ,g of a weighted join operator. The essential difficulty in this problem is unavailability of an explicit expression for the Hilbert space adjoint of W f ,g . As evident from Propositions 5.5 and 5.7, n-symmetric rank one extensions of weighted join operators exist in abundance. The problem of classifying all n-symmetric rank one extensions of weighted join operators remains unsolved. (b)

C ∗ -algebras Let T be a leafless rooted directed tree and let Wλu be a weighted (b)

join operator on T . Assume that Wλu is bounded and Vu = ∅ (see (2.2.1)). Recall that the essential spectrum of an orthogonal direct sum of two bounded operators A, B ∈ B(H) is a union of essential spectra of A and B. Also, since the essential spectrum is invariant under compact perturbations [23], Theorem 3.13 together with Proposition 2.12(iii) implies that ⎧ ⎪ ⎨σe (Dλu |2 (Desb [u]) ) ∪ {0} if b ∈ Des(u)\{∞}, ) = σe (Wλ(b) {0} if b = ∞, u ⎪ ⎩ σe (Dλu |2 (Des(u)) ) ∪ {0} otherwise. On the other hand, the essential spectrum of a normal operator is the complement of isolated eigenvalues of finite multiplicity in its spectrum [23]. Thus the weight system λu completely determines the essential spectra of bounded weighted join operators. (b) (b) (b) Let C ∗ (Wλu ) denote the C ∗ -algebra generated by Wλu . Since Wλu is essentially )/K can be identified normal (see Theorem 3.13), the quotient C ∗ -algebra C ∗ (Wλ(b) u (b)

with C(σe (Wλu )), where K is the C ∗ -algebra of compact operators and C(X ) denotes the C ∗ -algebra of continuous functions on a compact Hausdorff space X endowed with sup norm. We conclude this paper with another possible line of investigation. For b ∈ V , con(b) (b) sider the family Fb := {Wλu }u∈V of bounded linear weighted join operators Wλu on (b)

(b)

(b)

(b)

a directed tree T = (V , E). A routine verification shows that Wλu Wλv = Wλv Wλu if and only if λvw λuvw = λuw λvuw , w ∈ V .

The latter condition holds, in particular, for the constant weight systems λu , u ∈ V with value 1. Assume that the family Fb is commuting. By Theorems 3.13 and 6.6, the family Fb is essentially normal. Motivated by [47, Theorem 2.11], one may ask Reprinted from the journal

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whether the C ∗ -algebra C ∗ (Fb ) generated by Fb is completely determined by the directed tree T and weight systems λu , u ∈ V ? In case the answer is no, what are the complete invariants which determine C ∗ (Fb )? Acknowledgements The authors would like to thank Shailesh Trivedi and Soumitra Ghara for some stimulating conversations related to the subject of this paper. We also convey our sincere thanks to Sumit Mohanty for drawing our attention to the Refs. [30, 59], where respectively the notions of the graph with boundary and the spiral-like ordering, relevant to the present investigations, were introduced. Data Availability Statement No new data were created or analyzed in this study.

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Complex Analysis and Operator Theory (2023) 17:81 https://doi.org/10.1007/s11785-023-01383-3

Complex Analysis and Operator Theory

A Note on Joint Spectrum in Function Spaces Puyu Cui1 · Rongwei Yang2 Received: 5 May 2022 / Accepted: 4 June 2023 / Published online: 16 July 2023 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023

Abstract Given several bounded linear operators A1 , ..., An on a Hilbert space, their projective spectrum is the set of complex vectors z = (z 1 , ..., z n ) such that the multiparameter pencil A(z) = z 1 A1 + · · · + z n An is not invertible. This paper studies the projective spectrum of the shift operator T , its adjoint T ∗ and a projection operator P. Two spaces of concern are the classical Bergman space L a2 (D) and the L 2 space over the torus T2 . The projective spectra are completely determined in both cases. The results lead to new questions about Toeplitz operators. Keywords Toeplitz operator · Shift operator · Projective spectrum Mathematics Subject Classification Primary 47B35 · 47B20 · 32A36

In Memory of Jörg Eschmeier. Communicated by Mihai Putinar. The first-named author is supported by China Scholarship Council (201908210167). This research is supported by NNSF of China (11501277). This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz and Mihai Putinar.

B

Rongwei Yang [email protected] Puyu Cui [email protected]

1

Institute of Mathematics, Liaoning Normal University, Dalian 116029, People’s Republic of China

2

Deparetment of Mathematics and Statistics, University at Albany, The State University of New York, Albany, NY 12222, USA

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1 Introduction Normal operator has been fully described by its spectral decomposition. Operator theory in the past a few decades thus focused primarily on non-normal operators. One fundamental class of such operator is the shift operators on the classical Hardy or Bergman space over the unit disc D, whose study has served as a cornerstone for the development of analytic function theory [8], C ∗ -algebras [6], operator theory on analytic function spaces [22], functional model theory for contractions [12, 13, 19], as well as Hilbert modules of analytic functions [4, 5]. Indeed, this development has opened an extensive interplay among algebraic geometry, complex analysis, complex geometry, operator theory, operator algebras and topology, and the literature on these explorations has become quite extensive. Today, shift operators remain a rich source of examples and counter-examples in all related areas. On the other hand, in the front of multivariable operator theory, the notion of projective spectrum in Banach algebras was introduced in 2009 [20], and it has provided a new mechanism for studying several non-commuting operators, see for instance [3, 9–11]. It is thus an appealing question whether projective spectrum can be computed for pencils of shift operators in various settings. This paper is an effort to address this question. We believe this work will not only shed new light on the interplay between operator theory and analytic function theory but may also add a new direction to the theory of projective spectrum. The first half of the paper concerns with the classical Bergman space L a2 (D) where we study the projective spectrum related to the Bergman shift Tw : g(w) → wg(w) and the projection Q 0 : g → g(0). The second half of the paper concerns with the space L 2 (T2 ) on the torus where we investigate the projective spectrum related to the double shift T : f (θ1 , θ2 ) → eiθ1 eiθ2 f (θ1 , θ2 ), and the projection P defined by 

2π

P f (θ1 ) =

f (θ1 , θ2 )

0

dθ2 . 2π

A notable difference here is that Q 0 is of rank 1 but P has infinite rank. Thus the methods are quite different. We start by recalling the definition of projective spectrum. Let B be a complex unital Banach algebra and A = (A1 , . . . , An ) be a tuple of linearly independent elements in B. The multiparameter pencil A(z) := z 1 A1 + · · · + z n An is an important subject of study in numerous fields. Definition 1.1 For a tuple (A1 , . . . , An ) of elements in a unital Banach algebra B, its projective spectrum is defined as P(A) = {z ∈ Cn | A(z) is not invertible}. The projective resolvent set refers to the complement, P c (A) = Cn \P(A).

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2 On the Bergman Space L2a (D) Let L 2 (D) be the square-integrable functions on D with respected to the Lebesgue measure d A. The classical Bergman space L a2 (D) is the closed subspace in L 2 (D) consisting of analytic functions on D. The space L ∞ (D) is the collection of all essentially bounded measurable functions. Given a function f ∈ L ∞ (D), the Toeplitz operator T f with symbol f is defined as T f g = PB ( f g), g ∈ L a2 (D), where PB : L 2 (D) → L a2 (D) is the Bergman projection. It is well known that {ek = √ 2 k + 1w k }∞ The Toeplitz k=0 is an orthonormal basis of L a (D). In particular, e0 = 1. k+1 operator Tw is often called the Bergman shift. One checks that Tw ek = k+2 ek+1 . Bergman shift and Toeplitz operators in general have been well studied, and the literature has become quite rich [1, 14, 16, 17]. For more information on their spectral theory, we refer the readers to [2, 15, 22] and the references therein. Regarding the essential spectrum of Toeplitz operator on L a2 (D) with continuous symbol, the following fact can be found in [15]. Theorem 2.1 Let ϕ ∈ C(D). Then the essential spectrum σe (Tϕ ) = ϕ(T), and when Tϕ is Fredholm, its index ind(Tϕ ) = − wind(ϕ(T), 0), where wind stands for the winding number. However, unlike the situation in the classical Hardy space, the spectrum σ (Tϕ ) in the Bergman space setting is far from being well understood. In particular, for a general ϕ ∈ C(D), it is not true that ϕ(D) ⊂ σ (Tϕ ) or σ (Tϕ ) ⊂ ϕ(D). Some examples are given in [2, 7, 21]. But for simple functions, the following description still holds [21]. Theorem 2.2 Let ϕ(z) = w + bw + c, where b, c ∈ C. Then σ (Tϕ ) = ϕ(D). In terms of operator pencil, we can write Tϕ = cI + bTw + Tw∗ . Hence the theorem above completely describes the projective spectrum of the pencil. The summand Tw∗ makes the pencil interesting because Tw and Tw∗ do not commute. Let H be a separable complex Hilbert space and B(H ) be the Banach algebra of all bounded linear operator on H . For two nonzero vectors e1 , e2 ∈ H , we let e1 ⊗ e2 denote the rank-1 operator such that e1 ⊗ e2 (x) = x, e2 e1 . Clearly, if e = 1 then e ⊗ e is the orthogonal projection from H onto the 1-dimensional subspace spanned by e. For simplicity, we shall denote it by Q e in the sequel. On L a2 (D), Q 0 := e0 ⊗ e0 is the orthogonal projection onto the space of constants, namely, Q 0 (g) = g(0), and it does not seem to have any intrinsic connection with Tw . It is thus a tantalizing question whether the projective spectrum of the pencil B(z) = z 0 + z 1 Q 0 + z 2 Tw + z 3 Tw∗ , z = (z 0 , ..., z 3 ) ∈ C4 , can be determined. Setting  χ0 (w) =

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1

w=0

0

w = 0,

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we can write Q 0 = Tχ0 and consequently B(z) = Tϕ , where ϕ(w) = z 0 + z 1 χ0 (w) + z 2 w + z 3 w. Since Q 0 is of rank 1, the following general lemma is instrumental. Lemma 2.3 Let G be an invertible operator in B(H ), and assume e ∈ H with e = 1. Then the operator G + z 1 Q e is not invertible if and only if 1 + z 1 G −1 e, e = 0. Proof It is clear that z 1 = 0 in this theorem. Observe that G + z 1 Q e is Fredholm with index 0. If it is not invertible then there exists a nonzero h ∈ H such that (G + z 1 Q e )h = Gh + z 1 h, ee = 0. It follows that h, e = 0 and h + z 1 h, eG −1 e = 0. Taking the inner product of the left-hand side with e, we have

h, e(1 + z 1 G −1 e, e) = 0 which implies 1 + z 1 G −1 e, e = 0. For the other direction, if 1 + z 1 G −1 e, e = 0, then we set h = −z 1 G −1 e and check readily that (G + z 1 Q e )h = −z 1 (1 + z 1 G −1 e, e)e = 0.   Hence, for the pencil B(z), if we set G(z) = z 0 + z 2 Tw + z 3 Tw∗ , then the above lemma requires the knowledge of G −1 (z)e0 , e0  in order to determine the projective spectrum of B(z) = G(z) + z 1 Q 0 . To address this issue, we first recall the following well-known lemma which gives an explicit description of Tw∗ , see [18]. Lemma 2.4 Given any g ∈ L a2 (D), we have Tw∗ g(w) =

1 w2



w

tg  (t)dt.

0

Lemma 2.5 Consider ϕ = w + bw + c, b = 0. If Tϕ is invertible then (Tϕ−1 1)(0)

  ∞  k  (i + 1)(i + 3) (bw0 )k+1 2 1+ , = 2c + 3bw0 (3 + i)c + (5 + 2i)bw0 (k + 1)!

(1)

k=0 i=0

where w0 is the unique zero of bw 2 + cw + 1 inside D. Proof Since Tϕ is invertible, its Fredholm index is 0. On T we can write ϕ(w) := (bw2 + cw + 1)/w. Hence, Theorem 2.1 implies that wind(ϕ(T), 0) = 0, or equivalently, the equation (2) bw 2 + cw + 1 = 0

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has exactly one zero w0 in D. We set u = Tϕ−1 1 and write Tϕ u = (c + bTw + Tw∗ )u = 1. A calculation using Lemma 2.4 gives



2 bw + cw + 1 u  (w) + 2c + 3bw u(w) = 2, w ∈ D.

(3)

2 . Taking the kth derivative of Plugging in w = w0 in Eq. (3), we get u(w0 ) = 2c+3bw 0 Eq. (3) and evaluating at w0 , we obtain the iteration relation

k u

(k+1)

(w0 ) = −

j=0 (3 + 2 j)b

(3 + k)z 0 + (5 + 2k)bw0

= (−1)k+1

k  i=0

u (k) (w0 )

(i + 1)(i + 3) bk+1 u(w0 ). (3 + i)c + (5 + 2i)bw0

Therefore, in a neighborhood of w0 it holds that u(w) = u(w0 ) +

∞  u (k+1) (w0 ) k=0

= u(w0 ) 1 +

(k + 1)!

∞  k=0

(−1)

k+1

(w − w0 )k+1 k  i=0

(i + 1)(i + 3) bk+1 (w − w0 )k+1 . (3 + i)c + (5 + 2i)bw0 (k + 1)! (4)

Since u(w) is analytic in D, the series (4) converges normally in D. In particular, we obtain

∞  k  (i + 1)(i + 3) (bw0 )k+1 2 1+ . (5) u(0) = 2c + 3bw0 (3 + i)c + (5 + 2i)bw0 (k + 1)! k=0 i=0

  However, it is somewhat unsatisfactory that, although ϕ is simple enough, the value of (Tϕ−1 1)(0) does not show an intuitive connection with the function ϕ. We thus pose the following general question. Question 2.6 Assume ϕ ∈ C(D) and Tϕ is invertible. How to determine the value (Tϕ−1 1)(0)? The following is the main result of this section. Theorem 2.7 Suppose z 2 z 3 = 0. Then P c (B) = {z ∈ C4 : z 0 + z 2 w + z 3 w = 0, w ∈ D and 1 + z 1 u(0) = 0}, where u(0) is determined by Lemma 2.5 for c = z 0 /z 3 , b = z 2 /z 3 .

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Proof In view of Lemma 2.3 and Theorem 2.2, it is sufficient to show that if B(z) is invertible then G(z) = z 0 + z 2 Tw + z 3 Tw∗ is invertible. We will use the same method as that in [21]. Since B(z) is invertible, and Q 0 is compact, G(z) is Fredholm with ind G(z) = 0. It follows from Theorem 2.1 that υ(w) := z 0 + z 2 w + z 3 w = on T, and the equation

z 2 w2 + z 0 w + z 3

= 0 w

z 2 w2 + z 0 w + z 3 = 0

(6)

has exactly one zero w0 in D. If G(z) is not invertible, then there is g ≡ 0 such that G(z)g(w) = (z 0 I + z 2 Tw + z 3 Tw∗ )g(w)  z3 w  tg (t)dt ≡ 0. = (z 0 + z 2 w)g(w) + 2 w 0 Multiplying the both sides of the last equality by w2 and then taking the derivative, we arrive at 2z 0 + 3z 2 w g  (w) =− , w = w0 . g(w) z 2 w2 + z 0 w + z 3

(7)

We shall deduce a contradiction in two cases. 0 Case 1: when 2z 0 + 3z 2 w0 = 0. This implies w0 = − 2z 3z 2 . Then the other root of z0 Eq. (6) is w1 = − 3z 2 which is also in D. This contradicts the fact that Eq. (6) has only one root in D. Case 2: when 2z 0 + 3z 2 w0 = 0. Then using the residue theorem we have −

1 2πi

 γ

 1 g  (w) 2z 0 + 3z 2 w dw = dw g(w) 2πi γ z 2 w 2 + z 0 w + z 3   2z 0 + 3z 2 w , w = Res 0 z 2 w2 + z 0 w + z 3 2z 0 + 3z 2 w0 = 2z 2 w0 + z 0 1 =2− , 2 + z 2zw0 0

where γ ⊂ D is a piece-wise smooth simple contour enclosing all zeros of g. It follows that 2+

z0 1 = z 2 w0 n 350

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for some integers n ≥ 2, and hence w0 = Eq. (6) is w1 =

n z0 1−2n z 2 .

Consequently, the other root of

n − 1 z0 . 1 − 2n z 2

It is obvious that |w1 | < |w0 |, which again contradicts the fact that Eq. (6) has only one zero in D. This shows that G(z) is invertible, and thus the theorem follows from Lemmas 2.3 and 2.5.   It is meaningful to look at a special case of Theorem 2.7 in which the value of (Tϕ−1 1)(0) is more explicit. Corollary 2.8 Suppose B1 (z) = z 0 + z 1 Q 0 + z 2 Tw . Then P c (B1 ) = {(z 0 , z 1 , z 2 ) : |z 2 | < |z 0 | and z 0 + z 1 = 0}. Proof First, if z 0 + z 2 Tw is not invertible, then either it is not Fredholm (|z 0 | = |z 2 |) or Fredholm with index −1 (|z 0 | < |z 2 |). Since Q 0 is compact, the sum z 0 +z 1 Q 0 +z 2 Tw remains non-invertible. It is known that z 0 + z 2 Tw = Tz 0 +z 2 w is invertible if and only if z 0 + z 2 w = 0 on D, i.e., |z 2 | < |z 0 |, in which case we can write (z 0 + z 2 Tw )−1 =

∞  z 2k k (−1)k k+1 Tw . z0 k=0

Then direct calculations show that   z +z

−1 0 1 1 + z 1 z 0 + z 2 Tw e0 , e0 = . z0  

The corollary then follows from Lemma 2.3.

Note again that a unified way to write the conditions |z 2 | < |z 0 | and z 0 + z 1 = 0 in Corollary 2.8 is ψ(w) := z 0 + z 1 χ0 (w) + z 2 w = 0, w ∈ D. Thus the Toeplitz operator Tψ is invertible if and only if ψ does not vanish on D. The following is a particular case of Corollary 2.8. Corollary 2.9 For a complex number α, the classical spectrum  σ (Tw + α Q 0 ) =

D

|α| ≤ 1

D ∪ {α}

|α| > 1.

The next corollary is basically a restatement of Corollary 2.8. Corollary 2.10 Suppose B2 (z) = z 0 + z 1 Q 0 + z 3 Tw∗ . Then P c (B2 ) = {(z 0 , z 1 , z 3 ) : |z 3 | < |z 0 | and z 0 + z 1 = 0}.

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3 On the L2 Space over the Torus T2 Having studied the projective spectrum related to the Bergman shift on L a2 (D), it makes sense to consider a case of two variable nature, as this will give us more insight on how projective spectrum is dependent on the number of variables in concern. The richness of function space provides a unique advantage to address this question. In this section, we shall consider the Hilbert space H = L 2 (T2 ) and two natural operators on H. The first is the “two variable shift” operator T defined by T f (θ1 , θ2 ) = eiθ1 eiθ2 f (θ1 , θ2 ), and the other is the projection P defined by 

2π

P f (θ1 ) =

f (θ1 , θ2 )

0

dθ2 . 2π

Clearly, the operator T is a unitary. If we regard L 2 (T2 ) as the tensor product L 2 (T) ⊗ L 2 (T) then P is the orthogonal projection from L 2 (T2 ) to L 2 (T) ⊗ 1. In other words, given f ∈ L 2 (T2 ) we may write f (θ1 , θ2 ) =



eikθ2 f k (θ1 ),

(8)

k∈Z

 where every f k ∈ L 2 (T) and f 2 = k∈Z f k 2 . Then P f = f 0 . Hence P is of infinite rank. Consider the pencil A(z) = z 0 I +z 1 T +z 2 P, where z = (z 0 , z 1 , z 2 ) ∈ C3 . We shall determine the projective joint spectrum P(A). Observe that since P is a projection of infinite rank, Lemma 2.3 does not apply here. Indeed, this section requires some detailed analysis on the norm of the functions f k in the series (8). We shall make an effort to streamline the process. First of all, to check whether a given operator T is non-invertible we shall follow two routine steps. 1. Determine if T or T ∗ has a nontrivial kernel. 2. In the case both T and T ∗ are injective, we proceed to check if T has closed range. A description of P(A) will then be achieved through a few technical lemmas. To start the analysis, we shall now determine the set of z on which the injectivity of either A(z) or A∗ (z) fails. Suppose f ∈ ker A∗ (z). We write f as in (8) and compute that 0 = (z 0 I + z 1 T ∗ + z 2 P) f = z 0 f + z 1 e−iθ1 e−iθ2 f + z 2 f 0 = (z 0 f 0 + z 2 f 0 + z 1 e−iθ1 f 1 ) +



eikθ2 (z 0 f k + z 1 e−iθ1 f k+1 ),

k =0

which means

(z 0 + z 2 ) f 0 + z 1 e−iθ1 f 1 = 0

352

(9)

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and

z 0 f k + z 1 e−iθ1 f k+1 = 0, ∀k = 0.

(10)

Lemma 3.1 Consider the linear pencil A(z) = z 0 I + z 1 T + z 2 P. Then (a) If z ∈ {|z 1 | > |z 0 |} ∪ {|z 0 | > |z 1 |, z 0 + z 2 = 0}, then both A(z) and A∗ (z) are injective. (b) If z ∈ {|z 0 | > |z 1 |, z 0 + z 2 = 0}, then dim ker A(z) = dim ker A∗ (z) = ∞. Proof There are three cases to consider. Case i: |z 1 | > |z 0 |. First consider the subcase z 0 = 0. Then (10) implies f k =    |k|  z1     z 0  f k+1 and hence f k =  zz01  f 0 for k ≤ −1. But this is possible only if f 0 = 0 because  f k 2 = f 2 < ∞. (11) k∈Z

In turn one has f k = 0 for every k ≤ 0. Using (9) one also has f 1 = 0 and consequently f k = 0 for every k ≥ 1 by (10). This shows that f = 0. If z 0 = 0, then (10) shows that f k = 0 for every k = 1. It then follows from (9) that f 1 is also equal to 0. In conclusion, one has ker A∗ (z) = {0} in the case |z 1 | > |z 0 |. Case ii: |z 0 | > |z 1 | and z 0 + z 2 = 0. First, we consider the subcase z 1 = 0. Then by  k   induction, (10) implies f k+1 =  zz01  f 1 for k ≥ 1, and it follows from (11) that f k = 0 for every k ≥ 1. Since z 0 + z 2 = 0, one has f 0 = 0 by (9), and consequently f k = 0 for every k ≤ 0 by (10). If z 1 = 0, then A(z) = z 0 I + z 2 P which is invertible since σ (P) = {0, 1}. In conclusion one has ker A∗ (z) = {0} in the case |z 0 | > |z 1 | and z 0 + z 2 = 0. Case iii: |z 0 | > |z 1 | and z 0 + z 2 = 0. First we consider the case z 1 = 0. Then by (9) one has f 1 = 0, and consequently f k = 0 for every k ≥ 1 by (10). On the other hand, for k ≤ 0, one has   z 1 |k| ikθ1 e f0 . fk = − z0 This implies that ker A∗ (z) contains the elements f (θ1 , θ2 ) =

 k≤0

e

ikθ2

|k|   z1 −i(θ1 +θ2 ) − e f k (θ1 ) = f 0 z0 k≤0

z 0 f 0 (θ1 ) = , z 0 + z 1 e−i(θ1 +θ2 ) where f 0 ∈ L 2 (T). Moreover, if z 1 = 0 then A∗ (z) f = (z 0 I + z 2 P) f 0 = (z 0 + z 2 ) f 0 = 0 for every f 0 ∈ L 2 (T). So in conclusion dim ker A∗ (z) = ∞ in the case |z 0 | > |z 1 |   and z 0 + z 2 = 0. The case for ker A(z) can be proved by similar arguments.

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A bounded linear operator R on a Hilbert space H is said to be bounded below if there exists  > 0 such that Rx ≥  x for all nonzero vector x ∈ H . It is easy to see that R is bounded below if and only if R is injective and has closed range. Lemma 3.2 Consider the linear pencil A(z) = z 0 I +z 1 T +z 2 P. Then A(z) is bounded below if and only if z falls in the following two cases. (a) |z 1 | > |z 0 |; (b) |z 0 | > |z 1 | and z 0 + z 2 = 0. Proof The proof of (a) is divided into three cases. Case i: |z 0 | = |z 1 |. First, we prove by contradiction that if |z 0 | = |z 1 | then A(z) is not bounded below. The subcase |z 0 | = |z 1 | = 0 is trivial because the projection P has nontrivial kernel. Suppose |z 0 | = |z 1 | = 0 and A(z) were bounded below. Then there would exsit  > 0 such that for all f ∈ H with f = 1 one has A(z) f ≥ . Since the spectrum σ (T ) = T, and ker T = ker T ∗ = {0}, the pencil z 0 + z 1 T is not bounded below. So in this case one must have z 2 = 0. Therefore, for every natural number N ≥ 3 there exists f ∈ H with f = 1, written as in (8), such that (z 0 + z 1 T ) f 2 =



z 0 f k + z 1 eiθ1 f k−1 2
z 0 f 0 − . N N

(14)

We shall pick a large N later to derive a contradiction. Meanwhile, one has     z 0 f k + z 1 eiθ1 f k−1 eikθ2 . A(z) f = z 0 f 0 + z 2 f 0 + z 1 eiθ1 f −1 +

(15)

k =0

Since A(z) f ≥ , one has z 0 f 0 + z 2 f 0 + z 1 eiθ1 f −1 2 +



z 0 f k + z 1 eiθ1 f k−1 2 ≥  2 .

(16)

k =0

In view of (12), this implies  z 0 f 0 + z 2 f 0 + z 1 e

iθ1

f −1 >  1 −

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1 , N2

(17)

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and hence by (13), we have 

1 z 2 f 0 >  1 − 2 − z 0 f 0 + z 1 eiθ1 f −1 >  N



1 1 1− 2 − N N

 .

(18)

The inequality (14) is meaningful only when z 0 f 0 − k N > 0. In view of (18), to get the inequality one may choose k to satisfy      z0   k <   N2 − 1 − 1 N z2 N          or the stronger but simpler inequality k <  zz 02  (N − 2). Let t =  zz 02  (N − 2) be     the largest integer less than or equal to  zz 02  (N − 2). Then combining (14) and (18), one has       z 0   2  z 0 f k > (19) N − 1 − 1 − k > (t − k).   N z2 N It follows that |z 0 |2 = |z 0 |2 f 2 ≥

t 

z 0 f k 2

k=1

>

t  k=1

Since lim N →∞

t N

2 N

(t − k)2 = 2

 2 t(t − 1)(2t − 1) . 6N 2

    =  zz 02  > 0, there exists N big enough such that the right-hand side

of the above line is greater than |z 0 |2 , and this is a contradiction. Case ii: |z 0 | = |z 1 |. In this case the pencil z 0 + z 1 T is invertible. Hence there exists  > 0 such that (20) (z 0 + z 1 T ) f ≥  for every function f ∈ H with f = 1. Writing f as in (8), one verifies that   (z 0 + z 1 T ) f = z 0 f k (θ1 ) + z 1 eiθ1 f k−1 (θ1 ) eikθ2 , k∈Z

and hence by (20) one has 

z 0 f k + z 1 eiθ1 f k−1 2 ≥  2 .

(21)

k∈Z

Suppose A(z) were not bounded below. Then for every integer N > 0 there would exist f ∈ H with f = 1 such that A(z) f
z 1 f k−1 − N z0 N    2     z1  z1  1 +   ≥   z 0 f k−2 − z N z 0

.. .

0

  k  k−1     z1   z1   z1   ≥   z 0 f 0 − 1 +   + · · · +   z0 N z0 z0  k  z1   k  z1    z0  − 1     =   z 0 f 0 − z0 N  z 1  − 1 z0 ⎞ ⎛  k  z1  1  ⎠ +   1 . =   ⎝ z 0 f 0 −     z0 N  z1  − 1 N  z 1  − 1 z0

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z0

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 k   Denote the coefficient of  zz01  in the line above by M N . Then in view of (27),      z0  |z 0 | 1 1  1− 2 − M N ≥    − . z2 N N N |z 1 | − |z 0 |

(28)

Observe that the right-hand side  of (28) does not dependent on the choice of f in (22),   and its limit as N → ∞ is   zz 02 . Summarizing the above observations, one sees that for z ∈ C3 satisfying |z 1 | > |z 0 | > 0, if A(z) were bounded below then there would exist a positive integer N and a f ∈ L 2 (T2 ) with f = 1 such that (22) and (27) |z 0 | . But in view of the the sequence of inequalities above hold, and moreover M N > 2|z 2|

|z 1 | > 1, this would imply that f ≥ f k > 1 for some big k which is and the fact |z 0| a contradiction. The subcase z 1 = 0 but z 0 = 0 is simpler. In this case since T is unitary, one has (z 0 + z 1 T ) f = |z 1 | f = |z 1 |, e.g., we may take = |z 1 | in (20). Then (25)  implies k =0 f k−1 2 < N12 , and therefore f −1 > 1 − N12 . On the other hand,

(25) in particular implies f 0
z 0 f k+1 − N    z0   =   z 1 f k+1 − z1 N  2     z0    z 0    + 1 =   z 1 f k+2 − z1 N  z1  .. .     |k|−1  z0   z 0 |k|−2    =   z 1 f −1 − + ··· + 1 . z1 N  z1 

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After evaluating the finite sum above, we obtain ⎞ ⎛  |k|−1  z0  1  ⎠ +   1 ⎝ z 1 f −1 −   . z 1 f k >   z1 N  z 0  − 1 N  z 0  − 1 z1 z1

(30)

Since z 0 + z 2 = 0, combining (24) and (27), one has  z 1 f −1 ≥ |z 0 + z 2 | f 0 −  N  |z 0 + z 2 | 1 1  ≥ 1− 2 − − . |z 2 | N N N  |k|−1   It follows that the limit of the coefficient of  zz01  in (30), as N → ∞, is greater

2| than or equal to |z|z0 +z . This would imply that f k → ∞ as N → −∞ which 2| is impossible. In summary, A(z) is bounded below in the case 0 < |z 1 | < |z 0 | and z 0 +z 2 = 0. Further, if z 1 = 0 then since σ (P) = {0, 1},A(z) = z 0 I +z 2 P is invertible in this case and hence is bounded below. Finally, if |z 1 | < |z 0 | but z 0 + z 2 = 0, then A(z) has nontrivial kernel by Lemma 3.1 and hence not bounded below. This completes the proof.  

Lemmas 3.1 and 3.2 then give us the following main result of this section. Theorem 3.3 Consider the linear pencil A(z) = z 0 I + z 1 T + z 2 P on L 2 (T2 ). Then P(A) = {|z 0 | = |z 1 |} ∪ {|z 0 | > |z 1 |, z 0 + z 2 = 0}. The following is a direct consequence of the theorem, and it resembles Corollary 2.9. Corollary 3.4 For a complex number α, the classical spectrum  T |α| ≤ 1 σ (T + α P) = T ∪ {α} |α| > 1.

4 Concluding Remark Toeplitz operators play a fundamental role in operator theory, and this paper exlpores the possibility of computing their projective spectrum. Although shift operators are simple Toeplitz operators, the results here may shed light on more complicated cases. Such explorations have a potential to extend the lanscape of Toeplitz operator theory to multivariable operator theory, thereby intigating new questions and inspiring common interests. To conclude this paper, we would like to pose the following question motivated by Question 2.6, Corollaries 2.9 and 3.4. Question 4.1 Let T f be a Toeplitz operator on the Hardy space H 2 (T) with continuous symbol, and let Q : h → h(0) be the evaluation operator. How to compute the spectrum σ (T f + Q)?

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Since Toeplitz operators on the Hardy space is much better understood, this question may have a clean answer. Data Statement Data management is not required for this paper.

References 1. Abrahamse, M.B.: Subnormal Toeplitz operators and functions of bounded type. Duke Math 43, 597– 604 (1976) 2. Axler, S., Conway, J. B., McDonald, G.: Toeplitz operators on Bergman spaces. Can. J. Math. XXXIV(2), 466-483 (1982) 3. Chagouel, I., Stessin, M., Zhu, K.: Geometric spectral theory for compact operators. Trans. Am. Soc. 368(3), 1559–1582 (2016) 4. Chen, X., Guo, K.: Analytic Hilbert Modules. Chapman & Hall/CRC, Boca Raton (2003) 5. Douglas, R.G., Paulsen, V.I.: Hilbert Modules over Function Algebras. In: Pitman Research Notes in Mathematics Series, vol. 217. Harlow: Longman Scientific & Technical; New York: Wiley (1989) 6. Douglas, R.G.: Banach Algebra Techniques in Operator Theory, 2nd edn. Springer, New York (1998) 7. Duan, N.X., Zhao, X.F.: Invertibility of Bergman Toeplitz operators with harmonic polynomial symbols. Sci. China Math. 63(5), 965–978 (2020) 8. Garnett, J.B.: Bounded Analytic Functions. Academic Press, New York (1981) 9. Goldberg, B., Yang, R.: Self-similarity and spectral dynamics, to appear in J. Oper. Theory 10. Grigorchuk, R., Yang, R.: Joint spectrum and the infinite dihedral group. In: Proceedings of the Steklov Institute of Math. 297, 145-178 (2017) 11. He, W., Wang, X., Yang, R.: Projective Spectrum and Kernel Bundle(II). J. Oper. Theory 78(2), 417–433 (2017) 12. Rosenblum, M., Rovnyak, J.: Hardy Classes and Operator Theory. Oxford University Press, NewYork (1985) 13. Rota, G.-C.: On models for linear operators. Commun. Pure Appl. Math. 13, 469–472 (1960) 14. Shields, A.L.: Weighted shift operators and analytic function theory. Mathematical Surveys. Am. Math. Soc. 13, 49–128 (1974) 15. Stroethoff, K., Zheng, D.: Toeplitz and Hankel operators on Bergman spaces. Trans. Am. Math. Soc. 329, 773–794 (1992) 16. Sun, S., et al.: Multiplication operators on the Bergman space and weighted shifts. J. Oper. Theory 592(2), 435–454 (2008) 17. Sun, S.: Bergman shift is not unitarily equivalent to a Toeplitz operator. Mon. J. Sci. 28(8), 1027–1027 (1983) 18. Sundberg, C., Zheng, D.: The spectrum and essential spectrum of Toeplitz operators with harmonic symbols. Indiana Univ. Math. J. 59, 385–394 (2010) 19. Sz.-Nagy, B., Foias, C., Bercovici, H., Kerchy, L.: Harmonic analysis of operators on Hilbert space. Universitext, Springer, New York (2010) 20. Yang, R.: Projective spectrum in Banach algebras. J. Topol. Anal. 1(3), 289–306 (2009) 21. Zhao, X.F., Zheng, D.C.: The spectrum of Bergman Toeplitz operators with some harmonic symbols. Sci. China Math. 59(4), 731–740 (2016) 22. Zhu, K.: Operator Theory in Function Spaces. Marcel Dekker, New York (1990) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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Complex Analysis and Operator Theory (2022) 16:109 https://doi.org/10.1007/s11785-022-01295-8

Complex Analysis and Operator Theory

Left-Invertibility of Rank-One Perturbations Susmita Das1 · Jaydeb Sarkar1 Received: 1 March 2022 / Accepted: 14 October 2022 / Published online: 28 October 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract For each isometry V acting on some Hilbert space and a pair of vectors f and g in the same Hilbert space, we associate a nonnegative number c(V ; f , g) defined by c(V ; f , g) = ( f 2 − V ∗ f 2 )g2 + |1 + V ∗ f , g|2 . We prove that the rank-one perturbation V + f ⊗ g is left-invertible if and only if c(V ; f , g) = 0. We also consider examples of rank-one perturbations of isometries that are shift on some Hilbert space of analytic functions. Here, shift refers to the operator of multiplication by the coordinate function z. Finally, we examine D + f ⊗ g, where D is a diagonal operator with nonzero diagonal entries and f and g are vectors with nonzero Fourier coefficients. We prove that D + f ⊗ g is left-invertible if and only if D + f ⊗ g is invertible. Keywords Left-invertible operators · Rank-one perturbations · Shifts · Isometries · Diagonal operators · Reproducing kernel Hilbert spaces Mathematics Subject Classification 47A55 · 47B37 · 30H10 · 47B32 · 46B50 · 47B07

Dedicated to the memory of Jörg Eschmeier, a wonderful person and a great mathematician. Communicated by Raul Curto. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

B

Jaydeb Sarkar [email protected]; [email protected] Susmita Das [email protected]

1

Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560059, India

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Contents 1 Introduction . . . . . 2 Proof of Theorem 1.1 3 Analytic Operators . . 4 Diagonal Operators . 5 An Example . . . . . 6 Concluding Remarks . References . . . . . . . .

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2 4 10 14 18 19 21

1 Introduction Rank-one operators are the simplest as well as easy to spot among all bounded linear operators on Hilbert spaces. Indeed, for each pair of nonzero vectors f and g in a Hilbert space H, one can associate a rank-one operator f ⊗ g ∈ B(H) defined by ( f ⊗ g)h = h, g f

(h ∈ H).

These are the only operators whose range spaces are one-dimensional. Here B(H) denotes the algebra of all bounded linear operators on H. All Hilbert spaces in this paper are assumed to be infinite dimensional, separable, and over C. Note that finiterank operators, that is, linear sums of rank-one operators are norm dense in the ideal of compact operators, where one of the most important and natural examples of a noncompact operator is an isometry: A linear operator V on H is an isometry if V h = h for all h ∈ H, or equivalently V ∗ V = IH . Along this line, left-invertible operators (also known as, by a slight abuse of terminology, “operators close to an isometry” [20]) are also natural examples of noncompact operators: T ∈ B(H) is left-invertible if T is bounded below, that is, there exists  > 0 such that T h ≥ h for all h ∈ H, or equivalently, there exists S ∈ B(H) such that ST = IH . The intent of this paper is to make a modest contribution to the delicate structure of rank-one perturbations of bounded linear operators [15]. More specifically, this paper aims to introduce some methods for the left-invertibility of rank-one perturbations of isometries and, to some extent, diagonal operators. The following is the central question that interests us: Question 1 Find necessary and sufficient conditions for left-invertibility of the rankone perturbation V + f ⊗ g, where V ∈ B(H) is an isometry or a diagonal operator and f and g are vectors in H. The answer to this question is completely known for isometries. Given an isometry V ∈ B(H) and vectors f , g ∈ H, the perturbation X = V + f ⊗ g is an isometry

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if and only if there exist a unit vector h ∈ H and a scalar α of modulus one such that f = (α − 1)h and g = V ∗ h. In other words, a rank-one perturbation X of the isometry V is an isometry if and only if there exists a unit vector f ∈ H and a scalar α of modulus one such that X = V + (α − 1) f ⊗ V ∗ f .

(1.1)

This result is due to Nakamura [16, 17] (and also see [19]). For more on rank-one perturbations of isometries and related studies, we refer the reader to [2, 4, 5, 12, 14] and also the recent paper [13]. In this paper, we extend the above idea to a more general setting of left-invertibility of rank-one perturbations of isometries. In this case, however, left-invertibility of rank-one perturbations of isometries completely relies on certain real numbers. More specifically, given an isometry V ∈ B(H) and a pair of vectors f and g in H, we associate a real number c(V ; f , g) defined by c(V ; f , g) = ( f 2 − V ∗ f 2 )g2 + |1 + V ∗ f , g|2 .

(1.2)

This is the number which precisely determine the left-invertibility of V + f ⊗ g: Theorem 1.1 Let V ∈ B(H) be an isometry, and let f and g be vectors in H. Then V + f ⊗ g is left-invertible if and only if c(V ; f , g) = 0. Note that since V is an isometry, we have V ∗ f  ≤  f , and hence, the quantity c(V ; f , g) is always nonnegative. Therefore, the condition c(V ; f , g) = 0 in the above theorem can be rephrased as saying that c(V ; f , g) > 0, or equivalently, V ∗ f  <  f  or V ∗ f , g = −1. However, in what follows, we will keep the constant c(V ; f , g) in our consideration. Not only c(V ; f , g) plays a direct role in the proof of the above theorem but, as we will see in Remark 2.1, this quantity also appears in the explicit representation of a left inverse of a left-invertible perturbation. The following conclusion is now easy: Corollary 1.2 Let V ∈ B(H) be an isometry, and let f and g be vectors in H. Then V + f ⊗ g is not left-invertible if and only if V ∗ f  =  f  and V ∗ f , g = −1. The above theorem also provides us with a rich source of natural examples of leftinvertible operators. For instance, let us denote by D the open unit disc in C. Consider

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the shift Mz on the E-valued Hardy space HE2 (D) over D, where E is a Hilbert space. Then for any η ∈ ker Mz∗ = E ⊆ HE2 (D), and nonzero vector g ∈ HE2 (D), the rank-one perturbation Mz +η⊗g is left-invertible. A similar conclusion holds if f , g ∈ H 2 (D) and Mz∗ f , g = −1. Section 2 contains the proof of the above theorem. In Sect. 3, we discuss a followup question: Characterizations of shifts that are rank-one perturbations of isometries. Here a shift refers to the multiplication operator Mz on some Hilbert space of analytic functions (that is, a reproducing kernel Hilbert space) on a domain in C. Note, however, that our analysis will be mostly limited to the level of elementary examples. In Sect. 4, we study rank-one perturbations of diagonal operators. It is well known that the structure of rank-one perturbations of diagonal operators is also complicated (cf. [1, 11, 13, 14]). Moreover, comparison between perturbations of diagonal operators and that of isometries is perhaps inevitable if one views diagonals as normal operators and isometries as one of the best tractable non-normal operators. Here we consider D + f ⊗ g on some Hilbert space H, where D is a diagonal operator with nonzero diagonal entries with respect to an orthonormal basis {en }∞ n=0 of H. We also assume that the Fourier coefficients of f and g with respect to {en }∞ n=0 are nonzero. In Theorem 4.6, we prove: Theorem 1.3 D + f ⊗ g is left-invertible if and only if D + f ⊗ g is invertible. In Sect. 5, we observe that the parameterized spaces considered in the work of Davidson, Paulsen, Raghupathi and Singh [6] is connected to rank-one perturbations of isometries. In the final section, Sect. 6, we compute c(V ; f , g) when V + f ⊗ g is an isometry and make some further comments on rank-one perturbations of diagonal operators. Finally, we remark that the last two decades have witnessed more intense interest in the theory of left-invertible operators starting from the work of Shimorin [20]. For instance, see [18] and references therein. For a more recent account of Shimorin’s approach in the context of analytic model theory, invariant subspaces, and wandering subspaces in several variables, we refer the reader to Eschmeier [7] (also see [3] as part of the motivation), Eschmeier and Langendörfer [9], and Eschmeier and Toth [8]. Also see the monograph by Eschmeier and Putinar [10] for the general framework and motivation.

2 Proof of Theorem 1.1 In this section, we present the proof of the left-invertibility criterion of rank-one perturbations of isometries. First note that by expanding the right-hand side of (1.2),

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we have c(V ; f , g) = 1 +  f 2 g2 + 2ReV ∗ f , g + |V ∗ f , g|2 − V ∗ f 2 g2 . (2.1) Next, we make a list of the most commonly used rank-one operator arithmetic, which will be used several times in what follows. Let f , g ∈ H and let T ∈ B(H). The following holds true: (1) (2) (3) (4) (5)

( f ⊗ g)∗ = g ⊗ f . α( f ⊗ g) = (α f ) ⊗ g = f ⊗ (αg) ¯ for all α ∈ C. ( f ⊗ g)( f 1 ⊗ g1 ) =  f 1 , g f ⊗ g1 for all f 1 , g1 ∈ H. T ( f ⊗ g) = (T f ) ⊗ g and so ( f ⊗ g)T = f ⊗ (T ∗ g).  f ⊗ g =  f g.

Although these facts are well known, we prove them for the sake of completeness. Fix h, h˜ ∈ H. Recall that ( f ⊗ g)h = h, g f . For (1), we compute ˜ = h, h, ˜ g f  = g, hh, ˜ ˜ = (g ⊗ f )h, h, ˜ f  = h, f g, h ( f ⊗ g)∗ h, h which implies ( f ⊗ g)∗ = g ⊗ f . Of course, part (2) is a particular case of part (4) (consider T = α IH ). Part (3) follows from the fact that ( f ⊗ g)( f 1 ⊗ g1 )h = h, g1  f 1 , g f = ( f 1 , g f ⊗ g1 )h. Now we turn to part (4). Observe that T ( f ⊗ g)h = T (h, g f ) = h, gT f = (T f ⊗ g)h, which implies T ( f ⊗ g) = (T f ) ⊗ g, whereas ( f ⊗ g)T h = T h, g f = h, T ∗ g f = ( f ⊗ T ∗ g)h, implies that ( f ⊗ g)T = f ⊗ (T ∗ g). To prove (5), we observe that ( f ⊗ g)h = |h, g| f  ≤  f gh, and hence conclude that  f ⊗ g ≤  f g. On the other hand  1 g = g f , ( f ⊗ g) g 

implies that  f ⊗ g ≥ g f  and completes the proof of part (5).

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Finally, we note that T ∈ B(H) is left-invertible if and only if T ∗ T is invertible. Indeed, if T is left-invertible, then T ∗ T is an injective positive operator. Since T is bounded below, we know that T ∗ T is also bounded below and hence of closed range. Therefore, T ∗ T is invertible. Conversely, suppose X is the inverse of T ∗ T . Then (X T ∗ )T = I implies that T is left-invertible. We are now ready for the proof of the theorem. Proof of Theorem 1.1: The statement trivially holds for f = 0 or g = 0. So assume that both f and g are nonzero vectors. Suppose that V + f ⊗ g on H is left-invertible. Then (V + f ⊗ g)∗ (V + f ⊗ g) is invertible with the inverse, say L. We have I = L(V + f ⊗ g)∗ (V + f ⊗ g) = L(V ∗ + g ⊗ f )(V + f ⊗ g). Since V ∗ V = I , it follows that I = L(V ∗ + g ⊗ f )(V + f ⊗ g) = L(I + V ∗ f ⊗ g + g ⊗ V ∗ f +  f 2 g ⊗ g) = L + (L V ∗ f ) ⊗ g + Lg ⊗ V ∗ f +  f 2 Lg ⊗ g. In particular, evaluating both sides on the vector V ∗ f and g, respectively, we get V ∗ f = L V ∗ f + V ∗ f , gL V ∗ f + V ∗ f 2 Lg +  f 2 V ∗ f , gLg = (V ∗ f , g + 1)L V ∗ f + (V ∗ f 2 +  f 2 V ∗ f , g)Lg, and g = Lg + g2 L V ∗ f + g, V ∗ f Lg +  f 2 g2 Lg = g2 L V ∗ f + (1 + g, V ∗ f  +  f 2 g2 )Lg = g2 L V ∗ f + αLg, where α = 1 + g, V ∗ f  +  f 2 g2 . The latter equality implies that LV ∗ f =

1 (I − αL)g. g2

Now plug the value for L V ∗ f into the expression for V ∗ f above to get V∗ f =

1 (1 + V ∗ f , g)(I − αL)g + (V ∗ f 2 +  f 2 V ∗ f , g)Lg. g2

A little rearrangement then shows that  1  ∗ 1 + V f , g g g2  + V ∗ f 2 +  f 2 V ∗ f , g −

V∗ f =

366

 α ∗ (1 + V f , g) Lg. g2

(2.2)

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We compute α(1 + V ∗ f , g) = (1 + V ∗ f , g)(1 + g, V ∗ f  +  f 2 g2 ) = V ∗ f , g f 2 g2 + 2ReV ∗ f , g + |V ∗ f , g|2 +  f 2 g2 + 1 = V ∗ f , g f 2 g2 + V ∗ f 2 g2 + c(V ; f , g), where the last equality follows from the definition of c(V ; f , g) as in (2.1). Now we simplify the coefficient of Lg, say a, in the right-hand side of (2.2) as follows: a = V ∗ f 2 +  f 2 V ∗ f , g  1  ∗ 2 2 ∗ 2 2 − V f , g f  g + V f  g + c(V ; f , g) g2 1 =− c(V ; f , g). g2 Consequently, by (2.2), we have V∗ f =

1 1 (1 + V ∗ f , g)g − c(V ; f , g) Lg. g2 g2

Suppose if possible that c(V ; f , g) = 0. Then V∗ f =

1 (1 + V ∗ f , g)g, g2

and so 1 (1 + V ∗ f , g)g, g g2 = 1 + V ∗ f , g,

V ∗ f , g =

which is absurd. This contradiction proves that c(V ; f , g) = 0. Conversely, suppose that c := c(V ; f , g) = 0. Set R = (1 + g, V ∗ f )V ∗ f ⊗ g, and let 1 X = I + {g2 V ∗ f ⊗ V ∗ f + (V ∗ f 2 −  f 2 )g ⊗ g − (R + R ∗ )}. c We claim that X (V + f ⊗ g)∗ is a left inverse of V + f ⊗ g, that is X (V + f ⊗ g)∗ (V + f ⊗ g) = I .

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Indeed, the left hand side of the above simplifies to X (V + f ⊗ g)∗ (V + f ⊗ g) = X (V ∗ + g ⊗ f )(V + f ⊗ g) = X (I + g ⊗ V ∗ f + V ∗ f ⊗ g +  f 2 g ⊗ g)  1 = I + {g2 V ∗ f ⊗ V ∗ f + (V ∗ f 2 −  f 2 )g ⊗ g c   −(R + R ∗ )} I + g ⊗ V ∗ f + V ∗ f ⊗ g +  f 2 g ⊗ g . Hence there exists scalars a1 , a2 , a3 , and a4 such that X (V + f ⊗ g)∗ (V + f ⊗ g) = I + a1 g ⊗ g + a2 V ∗ f ⊗ g + a3 g ⊗ V ∗ f +a4 V ∗ f ⊗ V ∗ f . It is now enough to show that a1 = a2 = a3 = a4 = 0. Before getting to the proof of this claim, let us observe that ¯ ∗ f ⊗ g + (1 + β)g ⊗ V ∗ f , R + R ∗ = (1 + β)V where β := V ∗ f , g. Now we prove that a1 = 0: a1 = coefficient of g ⊗ g 1 ¯ f 2 ) + (V ∗ f 2 −  f 2 ) =  f 2 + −(1 + β)(V ∗ f 2 + β c   × (1 + β) +  f 2 g2 1 ¯ + β) f 2 + V ∗ f 2  f 2 g2 −β(1 =  f 2 + c −(1 + β) f 2 −  f 4 g2

 f 2  ¯ + β) + V ∗ f 2 g2 − (1 + β) −  f 2 g2 −β(1 c  f 2 (−c) = 0, =  f 2 + c

=  f 2 +

where the last but one equality follows from (2.1). Next we compute a2 : a2 = coefficient of V ∗ f ⊗ g   1 ¯ f 2 ) − (1 + β) ¯ (1 + β) +  f 2 g2 g2 (V ∗ f 2 + β =1+ c 1 ∗ 2 V f  g2 − |1 + β|2 −  f 2 g2 =1+ c = 0, 368

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as β = V ∗ f , g. We turn now to compute a3 : a3 = coefficient of g ⊗ V ∗ f 1 −(1 + β)β¯ − (1 + β) + (V ∗ f 2 −  f 2 )g2 =1+ c 1 −|1 + β|2 + (V ∗ f 2 −  f 2 )g2 =1+ c = 0, and, finally a4 = coefficient of V ∗ f ⊗ V ∗ f 1 2 ¯ − (1 + β)g ¯ = − g2 (1 + β) c = 0. This completes the proof of the fact that V + f ⊗g is left-invertible with X (V + f ⊗g)∗ as a left inverse. 

Remark 2.1 From the definition of X in (2.3), it is clear that if V + f ⊗g is left-invertible for some isometry V ∈ B(H) and vectors f and g in H, then  L=

 1 2 ∗ ∗ ∗ 2 2 ∗ I + {g V f ⊗ V f +(V f  −  f  )g ⊗ g−(R + R )} (V + f ⊗ g)∗ , c

is a left-inverse of V + f ⊗ g, where c = c(V ; f , g), and R = (1 + g, V ∗ f )V ∗ f ⊗ g. It is worthwhile to observe that for an isometry V ∈ B(H) and a vector f ∈ H, we have V ∗ f  =  f  if and only if f ∈ ranV . In particular, Theorem 1.1 yields the following: Corollary 2.2 Let V ∈ B(H) be an isometry and let f and g are nonzero vectors in H. If f ∈ / ranV , then V + f ⊗ g is left-invertible.

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3 Analytic Operators Recall that an isometry V ∈ B(H) is called a pure isometry if ∞

V n H = {0}.

n=0

As we will see soon, this is also known as the analytic property of V . It is known that an isometry V ∈ B(H) is pure if and only if V is unitarily equivalent to Mz on 2 (D), where W = ker V ∗ is the wandering subspace the W-valued Hardy space HW corresponding to V . Here Mz denotes the multiplication operator by the coordinate 2 (D) (see (3.1) below). Rank-one perturbations of isometries (or function z on HW pure isometries) that are pure isometries form a rich class of operators and are fairly complex in nature [17]. The methods involve heavy machinery of H ∞ (D)-function theory, which is mostly unavailable for general function spaces (see [2, 4, 12, 14, 19]). In this section we discuss some examples of rank-one perturbations of isometries that are shift or simply analytic. We begin with a brief introduction to shift operators on reproducing kernel Hilbert spaces. Let E be a Hilbert space and  be a domain in C. Let H be a Hilbert space of E-valued analytic functions on . Suppose the evaluation map evw ( f ) = f (w)

( f ∈ H),

defines a bounded linear operator evw : H → E for all w ∈ . Then the kernel function k :  ×  → B(E) defined by ∗ k(z, w) = evz ◦ evw

(z, w ∈ ),

is positive definite, that is, n 

k(z i , z j )η j , ηi E ≥ 0,

i, j=1 n n ⊆ , {ηi }i=1 ⊆ E and n ≥ 1. Moreover, k is analytic in the first variable for all {z i }i=1 and satisfies the reproducing property

evw ( f ), ηE =  f (w), ηE =  f , k(·, w)ηH , for all f ∈ H, w ∈  and η ∈ E. We denote the space H by Hk and call it analytic Hilbert space. The shift operator Mz on Hk is defined by (Mz f )(w) = w f (w)

370

( f ∈ Hk , w ∈ ).

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We always assume that Mz is a bounded linear operator on Hk (equivalently, zHk ⊆ Hk ). It is easy to see that if Mz is a shift on some Hk , then ∞

Mzn Hk =

n=0

∞

z n Hk = {0}.

n=0

This is the property which bridges the gap between left-invertible operators and leftinvertible shifts. More precisely, following the ideas of Shimorin [20], a bounded linear operator T on H is called analytic if ∞

T n H = {0}.

n=0

If T ∈ B(H) is a left-invertible analytic operator, then there exists an analytic Hilbert space Hk such that T and the shift Mz on Hk are unitarily equivalent [20]. Therefore, up to unitary equivalence, analytic left-invertible operators are nothing but left-invertible shifts. The following proposition collects some examples of analytic and shift operators. Proposition 3.1 Let V ∈ B(H) be a pure isometry, m, n ∈ Z+ , and let f 0 ∈ ker V ∗ . If S = V + V m f 0 ⊗ V n f 0 , then the following holds: (1) S is analytic whenever m > n. (2) S is a shift whenever m > n + 1. Proof For simplicity, for each t ∈ Z, we set ft =

V t f0 if t ≥ 0 V ∗−t f 0 if t < 0.

Since f 0 ∈ ker V ∗ , it follows that f t = 0 for all t < 0. Suppose m > n. Observe that  f m , f n  = V m f 0 , V n f 0  = 0, and hence S 2 = V 2 + f m+1 ⊗ f n + f m ⊗ f n−1 +  f m , f n  f m ⊗ f n = V 2 + f m ⊗ f n−1 + f m+1 ⊗ f n . Then, by induction, we have S k+1 = V k+1 + f m ⊗ f n−k + f m+1 ⊗ f n−k+1 + · · · + f m+k−1 ⊗ f n−1 + f m+k ⊗ f n , that is S k+1 = V k+1 +

k  j=0

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f m+ j ⊗ f n−k+ j ,

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for all k ≥ 1. In particular, if k = n + j and j ≥ 1, then it follows that S n+ j+1 = V n+ j+1 + f m ⊗ f − j + f m+1 ⊗ f − j+1 + · · · + f m+n+ j−1 ⊗ f n−1 + f m+n+ j ⊗ f n . At this point, we note that f − p = 0 for all p > 0, and hence, by property (4), listed at the beginning of Sect. 2, we have S n+ j+1 = V n+ j+1 + f m+ j ⊗ f 0 + f m+ j+1 ⊗ f 1 + · · · + f m+ j+n−1 ⊗ f n−1 + f m+n+ j ⊗ f n = V n+ j+1 (I +

n 

f m−n−1+i ⊗ f i ),

i=0

as m > n. This implies that S n+ j+1 H ⊆ V n+ j+1 H From here we see that

r ≥0

Sr H ⊆



Sr H ⊆

r ≥n+1

( j ≥ 1).



V r H = {0},

r ≥n+1

where the last equality follows from the fact that V is pure. To prove (2), we compute the value of c(V ; f , g) with f = V m f 0 and g = V n f 0 : c(V ; f , g) = ( f 2 − V ∗ f 2 )g2 + |1 + V ∗ f , g|2 = (V m f 0 2 − V m−1 f 0 2 )V n f 0 2 + |1 + V m−1 f 0 , V n f 0 |2 = 0 ×  f 0 2 + |1 + 0| = 1, where the last but one equality follows because m − n − 1 > 0 implies V ∗n V m−1 f 0 , f 0  = 0. The first part along with Shimorin’s analytic model [20] and Theorem 1.1 then completes the proof of part (2). 

The above observation is fairly elementary. The general classification of rank-one perturbations of isometries (or pure isometries) that are shift on some reproducing kernel Hilbert space is an open problem. However, see [16, Theorem 1] and [17] in the context of classifications of rank-one perturbations of isometries that are pure isometry. The following is also a simple class of examples of analytic operators. Proposition 3.2 Let V ∈ B(H) be a pure isometry, f and g be vectors in H, and suppose V ∗ g + g, f g = 0. Then V + f ⊗ g is analytic.

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Proof If we set S := V + f ⊗ g, then S 2 = V 2 + V f ⊗ g + f ⊗ (V ∗ g + g, f g) = V S. Therefore S n+1 = V n S

(n ≥ 1),

can be proved analogously by induction. In particular S n+1 H = V n SH ⊆ V n H

(n ≥ 0),

and hence, by using the fact that V is a pure isometry, it follows that ∞

(V + f ⊗ g)n+1 H ⊆

n=0

∞

V n H = {0},

n=0

that is, V + f ⊗ g is analytic.



Note that V ∗ g+g, f g = 0 is equivalent to the condition that g ∈ ker(V + f ⊗g)∗ . Recall that the scalar-valued Hardy space H 2 (D) is a reproducing kernel Hilbert space corresponding to the Szegö kernel S : D × D → C, where S(z, w) = (1 − z w) ¯ −1

(z, w ∈ D).

For each w ∈ D, consider the analytic function S(·, w) : D → C defined by (also known as the kernel function, see the discussion at the beginning of this section) (S(·, w))(z) = S(z, w)

(z ∈ D).

Example 3.3 The following examples illustrate some direct application of the above propositions. ¯ w). Choose (1) Fix w ∈ D, and set g = S(·, w). We know that Mz∗ S(·, w) = wS(·, −1 n 2 f ∈ H (D) such that g, f  H 2 (D) = −w¯ (for instance, f = w¯ n−1 z for some n ≥ 1). Evidently Mz∗ g + g, f g = 0, and hence Mz + f ⊗ S(·, w) is an analytic operator. (2) Consider f = z and g = 1 in H 2 (D). Then c(Mz ; f , g) = 2 = 0, and hence Mz + f ⊗ g is a shift.

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(3) Consider f = z and g = −1 in H 2 (D). Then c(Mz ; f , g) = 0, and hence Mz + f ⊗ g not left-invertible, but analytic by Proposition 3.1. Note that the rank-one perturbation Mz + z 2 ⊗ z is similar to Mz on H 2 (D). Here the similarity follows easily from the fact that Mz + z 2 ⊗ z is a weighted shift with the weight sequence {1, 2, 1, 1, . . .}. This implies, of course, that Mz + z 2 ⊗ z is analytic, where on the other hand Mz∗ z + z, z 2 z = 1 = 0. Therefore, Mz + z 2 ⊗ z is an example of an analytic rank-one perturbation of Mz which does not satisfy the hypothesis of Proposition 3.2.

4 Diagonal Operators In this section, we examine rank-one perturbations of diagonal operators. We prove that all the interesting left-invertible rank-one perturbations of diagonal operators are invertible. ∞ Throughout this section, we fix

a∞Hilbert space H with

∞orthonormal basis {en }n=0 of H. We also fix vectors f = n=0 an en and g = n=0 bn en in H and diagonal operator D ∈ B(H) with diagonal entries {αn }n≥0 . Also, we set T = D + f ⊗ g. We will assume throughout this section that αn , an , bn = 0

(n ≥ 0),

as this is the class of perturbations we all are most interested in (cf. [14]). Furthermore, we define the quantity r := 1 +

∞  an b¯n n=0

αn

.

The following result is from Ionascu [14, Proposition 2.4]: Proposition 4.1 T admits zero as an eigenvalue if and only if r = 0 and { αann }n≥0 is a square summable sequence. The key to our analysis lies in the following observation which is also a result of independent interest.

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Proposition 4.2 {en }n≥0 ⊆ ranT if and only if r = 0 and { αann }n≥0 is a square summable sequence. Proof Assume

that e j ∈ ranT for some arbitrary but fixed integer j ≥ 0. Then there exists x = ∞ n=0 cn en ∈ H such that T x = (D + f ⊗ g)x = e j . Therefore ∞ ∞   ej = (cn αn )en + x, g an en . n=0

(4.1)

n=0

Note that x, g = 0. Indeed, if x, g = 0, then cn =

1 αj

if n = j

0

otherwise,

and hence x = α1j e j . Since g = ∞ n=0 bn en , using x, g = 0, we have b j = 0. This contradiction shows, as promised, that x, g = 0. Now equating the coefficients of terms on either side of (4.1), we have cn =

(1 − a j x, g) an − αn x, g 1 αj

if n = j otherwise.

In particular, { αann }n≥0 is a square summable sequence, and, as x, g = we have x, g = −x, g

∞  an b¯n n=0

αn

+

∞

¯

n=0 cn bn ,

b¯ j , αj

which implies  x, g 1 +

∞  an b¯n n=0



αn

= x, gr =

b¯ j , αj

and hence r = 0. Conversely, assume that r = 0 and { αann }n≥0 is a square summable sequence. Fix an integer j ≥ 0. Then b¯ j y=− rαj

∞   an 1 en + e j , αn αj n=0

is a vector in H. Note that y, g = −

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Using the representation f =

∞

n=0 an en ,

T y = (D + f ⊗ g)y = −

we deduce from the above that

∞ b¯ j  an en + e j + y, g f = e j . rαj n=0

This implies that e j ∈ ranT for all j ≥ 0 and completes the proof of the proposition. 

We also need the following lemma: Lemma 4.3 If T is bounded below, then D is invertible. Proof Assume by contradiction that {αn k } is a subsequence of the sequence {αn }, which converges to zero. Now T en k = (D + f ⊗ g)en k = αn k en k + en k , g f = αn k en k + bn k f , implies T en k  ≤ |αn k | + |bn k | f . This shows that {T en k } converges to zero for the sequence of unit vectors {en k }. But this contradicts the fact that T is bounded below. Therefore the sequence {αn } has no subsequence that converges to zero. Consequently, there exists M > 0 such that |αn | > M

(n ≥ 0),

and hence { α1n } is a bounded sequence. We conclude that D is invertible.



The converse is not true. For example, choose f , g ∈ H such that  f , g = −1. Then, by Proposition 4.1, I + f ⊗ g is not injective, and hence I + f ⊗ g is not left-invertible. However, under the assumption that D + f ⊗ g is injective, we have the following: Proposition 4.4 If D is bounded below and T is injective, then T is left-invertible. Proof Assume by contradiction that T = D + f ⊗ g is not bounded below. Then there is a sequence {h n } ⊆ H with h n  = 1 such that T h n → 0. By the compactness of f ⊗ g, there exists a subsequence {h n k } of {h n } such that ( f ⊗ g)h n k converges. Then Dh n k = (T − f ⊗ g)h n k , converges. But since D is bounded below, this gives us h n k → h˜ for some h˜ ∈ H. In ˜ = 1. On the other hand, since T is a bounded linear operator, particular, we have h we have T h˜ = lim T h n k = 0, k→∞

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that is, h˜ ∈ ker T . But, ker T = {0} by our assumption, and hence h˜ = 0, which ˜ = 1. Therefore, T is bounded below. contradicts the fact that h 

Although Proposition 4.4 is not directly related to the main result of this section, but perhaps fits appropriately with our present context. The following result and its proof are also along the same line and perhaps of independent interest. Proposition 4.5 If D has a closed range, then T also has a closed range. Proof Suppose N = ker T , and suppose that ranD is closed. Then T |N ⊥ is injective. Assume by contradiction that ranT is not closed. Then X := T |N ⊥ is not left-invertible. Proceeding exactly as in the proof of Proposition 4.4 (by replacing the role of T by X ), we will find a similar contradiction. 

We come now to the main result on left-invertibility of rank-one perturbations. Theorem 4.6 D + f ⊗ g is left-invertible if and only if D + f ⊗ g is invertible. Proof For the nontrivial direction, assume that T = D + f ⊗ g is left-invertible. Assume by contradiction that T is not invertible. Since, in particular, ranT is closed, {en }n≥0  ranT . Now by Proposition 4.1, either r = 0 or the sequence { αann }n≥0 is not square summable. On the other hand, we know from Lemma 4.3 that D is invertible, and hence D −1 f =

∞  an en ∈ H. αn n=0

This implies, of course, that { αann }n≥0 is a square summable sequence, and hence r = 0. As a consequence, we can apply Proposition 4.2 to T : the basis vectors {en }n≥0 ⊆ ranT ; which is a contradiction. This proves that T is invertible. 

If we know that D is invertible (which anyway follows from Lemma 4.3) and r = 0, then the surjectivity of T = D + f ⊗ g in the above proof also can be obtained as follows: Observe that 1 + D −1 f , g = 1 +

∞  an b¯n n=0

αn

= r.

Then for each y ∈ H, we consider 1 x = D −1 y − D −1 y, gD −1 f . r We deduce easily that T x = y, which completes the proof of the fact that T is onto.

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5 An Example Let T be a bounded linear operator on H 2 (D). Suppose ⎡

⎤ 0 ... ⎢ . ⎥ ⎢a01 0 0 . . ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ .⎥ [T ] = ⎢a02 a12 0 ⎥, ⎢ ⎥ . ⎢a03 a13 a23 . . ⎥ ⎣ ⎦ .. . . . . . . . . . . 0

0

the matrix representation of T with respect to the standard orthonormal basis {z n , n ≥ 0} of H 2 (D). Clearly, T (z n ) ⊆ z n+1 H 2 (D), and hence T n (H 2 (D)) ⊆ z n H 2 (D)

(n ≥ 0).

It follows that ∞

T n H 2 (D) ⊆

n=0

∞

z n H 2 (D) = {0},

n=0

that is, T is analytic. In particular, for each α and β in C, the matrix operator ⎤ 0 0 0 0 ... ⎢ . ⎥ ⎢α 0 0 0 . . ⎥ ⎥ ⎢ ⎥ ⎢ ⎢β 0 0 0 . . . ⎥ ⎥ ⎢ [Tα,β ] = ⎢ ⎥, ⎢0 1 0 0 . . .⎥ ⎥ ⎢ ⎥ ⎢ ⎢0 0 1 0 . . .⎥ ⎦ ⎣ .. . . . . . . . . . . . . . ⎡

defines an analytic operator Tα,β on H 2 (D). Moreover, one can show that Tα,β = Mz2 + (αz + (β − 1)z 2 ) ⊗ 1, that is, Tα,β is a rank-one perturbation of the shift Mz2 on H 2 (D). Next, we compute c(Tα,β ; f , g), where f = αz + (β − 1)z 2 and g = 1. Since Mz∗2 f , g H 2 (D) = β − 1, and Mz∗2 f 2 = |β − 1|2 , and  f 2 = |α|2 + |β − 1|2 , it follows that c(Tα,β , αz + (β − 1)z 2 , 1) = |α|2 + |β|2 .

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In particular, if (α, β) ∈ C2 \ {(0, 0)}, then c(Tα,β , αz + (β − 1)z 2 , 1) = 0, and hence, by Theorem 1.1 and the fact that Tα,β is analytic, it follows that Tα,β is a shift on H 2 (D). Thus we have proved: Proposition 5.1 Let (α, β) ∈ C2 \ {(0, 0)}, and suppose f = αz + (β − 1)z 2 and g = 1. Then: (1) Tα,β is a shift on H 2 (D), (2) Tα,β = Mz2 + f ⊗ g, (3) c(Mz2 ; f , g) = |α|2 + |β|2 . We recall in passing that Tα,β is a shift means the existence of an analytic Hilbert space Hk and a unitary U : H 2 (D) → Hk such that Tα,β = U ∗ Mz U (see the discussion preceding Proposition 3.1). We continue with the matrix representation [Tα,β ]. It is immediate that Tα,β is an isometry if and only if |α|2 + |β|2 = 1. 2 (D) the closed codimension one subspace of H 2 (D) with orthonormal Denote by Hα,β 2 (D) is an invariant subspace of M 2 . One can basis {α + βz, z 2 , z 3 , . . .}. Clearly, Hα,β z 2 (D) defined by verify straightforwardly that the map U : H 2 (D) → Hα,β

Uz = n

α + βz if n = 0 z n+1 otherwise,

is a unitary operator and U Tα,β = Mz2 U , that is, Tα,β on H 2 (D) and Mz2 | H 2

α,β (D)

2 (D) are unitarily equivalent. The operon Hα,β

2 (D), for (α, β) ∈ C2 such that |α|2 + |β|2 = 1, has been ator Mz2 | H 2 (D) on Hα,β α,β considered in [6] in the context of invariant subspaces and a constrained NevanlinnaPick interpolation problem. Clearly, in the context of perturbation theory, it is worth exploring and explaining the results of [6].

6 Concluding Remarks We begin by computing c(V ; f , g) for rank-one perturbations that are isometries. Suppose V ∈ B(H) is an isometry and f and g are vectors in H. It is curious to observe that c(V ; f , g) = 1,

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whenever V + f ⊗ g is an isometry. Indeed, in the present case, by (1.1), there exist a unit vector h ∈ H and a scalar α of modulus one such that f = (α −1)h and g = V ∗ h. Then (2.1) yields c(V ; f , g) − 1 = |α − 1|2 V ∗ h2 + 2(Re(α − 1))V ∗ h2 + |α − 1|2 V ∗ h4 (1 − 1) = (|α − 1|2 + 2Re(α − 1))V ∗ h2 = 0,

as |α| = 1. This completes the proof of the claim. It would be interesting to investigate the nonnegative number c(V ; f , g) in terms of analytic and geometric invariants, if any, of rank-one perturbations of isometries. This is perhaps a puzzling question for which we do not have any meaningful answer or guess at this moment. We conclude this paper by making some additional comments on (non-analytic features of) perturbations of diagonal operators. The following easy-to-prove proposition says that rank-one perturbations of common diagonal operators do not fit well with shifts on reproducing kernel Hilbert spaces. Proposition 6.1 Let D ∈ B(H) be a Fredholm diagonal operator, and let f , g ∈ H. Then D + f ⊗ g cannot be represented as shift. Proof Assume the contrary, that is, assume that D + f ⊗ g is unitarily equivalent to Mz on some reproducing kernel Hilbert space Hk . Since D is Fredholm, and Mz and D + f ⊗ g are unitarily equivalent, we have ind(Mz ) = ind(D) = 0. On the other hand, since Mz is injective, it follows that ind(Mz ) = dim ker Mz − dim ker Mz∗ < 0, 

which is a contradiction.

In the context of Theorem 4.6, we remark that rank-one perturbations of diagonal operators need not be left-invertible: Consider a compact diagonal operator D (for instance, consider D with diagonal entries { n1 }n≥1 ). Then a rank-one perturbation of D is also compact, and hence the perturbed operator cannot be left-invertible. In Lemma 4.3, we prove that if D + f ⊗ g is bounded below, then D is invertible. This was one of the key tools in proving Theorem 4.6: D + f ⊗ g is left-invertible if and only if D + f ⊗ g is invertible. Of course, we assumed that the Fourier coefficients of f and g are nonzero. Here, we would like to point out that rank-one perturbation of an invertible operator need not be invertible. In fact, the invertibility property of rank-one perturbations of invertible operators can be completely classified (see [14, Lemma 2.7]): Let D be an invertible diagonal operator. Then D + f ⊗ g is invertible if and only if 1 + D −1 f , g = 0.

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Left-Invertibility of Rank-One Perturbations

Finally, in the context of left-invertibility, consider D = IH and choose f and g from H such that  f , g = −1. It is easy to see that c(D; f , g) = 0, and hence, D + f ⊗ g is not left-invertible. Acknowledgements The authors thank the anonymous referee for carefully reading the article and for many useful comments. The research of the second named author is supported in part by Core Research Grant, File No: CRG/2019/000908, by the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India. Data Availability Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References 1. Albrecht, E., Chevreau, B.: Compact perturbations of scalar type spectral operators. J. Oper. Theory 86, 163–188 (2021) 2. Benhida, C., Timotin, D.: Functional models and finite-dimensional perturbations of the shift. Integral Equ. Oper. Theory 29, 187–196 (1997) 3. Bhattacharjee, M., Eschmeier, J., Keshari, D., Sarkar, J.: Dilations, wandering subspaces, and inner functions. Linear Algebra Appl. 523, 263–280 (2017) 4. Choi, M., Wu, P.: Finite-rank perturbations of positive operators and isometries. Studia Math. 173, 73–79 (2006) 5. Clark, D.: One dimensional perturbation of restricted shifts. J. Anal. Math. 25, 169–191 (1972) 6. Davidson, K., Paulsen, V., Raghupathi, M., Singh, D.: A constrained Nevanlinna–Pick interpolation problem. Indiana Univ. Math. J. 58, 709–732 (2009) 7. Eschmeier, J.: Bergman inner functions and m-hypercontractions. J. Funct. Anal. 275, 73–102 (2018) 8. Eschmeier, J., Toth, S.: K -inner functions and K -contractions. In: Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications, vol. 282, pp. 157–181. Birkhäuser/Springer, Cham (2021) 9. Eschmeier, J., Langendörfer, S.: Multivariable Bergman shifts and Wold decompositions. Integral Equ. Oper. Theory 90, 56 (2018) 10. Eschmeier, J., Putinar, M.: Spectral Decompositions and Analytic Sheaves. London Mathematical Society Monographs. New Series, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1996) 11. Fang, Q., Xia, J.: Invariant subspaces for certain finite-rank perturbations of diagonal operators. J. Funct. Anal. 263, 1356–1377 (2012) 12. Fuhrmann, P.A.: On a class of finite dimensional contractive perturbations of restricted shifts of finite multiplicity. Israel J. Math 16, 162–176 (1973) 13. Gallardo-Gutiérrez, E., González-Doña, F.: Finite rank perturbations of normal operators: spectral subspaces and Borel series. J. Math. Pures Appl. 162, 23–75 (2022) 14. Ionascu, E.: Rank-one perturbations of diagonal operators. Integral Equ. Oper. Theory. 39, 421–440 (2001) 15. Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1995). Reprint of the 1980 edition 16. Nakamura, Y.: One-dimensional perturbations of the shift. Integral Equ. Oper. Theory 17, 373–403 (1993) 17. Nakamura, Y.: One-dimensional perturbations of isometries. Integral Equ. Oper. Theory 9, 286–294 (1986) 18. Pietrzycki, P.: Generalized multipliers for left-invertible operators and applications. Integral Equ. Oper. Theory. 92(5), 41 (2020) 19. Serban, I., Turcu, F.: Compact perturbations of isometries. Proc. Am. Math. Soc. 135, 1175–1180 (2007) 20. Shimorin, S.: Wold-type decompositions and wandering subspaces for operators close to isometries. J. Reine Angew. Math. 531, 147–189 (2001)

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Complex Analysis and Operator Theory (2022) 16:71 https://doi.org/10.1007/s11785-022-01243-6

Complex Analysis and Operator Theory

A-Isometries and Hilbert-A-Modules Over Product Domains Michael Didas1 Received: 15 March 2022 / Accepted: 9 May 2022 / Published online: 18 June 2022 © The Author(s) 2022

Abstract For a compact set K ⊂ Cn , let A ⊂ C(K ) be a function algebra containing the polynomials C[z 1 , · · · , z n ]. Assuming that a certain regularity condition holds for A, we prove a commutant-lifting theorem for A-isometries that contains the known results for isometric subnormal tuples in its different variants as special cases, e.g., Mlak (Studia Math. 43(3): 219–233, 1972) and Athavale (J. Oper. Theory 23(2): 339–350, 1990; Rocky Mt. J. Math. 48(1): 2018; Complex Anal. Oper. Theory 2(3): 417–428, 2008; New York J. Math. 25: 934–948, 2019). In the context of Hilbert-A-modules, our result implies the existence of an extension map ε : Hom A (S1 , S2 ) → HomC(∂ A ) (H1 , H2 ) for hypo-Shilov-modules Si ⊂ Hi (i = 1, 2). By standard arguments, we obtain an identification Hom A (S1 , S2 ) ∼ = Hom A (H1  S1 , H2  S2 ) where Hi is the minimal C(∂ A )-extension of Si (i = 1, 2), provided that H1 is projective and S2 is pure. Using embedding techniques, we show that these results apply in particular to the domain algebra A = A(D) = C(D)∩O(D) over a product domain D = D1 ×· · ·× Dk ⊂ Cn where each factor Di is either a smoothly bounded, strictly pseudoconvex domain or a bounded symmetric and circled domain in some Cdi (1 ≤ i ≤ k). This extends known results from the ball and polydisc-case, Guo (Studia Math. 135(1): 1–12, 1999) and Chen and Guo (J. Oper. Theory 43: 69–81, 2000). Keywords Commutant lifting for subnormal isometric tuples · Analytic Hilbert modules over product domains Mathematics Subject Classification 47A13 · 47B20 · 46H25 Dedicated to the memory of Jörg Eschmeier, in deep gratitude for his guidance and inspiration. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

A previous version of this chapter was published Open Access under a Creative Commons Attribution 4.0 International License at http://link.springer.com/10.1007/s11785-022-01243-6.

B 1

Michael Didas [email protected] Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH, Oktavie-Allee, 66687 Wadern, Germany

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1 A-Isometries Let H be a separable complex Hilbert space. A spherical isometry is a commuting tuple T = (T1 , · · · , Tn ) ∈ B(H)n of bounded linear operators on H that satisfies the condition T1∗ T1 + · · · + Tn∗ Tn = 1H . Athavale proved in [3] that spherical isometries are subnormal with normal spectrum σn (T ) conained in ∂Bn , a fact which provides them with a natural A(Bn )-functional calculus, where A(Bn ) = { f ∈ C(Bn ) : f |Bn is holomorphic} denotes the ball algebra. Replacing A(Bn ) with a general function algebra A ⊂ C(K ) on a compact set K ⊂ Cn and ∂Bn with its Shilov-boundary ∂ A , one arrives at the notion of A-isometry as introduced by Eschmeier in [16] (see below for a precise definition). It is this general setting in which we will formulate most of our results. We use the same notations as in [14] and [17]. A reader familiar with one of these works may readily proceed with Sect. 2. Subnormal tuples A tuple T ∈ B(H)n is called subnormal, if there is a commuting tuple of normal  ⊃ H such that H is invariant for the  n on a Hilbert space H operators U ∈ B(H) components of U and T = U |H. We suppose in the following that U is minimal  The normal in the sense that the only reducing subspace for U containing H is H. spectrum of T is defined as σn (T ) = σ (U ), where σ (U ) denotes the Taylor spectrum of U . This is independent of the special choice of U . By a result of Putinar, the spectral inclusion σn (T ) ⊂ σ (T ) holds. Spectral theory for normal tuples asserts the existence of a so-called scalar spectral measure, i. e., a finite positive Borel measure μ on σ (U ) with the property that there exists an isomorphism of von Neumann algebras  mapping z i → Ui (i = 1, · · · , n), U : L ∞ (μ) → W ∗ (U ) ⊂ B(H) called the L ∞ -functional calculus of U . (Note that such a measure μ is unique up to mutual absolute continuity.) One then defines the restriction algebra RT = { f ∈ L ∞ (μ) : U ( f )H ⊂ H}, which is a w ∗ -closed subalgebra of L ∞ (μ) containing the polynomials C[z] in n complex vairables z = (z 1 , · · · , z n ). By [10, Proposition 1.1], the induced w ∗ -continuous algebra homomorphism γT : RT → B(H),

f → U ( f )|H

is isometric again. It satisfies the following uniqueness property: γT ( f |σn (T )) = f (T )

(whenever f ∈ O(W ), W ⊂ Cn open and W ⊃ σ (T )), (1.1)

where the symbol f (T ) on the right-hand side is meant in the sense of Taylor’s holomorphic functional calculus (see, e.g., [18], Chapter 2). To see this,

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recall that the L ∞ -calculus U satisfies a spectral mapping theorem of the form σ (U ( f 1 ), · · · , U ( f k )) = F(σ (U )) for every k-tuple F = ( f 1 , · · · , f k ) ∈ C(K )k with k ∈ N (see, e. g., [29], Section 0.3 and the references therein). A well-known uniqueness result for Taylor’s functional calculus (cp. Theorem 5.2.4 in [18]) then guarantees that U ( f ) = f (U ) for every f ∈ O(W ) if W ⊂ Cn is any open set containing σ (U ). If we even assume that W ⊃ σ (T ), we can apply Lemma 2.5.8  as intertwiner of T and U to deduce that in [18] to the inclusion map i : H → H f (U )|H = f (T ) for f ∈ O(W ). Hence the asserted uniqueness (1.1) follows. A-subnormal tuples, A-isometries Fix a compact set K ⊂ Cn and let C(K ) denote the algebra of all continuous complexvalued functions equipped with the supremum norm. For a closed subalgebra A ⊂ C(K ), we write ∂ A for the Shilov boundary of A, i.e. the smallest closed subset of K such that f ∞,K = f ∞,∂ A . Following Eschmeier, [16], we define: Definition 1.1 Suppose that A ⊂ C(K ) is a closed subalgebra containing C[z]. A subnormal tuple T ∈ B(H )n is said to be A-subnormal, if σn (T ) ⊂ K and RT ⊃ A. If, in addition, σn (T ) ⊂ ∂ A , then T is said to be an A-isometry. An A-isometry consisting of normal operators is called A-unitary. Natural choices for A to consider are algebras of disc-algebra type, that is, A(D) = { f ∈ C(D) : f |D is holomorphic} ⊂ C(D), where D ⊂ Cn is a suitably chosen bounded open set. As pointed out above, setting A = A(Bn ) yields exactly the class of spherical isometries, whereas A(Dn )-isometries (D ⊂ C stands for the open unit disc) are precisely commuting tuples whose components are isometric operators on H. Let now T ∈ B(H)n be an A-subnormal tuple for A ⊂ C(K ) as in the definition, and fix a scalar spectral measure μ of its minimal normal extension. Via trivial extension, we may regard μ as an element of M + (K ), the set of all finite regular Borel measures on K . Since RT is w ∗ -closed and assumed to contain A, the restriction of γT to the algebra H A∞ (μ) = A

w∗

⊂ L ∞ (μ)

is well defined. By the properties of γT stated above, it therefore induces an isometric and w ∗ -continuous isomorphism onto its image Ta (T ) = γT (H A∞ (μ)) (the set of all so-called analytic T -Toeplitz operators). To keep the notation simple, we denote the induced map again by γT , and call it the canonical H ∞ -functional calculus of T : γT : H A∞ (μ) → Ta (T ) ⊂ B(H ),

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A-unitary dilations  n. Let again T ∈ B(H)n be A-subnormal with minimal normal extension U ∈ B(H)  As a ∗-homomorphism, the map U |C(K ) : C(K ) → B(H) is completely contractive, and hence so is γT |A : A → B(H), if A is regarded as an operator subspace of C(K ). Since both embeddings A ⊂ C(K ) and A ⊂ C(∂ A ) induce the same operator space structure on A (see [27, Theorem 3.9]), a theorem of Arveson (see, e.g., [27, Corollary 7.7]) asserts that there exists an A-unitary tuple V ∈ B(K)n on a Hilbert space K ⊃ H such that γT ( f ) = PH V ( f )|H

( f ∈ A).

We call such a tuple an A-unitary dilation of T . When restricted to the smallest closed subspace that contains H and reduces V , it is said to be a minimal A-unitary dilation.

2 Regularity If T ∈ B(H)n is an A-subnormal tuple, then one can use its canonical functional calculus γT to carry over approximation properties of the underlying function algebra A to the algebra Ta (T ) of analytic Toeplitz operators. As it turns out, of particular interest are approximation properties of A that are related to the so-called abstract inner function problem. In its classical form, the inner function problem asks if there are non-constant bounded analytic functions θ : Bn → C on the unit ball Bn such ˜ = 1 σ -a.e. (so-called inner functions). that their radial limit θ˜ : ∂Bn → C satisfies |θ| Here, σ denotes the normalized surface measure on ∂Bn . In his celebrated work [1], Aleksandrov succeded to solve the inner function problem for what he called “regular triples” (A, K , μ) (cf. Corollary 29 therein and Proposition 2.6 below for a suitable formulation in our context). The triple (A(Bn ), Bn , σ ) models the classical context. The precise definition of regularity reads as follows: Definition 2.1 Let K ⊂ Cn compact and A ⊂ C(K ) a closed subalgebra. (a) For μ ∈ M + (K ), the triple (A, K , μ) is called regular (in the sense of Aleksandrov [1]), if for every ϕ ∈ C(K ) with ϕ > 0, there is a sequence ( f k ) in A satisfying | f k | < ϕ (on K ) for all k ∈ N and limk→∞ | f k | = ϕ (μ-a.e. on K ). (b) We call the algebra A ⊂ C(K ) itself regular, if (A, K , μ) is regular for every measure μ ∈ M + (K ) with supp(μ) ⊂ ∂ A . Note that the condition on the support of μ in part (b) of the preceding definition results from the fact that the stated inclusion necessarily holds for regular triples. A concrete source of regular triples is the following embedding criterion due to Aleksandrov: Theorem 2.2 (Aleksandrov [2, Theorem 3]) Let A ⊂ C(K ) be a function algebra on a compact set K . Suppose that, for some m ∈ N, there exists an injective map F ∈ Am such that F(∂ A ) ⊂ ∂Bm . Then, A ⊂ C(K ) is regular. Using this criterion, one can show that the algebra A(D) is regular for various kinds of open sets D ⊂ Cn , among them bounded symmetric and circled domains, and

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relatively compact, strictly pseudoconvex open subsets of Cn , or even of a Stein submanifold of Cn (cp. [1, 2] and [11, Corollary 2.1.3 (e)]). By constructing a suitable embedding, we show the following result for product domains: Theorem 2.3 Let k ∈ N and Di ⊂ Cdi be open sets with di ∈ N for 1 ≤ i ≤ k, each of which is either a strictly pseudoconvex domain with C 2 boundary or a bounded symmetric and circled domain. Then the algebra A(D1 ×· · ·×Dk ) ⊂ C(D 1 ×· · ·×D k ) is regular. We first recall some basic facts about bounded symmetric domains: By definition, a bounded domain D ⊂ Cd is symmetric if, for each z ∈ D, there exists a biholomorphic map sz : D → D possessing z as isolated fixed point such that sz ◦ sz = id D . We assume that D is circled at the origin, i. e., 0 ∈ D and z D ⊂ D for z ∈ C with |z| = 1. Every set of this type is convex [24, Corollary 4.6] and the Shilov boundary of A(D) consists precisely of those points in D with maximal Euclidean distance from the origin in Cd [24, Theorem 6.5]. Since every finite product of bounded symmetric and circled domains is again of this type, we only have to consider one such factor in Theorem 2.3. To keep the notations in the proof simple, we restrict ourselves to the case D = D1 × D2 × D3 , where D1 and D2 are strictly pseudoconvex sets and D3 is the circled and bounded symmetric factor. This is no restriction, since the result is well-known for k = 1 (see the references given above), whereas the cases k = 2 and k > 3 can be handled by straight-forward modifications of our construction described below. In the following, we write S D = ∂ A(D) . To start with, let us fix real numbers r1 , r2 , r3 > 0 such that D i ⊂ ri · Bdi (1 ≤ i ≤ 2) and D3 ⊂ r3 · Bd3 , S D3 = D 3 ∩ (r3 · ∂Bd3 ). We first calculate the Shilov boundary of D. From the existence of peaking functions for every boundary point of a strictly pseudoconvex domain (see Range [28, Corollary VI.1.14]), it follows that S Di = ∂ Di for the strictly pseudoconvex factors i = 1, 2. Lemma 2.4 S D = ∂ D1 × ∂ D2 × S D3 . Proof Fix an arbitrary z = (z 1 , z 2 , z 3 ) ∈ D1 × D2 × D3 . Since f (·, z 2 , z 3 ) ∈ A(D1 ), there is a w1 ∈ ∂ D1 such that | f (z 1 , z 2 , z 3 )| ≤ | f (w1 , z 2 , z 3 )|. Repeating the argument twice, we obtain w2 ∈ ∂ D2 and w3 ∈ S D3 such that | f (z 1 , z 2 , z 3 )| ≤ | f (w1 , w2 , w3 )|. This shows that ∂ D1 × ∂ D2 × S D3 is a boundary for A(D). Now fix w = (w1 , w2 , w3 ) in the latter product set. To finish the proof, it suffices to observe that h(z 1 , z 2 , z 3 ) = h 1 (z 1 ) · h 2 (z 2 ) · h 3 (z 3 ) is a peaking function for w, if h i ∈ A(Di ) is a peaking functions for wi (i = 1, 2, 3). For the strictly pseudoconvex factors, such functions exist by the reference given above. For D3 , we may take the restriction of a   peaking function for w3 in A(r3 · Bd3 ).

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The key ingredient in the proof of Theorem 2.3, is an embedding theorem of Løw [25, Theorem 3]. Reduced to what we need, this theorem says the following: Given a strictly pseudoconvex domain D ⊂ Cn with C 2 -boundary and a strictly positive continuous function φ : ∂ D → (0, ∞), there exists a mapping g ∈ A(D)l for some natural number l such that |g(z)| = φ(z) for every z ∈ ∂ D. Proof of Theorem 2.3 Since, by assumption, D i ⊂ ri · Bdi for i = 1, 2, the functions 1 φi (z) = (ri2 − |z|2 ) 2 (z ∈ ∂ Di , i = 1, 2) are strictly positive if | · | denotes the li Euclidean norm on Cdi . By the cited theorem of Løw,  we find gi ∈ A(Di ) (i = 1, 2) with |gi (z)| = φi (z) (z ∈ ∂ Di ). Then we set r =   F(z) = (1/r ) · z, g1 (z 1 ), g2 (z 2 ) ∈ Cm

r12 + r22 + r32 and define

(z = (z 1 , z 2 , z 3 ) ∈ D 1 × D 2 × D 3 = D),

to obtain an injective mapping F ∈ A(D)m with m = d1 + d2 + d3 + l1 + l2 which, for z = (z 1 , z 2 , z 3 ) ∈ ∂ D1 × ∂ D2 × r3 · ∂Bd3 , satisfies   |F(z)|2 = (1/r 2 ) · |z 1 |2 + |z 2 |2 + r32 + (r12 − |z 1 |2 ) + (r22 − |z 2 |2 ) = 1. Together with Lemma 2.4 and the fact that S D3 ⊂ r3 · ∂Bd3 , this yields the inclusion F(S D ) ⊂ ∂Bm . The desired regularity now follows as an application of Aleksandrov’s embedding criterion stated as Theorem 2.2 above.   Remark 2.5 Let F ∈ A(D)m , and let | · | denote the Euclidean norm and ·, · the Euclidean scalar-product on Cm . By considering scalar-valued functions of the form F(·), F(w) with suitably chosen w ∈ D, one can show that |F| takes its maximum value on the Shilov boundary S D , and that F must be constant if |F| takes its maximum at some point inside D. In view if this, we can – for further reference – state some additional properties of the map F constructed in the proof of Theorem 2.3, namely F(D) ⊂ Bm and F(D) ⊂ Bm . Moreover, we remark that the map F : D → F(D) is a homeomorphism, as it is a continuous bijection from a compact space onto a Hausdorff space. As for the scope of Theorem 2.3, we should remark that every bounded open set D ⊂ C with C 2 boundary is strictly pseudoconvex, cp. the remark at the end of Section 1.5 in [22]. So, in particular A(D) ⊂ C(D), is regular for polydomains D = D1 × · · · × Dk with C 2 -bounded domains Di ⊂ C (i = 1, · · · , k). Now, let (A, K , μ) be a regular triple. In analogy with the classical case we define a μ-inner function to be an element of the set  Iμ = θ ∈ H A∞ (μ) : |θ | = 1 (μ-a.e. on ∂ A ) . Our study of A-isometries in the following sections relies on a density result for such abstract inner functions taken from [14]. It is a slight variation of Aleksandrov’s corresponding result [1, Corollary 29]. In its original form, it says that, given the regularity of (A, K , μ), the w∗ -closure of the set Iμ in L ∞ (μ) contains the set

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{[ f ]μ : f ∈ A, | f | ≤ 1 everywhere on K } supposed that μ is a continuous measure. Continuity means in this context that every one-point set has μ-measure zero. The key point here is that the following result holds without any continuity assumption on μ. Proposition 2.6 (see [14, Proposition 2.4 and Corollary 2.5]) If (A, K , μ) is a regular triple and A ⊃ C[z]. Then we have H A∞ (μ) = L H where L H

w∗

w∗

(Iμ )

and

L ∞ (μ) = L H

w∗

({θη : θ, η ∈ Iμ }),

denotes the w ∗ -closed linear hull.

Note that, in the context of the preceding proposition, the first of these density relations implies the second one: The w∗ -closed linear hull on the right is easily seen to be a w ∗ -closed subalgebra of L ∞ (μ) that is closed under complex conjugation. Since it contains the restrictions of all polynomials (by the first density relation), it actually must be all of L ∞ (μ). It has turned out (cp. [14, 17, 19]) that this kind of density is exactly what is needed for operator-theoretic applications. So we define: Definition 2.7 Let K ⊂ Cn compact, A ⊂ C(K ) a closed subalgebra. (a) Given a measure μ ∈ M + (K ), the triple (A, K , μ) will be called w ∗ -regular, if w∗ w∗ H A∞ (μ) = L H (Iμ ), and L ∞ (μ) = L H ({θ η : θ, η ∈ Iμ }) hold. (b) We simply call the algbera A itself w ∗ -regular, if the triple (A, K , μ) is w∗ -regular for every measure μ ∈ M + (K ) with supp(μ) ⊂ ∂ A . (c) If A ⊃ C[z], then an A-isometry is called w∗ -regular, if the triple (A, K , μ) is w∗ -regular, where μ ∈ M + (∂ A ) denotes the scalar-valued spectral measure of a minimal normal extension of T . Theorem 2.3 provides natural examples of w ∗ -regular function algebras. Further examples are the so called unit-modulus algebras introduced by Guo and Chen in [9, Definition 2.4].

3 Commutant Lifting for A-Isometries The aim of this section is to prove the following commutant-lifting theorem that applies by the preceding section in particular to all algebras of the form A(D1 × · · · × Dn ) where each of the factors Di is either a bounded strictly pseudoconvex domain with C 2 -boundary, or a bounded sysmmetric and circled domain. Theorem 3.1 Let K ⊂ Cn be a compact set and A ⊂ C(K ) a closed, w ∗ -regular subalgebra that contains C[z]. Suppose that T1 ∈ B(H1 )n is an A-subnormal tuple  1 )n and that T2 ∈ B(H2 )n is an A-isometry with minimal A-unitary dilation U1 ∈ B(H  2 )n . Then every operator X ∈ B(H1 , H2 ) with minimal normal extension U2 ∈ B(H satisfying X γT1 ( f ) = γT2 ( f )X

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1, H  2 ) with the property that possesses a unique extension to an operator  X ∈ B(H  X X U1 ( f ) = U2 ( f ) 

( f ∈ C(∂ A )).

This extension satisfies  X = X . If X has dense range, then so has  X . If U1 is X. even an A-unitary extension of T1 and, in addition, X is isometric, then so is  The above theorem contains several known intertwining results for subnormal isometric tuples as special cases (note that all but the last one require T1 to be an A-isometry): • A result of Guo (see [21, Lemma 3.6]) from the context of normal Hilbert modules. Actually our proof is inspired by Guo’s idea. But his hypotheses are rather special: Roughly speaking, both tuples are assumed to be spherical isometries, and the spectral measures of their minimal normal extensions are assumed to be absolutely continuous with respect to the surface measure of the sphere. • An intertwining result for spherical isometries by Athavale (Proposition 8 in [3]). More generally, the analogous result for so called ∂ D-isometries introduced by Athavale in [4], where D ⊂ Cn is a relatively compact strictly pseudoconvex domain: Our Theorem 3.1 implies Athavale’s Theorem 3.2. We carry out the details for the convenience of the reader: First of all, note that A(D) is w ∗ -regular according to the remark preceding Theorem 2.3. Now, if we choose a domain  ⊃ D in such a way that O()|D is dense in A(D) (called HKL-superdomain in [4], see Remark 2.1 therein; for the existence, cp. [11, Corollary 2.1.3 (b)] and the references therein), then every ∂ D-isometry with σ (T ) ⊂  is actually an A(D)-isometry: To see this, note that RT contains O(), since γT extends the O()-functional calculus for T as observed in (1.1), Sect. 1, and hence RT ⊃ A(D) for density reasons. In Athavales result, two ∂ D-isometries Tk = (Tk,1 , · · · , Tk,n ) ∈ B(Hk )n (k = 1, 2) are assumed to satisfy the inclusion σ (T1 ) ∪ σ (T2 ) ⊂  and an intertwining relation of the form X T1,i = T2,i X

(i = 1, · · · , n).

To apply our theorem, it remains to check that X γT1 ( f ) = γT2 ( f )X for every f ∈ A(D) hold in this case. But the assumed intertwining relation for the components immediately implies that X f (T1 ) = f (T2 )X for f ∈ O(), where f (Tk ) is built using the holomorphic functional calculus. Again, the uniqueness of the canonical H ∞ -calculi of T1 and T2 then implies that X γT1 ( f ) = γT2 ( f )X for all f ∈ O(). So by the density of O()|D in A(D), the hypothesis of Theorem 3.1 is indeed satisfied. • An intertwining result for so-called S -isometries [6, Theorem 2.1]. We sketch the main arguments: For a relatively compact, convex set  ⊂ Cn with 0 ∈ , it follows from the Oka-Weil theorem and radial approximation that C[z] is dense in A(). So in this case, the condition RT ⊃ A() is always fulfilled. This shows in particular that S -isometries (where  is a Cartan domain of type I, II, III, or IV) as defined by Athavale in [6] are A-isometries with A = A(), which is known to be regular for bounded symmetric and circled domains. Again, for uniqueness

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reasons of the involved functional calculi, a componentwise intertwining relation implies the one assumed in our theorem. For another class of convex domains, where the domain algebra is regular, cp. [5] (see Proposition 2.5 therein for the regularity, and Proposition 4.6 for the corresponding commutant-lifting result). The sets considered there are special complex ellipsoids. • Réolon’s analogous result for K -isometries (cp. Proposition 3.5.18 in [29]) which are by definition A-isometries with A = C[z] ⊂ C(K ), where K ⊂ Cn compact and A is assumed to be regular in our sense. This class contains in particular finite commuting tuples of isometries, which were first treated by Mlak [26, Proposition 5.2]. We should further remark that all but the first cited result rely on a lemma of Mlak on spectral dilations [26, Lemma 4.1]. In contrast, we give an alternative and elementary proof by writing down an explicit formula for  X , based on an idea of Guo [21, Lemma 3.6]. For the proof of Theorem 3.1, we need an operator-theoretic version of the measuretheoretic density assumption from Definition 2.7. To formulate it appropriately, let us fix the following notation: Given two measures μ, ν ∈ M + (∂ A ) with μ  ν, let rμν : L ∞ (ν) → L ∞ (μ) denote the canonical map defined by [ f ]ν → [ f ]μ for every bounded measurable function f : ∂ A → C. Clearly, every rμν is a w ∗ -continuous and contractive *-homomorphism.  n with scalar-valued spectral measure μ  ν, Given an A-unitary tuple U ∈ B(H) ν we denote by U the composition  Uν = U ◦ rμν : L ∞ (ν) → B(H). As a consequence of Proposition 2.6, we have:  n an A-unitary tuple Lemma 3.2 Let (A, K , ν) be a w ∗ -regular triple and U ∈ B(H) + with scalar-valued spectral measure μ ∈ M (∂ A ) satisfying μ  ν. Then, for every  the set subspace H ⊂ H,  0 = L H { ν (θη)h : θ, η ∈ Iν , h ∈ H} ⊂ H  H U  that contains H and reduces U . (Here, L H stands is the smallest closed subspace of H  for the norm-closed linear hull in H.)  0 reduces U . Towards this, let θ0 , η0 ∈ Iν . Obviously, Proof We first show that H ν U (θ 0 η0 ) maps the set {Uν (θη)h : θ, η ∈ Iν , h ∈ H} into itself. By linearity and continuity, we may pass to L H , so 0 ⊂ H 0 Uν (θη)H Reprinted from the journal

(whenever θ, η ∈ Iν ).

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Since invariant subspaces are preserved under w ∗ -limits, the w ∗ -regularity assumption 0 ⊂ H  0 for every f ∈ L ∞ (ν). (cf. Definition 2.7) allows us to conclude that Uν ( f )H  0 is actually reducing for U . By taking f to be z i and z i (i = 1, · · · , n), we see that H Standard arguments based on the theorems of Stone-Weierstraß and Lusin yield the stated minimality.   Besides the above density result, an essential ingredient of the proof is the simple observation that, for every θ ∈ Iν , the operator Uν (θ ) is isometric, since it evidently satisfies Uν (θ )∗ Uν (θ ) = Uν (θθ) = 1H . (Actually, it is even unitary as θ is the inverse of θ in L ∞ (ν), but isometry will suffice for our purposes.) Proof of Theorem 3.1 For k = 1, 2, let μk ∈ M + (K ) denote the trivial extensions of scalar spectral measures of Uk to elements in M + (K ). The proof will actually work if we replace the hypothesis on A to be a w∗ -regular function algebra with the following weaker but more technical one: There exists a measure ν ∈ M + (∂ A ) such that (A, K , ν) is w∗ -regular and μ1 , μ2  ν. Let us fix such a measure ν for the rest of the proof. (Under the original hypothesis of the theorem, ν = μ1 + μ2 evidently has the desired properties.) Throughout the proof, we use the abbreviations  k ) and γ ν = γT ◦ r ν : H ∞ (ν) → B(H) for Uν k = Uk ◦ rμν k : L ∞ (ν) → B(H μk k A Tk k = 1, 2.  1 (cf. Lemma 3.2) in the obvious way, We first define  X on a dense subset of H namely

 X



Uν 1 (θ i ηi )h i

=

i∈F



Uν 2 (θ i ηi )X h i ,

(3.1)

i∈F

where F ⊂ N denotes a finite set and θi , ηi ∈ Iν , h i ∈ H are arbitrary elements for i ∈ F. To see that

this is well defined, we estimate the norm of the right-hand side. If we define π = i∈F θi , then π ∈ Iν , and so the operator Uν 2 (π ) is isometric. This yields             ν ν U2 (θ i ηi )X h i  =  U2 (π θ i ηi )X h i       i∈F

(3.2)

i∈F

From the very definition of π , the functions π θ i belong to Iν for every choice of i ∈ F. Since, in addition, X maps into H2 , and U2 is an A-unitary extension of T2 , we may replace Uν 2 with γTν2 . Then we use the intertwining relation γT2 ( f )X = X γT1 ( f ) = X PH U1 ( f )|H ( f ∈ A) from the hypothesis. We restate this as γT2 ( f )X h, k = U1 ( f )h, X ∗ k

( f ∈ A, h, k ∈ H).

to see that it extends by w∗ -WOT-continuity to γTν2 ( f )X = X PH Uν 1 ( f )|H 392

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These considerations allow us to continue expression (3.2) as follows             ν ν γT2 (π θ i ηi )X h i  ≤ X ·  U1 (π θ i ηi )h i  ··· =      i∈F i∈F       = X ·  Uν 1 (θ i ηi )h i  ,  

(3.3)

i∈F

where we again used the isometry of Uν 1 (π ) in the last step. First of all, this shows that  X is indeed well defined (and then, by construction, linear) on the space F of all finite linear combinations of vectors of the form Uν 1 (θ i ηi )h i with θi , ηi ∈ Iν , h i ∈ H1 . Moreover, the estimate derived in (3.2)–(3.3) briefly says that

 X x ≤ X · x

(x ∈ F).

(3.4)

 1 to Cauchy-sequences in H  2 and In particular,  X maps Cauchy-sequences in F ⊂ H thus can be continuously extended to a linear map on the closure F, which coincides  1 by Lemma 3.2. This map has obviuously norm less than X , and a look at with H the assumed density (Definition 2.7 (b)) and the defining formula (3.1) for  X shows that it actually has the desired intertwining property. In view of Lemma 3.2 and (3.1),  X has dense range if so has X . Moreover, if U1 is even an A-unitary extension of T1 , then we may drop the operator PH in the considerations preceding estimate (3.3). Thus, if X is isometric, then equality holds in (3.3) and (3.4) with X = 1. To finish the proof, note that an operator  X satisfying the intertwining relation X , because from the statement of the theorem necessarily fulfills  X Uν 1 ( f ) = Uν 1 ( f )  ∗ ∞ C(∂ A ) is w -dense in L (ν). Consequently, it satisfies (3.1), and therefore coincides with the extension constructed above.   When specialized to one A-isometry, Theorem 3.1 reads as follows: Corollary 3.3 Let T ∈ B(H)n be a w ∗ -regular A-isometry with minimal normal exten n . Then, every element X ∈ Ta (T ) possesses a unique extension to sion U ∈ B(H) an element  X ∈ (U ) . Combining Theorem 3.1 and Lemma 1 from [3]), we can draw another standard conclusion in this context (cp. Athavale [3, Proposition 9]): Corollary 3.4 If two w∗ -regular A-isometries are quasi-similar, then their minimal unitary extensions are unitarily equivalent.

4 Lifting of Module Homomorphisms Let A ⊂ C(K ) be function algebra, i. e., a closed subalgebra of C(K ) that separates the points of K . Recall that a Hilbert-A-module is a Hilbert space H together with a

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continuous bilinear map A × H → H, ( f , h) → f · h turning H into an A-module in the algebraical sense. Given f ∈ A, we write M f (or M H f , if we want to emphasize the underlying space) for the continuos linear multiplication operator H → H, h → f · h with f ∈ A. The continuous linear map H : A → B(H), f → M f will be referred to as the underlying representation of H. The Hilbert module H is called contractive, if so is H . Definition 4.1 Let A ⊂ C(∂ A ) be a function algebra. A Hilbert-A-module S is called a hypo-Shilov-module, if there exists a Hilbert-C(∂ A )-module H such that S ⊂ H is an A-submodule. In this case, H is called a C(∂ A )-extension of H. If H is contractive, S is called a Shilov module over A. Given this definition, the following further notations are natural: The extension H of a (hypo-)Shilov-module is called minimal, if C(∂ A ) · S is dense in H. A hypoShilov-module is called reductive, if it is a C(∂ A )-submodule of H, and pure, if no non-zero subspace of it is reductive. The key observation for applying Theorem 3.1 in the context of Hilbert modules is the following simple lemma which says that A-isometries and Shilov- A-modules are essentially the same. Lemma 4.2 Let A ⊂ C(K ) be a closed subalgebra containing C[z]. (a) If S ⊂ H is a Shilov-A-module with minimal contractive C(∂ A )-extension H, then Mz ∈ B(S)n is an A-isometry with minimal normal extension Mz ∈ B(H)n . (b) Conversely, if T ∈ B(S)n is an A-isometry with minimal normal extension U ∈ B(H)n , then, introducing the multiplications A × S → S, ( f , h) → f · h = γT ( f )h on H and C(∂ A ) × H → H, ( f , h) → f · h = U ( f )h on H turns S ⊂ H into a Shilov module over A with minimal C(∂ A )-extension H. Proof Since, in part (a), the representation H : C(∂ A ) → B(H), f → M f is assumed to be contractive, it is a ∗-homomorphism (see [15, Theorem 1.12]) and thus gives rise to a normal tuple Mz = (Mz 1 , · · · , Mz n ) ∈ B(H)n with Taylor spectrum σ (Mz ) ⊂ ∂ A . Since S is an A-submodule of H, we have the inclusion MfS ⊂ S

(for every f ∈ A).

(4.1)

This implies at first that T = Mz |S is subnormal with normal extension U = Mz ∈ B(H)n . Since, by assumption, C(∂ A )·S is dense in H, it follows that U is a minimal normal extension of T . By a Stone-Weiertraß argument, the identity H ( f ) = U ( f ), where U is the canonical L ∞ -calculus of U (see Sect. 1) holds for all f ∈ C(∂ A ). Hence the inclusion (4.1) from above actually says that RT ⊃ A as required in the definition of an A-isometry. This observation completes the proof of part (a). Part (b) is obvious.  

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By an A-module homomorphism X : H → K between two Hilbert-A-modules we mean a bounded linear map that respects the module multiplication in the sense that X ( f · h) = f · X h whenever f ∈ A and h ∈ H. We write Hom A (H, K) to denote the set of all A-module homomorphisms from H to K. Let now two Shilov modules S1 , S2 with minimal C(∂ A )-extensions H1 , H2 be k given. Using the identifications from the preceding proof, namely γTk ( f ) = M H f |Sk

k ( f ∈ A) and Uk ( f ) = M H ( f ∈ C(∂ A )), the following identities are evident: f

Hom A (S1 , S2 ) = {X ∈ B(S1 , S2 ) : γT1 ( f )X = X γT2 ( f ) : f ∈ A} HomC(∂ A ) (H1 , H2 ) = {X ∈ B(H1 , H2 ) : U1 ( f )X = X U2 ( f ) : f ∈ C(∂ A )}. As a last ingredient for the extension theorem we aim for (Theorem 4.4), we need the following well-known fact: Theorem 4.3 (cp. [15, Theorem 1.9]) Let H be a Hilbert-C(K )-module. Then H is similar to a contractive C(K )-module K, i. e., there exists an invertible C(K )-module map (a similarity map) X : H → K. Now we are ready to prove the main result of this section. In the case A = A(Bn ), this appears as Lemma 3.6 in [21] (where it is stated for Shilov-modules only, and under an additional continuity assumption). For a unit modulus algebra A, the corresponding result appears as Proposition 2.5 in [9]. Our theorem contains both as special cases. Theorem 4.4 Let Sk be hypo-Shilov modules over A with C(∂ A )-extensions Hk (k = 1, 2). Suppose that A ⊃ C[z] is w ∗ -regular. Then there exists a map ε : Hom A (S1 , S2 ) → HomC(∂ A ) (H1 , H2 ) with ε(X )|S1 = X . If H1 is minimal, then ε is unique. Proof The proof is divided into three steps. Step I: Reduction to the minimal case. To prove the existence of ε, we may assume that Hk are minimal C(∂ A )-extensions of Sk (k = 1, 2). To see this, let Kk are arbitrary C(∂ A )-extensions of Sk , and set Hk = C(∂ A ) ·Kk Sk , which is minimal. Then H1 is a reducing submodule of K1 , whence the projection PH1 ∈ B(K1 ) is a C(∂ A )module homomorphism, as is the inclusion map i K2 : H2 → K2 (for trivial reasons). Assuming the existence-assertion of the theorem to hold for the minimal extensions Hk , we may set ε (X ) = i K2 ◦ ε(X ) ◦ PK1 ∈ HomC(∂ A ) (K1 , K2 )

(X ∈ Hom A (S1 , S2 ))

to obtain an extension mapping ε : Hom A (S1 , S2 ) → HomC(∂ A ) (K1 , K2 ) in the general case. To finish the first step, we have to justify that replacing K2 with H2 does not impose any restriction on the uniqueness assertion either: But if H1 is assumed to be minimal (as it is in the uniqueness part), then every  X ∈ HomC(∂ A ) (H1 , K2 ) satisfies  X (C(∂ A ) · S1 ) ⊂ C(∂ A ) · S2 and thus has range in H2 = C(∂ A ) · S2 , and may

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actually considered an element of HomC(∂ A ) (H1 , H2 ). So in what follows, we may and shall assume Hk to be a minimal extension of Sk (k = 1, 2). Step II: Reduction to the Shilov-case. According to Theorem 4.3, we find invertible Hilbert-C(∂ A )-module maps Sk : Hk → Kk

(k = 1, 2)

onto contractive Hilbert-C(∂ A )-modules. Since restrictions Sk |Sk : Sk → Sk Sk are also invertible A-module maps, the vertical maps in the following diagram Hom A (S1 , S2 )

ε

HomC(∂ A ) (H1 , H2 )

C

Hom A (S1 S1 , S2 S2 )

εS

 C

HomC(∂ A ) (K1 , K2 ),

which are defined as “conjugations” C(X ) = (S2 |S2 ) ◦ X ◦ (S1 |S1 )−1  X ◦ S1−1 C( X ) = S2 ◦ 

(X ∈ Hom A (S1 , S2 )) ( X ∈ HomC(∂ A ) (H1 , H2 ))

are bijections. Assuming the theorem to hold for Shilov-modules and their minimal extensions, there is an extension map ε S from Hom A (S1 S1 , S2 S2 ) to HomC(∂ A ) (K1 , K2 ) as stated in the theorem, which we take as the lower horizontal map of the diagram. We define the upper horizontal map so to make the diagram commutative. To state it explicitly,   ε(X ) = S2−1 ◦ ε S (S2 |S2 ) ◦ X ◦ (S1 |S1 )−1 ◦ S1 (X ∈ Hom A (S1 , S2 )). The stated extension property ε(X )|S1 = X for X ∈ Hom A (S1 , S2 ) is obvious. Moreover, using the diagram, it is easy to see that ε is unique with the stated extension property if and only if so is ε S with the corresponding property. Step III: The Shilov-case with minimal extensions. So we may finally assume that Sk are Shilov-modules with minimal C(∂ A )-extensions Hk for k = 1, 2. By Lemma 4.2 (a) and the subsequent remarks on A-module homomorphisms, the statement of the theorem is nothing but a reformulation of Theorem 3.1. Thus, the proof is complete.   The existence of a lifting map of the above type implies by standard arguments the existence of an “Hom-isomorphism theorem” for hypo-Shilov-modules and their complements (cp. [21] and [9]). To give an appropriate formulation (see Theorem 6.1 below), we need the concept of projectivity, which is introduced and studied for A(D)modules over product domains in the next section.

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5 Projectivity and Injectivity Over Product Domains Let H A denote the category of Hilbert-A-modules. An object H ∈ H A is called projective, if for every pair of objects H1 , H2 ∈ H A together with A-module maps X 2 : H → H2 and X 21 : H1 → H2 with X 21 onto, there exists an A-module map X 1 : H → H1 such that the following diagram is commutative

∃X 1

H

X2

H1 X 21

H2 .

A module H ∈ H A is called injective, if for every pair H1 , H2 ∈ H A with A-module maps Y1 : H1 → H and Y12 : H1 → H2 such that Y12 is injective with closed range, there exists an A-module map Y2 : H2 → H such that Y1 = Y2 ◦ Y12 . We refer to [7] for the homological background. Proposition 2.1.5 therein relates the characterization of injective and projective objects to the vanishing of cohomology groups Ext1H A (K, H) defined as follows: Two short exact sequences A

A

B

B

E : 0 −→ H −→ J −→ K −→ 0 and E  : 0 −→ H −→ J −→ K −→ 0 in the category H A are called equivalent if there exists a morphism X ∈ Hom A (J, J ) making the diagram E :0

H

A

J

A

J

B

K

0

K

0

X

E

:0

H

B

commutative. The first cohomology group is then defined by  A B Ext1H A (K, H) = [E]; E : 0 → H → J → K → 0 is an exact sequence in H A . The zero element of Ext1H A (K, H) is the split extension iH

PK

0 −→ H −→ H ⊕ K −→ K −→ 0, where i H and PK denote the canonical inclusion and projection, respectively. In terms of Ext-groups, an element K ∈ H A is projective, if and only if Ext1H A (K, H) = 0 for every H ∈ H A , and injective, if and only if Ext1H A (H, K) = 0 for every H ∈ H A . It was shown by Carlson and Clark [8] for D = Dn , by Guo [21] for D = Bn (under an additional continuity assumption on the underlying modules related to the surface measure on ∂Bn ) and by Eschmeier and the author [13] in full generality for every strictly pseudoconvex bounded open set and every bounded symmetric domain

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D ⊂ Cn that Ext1H A (K, H) = 0 and Ext1H A (H, K) = 0, whenever H is a HilbertC(S D )-module. The main result of this section says that this remains true in the product setting we studied in Sect. 2: Theorem 5.1 Let k ∈ N and Di ⊂ Cdi be bounded open sets with di ∈ N for 1 ≤ i ≤ k, each of which is either a strictly pseudoconvex domain with C 5 boundary or a bounded symmetric and circled domain, and let D = D1 × · · · × Dk be the product domain. If H is a Hilbert-C(S D )-module, then we have Ext1H A(D) (K, H) = 0

and

Ext1H A(D) (H, K) = 0

for every Hilbert-A(D)-module K. To prove this, we basically use an emedding argument similar to the one used to deduce the bounded symmetric case from the strictly pseudoconvex one in [13] (see the remarks preceding Corollary 3.2 therein). While the embedding used in the latter case was the trivial one (namely, the inclusion of a bounded symmetric and circled domain into its envelopping ball), the construction of a suitable embedding is the non-trivial part in the product setting under consideration. In preparation of the main construction we show the following auxiliary lemma: Lemma 5.2 Let D1 ⊂ Cm and D2 ⊂ Ck be bounded open sets and assume that D2 is convex. Then, the mapping S : A(D1 × D2 ) −→ A(D2 , A(D1 )),

(S f )(z 2 ) = f (·, z 2 ) for z 2 ∈ D2

is an isometric isomorphism. Here, A(D2 , A(D1 )) ⊂ C(D 2 , A(D1 )) is the closed subspace consisting of those functions that are analytic on D2 , viewed as functions with values in the Banach space A(D1 ). Proof It is elementary to check that S, viewed as a map A(D1 × D2 ) → C(D 2 , A(D1 )) is a well-defined isometry. To see that it actually maps into A(D2 , A(D1 )), we have to make sure that S f |D2 is analytic as a function with values in A(D1 ). Since A(D1 ) isometrically embeds as a subspace into H ∞ (D1 ) (by restriction), it suffices to check the holomorphy of S f |D2 , viewed as a map D2 → H ∞ (D1 ). But this follows from [12, Lemma 5.4], which settles the corresponding identification H ∞ (D1 × D2 ) ∼ = H ∞ (D2 , H ∞ (D1 )). So the map S from the statement of the theorem is actually a well-defined isometry. To conclude the proof, it suffices to check that S is surjective. Towards this, let f ∈ A(D2 , A(D1 )) be given. Applying a suitable translation, we may assume that D2 contains the origin. Using the uniform continuity of f , one shows that, for r ↑ 1 (0 < r < 1), the functions fr defined by fr (z) = f (r z) which are holomorphic on the open set Dr = (1/r ) · D2 ⊃ D 2 converge to f uniformly on D 2 . Thus we can choose a sequence (rn ) in such a way that the functions f n = frn ∈ O(Drn , A(D1 )) satisfy f − f n ∞,D 2
1). The latter case becomes more interesting because of two reasons – the holomorphic function theory is much more complicated for N > 1 and the Beurling theorem characterizing invariant subspaces is no longer true. The structure of the paper is as follows. In Sect. 2, we present some natural conjugations in L 2 (T N ) and give their basic properties. In Sect. 3, the characterization of conjugations intertwining with multiplication operators and its adjoints is settled. Sect. 4 presents a characterization of conjugations commuting with multiplication operators. Sects. 5, 6, 7 are devoted to conjugations and invariant subspaces. In Sect. 5, it is shown that there are no conjugations intertwining multiplication operators and its adjoint and leaving any subspace of H 2 (D N ) invariant. In Sects. 6 and 7, we characterize all conjugations commuting with multiplication operators and leaving invariant given invariant subspace of H 2 (D N ). Sect. 6 deals with invariant subspaces of type θ H 2 (D N ), where θ is an inner function. In Sect. 7, we consider subspaces fulfilling property (*), introduced in [6], which are different from those considered in Sect. 6.

2 Preliminaries For a vector z = (z 1 , . . . , z N ) ∈ C N , we define its conjugation by z¯ = (z 1 , . . . , z N ).

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In the space L 2 (T N ) there are two conjugations, which seems very natural. First of them is J , which is simply a conjugation of the function, i.e., J f = f¯,

f ∈ L 2 (T N ).

(2.1)

The second one is J # , which derives from conjugating Fourier coefficients of functions in L 2 (T N ). Precisely, J # f = f #,

f ∈ L 2 (T N ),

(2.2)

where f # (z) = f (¯z ), z ∈ T N . The conjugation J # keeps the Hardy space invariant (J # H 2 (D N ) = H 2 (D N )). On the other hand, J H 2 (D N ) = H 2 (D N ). Let M be a subspace of H 2 (D N ). We denote the set J # M by M# . Moreover, if M is closed, then M is called invariant if z i M ⊂ M for i = 1, . . . , N . Let us summarize some basic properties below. Proposition 2.1 Let M be a subspace of H 2 (D N ). Then: (a) M# ⊂ H 2 (D N ), (b) If M is invariant, then M# is also invariant, (c) J M ⊂ H 2 (D N ). The behaviour of J and J # with respect to the multiplication operators Mzi , i = 1, . . . , N , is also completely different. Precisely, J Mzi = Mz¯i J

and

J # Mzi = Mzi J # for i = 1, . . . , N .

(2.3)

Let ψ ∈ L ∞ (T N ). Then (Mψ J )Mzi = Mz¯i (Mψ J ) and (Mψ J # )Mzi = Mzi (Mψ J # ) for i = 1, . . . , N . (2.4) A unimodular function ψ ∈ L ∞ (T N ) is called symmetric if ψψ # = 1 or equivalently ψ(z) = ψ(¯z ) for z ∈ T N . Now, we set apart functions ψ ∈ L ∞ (T N ) such that operators Mψ J and Mψ J # are conjugations. Proposition 2.2 Let ψ ∈ L ∞ (T N ). Then: (a) C = Mψ J is a conjugation if and only if ψ is a unimodular function, (b) C = Mψ J # is a conjugation if and only if ψ is a unimodular symmetric function.

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Proof We give a proof only for (b). Assume that C = Mψ J # is a conjugation. Then, for any f , g ∈ L 2 (T N ), we have  g f¯ dm = g, f  = C f , Cg = ψ f # , ψ g #   = |ψ(z)|2 f (¯z )g(¯z ) dm(z)  = |ψ(¯z )|2 f (z)g(z) dm(z), z ∈ T N . Hence |ψ| = 1 a.e. on T N . For all f ∈ L 2 (T N ) we have f = C 2 f = Mψ J # Mψ J # f = Mψ J # (ψ f # ) = ψψ # f , which implies that ψψ # = 1 a.e. on T N . Therefore, ψ is symmetric, i.e., ψ(z) = ψ(¯z )  a.e. on T N . The reverse implication is clear. Let α ∈ L ∞ (T N ) be a unimodular function. If α ∈ H 2 (D N ), then we say that α is inner. We say that an inner function α is irreducible if it is not a product of two non-constant inner functions. For any inner functions α, β, we write α ≤ β if βα is inner. In such case β H 2 (D N ) ⊂ α H 2 (D N ). Finally, we say that α is self–reflected if α ≤ α # and α # ≤ α.

3 Characterization of Mz –Conjugations Consider a conjugation C on L 2 (T N ). It is called Mz –conjugation1 if Mzi C = C Mz¯i , for i = 1, . . . , N . Observe that the natural conjugation J on L 2 (T N ) defined by (2.1) is an Mz -conjugation. Let us define a space L∞ (T N ) = {Mϕ : ϕ ∈ L ∞ (T N )}. It will be used in the proof of the next theorem, which completely characterizes all Mz –conjugations on L 2 (T N ). Theorem 3.1 Let C be a conjugation on L 2 (T N ). The following conditions are equivalent: (a) (b) (c) (d)

C is Mz –conjugation, Mϕ C = C Mϕ¯ for all ϕ ∈ L ∞ (T N ), there is a unimodular function ψ ∈ L ∞ (T N ) such that C = J Mψ , there is a unimodular function ψ  ∈ L ∞ (T N ) such that C = Mψ  J .

Note that function ψ  from condition (d) corresponds to function ψ¯ from condition (c). Proof Equivalence of (a) and (b) is a consequence of standard approximation procedure. Condition (c) is equivalent to (d). In fact, it is enough to show that (b) implies (c). 1 We adopt the notions of M -conjugation and M –commuting conjugation from [4] and [5] for multidiz z

mensional case.

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If ϕ ∈ L ∞ (T N ), then J C Mϕ = J Mϕ¯ C = Mϕ J C. Thus J C is a bounded linear operator and J C commutes with all Mϕ ∈ L∞ (T N ). By [10, Theorem 1.20], L∞ (T N ) is a m.a.s.a. (maximal abelian selfadjoint algebra) and as a consequence J C ∈ L∞ (T N ). Therefore, there is a function ψ ∈ L ∞ (T N ) such that J C = Mψ . Finally, C = J Mψ . 

4 Characterization of Mz –Commuting conjugations Let C be a conjugation on L 2 (T N ). C is called Mz –commuting conjugation if Mzi C = C Mzi , i = 1, . . . , N . Notice that the natural conjugation J # on L 2 (T N ) defined by (2.2) is Mz –commuting conjugation. The theorem below completely characterizes all Mz –commuting conjugations on L 2 (T N ). Theorem 4.1 Let C be a conjugation on L 2 (T N ). The following conditions are equivalent: (a) (b) (c) (d)

C is an Mz –commuting conjugation, Mϕ C = C Mϕ # for all ϕ ∈ L ∞ (T N ), there is a symmetric unimodular function ψ ∈ L ∞ (T N ) such that C = J # Mψ , there is a symmetric unimodular function ψ  ∈ L ∞ (T N ) such that C = Mψ  J # .

Note that function ψ  from condition (d) corresponds to function ψ # from condition (c). Proof Assume (a) and note that Mϕ C = C Mϕ # if ϕ is a polynomial. Now, equivalence of (a) and (b) is a consequence of standard approximation procedure. Clearly, condition (c) is equivalent to (d). It remains to show the implication from (b) to (c). If ϕ ∈ L ∞ (T N ), then J # C Mϕ = J # Mϕ # C = Mϕ J # C. Using again the fact that L∞ (T N ) is a m.a.s.a., we get J # C ∈ L∞ (T N ). Therefore, there is a function ψ ∈ L ∞ (T N ) such that J # C = Mψ . Hence C = J # Mψ = Mψ # J # . By Proposition 2.2, the function ψ is unimodular and symmetric. 

5 Invariant Subspaces of H 2 (DN ) and Mz –Conjugations In this section, we study the image of a given invariant subspace of the Hardy space under Mz –conjugations. Theorem 5.1 Let M ⊂ H 2 (D N ) be a non–zero invariant subspace. Then there does not exist Mz –conjugation C such that CM ⊂ H 2 (D N ).

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Proof Let C be an Mz –conjugation such that CM ⊂ H 2 (D N ). Let n 1 , . . . , n N be non–negative integers. Since z i M ⊂ M, we get z 1n 1 · · · z nNN M ⊂ M. From the fact that C is an Mz –conjugation, we deduce   z¯ 1n 1 · · · z¯ nNN CM = C z 1n 1 · · · z nNN M ⊂ CM ⊂ H 2 (D N ). Finally, CM ⊂ z 1n 1 · · · z nNN H 2 (D N ). Therefore, CM ⊂



z 1n 1 · · · z nNN H 2 (D N ) = {0}

(n 1 ,...,n N )∈N N

and as a consequence M = {0}.



Corollary 5.2 Let M ⊂ H 2 (D N ) be a non–zero invariant subspace. Then there does not exist Mz –conjugation C such that CM ⊂ M. Corollary 5.3 There are no Mz –conjugations on L 2 (T N ) which preserve H 2 (D N ).

6 Invariant Subspaces of Type ˛H 2 (DN ) and Mz –Commuting Conjugations Let α ∈ H 2 (D N ) be an inner function. Then α H 2 (D N ) is invariant. We will now investigate conjugations in L 2 (T N ) which preserve subspaces of this form. Let α, β be two inner functions. Since H 2 (D N ) is invariant for J # , the operator Mβ J # Mα¯ = Mβα # J # : L 2 (T N ) → L 2 (T N ) is an antilinear isometry which maps α H 2 (D N ) onto β H 2 (D N ) and commutes with Mzi , 1, . . . , N . However, this operator does not have to be an involution. By Proposition 2.2, the necessary and sufficient condition is that βα # is symmetric (or equivalently αα # = ββ # ). The theorem below characterizes all Mz –commuting conjugations mapping one invariant subspace of the considered type into another. Theorem 6.1 Let θ , α ∈ H 2 (D N ) be two inner functions. The following conditions are equivalent: (a) there exists an Mz −commuting conjugation C such that C(α H 2 (D N )) ⊂ θ H 2 (D N ),

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(6.1)

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(b) there exists an inner function β ∈ H 2 (D N ) such that θ ≤ β, αα # = ββ # . Moreover, every conjugation C fullfiling condition (a) has a form C = Mα # β J # for some β ∈ H 2 (D N ) such that θ ≤ β, αα # = ββ # , and C(α H 2 (D N )) = β H 2 (D N ).

Proof Assume firstly that C is an Mz –commuting conjugation satisfying (6.1). By Theorem 4.1, C = Mψ J # for some unimodular symmetric function ψ ∈ L ∞ (T N ). In particular, θ H 2 (D N ) C(α) = Mψ J # α = ψα # , and there exists u ∈ H 2 (D N ) such that ψα # = θ u. Note that u must be inner and so ψ = βα # with β = θ u, θ ≤ β. Hence C = Mα # β J # . Moreover, βα # is symmetric, i.e., ββ # = αα # . Now, let us assume that (b) holds. We define

C = Mα # β J # . The condition ββ # = αα # guarantees that C is a conjugation. Then, for any f ∈ H 2 (D N ), since θ ≤ β, we have C(α f ) = Mβα # J # (α f ) = Mβα # (α # f # ) = β f # ∈ β H 2 (D N ) ⊂ θ H 2 (D N ).  H 2 (D N )

which are The corollary below characterizes all possible subspaces of images of α H 2 (D N ) under some Mz –commuting conjugation. We can take θ = 1 in Theorem 6.1 (b) to get the following. Corollary 6.2 Let α ∈ H 2 (D N ) be an inner function and let M ⊂ H 2 (D N ) be a closed subspace. Then the following conditions are equivalent: (a) there exists an Mz –commuting conjugation C such that C(α H 2 (D N )) = M, (b) there exists an inner function β ∈ H 2 (D N ) such that ββ # = αα # and M = β H 2 (D N ) The another consequence of Theorem 6.1 is the following corollary. Corollary 6.3 Let α be an inner function and let C be an Mz –commuting conjugation in L 2 (T N ). Then: (a) C(α H 2 (D N )) ⊂ α H 2 (D N ) if and only if C = Mαα # J # , (b) C(α H 2 (D N )) ⊂ α # H 2 (D N ) if and only if C = J # . Moreover, if α is self-reflected, then C(α H 2 (D N )) ⊂ α H 2 (D N ) if and only if C = J # . Proof By Theorem 6.1, C(α H 2 (D N )) ⊂ α H 2 (D N ) if and only if there exists an inner function β such that α ≤ β and ββ # = αα # . Thus β = uα for some inner function u and uαu # α # = αα # . Hence uu # = 1. This is only possible if u = λ ∈ T since  u ∈ H 2 (D N ). This gives (a). The proof of (b) is similar. In particular, we have the following fact.

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Corollary 6.4 Let C be an Mz –commuting conjugation on L 2 (T N ). Then C(H 2 (D N )) ⊂ H 2 (D N ) if and only if C = J # . Remark 6.5 Let α and θ be two inner functions satisfying condition θ H 2 (D N )  α H 2 (D N ). Then there does not exist Mz –commuting conjugation C such that C(α H 2 (D N )) ⊂ θ H 2 (D N ). Indeed, if so, C(α H 2 (D N )) ⊂ θ H 2 (D N ) ⊂ α H 2 (D). Thus, by Corollary 6.3(a), C = Mαα # J # and θ H 2 (D N ) = α H 2 (D). This is a contradiction. Compare this remark with Example 7.5. Notice that, for given inner functions α, θ , the number of existing Mz –commuting conjugations fulfilling (6.1) can be different. We provide three examples. First of them is based on [11, Theorem 5.5.2.]. Example 6.6 Let c ∈ (0, 18 ) and let α ∈ H 2 (T2 ) be an inner function defined by α(z 1 , z 2 ) =

2z 12 z 2 − z 1 + c 2 − z 1 z 2 + cz 12 z 2

.

Then α # = α. Suppose that C is an Mz –commuting conjugation such that C(α H 2 (T2 )) ⊂ θ H 2 (T2 )

(6.2)

for some inner function θ ∈ H 2 (T2 ). By Theorem 6.1, there is an inner function β such that θ ≤ β and α 2 = ββ # . Note that α is continuous on D2 . Then, by [12, Theorem 2.3.], α 2 has the unique Rudin-Ahern factorization. Since α is irreducible and α 2 = ββ # , β = β # = α. Hence there is only one possible conjugation C fulfilling (6.2), namely C = Mαα # J # = J # . On the other hand, there are only two functions θ (θ = 1 or θ = α) which can satisfy (6.2), since θ ≤ β = α. Example 6.7 Let α1 and α2 be two different inner functions such that • α1 and α2 are irreducible, • α #j = αk for j, k ∈ {1, 2}, • both α1 (z 1 , ·) and α2 (z 1 , ·) are rational for z 1 in some subset of positive measure in T.2 2 Let us recall that f (z , ·) denotes a boundary section of f , i.e., holomorphic function defined as a 1 lim f (r z 1 , ·) in topology of locally uniform convergence on D N . r →1

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We can consider α j (z 1 , z 2 ) =

z 1 z 2 −λ j 1−λ j z 1 z 2

for j = 1, 2, where λ1 , λ2 ∈ D \ R and

λ1 = λ2 , λ1 = λ¯ 2 . These functions satisfy the above three condition. The conditions will allow us to apply [12, Theorem 2.3]. Now, let us take α = α1 α2 and θ = α2# . Note that α H 2 (D) ⊂ θ H 2 (D). We would like to find all Mz –commuting conjugations such that C(α H 2 (D)) ⊂ θ H 2 (D).

(6.3)

Realize that, by the uniqueness of the Rudin-Ahern factorization, there are exactly two inner function satisfying condition (b) in Theorem 6.1: β1 = α1 α2# and β2 = α1# α2# . Applying this theorem we obtain only two conjugations C1 = Mα α # J # and C2 = J # . 1 1

Example 6.8 Consider α as above and θ = 1, i.e., condition C(α H 2 (D)) ⊂ H 2 (D). Then there are exactly four inner functions satisfying condition (b) in Theorem 6.1: β1 = α1 α2 , β2 = α1# α2 , β3 = α1 α2# , and β4 = α1# α2# . Hence, we obtain four possible conjugations: C1 = Mα α α # α # J # , C2 = Mα α # J # , C3 = Mα α # J # , and C4 = J # . 1 2 1 2

2 2

1 1

7 Invariant Subspaces With Condition (*) and Mz –Commuting Conjugations The aim of this section is to study different invariant subspaces fulfilling condition (*) introduced in [6]. This class of subspaces is completely different from those one considered in Sect. 6 if N > 1. Following [11, Chapter 2], we introduce some function theory notation. We define multivariable Poisson kernel by the formula P(z, w) =

N 

Pri (θi − ϕi ), z j = r j eiθ j , w j = eiϕ j ,

i=1

where Pr denotes one dimensional Poisson kernel Pr (θ ) =

1 − r2 . 1 − 2r cos θ + r 2

For a complex Borel measure μ on T N we define its Poisson integral by the formula  P[μ](z) =

TN

P(z, w)dμ(w), z ∈ D N .

Let f ∈ H 2 (D N ). Then there is a real singular measure σ f on T N such that the least harmonic majorant of function log | f |, denoted by u(log | f |), is given by u(log | f |)(z) = P[log | f |dm N + dσ f ](z), z ∈ D N .

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It is known that dσ f ≤ 0 for every f ∈ H 2 (D N ). For the invariant subspace M, following [6], we define the zero set Z (M) = {z ∈ D N : f (z) = 0 for all f ∈ M} and the measure Z ∂ (M) = inf{−dσ f : f ∈ M, f = 0}. The invariant subspace M is said to satisfy condition (*) if Z ∂ (M) = 0 and the real (2N − 2)–dimensional Hausdorff measure of Z (M) is 0. Proposition 7.1 Let M ⊂ H 2 (D N ) be an invariant subspace. If M satisfies condition (*), then M# satisfies condition (*). Proof Let μ be a measure on T N and let μc be defined as follows μc (ω) = μ({w¯ : w ∈ ω}). Notice that m cN = m N . Moreover, 

(r )

h # (z) = = = =

T T T

P N

P N

P N

TN

P

z r z r z r  z¯ r

 , w log | f # (r w)|dm N (w)  , w log | f (r w)|dm ¯ N (w)  , w¯ log | f (r w)|dm N (w)  , w log | f (r w)|dm N (w) = h (r ) (¯z ).

From [11, Theorem 3.2.4] we get the connection between least N –harmonic majorants ) u(log | f |)(z) = lim h (r ) (z) = lim h (r z ) = u(log | f # |)(¯z ). # (¯ r →1−

r →1−

Hence by (7.1) 

 TN

P(z, w)dσ f # (w) =

TN

P(z, w)dσ cf (w), z ∈ D N .

Consequently, by uniqueness of the measure, σ f # = σ cf . Now, one can easily see that Z ∂ (M) = Z ∂ (M# ). The easy observation Z (M# ) = Z (M) finishes the proof.  Theorem 7.2 Let C be an Mz –commuting conjugation on L 2 (T N ). Let M ⊂ H 2 (D N ) be an invariant subspace fulfilling condition (*). If CM ⊂ H 2 (D N ) then there is a symmetric inner function ψ such that C = Mψ J # and CM = ψM# .

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Proof Since C is Mz –commuting conjugation thus by Theorem 4.1 there is a unimodular symmetric function ψ ∈ L ∞ (T N ) such that C = Mψ J # = J # ψ # . Thus ψ # M = J # J # ψ # M = J # CM ⊂ J # H 2 (D N ) = H 2 (D N ). The subspace M is invariant subspace fulfilling condition (*). Moreover ψ # M ⊂ H 2 (D N ). Hence by [6, Theorem 1] function ψ # is holomorphic and in particular inner. Therefore ψ is a symmetric inner function.  Notice that the above theorem can also be proved using Proposition 7.1. Now, for any α ∈ D N , let Hα2 be defined as Hα2 = { f ∈ H 2 (D N ) : f (α) = 0}. In [13, Example 2.9] it was shown that Hα2 is invariant and fulfills property (*). As a consequence of Theorem 7.2 and the fact that J # Hα2 = Hα¯2 for any α ∈ D N , we get the following. Theorem 7.3 Let C be an Mz –commuting conjugation in L 2 (T N ). Let α ∈ D N . If C Hα2 ⊂ H 2 (D N ) then there is an inner symmetric function ψ such that C = Mψ J # and C Hα2 = ψ Hα¯2 . To illustrate the above theorem we give two examples, showing that there exists a lot of conjugations leaving invariant space Hα2 . Example 7.4 Let α ∈ D N ∩ R N . Then there are infinitely many conjugations C on L 2 (T N ) such that C Hα2 ⊂ Hα2 . In fact, there is a one-to-one correspondence between such conjugations and the set of all inner symmetric functions. Example 7.5 Let α = (α1 , . . . , α N ) ∈ D N \ R N . Then there are infinitely many conjugations C on L 2 (T N ) such that C Hα2 ⊂ Hα2 . If ψ is any symmetric inner function such that ψ(α) = 0, then C = Mψ J # is a conjugation such that C Hα2 ⊂ Hα2 . It is clear that, in this case, the inclusion is strict. Compare this example with Remark 6.5. To give a concrete example of ψ, denote by Bα the Blaschke product, i.e., Bα (z) =

N  z j − αj 1 − α¯j z j j=1

for z = (z 1 , . . . , z N ) ∈ D N . Now, take a function ψ defined by ψ(z) = Bα (z)Bα (z)ϕ, z ∈ D N , where ϕ is a symmetric inner function (e.g. ϕ ≡ 1). It is easily seen that ψ is symmetric and C = Mψ J # satisfies C Hα2 ⊂ Hα2 . Data Availability Statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Declarations Conflicts of interest There is no conflicts of interest.

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P. Dymek et al. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

References 1. Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Reports on Progress in Physics 70(6), 947–1018 (2007). https://doi.org/10.1088/0034-4885/70/6/r03 2. Bender, C.M., Boettcher, S.: Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry. Phys. Rev. Lett. 80(24), 5243–5246 (1998). https://doi.org/10.1103/PhysRevLett.80.5243 3. Caliceti, E., Graffi, S., Maioli, M.: Perturbation theory of odd anharmonic oscillators. Communications in Mathematical Physics 75(1), 51–66 (1980). https://doi.org/10.1007/BF01962591 4. Câmara, C., Kli´s-Garlicka, K., Ptak, M.: Asymmetric truncated Toeplitz operators and conjugations. Filomat 33, 3697–3710 (2019). https://doi.org/10.2298/FIL1912697C 5. Câmara, C., Kli´s-Garlicka, K., Łanucha, B., Ptak, M.: Conjugations in L2 and their invariants. Anal. Math. Phis. 10(2), 1–14 (2020). https://doi.org/10.1007/s13324-020-00364-5 6. Douglas, R.G., Yan, K.: On the rigidity of Hardy submodules. Integr. Equat. Op. Theory 13(3), 350–363 (1990). https://doi.org/10.1007/BF01199890 7. Dymek, P., Płaneta, A., Ptak, M.: Conjugations on L 2 space on the real line. In: preprint (2021) 8. Garcia, S.R., Prodan, E., Putinar, M.: Mathematical and phisical aspects of complex symmetric operators. J. Phis. A, Math Theor. 47, 1–54 (2014). https://doi.org/10.1090/S0002-9947-05-03742-6 9. Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. Trans. Amer. Math. Soc. 358, 1285–1315 (2006). https://doi.org/10.1007/s13324-020-00364-5 10. Radjavi, H., Rosenthal, P.: Invariant Subspaces. Springer, Berlin, Heidelberg (1973) 11. Rudin, W.: Function theory in polydisc. W. A. Benjamin Inc., New York, Amsterdam (1969) 12. Sawyer, E.: Good-irreducible inner functions on a polydisc. Annales de l’institut Fourier 29(2), 185– 210 (1979). https://doi.org/10.5802/aif.746 13. Yang, R.: A brief survey of operator theory in H 2 (D2 ). Vol. Handbook of analytic operator theory. Chapman and Hall/CRC, 2019, pp. 223–258. https://doi.org/10.1201/9781351045551 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Complex Analysis and Operator Theory (2022) 16:51 https://doi.org/10.1007/s11785-022-01220-z

Complex Analysis and Operator Theory

Best Approximations in a Class of Lorentz Ideals Quanlei Fang1 · Jingbo Xia2 Received: 24 December 2021 / Accepted: 14 February 2022 / Published online: 2 April 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract +(0) be the We consider the family of Lorentz ideals C + p , 1 ≤ p < ∞. Let C p + +  ·  p -closure of the collection of finite-rank operators in C p . It is well known that

+(0) = C + is proximinal in C + C +(0) p p . We show that C p p . We further show that a classic approximation for Hankel operators (Axler et al. in Ann Math (2) 109, 601–612, 1979) does not generalize to this new context.

Keywords Lorentz ideal · Best approximation Mathematics Subject Classification 41A50 · 47B10 · 47B35

1 Introduction Let X be a Banach space and let M be a closed linear subspace of X . An element x ∈ X is said to have a best approximation in M if there is an m ∈ M such that x − m ≤ x − a for every a ∈ M. The subspace M is said to be proximinal in X if every x ∈ X has a best approximation in M.

Dedicated to the memory of Jörg Eschmeier. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz and Mihai Putinar.

B

Jingbo Xia [email protected] Quanlei Fang [email protected]

1

Department of Mathematics and Computer Science, Bronx Community College, CUNY, Bronx, NY 10453, USA

2

Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA

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One of the most familiar and significant examples of such a pair is the case of X = B(H) and M = K(H), where H is a Hilbert space, B(H) is the collection of bounded operators on H, and K(H) is the collection of compact operators on H. It is well known that K(H) is proximinal in B(H), which is a result in the influential book [11] by Gohberg and Krien. Given any A ∈ B(H), to find its best approximation in K(H), one takes the polar decomposition A = U |A|, where |A| = (A∗ A)1/2 and U is a partial isometry. Then from the spectral decomposition of |A| one easily finds the best compact approximation to A. The moral of this example is that when looking for best approximations for operators on a Hilbert space, one should take advantage of spectral decomposition, which is not available on other Banach spaces. The relation between B(H) and K(H) is that the latter is the closure of the collection of finite-rank operators in the former. On a Hilbert space H, there are many pairs that fit this description, but with different norms. In particular, the norm ideals of Robert Schatten [16] are a good source for interesting examples of X and M. Before getting to these examples, it is necessary to give a general introduction for norm ideals. For this we follow the approach in [11, 19], because it offers the level of generality that is suitable for this paper. As in [11], we write cˆ for the linear space of sequences {a j } j∈N , where a j ∈ R and for every sequence the set { j ∈ N : a j = 0} is finite. A symmetric gauge function is a map  : cˆ → [0, ∞) that has the following properties: (a)  is a norm on c. ˆ (b) ({1, 0, …, 0, . . . }) = 1. (c) ({a j } j∈N ) = ({|aπ( j) |} j∈N ) for every bijection π : N → N. See [11,p. 71]. Each symmetric gauge function  gives rise to the symmetric norm A = sup ({s1 (A), . . . , s j (A), 0, . . . , 0, . . . }) j≥1

for bounded operators, where s1 (A), . . . , s j (A), . . . are the singular numbers of A. On any separable Hilbert space H, the set of operators C = {A ∈ B(H) : A < ∞}

(1.1)

is a norm ideal [11,p. 68]. That is, C has the following properties: • For any B, C ∈ B(H) and A ∈ C , B AC ∈ C and B AC ≤ BA C. • If A ∈ C , then A∗ ∈ C and A∗  = A . • For any A ∈ C , A ≤ A , and the equality holds when rank(A) = 1. • C is complete with respect to  ·  . (0) Given a symmetric gauge function , we define C to be the closure with respect to the norm  ·  of the collection of finite-rank operators in C .

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Both ideals C and C are important in operator theory and operator algebras. For example, if one considers the problem of diagonalization under perturbation for single self-adjoint operators [13, 14] or for commuting tuples of self-adjoint operators (0) [2, 18–21], then the natural perturbing operators come from ideals of the form C . If one studies Toeplitz operators or Hankel operators on various reproducing-kernel Hilbert spaces, then a natural question is the membership of these operators in ideals of the form C [10, 12, 22]. (0) For many symmetric gauge functions, we simply have C = C . For example, if we take any 1 ≤ p < ∞ and consider the symmetric gauge function  p ({a j }) =

 ∞

1/ p |a j | p

, {a j } ∈ c, ˆ

j=1

then the norm ideal C p defined according to (1.1) is simply the familiar Schatten (0)

p-class. It is well known and obvious that C p = C p . From [11] we know that there also are many symmetric gauge functions for which (0) C = C . The most noticeable of such examples is the symmetric gauge function ∞ ({a j }) = sup |a j |, {a j } ∈ c. ˆ j∈N

Obviously, the norm  · ∞ is none other than the ordinary operator norm. Therefore (0) (1.1) gives us C∞ = B(H). It is also obvious that C∞ = K(H). Thus the classic (0)

result that K(H) is proximinal in B(H) can be rephrased as the statement that C∞ is proximinal in C∞ . Once one realizes that, it does not take too much imagination to propose (0)

Problem 1.1 For a general symmetric gauge function  with the property C = C , (0) is C proximinal in C ? In such generality, Problem 1.1 does not appear to be easy, for it simply covers too many ideals of diverse properties. It is not too hard to convince oneself that to determine (0) whether or not C is proximinal in C , one needs to know the specifics of . At this point, we do not see how to get a general answer using only properties (a)-(c) listed (0) above plus the condition C = C . But we are pleased to report that there is a family of symmetric gauge functions of common interest for which we are able to solve Problem 1.1 in the affirmative. Let us introduce these symmetric gauge functions and the corresponding ideals. For each 1 ≤ p < ∞, let + p be the symmetric gauge function defined by the formula |aπ(1) | + |aπ(2) | + · · · + |aπ( j) | , {a j } j∈N ∈ c, ˆ −1/ p + 2−1/ p + · · · + j −1/ p j≥1 1

+ p ({a j } j∈N ) = sup

where π : N → N is any bijection such that |aπ(1) | ≥ |aπ(2) | ≥ · · · ≥ |aπ( j) | ≥ · · · , which exists because each {a j } j∈N ∈ cˆ only has a finite number of nonzero terms. The

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ideal C+p , which is defined by (1.1) using + p , is often called a Lorentz ideal. It is well (0)

known that C+p = C+ [11]. The ideal C+ deserves special mentioning, because it 1

p

is the domain of the Dixmier trace [4, 6], which has wide-ranging connections [3, 7, 8, 12, 17]. (0) The ideals C+p and C + , 1 ≤ p < ∞, are the main interest of this paper. Since they p

will appear so frequently in the sequel, let us introduce a simplified notation. From now on we will write (0)

C+ , C +(0) = C+ and  · + p = C+ p p =  · + p p p

(1.2)

for 1 ≤ p < ∞. Here is our main result: Theorem 1.2 For every 1 ≤ p < ∞, C +(0) is proximinal in C + p p. The result that K(H) is proximinal in B(H) has refinements within specific classes of operators [1]. One such class of operators are the Hankel operators H f : H 2 → L 2 , where H 2 is the Hardy space on the unit circle T ⊂ C. We recall the following: Theorem 1.3 [1,Theorem 3] For each f ∈ L ∞ , the best compact approximation to the Hankel operator H f can be realized in the form of a Hankel operator Hg . In other words, Theorem 1.3 says that H f has a best compact approximation that is of the same kind, a Hankel operator. Using the method in [1], Theorem 1.3 can be easily generalized to Hankel operators on the Hardy space on the unit sphere in Cn . +(0) Since Theorem 1.2 tells us that each C p is proximinal in C + p , we can obviously ask a more refined question along the line of Theorem 1.3: Suppose that we have an operator A in a natural class N , and suppose we know that A ∈ C + p , can we find a +(0)

best C p -approximation to A in the same class N ? In particular, what if N consists of Hankel operators? As we will see, the answer to this last question turns out to be negative. The rest of the paper is organized as follows. We prove Theorem 1.2 in Sect. 2. Then in Sect. 3, we present the above-mentioned negative answer. Namely, we give an example of a Hankel operator on the unit sphere in C2 which is in the ideal C4+ +(0) and which does not have any Hankel operator as its best C4 -approximation. This example requires some explicit calculation, which may be of independent interest.

2 Existence of Best Approximation Recall that the starting domain for every symmetric gauge function  is the space c, ˆ which consists of real sequences whose nonzero terms are finite in number. Our first order of business is to follow the standard practice to extend the domain of  to include every sequence. That is, for any sequence ξ = {ξ j } of complex numbers, we define (ξ ) = sup ({|ξ1 |, |ξ2 |, . . . , |ξk |, 0, . . . , 0, . . . }). k≥1

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It is well known that the properties of  imply that if |a j | ≤ |b j | for every j, then ({a j }) ≤ ({b j }). This fact will be used in many of our estimates below. We will focus exclusively on the symmetric gauge functions + p , 1 ≤ p < ∞. For the rest of the paper, p will always denote a positive number in [1, ∞). Definition 2.1 (1) Write c+ p for the collection of sequences ξ satisfying the condition (ξ ) < ∞. + p + ˆ in c+ (2) Let c+ p (0) denote the  p -closure of {a + ib : a, b ∈ c} p. + + + (3) For each ξ ∈ c p , denote  p,ess (ξ ) = inf{ p (ξ − η) : η ∈ c+ p (0)}. + satisfying the conditions for the collection of sequences x = {x } in c (4) Write d + j p p that x j ≥ 0 and that x j ≥ x j+1 for every j ∈ N. + In other words, d + p consists of the non-negative, non-increasing sequences in c p . Proposition 2.2 For every ξ = {ξ j } ∈ c+ p , we have + + p,ess (ξ ) = lim  p ({ξm+1 , ξm+2 , . . . , ξm+k , . . . }). m→∞

In particular, ξ = {ξ j } ∈ c+ p (0) if and only if lim + p ({ξm+1 , ξm+2 , . . . , ξm+k , . . . }) = 0.

m→∞

Proof From the above definitions it is obvious that + + p,ess (ξ ) ≤ lim  p ({ξm+1 , ξm+2 , . . . , ξm+k , . . . }). m→∞

On the other hand, for any a, b ∈ c, ˆ there exist a ν ∈ N and ζ1 , . . . , ζν ∈ C such that ξ − a − ib = {ζ1 , . . . , ζν , ξν+1 , ξν+2 , . . . , ξν+k , . . . }. Thus for every m ≥ ν we have + + p (ξ − a − ib) ≥  p ({ξm+1 , ξm+2 , . . . , ξm+k , . . . }). + Since c+ ˆ it follows that p (0) is the  p -closure of {a + ib : a, b ∈ c}, + + p,ess (ξ ) ≥ lim  p ({ξm+1 , ξm+2 , . . . , ξm+k , . . . }). m→∞



This completes the proof.

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Proposition 2.3 For every x = {x j } ∈ d + p we have + p,ess (x) = lim sup j→∞

x1 + x2 + · · · + x j . 1−1/ p + 2−1/ p + · · · + j −1/ p

Proof For x = {x j } ∈ d + p , (2.1) trivially holds if ∞ x = ∞. Then for every m ∈ N we have j j=1 lim

j→∞

∞

j=1

(2.1)

x j < ∞. Suppose that

xm+1 + · · · + xm+ j x1+ j + · · · + xm+ j x1 + · · · + xm = 1 − lim + lim = 1. j→∞ x 1 + · · · + x j j→∞ x1 + · · · + x j x1 + · · · + x j

Therefore lim sup j→∞

x1 −1/ 1 p

+ ··· + xj ≤ + p ({x m+1 , x m+2 , . . . , x m+k , . . . }) + · · · + j −1/ p

for every m ∈ N. By Proposition 2.2, this means lim sup j→∞

x1 −1/ 1 p

+ ··· + xj ≤ + p,ess (x). + · · · + j −1/ p

To prove the reverse inequality, note that for each m ∈ N, there is a k(m) ∈ N such that + p ({x m+1 , x m+2 , . . . , x m+k , . . . }) ≤

xm+1 + · · · + xm+k(m) 1 + . (2.2) 1−1/ p + · · · + {k(m)}−1/ p m

If there is a sequence m 1 < m 2 < · · · < m i < · · · in N such that k(m i ) → ∞ as i → ∞, then from Proposition 2.2 and (2.2) we obtain + p,ess (x) ≤ lim sup i→∞

x1 + · · · + x j x1 + · · · + xk(m i ) ≤ lim sup −1/ p . 1−1/ p + · · · + {k(m i )}−1/ p 1 + · · · + j −1/ p j→∞

The only other possibility is that there is an N ∈ N such that k(m) ≤ N for every m ∈ N. Obviously, the membership x ∈ d + p implies lim j→∞ x j = 0. Thus in the case k(m) ≤ N for every m ∈ N, from Proposition 2.2 and (2.2) we obtain + + p,ess (x) = lim  p ({x m+1 , x m+2 , . . . , x m+k , . . . }) = 0 m→∞

≤ lim sup j→∞

x1 −1/ 1 p

+ ··· + xj . + · · · + j −1/ p 

This completes the proof. Proposition 2.4 Let a ξ = {ξ j } ∈ c+ p be given and denote Ni = card{ j ∈ N : 2−i/ p < |ξ j | ≤ 2−(i−1)/ p }

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for every i ∈ N. If lim 2−i Ni = 0,

i→∞

(2.3)

then ξ ∈ c+ p (0). Proof By (2.3), for every k ∈ N, there is a natural number i(k) > k + 3 such that Ni ≤ 2i−k for every i ≥ i(k).

(2.4)

For every i ≥ i(k), we also have card{ j ∈ N : 2i−k ≤ j < 2i+1−k } = 2i+1−k − 1 − 2i−k + 1 = 2i−k . That is, card{ j ∈ N : 2−(i−1)/ p < 2−(k−2)/ p j −1/ p ≤ 2−(i−2)/ p } = 2i−k when i ≥ i(k). Combining this with (2.4), we see that + −(k−2)/ p −1/ p j } j∈N ) = 2−(k−2)/ p . + p,ess (ξ ) ≤  p ({2 + Since this holds for every k ∈ N, we conclude that + p,ess (ξ ) = 0, i.e., ξ ∈ c p (0). 

Proposition 2.5 For each x ∈ d + p , there is a decomposition x = y + z such that with y ∈ d+ p + + p (y) =  p,ess (x)

and z = {z j } ∈ c+ p (0), where z j ≥ 0 for every j ∈ N. + Proof Obviously, it suffices to consider x ∈ d + p with  p,ess (x) = 1. Write x = {x j }. + By the definition of d p , we have x j ≥ 0 and x j ≥ x j+1 for every j ∈ N. We define the desired sequences y = {y j } and z = {z j } inductively, starting with j = 1. If x1 ≤ 1, we define y1 = x1 and z 1 = x1 − y1 = 0. If x1 > 1, we define y1 = 1 and z 1 = x1 − y1 = x1 − 1 > 0. Let ν ≥ 1 and suppose that we have defined 0 ≤ y j ≤ x j and z j = x j − y j for every 1 ≤ j ≤ ν such that the following hold true: for every 1 ≤ j ≤ ν we have

y1 −1/ 1 p

+ · · · + yj ≤ 1, + · · · + j −1/ p

and for each j ∈ {1, . . . , ν} with the property y j < x j we have y1 + · · · + y j = 1. 1−1/ p + · · · + j −1/ p Reprinted from the journal

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Then we define yν+1 and z ν+1 as follows. Suppose that 1−1/ p

y1 + · · · + yν + xν+1 ≤ 1. + · · · + ν −1/ p + (ν + 1)−1/ p

In this case, we define yν+1 = xν+1 and z ν+1 = xν+1 − yν+1 = 0. Suppose that y1 + · · · + yν + xν+1 > 1. 1−1/ p + · · · + ν −1/ p + (ν + 1)−1/ p Since we know that y1 −1/ 1 p

+ · · · + yν ≤ 1, + · · · + ν −1/ p

these two inequalities imply that there is a yν+1 ∈ (0, xν+1 ) such that y1 + · · · + yν + yν+1 = 1. 1−1/ p + · · · + ν −1/ p + (ν + 1)−1/ p This defines yν+1 . We then define z ν+1 = xν+1 − yν+1 , which is greater than 0 in this case. Thus we have inductively defined the sequences y = {y j } and z = {z j } with the properties that y j ≥ 0, z j ≥ 0 and y j + z j = x j for every j ∈ N. That is, x = y + z. The construction above ensures y1 + · · · + y j ≤1 1−1/ p + · · · + j −1/ p

(2.5)

for every j ∈ N. Moreover, the construction ensures that the equality y1 + · · · + y j =1 1−1/ p + · · · + j −1/ p

(2.6)

holds for each j ∈ N with the property y j < x j . Next we show that y = {y j } is a non-increasing sequence, i.e., y j ≥ y j+1 for every j ∈ N. First, consider the case j = 1. If x1 ≤ 1, then by definition y1 = x1 ≥ x2 ≥ y2 , since x1 ≥ x2 . If x1 > 1, then y1 = 1 by definition, and (2.5) gives us 1 + y2 ≤ 1−1/ p + 2−1/ p . Thus in the case x1 > 1 we have y2 ≤ 2−1/ p < 1 = y1 . Now consider any j ≥ 2. If y j = x j , then we again have y j = x j ≥ x j+1 ≥ y j+1 , since x is a non-increasing sequence. If y j < x j , then (2.5) and (2.6) imply y1 + · · · + y j−1 ≤ 1−1/ p + · · · + ( j − 1)−1/ p , y1 + · · · + y j−1 + y j = 1−1/ p + · · · + ( j − 1)−1/ p + j −1/ p , y1 + · · · + y j−1 + y j + y j+1 ≤ 1−1/ p + · · · + ( j − 1)−1/ p + j −1/ p + ( j + 1)−1/ p , from which we deduce y j ≥ j −1/ p > ( j + 1)−1/ p ≥ y j+1 .

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Thus y = {y j } is indeed a non-increasing sequence. Combining this fact with (2.5), + + we conclude that y ∈ d + p with  p (y) ≤ 1. What remains is to show that z ∈ c p (0). + To prove that z ∈ c p (0), we first consider the case where 1 < p < ∞. Define Ni = card{ j ∈ N : z j ≥ 2−i/ p } for each i ∈ N. By Proposition 2.4, to prove that z ∈ c+ p (0), it suffices to show that lim 2−i Ni = 0.

(2.7)

i→∞

Suppose that this failed. Then there would be a δ > 0 and an increasing sequence i1 < i2 < · · · < iν < · · · of natural numbers such that 2−iν Niν ≥ δ for every ν ∈ N.

(2.8)

We will show that this leads to a contradiction. −i ν / p . Write a = + p (x). Let ν and j be any pair of natural numbers such that z j ≥ 2 −i / p Since z j = x j − y j and y j ≥ 0, we have x j ≥ 2 ν . On the other hand, since x is a non-increasing sequence, we have x1 + · · · + x j jxj ≤ −1/ p ≤ a. 1−1/ p + · · · + j −1/ p 1 + · · · + j −1/ p Writing C p = p/( p − 1), the above facts lead to the inequality j2−iν / p ≤ j x j ≤ C p a j ( p−1)/ p . That is, if z j ≥ 2−iν / p , then j ≤ (C p a) p 2iν . For each ν ∈ N, let jν = max{ j ∈ N : z j ≥ 2−iν / p }. By (2.6), the fact z jν > 0 forces y1 + · · · + y jν 1−1/ p

−1/ p

+ · · · + jν

= 1.

Thus x1 + · · · + x jν 1−1/ p

+ ··· +

−1/ p jν

=

y1 + · · · + y jν 1−1/ p

≥1+

+ ··· + Niν 2−iν / p

−1/ p jν

( p−1)/ p

+

z 1 + · · · + z jν 1−1/ p

C p jν Ni ν δ = 1 + p p−1 ≥ 1 + p p−1 , i ν Cpa 2 Cpa Reprinted from the journal

427

−1/ p

+ · · · + jν Niν 2−iν / p ≥1+ C p {(C p a) p 2iν }( p−1)/ p

Q. Fang, J. Xia

where the last ≥ follows from (2.8). Since the above inequality supposedly holds for every ν ∈ N, by Proposition 2.3, it contradicts the condition + p,ess (x) = 1. This proves (2.7). Applying Proposition 2.4, in the case 1 < p < ∞ we have z ∈ c+ p (0). Now consider the case p = 1, which is much more complicated. To prove z ∈ c1+ (0) in this case, pick an > 0. Define the sequences u = {u j } and v = {v j } by the formulas

uj =

⎧ ⎨z j ⎩

0

if z j > x j if z j ≤ x j

and v j =

⎧ ⎨0 ⎩

zj

if z j > x j if z j ≤ x j

,

+ j ∈ N. We have z = u + v by design. Then note that + 1 (v) ≤ 1 (x). Since > 0 + is arbitrary, it suffices to show that u ∈ c1 (0). To prove that u ∈ c1+ (0), consider the set N = { j ∈ N : u j > 0}. If card(N ) < ∞, then we certainly have the membership u ∈ c1+ (0). Suppose that card(N ) = ∞. Then we enumerate the elements in N as a sequence

j(1) < j(2) < · · · < j(k) < · · · . Keep in mind that z j(k) > x j(k) for every k ∈ N. Claim 1 If k1 < k2 < · · · < kν < · · · are natural numbers such that log kν ≥

1 log j(kν ) 2

(2.9)

for every ν ∈ N, then

lim

ν→∞

x j(1) + x j(2) + · · · + x j(kν ) 1−1 + 2−1 + · · · + kν−1

= 0.

(2.10)

Indeed for each ν ∈ N, since z j(kν ) > 0, i.e., y j(kν ) < x j(kν ) , we have x1 + x2 + · · · + x j(kν ) + 2−1 + · · · + { j(kν )}−1 y1 + y2 + · · · + y j(kν ) z 1 + z 2 + · · · + z j(kν ) = −1 + −1 1 + 2−1 + · · · + { j(kν )}−1 1 + 2−1 + · · · + { j(kν )}−1 z j(1) + z j(2) + · · · + z j(kν ) x j(1) + x j(2) + · · · + x j(kν ) ≥ 1 + −1 ≥ 1 + −1 −1 −1 1 + 2 + · · · + { j(kν )} 1 + 2−1 + · · · + { j(kν )}−1 x j(1) + x j(2) + · · · + x j(kν ) 1−1 + 2−1 + · · · + kν−1 = 1 + · −1 · . −1 −1 1 + 2 + · · · + { j(kν )} 1−1 + 2−1 + · · · + kν−1

1−1

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Combining this with (2.9), we find that for ν ≥ 3,

1−1

x j(1) + x j(2) + · · · + x j(kν ) x1 + x2 + · · · + x j(kν ) log kν · ≥1+ · −1 −1 + 2 + · · · + { j(kν )} 2 log j(kν ) 1−1 + 2−1 + · · · + kν−1 x j(1) + x j(2) + · · · + x j(kν ) ≥1+ · . 4 1−1 + 2−1 + · · · + kν−1 (2.11)

It follows from the condition + 1,ess (x) = 1 and Proposition 2.3 that lim sup ν→∞

x1 + x2 + · · · + x j(kν ) ≤ 1. 1−1 + 2−1 + · · · + { j(kν )}−1

(2.12)

Obviously, (2.10) follows from (2.11) and (2.12). This proves Claim 1. Claim 2 Let E 1 , . . . , E s , . . . be finite subsets of N such that lim card(E s ) = ∞.

(2.13)

s→∞

Suppose that log k
k 2 for every k ∈ ∪∞ s=1 E s . Therefore card(E s \Fs ) ≤ m for every s. Thus it follows from (2.13) that

lim

s→∞

  k∈E s

x j(k)

card(E

card(F s ) 1    s ) 1  − = 0. x j(k) i i i=1

k∈Fs

i=1

Since m ∈ N is arbitrary, (2.15) will follow if we can show that

lim sup s→∞

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 k∈Fs

x j(k)

card(F s ) i=1

429

+ (x) 1 ≤ 1 . i m

(2.16)

Q. Fang, J. Xia

For each s ∈ N, since j(k) ≥ mk for every k ∈ Fs and since the sequence x is non-increasing, we have 

x j(k) ≤

k∈Fs



xmk ≤

card(F s )

k∈Fs

xmi ≤

i=1

1 m

mcard(F  s)

xi ≤

i=1

1 m

 mcard(F  s) 1  + 1 (x). i i=1

That is,  k∈Fs

x j(k)

card(F s ) i=1

1 1 ≤ i m

 mcard(F  s ) 1 card(F s ) 1 + 1 (x) i i i=1

(2.17)

i=1

for every s ∈ N. Since card(Fs ) → ∞ as s → ∞, (2.17) implies (2.16). This completes the proof of Claim 2. Having proved Claims 1 and 2, we are now ready to prove the membership u ∈ c1+ (0). Recall that for every j for which u j = 0, we have u j = z j ≤ x j , and that the elements in N = { j ∈ N : u j > 0} are listed as j(1) < j(2) < · · · < j(k) < · · · . Since x is non-increasing, by Proposition 2.3, the membership u ∈ c1+ (0) will follow if we can show that x j(1) + x j(2) + · · · + x j(k) = 0. 1−1 + 2−1 + · · · + k −1

lim

k→∞

Suppose that this limit did not hold. Then there would be a sequence n1 < n2 < · · · < ns < · · · of natural numbers such that lim

s→∞

x j(1) + x j(2) + · · · + x j(n s ) 1−1 + 2−1 + · · · + n −1 s

=b

(2.18)

for some b > 0. Again, we will show that this leads to a contradiction. For each s ∈ N, define As = {k ∈ {1, 2, . . . , n s } : log k ≥ (1/2) log j(k)}. If s is such that As = ∅, we define G s = {1, 2, . . . , n s }. If s is such that As = ∅, we let ks be the largest element in As and we define G s = {1, 2, . . . , n s }\{1, 2, . . . , ks }.

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Note that for each s, the definition of G s guarantees that log k < (1/2) log j(k) if k ∈ G s . Denote = {s ∈ N : As = ∅}. For each s ∈ , define αs =

1−1 + 2−1 + · · · + ks−1

1−1 + 2−1 + · · · + n −1 s

card(G s )

and βs =

i −1 i=1 , 1−1 + 2−1 + · · · + n −1 s

where βs is understood to be 0 in the case G s = ∅. For s ∈ with G s = ∅, we have  x j(1) + x j(2) + · · · + x j(ks ) + k∈G s x j(k) x j(1) + x j(2) + · · · + x j(n s ) = 1−1 + 2−1 + · · · + n −1 1−1 + 2−1 + · · · + n −1 s s  x j(1) + x j(2) + · · · + x j(ks ) k∈G s x j(k) = αs + β . s  card(G s ) −1 1−1 + 2−1 + · · · + ks−1 i i=1 (2.19) Suppose that = ∅. Then = {s ∈ N : s ≥ } for some ∈ N. Thus there is a sequence s1 < s2 < · · · < sr < · · · contained in such that both limits lim αsr and

lim βsr

r →∞

r →∞

exist. By definition, log ks ≥ (1/2) log j(ks ). By Claim 1, we have lim

r →∞

x j(1) + x j(2) + · · · + x j(ksr ) 1−1 + 2−1 + · · · + ks−1 r

= 0 in the event

lim αsr = 0.

r →∞

Recall that if k ∈ G s , then log k < (1/2) log j(k). By Claim 2, we have  k∈G s x j(k) lim card(Gr ) = 0 in the event lim βsr = 0. sr r →∞ r →∞ i −1 i=1 Combining these facts with (2.19), we find that lim

r →∞

x j(1) + x j(2) + · · · + x j(n sr ) 1−1 + 2−1 + · · · + n −1 sr

= 0,

which contradicts (2.18) in the case = ∅. Suppose that = ∅. Then by definition we have G s = {1, 2, . . . , n s } for every s ∈ N. Thus we can apply Claim 2 to conclude that lim

s→∞

x j(1) + x j(2) + · · · + x j(n s ) 1−1

+ 2−1

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+ · · · + n −1 s

= lim

s→∞

431

 k∈G s

x j(k)

card(G s) i=1

1 = 0. i

Q. Fang, J. Xia

Thus (2.18) is also contradicted in the case = ∅. This completes the proof of the proposition.  Having only dealt with sequences so far, we now apply the above results to operators, which are the main interest of the paper. Let H be a Hilbert space. For any u, v ∈ H, the notation u ⊗ v denotes the operator on H defined by the formula u ⊗ v f =  f , vu,

f ∈ H.

It is well known that if A is a compact operator on an infinite-dimensional Hilbert space H, then it admits the representation A=

∞ 

s j (A)u j ⊗ v j ,

j=1

where {u j : j ∈ N} and {v j : j ∈ N} are orthonormal sets in H. See, e.g., [5, 11]. We remind the reader of our notation (1.2). For each each A ∈ C + p , we define + +(0) A+ }. p,ess = inf{A − K  p : K ∈ C p + We think of A+ p,ess as the essential  ·  p -norm of A, hence the notation.

Proposition 2.6 For every operator A ∈ C + p , we have + A+ p,ess =  p,ess ({s j (A)}) = lim sup j→∞

s1 (A) + s2 (A) + · · · + s j (A) . 1−1/ p + 2−1/ p + · · · + j −1/ p

Proof For an A ∈ C + p , there are orthonormal sets {u j : j ∈ N} and {v j : j ∈ N} such that A=

∞ 

s j (A)u j ⊗ v j .

j=1 + Therefore it is obvious that A+ p,ess ≤  p,ess ({s j (A)}). To prove the reverse inequality, for every k ∈ N we define the orthogonal projection

Ek =

∞ 

u j ⊗ u j.

j=k

If F is a finite-rank operator, then E k F+ p → 0 as k → ∞. Therefore + + A − F+ p ≥ lim sup E k (A − F) p = lim sup E k A p k→∞

= lim

k→∞

k→∞ +  p ({sk (A), sk+1 (A), . . . , sk+ j (A), . . . })

432

= + p,ess ({s j (A)}),

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where the last = follows from Proposition 2.2. Since this inequality holds for every + finite-rank operator F, we conclude that A+ p,ess ≥  p,ess ({s j (A)}). Recalling Proposition 2.3, the proof is complete.  With the above preparation, we now prove Theorem 1.2 in a more explicit form: +(0)

Theorem 2.7 For each A ∈ C + p , there is a K ∈ C p + A − K + p = A p,ess = lim sup j→∞

such that

s1 (A) + s2 (A) + · · · + s j (A) . 1−1/ p + 2−1/ p + · · · + j −1/ p

Proof Given an A ∈ C + p , we again represent it in the form A=

∞ 

s j (A)u j ⊗ v j ,

j=1

where {u j : j ∈ N} and {v j : j ∈ N} are orthonormal sets. Applying Proposition 2.5 + to the sequence {x j } = {s j (A)}, we obtain y = {y j } ∈ d + p and z = {z j } ∈ c p (0) such that s j (A) = y j + z j

(2.20)

+ for every j ∈ N and + p (y) =  p,ess ({s j (A)}). Define

K =

∞ 

z ju j ⊗ vj.

j=1 +(0) . From (2.20) we obtain The condition z ∈ c+ p (0) obviously implies K ∈ C p

A−K =

∞ 

yju j ⊗ vj.

j=1

Therefore + + A − K + p =  p (y) =  p,ess ({s j (A)}).

Now an application of Proposition 2.6 completes the proof.



3 A Contrast to the Classic Case As we mentioned in the Introduction, the result that K(H) is proximinal in B(H) has refinements within specific classes of operators. One such class of operators are

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the Hankel operators H f : H 2 → L 2 , where H 2 is the Hardy space on the unit circle T ⊂ C. Specifically, [1,Theorem 3] tells us that for f ∈ L ∞ , the best compact approximation to the Hankel operator H f : H 2 → L 2 can be realized in the form of a Hankel operator Hg . In other words, [1,Theorem 3] says that H f has a best compact approximation that is of the same kind, a Hankel operator. Using the method in [1], this result of best compact approximation can be easily generalized to Hankel operators on the Hardy space H 2 (S) on the unit sphere S ⊂ Cn . The fact that each C +(0) is proximinal in C + p p raises an obvious question: Suppose that we have an operator A in a natural class N , and suppose we know that A ∈ C + p, +(0)

can we find a best C p -approximation to A in the same class N ? In particular, what if N is the class of Hankel operators on H 2 (S)? In this section we show that the answer to the last question is negative. This negative answer provides a sharp contrast to the classic result [1,Theorem 3]. For the rest of the paper we assume n ≥ 2. Let S denote the unit sphere {z ∈ Cn : |z| = 1} in Cn . Write dσ for the standard spherical measure on S with the normalization σ (S) = 1. Recall that the Hardy space H 2 (S) is the norm closure of the analytic polynomials C[z 1 , . . . , z n ] in L 2 (S, dσ ) [15]. Let P : L 2 (S, dσ ) → H 2 (S) be the orthogonal projection. Then the Hankel operator H f : H 2 (S) → L 2 (S, dσ ) is defined by the formula H f h = (1 − P)( f h), h ∈ H 2 (S). For these Hankel operators, let us recall the following results: + Proposition 3.1 [9,Proposition 7.2] If f is a Lipschitz function on S, then H f ∈ C2n .

Proposition 3.2 When the complex dimension n is at least 2, for any f ∈ L 2 (S, dσ ), +(0) / C2n . if H f is bounded and if H f = 0, then H f ∈ Proof We apply [9,Theorem 1.6] which tells us that for f ∈ L 2 (S, dσ ), if H f is bounded and if H f = 0, then there is an > 0 such that s1 (H f ) + · · · + sk (H f ) ≥ k (2n−1)/2n + for every k ∈ N. Thus it follows from Proposition 2.6 that H f + 2n,ess > 0, if H f 2n +(0) is finite to begin with. In any case, we have H f ∈ / C2n .



As usual, we write z 1 , . . . , z n for the complex coordinate functions. Here is the main technical result of the section: + Theorem 3.3 When the complex dimension n equals 2, we have Hz¯ 1 + 4 > Hz¯ 1 4,ess .

This leads to the negative answer promised above: Example 3.4 Let the complex dimension n be equal to 2. By Theorem 2.7, Hz¯ 1 has a best +(0) approximation in C4 . On the other hand, it follows from the inequality Hz¯ 1 + 4 > 434

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Hz¯ 1 + 4,ess that if K ∈ C4 +(0) C4

is a best approximation of Hz¯ 1 , then K = 0. The

implies that K is not a Hankel operator, for Proposition 3.2 membership K ∈ +(0) tells us that C4 does not contain any nonzero Hankel operators on H 2 (S) in the case S ⊂ C2 . Thus for the class of Hankel operators on the Hardy space H 2 (S), S ⊂ C2 , +(0) the analogue of Theorem 1.3 does not hold for the pair C4+ and C4 , even though +(0) + is proximinal in C4 . C4 Having presented the principal conclusion of the section, we now turn to the proof of Theorem 3.3, which requires some calculation. We begin with the generality n ≥ 2, and then specialize to the complex dimension n = 2. We need to make one use of Toeplitz operators, whose definition we now recall. Given an f ∈ L ∞ (S, dσ ), the Toeplitz operator T f is defined by the formula T f h = P( f h), h ∈ H 2 (S). We need the following relation between Hankel operators and Toeplitz operators: We have H ∗f H f = T| f |2 − T f¯ T f

(3.1)

for every f ∈ L ∞ (S, dσ ). We follow the usual multi-index convention [15,p. 3]. Then the standard orthonormal basis {eα : α ∈ Zn+ } for H 2 (S) is given by the formula  eα (z) =

(n − 1 + |α|)! (n − 1)!α!

1/2

z α , α ∈ Zn+ .

Consider the symbol function z¯ 1 . Straightforward calculation using (3.1) shows that Hz¯∗1 Hz¯ 1 eα , eβ  = 0 if α = β and that Hz¯∗1 Hz¯ 1 eα , eα  =

n − 1 + |α| − α1 for every α ∈ Zn+ , (n − 1 + |α|)(n + |α|)

(3.2)

where α1 denotes the first component of α. Thus Hz¯∗1 Hz¯ 1 is a diagonal operator with respect to the standard orthonormal basis {eα : α ∈ Zn+ }, and the above are the snumbers of Hz¯∗1 Hz¯ 1 . Consequently, the s-numbers of Hz¯ 1 are a descending arrangement of 

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1/2 , α ∈ Zn+ .

Q. Fang, J. Xia

Lemma 3.5 In the case where the complex dimension n equals 2, we have −1/4 . Hz¯ 1 + 4,ess = 6

Proof For α = (α1 , α2 ) ∈ Z2+ , note that |α| − α1 = α2 . Thus from (3.2) we obtain (Hz¯∗1 Hz¯ 1 )1/2

=

  α∈Z2+

1 + α2 (1 + |α|)(2 + |α|)

1/2 eα ⊗ eα .

+(0)

It is also easy to see that (Hz¯∗1 Hz¯ 1 )1/2 = Y + Z , where Z ∈ C4 

Y =

α∈Z2+ \{0}

and

√ α2 eα ⊗ eα . |α|

+ ∗ 1/2 + Hence Hz¯ 1 + 4,ess = (Hz¯ 1 Hz¯ 1 ) 4,ess = Y 4,ess , and we need to figure out the latter. 2 To find Y + 4,ess , consider Q = {(x, y) ∈ R : x ≥ 0 and y ≥ 0}, the first quadrant in the x y-plane. For each a > 0, define

E a = {(x, y) ∈ Q : ay ≥ (x + y)2 }. Solving the inequality ay ≥ (x + y)2 in Q, we find that E a = {(x, y) ∈ Q : 0 ≤ y ≤ a and 0 ≤ x ≤

√ ay − y}.

Let m 2 denote the natural 2-dimensional Lebesgue measure on Q. Then

m 2 (E a ) = 0

a

a2 √ ( ay − y)dy = . 6

For each r > 1 we define N (r ) = card{α ∈ Z2+ \{0} :

√

α2 /|α| > 1/r }.

To each α ∈ Z2+ we associate the square α + I 2 , where I 2 = [0, 1] × [0, 1]. From this association we see that 1 N (r ) = m 2 (Er 2 ) + o(r 4 ) = r 4 + o(r 4 ). 6 436

(3.3)

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We have   √ 

r2 r y dx √ dy = dy y log x + y y 0 0 0

1

1 √ 2 1 1 4 1 = r3 u log √ du = 2r 3 t 2 log dt = r 3 = · r 3 . t 9 3 6 u 0 0

√

Er 2

y d xd y = x+y

r2

√

Denote Ar = {α ∈ Z2+ \{0} :  √α2 s j (Y ) = |α| j=1 α∈Ar 

=



r

√

y−y

y

√

α2 /|α| > 1/r }. Then

N (r ) 

α+I 2

α∈Ar



= Er 2

√ √  

y y α2 d xd y + − d xd y x+y |α| x+y α+I 2 √

√

α∈Ar

y 4 1 d xd y + o(r 3 ) = · r 3 + o(r 3 ). x+y 3 6

On the other hand, from (3.3) we obtain N (r ) 

j −1/4 =

j=1

4 4 {N (r )}3/4 + o({N (r )}3/4 ) = 3 3

 3/4 1 r 3 + o(r 3 ). 6

Combining these two identities, we find that  N (r ) j=1

s j (Y )

j=1

j −1/4

lim  N (r ) r →∞

= lim

r →∞

· 16 r 3 + o(r 3 ) =   4 1 3/4 3 3) r + o(r 3 6 4 3

 1/4 1 . 6

Thus the proof of the lemma will be complete if we can show that Y + 4,ess

 N (r ) j=1

s j (Y )

j=1

j −1/4

= lim  N (r ) r →∞

.

(3.4)

To prove (3.4), first note that by Proposition 2.6, the left-hand side is greater than or equal to the right-hand side. Thus we only need to prove the reverse inequality. But for the reverse inequality, note that (3.3) gives us N (ν + 1) − N (ν) = o(ν 4 ), ν ∈ N. Hence N (ν+1) j=N (ν)+1

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1 (N (ν + 1) − N (ν)) = o(ν 3 ). ν

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Q. Fang, J. Xia

For a large k ∈ N, there is a ν(k) ∈ N such that N (ν(k)) ≤ k < N (ν(k) + 1). Thus  N (ν(k)+1) s j (Y ) s1 (Y ) + · · · + sk (Y ) j=1 ≤  N (ν(k)) −1/4 −1/4 1 + ··· + k j −1/4 j=1  N (ν(k)+1)  N (ν(k)) s j (Y ) j=1 j=N (ν(k))+1 s j (Y ) =  N (ν(k)) +  N (ν(k)) j −1/4 j −1/4 j=1 j=1  N (ν(k)) s j (Y ) o({ν(k)}3 ) j=1 . =  N (ν(k)) + (4/3){N (ν(k))}3/4 + o({N (ν(k))}3/4 ) j −1/4 j=1 Using (3.3) again, we find that  N (r ) s1 (Y ) + · · · + sk (Y ) j=1 s j (Y ) lim sup −1/4 ≤ lim  N (r ) . −1/4 r →∞ + ··· + k k→∞ 1 j −1/4 j=1

Thus, by Proposition 2.6, the left-hand side of (3.4) is less than or equal to the righthand side as promised. This completes the proof of the lemma.  Proof of Theorem 3.3. Under the assumption n = 2, (3.2) gives us Hz¯ 1 12 = 1/2. −1/2 . On the other hand, Lemma 3.5 tells us that Thus Hz¯ 1 + 4 ≥ Hz¯ 1  ≥ 2 + + −1/4 −1/2 . Since 2 > 6−1/4 , it follows that Hz¯ 1 + Hz¯ 1 4,ess = 6 4 > Hz¯ 1 4,ess .  We choose to present Lemma 3.5 separately because its proof is more elementary than the general case. But the calculation in Lemma 3.5 can be generalized to all complex dimensions n ≥ 2, which may be of independent interest: Proposition 3.6 In each complex dimension n ≥ 2, we have Hz¯ 1 + 2n,ess

 =

1 n−1 · n! 2n − 1

1/(2n) .

Proof We begin with some general volume calculation. For j ≥ 1, let v j denote the (real) j-dimensional volume measure. Let k ≥ 2 and define k (t) = {(x1 , . . . , xk ) ∈ Rk : x1 ≥ 0, . . . , xk ≥ 0 and x1 + · · · + xk = t} for t ≥ 0. Elementary calculation shows that vk−1 (k (1)) = {(k − 1)!}−1 k 1/2 . Hence √ k t k−1 vk−1 (k (t)) = (k − 1)!

(3.5)

for all t > 0.

438

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Consider the “first quadrant” Q n = {(x1 , . . . , xn ) ∈ Rn : x1 ≥ 0, . . . , xn ≥ 0} in Rn . We write the elements in Q n in the form (x, y), where x ≥ 0 and y = (y1 , . . . , yn−1 ) with y j ≥ 0 for 1 ≤ j ≤ n − 1. For such a y, we denote |y| = y1 + · · · + yn−1 in this proof. Adapting the proof of Lemma 3.5 to general n ≥ 2, we now define E a = {(x, y) ∈ Q n : a|y| ≥ (x + |y|)2 } for a > 0. We claim that vn (E a ) =

an n − 1 · . n! 2n − 1

(3.6)

To prove this, note that the condition a|y| ≥ (x + |y|)2 implies a ≥ x + |y| and, consequently, a ≥ x and a ≥ |y|. For each 0 ≤ t ≤ a, define

a (t) = n (t) ∩ E a = {(x, y) ∈ Q n : x + |y| = t and a|y| ≥ t 2 }. Obviously,

a (t) = {(t − ρ, y) ∈ Q n : t 2 /a ≤ ρ ≤ t and |y| = ρ}. For any λ, μ ∈ [t 2 /a, t], the distance between the slices {(t − λ, y) ∈ Q n : |y| = λ} and {(t − μ, y) ∈ Q n : |y| = μ} is easily seen to be  1/2  n {μ − λ}2 + (n − 1){(n − 1)−1 λ − (n − 1)−1 μ}2 |λ − μ|. = n−1 Combining this fact with (3.5), when n ≥ 3 we have  vn−1 ( a (t)) =  =

n n−1 n n−1

t

t 2 /a

t t 2 /a

vn−2 ({(t − ρ, y) ∈ Q n : |y| = ρ})dρ √

√ n n − 1 n−2 ρ {t n−1 − (t 2 /a)n−1 }. dρ = (n − 2)! (n − 1)!

When n = 2, we can omit the first two steps above and the last = trivially holds. Let u be the unit vector (n −1/2 , . . . , n −1/2 ) in Rn . For s, t ∈ [0, ∞), if su ∈ n (t), then

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Q. Fang, J. Xia

n 1/2 s = t. Since x + |y| ≤ a for (x, y) ∈ E a , integrating along the “u-axis" in Rn , we have

a 1 vn−1 ( a (n 1/2 s))ds = √ vn−1 ( a (t))dt n 0 0  

a 1 1 an 1 n−1 −(n−1) 2n−2 = (t −a t )dt = − . (n − 1)! 0 (n − 1)! n 2n − 1

vn (E a ) =

n −1/2 a

Then an obvious simplification of the right-hand side proves (3.6). Let us again write each α ∈ Zn+ in the form α = (α1 , α2 ), but keep in mind that this time we have α2 ∈ Zn−1 + . Accordingly, |α| − α1 = |α2 |. Thus from (3.2) we obtain (Hz¯∗1 Hz¯ 1 )1/2

=

  α∈Zn+

n − 1 + |α2 | (n − 1 + |α|)(n + |α|) +(0)

Again, (Hz¯∗1 Hz¯ 1 )1/2 = Y + Z , where Z ∈ C2n Y =

1/2 eα ⊗ eα .

and

√



|α2 | eα ⊗ eα . |α|

α∈Zn+ \{0}

+ ∗ 1/2 + Hence Hz¯ 1 + 2n,ess = (Hz¯ 1 Hz¯ 1 ) 2n,ess = Y 2n,ess , and we need to compute Y + 2n,ess . For each large r > 1 we define the set

Ar = {α ∈ Zn+ \{0} :



|α2 |/|α| > 1/r }.

To each α ∈ Zn+ we associate the cube α + I n , where I n = {(x1 , . . . , xn ) ∈ Rn : 0 ≤ x j ≤ 1 for j = 1, . . . , n}. Obviously, there is a constant 0 < C < ∞ such that for any α ∈ Zn+ \{0} and any (x, y) ∈ α + I n , we have  √ √  |α2 | C |y|    |α| − x + |y|  ≤ (1 + |α |1/2 )|α| . 2

(3.7)

Write N (r ) = card(Ar ). From (3.7) it is easy to deduce that N (r ) = vn (Er 2 )+o(r 2n ). Combining this fact with (3.6), we obtain N (r ) = γn r 2n + o(r 2n ),

(3.8)

where we denote γn =

1 n−1 · . n! 2n − 1 440

(3.9)

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Best Approximations in a Class of Lorentz Ideals

For convenience let us write dy = dy1 · · · dyn−1 on Rn−1 . We have

Er 2

√

∞  |y| d xd y = vn ({(x, y) ∈ Er 2 : |y|/(x + |y|) > t})dt x + |y| 0

∞

∞ 1 1 1 vn (E 1/t 2 )dt + vn (Er 2 ) = γn dt + γn r 2n = 2n r t r 1/r 1/r 2n 2n−1 = , γn r 2n − 1

where the third = follows from (3.6) and (3.9). Thus N (r ) 

 √|α2 | |α| α∈Ar



s j (Y ) =

j=1

=

α∈Ar



= Er 2

α+I n

√ √ √  

|y| |α2 | |y| d xd y + − d xd y x + |y| |α| x + |y| α+I n α∈Ar

√ |y| 2n d xd y + o(r 2n−1 ) = γn r 2n−1 + o(r 2n−1 ). x + |y| 2n − 1

On the other hand, from (3.8) we obtain N (r ) 

j −1/(2n) =

j=1

=

2n {N (r )}(2n−1)/(2n) + o({N (r )}(2n−1)/(2n) ) 2n − 1 2n (2n−1)/(2n) 2n−1 γn r + o(r 2n−1 ). 2n − 1

Combining these two identities, we find that  N (r ) j=1

lim  N (r ) r →∞ j=1

s j (Y )

j −1/(2n)

2n 2n−1 + o(r 2n−1 ) 2n−1 γn r r →∞ 2n γ (2n−1)/(2n) r 2n−1 + o(r 2n−1 ) 2n−1 n

= lim

1/(2n)

= γn

.

Recalling (3.9), the proof of the proposition will be complete if we can show that Y + 2n,ess

 N (r ) j=1

= lim  N (r ) r →∞ j=1

s j (Y )

j −1/(2n)

.

(3.10)

As in the proof of Lemma 3.5, we first note that by Proposition 2.6, the left-hand side of (3.10) is greater than or equal to the right-hand side. Thus we only need to prove the reverse inequality. But for the reverse inequality, note that (3.8) gives us N (ν + 1) − N (ν) = o(ν 2n ),

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Q. Fang, J. Xia

ν ∈ N. Hence N (ν+1) j=N (ν)+1

s j (Y ) ≤

1 (N (ν + 1) − N (ν)) = o(ν 2n−1 ). ν

Once we have this, by the argument at the end of the proof of Lemma 3.5, the righthand side of (3.10) is greater than or equal to the left-hand side. This completes the proof.  The point that we try to make with Proposition 3.6 is that it is not easy to come + up with functions f on S ⊂ Cn such that H f + 2n > H f 2n,ess . Theorem 3.3 says 2 that the function z¯ 1 on S ⊂ C has this property. So what about the function z¯ 1 on S ⊂ C3 ? In the case n = 3, Proposition 3.6 gives us Hz¯ 1 + 6,ess

 =

1 15

1/6 .

On the other hand, the obvious lower bound that we obtain from (3.2) in the case n = 3 −1/2 . Since 3−1/2 < 15−1/6 , this is of no use to us. The difficulty here is Hz¯ 1 + 6 ≥3 is to obtain an estimate of Hz¯ 1 + 2n that is close to its true value. In view of this, it is somewhat surprising that we can actually calculate the essential norm Hz¯ 1 + 2n,ess . −1/2 is to In the case n = 2, we do not know how close the lower bound Hz¯ 1 + ≥ 2 4 the true value of Hz¯ 1 + 4 . So it is really a matter of luck that the apparently crude lower −1/2 in the case n = 2 is good enough to give us Example 3.4, bound Hz¯ 1 + 4 ≥ 2 which is the main purpose of the section. As of this writing, Example 3.4 is the only example of its kind that we are able to produce. Data availability Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

References 1. Axler, S., Berg, I., Jewell, N., Shields, A.: Approximation by compact operators and the space H ∞ +C. Ann. Math. (2) 109, 601–612 (1979) 2. Berg, I.: An extension of the Weyl–von Neumann theorem to normal operators. Trans. Am. Math. Soc. 160, 365–371 (1971) 3. Connes, A.: The action functional in noncommutative geometry. Commun. Math. Phys. 117, 673–683 (1988) 4. Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994) 5. Conway, J.: A Course in Functional Analysis. Graduate Texts in Mathematics, vol. 96, 2nd edn. Springer, New York (1990) 6. Dixmier, J.: Existence de traces non normales. C. R. Acad. Sci. Paris Sér. A-B 262, A1107–A1108 (1966) 7. Engliš, M., Guo, K., Zhang, G.: Toeplitz and Hankel operators and Dixmier traces on the unit ball of C n . Proc. Am. Math. Soc. 137, 3669–3678 (2009) 8. Engliš, M., Zhang, G.: Hankel operators and the Dixmier trace on the Hardy space. J. Lond. Math. Soc. (2) 94, 337–356 (2016)

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Best Approximations in a Class of Lorentz Ideals 9. Fang, Q., Xia, J.: Schatten class membership of Hankel operators on the unit sphere. J. Funct. Anal. 257, 3082–3134 (2009) 10. Fang, Q., Xia, J.: Hankel operators on weighted Bergman spaces and norm ideals. Complex Anal. Oper. Theory 12, 629–668 (2018) 11. Gohberg, I., Krein, M.: Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society Translations of Mathematical Monographs, vol. 18. Providence (1969) 12. Jiang, L., Wang, Y., Xia, J.: Toeplitz operators associated with measures and the Dixmier trace on the Hardy space. Complex Anal. Oper. Theory 14(2), Paper No. 30 (2020) 13. Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1976) 14. Kuroda, Sh.: On a theorem of Weyl–von Neumann. Proc. Jpn. Acad. 34, 11–15 (1958) 15. Rudin, W.: Function Theory in the Unit Ball of Cn . Springer, New York (1980) 16. Schatten, R.: Norm Ideals of Completely Continuous Operators. Springer, Berlin (1970) 17. Upmeier, H., Wang, K.: Dixmier trace for Toeplitz operators on symmetric domains. J. Funct. Anal. 271, 532–565 (2016) 18. Voiculescu, D.: Some results on norm-ideal perturbations of Hilbert space operators. J. Oper. Theory 2, 3–37 (1979) 19. Voiculescu, D.: On the existence of quasicentral approximate units relative to normed ideals, I. J. Funct. Anal. 91, 1–36 (1990) 20. Xia, J.: Diagonalization modulo norm ideals with Lipschitz estimates. J. Funct. Anal. 145, 491–526 (1997) 21. Xia, J.: A condition for diagonalization modulo arbitrary norm ideals. J. Funct. Anal. 255, 1039–1056 (2008) 22. Xia, J.: Bergman commutators and norm ideals. J. Funct. Anal. 263, 988–1039 (2012) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Complex Analysis and Operator Theory (2022) 16:45 https://doi.org/10.1007/s11785-022-01218-7

Complex Analysis and Operator Theory

Szegö’s Theorem on Hardy Spaces Induced by Rotation-Invariant Borel Measures Kunyu Guo1 · Qi Zhou1 Received: 7 December 2021 / Accepted: 6 February 2022 / Published online: 25 March 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract It is shown in this paper that under a mild condition, an analogous version of Szegö’s theorem on Hardy spaces induced by rotation-invariant measures on the closed unit disk is true. This leads to a natural connection between cyclic vectors on these spaces and function-theoretic invariants involving these spaces. Keywords Szegö’s theorem · Cyclic vectors · Hardy space · Rotation-invariant Borel measure Mathematics Subject Classification 30H10 · 47A16 · 60B05

1 Introduction Let D be the open unit disk on the complex plane C with the boundary T. We denote by A(D) the disk algebra, which is the class of all continuous functions on the closure D of D whose restrictions to D are analytic. Let A0 (D) be the functions in A(D) whose evaluation at the origin are 0. Recall that there is a well known theorem first given by G. Szegö in 1939 [13].

Dedicated to the memory of Jörg Eschmeier. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz and Mihai Putinar.

B

Kunyu Guo [email protected] Qi Zhou [email protected]

1

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

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K. Guo, Q. Zhou dθ dθ Theorem 1.1 If K is a nonnegative function in L 1 ( 2π ) with log K ∈ L 1 ( 2π ), then



2π

inf

p∈A0 (D) 0

|1 − p|2 K

dθ = exp 2π



2π

log K 0

dθ 2π

 .

(1.1)

This theorem is among the most celebrated results in the theory of orthogonal polynomials on the unit circle, which has a profound influence in many areas, including theoretical physics, stochastic processes, and numerical analysis. dθ Recall that M is an invariant subspace of the Hardy space H 2 ( 2π ) if M is closed and z M ⊂ M. We will use [ f ] for the invariant subspace generated by f which is the dθ ), we call f a cyclic vector smallest invariant subspace containing f . If [ f ] = H 2 ( 2π dθ 2 in H ( 2π ). It turns out that there are some highly nontrivial connections between Szegö’s theorem, invariant subspaces, and cyclic vectors in the following sense. First note that as shown in [[7], page 53] the necessary and sufficient condition for dθ dθ ) with log K ∈ L 1 ( 2π ) is K = | f |2 for some function a non-negative K ∈ L 1 ( 2π dθ 2 f ∈ H ( 2π ). Therefore Szegö’s theorem can be restated as follows,  inf

p∈A0 (D) 0

2π

|1 − p| | f | 2

2 dθ

2π



2π

= exp

log | f |

0

2 dθ

2π

 ,

(1.2)

dθ for every nonzero f ∈ H 2 ( 2π ). Actually, it shows that the geometric quantity, the distance from f to the invariant subspace z[ f ], is equal to a function-theoretic quantity M( f ), the Mahler measure of f . The Mahler measures of polynomials have received considerable attention in number theory for which there is a related long-standing open problem known as “Lehmer’s problem” or “Lehmer’s conjecture”. One can refer to [6] for more about Mahler measures and Szegö’s theorem. dθ ) has an innerHere is a simple way to see (1.2). Since every nonzero f ∈ H 2 ( 2π dθ dθ 2 ) is outer factorization, say f = ηF, where η ∈ H ( 2π ) is inner, and F ∈ H 2 ( 2π dθ 2 2 2 outer, then | f | = |F| . Since F is outer and therefore cyclic in H ( 2π ), one sees easily that

 inf

p∈A0 (D) 0

2π

|1 − p| |F| 2

2 dθ

2π

 = |F(0)| = exp 2

0

2π

log |F|

2 dθ

2π

 ,

which leads to (1.2). Noticing that K in (1.1) is exactly the square of the modulus of dθ a cyclic vector F in H 2 ( 2π ), this is the simplest case for existing connection between Szegö’s theorem and cyclic vectors on the Hardy space. One of aims of this paper is to discuss the relationship between cyclic vectors and Szegö’s theorem in general Hardy spaces. In this paper, we consider the Hardy space induced by a general rotation-invariant probability measure dμ on the closed unit disk D. A Borel measure dμ on the closed unit disk D is called rotation-invariant if μ(E) = μ(ξ E) for every Borel set E ⊂ D and ξ ∈ T. Define ρ = sup{|z| : z ∈ support(dμ)}, and in this paper, we always

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Szegö’s Theorem on Hardy Spaces Induced by Rotation-Invariant... dθ assume ρ = 1. The simplest examples of such measures are 2π , and the weighted area cα measure d Aα on D, where d Aα = π (1 − r 2 )α r dr dθ , α > −1 and cα = α + 1. dθ We denote by H 2 (dμ) the norm-closure of A(D) in L 2 (dμ). In the cases dμ = 2π dθ 2 2 and dμ = d Aα , the resulting spaces H (dμ) are the Hardy space H ( 2π ) and the dθ ) weighted Bergman space L a2 (d Aα ), respectively. As done for the Hardy space H 2 ( 2π dθ 2 on the unit circle, one pulls functions in H ( 2π ) back to the unit disk D via the Poisson integral which yields the Hardy space H 2 (D) on the unit disk D. As a result, the dθ ) is identified with the Hardy space H 2 (D) by Fatou’s theorem Hardy space H 2 ( 2π [[4] Theorem 2.6]. Similarly, H 2 (dμ) can be identified with the space H 2 (D, dμ) of analytic functions on the unit disk D, which is the set of all analytic functions g on D with  |gr (z)|2 dμ(z) < ∞, g2H 2 (D,dμ) = sup 0 −1. In [14], Dr. P. Wang obtained some similar results by considering when operators Vw defined on analytic function spaces on the unit ball Bd of Cd by Vw ( f ) = ( f ◦ ϕw ) kw are unitary operators. In fact, in dimension d = 1, we see from the proof of Theorem 2.3 that Vw are unitary operators only on the Hardy space and weighted Bergman space of the unit disk. Recall that H 2 (D, dμ) is defined to be the space consisting of all analytic functions g on D with  g2H 2 (D,dμ) = sup |gr (z)|2 dμ(z) < ∞. 0 r0 , we have  f − f˜r 2 ≤  f − q2 + q − qr 2 + qr − f˜r 2 ≤ 3ε1/2 . It follows from the above reasoning that    | f (z)|2 dμ(z) = lim | f˜r (z)|2 dμ(z) = sup | f˜r (z)|2 dμ(z). D

r →1 D

0 j ≥ 1. We say that T is quasitriangular if T = T0 + K , where T0 ∈ B(H) is triangular and K is compact – we write T ∈ (QT) – and that T is biquasitriangular if both T and T ∗ are quasitriangular, in which case we write T ∈ (BQT). The orthonormal bases which “quasitriangularise” T and T ∗ need not be the same! Let us recall that an operator T ∈ B(H) is said to be semi-Fredholm if its range is closed, and at least one of nul T (the nullity of T ) and nul T ∗ (the nullity of T ∗ ) is finite, in which case we define the semi-Fredholm index ind T := nul T − nul T ∗ ∈ Z ∪ {−∞, ∞}. (It is understood that m − ∞ := −∞ and ∞ − m := ∞ for all m ∈ N.) This generalises the notion of a Fredholm operator, which may be defined as a semiFredholm operator with finite index. Whereas an operator R ∈ B(H) is Fredholm if and only if its image π(R) in the Calkin algebra under the canonical quotient map π : B(H) → B(H)/K(H) is invertible, it can be shown that an operator T ∈ B(H) is semi-Fredholm if and only if π(T ) is either left- or right-invertible (or both). The 1 Halmos explains that his motivation in asking the question in this form was his desire to express all of his questions in such a way as to require a “yes” or “no” answer.

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semi-Fredholm domain of an operator T ∈ B(H) is the set ρs F (T ) := {α ∈ C : T − α I is semi-Fredholm}. We refer the reader to the monograph of Caradus et al. [8] for more information regarding semi-Fredholm operators. It was Douglas and Pearcy [16] who first observed that the semi-Fredholm index provided an obstruction to membership in (QT); indeed, they observed that the unilateral forward shift S could not be a limit of triangular operators because ind S := nul S − nul S ∗ = −1. A fortiori, they observed that if there exists λ ∈ C such that ind (T − λI ) < 0, then T ∈ / (QT). Shortly thereafter, Pearcy conjectured that index was the only obstruction to membership in (QT). He would eventually be proven correct by Apostol et al. [3] (see also [17]). The relevance of quasitriangularity to Halmos’ seventh problem is that every nilpotent operator is quasitriangular, and that the set (QT) of quasitriangular operators is norm-closed. Furthermore, the set Nil(B(H)) of nilpotent operators in B(H) is selfadjoint, and as a consequence we see that if T ∈ Nil(B(H)), then ind(T − λ) = 0 whenever T − λI is semi-Fredholm. Two other necessary conditions for membership in Nil(B(H)) arise from the uppersemicontinuity of the spectrum and the fact that the invertible elements form an open set in any unital Banach algebra A; if t ∈ Nil(A) and K
A is any closed, two-sided ideal of A, then σA/K (πK (t)) is connected and contains the origin, where πK : A → A/K denotes the canonical quotient map. Six years after the question was originally raised by Halmos, as part of a seemingly infinite series of papers by an uncountable number of authors, Apostol, Foia¸s and Voiculescu [4] proved that the above necessary conditions were also sufficient: Theorem (Apostol, Foia¸s and Voiculescu) An operator T ∈ B(H) is a limit of nilpotent operators if and only if (i) the spectrum σ (T ) of T is connected and contains the origin; (ii) the essential spectrum σe (T ) := σB(H)/K(H) (πK(H) (T )) of T is connected and contains the origin; and (iii) ind (T − λI ) = 0 for all λ ∈ C for which T − λI is semi-Fredholm; equivalently, T is biquasitriangular. One of the key steps along the way to this general solution was earlier provided by D.A. Herrero [24], who proved that if N ∈ B(H) is a normal operator, then N ∈ Nil(B(H)) if and only if σ (N ) is connected and contains the origin. Of course, Halmos’ question makes sense in any Banach algebra, and more specifically in any C ∗ -algebra, which leads us to ask the following questions. (Since the only quasinilpotent element of a commutative C ∗ -algebra is 0, the questions below are only interesting when A is non-commutative.) Problem 1 Let A be a C ∗ -algebra. Characterise the norm-closure of the set Nil(A) of nilpotent elements of A. It is to be expected that the solution to the problem will depend upon the nature of the C ∗ -algebra in question, meaning that it is in effect a collection of questions. As always, specific examples worth investigation might include UHF C ∗ -algebras, simple AF C ∗ -algebras, the Cuntz algebras On , etc..

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Naturally, Halmos’ original question also makes sense in this context. Problem 1.1 Let A be a C ∗ -algebra, and q ∈ A be quasinilpotent. Is q a limit of nilpotent elements of A? Phrased differently: does there exist a C ∗ -algebra A and a quasinilpotent element q ∈ A that is not a limit of nilpotent elements of A? If so, can we characterise the C ∗ -algebras for which this can happen? In the hope that Herrero’s characterisation of normal operators lying in the closure of Nil(B(H)) has an extension to the C ∗ -algebra setting which proves as useful as the original result did in the Hilbert space setting, we are led to ask the following. Problem 1.2 Let A be a C ∗ -algebra. Characterise the normal elements of A which lie in the closure of Nil(A). The presence of a (faithful) tracial state τ acting on the C ∗ -algebra makes life more interesting, and potentially more difficult. For example, if N ∈ B(H) is an hermitian operator whose spectrum is [0, 1], then N ∈ Nil(B(H)) by Herrero’s result. If A is a UHF C ∗ -algebra with tracial state τ , for example, and h = h ∗ ∈ A satisfies σ (h) = [0, 1], then – by the faithfulness of the trace – τ (h) > 0, and so (by the continuity of τ ) we find that h ∈ / Nil(A). It gets worse. If h = h ∗ , σ (h) = [−1, 1] / Nil(A), from which we again and even if τ (h) = 0, then σ (h 2 ) = [0, 1], and so h 2 ∈ conclude that h ∈ / Nil(A). More generally, if p ∈ C[z] is a polynomials satisfying p(0) = 0 and if n ∈ A is normal, then τ ( p(n)) = 0 implies that p(n) ∈ / Nil(A), whence n ∈ / Nil(A). By Gelfand Theory, the trace τ on A corresponds to a positive, regular Borel measure μτ on C ∗ (n)  C(σ (n)), in that for all f ∈ C(σ (n)),  τ ( f (n)) = f (z)dμτ . σ (n)

In effect, the question may be measure-theoretic in that we are looking for measures on σ (n) which annihilate the polynomials which vanish at zero. This may not be the only property of the measure we require, but it does suggest that perhaps (an analogue of) a certain theorem of F. Riesz and M. Riesz [26] may have a role to play here. The interested reader should also see the paper of Skoufranis [45, Question 5.9]. Our next question is the restriction of the previous question to a specific C ∗ -algebra, but may prove to be the litmus test for any conjectures for more general C ∗ -algebras equipped with one or more tracial states. Problem 1.3 Let A be a simple, unital AF C ∗ -algebra with a unique tracial state. Characterise the normal elements of A which lie in closure of Nil(A). Paul Skoufranis [46, Theorem 3.5] has shown that there exists a non-simple, AFembeddable C ∗ -algebra A such that Nil(A) contains a positive element. Of course, such a C ∗ -algebra fails to admit a faithful tracial state. At the other end of the spectrum (metaphorically speaking) are C ∗ -algebras which do not admit a trace. Again, we refer the reader to the same very interesting paper of

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Skoufranis [46] who proved (Theorem 2.6 of that paper) that if A is unital, simple and purely infinite, then a normal element n ∈ A is a limit of nilpotent elements of A if and only if σ (n) is connected and contains the origin, and λ1A − n lies in the connected component of the identity in the set of invertible elements of A for all λ ∈ C\σ (n). The latter condition may be viewed as the C ∗ -analogue of the index obstruction observed in the B(H) setting. There may prove to be more “room” to work in the setting of von Neumann algebras. Again, we refer to Skoufranis [45] for more information here. It would be interesting to know the answers to the following subquestions. Problem 1.4 (a) Let M be a type II1 factor. Characterise the set Nil(M), as well as the set of normal elements of that set. (b) Let N be a type II∞ factor. Characterise the set Nil(M), as well as the set of normal elements of that set.

2 Algebraic Elements of C ∗ -algebras As mentioned above, the notion of biquasitriangularity plays a pivotal role in the characterisation of the closure of the set of nilpotent operators in B(H). The characterisation of biquasitriangularity in terms of index provides a simple and effective way of determining whether or not a given operator lies in the set (BQT). There is, however, a third characterisation, due to Voiculescu [49], which is easily expressed in a more general setting. We shall say that an element a of a Banach algebra A is algebraic if there exists a non-zero polynomial p such that p(a) = 0. We denote by Alg(A) the set of algebraic elements of A. Theorem (Voiculescu) Let H be an infinite-dimensional, complex, separable Hilbert space. An operator T ∈ B(H) is biquasitriangular if and only if T is a limit of algebraic operators. This invites the question: Problem 2 Let A be a unital C ∗ -algebra. Characterise the closure of the set Alg(A) of algebraic elements of A. Let T ∈ B(H) be arbitrary, and let N ∈ B(H) be a normal operator such that σ (N ) = σe (N ) = σ (T ). It is not hard to verify that the semi-Fredholm domain of N ⊕ T is exactly the common resolvent ρ(T ) = ρ(N ) of T and N , and as a consequence, (N ⊕ T ) − λI is invertible (and thus has index zero) for all λ ∈ ρs F (N ⊕ T ). In other words, N ⊕ T ∈ (BQT). By Voiculescu’s Theorem above, N ⊕ T is a limit of algebraic operators. Problem 2.1 Let A be a simple, unital C ∗ -algebra. Suppose that t, n ∈ A, n is normal and σ (n) ⊇ σ (t). Is t ⊕ n a limit of algebraic elements in M2 (A) (or, given k ≥ 3 and a normal element m ∈ Mk−1 (A), is t ⊕ m a limit of algebraic elements in Mk (A))? (In the case where A is not simple, one may have to modify the above question to ensure that the spectrum of n should agree with the spectrum of its image in any quotient of A modulo a maximal ideal.)

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3 Commutators in C ∗ -algebras If A is any algebra and a, b ∈ A, we shall refer to [a, b] := ab−ba as the commutator of a and b. We denote the set of all commutators of elements of A by C(A) := {[a, b] : a, b ∈ A}. Let n ∈ N. It is a standard Theorem of Shoda [43] (see also [1]) that C(Mn (C)) = {T ∈ Mn (C) : Tr(T ) = 0}. Of course, Mn (C)  B(Cn ). Let H be an infinite-dimensional, separable Hilbert space. We recall the following remarkable extension of Shoda’s result by Brown and Pearcy [6]. Theorem (Brown and Pearcy) If H is infinite-dimensional, separable Hilbert space. Then C(B(H)) := {T ∈ B(H) : T ∈ / {α I + K : 0 = α ∈ C, K ∈ K(H)}}. Once again, the necessity of this condition is relatively simple to demonstrate. A standard result in Banach algebra theory states that if A is a unital Banach algebra and x, y ∈ A, then σ (x y) ∪ {0} = σ (yx) ∪ {0}. Using this, we see that the equation x y − yx = α1 implies that σ (x y) = α + σ (yx), which contradicts the compactness of the spectrum of σ (x y) if α = 0. In particular, if T ∈ C(B(H)) and π : B(H) → B(H)/K(H) is the quotient map, then π(T ) = α π(I ) for any α = 0, which implies that T ∈ / {α I + K : 0 = α ∈ C, K ∈ K(H)}. All of the work of Brown and Pearcy therefore goes into proving the sufficiency of this condition. Problem 3 Let A be a C ∗ -algebra. Characterise C(A) := {[a, b] : a, b ∈ A}. We stake no claim to authorship of this question, as it has been examined by a number of people in various specific C ∗ -algebras and algebras of (Hilbert and Banach space) operators. For example, Dykema and Skripka have investigated commutators in type II1 factors; Dykema, Figiel, Weiss and Wodzicki have done a detailed study of commutators in operator ideals [18]; and Dykema and Krishnaswamy-Usha [19] have studied nilpotent compact operators as commutators of compact operators. Recently, Dosev [10] and Dosev and Johnson [11] have classified commutators in B(1 ) and in B(∞ ) respectively (see also the work of Dosev et al. [12]). Again, there is a dichotomy of strategies depending upon whether or not the algebra A admits a faithful tracial state τ . If it does, then clearly C(A) ∈ ker τ . This is the case, therefore, when A is a UHF C ∗ -algebra. In that case, ker τ has co-dimension 1, and it is not hard to show that every element of trace zero in A is a limit of commutators. (Indeed, if z ∈ ker τ , then z = limn an , where an ∈ An  Mk(n) (C) for some k(n) ∈ N. The continuity of the trace implies that αn := τ (z n ) converges to 0 as n tends to infinity, and thus an − αn 1 ∈ ker τ converges to z.) Furthemore, we have shown that every element of A is a sum of at most two commutators [32]. Despite this, the question of characterising the class C(A) has

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eluded us. (That C(A) consists of sums of two commutators in A happens in quite a few (simple) C ∗ -algebras. We direct the reader to [32, 33] for further details.) Also, P.W. Ng has shown that every trace zero element in the reduced C ∗ -algebra Cr∗ (F∞ ) of the free group in infinitely many generators is a sum of three commutators [37], and that every trace zero element of the Jiang-Su algebra is a sum of at most five commutators [36]. The question of whether or not an operator in a C ∗ -algebra is a commutator may be easier to solve for certain specific subclasses of operators. Consider, for example, the following question. Problem 3.1 Let A be a C ∗ -algebra. Which normal (or nilpotent) operators in A are commutators? The next question stems from the seminal work of Fack [20] and Thomsen [48] on spans of commutators in C ∗ -algebras. Fack showed that if A is a simple, unital AF C ∗ -algebra with unique trace (or if A is a simple, unital, infinite C ∗ -algebra), then every trace zero element is a sum of no more than 14 commutators in A, while Thomsen extended Fack’s first theorem to certain unital, simple, homogeneous C ∗ algebras (with a bound on the number of commutators required which depends upon the covering dimension of the spaces involved). As a consequence of their results, the linear span of the commutators in the C ∗ -algebras they consider is closed. Problem 3.2 Characterise those C ∗ -algebras A for which the set [A, A] := span C(A) is closed. If T ∈ Mn (C) has trace zero (equivalently, if T ∈ C(Mn (C))), then there exists an orthonormal basis {ek }nk=1 relative to which the diagonal of the matrix of T is zero; i.e. T ek , ek  = 0, 1 ≤ k ≤ n. Problem 3.3 Let a ∈ C(A). Does there exist a masa M (which depends upon a) in A and a conditional expectation ϕ of A onto M such that ϕ(a) = 0? If A admits a unique tracial state τ and b ∈ A satisfies τ (b) = 0, do such a masa (again, dependent upon b) and conditional expectation exist? (In this last question, we do not assume that b is itself a commutator.) There is an strong connection between commutators, nilpotents, idempotents and projections in C ∗ -algebras [33]. For example, using the fact that every trace-zero element of certain simple, unital C ∗ -algebras A of real rank zero is a sum of two commutators in the algebra, we were able to show [32] that every every element of A is a linear combination of at most 113 projections in A. This estimate has virtually no chance of being sharp. Can we do better? In his paper [5], Blackadar introduced several notions of comparability of projections in simple, unital C ∗ -algebras possessing a trace. To state our next question, we require the notion he refers to as the “Fundamental Comparability Question 2”: a simple, unital C ∗ -algebra A is said to satisfy FCQ2 if for all projections p, q ∈ A, we have that p ≺ q if τ ( p) < τ (q) for all traces τ on A. As usual, p ≺ q means that there exists a partial isometry v ∈ A such that p = v ∗ v, and q − vv ∗ > 0.

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Problem 3.4 Let A be a simple, unital C ∗ -algebra A of real rank zero, satisfying Blackadar’s FCQ2 and admitting a unique tracial state. Find the minimum positive integer n such that every element of A may be written as a linear combination of at most n projections in A. We digress temporarily to mention that Paskiewicz [38] has shown that every selfadjoint element of B(H) is a linear combination of at most five projections, from which it trivially follows that every T ∈ B(H) is a linear combination of at most ten. It would be interesting to know what the minimum number of projections required for an arbitrary T ∈ B(H) setting really is.

4 Specht’s Theorem in C ∗ -algebras Let n ≥ 1. The two most important equivalence relations on B(Cn )  Mn (C) are unitary equivalence and similarity. Of course, two operators A, B ∈ B(Cn ) are unitarily equivalent (and we write A  B), if there exists a unitary operator U ∈ B(Cn ) such that B = U ∗ AU , while A and B are similar (and we write A ∼ B) if there exists an invertible operator S ∈ B(Cn ) such that B = S −1 AS. A standard and elementary result from linear algebra shows that two matrices A, B ∈ B(Cn ) are similar if and only if they admit the same Jordan canonical form. Less well known, but equally important, is the following result of Specht [47]: Theorem (Specht) Two operators A, B ∈ B(Cn ) are unitarily equivalent if and only if Tr(w(A, A∗ )) = Tr(w(B, B ∗ )) whenever w(x, y) is a word in two non-commuting variables. We shall refer to the “only if” half of this theorem as “Specht’s trace condition”. It is interesting to note that Pearcy [39] has shown that one need only consider words of length at most 2n 2 . Observe that the group of unitary operators in B(Cn ) is compact, which is easily seen to imply that the unitary orbit U(T ) := {U ∗ T U : U ∈ Mn (C) unitary} of an element T ∈ Mn (C) is closed. If A is a C ∗ -algebra which admits a tracial state τ , and w(x, y) is a word in two non-commuting variables, then a routine calculation shows that τ (w(a, a ∗ )) = τ (w(b, b∗ )) whenever b belongs to the closure of U(a) := {u ∗ au : u ∈ A unitary}. In fact, the notion that b ∈ U(a) defines an equivalence relation on A referred to as approximate unitary equivalence of a and b, and is denoted by writing a a b. In light of this, if one were to try to extend Specht’s Theorem to more general C ∗ -algebras A (which admit a tracial state), then the best one could hope for is a

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result along the lines of: a, b ∈ A are approximately unitarily equivalent if and only if τ (w(a, a ∗ )) = τ (w(b, b∗ )) for all words w(x, y) in two non-commuting variables. A natural first candidate to consider in extending Specht’s Theorem to the setting of C ∗ -algebras is a UHF C ∗ -algebra. Since any such algebra A = ∪n An is the normclosure of an increasing union of algebras An , each ∗ -isomorphic to a full matrix algebra, it is clear that Specht’s Theorem hold for pairs a, b in the dense subset of ∪n An of A. Modulo a technical assumption (required to enable us to use a powerful result due to Schafhauser [42]), Y. Zhang and the present author [35] obtained a positive answer in the universal UHF C ∗ -algebra Q. (The universal UHF C ∗ -algebra Q is the one whose supernatural number is divisible by any positive integer.) Theorem (Marcoux and Zhang) Let Q be the universal UHF C ∗ -algebra with tracial state τ , and let a, b ∈ Q. Suppose furthermore that C ∗ (a) satisfies the universal coefficient theorem (UCT). Then a and b are approximately unitarily equivalent if and only if τ (w(a, a ∗ )) = τ (w(b, b∗ )) whenever w(x, y) is a word in two non-commuting variables. Perhaps surprisingly, there is a K -theoretic obstruction which prevents this generalisation of Specht’s Theorem from holding in all UHF C ∗ -algebras. In fact, one can find a, b in the CAR algebra M2∞ such that C ∗ (a) and C ∗ (b) both satisfy the UCT, a and b satisfy Specht’s trace condition, and yet a and b are not approximately unitarily equivalent in M2∞ . Hence we are left with the following question: Problem 4 Determine necessary and sufficient conditions on a C ∗ -algebra A admitting a tracial state τ so that if a, b ∈ A satisfy τ (w(a, a ∗ )) = τ (w(b, b∗ )) for all words w(x, y) in two non-commuting variables, then a and b are approximately unitarily equivalent. If A admits multiple tracial states, we require that Specht’s trace condition should hold for each of them. In the case where A is an arbitrary UHF C ∗ -algebra and a ∈ A is normal, the hypothesis that C ∗ (a) satisfies the UCT is automatic, and the K -theoretic obstruction vanishes. Indeed, if b ∈ A and a, b satisfies Specht’s trace condition, then a a b. Problem 4.1 Determine necessary and sufficient conditions on a C ∗ -algebra A admitting a tracial state τ so that if m, n ∈ A are normal and satisfy τ (w(m, m ∗ )) = τ (w(n, n ∗ )) for all words w(x, y) in two non-commuting variables, then m and n are approximately unitarily equivalent. In the paper [34] of Mastnak, Radjavi and the present author, it was shown that two matrices A and B ∈ Mn (C) are unitarily equivalent if and only if Tr(| p(A, A∗ )|) = Tr(| p(B, B ∗ )|)

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for all polynomials p in two non-commuting variables. (That absolute values of words w(x, y) in two non-commuting variables is not sufficient is easily seen by taking A = In and B any other unitary operator.) More generally, given a C ∗ -algebra A and a, b ∈ A, we shall say that a and b satisfy the approximate absolute value condition (AAVC) if | p(a, a ∗ )| is approximately unitarily equivalent to | p(b, b∗ )| for all polynomials p in two non-commuting variables. Proposition 4.3 of [35] asserts that if A admits a tracial state and a, b ∈ A satisfy the AAVC, then a and b satisfy Specht’s tracial condition. In this sense, we may view the AAVC as an extension of Specht’s tracial condition to C ∗ -algebras which do not necessarily admit a tracial state. This allows us to reformulate Problem 3 above to a much larger class of C ∗ -algebras: Problem 4.2 Determine necessary and sufficient conditions on a C ∗ -algebra A such that if a, b ∈ A satisfy the AAVC, then a and b are approximately unitarily equivalent. It is worth noting that two operators A, B ∈ B(H) satisfy the AAVC if and only if they are approximately unitarily equivalent [35, Theorem 4.7]. Problem 4.3 Determine necessary and sufficient conditions on a C ∗ -algebra A so that if a, b ∈ A satisfy the AAVC and a is normal, then a and b are approximately unitarily equivalent. We conclude this section by mentioning that the question of extending Specht’s Theorem to the von Neumann algebra setting has been considered before; for the sake of brevity, we cite only the papers of Pearcy [39] and of Pearcy and Ringrose [40].

5 Closures of Intermediate Similarity Orbits in C ∗ -algebras The monumental task of describing the closure of the similarity orbit S(T ) := {R −1 T R : R ∈ B(H) invertible} of (virtually) every Hilbert space operator T was undertaken by (amongst others) Apostol et al. [2], and in that volume they also explore the characterisation of other sets of operators invariant under similarity conjugation. The closure of similarity-invariant subsets of B(H) is typically described in terms of spectral conditions (spectrum, essential spectrum, dimensions of Riesz subspaces, index), and in the infinite-dimensional setting, is usually far more tractable than a characterisation of the sets themselves. We are led to the following question. Problem 5 Let A be a C ∗ -algebra and a ∈ A. Characterise S(a). More generally, if

⊆ A is a set which is invariant under conjugation by similarities, characterise . As in the B(H) setting, this lends itself to a multitude of subproblems. Problem 5.1 Let A be a unital C ∗ -algebra. Characterise the norm-closure of each of the following sets: (a) K σ := {a ∈ A : σ (a) = K }, where K ⊆ C is a fixed, non-empty compact set. (b) C(A) := {[a, b] := ab − ba : a, b ∈ A}; (c) CE (A) := {[e, f ] := e f − f e : e, f ∈ A, e = e2 , f = f 2 }. Reprinted from the journal

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(d) Neg(A) := {a ∈ A : a ∼ −a}. (e) SimNor(A) := {a ∈ A : a ∼ n for some normal element n ∈ A}. The perspicacious reader (hopefully you) will have noticed that there is non-trivial intersection between this general problem and some of the problems we have mentioned earlier. In particular, by choosing K σ = {0} in part (a), we are looking for a characterisation of the norm-closure of the quasinilpotent elements of A. The set CE (A) actually lies in Neg(A), and one question is whether or not their closures are equal. Our interest in SimNor(A) stems from the fact that when A = B(H), it is known [25] that the norm-closures of SimNor(B(H)) and Alg(B(H)) coincide. Is this a general phenomenon? In connection with the question of determining the closure of the block-diagonal nilpotent operators (as first raised by L.R. Williams [51]), Herrero introduced a new orbit which he referred to as the (U + K)-orbit of T , defined as: (U + K)(T ) := {R −1 T R : R = U + K invertible U ∈ U(H), K ∈ K(H)}. This provides us with a host of corresponding problems in the C ∗ -algebra setting. Problem 5.2 Let A be a unital C ∗ -algebra and J
A be a closed, two-sided ideal of A. Given a ∈ A, characterise (U + J )(a), where (U + J )(a) := {r −1 ar : r ∈ A invertible and r = u + j, u ∈ U(A), j ∈ J }. When J = {0}, we are asking for a characterisation of the closure of the unitary orbit of a. At the other extreme, when J = A, we are asking for a characterisation of the closure of the similarity orbit of a. The (U + K)(T )-orbit of T ∈ B(H) has been characterised in a small number of cases: • • • •

T T T T

is normal [21]; is the unilateral shift [21]; is “shift-like”, meaning that it has the same spectral picture as the shift [31]; is an “essentially normal model” [13, 14, 27–29, 52–54].

Each case mentioned above consists of essentially normal operators. The decision to restrict to this class results from the desire to take advantage of the fact that conjugating by an invertible operator of the form R = U + K in B(H) translates to conjugating by the unitary π(U ) in the Calkin algebra, and unitary equivalence in the Calkin algebra is best understood for normal elements there. This suggests that one might study the question: Problem 5.3 Let A be a unital C ∗ -algebra and n ∈ A be normal. Characterise (U + J )(n). The work of Skoufranis [44] investigating the norm-closures of unitary and similarity orbits of normal operators in purely infinite C ∗ -algebras is relevant here. For example, Skoufranis characterises which normal operators sitting in a unital, simple,

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purely infinite C ∗ -algebra A lie in the norm-closure of the similarity orbit of a second normal operator in A. Can one extend such a result to the setting of the more general orbits described above? While the number of operators T ∈ B(H) for which (U + K)(T ) has been classified is rather small, the evidence nonetheless supports the conjecture that (U + K)(T ) = S(T ) ∩ U(T ) + K(H). We finish with the following problem. Problem 5.4 Let A be a unital C ∗ -algebra. If J
A is a closed, two-sided ideal of A and a ∈ A, is (U + J )(a) = S(a) ∩ U(a) + J ? We would like to thank the referee for a careful reading of the paper, and for a number of helpful suggestions.

References 1. Albert, A.A., Muckenhoupt, B.: On matrices of trace zero. Michigan Math. J. 3, 1–3 (1957) 2. Apostol, C., Fialkow, L.A., Herrero, D.A., Voiculescu, D.: Approximation of Hilbert space operators II, volume 102 of Research Notes in Math. Pitman Advanced Publishing Program, Boston (1984) 3. Apostol, C., Foia¸s, C., Voiculescu, D.: Some results on nonquasitriangular operators. IV. Rev. Roumaine Math. Pures Appl. 18, 487–514 (1973) 4. Apostol, C., Foia¸s, C., Voiculescu, D.: On the norm-closure of the nilpotents II. Rev. Roumaine Math. Pures Appl. 19, 549–557 (1974) 5. Blackadar, B.E.: Comparison theory for simple C ∗ -algebras. In Operator algebras and applications, volume 1 of London Math. Soc. Lecture Note Ser., pp. 21–54. Cambridge University Press, Cambridge(1988) 6. Brown, A., Pearcy, C.: Structure of commutators of operators. Ann. Math. 82, 112–127 (1965) 7. Brown, L.G., Douglas, R.G., Fillmore, P.A.: Unitary equivalence modulo the compact operators and extensions of C ∗ -algebras. In: Proceedings of a Conference on Operator Theory, Halifax, Nova Scotia, volume 23 of Lecture Notes in Math., pp. 58–128. Springer, Berlin (1973) 8. Caradus, S.R., Pfaffenberger, W.E., Yood, B.: Calkin algebras and algebras of operators on Banach spaces, volume 9 of Lecture Notes Pure Appl. Math. Marcel Dekker, Inc., New York (1974) 9. Davidson, K.R.: C ∗ -algebras by example, volume 6 of Fields Institute Monographs. Amer. Math. Soc., Providence, RI (1996) 10. Dosev, D.T.: Commutators on 1 . J. Funct. Anal. 256, 3490–3509 (2009) 11. Dosev, D.T., Johnson, W.B.: Commutators on ∞ . Bull. London Math. Soc. 42, 155–169 (2010) 12. Dosev, D.T., Johnson, W.B., Schechtman, G.: Commutators on l p , 1 ≤ p < ∞. J. Am. Math. Soc. 26, 101–127 (2013) 13. Dostál, M.: Closures of (U + K)-orbits of essentially normal models. PhD thesis, University of Alberta (1998) 14. Dostál, M.: (U + K)-orbits, a block tridiagonal decomposition technique and a model with multiply connected spectrum. J. Oper. Th. 46, 491–516 (2001) 15. Douglas, R.G.: Banach Algebra Techniques in Operator Theory. Academic Press, New York (1972) 16. Douglas, R.G., Pearcy, C.M.: A note on quasitriangular operators. Duke Math. J. 37, 177–188 (1970) 17. Douglas, R.G., Pearcy, C.M.: Invariant subspaces of non-quasitriangular operators. Proc. Conf. Operator Theory 345, 13–57 (1973) 18. Dykema, K., Figiel, T., Weiss, G., Wodzicki, M.: Commutator structure of operator ideals. Adv. Math. 185, 1–79 (2002)

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L. W. Marcoux 19. Dykema, K., Krishnaswamy-Usha, M.: Nilpotent elements of operator ideals as single commutators. Proc. Am. Math. Soc. 146, 3031–3037 (2018) 20. Fack, Th.: Finite sums of commutators in C ∗ -algebras. Ann. Inst. Fourier, Grenoble 32, 129–137 (1982) 21. Guinand, P.S., Marcoux, L.W.: Between the unitary and similarity orbits of normal operators. Pac. J. Math. 159, 299–335 (1993) 22. Halmos, P.R.: Ten problems in Hilbert space. Bull. Am. Math. J. 76, 887–933 (1970) 23. Halmos, P.R.: Ten year in Hilbert space. Integral Equ. Oper. Theory 2, 529–564 (1979) 24. Herrero, D.A.: Normal limits of nilpotent operators. Indiana Univ. Math. J. 23, 1097–1108 (1973/74) 25. Herrero, D.A.: Approximation of Hilbert space operators I, volume 224 of Pitman Research Notes in Math. Longman Scientific and Technical, Harlow, New York, second edition (1989) 26. Hoffman, K.: Banach spaces of analytic functions. Reprint of the 1962 original. Dover Publications, Inc., New York (1988) 27. Ji, Y., Jiang, C., Wang, Z.: The (U + K)-orbit of essentially normal operators and compact perturbation of strongly irreducible operators. In: Functional Analysis in China, volume 356 of Mathematics and its applications, pages 307–314, Dordrecht-Boston, 1996. Kluwer Academic Publishers 28. Ji, Y., Jiang, C., Wang, Z.: The (U +K)-orbit of essentially normal operators and compact perturbations of strongly irreducible operators. Chinese Ann. Math. Ser. B 21, 237–248 (2001) 29. Ji, Y., Li, J.: The quasiapproximate (U + K)-invariants of essentially normal operators. Integ. Equ. Oper. Th. 50, 255–278 (2004) 30. Lomonosov, V.J.: Invariant subspaces for operators commuting with compact operators (Russian). Funkcional. Anal. i. Prilo˘zen 7, 55–56 (1973) 31. Marcoux, L.W.: The closure of the (U + K)-orbit of shift-like operators. Indiana Univ. Math. J. 41, 1211–1223 (1992) 32. Marcoux, L.W.: Sums of small numbers of commutators. J. Oper. Th. 56, 111–142 (2006) 33. Marcoux, L.W.: Projections, commutators and lie ideals in C ∗ -algebras. Math. Proc. R. Irish Acad. 110A, 31–55 (2010) 34. Marcoux, L.W., Mastnak, M., Radjavi, H.: An approximate, multivariable version of Specht’s theorem. Linear Mutlilinear Alg. 55, 159–173 (2007) 35. Marcoux, L.W., Zhang, Y.: On Specht’s Theorem in UHF C ∗ -algebras. J. Funct. Anal. 280, 28 (2021) 36. Ng, P.W.: Commutators in the Jiang-Su algebra. Int. J. Math. 23, 29 (2012) 37. Ng, P.W.: Commutators in Cr∗ (F∞ ). Houston J. Math. 40, 421–446 (2014) 38. Paszkiewicz, A.: Any selfadjoint operator is a finite linear combination of projectors. Bull. Acad. Polon. Sci. Sér. Sci. Math. 7–8, 337–345 (1981) 39. Pearcy, C.: A complete set of unitary invariants for operators generating finite W ∗ -algebras of Type I. Pac. J. Math. 12, 1405–1416 (1962) 40. Pearcy, C., Ringrose, J.R.: Trace-preserving isomorphisms in finite operator algebras. Am. J. Math. 90, 444–455 (1968) 41. Rickart, C.E.: General theory of Banach algebras. Robert Krieger Publishing Co. Inc., Huntington, N.Y. (1974) 42. Schafhauser, C.: Subalgebras of simple AF-algebras. Ann. Math. 192, 309–352 (2020) 43. Shoda, K.: Einige Sätze über Matrizen. Jpn. J. Math. 13, 361–365 (1936) 44. Skoufranis, P.: Closed unitary and similarity orbits of normal operators in purely infinite C ∗ -algebras. J. Funct. Anal. 265, 474–506 (2013) 45. Skoufranis, P.: Normal limits of nilpotent operators in von Neumann algebras. Integr. Equ. Oper. Theory 77, 407–439 (2013) 46. Skoufranis, P.: Normal limits of nilpotent operators in C ∗ -algebras. J. Oper. Th. 72, 135–158 (2014) 47. Specht, W.: Zur Theorie der Matrizen. II. Jahresber. Dtsch. Math-Ver. 50, 19–23 (1940) 48. Thomsen, K.: Finite sums and products of commutators in inductive limit C ∗ -algebras. Ann. Inst. Fourier, Grenoble 43, 225–249 (1993) 49. Voiculescu, D.: Norm-limits of algebraic operators. Rev. Roumaine Math. Pures Appl. 19, 371–378 (1974) 50. Voiculescu, D.: A non-commutative Weyl-von Neumann theorem. Rev. Roumaine Math. Pures Appl. 21, 97–113 (1976) 51. Williams, L.R.: On quasisimilarity of operators on Hilbert space. PhD thesis, University of Michigan (1976)

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Five Hilbert Space Problems in Operator Algebras 52. Zhai, F.H.: The closures of (U + K)-orbits of class essentially normal operators. Houston J. Math. 30, 1177–1194 (2004) 53. Zhai, F.H., Guo, X.Z.: The closures of (U +K)-orbits of essentially normal triangular operator models. Sci China Math 53, 1045–1066 (2010) 54. Zhai, F.H., Zhao, J.: A new characterization of the closure of the (U + K)-orbit of certain essentially normal operators. Oper. Matrices 2, 455–463 (2008) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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Complex Analysis and Operator Theory (2022) 16 : 76 https://doi.org/10.1007/s11785-022-01250-7

Complex Analysis and Operator Theory

Schanuel’s Lemma for Exact Categories Martin Mathieu1

· Michael Rosbotham1

Received: 22 January 2022 / Accepted: 20 May 2022 © Crown 2022

Abstract We prove an injective version of Schanuel’s lemma from homological algebra in the setting of exact categories. Keywords Cohomological dimension · Injective object · Exact structures Mathematics Subject Classification 18A20 · 18G20 · 18G50

1 Introduction Schanuel’s lemma is a useful tool in homological algebra and category theory. It appears to have come about as a response to a question by Kaplansky, see [4, p. 166], and simplifies the definition of the projective (or, injective) homological dimension in module categories, hence in abelian categories. The typical categories that arise in functional analysis are not abelian but lately, the use of exact structures on additive categories of Banach modules and related ones has been suggested and indeed been exploited successfully. In [3], Bühler develops homological algebra for bounded cohomology in the setting of Quillen’s exact categories. In [1], exact categories of sheaves of operator modules To the memory of Jörg Eschmeier (1956–2021) who was fond of the use of homology in Functional Analysis. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

A previous version of this chapter was published Open Access under a Creative Commons Attribution 4.0 International License at http://link.springer.com/10.1007/s11785-022-01250-7.

B

Martin Mathieu [email protected] Michael Rosbotham [email protected]

1

Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast BT7 1NN, Northern Ireland

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over C* -ringed spaces are studied. Relative cohomology and cohomological dimension for (not necessarily self-adjoint) operator algebras is the topic of [6], see also [8]. In view of this, it seems beneficial to establish an injective version of Schanuel’s lemma for exact categories and show how it yields the injective dimension theorem. When we equip an additive category A with an exact structure we fix a pair (M , P) consisting of a class of monomorphisms M and a class of epimorphisms P such that each μ ∈ M and π ∈ P form a kernel-cokernel pair which we write as μ

E

F

π

G

where E, F and G are objects in A . We require that M and P contain all identity morphisms and are closed under composition, and term their elements as admissible monomorphisms and admissible epimorphisms, respectively. Furthermore, the pushout of an admissible monomorphism along an arbitrary morphism exists and yields an admissible monomorphism, and, likewise, the pull-back of an admissible epimorphism along an arbitrary morphism exists and yields an admissible epimorphism. If these conditions are fulfilled and (M , P) is invariant under isomorphisms, (M , P) is called an exact structure on A and will typically be denoted by Ex. The pair (A , Ex) is said to be an exact category. Unlike in abelian categories not every morphism in an exact category has a canonical factorisation into an epimorphism followed by a monomorphism. One therefore has to restrict to admissible morphisms which are those that arise as μ ◦ π for some μ ∈ M and π ∈ P. (It is easy to check that, once such factorisation exists, it is unique up to unique isomorphism.) The kernel-cokernel pairs replace the usual short exact sequences in abelian categories while long exact sequences are built from admissible morphisms. A very readable introduction into exact categories is given in [2]. In this note, we provide the details of how Schanuel’s lemma works in general exact categories and establish the Injective Dimension Theorem (Theorem 3.5).

2 Preliminaries We include here the necessary terminology and initial results, for a fixed exact category (A , Ex), where Ex = (M , P). Definition 2.1 An object I in an exact category (A , Ex) is M -injective if, when given μ

F and a morphism f ∈ MorA (E, I ), for objects E, F ∈ A , there exists a E morphism g ∈ MorA (F, I ) making the following diagram commutative E

μ

F

f

I

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The exact category has enough M -injectives if, for every E ∈ A , there exist an I. M -injective object I and an admissible monomorphism E We will also make use of the following characterisations of M -injective objects. Proposition 2.2 Let E be an object in an exact category (A , Ex). The following are equivalent. (i) E is M -injective; F, for F ∈ A , has a left inverse; (ii) Every admissible monomorphism E (iii) There exist an M -injective object I ∈ A and a morphism E −→ I with a left inverse (i.e., E is a retract of an M -injective object). The arguments are standard. As exact categories are additive, we can form the product of any two objects (and thus, of any finite number of objects). Proposition 2.3 Let E, F, G be objects in an additive category A . The following are equivalent: (i) F is a product of E and G; (ii) F is a coproduct of E and G; (iii) There exist a kernel-cokernel pair in A , E

μ

F

π

G

(2.1)

and morphisms  μ ∈ MorA (F, E) and  π ∈ MorA (G, F) such that  μ ◦ μ = id E and π ◦  π = idG , and μ ◦  μ+ π ◦ π = id F ; (vi) There exist a kernel-cokernel pair in A , E

μ

F

π

G

(2.2)

and a morphism  μ ∈ MorA (F, E) such that  μ ◦ μ = id E , the identity morphism on E. Moreover, if these equivalent conditions are met, the kernel-cokernel pair in Diagram (2.2) will belong to every exact structure that can be placed on A . Proof Finite products, coproducts and biproducts coincide in an additive category (see, e.g., [7, Proposition 7.1–Corollary 7.3.]), and condition (iii) is just the definition of F being a biproduct of E and G. That condition (iii) is equivalent to condition (iv) can be proven in the exact same way as the ‘Splitting Lemma’ in module theory (see, e.g., [5, Proposition 4.3.]). The final statement of this proposition is a direct consequence of the conditions required for monomorphisms and epimorphisms to be admissible; see [2, Lemma 2.7.] for details. Kernel-cokernel pairs satisfying condition (iii) of Proposition 2.3 are said to be split. For objects E and F in A we will denote their (co)product by E ⊕ F.

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Proposition 2.4 Suppose E

μ

F

π

G is a kernel-cokernel pair in Ex.

(i) For any A ∈ A , there is a kernel-cokernel pair in Ex, F⊕A

E⊕A

G

(ii) If F ∼ = E ⊕ G, then F is M -injective if and only if both E and G are M -injective. Proof We first prove (i). For A ∈ A , there exist split kernel-cokernel pairs E

ι

E⊕A

ρ

and

A

A

θ

F⊕A

τ

F

and π ◦ τ ∈ MorA (F ⊕ A, G) is an admissible epimorphism, as a compostion of morphisms in P. Define ϕ ∈ MorA (E ⊕ A, F ⊕ A) by ϕ = τ ◦ μ ◦ ι+θ ◦ρ using the same notation as in Proposition 2.3. Then (ϕ, π ◦ τ ) is the desired kernelcokernel pair. To show this, it is enough to demonstrate that ϕ is a kernel of π ◦ τ . First note the composition ι + π ◦ (τ ◦ θ ) ◦ ρ = 0. (π ◦ τ ) ◦ ϕ = id F ◦ (π ◦ μ) ◦ Now suppose there exist B ∈ A and a morphism f ∈ MorA (B, F ⊕ A) such that (π ◦ τ ) ◦ f = 0. As μ is a kernel for π, there exists a unique morphism g  ∈ MorA (B, E) such that μ ◦ g  = τ ◦ f . Define g ∈ MorA (B, E ⊕ A) by ◦  θ ◦ f. g = ι ◦ g + ρ Then ϕ ◦ g = ( τ ◦ τ ) ◦ f + (θ ◦  θ ) ◦ f = id F⊕A ◦ f = f . To finish the proof of (i), we show that there is no other morphism h ∈ MorA (B, E ⊕ A) such that ϕ ◦ h = f . Suppose we have such a morphism h. Then,  θ◦ f = θ ◦ ϕ ◦ h = ρ ◦ h, and μ ◦ g  = τ ◦ f = τ ◦ ϕ ◦ h = μ ◦ ι ◦ h, and therefore  ι ◦ h. Combining these facts gives: g = ι+ρ  ◦ ρ) = ι ◦ g  + ρ ◦  θ ◦ f = g, h = id E⊕A ◦ h = (ι ◦ as required. For assertion (ii) suppose F ∼ μ ∈ = E ⊕ G. Then there exist morphisms  π ∈ MorA (G, F) such that  μ ◦ μ = id E and π ◦  π = idG , MorA (F, E) and  and μ ◦  μ+ π ◦ π = id F . In particular, E and G are retracts of F. By Proposition 2.2, if F is M -injective so are E and G. Finally, suppose E and G are M -injective and there is an admissible monomorphism F

f

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where B ∈ A . Because E and G are M -injective, there exist g E ∈ MorA (B, E) such that  μ = g E ◦ f and gG ∈ MorA (B, G) such that and π = g F ◦ f . Let π ◦ gG , then g is a left inverse of f , indeed: g = μ ◦ gE +  π ◦ (gG ◦ f ) = μ ◦  μ+ π ◦ π = id F . g ◦ f = μ ◦ (g E ◦ f ) +  Hence, by Proposition 2.2, F is M -injective.



3 Schanuel’s Lemma Fix an exact category (A , Ex). The following is the injective version of Schanuel’s lemma for exact categories. μ

μ

π

π

Proposition 3.1 Suppose E I F and E I F  are kernel-cokernel pairs in Ex, and that I , I  are M -injective objects. Then I ⊕ F  ∼ = I  ⊕ F in A . Proof First, by the axioms of an exact structure, we can form the following push-out, μ

E

I

μ

(3.1)

h

I

C

h

where every morphism is an admissible monomorphism. Extending this diagram to include the given cokernels, and adding in some zero morphisms, we get the following commutative diagram:

μ

E

0 μ

π

I

F

(3.2)

h

I

h

π

C 0

F

By the universal property of push-outs, there are a unique morphism p ∈ Mor(C, F) such that ph  = 0 and ph = π , and a unique morphism p  ∈ Mor(C, F  ) such that p  h = 0 and p  h  = π  . Hence, we have the following commutative diagram:

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M. Mathieu, M. Rosbotham μ

E

π

I

μ

F id F

h

I

C

h

(3.3)

F

p p

π

F

F

id F 

The result will follow if the middle row and middle column are both split kernelcokernel pairs. As h, h  ∈ M and I , I  are M -injective, this will be the case if both (h  , p) and (h, p  ) are kernel-cokernel pairs. We deal with (h  , p), the other pair is done in the exact same way. To show that (h  , p) is a kernel-cokernel pair, it is enough to verify that p is a cokernel of h  . Suppose there exist an object G ∈ A and a morphism q ∈ Mor(C, G) such that qh  = 0. We are done if we find a unique morphism ψ ∈ Mor(F, G) such that the following diagram is commutative: 0

I

C

h

q 0

F

p

(3.4)

ψ

G

We have (qh)μ = q(hμ) = q(h  μ ) = 0 and, because (μ, π ) is a kernel-cokernel pair, there exists a unique morphism t ∈ Mor(F, G) such that tπ = qh. Therefore, the following diagram is commutative: E

μ

I

μ

h

I

tπ

(3.5)

C

h

q

G

0

By the universal property of push-outs, q is the unique morphism C → G that makes Diagram (3.5) commutative. However, (t p)h = t( ph) = tπ and (t p)h  = t( ph  ) = 0. So, q = t p and setting ψ = t makes Diagram (3.4) commutative. Finally, suppose there also exists t  ∈ Mor(F, G) such that q = t  p. Recalling from Diagram (3.3) that π = ph, we have t  π = t  ( ph) = (t  p)h = (t p)h = t( ph) = tπ, and, because π is an epimorphism, t  = t. Thus, uniqueness has been verified.

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Corollary 3.2 Suppose there is a diagram of morphisms in an exact category (A , Ex) of the form E

I

F

I

F

∼ =

E

such that I and I  are M -injective, the horizontal lines are in Ex and the vertical arrow is an isomorphism. Then I ⊕ F  ∼ = I  ⊕ F in A . We extend Schanuel’s lemma to injective resolutions in Proposition 3.4 below. Recall that a morphism is admissible if it is the composition μ ◦ π for some μ ∈ M and π ∈ P. Such factorisation is unique up to unique isomorphism ([2, Lemma 8.4]). Definition 3.3 For an object E ∈ A , an M -injective resolution of E is a sequence of admissible morphisms of the form: ···

I0

E

···

In

I n−1

∼ =

G0

G1

Gn

G n−1

G n+1

such that, for each n ≥ 0, the object I n is M -injective, and Gn

In

G n+1

forms a kernel-cokernel pair in Ex (this is the exactness condition at I n ). If A has enough M -injectives, we can build an injective resolution for every object in A . Proposition 3.4 Suppose we have the following M -injective resolutions of E, with the factorisation of each admissible morphism included: ···

I0

E

···

In

I n−1

∼ =

G0

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G n−1

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and ···

J0

E

···

Jn

J n−1

∼ =

H0

H1

Hn

H n−1

H n+1

Then, for each n ≥ 1, we have isomorphisms I 0 ⊕J 1 ⊕I 2 ⊕ · · · ⊕J 2n−1 ⊕G 2n ∼ = J 0 ⊕I 1 ⊕J 2 ⊕ · · · ⊕I 2n−1 ⊕H 2n and I 0 ⊕J 1 ⊕I 2 ⊕ · · · ⊕J 2n−1 ⊕I 2n ⊕H 2n+1 ∼ = J 0 ⊕I 1 ⊕J 2 ⊕ · · · ⊕I 2n−1 ⊕J 2n ⊕G 2n+1 . Proof We prove this by induction. For n = 1, first note that Corollary 3.2, applied to the diagram G0

I0

G1

J0

H1

∼ =

H0

gives I 0 ⊕H 1 ∼ = J 0 ⊕G 1 . By Proposition 2.4, there is a diagram of the form I 0 ⊕H 1

I 0 ⊕J 1

H2

J 0 ⊕I 1

G2

∼ =

J 0 ⊕G 1

and Corollary 3.2 gives I 0 ⊕J 1 ⊕G 2 ∼ = J 0 ⊕I 1 ⊕H 2 . To finish the proof for n = 1, we again apply Proposition 2.4 followed by Corollary 3.2, to get a diagram I 0 ⊕J 1 ⊕G 2

I 0 ⊕J 1 ⊕I 2

G3

J 0 ⊕I 1 ⊕J 2

H3

∼ =

J 0 ⊕I 1 ⊕H 2

and an isomorphism I 0 ⊕J 1 ⊕I 2 ⊕H 3 ∼ = J 0 ⊕I 1 ⊕J 2 ⊕G 3 .

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Assume the result holds some n ≥ 1. By Proposition 2.4, there is a diagram of the form I 0 ⊕ · · · ⊕ I 2n ⊕H 2n+1

I 0 ⊕ · · · ⊕ I 2n ⊕J 2n+1

H 2(n+1)

J 0 ⊕ · · · ⊕ J 2n ⊕I 2n+1

G 2(n+1)

∼ =

J 0 ⊕ · · · ⊕ J 2n ⊕G 2n+1 and Corollary 3.2 gives

I 0 ⊕J 1 ⊕I 2 ⊕ · · · ⊕J 2(n+1)−1 ⊕G 2(n+1) ∼ = J 0 ⊕I 1 ⊕J 2 ⊕ · · · ⊕I 2(n+1)−1 ⊕H 2(n+1) . One final application of Proposition 2.4 yields the following diagram: I 0 ⊕J 1 ⊕I 2 ⊕ · · · ⊕J 2n+1 ⊕G 2(n+1)

∼ =

J 0 ⊕I 1 ⊕J 2 ⊕ · · · ⊕I 2n+1 ⊕H 2(n+1)

I 0 ⊕J 1 ⊕I 2 ⊕ · · · ⊕J 2n+1 ⊕I 2(n+1)

J 0 ⊕I 1 ⊕J 2 ⊕ · · · ⊕I 2n+1 ⊕J 2(n+1)

G 2(n+1)+1

H 2(n+1)+1

By Corollary 3.2, I 0 ⊕J 1 ⊕I 2 ⊕ · · · ⊕J 2(n+1)−1 ⊕I 2(n+1) ⊕H 2(n+1)+1 ∼ = J 0 ⊕I 1 ⊕J 2 ⊕ · · · ⊕I 2(n+1)−1 ⊕J 2(n+1) ⊕G 2(n+1)+1 

as required. We can now prove the Injective Dimension Theorem.

Theorem 3.5 Let M be the class of admissible monomorphisms in an exact category (A , Ex). Suppose A has enough M -injectives. The following are equivalent for n ≥ 1 and every E ∈ A . (i) If there is an exact sequence of admissible morphisms E

I0

···

I n−1

F

with each I m , 0 ≤ m ≤ n − 1 injective, then F must be injective;

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(ii) There is an exact sequence of admissible morphisms ···

I0

E

I n−1

(3.7)

In

with each I m , 0 ≤ m ≤ n injective. Proof Let E ∈ A . First we show (i) implies (ii). As A has enough M -injectives, we can build an M -injective resolution of E: ···

J0

E

Jn

J n−1

···

∼ =

G0

G1

Gn

G n−1

Relabel J k as I k for all 0 ≤ k ≤ n − 1 and G n as I n , this gives an exact sequence as in Diagram (3.7), and I n must be M -injective, by condition (i). Now suppose that condition (ii) holds. There must exist an injective resolution of E of the form ···

J0

E

Jn

J n−1

∼ =

···

∼ =

H0

H1

Jn

H n−1

and for any exact sequence as in Diagram (3.6), with each I n injective, there exists an injective resolution ···

I0

E

In

I n−1

···

∼ =

G0

G1

Gn

G n−1

with G n = F. By Proposition 3.4, there exists a kernel-cokernel pair F

μ

I

π

G

and a morphism  μ ∈ MorA (I , F) such that  μ ◦ μ = id F , and I is a finite product of M -injective objects. Then by Proposition 2.4, I is injective and  μ is a left inverse for μ, hence, by Proposition 2.2, F is M -injective. Definition 3.6 Let M be the class of admissible monomorphisms in an exact category (A , Ex). We say E ∈ A has finite M -injective dimension if there exists an exact sequence of admissible morphisms as in Diagram (3.7) with all I m M -injective. If E

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is of finite M -injective dimension we write InjM -dim (E) = 0 if E is M -injective and InjM -dim (E) = n if E is not M -injective and n is the smallest natural number such that there exists an exact sequence of admissible morphisms as in Diagram (3.7) where every I m is M -injective. If E is not of finite M -injective dimension, we write InjM -dim (E) = ∞. The global dimension of the exact category (A , Ex) is   sup InjM -dim (E) | E ∈ A ∈ N0 ∪ {∞}. Remark 3.7 The M -injective dimension of an object E in an exact category (A , Ex) can be obtained by examining any of its M -injective resolutions. Indeed, suppose the following is an M -injective resolution of E (with the factorisation of each admissible morphism included): ···

J0

E

Jn

J n−1

···

∼ =

G0

G1

G n−1

Gn

Then, by Theorem 3.5, InjM -dim (E) ≤ n if and only if G n is M -injective. The original version of Schanuel’s lemma is formulated for projective resolutions, see, e.g., [4, Lemma 5.1] or [9, Theorem 3.41]. An analogous version using the epimorphisms in the class P can be obtained in any exact category with exact structure (M , P). Data Availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

References 1. Ara, P., Mathieu, M.: Sheaf cohomology for C ∗ -algebras. Memoir in preparation 2. Bühler, T.: Exact categories. Expo. Math. 28(1), 1–69 (2010) 3. Bühler, T.: On the algebraic foundations of bounded cohomology. Mem. Amer. Math. Soc., 214(1006):xxii+97, (2011) 4. Lam, T.Y.: Lectures on modules and rings. Graduate Texts in Mathematics, vol. 189. Springer-Verlag, New York (1999) 5. Mac Lane, S.: Homology. Classics in Mathematics. Springer-Verlag, Berlin, (1995). Reprint of the 1975 edition

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M. Mathieu, M. Rosbotham 6. Mathieu, M., Rosbotham, M.: Exact structures for operator modules. Canad. J. Math., to appear, arXiv:2105.05006 7. Osborne, M.S.: Basic homological algebra. Graduate Texts in Mathematics, vol. 196. Springer-Verlag, New York (2000) 8. Rosbotham, M.: Cohomological dimension for C ∗ -algebras. PhD. Thesis, Queen’s University Belfast, Belfast, (2021) 9. Rotman, J.J.: Notes on homological algebras. Van Nostrand Reinhold Mathematical Studies, No. 26. Van Nostrand Reinhold Co., New York-Toronto-London, (1970) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Complex Analysis and Operator Theory (2023) 17:40 https://doi.org/10.1007/s11785-023-01346-8

Complex Analysis and Operator Theory

Riesz–Kolmogorov Type Compactness Criteria in Function Spaces with Applications Mishko Mitkovski1 · Cody B. Stockdale1 Brett D. Wick2

· Nathan A. Wagner2 ·

Received: 29 April 2022 / Accepted: 8 March 2023 / Published online: 25 March 2023 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023

Abstract We present forms of the classical Riesz–Kolmogorov theorem for compactness that are applicable in a wide variety of settings. In particular, our theorems apply to classify the precompact subsets of the Lebesgue space L 2 , Paley–Wiener spaces, weighted Bargmann–Fock spaces, and a scale of weighted Besov–Sobolev spaces of holomorphic functions that includes weighted Bergman spaces of general domains as well as the Hardy space and the Dirichlet space. We apply the compactness criteria to characterize the compact Toeplitz operators on the Bergman space, deduce the compactness of Hankel operators on the Hardy space, and obtain general umbrella theorems. Keywords Compactness · Framed spaces · Spaces of holomorphic functions · Toeplitz operators · Hankel operators Mathematics Subject Classification Primary 46B50; Secondary 46E15 · 46E30 · 47B35

Communicated by Raul curto. This article is part of the topical collection “Multivariable Operator Theory. The J"org Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

B

Cody B. Stockdale [email protected] Mishko Mitkovski [email protected] Nathan A. Wagner [email protected] Brett D. Wick [email protected]

1

School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC 29634, USA

2

Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, MO 63130, USA

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1 Introduction The Riesz–Kolmogorov theorem is a fundamental result in analysis that characterizes the precompact subsets of L p (Rn ). The statement is as follows. Theorem A. Let p ∈ [1, ∞). A set F ⊆ L p (Rn ) is precompact if and only if  lim sup

R→∞ f ∈F

|x|>R

| f (x)| p d x = 0

and  lim sup

|h|→0 f ∈F

Rn

| f (x − h) − f (x)| p d x = 0.

Theorem A is classically presented with the additional condition of F being a bounded subset of L p (Rn ), however this condition is redundant as it is implied by the other two conditions of the theorem, see [21]. The Riesz–Kolmogorov criterion is named after the work of Kolmogorov and Riesz from [27] and [34], respectively. In [27], Kolmogorov proved a version of Theorem A in the case when 1 < p < ∞ and all functions in F are supported on a common bounded set. Riesz independently discovered a version of Theorem A in [34] in the case 1 ≤ p < ∞. See [20] for a more detailed historical accounting of this topic. The Riesz–Kolmogorov characterization has been adapted to handle many other situations. For example, Fréchet proved a version of the theorem that includes arbitrary p > 0 in [12], Phillips characterized precompact subsets of L p with respect to arbitrary measure spaces in [32], Weil obtained a version of the theorem in the setting of locally compact groups in [41], and Takahashi proved a version of the theorem for Orlicz spaces in [37]. There are also versions of the precompactness criterion for weighted settings in [7, 19] and matrix weighted settings in [29]. See [3, 5, 6, 10, 11, 14–17, 25, 26, 31, 33] for further references. As shown in [21, Theorem 4] or [4, p. 466] the Riesz–Kolmogorov theorem can be proved using the following more abstract compactness criterion of Mazur. Theorem B. Let X be a Banach space and suppose that {Tn }∞ n=1 is a sequence of compact operators on X that converges to the identity in the strong operator topology; that is, limn→∞ Tn f − f X = 0 for all f ∈ X . A bounded set F ⊆ X is precompact if and only if lim sup Tn f − f X = 0.

n→∞ f ∈F

In [32, Theorem 3.7], Phillips proved a very similar theorem and applied it to characterize the precompact subsets of L p with respect to arbitrary measure spaces. In [36], Sudakov showed that if at least one of the operators Tn does not have 1 as an eigenvalue, then the boundedness condition on F in Theorem B is not needed (see also [21, p. 90–91]).

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Proof of Theorem B First suppose that F is precompact. By the uniform boundedness principle, B := supn∈N Tn X →X < ∞. Let ε > 0. Since F is precompact, there exists a finite subset { f 1 , . . . , f K } ⊆ F such that for each f ∈ F there exists 1 ≤ j ≤ K with  f j − f X < 3ε min(B, 1). Choose N so that Tn f j − f j X < 3ε for all n ≥ N and all 1 ≤ j ≤ K . For f ∈ F and n ≥ N , let 1 ≤ j ≤ K be such that  f j − f X < 3ε and note Tn f − f X ≤ Tn ( f − f j )X + Tn f j − f j X +  f j − f X < ε. Assuming the uniform strong operator topology convergence of Tn to the identity, we have that for any ε > 0 there exists N ∈ N such that dist( f , TN F) < ε for all f ∈ F. Since F is bounded and TN is compact, TN F is precompact. The precompactness of F follows.

We observe that a slight strengthening of Mazur’s Theorem B can be obtained in a Hilbert space setting by relaxing the norm conditions involving Tn f − f X to quadratic form conditions. This result is likely already known, but we were unable to find a reference. Theorem 1.1 Let H be a Hilbert space and suppose that {Tn }∞ n=1 is a sequence of compact operators on H such that limn→∞ Tn f − f , f H = 0 for all f ∈ H. A bounded set F ⊆ H is precompact if and only if lim sup |Tn f − f , f H | = 0.

n→∞ f ∈F

The usual way to derive the Riesz–Kolmogorov theorem when all functions in F are supported on a common bounded set from Mazur’s Theorem B is to use the averaging operators 

1

Tn f (x) =

V (B(x,

1 n ))

B(x, n1 )

f (y) dy = f ∗

1 V (B(0, n1 ))

χ B(0, 1 ) (x), n

where V denotes the Lebesgue measure, see [21, 36]. Loosely speaking, the Riesz– Kolmogorov theorem says that for a set F to be compact, all of its elements need to have uniformly small tails on the spatial side (first condition) and on the frequency side (second condition). Therefore, to use Mazur’s theorem to derive a compactness criterion of Riesz–Kolmogorov type, one must use operators Tn that “truncate" in both of the spatial and frequency domains. The simplest application of this idea gives the following theorem. Theorem C. A bounded set F ⊆ L 2 (Rn ) is precompact if and only if  lim sup

R→∞ f ∈F

 | f (x)| d x = 0 and 2

|x|>R

lim sup

R→∞ f ∈F

|ξ |>R

| fˆ(ξ )|2 dξ = 0.

Theorem C inspired the work of Dörfler, Feichtinger, and Gröchenig in [9] where they derived compactness criteria for modulation spaces and co-orbit spaces using

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the short-time Fourier transform. Recall that the short-time Fourier transform Sφ : 2 n 2 2n function φ ∈ L 2 (Rn ) is defined by Sφ f (a, b) :=  L (R ) with a window L (R ) → 2πibx φ(x − a). The most classical window φ is the f , φ(a,b) , where φ(a,b) (x) := e Gaussian window. The following is the compactness characterization in terms of the short-time Fourier transform obtained in [9]. Theorem D. A bounded set F ⊆ L 2 (Rn ) is precompact if and only if  lim sup |Sφ f (a, b)|2 dadb = 0. R→∞ f ∈F

R2n \[−R,R]2n

Since that Gabor basis simultaneously respects both the spatial and the frequency behavior, only one uniform decay condition is needed in Theorem D. Our first main result is a direct generalization of Theorem D. It turns out that one can replace the Gabor system {φ(a,b) : (a, b) ∈ R2n } with any continuous Parseval frame. Recall that for a Hilbert space H, a collection {k x } ⊆ H indexed by a measure space (X , μ) is a continuous Parseval frame for H if  2  f H = | f , k x H |2 dμ(x) X

for each f ∈ H. If {k x }x∈X is a continuous Parseval frame for a Hilbert space H, then  f =  f , k x H k x dμ(x) X

for each f ∈ H. By an exhaustion for X we mean a sequence of subsets of X , {Fn }∞ n=1 , F = X . such that Fn ⊆ Fn+1 for each n and ∞ n=1 n Theorem 1.2 Let H be a Hilbert space with a continuous Parseval frame {k x } indexed by a measure space (X , μ). Suppose that supx∈X k x H < ∞ and that X has an exhaustion {Fn }∞ n=1 such that μ(Fn ) < ∞ for all n ∈ N. A bounded set F ⊆ H is precompact if and only if  | f , k x H |2 dμ(x) = 0. lim sup n→∞ f ∈F

X \Fn

Assuming more on the the frame {k x }x∈X , we may relax the finite measure assumption of Theorem 1.2. The following frame-theoretic statement relies on Theorem 1.1. Theorem 1.3 Let H be a Hilbert space equipped with a continuous Parseval frame {k x } indexed by an unbounded metric measure space (X , d, μ) satisfying for some w : X → (0, ∞)  −1 |k x , k y H |w(x) dμ(x) < ∞, sup w(y) y∈X

X

lim sup w(y)

R→∞ y∈X

−1

 X \B(y,R)

|k x , k y H |w(x) dμ(x) = 0, and

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|k x , k y H | → 0 as d(x, y) → ∞. Suppose that X has an exhaustion {Fn }∞ n=1 such that lim

d(y,y0 )→∞

μ(Fn ∩ B(y, R)) = 0

for some (any) y0 ∈ X , some (any) R > 0, and all n ∈ N. A bounded set F ⊆ H is precompact if and only if  lim sup

n→∞ f ∈F

| f , k x H |2 dμ(x) = 0.

X \Fn

A version of Theorem 1.2 also holds in appropriate Banach space settings. For a Banach space X , p ∈ [1, ∞), and a measure space (X , μ), we say ({ f x }x∈X , { f x∗ }x∈X ) ⊆ X × X ∗ is a continuous frame for X with respect to L p (X , μ) if (1) sup  f x∗ X →C < ∞, x∈X

(2) the function x →  f , f x∗ is in L p (X , μ) for all f ∈ X , (3) there exist c, C > 0 such that c f X ≤  f , f x∗  L p (X ,μ) ≤ C f X for all f ∈ X , and (4) each f ∈ X satisfies  f = X

 f , f x∗ f x dμ(x).

Note that, unlike in the Hilbert space setting, the existence of f x ∈ X such that (4) holds is not guaranteed from condition (3) in general Banach spaces, so their existence is assumed. Theorem 1.4 Let p ∈ [1, ∞) and X be a reflexive Banach space equipped with a continuous frame ({ f x }, { f x∗ }) with respect to L p (X , μ). Suppose that X has an exhaustion {Fn }∞ n=1 such that μ(Fn ) < ∞ for all n ∈ N. A bounded set F ⊆ X is precompact if and only if  lim sup

n→∞ f ∈F

X \Fn

| f , f x∗ | p dμ(x) = 0.

We next extend our compactness criterion to function spaces which are not necessarily framed spaces. More precisely, we consider Banach function spaces consisting of functions defined on a metric measure space (X , d, μ) with a Radon measure μ.

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Theorem 1.5 Let X be a Banach space of functions on a metric measure space (X , d, μ) with a compact exhaustion {Fn }∞ n=1 . Let p ∈ [1, ∞) and suppose that there is a point x0 ∈ X and linear maps D j : X → C(X ), j = 1, 2, . . . , N + M, such that p

 f X =

N   j=1

|D j f (x)| p dμ(x) + X

N +M

|D j f (x0 )| p

j=N +1

for all f ∈ X . Suppose also that (i) F ⊆ X is bounded, (ii) for each set Fn and 1 ≤ j ≤ N , the collection of functions {D j f : f ∈ F} is equicontinuous on Fn , and (iii) for each x ∈ X and 1 ≤ j ≤ N + M, sup f ∈F |D j f (x)| < ∞. Then F is precompact if and only if

lim sup

N  

n→∞ f ∈F j=1

X \Fn

|D j f (x)| p dμ(x) = 0.

Note that Theorem 1.5 generalizes our Theorem 1.2 in the case when x → k x is continuous by taking N = 1, M = 0, p = 2, and D f (x) =  f , k x H . 1.1 Compactness Criteria in Function Spaces We now show how our results can be used to establish compactness criteria in various function spaces including the Lebesgue space L 2 (Rn ), Paley–Wiener spaces, weighted Bargmann-Fock spaces, and a scale of weighted Besov–Sobolev spaces that includes weighted Bergman spaces, the Hardy space, and the Dirichlet space. This list of applications is certainly not exhaustive—we only mention a focused selection of well-known examples in which our results apply. 1.1.1 The Lebesgue Space L2 (Rn ) We already presented several alternative compactness characterizations in L 2 (Rn ) besides the classical Riesz–Kolmogorov theorem. Our Theorem 1.2 shows that every continuous Parseval frame provides a new compactness criterion. For example, if we use the continuous Parseval frame of wavelets indexed as usual by the ax + b group := (0, ∞) × Rn equipped with the usual hyperbolic measure and metric, we Rn+1 + obtain a compactness characterization in terms of the continuous wavelet transform. Namely, a bounded set F ⊆ L 2 (Rn ) is compact if and only if the continuous wavelet transforms of all the elements of F have uniformly null tails. This fact seems to have been first noticed in [9, Theorem 3].

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1.1.2 Paley–Wiener Spaces Recall that for a Borel measurable set E ⊆ Rn with finite Lebesgue measure, the Paley–Wiener space PW(E) is the subspace of L 2 (Rn ) consisting of functions whose Fourier transform is supported in E. In the case when E = [−a, a]n , all elements of PW(E) can be extended to entire functions with exponential type no greater than a. Every Paley–Wiener space is a reproducing kernel Hilbert space, and an application of the Plancherel theorem shows that the normalized reproducing kernels form a continuous Parseval frame for PW(E). Therefore our Theorem 1.2 immediately gives the following simple criterion for compactness in Paley–Wiener spaces. Theorem 1.6 A bounded set F ⊆ PW(E) is precompact if and only if  | f (x)|2 d x = 0. lim sup R→∞ f ∈F

|x|>R

We remark that in the classical case E = [−a, a]n this fact is also immediate from Theorem C since the second condition of that theorem is automatically satisfied by a family of functions in the Paley–Wiener space PW([−a, a]n ). 1.1.3 Weighted Bargmann–Fock spaces The weighted Bargmann-Fock space Fφ (Cn ) is the space of all entire functions f : Cn → C satisfying the integrability condition   f 2φ := | f (z)|2 e−2φ(z) d V (z) < ∞, Cn

where φ : Cn → R is a plurisubharmonic function such that for all z ∈ Cn ¯  i∂ ∂|z| ¯ 2, i∂ ∂φ in the sense of positive currents. The classical Bargmann-Fock space F(Cn ) is an important special case obtained when φ(z) = π2 |z|2 . Equipped with the norm ·φ , the weighted Bargmann-Fock space Fφ (Cn ) is a φ reproducing kernel Hilbert space. We will denote its reproducing kernel at z by K z . It is easy to see that the normalized reproducing kernels indexed by the metric measure φ space (Cn , Vφ , d), where d Vφ (z) := K z 2φ e−2φ(z) d V (z) and d is the usual Euclidean n metric on C , form a continuous Parseval frame. A straightforward application of our Theorem 1.2 gives the following criterion for compactness in weighted BargmannFock spaces. Theorem 1.7 A bounded set F ⊆ Fφ (Cn ) is precompact if and only if  lim sup

R→∞ f ∈F

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|z|>R

| f (z)|2 e−2φ(z) d V (z) = 0.

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1.1.4 Weighted Besov–Sobolev Spaces Let D ⊆ Cn be a bounded domain, p ∈ [1, ∞),  and σ be an integrable weight on D, that is, σ is positive almost everywhere and D σ d V < ∞. In order for our spaces to be Banach spaces, we additionally suppose that for any compact K  D, there exists a constant C K , p,σ > 0 such that  | f (z)| p ≤ C K , p,σ

| f |pσ d V

(1)

D

 for every z ∈ K and all holomorphic functions f for which D | f | p σ d V < ∞. For J ∈ N and such D ⊆ Cn , p ∈ [1, ∞), and such integrable weights σ , we define the p,J weighted Besov–Sobolev space Bσ (D) to be the space of holomorphic f : D → C such that ⎛

 f B p,J (D) σ



p  ∂α f   ∂α f





:= ⎝ (z ) +

∂z α 0

α D ∂z |α| δ} and notice that D0 = D. The following is a consequence of Theorem 1.5. p,J

Theorem 1.8 A bounded set F ⊆ Bσ (D) is precompact if and only if lim sup

 

δ→0+ f ∈F |α|=J

α

∂ f

α D\Dδ ∂z

p

σ d V = 0.

Theorem 1.8 immediately gives compactness criteria for various function spaces including weighted Bergman spaces, the Hardy space, and the Dirichlet space. Let D be a strongly pseudoconvex domain with a C 2 defining function ρ, that is, ρ is a C 2 plurisubharmonic function such that with D = {z ∈ Cn : ρ(z) < 0} and ∇ρ(z) = 0 for z ∈ ∂ D. For p ∈ [1, ∞) and t > −1, define the weighted Bergman p space of D, At (D), to be the space of holomorphic f : D → C such that

  f Atp (D) :=

1/ p | f | p (−ρ)t d V

< ∞.

D

Note that these spaces generalize the radially weighted Bergman spaces of the unit p ball Bn ⊆ Cn with weight (1 − |z|2 )t . We denote A p (D) := A0 (D). p

Corollary 1.9 A bounded set F ⊆ At (D) is precompact if and only if  | f | p (−ρ)t d V = 0.

lim sup

δ→0+ f ∈F

D\Dδ

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We also apply Theorem 1.8 to weighted Bergman spaces with respect to B p weights; see [39] for a definition of B p weights on C 2 domains. For a strongly pseudoconvex C 2 domain D, p ∈ [1, ∞), and σ ∈ B p , define the weighted Bergman space of D p with respect to σ , Aσ (D), to be the space of holomorphic f : D → C such that

 f

p Aσ (D)

1/ p

:=

| f | σ dV

< ∞.

p

D

p

Notice that if σ ≡ 1, then Aσ (D) = A p (D). p

Corollary 1.10 A bounded set F ⊆ Aσ (D) is precompact if and only if  lim sup | f | p σ d V = 0. δ→0+ f ∈F

D\Dδ

Remark 1.11 The hypothesis that F is bounded can be removed in both Corollary 1.9 and Corollary 1.10 since the boundedness of F is implied by the uniformly vanishing integral condition. We illustrate the proof when σ is a B p weight and note that the obvious modifications can be made when the weight is as in Corollary 1.9. Take ε = 1 and fix the corresponding δ as in the proof of Theorem 1.8 (see Sect. 2). It suffices to show that  sup | f | p σ d V < ∞. f ∈F

Dδ

We claim that the functions in F are uniformly bounded on the compact set ∂ Dδ/2 . Indeed, if z ∈ ∂ Dδ/2 , then the Euclidean ball B(z, δ/4) is contained in D \ Dδ . We then estimate for such a point z and f ∈ F as follows  1 | f | dV | f (z)| ≤ V (B(z, δ/4)) B(z,δ/4)  ≤ Cδ | f | dV D\Dδ

 ≤ Cδ

1/ p 

σ −1/( p−1) d V

| f |pσ d V D\Dδ

1/ p

D

≤ Cδ, p,σ . By the maximum principle, the functions in F are uniformly bounded on Dδ , and thus the above inequality holds. Remark 1.12 We note that compactness criteria for A p (Bn ) follow from either of Theorem 1.8 or Theorem 1.4. An application of Theorem 1.8 with D = Bn , p ∈ 1 , and J = 0 shows that F ⊆ A p (Bn ) is precompact if and only if [1, ∞), σ ≡ V (B n)  lim sup

r →1− f ∈F

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| f (w)| p dv(w) = 0,

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where dv represents normalized Lebesgue measure on the unit ball. On the other hand, if p ∈ (1, ∞), then A p (Bn ) is a reflexive Banach space with a continuous frame ( p)

( p )

( p)

{kw , kw } with respect to L p (Bn , dλ), where kw (z) :=

n+1 

(1−|w|2 ) p denotes the “ p(1−zw)n+1 2 −(n+1) |w| ) dv(w) denotes

normalized” reproducing kernel at w and dλ(w) := (1 − the hyperbolic measure on Bn . Theorem 1.4 gives that F ⊆ A p (Bn ) is precompact if and only if  lim sup

r →1− f ∈F



Bn \r Bn

(p ) p | f , kw | dλ(w) = 0.

We also remark that Theorems 1.2 and 1.3 both apply in the Hilbert space case p = 2. Remark 1.13 Both Corollary 1.9 and Corollary 1.10 apply in the case of weighted Bergman spaces of Bn with radial weights σ (z) = (1 − |z|2 )t for t ∈ (−1, p − 1), since σ is a B p weight for this range of t. Corollary 1.9 extends this fact to all t > −1, and Corollary 1.10 generalizes the result to arbitrary B p weights. The Hardy space, H2 (Bn ), is the space of holomorphic f : Bn → C such that

  f H2 (Bn ) := sup

0 0, set D(z, r ) := {w ∈ Bn : β(z, w) < r }. It is well-known that for any r > 0, there exists Cr > 0 such that Cr−1 ≤

K z A2 (Bn ) K w A2 (Bn )

≤ Cr

(2)

for all z, w ∈ Bn with β(z, w) < r , see [42, Lemma 2.20]. It is also well-known that λ(D(z, r )) = λ(D(w, r )) for all z, w ∈ Bn and r > 0, see [42, Lemma 1.24]. The following corollary is immediate from Theorem 1.4 or Theorem 1.8 (see Remark 1.12). Corollary 3.1 Let p ∈ (1, ∞) and T be a bounded operator on A p (Bn ). Then T is compact on A p (Bn ) if and only if  ( p ) lim sup |T f , kw A2 (Bn ) | p dλ(w) = 0. R→∞ f ∈A p (Bn ) Bn \D(0,R)  f A p (Bn )≤1

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Proposition 3.2 Let p ∈ (1, ∞) and T be a bounded operator on A p (Bn ) such that 

 sup

|T ∗ k z , kw A2 (Bn ) |

Bn

z∈Bn

K z A2 (Bn )

1− 2δ( p−1)2 p(n+1)

dλ(w) < ∞

K w A2 (Bn )

(3)

for some δ > 0. If 

 lim sup

R→∞ z∈Bn

Bn \D(0,R)

|T k z , kw A2 (Bn ) |

K z A2 (Bn )

1− 2δ( p−1)2 p(n+1)

dλ(w) = 0, (4)

K w A2 (Bn )

then T is compact on A p (Bn ). Proof We verify the condition of Corollary 3.1. Let f ∈ A p (Bn ) with  f A p (Bn ) ≤ 1. Then, by Fubini’s theorem     ( p ) ( p ) A2 (Bn ) = T  f , k z A2 (Bn ) k z dλ(z) , kw T f , kw Bn

 =

( p )

Bn

 f , kz

A2 (Bn )



( p)

(p ) A2 (Bn ) T k z , kw A2 (Bn ) dλ(z).

By Hölder’s inequality and the assumption (3), we have 

(p ) |T f ,kw A2 (Bn ) | p ≤

⎛ ⎜ ≤⎝





( p )

Bn

| f , k z



( p)

(p ) |T k z , kw A2 (Bn ) |

Bn

⎛  ⎜ ×⎜ ⎝

Bn

p ( p) ( p  ) A2 (Bn ) ||T k z , kw A2 (Bn ) | dλ(z) 2 2δ p − p (n+1) w A2 (B ) n 2δ 2− p2 − p (n+1) z A2 (B ) n

K 

⎞ pp ⎟ dλ(z)⎠

K 

⎞ ⎞ pp 2 2δ p − p (n+1)  K ⎟ z A2 (B ) ⎜ ⎟ ( p) ( p  ) ( p ) n |T k z , kw A2 (Bn ) || f , k z A2 (Bn ) | p ⎝ ⎠ dλ(z)⎟ 2δ ⎠ 2− p2 − p (n+1) K w A2 (B ) ⎛

n

≤C

p p

 Bn

⎛ ( p)



( p )

(p ) |T k z , kw A2 (Bn ) || f , k z

2 2δ p − p (n+1) z A2 (B ) n 2δ 2− p2 − p (n+1) w A2 (B ) n

⎜ K  A2 (Bn ) | p ⎝ K 

( p)

⎞ pp ⎟ ⎠

1−

dλ(z),

2 

p where C is the finite constant from (3). Using k z A2 (Bn ) = K z A2 (B , one has n)   (p ) that for any R > 0, Bn \D(0,R) |T f , kw A2 (Bn ) | p dλ(w) can be controlled above by

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Riesz–Kolmogorov Type Compactness Criteria...

C

p p



 Bn \D(0,R) Bn

=C

p p

 Bn

⎛

( p)

( p )



(p ) |T k z , kw A2 (Bn ) || f , k z

( p ) | f , k z A2 (Bn ) | p

⎞ pp 2 2δ p − p (n+1) K  z ⎜ ⎟ A2 (Bn ) A2 (Bn ) | p ⎝ ⎠ dλ(z)dλ(w) 2δ 2− p2 − p (n+1) K w A2 (B ) 

 Bn \D(0,R)

|T k z , kw A2 (Bn ) |

n

K z A2 (Bn )

1− 2δ( p−1) p (n+1)

dλ(w)dλ(z).

K w A2 (Bn )

Let ε > 0 be given. Apply the assumption (4) to get a constant R > 0 such that 1− 2δ( p−1)2

p(n+1)  K z A2 (B ) n dλ(w) < εp for all z ∈ Bn . Bn \D(0,R) |T k z , kw A2 (Bn ) | K w  2 A (Bn )

C

Then 



Bn \D(0,R)

|T f | p dv =

p



Bn \D(0,R)

(p ) |T f , kw A2 (Bn ) | p dλ(w) < C p

p

ε C



p p

Bn

| f | p dv ≤ ε.



Therefore T is compact by Corollary 3.1.

We now characterize the compact operators within a class of bounded and localized operators on A p (Bn ). Theorem 3.3 Let p ∈ (1, ∞) and T be a bounded operator on A p (Bn ) satisfying (3) and 

 lim sup

R→∞ z∈Bn

Bn \D(z,R)

|T k z , kw A2 (Bn ) |

K z A2 (Bn )

1− 2δ( p−1)2 p(n+1)

dλ(w) = 0. (5)

K w A2 (Bn )

(z) → 0 as z → 1− . Then T is compact on A p (Bn ) if and only if T Proof The forward direction is clear (in particular, use the well-known fact that the ( p) p-normalized kernels k z converge weakly to 0 in A p (Bn ) as z → 1− ), so we only consider the reverse direction. We will verify condition (4) and establish the theorem by applying Proposition 3.2. We will use the following condition which is implied by the vanishing Berezin transform hypothesis (see [23]): for each R > 0, we have lim

sup

z→1− w∈D(z,R)

|T k z , kw A2 (Bn ) | = 0.

(6)

Let ε > 0 be given. Apply assumption (5) to find R0 > 0 such that 

 Bn \D(z,R0 )

|T k z , kw A2 (Bn ) |

K z A2 (Bn ) K w A2 (Bn )

1− 2δ( p−1)2 p(n+1)

dλ(w)
0 such that ε for all z ∈ Bn \D(0, N0 ) and w ∈ |T k z , kw A2 (Bn ) | < 2δ( p−1)2 1−

2C R

0

p(n+1)

λ(D(0,R0 ))

D(z, R0 ), where C R0 is the constant given in (2). Set R = N0 + R0 .

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Write 

 Bn \D(0,R)

|T k z , kw A2 (Bn ) |

K z A2 (Bn )

1− 2δ( p−1)2 p(n+1)

dλ(w)

K w A2 (Bn ) 

 =

Bn \(D(0,R)∪D(z,R0 ))

|T k z , kw A2 (Bn ) |

K z A2 (Bn ) 

(Bn \D(0,R))∩D(z,R0 )

p(n+1)

dλ(w)

K w A2 (Bn )

 +

1− 2δ( p−1)2

|T k z , kw A2 (Bn ) |

K z A2 (Bn )

1− 2δ( p−1)2 p(n+1)

dλ(w).

K w A2 (Bn )

The first term above is controlled by the choice of R0 : 

 Bn \(D(0,R)∪D(z,R0 ))

|T k z , kw A2 (Bn ) |

1− 2δ( p−1)2

K z A2 (Bn )

p(n+1)

dλ(w)
1.

Let us point out that the value of the integral does not only depend on the location of the point of discontinuity ξ0 := eit0 , but additionally on the argument t0 defining ξ0 (due to the factor t0 /2). It is also worth to mention that the function  a →

√ z dz z−a t0

|z|=1

is holomorphic outside T. So (in case |a| < 1) the right hand side, which a priori is only holomorphic on the slitted disk, has an analytic continuation to D. The reason is that the function

1−w h : D → C, w → w log 1+w



√ is an even function, and so h( z) has this extension. Finally, for later purposes involving the principal value of the Cauchy integral (Remark 4), we mention that in the case |a| < 1 and t0 < arg a < t0 + 2π , √



√ −it0 /2 √ √ √ ae −1 1 − a e−it0 /2 + 2 a iπ −4eit0 /2 − 2 a log √ −it /2 = −4eit0 /2 − 2 a log √ −it /2 0 0 1+ a e ae +1

where we have used log(−1) = −iπ . Hence 1 2πi

 |z|=1

√ √ z a + R(a, t0 ) if |a| < 1 dz = z−a 0 + R(a, t0 ) if |a| > 1 t0

for some “error” term. Proposition 2 Let a = eiα . If t0 < α < t0 + 2π , then1  PV

t0

√ −it0 /2 √ z √ √ ae −1 + a iπ. dz = −4eit0 /2 − 2 a log √ −it /2 0 z−a ae +1 t0

|z|=1

For stylistic reasons, we will skip in the sequel the upper index t0 when writing √ z. No ambiguities will arise.

1 The notion PV



will be defined later in (5.1).

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Fig. 1 The integration countour

2 Using Cauchy’s Integral Theorem √ √ Step 1 Recall that for −π ≤ θ < π, z = |z|eiθ/2 is the main branch of the square root of z = |z|eiθ . Let ε,ρ be the curve given by the boundary of {z ∈ C : ρ ≤ |z| ≤ 1, | arg z| < π − ε}. (see Fig. 1). Suppose that 0 < a < 1. Let f (z) =

ρ→0 ε,ρ ε→0

z z−a .



 lim

√

f (z)dz =

Then

 |z|=1

f (z)dz +

−1

 =

|t|1/2 i dt + t −a

0

 |z|=1

f (z)dz + 2i

0

−1



−1 0

|t|1/2 (−i) dt t −a

|t|1/2 dt. t −a

Hence, by substituting t = −s, 



|z|=1

1

f (z)dz = 2πi Res ( f , a) + 2i 0

Thus, by substituting s 1/2 = x, and for 0 < a < 1 and 0 <  |z|=1

√ f (z)dz = 2πi a + 4i



s 1/2 ds. s+a

(2.1)

√ a < 1,

1

x2 dx 2 0 x +a  1 √ a dx 1− 2 = 2πi a + 4i x +a 0 √ √ √ = 2πi a + 4i − 4i a arctan(1/ a).

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The Cauchy Transform of the Square Root Function on the…

Using that arctan x + arctan(1/x) = π/2 for x > 0, we obtain for 0 < a < 1  |z|=1

√ √ f (z)dz = 4i + 4i a arctan a.

(2.2)

If a > 1, then Res ( f , a) = 0 and so (by (2.1))  |z|=1

Since for |x| < 1,

√ √ f (z)dz = 4i − 4i a arctan(1/ a).

√ arctan x √ x

holomorphy of a → g(a) :=

= 

∞

n 1 n n=0 (−1) 2n+1 x =: G(x), we deduce from √ z |z|=1 z−a dz in D that for every a ∈ D, or |a| > 1,

 g(a) =

the

4i + 4ia G(a) if |a| < 1 4i − 4i G(1/a) if |a| > 1.

√ √ Step 2 Next we deal with the branch given by z := |z|eit/2 , where t0 ≤ t < t0 + 2π , for t0 ∈ R, so that the (unique) point of discontinuity of the integrand is at eit0 . Then, by using the substitution θ := t − t0 − π , we obtain  |z|=1

√  π z ei(θ+t0 +π )/2 i(θ+t0 +π ) dz = i e dθ i(θ+t 0 +π ) − a z−a −π e  π eiθ/2 i(t0 +π )/2 ieiθ dθ =e iθ −i(t0 +π ) −π e − ae √  z i(t0 +π )/2 =e dz |z|=1 z − b = ei(t0 +π )/2 g(b)

with b := ae−i(t0 +π ) = −ae−it0 . Using that for the main branch of the logarithm and z∈D

 ∞ 1 − iz (−1)n 2n+1 i = z = zG(z 2 ), arctan z = log 2 1 + iz 2n + 1 n=0

we can rewrite for a ∈ D the value of   g (a) :=

|z|=1

as

Reprinted from the journal

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√ z dz z−a

R. Mortini, R. Rupp

   g (a) = ieit0 /2 4i + 4ibG(b) = −4eit0 /2 + 4e−it0 /2 a G(ae−i(t0 +π ) ) √

√ 1 − a e−it0 /2 it0 /2 = −4e − 2 a log √ 1 + a e−it0 /2

(2.3)

√ where a is a fixed value of the square root of a ∈ D (implying that the point of the form (1 − w)/(1 + w) at which we evaluate the logarithm, belongs to the right half plane). For |a| > 1 we get    g (a) = ieit0 /2 4i − 4i G(1/b) ei(t0 +π ) = −4eit0 /2 + 4eit0 /2 G a √ −it0 /2

√ a e +1 it0 /2 = −4e + 2 a log √ −it /2 , a e 0 −1 as this time the point (w + 1)/(w − 1) belongs to the right half plane.  g (a) = g(a) and that for t0 = 0 and |a| < 1, Also observe that for t0 = −π , we get  we have  |z|=1

√

√ z √ 1+ a . dz = −4 + 2 a log √ z−a 1− a

3 Approach Using Series For β > 0, t0 ∈ [0, 2π [ and t0 < arg z < t0 + 2π let logt0 (z) := log |z| + i arg z be the holomorphic branch of the logarithm on the domain C\{r eit0 : r ≥ 0}. Put z β := exp(β logt0 z). Then, for a ∈ D,  |z|=1

zβ dz = z−a



t0 +2π

eiβt ieit dt = i it e −a t0  t0 +2π ∞  =i an ei(β−n)t dt n=0



t0 +2π

t0

eiβt dt 1 − ae−it

t0

⎧ ∞  a n −int0 ⎪ ⎨eiβt0 (e2πβi − 1) if β ∈ /N e β −n = n=0 ⎪ ⎩ 2πi a β if β ∈ N.

580

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The Cauchy Transform of the Square Root Function on the…

Now for β = 1/2, one deduces  |z|=1

√ ∞ ∞ z   (ae−it0 )n (ae−it0 )n+1 it0 /2 it0 /2 it0 /2 dz = 4e = −4e + 4e z−a 2n − 1 2n + 1 n=0

n=0

= −4eit0 /2 + 4ae−it0 /2

∞  (ae−it0 )n n=0

2n + 1

= −4eit0 /2 + 4ae−it0 /2 G(−ae−it0 ). This is (2.3). Similarly for |a| > 1.



4 Third Approach by a Change of the Variable Suppose that |a| = 1 and that z = eit ∈ T. By putting w := eit/2 for t0 ≤ t < t0 +2π , we have w 2 = z. Now the image curve  ⊆ T of the "broken" circle eit , t0 < t < √ t0 + 2π , with respect to z is a half-circle on T with starting-point η0 := eit0 /2 and i(t +2π )/2 = −η0 . end-point η1 := e 0 Hence  I :=

|z|=1

√  z 2w 2 dz = dw. 2 z−a  w −a

Depending on whether |a| < 1 or |a| > 1, we will now determine a primitive of √ 2 q(w) := w2w a be a fixed square root of the 2 −a in suitable neighborhoods of . Let complex number a. We consider on D− := C \] − ∞, 0] the principal branch of the logarithm log z = log |z| + i arg z with −π < arg z < π . Then log 1 = 0 and log z is a bijective holomorphic map of D− onto the strip −π < Im z < π . Now, on the one hand,



a a w2 = 2 1+ 2 =2 1− 2 2 w −a w −a a − w2 √

1 a 1 , = 2 1− +√ √ 2 a−w a+w and on the other hand

2

w2 a = 2 1 + w2 − a w2 − a √

1 a 1 . = 2 1+ √ − √ 2 w− a w+ a

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Formally, we obtain the primitives

√ √ a+w , P1 (w) := 2w − a log √ a−w and P2 (w) := 2w −

√

√ w+ a . a log √ w− a

To √ defined, we have to check for which w the data 1 (w) := √ make sure these are well a+w w+ a √ belong to D− , so that log j (w) defines a holomorphic √ or

(w) := 2 a−w w− a function. √ In fact, if |a| > 1, then the numerator of the real part of 1 (w) = √a+w equals a−w √ 2 |a| − |w| and 1 maps {|w| < |a|} onto the right-half plane. √ a √ equals |w|2 − |a| If |a| < 1, then the numerator of the real part of 2 (w) = w+ w− a √ and 2 maps {|w| > |a|} onto the right-half plane. of q on the Thus in both √ √ cases log j (w) is well defined and so P1 is a primitive disk {|w| < |a|} for |a| > 1 and P2 is a primitive of q on {|w| > |a|} for |a| < 1. Now we restrict to |w| = 1. Case 1 |a| > 1.  |z|=1

√  z 2w 2 dz = dw 2 z−a  w −a = P1 (η1 ) − P1 (η0 ) = P1 (−η0 ) − P1 (η0 )

√ −it0 /2 √ ae +1 . = −4eit0 /2 + 2 a log √ −it /2 a e 0 −1

Here we have used that for points u, v ∈ {ξ ∈ C : Re ξ > 0} we have log u − log v = log(u/v),

(4.1)

as u/v ∈ D− . Case 2 |a| < 1. Then, by (4.1) again,  |z|=1

√  z 2w 2 dz = dw 2 z−a  w −a = P2 (η1 ) − P2 (η0 ) = P2 (−η0 ) − P2 (η0 ) √

√ 1 − a e−it0 /2 it0 /2 . = −4e − 2 a log √ 1 + a e−it0 /2

Remark 3 This approach via primitives also allows us to calculate the integral in case √ z is a function defined outside a Jordan arc connecting 0 and infinity, and which

582

Reprinted from the journal

The Cauchy Transform of the Square Root Function on the… a=exp(i ) a

2 a 1

exp(it_0)

Fig. 2 The integration curve

intersects the unit circle T exactly once at eit0 . We may take for example, an infinite spiral.

5 The Case |a| = 1 √ z dz does not exist whenever a ∈ T. Using Cauchy’s |z|=1 z − a principal value though, which is given by 

Of course, the integral

 PV |ξ |=1

f (ξ ) dξ := lim ε→0 ξ −a



|ξ |=1 | arg ξ −arg a|>ε

f (ξ ) dξ, ξ −a

(5.1)

the orientation of the circle being, as usual, counterclockwise, one obtains a positive result: if f ∈ A(D) (where A(D) denotes the disk-algebra of all functions continuous on the closure of D and holomorphic in D), then  PV

|ξ |=1

f (ξ ) dξ = iπ f (a) ξ −a

(see [2, Proposition 4.79]). √ √ Suppose now that our branch of the square root is given by z := |z|eit/2 , where t0 < t < t0 +2π , for t0 ∈ R, so that the (unique) point of discontinuity of the integrand is at eit0 . Let a = eiα , t0 < α < t0 + 2π , a1 = ei(α−ε) , a2 = ei(α+ε) , ε > 0 small (see Fig. 2). √ √ Since for |w| = 1, but w = ± a, the number ξ := √a+w lies on the imaginary a−w axis, log ξ is well defined for the principal branch of the logarithm. Moreover, since √ a+w √ ∈ D− whenever a−w  √  √ w ∈  := C\ {r a : r ≥ 1} ∪ {s a : s ≤ −1} ,

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the function √

√ a+w P1 (w) := 2w − a log √ , w ∈ , a−w

f (w) d is a primitive for w2w 2 −a (use that dw log f (w) = f (w) whenever f is holomorphic and f (w) ∈ D− ). it/2 Thus t → P1 (eit/2 ) is a primitive (in t) for q(t) := eeit −a ieit . Consequently, 2

 PV |z|=1

 t0 +2π

√  α−ε z dz = lim q(t)dt + q(t)dt ε→0 z−a α+ε t0





= lim P1 ei(t0 +2π)/2 − P1 ei(α+ε)/2 + P1 ei(α−ε)/2 − P1 eit0 /2 ε→0

Now 

eiα/2 + ei(α+ε)/2 log iα/2 e − ei(α+ε)/2



1 + eiε/2 1 − eiε/2   

= log

:=x

and 

eiα/2 + ei(α−ε)/2 log iα/2 e − ei(α−ε)/2



1 + e−iε/2 = log 1 − e−iε/2   

:=y

Note that x/y = −1 , that x, y are purely imaginary and that y = x. More precisely, x = i cot(ε/4). So, as our arguments for the principal branch of log z belong to ] − π, π [, we get arg y = − arg x and so log x − log y = 2i arg x = 2iπ/2 = iπ . Consequently,  PV |z|=1

√ z √ dz = P1 (ei(t0 +2π )/2 ) − P1 (eit0 /2 ) + a iπ z−a

√ −it0 /2 √ √ ae −1 + a iπ. = −4eit0 /2 − 2 a log √ −it /2 ae 0 + 1 

Remark 4 This result in Proposition 2 is not a surprise, as a quite general theorem due to Privalov (see [1,p. 431]) tells us that for almost all a ∈ T  lim

z∈D→a∈T T n.t.

f (ξ ) dξ = P V ξ −z

584

 T

f (ξ ) + iπ f (a) ξ −a

(5.2)

Reprinted from the journal

The Cauchy Transform of the Square Root Function on the…

and 

f (ξ ) dξ = P V ξ −z

lim

z∈C\D→a∈T T n.t.

 T

f (ξ ) − iπ f (a) ξ −a

whenever f is Lebesgue integrable on T. Here n.t. denotes the non-tangential limit ([2, Chap. 27]). √  z Observation 5 If a = eit0 , then P V dz does not exist. z−a |z|=1

Proof In fact, if α := arg a = t0 ,  PV |z|=1

√  α+2π −ε z dz = lim q(t)dt z−a ε→0 α+ε

P1 (ei(α+2π −ε)/2 ) − P1 (ei(α+ε)/2 ) ε→0      1 − e−iε/2 1 + eiε/2 iα/2 iα/2 = −4e −e lim log − log ε→0 1 + e−iε/2 1 − eiε/2

= lim



= −4eiα/2 + eiα/2 lim log cot 2 (ε/4) unbounded. ε→0



6 Appendix √ z is holomorphic on  := C\{r eit0 : r ≥ 0}. √ Proof We first note that h : z = |z|eit → |z|eit/2 is continuous as a function of its polar-coordinates r > 0 and t ∈ ]t0 , t0 + 2π [, hence continuous in z. If q(w) = w 2 , then q ◦ h = id on . Therefore, for z, z 0 ∈ , z = z 0 , w := h(z), w0 := h(z 0 ), the continuity and injectivity of h imply that

Appendix 6 The function z →

t0

h(z) − h(z 0 ) = z − z0 =

1 q(h(z))−q(h(z 0 )) h(z)−h(z 0 )

1

−→

q(w)−q(w0 ) w→w0 w−w0

1 . 2w0

t0 √ Here are some additional features of the function z. By definition, we obtain for t0 √ each t0 ∈ [0, 4π [ a different branch, where for t0 ∈ [2π, 4π [, the branch z is the

opposite of that for t0 − 2π . Note that for t0 ∈ {4kπ − π : k ∈ Z} (and only for these t0 √ √ points) we have z = z. In fact, if t0 = 4kπ − π and 4kπ − π < s < 4kπ + π then with t := s − 4kπ we have −π < t < π and so z = eis = eit and eis/2 = eit/2 . t0 √ √ Hence z = z. Similarly, t0 +4π

Reprinted from the journal

t √ z = 0 √ z.

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Hence the branches display a 4π -periodic behaviour. Under these normalizations, t0 √ 1 = 1 if t < 2kπ < t + 2π (k even) , t0 √ 1 = −1 if t < 2kπ < t + 2π (k 0 0 0 0 t0 √  k 1 = (−1) if t ∈ {2kπ : k ∈ Z}. odd), and 0

By noticing that t0 < 2kπ < t0 + 2π ⇐⇒ 2π(k − 1) < t0 < 2kπ , we also t0 √ have that this branch z of the square-root function maps  bijectively onto the half-plane   t0 t0 w∈C: < arg w < + π (mod 2π ) , 2 2 and that the function Q : w → w2 restricted to this half-plane is its inverse. The other right-inverse (defined on ) of Q is of course the opposite of this branch.

Acknowledgements We thank the referee for a careful reading of the manuscript. Data Availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References 1. Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable. AMS, Providence (1969) 2. Mortini, R., Rupp, R.: Extension Problems and Stable Ranks, A Space Odyssey, p. 2200. Birkhäuser, Cham (2021) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Complex Analysis and Operator Theory (2022) 16: 92 https://doi.org/10.1007/s11785-022-01262-3

Complex Analysis and Operator Theory

Function Theory from Tensor Algebras Paul S. Muhly1

· Baruch Solel2

Received: 26 March 2022 / Accepted: 6 July 2022 / Published online: 18 August 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract We survey connections between our work on tensor algebras over C ∗ -correspondences and free analysis. The goal is to expose the fundamental features that underwrite both N C-function theory and our tensorial function theory: ‘Respecting intertwiners’ and ‘local uniform boundedness’. Keywords Free analysis · Noncommutative function theory · Tensor algebras · Matricial sets · Matricial functions Mathematics Subject Classification Primary 46L15 · 47L30 · 47D99 · 15A72 · 16S20; Secondary 15A24 · 16D20 · 16E99 · 16G20 · 30A98 · 46J15 · 46K50 · 46T25

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . 2 Algebraic Background and Notational Conventions 2.1 Rings and Things . . . . . . . . . . . . . . . 2.2 Bimodules . . . . . . . . . . . . . . . . . . . 2.3 Categories . . . . . . . . . . . . . . . . . . . 3 Quivers and Tensors . . . . . . . . . . . . . . . . 3.1 Quiver Algebras as Tensor Algebras . . . . .

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To the memory of Jörg Eschmeier. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

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Paul S. Muhly [email protected] Baruch Solel [email protected]

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Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA

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Department of Mathematics, Technion, 32000 Haifa, Israel

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P. S. Muhly, B. Solel 3.2 Representations of Tensor Algebras . . . . . . . . . . . . . . . . . . 4 The Representation of T A (M) as Functions  . . π. . . . . . . . . . . . . . 4.1  The End(V )-valued Functions on π ∈ A  M Determined by T A (M) π 4.2 π ∈ A  M as a Function of π . . . . . . . . . . . . . . . . . . . . 5 The Analytic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction The broad general setting of the work we describe here may be formulated as follows. The Setting: Let R and S be two unital rings and let X denote the set of all the unital homomorphisms ϕ : R → S. Then each r ∈ R defines an S-valued function,  r : X → S via the equation  r (ϕ) := ϕ(r ),

r ∈ R.

(1.1)

Two natural questions are: What sort of function is r and what does the collection  tell us about R? of these functions, R, The reader may well react negatively to such a broad general formulation of the setting and its “natural questions”, thinking, perhaps: “What possibly can one conclude, of any significance, from studying such a broad framework?” Our response is to point out that, of course, one has to work with specific examples, but beyond that, in surprising generality, the setting provides what we like to regard as organizing principals for function theory, and has a long and storied history, dating back at least to 1882 when Dedekind and Weber, in their famous Theorie der algebraischen Functionen einer Veränderlichen treated the case when R is the function field of an algebraic curve, S is C (with ∞ adjoined to accommodate poles) and X is the Riemann surface for R. In fact, on page 236 of [9], they defined the Riemann surface to be all the homomorphisms from R into C ∪ {∞}. A mere 11 years later, Hilbert investigated the case where R is the ring of invariants for what we, today, would call a reductive algebraic group acting on a polynomial algebra in some number, n, of variables [20]. He proved that the ring is finitely generated and that X is the space of its maximal ideals realized as an algebraic variety in Cn . Along the way, for the purposes of his analysis, he proved the famed Nullstellensatz. Today we learn that if R is commutative, then the setting is that of affine schemes  is as described in [12, page 8 ff.]. The space X is the space of S-valued points and R the sheaf of regular functions on X . Noncommutative schemes are an active area of current research (see, e.g., [46, 47]), but is not as well developed as the commutative theory. Nevertheless, it is clear that an understanding of Eq. (1.1) plays an important role in the theory. Functional analysts recognize the setting as coming from the Gelfand theory of commutative Banach algebras. For the noncommutative theory, we want to call special attention to the papers of Masamichi Takesaki [48] and Klaus Bichteler [5] which describe how to reconstruct a C ∗ -algebra from its space of representations on Hilbert

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space. The fundamental idea behind what Takesaki calls an admissable operator field [48, p. 371] plays a central role in our work. These fields respect intertwiners in a sense that we will make precise in Definition 4.3. This notion seems to have occurred to Joseph Taylor also, about the same time. He used it to develop a spectral theory for tuples of not-necessarily-commuting operators on a Banach space [49–52]. Taylor’s theory went largely unnoticed until the end of the last century when interest in it exploded, thanks in large part to advances in free probability, matrix inequalities, control theory and related developments in operator theory. The number of contributors to the theory and the list of contributions now are too large to expatiate upon here. But we do want to point out that the principal features of Taylor’s theory were developed, refined and polished further by Dmitri Kalyusnyi-Verbovetski and Victor Vinnikov who, in [22], give a comprehensive introduction to Taylor’s theory that nowadays is universally known as Noncommutative (or NC-) Function Theory. Our principal contribution to the theory is [33], in which we developed a function theory based on tensor algebras over C ∗ -correspondences. We began to investigate such tensor algebras in [31]. Without going into detail at this point, the purely algebraic setting can be described roughly as follows. Let A be an algebra and let M be a bimodule over A. Then one may build the tensor algebra T A (M) over A determined  with by M in a well-known fashion, which we will review in Sect. 3. For each π ∈ A values in End(Vπ ) and for each F ∈ T A (M) one obtains a function π : M π → End(Vπ ), F where M π is a certain vector space of intertwiners between π and an induced representation determined by M and π . (See Definition 3.14 for the precise formulas.) It π behaves remarkably like a polynomial function that one obtains by turns out that F evaluating a polynomial at points in an appropriate affine space (provided one does not insist that the values commute). Further, as we showed in [33], it is possible to π to establish a apply the technology developed by Taylor op. cit. to our functions F bonafide theory of analytic built  from T A (M).  functions π and  The M We like to think of   End(Vπ ) as bundles fibred over A. π ∈ A π∈A  π bundle π ∈ A M is viewed as a generalized affine space the open subsets of which, U := π ∈ A U(π ), are where our analytic functions are defined. The functions f that we consider respect the fibration and so are families f = { f π }π ∈ A, with f π defined on U(π ) and having values in End(Vπ ). The basic properties of f on which the entire theory is built are 1. f respects intertwiners in the sense of Definition 4.3, and 2. f is locally uniformly bounded in a sense made precise in Eq. (5.7). Under these innocuous hypotheses the otherwise lifeless family of functions f admits  so as to make each fibre, f π , a bonafide a power series expansion that is fibred over A holomorphic function from U(π ) to End(Vπ ). In this article, we want to recapitulate and explain parts of [33], focusing on the special case when the C ∗ -correspondence is built from a finite quiver. Our hope is to stimulate algebraists to consider some of the problems and issues we have faced and, going in the other direction, we want to encourage the growing community of

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NC-function theorists to consider the enormous body of work devoted to quiver representation theory. To illustrate the value of such “cross-fertilization”, we note that our earliest contribution to the theory of tensor algebras, [31], was inspired in large part by the serendipitous receipt of a preprint of Mihai Pimsner’s fundamental paper [37] shortly after learning from [13] that in 1947 Gerhard Hochschild [21] had shown that every finite dimensional algebra over an algebraically closed field can be represented as a quotient of a tensor algebra built from a bimodule over a semi-simple algebra. Indeed, if B is the algebra in question, with radical R, then B/R is semisimple and R/R 2 is naturally a bimodule over B/R. So in this case, we have A = B/R, M = R/R 2 , and B is realized as a quotient of T A (M). Further, thanks to the paper by Cecil Nesbitt and William Scott [36] which, among several things, shows that a finite dimensional algebra over an algebraically closed field is Morita equivalent to a so-called basic algebra, i.e., one that is commutative modulo its radical, one may assume that B/R is a direct sum of copies of the field.1 Once this is done, one may unravel the structure of R/R 2 by calculating the dimensions of the intertwiner spaces of minimal idempotents in B/R. These observations led Peter Gabriel to introduce in [14] quivers to organize the calculations of the intertwiner dimensions and therewith to describe the structure of finite dimensional algebras. To be clear and to be a bit more accurate from the point of view of history, we note that quivers are nothing more than directed graphs. And beginning with Hochschild’s work in 1947, if not before, directed graphs were used to unravel the structure of finite dimensional algebras. The more poetic term “quiver” was advanced by Gabriel because quivers are what carry arrows and the entire analysis of the structure of algebras was reduced to chasing diagrams built out of arrows. It is also arguable that the reason that the algebra community adopted Gabriel’s terminology is the success he had with it in contributing to the proof of one of the Brauer-Thrall conjectures, an important problem in algebra of that time.

2 Algebraic Background and Notational Conventions 2.1 Rings and Things We follow the notation and terminology of [4], as much as possible, supplemented at times by [6] and [7]. Also, so as not to become too lost in technicalities, all of our rings, modules, etc. will have an underlying structure as vector spaces over the complex numbers C. In the case of rings, C is always contained in the center. Thus we use the terms “ring” and “algebra” (over C) interchangeably. Until Sect. 5, we shall ignore topological considerations or mention them only in passing. All rings will be unital. If V is a vector space, then the ring of linear transformations on V will be denoted End(V ) and the elements of End(V ) will act on the left.2 1 Of course, Morita’s notion of equivalence wasn’t formally presented by him until 1957. The presentation of Nesbitt and Scott proceeds directly by showing how to a produce a basic algebra from the given algebra and how to build an isomorphism between the module structure of the original algebra and the basic algebra. 2 So, to be in strict compliance with [4], we should be writing Endl (V ).

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We use the terms “representation” and “module” interchangeably in the following standard way. If A is an algebra and π : A → End(V ) is a representation of A on V , then we will think of V as a left A-module via π , i.e. π(a)ξ = a · ξ, and conversely given a (left) A-module V , we will read this equation from right to left to define π . This “module = representation” convention will be made in other situations, as well, but we shall identify them explicitly. If π and τ are two representations of an algebra A on the vector spaces Vπ and Vτ , then we write I(π, τ ) := {X ∈ HomC (Vπ , Vτ ) | τ (a)X = X π(a), a ∈ A}.

(2.1)

This space is called the intertwiner space from π to τ and its elements are called intertwiners. Note that I(π, τ ) is oriented; its elements are maps from the space of π , Vπ , to the space of τ , Vτ . Observe that I(π, π ) is known variously as the commutant of π and the centralizer of π . 2.2 Bimodules If A is a not-necessarily-commutative ring, then Aop will denote the set A with the same abelian group structure, but with the multiplication reversed. That is, if ◦ denotes (temporarily) the multiplication in A and ∗ the multiplication in Aop , then a ∗b = b ◦a for all a, b ∈ A. Definition 2.1 If A and B are two algebras, we shall say that the vector space M is an (A, B)-bimodule (also called a correspondence from A to B) in case there are two homomorphisms, ρ : A → End(M) and σ : B op → End(M), such that ρ(a)σ (b) = σ (b)ρ(a) for all a ∈ A, and b ∈ B. That is, M is a bimodule in case ρ and σ have commuting images. Thus, following our “module = representation” convention, we will write a · v · b = ρ(a)σ (b)v. If A = B, we will sometimes refer to M simply as a bimodule (or correspondence) over A. Often when discussing an (A, A)-bimodule M determined by two homomorphisms ρ and σ , the homomorphisms are suppressed. However, if it is important to keep them in mind, then we shall write ρ Mσ in place of M and A M A . When it is not necessary to name ρ or σ explicitly, we shall simply write ρ(a)ξ = a · ξ and σ (a)ξ = ξ · a. Note that if A is any algebra, then one may view A as a bimodule over itself. Here M is A viewed as a vector space and A acts on M via left and right multiplication, i.e., the homomorphisms ρ and σ are defined by the equations ρ(a)X = a X and σ (a)X = Xa. This bimodule is sometimes called the regular bimodule over A and is denoted A A A . An important example of (A, A)-bimodule is an A-ring.

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Definition 2.2 Suppose A and A are (not-necessarily commutative) unital algebras over C and that π : A → A is a unital homomorphism. Then A is called an A-ring (via π ). Evidently, the (A, A)-bimodule structure on A is obtained by composing π with the left and right actions of A on A. That is, ρ(a) · X := π(a)X , for a ∈ A and X ∈ A, while σ (a) · (X ) := X π(a).(Here, the \cdot, · , denotes the module product while the unalloyed product stands for the product in the A-ring.) Alternatively, ρ = l ◦ π and σ = r ◦ π , where l and r denote left and right multiplication in A. Note that the fact that σ is a homomorphism of Aop is reflected in the associativity of multiplication in A. The notion of an A-ring is an “upgrade” of the (perhaps more) familiar notion of an R-algebra, where R is a commutative ring: a not-necessarily-commutative ring B is called an R-algebra in case there is an embedding, α : R → Z(B), of R into the center, Z(B), of B. Often, mention of α is omitted, but note that different α’s may lead to different R-algebras. In a like manner the structure of the A-ring A depends on π . Note, too, that even if A is commutative, π need not carry A into the center of A. This fact is, perhaps, most easily seen using quivers. See Remarks 3.6. 2.3 Categories We shall be a bit informal about the categories that we consider here. This may lead to some ambiguities, but we hope the reader will perceive them as minor.  the category of all the representations of The most important category for us is A, 0  an algebra A. The object space, A , is the collection of all the representations; the 1 , are the intertwiners. The problem that we want pass over in silence morphisms, A 0  is that A is not a set. We could “make it into a set” by picking out a representative 0 , but we would like to allow for the of the isomorphism class of each element of A possibility of having two representations being isomorphic without being identical. Thus, if we were to be completely meticulous, we would specify in advance some set 0 that includes an exemplar of each representation of A. We will not pursue or for A make such specifications in this note. The same convention will be adopted for each of the other categories we consider. Also, we note in passing that in [33] it was advantageous to work with various  We shall not pursue those matters here, either. subcategories of A. The category of bimodules over an algebra A, will be denoted by M A . The objects of M A , M0A , are, of course, the (A, A)-bimodules that we just defined. The morphisms of M A , M1A , are the homomorphisms between bimodules that preserve all the structure, i.e., C-linear maps f : ρ Mσ → ρ  Mσ  such that ρ  (a) ◦ f = f ◦ ρ(a) and σ  (b) ◦ f = f ◦ σ (b) for all a, b ∈ A. We shall often refer to such maps simply as bimodule maps. The category of A-rings will be denoted by R A . The objects of R A , R0A , are the A-rings, themselves, while the morphisms, R1A , are the algebra homomorphisms that preserve the actions of the π ’s. That is, if for i = 1, 2, Bi is an A-ring via the homomorphism πi , then an A-ring morphism from B1 to B2 is an algebra homomorphism f : B1 → B2 such that f ◦ π1 = π2 ◦ f .

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Evidently, there is a natural functor F : R A → M A , the forgetful functor. It looks like the identity map: It takes an object B ∈ R A (with corresponding π ) and ignores “most” of its multiplicative structure, allowing only right and left multiplications by elements in the range of π . That then makes B a bimodule over A. Of course, A-ring homomorphisms become (A, A)-bimodule homomorphisms. However, there may be (A, A)-bimodule homomorphisms that do not come from A-ring homomorphisms. Our goal in Sect. 3 is to show that the process of building a tensor algebra, T A (M), from a bimodule M ∈ M A provides an adjoint to the forgetful functor in a sense discussed in [45, Definition 4.1.1] (See also [30, page 80].) “Adjunction” seems to underlie the general problem with which we began this paper.

3 Quivers and Tensors The term “quiver” is synonymous with “directed graph”, as we noted in the introduction. For the uninitiated, quivers provide excellent examples supporting the general theory of tensor algebras, which, in turn, may appear at first sight to be excessively picky – introducing distinctions without evident differences. So, while we have no intention to recapitulate the theory of tensor algebras ab ovo, we do want to survey some high points that will help make it clear how to think about a tensor algebra as an algebra of functions on its space of representations. Thus we introduce quiver algebras and tensor algebras simultaneously. 3.1 Quiver Algebras as Tensor Algebras Definition 3.1 A quiver is a quadruple Q := {Q 0 , Q 1 , r , s}, where Q 0 and Q 1 are sets called, respectively, the vertices and edges of Q; and r and s are two functions from Q 1 to Q 0 , called the range and source maps. While some study quivers whose vertices and edges may be infinite sets, we will restrict our attention here to those where the edges and vertices are finite in number. We shall write C(Q 0 ) for all the complex-valued functions on Q 0 . Under pointwise operations, C(Q 0 ) is a finite dimensional, commutative, semi-simple algebra over C. It also is a C ∗ -algebra, where the norm is the sup-norm3 , and the involution is point-wise complex conjugation. We shall also write E(Q 1 ) for the space of all complex-valued functions on Q 1 . By our assumption that Q is a finite quiver, E(Q 1 ) is a finite dimensional vector space under pointwise operations. It is also a bimodule over C(Q 0 ), where the left and right actions of C(Q 0 ) on E(Q 1 ) are given by the formula a · ξ · b(ε) := a(r (ε))ξ(ε)b(s(ε)),

a, b ∈ C(Q 0 ), ξ ∈ E(Q 1 ), ε ∈ Q 1 ; (3.1)

that is, in the notation of the Sect. 2.2, ρ(a) is multiplication by a ◦ r while σ (b) is multiplication by b ◦ s. The fact that E(Q 1 ) is a bimodule over C(Q 0 ) is an immediate verification. 3 We’ll emphasize the norm later, in Sect. 5.

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Suppose next, for the sake of comparison, that U and V are bimodules over the not-necessarily-commutative algebra, A. We want to consider first the vector space tensor product of U with V , U ⊗ V . We follow the standard point of view that that U ⊗ V is really a pair, (T , t), where T is a vector space and t is a bilinear map from U × V to T that satisfies this universal mapping property: Given a bilinear map b from U × V to a vector space W , then there is a unique linear map a from T to W such that b = a ◦ t. One can show that such a pair (T , t) always exists and is unique up to a linear isomorphism that respects the bilinear maps involved. It is customary to write t(u, v) as u ⊗ v. The set of all such u ⊗ v, i.e., the range of t, is called the set of decomposable tensors. Note that the set of decomposable tensors is not a linearly independent set. However, it does span U ⊗ V . In the quiver setting, as one can easily see, the underlying vector space of the tensor product of E(Q 1 ) with itself, E(Q 1 )⊗E(Q 1 ), may be taken to be the space of C-valued functions on Q 1 × Q 1 , with the bilinear map t : E(Q 1 ) × E(Q 1 ) → E(Q 1 ) ⊗ E(Q 1 ) defined by the formula t(ξ, η)(δ, ε) := ξ(δ)η(ε), (δ, ε) ∈ Q 1 × Q 1 . Returning to A and the bimodules U and V , it is important to note that while the vector space tensor product U ⊗ V is a bimodule over A in the obvious fashion (a · (u ⊗ v) · b := (a · u) ⊗ (v · b)), there is no a priori relation between u · a ⊗ v and u ⊗ a · v. In order to have associativity in the tensor algebra of a bimodule, which we are in the process of developing, it is important to have equality. So, we force it: Let R be the subbimodule of U ⊗ V spanned by the differences {(u · a ⊗ v) − (u ⊗ a · v) | u ∈ U , v ∈ V , a ∈ A}. The quotient (U ⊗ V )/R is a bimodule over A, denoted U ⊗ A V in which u · a ⊗ v = u ⊗ a · v.4 It is called the tensor product of U with V that is balanced over A. We note, for good measure, that in U ⊗ A V , the equations a · (u ⊗ v) = (a · u) ⊗ v and (u ⊗ v) · a = u ⊗ (v · a) still hold. In the case when U = V = E(Q 1 ), it is easy to see that R is complemented in E(Q 1 ) ⊗ E(Q 1 ) by the subspace E(Q 2 ) consisting of all complex-valued functions on Q 1 × Q 1 that are supported on Q 2 := {(δ, ε) ∈ Q 1 × Q 1 | s(δ) = r (ε)}, the so-called paths of length 2. Thus, we may view E(Q 1 ) ⊗C(Q 0 ) E(Q 1 ) as E(Q 2 ). The associativity of tensor products requires a pause. Given three bimodules over A, U ,V ,W , we may form the two expressions: (U ⊗ A V )⊗ A W and U ⊗ A (V ⊗ A W ). They are not, in general, equal. They are, however, isomorphic. Indeed, the map (u ⊗ v) ⊗ w → u ⊗ (v ⊗ w) extends to a well-defined isomorphism, φ, from (U ⊗ A V ) ⊗ A W to U ⊗ A (V ⊗ A W ). Further, φ is the only linear map from (U ⊗ A V ) ⊗ A W to U ⊗ A (V ⊗ A W ) that satisfies φ((u ⊗ v) ⊗ w) = u ⊗ (v ⊗ w) for all decomposable tensors (u ⊗ v) ⊗ w) [6, II.3.8]. Consequently, we shall abuse notation, and write (U ⊗ A V ) ⊗ A W = U ⊗ A (V ⊗ A W ). In fact we will do away with the parentheses altogether whenever it is convenient, writing U ⊗ A V ⊗ A W for (U ⊗ A V ) ⊗ A W . This abuse persists for decomposable tensors, u ⊗ v ⊗ w = (u ⊗ v) ⊗ w, and tensor products involving more than three (A, A)-bimodules. The bottom line, up to this point, is that if we have n(> 1) (A, A)-bimodules n and a map m from U1 × U2 × · · · × Un to the (A, A)-bimodule W that is {Ui }i=1 4 Of course, we are dropping any notational reference to the fact that U ⊗ V consists of cosets from A

(U ⊗ V )/R.

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additive in each variable separately and satisfies the equation m(u 1 , . . . , u k−1 , a · u k · b, u k+1 , . . . , u n ) = m(u 1 , . . . , u k−1 · a, u k , b · u k+1 , . . . , u n )

(3.2)

for all k and all a, b ∈ A, then there is a unique bimodule homomorphism μ : U1 ⊗ A U2 ⊗ A · · · ⊗ A Un → W such that m(u 1 , · · · , u n ) = μ(u 1 ⊗ u 2 ⊗ · · · ⊗ u n ), for all u i ∈ Ui , 1 ≤ i ≤ n. So, for an (A, A)-bimodule U we define U ⊗n = U ⊗ A U ⊗ A · · ·⊗ A U (n-times) and obtain the formula U ⊗(n+m) = U ⊗n ⊗ A U ⊗m . At the level of decomposable tensors, we have (u 1 ⊗ u 2 ⊗ · · · u n ) ⊗ (u n+1 ⊗ u n+2 ⊗ · · · u n+m ) = u 1 ⊗ u 2 ⊗ · · · ⊗ u n+m , for all positive integers n and m. We set U ⊗0 := A and extend these formulas in the obvious fashion, taking into account the balancing act involving A and the fact that we want the identity of A, 1 A , to act like the identity transformation on each U ⊗n . Thus, in particular, if u = u 1 ⊗ u 2 ⊗ · · · ⊗ u n , v = v1 ⊗ v2 · · · ⊗ vm , and a ∈ A, then u ⊗ a = u 1 ⊗ u 2 ⊗ · · · ⊗ (u n · a ⊗ 1 A ) = u 1 ⊗ u 2 ⊗ · · · ⊗ (u n · a), while a ⊗ v = (a ⊗ v1 ) ⊗ v2 · · · ⊗ vm = (a · v1 ) ⊗ v2 · · · ⊗ vm . We therefore may add the formulas (u · a) ⊗ v = (u ⊗ a) ⊗ v = u ⊗ (a ⊗ v) = u ⊗ (a · v) to our tensorial lexicon. The culmination of these computations is Definition 3.2 Let U be  a bimodule over the algebra A. The tensor algebra of U , ⊗n . The multiplication is defined on decomposable T A (U ), is the direct sum ∞ n=0 U tensors u = u 1 ⊗ u 2 ⊗ · · · ⊗ u n ∈ U ⊗n and v = v1 ⊗ v2 ⊗ · · · ⊗ vm ∈ U ⊗m by the formula u · v := u ⊗ v = u 1 ⊗ u 2 ⊗ · · · u n ⊗ v1 ⊗ · · · ⊗ vm , and is extended to all of T A (U ) by linearity. The identity of T A (U ) is simply the identity of A embedded in the zeroth summand defining T A (U ). The embedding is denoted π0 . In the setting of quivers, it is clear that E(Q 1 )⊗n should be expressed in terms of paths of length n: elements of Q n := {α := (α1 , α2 , . . . αn ) ∈ (Q 1 )n | s(αk ) = r (αk+1 ), 1 ≤ k ≤ n − 1}. We extend the maps r and s to Q n simply by declaring r (α) := r (α1 ) and s(α) := s(αn ). It is almost immediately seen that if E(Q n ) is defined to be all the complexvalued function on Q n , then E(Q n ) may be identified with E(Q 1 )⊗n in such a way that the isomorphism between E(Q n ) ⊗C(Q 0 ) E(Q m ) and E(Q n+m ) is implemented by the concatenation of paths. That is if α = (α1 , α2 , . . . αn ) and β = (β1 , β2 , . . . βm ) are in Q n and Q m , respectively, and if s(αn ) = r (β1 ), then we may concatenate α

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with β to obtain α ∗ β := (α1 , α2 , · · · αn , β1 , β2 , · · · βm ) ∈ Q n+m Further, if we declare E(Q 0 ) := C(Q 0 ) and extend the definition of r and s to Q 0 simply by letting them be the identity map on Q 0 , then we arrive at the so-called path category of Q. Q∗ :=



Qn

n≥0

The object space of Q∗ is Q 0 and for ε, δ ∈ Q 0 , Hom(ε, δ) := {α ∈ Q∗ | s(α) = ε, r (α) = δ}. Concatenation gives the product on the Hom sets, yielding Hom(δ, η) ∗ Hom(ε, δ) = Hom(ε, η). Further, still, if we let P(Q) :=

∞ 

E(Q n ),

(3.3)

n=0

then P(Q) is an algebra over C: Addition and multiplication by scalars are defined pointwise in the obvious fashion, and the algebra multiplication is defined by convolution, i.e., by the formula f ∗ g(α) :=



f (β)g(γ ),

f , g ∈ P(Q), α, β, γ ∈ Q∗ .

(3.4)

βγ =α

Note that multiplication is well-defined, since f and g have finite support in Q∗ by definition of P(Q) as the algebraic direct sum of the E(Q n ). It does not require much effort to prove directly that P(Q) T A (U ) where A = C(Q 0 ) and U = E(Q 1 ), but we want to provide a context for the proof that promotes the key features of both algebras. First, some examples are in order, as well as some remarks. Example 3.3 If Q has one vertex and one edge, then each Q n has only one element, the path of length n. In this event, Q∗ , as a category, is the non-negative integers N0 under addition. Further, since each E(Q n ) may be identified with C, P(Q) is the collection of finitely non-zero sequences indexed by N0 , Cc (N0 ), and the product (3.4) is the

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usual convolution product on Cc (N0 ). But in algebra texts, Cc (N0 ) is written C[x] (cf. eg. [11, Sect. 9.1]). The indeterminate x may be replaced by any element in any C-algebra, S-say, one might want. In so doing, one obtains an S-valued point. Example 3.4 Suppose Q again has one vertex, but now assume that Q 1 has d-edges, for a finite integer d. Then for n > 0, the elements of Q n are called words of length n chosen from the d elements of Q 1 , that is Q n may be identified with {1, 2, . . . , d}n . The path algebra P(Q) is ordinarily called the free algebra on d generators and is denoted C x1 , x2 , . . . , xd . Again, the indeterminants xi may be replaced by elements X i in any C-algebra S, yielding S-valued points. Example 3.5 Let Q be the quiver where Q 0 = {1, 2, . . . , n} and Q 1 consists of arrows, one between each consecutive pair of integers, pointing to the right. Pictorially, Q 1 looks like this: 1 −→ 2 −→ 3 · · · −→ n − 1 −→ n. It is not difficult to see that P(Q) is naturally isomorphic to the algebra of all upper triangular n × n matrices. In general, every finite, partially ordered set gives rise to a quiver Q where Q 0 consists of the points in the set, and there is an arrow from point p to point q in Q 1 precisely when p is an immediate predecessor of q. The resulting quiver algebra P(Q) is the so-called incidence algebra of the partially ordered set. Remarks 3.6 Let Q be an arbitrary quiver. 1. A cycle in Q is any path α of length at least 1 such that r (α) = s(α). If there are no cycles in Q, the quiver is called acyclic. It is easy to prove that P(Q) is finite dimensional if and only if Q is acyclic. We want to emphasize, however, that acyclic quivers are not the only way to generate finite dimensional algebras with quivers. One must pass to quivers with relations. The relations are paths which span a special kind of ideal J in P(Q) that makes P(Q)/J finite dimensional. As we noted on page 3, every finite dimensional algebra is Morita equivalent to such a quotient. See [10] (especially Chapter 3) or [13] for more about the representation theory of quivers with relations. 2. Clearly P(Q) is an A-ring in the sense of Defintion 2.2, where A = C(Q 0 ) and π is the identification of A with E(Q 0 ). However, P(Q) is not an A-algebra (i.e., A does not act centrally) unless r = s, in which case Q is the disjoint union of a number of loops, the number being the cardinality of Q 0 . 3. A vertex v ∈ Q 0 is called a source in case r −1 (v) = ∅. Similarly, v is called a sink in case s −1 (v) = ∅. Sources and sinks are called singular vertices. A quiver without singular vertices is called nonsingular. In a nonsingular quiver, all paths can be “extended in both directions”, i.e. if α is any path in Q, then there are paths β and γ of positive lengths such that βαγ ∈ Q ∗ . 4. There is a growing literature devoted to analyzing the singularities of quivers and to developing methods for resolving them. (cf. [35] and references cited there.)

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Evidently, if U is an (A, A)-bimodule, then T A (U ) is an A-ring in the sense of Definition 2.2. The homomorphism π in this case is simply what we called π0 , above: the identification of A with U ⊗0 . The fundamental feature of tensor algebras is that they are solutions to certain so-called universal mapping problems. In our situation, we may have a bimodule U over an algebra A and a map f from U to an algebra A that is a bimodule map, in the sense described in Sect. 2.3. Here, that would be expressed through the equations f ◦ ρ = (l ◦ π ) ◦ f , while f ◦ σ = (r ◦ π ) ◦ f . Then, as the following theorem asserts, we can extend it in a unique way to a homomorphism F : T A (U ) → A that reflects the original structures. Theorem 3.7 Let A be an algebra over C; let U = ρ Uσ be an (A, A)-bimodule; and let ψ : U → T A (U ) be the map that identifies U with U ⊗1 . If A is an A-ring via the homomorphism π : A → A and if f is a bimodule map from U to A, viewed as a bimodule over A in the fashion described above, then there exists a unique A-ring homomorphism F : T A (U ) → A such that f = F ◦ ψ. That is, the diagram U

ψ

T A (U ) F

f

A

commutes. Or better, it commutes uniquely. Proof Consider the map f ×2 : U × U → A defined by f ×2 (u, v) = f (u) f (v). Then f ×2 is additive in each variable separately and satisfies also f ×2 (a · u · b, c · v · d) = a · f ×2 (u · (bc), v) · d = a · f ×2 (u, (bc) · v) · d. Hence, by the properties of balanced tensor products, there is a unique linear map f ⊗2 : U ⊗2 → A such that f ×2 (u, v) = f ⊗2 (u ⊗ v) that preserves the left and right actions of A. Following the analysis that led to (3.2), we see that if we define f ×n (u 1 , u 2 , · · · , u 2 ) = f (u 1 ) f (u 2 ) · · · f (u n ), then f ×n may be expressed uniquely f ⊗0 := π0 and in terms of bimoldule map f ⊗n : U ⊗n → A. Then, if we call ⊗1 ⊗n yields := f , then with little difficulty one can see that defining F := ∞ f n=0 f the desired result.   The crucial feature of the relation between the (A, A)-bimodule U and its tensor algebra, T A (U ), that we want to secure is that it is functorial in U . Theorem 3.8 Let U and V be two (A, A)-bimodules and let θ : U → V be a bimodule map. Then there is a unique algebra homomorphism  : T A (U ) → T A (V ) such that  ◦ ψU = ψV ◦ θ , where ψU and ψV are the embeddings of U and V into T A (U ) and T A (V ) respectively. Further,  maps U ⊗n into V ⊗n .

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Proof The result follows from Theorem 3.7: Let ψV : V → T A (V ) be the map that identifies V with V ⊗1 . Then ψV ◦ θ is a bimodule map from U into T A (V ). So,  is simply the map F from that theorem. Further, since ψV ◦ θ maps U ⊗1 into V ⊗1 ,    maps U ⊗n into V ⊗n . 3.2 Representations of Tensor Algebras Suppose ρ : T A (M) → End(V ) is a representation of the tensor algebra of the (A, A)-bimodule M on the vector space V . Then, if we let π be the restriction of ρ to M ⊗0 where, recall, M ⊗0 is the copy of A embedded in T A (M) and if we let λ be the restriction of ρ to M ⊗1 , the copy of M, the pair (λ, π ) is a representation of M on V in the sense of the following definition. Definition 3.9 Let M be a bimodule over the algebra A. A representation of M on a vector space V is a pair (λ, π ) where π : A → End(V ) is representation of A (i.e., a ring homomorphism from A into End(V )) and λ : M → End(V ) is a C-linear map such that λ(a · m · b) = π(a)λ(m)π(b),

a, b ∈ A, m ∈ M.

Alternatively, recalling Sect. 2.2, λ is a bimodule map from M to End(V ) made into an (A, A)-bimodule via π . Evidently, this process is “reversed” by Theorem 3.7. That is, if (λ, π ) is a bimodule representation of the (A, A)-bimodule M on a vector space V , then there is a unique extension, ρ, of (λ, π ) to an algebra representation of T A (M) on V . This assertion is strengthened by the following theorem, Theorem 3.10, which, really, is a summary of all we have said to this point. To state it, recall, first, that in Sect. 2.3 we defined the so-called forgetful functor F from R A to M A : For B ∈ R0A , F(B) = B as sets; and for f ∈ HomR A (B1 , B2 ), F( f ) = f viewed as an element of HomM A (B1 , B2 ) for all Bi ∈ R A , i = 1, 2. Note, too, that we also now have the functor T A : M A → R A : For M ∈ M0A , T A (M) is the tensor algebra of M over A, as we defined it; and for two modules M1 and M2 in M0A , and f ∈ HomM A (M1 , M2 ), T A ( f ) is given by Theorem 3.8. Theorem 3.10 For every algebra A, every bimodule M ∈ M0A and every B ∈ R0A , there is a natural C-linear isomorphism HomR A (T A (M), B) HomM A (M, F(B)).

(3.5)

This theorem is sometimes cited as saying that the tensor algebra functor, T A is a left adjoint of the forgetful functor F, while F is a right adjoint of T A . Informally, the assertion that the isomorphism (3.5) is “natural” means that no special choices are made in its formation. This is akin to what happens in vector space theory. Every finite dimensional vector space, V , is naturally isomorphic to its second dual, V ∗∗ . However, no finite dimensional vector space V of positive dimension is naturally isomorphic to its dual, V ∗ , even though V is isomorphic to V ∗ ; every isomorphism depends on a

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choice of basis in V . For the precise meaning of the use of “natural” in describing Eq. (3.5), see [45, Section 4.1], especially Lemma 4.1.3.5 The proof of Theorem 3.10 is easy and straightforward from what we have done to this point, and so will be omitted. However, for emphasis and future reference we want to call attention to the following corollary which is obtained from Theorem 3.10 by specializing to the case when the A-ring B in its statement is specialized to B := End(V ) that is made into an A-ring by a homomorphism π : A → End(V ), i.e., a representation of A on V . Corollary 3.11 For every algebra A, every bimodule M ∈ M0A and every representation π of A on a vector space V , there is a natural isomorphism HomR A (T A (M), End(V )) HomM A (M, End(V )).

(3.6)

The proof is immediate from Theorem 3.10 once it is recognized that F(End(V )), where End(V ) is viewed as an A-ring, is End(V ) viewed as an (A, A)-bimodule. Note that while the isomorphism (3.6) is natural, it needs to be unpacked in order to help with the function theory that we will present, beginning in Sect. 4. So, for this purpose, we turn now to get a better understanding of the structure HomM A (M, End(V )). In the special setting considered in [31], we refer to a bimodule representation as a covariant representation. The reason for that terminology is not important for the present discussion, but we call attention to it because in Section 3 of that paper we connect covariant representations with induced representations in the sense of Marc Rieffel [44]. We require features of that connection here. Note that given an (A, A)-bimodule M, then we may reformulate the notion of a bimodule representation, (λ, π ), of M on the vector space V by first considering M as a right A-module and forming M ⊗π V – the tensor product of M, regarded as a right A-module, with V regarded as a left A-module via π that is balanced over A, i.e., m · a ⊗ v = m ⊗ π(a)v, m ∈ M, a ∈ A, and v ∈ V . When this is done, then λ may be promoted to a uniquely defined linear map  λ from M ⊗π V to V defined by  λ(m ⊗ v) := λ(m)(v),

m ∈ M, a ∈ A, v ∈ V .

(3.7)

This simply reflects the fact that m, v → λ(m)(v) is a bilinear map that is balanced over A and the definition of balanced tensor products. The uniqueness of  λ is a consequence of the fact that the collection of vectors m ⊗ v is a total set in M ⊗π V . Now let’s bring in the notation ρ and σ for the left and right actions of A on M, i.e., a · m · b := ρ(a)mσ (b). Also, for the sake of compacting the notation, let E := End(M A ). Then on the vector space M ⊗π V we have a representation of E, denoted π E , that is defined by the formula π E (X )(m ⊗ v) = (X m) ⊗ v,

X ∈ E, v ∈ V .

(3.8)

5 We also note in passing that adjoints are unique up to natural isomorphism [30, Corollary 1, page 85]

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Definition 3.12 The representation π E of E on M ⊗π V is called the induced representation of E determined by π . Since ρ is a homomorphism of A into E, we may compose π E with ρ, yielding ◦ ρ, which is a representation of A on M ⊗π V . Thus we have two representations of A: π which acts on V , and π E ◦ ρ which acts M ⊗π V . Further, and this is the key point, we have the following lemma, whose proof is immediate from the definitions. πE

Lemma 3.13 Let π be a representation of A on V . The Eq. (3.7) establishes a bijection between the collection of all linear maps λ from M into End(V ) such that (λ, π ) is a representation of the bimodule M in the sense of Definition 3.9 and the set of all linear maps  λ : M ⊗π V → V that satisfy  λπ E ◦ ρ = π λ,

(3.9)

i.e.,  λ((ρ(a)m) ⊗ v) = π(a) λ(m ⊗ v), for all m ⊗ v ∈ M ⊗π V and a ∈ A. This lemma, too, is really an expression of functorial adjunction. It is a special case of what Cohn calls adjoint associativity [7, Formulas (2.3.2) and (2.3.3), page 52]. Evidently, then, for each π , the collection of bimodule maps (λ, π ) may be identified with the intertwiner space I(π E ◦ ρ, π ). We adopt the somewhat more compact notation and terminology. Definition 3.14 We write M π := I(π E ◦ ρ, π ), and refer to it as the π -dual of M. The π -dual, M π , is also a bimodule, but not over A, in general. Rather, M π is a bimodule over the commutant of π(A). Also, of course, it is the endomorphism algebra of the left A-module V viewed as a left module over π(A). The left and right actions of π(A) on M π are defined by the formulas t · X := t X ,

t ∈ π(A) , X ∈ M π ,

(3.10)

and X · t := X (I M ⊗ t),

X ∈ M π , t ∈ π(A) .

(3.11)

Note that these formulas make sense. Since X is a map from M ⊗π V to V , and t is a map on V , t X , the product of these linear maps is a well-defined map from M ⊗π V to V . Likewise, I M ⊗ t is a linear map on M ⊗ V and the hypothesis that t ∈ π(A) guarantees that I M ⊗ t passes to the quotient M ⊗π V . The fact that M π is a bimodule over π(A) is now clear. We turn next to examining what M π looks like in the case of a quiver Q. Recall that in this case, M is E(Q 1 ) and A = C(Q 0 ). In general, the commutant of a representation of an algebra can be quite complicated, even in the finite dimensional setting. However, in the quiver setting, where A = C(Q 0 ), we are aided by the fact that A is the direct sum of a finite number of copies of C. This makes it easy to parametrize all of its representations and identify their commutants. Further, as will

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be clear, as well as important for us, the double commutant of π(A), π(A) equals π(A). By definition π(A) = {X ∈ End(V ) | X t = t X , t ∈ π(A) }. So, from the definition, π(A) must be contained in π(A) , equality need not hold, in general, even when the algebra in question is commutative. Equality does hold in the case of quivers and it does so in a very stark fashion. Let Q 0 be labeled as {v1 , v2 , · · · , vn }. Remember, our quivers are assumed to be finite. The labeling is for convenience only. Nothing important depends on it. If π is a representation of A := C(Q 0 ) on the vector space V , then V decomposes  as nk=1 π(δvk )V , where δvk is the function in C(Q 0 ) that is 1 at vk and 0 elsewhere. We write V (k) for π(δvk )V . Then, for all ϕ ∈ C(Q 0 ), π(ϕ) is the block diagonal matrix associated to the direct sum, whose k th block is ϕ(vk )I V (k) . Manifestly, then, n  π(A) = k=1 End(V (k)) and π(A) = π(A). It is fairly traditional to call the function d : Q 0 → N0 defined by d(vk ) := dim(V (k)), the dimension vector of the representation π . It is evident that two representations of C(Q 0 ) are similar or spatially isomorphic if and only if they have the same dimension vector. In fact, it is customary often to begin with the dimension π could just as easily be called M d . vector, d; build π ; and then n build M , which π Once V is written as k=1 V (k), M may be written as Mπ = =

n  k,l=1 n 

HomC(Q 0 ) (M ⊗C(Q 0 ) V (k), V (l))C(Q 0 ) HomC(Q 0 ) ((ρ(δl )Mσ (δk )) ⊗C(Q 0 ) V (k), V (l))C(Q 0 )

k,l=1

=



HomC (V (s(α)), V (r (α))),

(3.12)

α∈Q 1

exploiting the fact that ρ and σ are the maps that make M = E(Q 1 ) a bimodule over C(Q 0 ).  We note in passing that writing M π as α∈Q 1 HomC (V (s(α)), V (r (α))) is the traditional way of portraying M π . Observe that it makes M π ’s structure as a π(C(Q 0 )) -bimodule clear. It also makes the isomorphism classes of the representations of P(Q) transparent. Let d be the dimension vector associated to π and let G L(d) = v∈Q 0 G L(d(v), C). Then G L(d) is a reductive algebraic group that acts on  π α∈Q 1 HomC (V (s(α)), V (r (α))) as follows: Let T ∈ M and assume, without loss of generality, that T ∈ Hom(V (s(α)), V (r (α))) for an α ∈ Q 1 . Then for φ ∈ G L(d), φ · T := φ(r (α))−1 T φ(s(α)).

(3.13)

The orbits of this action parameterize the isomorphism classes of the bimodule representations in M π . We should add that understanding this orbit structure is a major focus of geometric invariant theory. One further remark may be helpful. In [32], we analyzed the structure of the bimodule M = E(Q 1 ) by letting A = C(Q 0 ) be the collection of diagonal matrices indexed

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by Q 0 . We then let C be the |Q 0 | × |Q 0 | matrix whose  (i, j) entry is the number of n arrows in Q 1 from v j to vi . From C we built E(C) := i, j=1 Hi, j where Hi, j is a Hilbert space of dimension Ci, j . (The fact that we used Hilbert spaces is not important here.) We fixed an orthonormal basis {ei(k) j | 1 ≤ k ≤ Ci j } for Hi j , so E(C) is the span (k)

of the ei j , and we viewed E(C) as an (A, A)-bimodule via the formulae: (k)

(k)

ei j · ell := δ jl ei j (k)

(k)

ell · ei j := δil ei j . While these formulas may seem opaque, it turns out that it is not hard to show that for any pair of N0 -valued matrices, C and B, the balanced tensor product E(C) ⊗ A E(B) is naturally isomorphic to E(C B). As a result, we find that if E(C 0 ) is identified with ∞ A, then n=0 E(C n ) is naturally isomorphic to P(Q). See [32, Lemma 5.1, and Corollary 5.2]. We used these results to identify a number of features of the modules over P(Q).

4 The Representation of TA (M) as Functions Throughout this section, M will be a fixed bimodule over the algebra A. Our first objective, treated in the first subsection, is to show that for each representation π of A on a vector space V , the space M π is the space of all End(V )-valued points for T A (M), and that the natural habitat for the function theory built from T A (M) is the set theoretic coproduct 

Mπ ,

 π∈A

 is the collection of all the representations of A.6 where A Our second objective, treated in the second subsection, is to show that there are relations among the M π that gives the entire coproduct an organicexistence that is reminiscent of an affine space. The key point that will emerge is that π ∈ A M π should  and that the functions coming from T A (M) be viewed as a set that is fibred over A interact with the fibration. Our goal is to show how this interaction takes place. From now on, we will denote the points of M π by lower-case Fraktur letters from the end of the alphabet. 4.1 The End(V)-valued Functions on



 ∈ AM

Determined by TA (M)

 acting on the vector space V , and z ∈ M π we write π ×z for Definition 4.1 For π ∈ A, the representation of T A (M) on V defined on the generators M ⊗0 = A and M ⊗1 = M by the formulas 6 Strictly speaking, we should be writing A 0 . However, we omit the superscript 0 since it weighs down the

notation and omitting it leaves no serious ambiguity.

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π × z(a)(v) := π(a)(v),

a ∈ A, v ∈ V

(4.1)

m ∈ M, v ∈ V ,

(4.2)

and π × z(m)(v) := z(m ⊗ v), and extended to all T A (M) via Theorem 3.7. The problem is how to organize the extension promised by Theorem 3.7. The key is to understand how π × z acts on decomposable tensors. We want the equation π × z(m 1 ⊗ m 2 ) = (π × z(m 1 ))(π × z(m 2 )) to be satisfied. Both π × z(m 1 ) and π × z(m 2 ) are operators on V defined by Eq. (4.2). So we “foil” the product acting on a vector v to obtain the equation: (π × z(m 1 ))(π × z(m 2 ))(v) = π × z(m 1 )((π × z(m 2 ))(v)) = π × z(m 1 )(z(m 2 ⊗ v)) = z(m 1 ⊗ (z(m 2 ⊗ v))) = z(I M ⊗ z(m 1 ⊗ m 2 ⊗ v)) = (z(I M ⊗ z))(m 1 ⊗ m 2 ⊗ v). Thus π × z(m 1 ⊗ m 2 ) as a linear map on V is given by the formula π × z(m 1 ⊗ m 2 )(v) = (z(I M ⊗ z))(m 1 ⊗ m 2 ⊗ v). The general formula for the π × z acting on M ⊗k is then readily seen to be (π × z)(m 1 ⊗ m 2 · · · m k )(v) = z(I M ⊗ z) · · · (I M ⊗(k−1) ⊗ z)(m 1 ⊗ m 2 ⊗ · · · m k ⊗ v).

(4.3)

Formula (4.3) reveals a natural splitting, or factoring, of (π ×z)(m 1 ⊗m 2 · · · m k )(v). We call the first factor, z(I M ⊗ z) · · · (I M ⊗(k−1) ⊗ z), the k th generalized power of z and denote it by z(k) . Evidently, z(k) is a C-linear map from M ⊗k ⊗ V to V . In fact, it is an intertwiner from π E (k) ◦ ρk to π , where E(k) := End(M A⊗k ), π E (k) is the induced representation of E(k) (see Definition 3.12), and ρk is left action of A on M ⊗k . Consider, now, the function Zk : M π → (M ⊗k )π defined by the formulas Zk (z) = z(k) ,

z ∈ M π , k > 0;

(4.4)

and Z0 (z) ≡ I V . Clearly, Zk is a polynomial map that satisfies the “power formula” Zk+l (z) = Zk (z)(I M ⊗k ⊗ Zl (z)),

z ∈ Mπ .

(4.5)

Further, for z and ζ in M π , one may form Zk (z + ζ ) − Zk (z) and expand it as a function of z and ζ using (4.5). If one keeps the terms that are linear in ζ , one is led to the algebraic derivative formula: (DZk )(z)[ζ ] :=

k−1 

Zl (z)(I M ⊗l ⊗ ζ )(I M ⊗(l+1) ⊗ Zk−l−1 (z))

(4.6)

l=0

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which is valid for all z, ζ ∈ M π . One can, of course, calculate higher derivatives, and we will take these up in a somewhat different context shortly. For the moment, we return to Eq. (4.3) and its consequences. A general element θk ∈ M ⊗k may be written as a sum of decomposable tensors m 1 ⊗ m 2 ⊗ · · · ⊗ m k all of the same degree, k. So, (4.3) leads to z ∈ M π , θk ∈ M ⊗k , v ∈ V .

(π × z)(θk )(v) = Zk (z)(θk ⊗ v),

(4.7)

For θk ∈ M ⊗k , let L θk be the linear map from V to M ⊗k ⊗ V defined by the formula L θk (v) = θk ⊗v, then we arrive at the following theorem that gives an explicit formula for π × z representing T A (M) on V and, therewith, formulas for the EndC (V )-valued functions on M π determined by elements of T A (M). Theorem 4.2 Suppose F = nk=0 θk is an element in T A (M), where θk ∈ M ⊗k and suppose z ∈ M π , then π (z) := π × z(F) = F

n 

Zk (z)L θk .

k=0

π given by π has an algebraic derivative D F Further, F π (z)[ζ ] = DF

n 

D(Zk (z))[ζ ]L θk .

k=0

π }, it may Before we investigate the role of π in the collection of functions { F be helpful to review Examples 3.3 and 3.4 in the current setting. We’ll focus on Example 3.4, treating Example 3.3 as a special sub-case. So, here, the algebra A is just C, and M = Cd for a prescribed positive integer d. The tensor algebra, T A (M), in this case may be identified with the path algebra P(Q), where Q is the quiver with one vertex, and d loops beginning and ending at that vertex. Elements of T A (M) = P(Q) may be viewed as linear combinations of functions f ∈ P(Q) of the form f = δα , where α = α1 α2 · · · αn for some n and αi is one of the d possible loops in Q 1 . Once n is fixed the collection of all such functions δα , where |α| = n is a basis for M ⊗n = E(Q n ). So a general element F ∈ T A (M) may be written as F=



a α δα .

α∈Q∗

A representation π of A on a vector space V is given by scalar multiplication and its isomorphism class is given by the dimension of V . We shall assume the dimension of V is n, 1 ≤ n < ∞. Then, of course, we may identify V with Cn and assume that π(c) = cIn , c ∈ C. The π -dual of M, M π , is easily seen to be HomC (Cd ⊗ Cn , Cn ), which in turn may be identified with all d-tuples of n × n matrices. When this identification is made, we see that if z = (Z 1 , Z 2 , · · · , Z d ) ∈ Mn (C)d = M π , then

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π (z) = F



aα Z α ,

(4.8)

α∈Q∗

where Z α = Z α1 Z α2 · · · Z αk and k = |α|. Thus we find that the collection of all π , as F ranges over T A (M) coincides with what is known as the ring of d-generic F n × n matrices which plays such a prominent role in the theory of polynomial identity algebras [40]. If d = 1, the functions in (4.8) are what arise in the polynomial functional calculus that is familiar from linear algebra. Further, the derivative formula in Theorem 4.2 reduces to the ordinary derivative of a polynomial. 4.2



 ∈ AM

as a Function of 

π with π fixed. Here we want to analyze the To this point, we have focused on the F dependence on π . The key relation is that if C ∈ I(π, τ ) and Cz = w(I M ⊗ C), then τ (w)C for all F ∈ T A (M). We express this π (z) = Cπ × z(F) = τ × w(F)C = F CF relation succinctly by writing: π (z), F τ (w)). I(π × z, τ × w) ⊆ I( F

(4.9)

It makes sense to apply this intertwining condition more broadly. Definition 4.3 A family { f π }π ∈ A functions such that f π : M π → End(Vπ ) for all  is said to respect intertwiners in case I(π × z, τ × w) ⊆ I( f π (z), f τ (w)), for π∈A  z ∈ M π , and w ∈ M τ . In this event, we call { f π }π ∈ A a matricial family all π, τ ∈ A, of functions. This terminology is taken from [33, Definition 1.2]. It is essentially the same as that used in [22] π (z) The amazing fact is that “respecting intertwiners” characterizes the functions F under very mild hypotheses. Theorem 4.4 Let { f π }π ∈ A be a matricial family of functions such that f π : M π →  Assume End(Vπ ) for all π ∈ A. 1. Each f π is a polynomial map7 from M π to End(Vπ ). 2. There is a uniform finite upper bound on the degrees of the f π ’s. π = f π for all π ∈ A.  Then there is an F ∈ T A (M) such that F This theorem is a mild upgrade of Theorem 6.8 in [22]. The methods of proof are essentially the same. We will see that in the setting of analytic functions, a similar theorem is true under far weaker hypotheses. We turn now to presenting an idea about how the proofs work. The key is to understand how f π ⊕τ is related to f π and f τ , when { f π }π ∈ A is a matricial family. 7 A function F : U → V is called a polynomial map in case for some choice of bases for U and V the components of F(u) with respect to the basis in V are polynomials in the coordinates of u with respect to the basis in U . [42, p. 132]

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To this end, let π and τ be two not-necessarily-distinct representations of A on  acts on Vπ ⊕ Vτ vector spaces Vπ and Vτ , respectively. Then the direct sum π ⊕ τ ∈ A π ⊕τ by definition. Recalling Definition 3.14, we write out M in terms of π and τ of A as follows: M π ⊕τ = I((π ⊕ τ )E ◦ ρ, π ⊕ τ ), where ρ gives the left

action z12  acting on M. This intertwiner space, in turn, is a set of operator matrices zz11 z 21 22 as transformations from Vπ E ◦ρ ⊕ Vτ E ◦ρ to Vπ ⊕ Vτ , where z11 ∈ I(π E ◦ ρ, π ), z12 ∈ I(τ E ◦ ρ, π ), z21 ∈ I(π E ◦ ρ, τ ), and z22 ∈ I(τ E ◦ ρ, τ ). Note, in particular that M π ⊕ M τ ⊆ M π ⊕τ ,

(4.10)

z11 z12 

where M π ⊕ M τ is represented as the collection of all the matrices z21 z22 where z12 and z21 vanish. The entire analysis rests on the following lemma which is due to Joe Taylor and may be found in [50, Proposition 6.11]. It forms the basis for the monograph by Dima Kaliuzhnyi-Verbovetskyi and Victor Vinnikov [22] and our work [33].  Lemma { f π }π ∈ A be a matricial family of functions on π ∈ A M π , and let

z u  4.5 πLet ⊕τ . Then there is a linear transformation  f π,τ (z, w) mapping Vτ to Vπ 0w ∈ M such that   z u f π (z)  f π,τ (z, w)(u) f π ⊕τ . (4.11) = 0 f τ (w) 0w Proof Write Iπ and Iτ for the identity transformations on Vπ and Vτ , respectively. Then we have the equations  

 I z u I ⊗ Iπ = π z (4.12) × M 0 0 0w and





 z u 0 Iτ × = w 0 Iτ . 0w

(4.13)

Since the family { f π }π ∈ A respects intertwiners, we may conclude that f π ⊕τ

   z u I I × π = π f π (z) 0 0 0w

(4.14)

and



0 Iτ f π ⊕τ



 

z u = f τ (w) 0 Iτ . 0w

(4.15)

 u 

a12  So, if we write f π ⊕τ 0z w = aa11 21 a22 , then (4.14) shows that a11 = f π (z) and a21 = 0, while (4.15) also shows that a21 = 0 as well as a22 = f τ (w). The remaining entry, a12 , we take as the definition of the expression  f π,τ (z, w)(u). How this

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expression depends on the variables involved is not at all clear. In particular, it is not at all clear why it should even be homogeneous in u. We shall show that a bit more is true: If b ∈ π(A) is such that z(I M ⊗ b) also equals bz, then  f π,τ (z, w)(bu) = b f π,τ (z, w)(u). To this end, simply note that since the f π ’s respect intertwiners, the equation

    0 b 0 z u z bu I M ⊗ b = 0 I M ⊗ Iτ 0 Iτ 0 w 0 w implies the equation  

  f (z)  f π,τ (z, w)(bu) b 0 b 0 f π (z)  f π,τ (z, w)(u) = π , 0 Iτ 0 f τ (w) 0 f τ (w) 0 Iτ which, in turn, shows that  f π,τ (z, w)(bu) = b f π,τ (z, w)(u), as was required. To show that  f π,τ (z, w)(u) is additive in u requires a bit more complicated “matricial trick” that we learned from Victor Vinnikov. Keeping the notation we already 

have,

z u1 +u2  z 0 u1 π ⊕τ ∈M , B := 0 z u2 we let u = u1 + u2 and form the matrices A := 0 w 00 w 

z 0 u2 2π ⊕τ . Note that C equals B except for the order of the u’s in and C = 0 z u1 ∈ M 00 w

the third columns. Split 2π ⊕ τ as π ⊕ (π ⊕ τ ); we shall apply what we have done in the first paragraph to calculate f π ⊕τ (A), f 2π ⊕τ (B), and f 2π ⊕τ (C). We obtain the equations:

 f π (z) π,τ (z, w)(u1 + u2 ) , 0 f τ (w) ⎡ ⎤ f π (z) x y f 2π ⊕τ (B) = ⎣ 0 f π (z) π,τ (z, w)(u2 )⎦ , 0 0 f τ (w) f π ⊕τ (A) =

and ⎡

⎤ f π (z) u v f 2π ⊕τ (C) = ⎣ 0 f π (z) π,τ (z, w)(u1 )⎦ . 0 0 f τ (w) The challenge now is to reveal relations among x, y, u and v that connect them to  f π ⊕τ (z, w)(ui ), i = 1, 2. To this end, let S be the 3×3 permutation matrix associated to the transposition (1, 2) and observe that C = S B S −1 = S B S. Since { f π }π ∈ A is a matricial family, we conclude that f 2π ⊕τ (C) = f 2π ⊕τ (S B S) = S f 2π ⊕τ (B)S. This

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equation, in turn, reveals that y =  f π,τ (z, w)(u1 ), and therefore, that ⎡

⎤ f π (z) x  f π,τ (z, w)(u1 ) f 2π ⊕τ (B) = ⎣ 0 f π (z) π,τ (z, w)(u2 ) ⎦ . 0 0 f τ (w)   Now observe that if D is the 2 × 3 matrix I0π I0π I0τ , then D B = AD. Since the f ’s respect intertwiners, we conclude that D f 2π ⊕τ (B) = f π ⊕τ (A)D. Equating the (1, 3) entries of each side of this equation yields the desired result:  f π,τ (z, w)(u1 + u2 ) =  f π,τ (z, w)(u1 ) +  f π,τ (z, w)(u2 )  

which completes the proof.

One should view  f π,τ (z, w)(·) as a difference operator. The corresponding differential operator is obtained by setting π = τ and z = w. The result is called the Taylor derivative of f at z and is denoted  f π (z). Notice that it is a linear  transformation on z w Vπ . Notice, too, that if we apply (4.11) to the matrix where z and w are in 0 z+w π M we obtain   z w f π (z)  f π,π (z, z + w)(w) f 2π . (4.16) = 0 f π (z + w) 0 z+w If we use the facts that



 z u 

−Iπ Iτ = z −I M ⊗ Iπ I M ⊗ Iτ 0 z+u and that the f π ’s respect intertwiners, we see immediately that f π (z + u) = f π (z) +  f π,π (z, z + u)(u),

(4.17)

which looks like the beginning of expansion of f π in a Taylor series. It is, in fact, but to see the full story requires substantially more “matricial gymnastics”. We refer the reader to [22] and to our paper [33, Theorem 5.12] for full details. However, we offer some highlights here. What are required are higher order difference operators n . To define them, we let E (i) π n = π0 ⊕ π1 ⊕ · · · ⊕ πn , we choose zi ’s in the M πi ’s and ui ∈ I(πi ◦ ρ, πi−1 ). Note that the indices on the πi run from 0 to n, while the indices on the ui run from 1 to n. Now consider the equation ⎡ z0 u1

0 ··· 0 z1 u2

⎢. ⎢ f π n ⎢ .. ⎣

0

0 0

⎤

⎡

⎢ ⎥ ⎢ . . . . .. ⎥ . . . ⎥ := ⎢ ⎢ ⎦ ⎣ .. .

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zn

f π0 (z0 ) a01 a02 0 f π1 (z1 ) a12

.. . .. .

0

609

...

...

···

..

.

a0n a1n

.. .

f πn−1 (zn−1 ) an−1n f πn (zn )

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(4.18)

P. S. Muhly, B. Solel

and focus on a0n . Using the “matricial gynmastics” alluded to above, we find that a0n depends multilinearly on the ui ’s. So, putting all the other ingredients into the picture we define n via the equation a0n (u1 , u2 , · · · , un ) = n f π0 ,π1 ,··· ,πn (z0 , z1 , · · · , zn )(u1 , u2 , · · · , un ). We call n the n th -order Taylor difference operator. To arrive at the n th -order differential operator, we set all the zi ’s equal to one z and we let all the πi ’s be equal, so that π n = (n + 1) · π for some prescribed π . We then evaluate f (n+1)·π at the (n + 1) × (n + 1) matrix ⎡

z w 0 ⎢ ⎢0 z . . . ⎢ ⎢ .. . . . . ⎢. . . ⎢ ⎢. .. ⎣ .. . 0 ··· ···

··· .. . .. .

0

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ z w ⎦ 0 z+w 0 .. .

to obtain the differential operators and Theorem 4.6 (Taylor’s Taylor Formula with Remainder [52]) If z, w ∈ M π , then f π (z + w) =

n−1 

l f π (z)(w, w, · · · , w)

l=0

+n f π,π,··· ,π (z, z, · · · , z + w)(w, w, · · · , w)

(4.19)

n−1 l Observe that l=0  f π (z)(w, w, · · · , w) is a polynomial map in the w’s whose degree can be as high as a fixed multiple of n − 1. So, if the degrees of the f π have an upper bound that is independent of π , the expansion (4.19) eventually yields f π (w) =

n−1 

l f π (0)(w, w, · · · , w),

(4.20)

l=0

after setting z = 0. This proves Theorem 4.4.

5 The Analytic Theory So far, we have proceeded purely at the algebraic level. We turn now to the analytic aspects of the theory. More accurately, we focus on one analytic aspect, viz. the one developed in [33]. Now, the algebra A becomes a C ∗ -algebra, and its representations π become C ∗ -representations on Hilbert space. So we shall change notation slightly and write Hπ for what has been written Vπ . We write B(Hπ ) for the algebra of all continuous linear operators on Hπ . Since we don’t insist our Hilbert spaces are finite

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dimensional, B(Hπ ) will be strictly smaller than End(Hπ ) precisely when Hπ has infinite dimension. The bimodule M also has a norm structure, but the interaction between this structure and the C ∗ -algebra A with its norm structure requires some development. First, M, viewed simply as a right A-module via σ , is assumed to have an A-valued inner product, denoted ·, ·, that is linear in the right-hand variable and satisfies ξ, η∗ = η, ξ , which makes it conjugate linear in the left-hand variable. Further it is required that

ξ, η · a = ξ, η · a. Further, still, it is required that ξ, ξ  is a nonnegative (i.e. a positive semidefinite) element in A that is zero only when ξ = 0. With these axioms one can show that a version of the Cauchy-Schwarz inequality is valid, η, ξ  ξ, η ≤ 1  ξ, ξ  η, η, from which it follows that ξ →  ξ, ξ  2 is a norm on M. It is required that M is complete in this norm. Before going further, we want to note that so far we have described what it means for M to be a right Hilbert C ∗ -module over A. We note in passing that the map (ξ, a) → σ (a)ξ = ξ a from M × A to M is continuous. In fact, ξ a M ≤ ξ  M a A . This is equivalent to asserting that σ , the map giving the right action, is non-expansive as a map from A to the Banach algebra B(M) of the continuous C-linear operators on M with the usual operator norm. In the future, we shall not distinguish among norms using subscripts unless there may be some ambiguity perceived from failing to do so. Because M is a Banach space in its norm, what it means for an endomorphism of M as a right A-module to be continuous makes perfectly good sense. However, normally one does not want to consider all continuous endomorphisms. Rather, one focuses on those that have adjoints in the following sense: A continuous endomorphism X of M A is called adjointable in case there is an endomorphism, X ∗ , of M A such that

X ξ, η = ξ, X ∗ η,

ξ, η ∈ M.

(5.1)

The endomorphism X ∗ is automatically continuous and uniquely determined by X , when it exists. The collection of all adjointable endomorphisms of M A is denoted L(M). The adjointable endomorphisms, L(M), is a C ∗ -algebra with the involution X → X ∗ and the operator norm on B(M). In fact, it is a substantial subalgebra of B(M): It contains the algebra K(M), known as the algebra of compact (adjointable) operators on M, which in turn is the norm closure of the linear span of the rank one operators, ξ ⊗ η∗ , ξ, η ∈ M, where ξ ⊗ η∗ (ζ ) := ξ η, ζ ,

ξ, η, ζ ∈ M.

The adjoint of ξ ⊗ η∗ is η ⊗ ξ ∗ . It is clear from the definition that the collection of all rank one operators acts transitively on M. Definition 5.1 A bimodule M = ρ Mσ over a C ∗ -algebra A is called a C ∗ correspondence from A to A, or over A, in case: 1. As a right A-module, Mσ is endowed with an A-valued inner product making M a right Hilbert C ∗ -module over A; and 2. ρ is a C ∗ -homomorphism mapping A into L(M).

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Quivers provide basic clarifying examples. When Q is a quiver, the algebra C(Q 0 ) is a C ∗ -algebra, with norm given by the formula a := supv∈Q 0 |a(v)|. The space M = E(Q 1 ) is given the C(Q 0 )-valued inner product defined by the formula

ξ, η(v) :=



ξ(α)η(α).

s(α)=v

It is straightforward to verify that E(Q 1 ) is a right Hilbert C ∗ -module over C(Q 0 ). It appears as if E(Q 1 ) is a direct sum of finite dimensional Hilbert spaces. And it is. However, it is not the Hilbert space direct sum. There is no global C-valued inner product provided with respect to which the summands are orthogonal. It seems best to think of E(Q 1 ) as a field of Hilbert spaces indexed by Q 0 . Another basic example may be constructed as follows: Let A be an arbitrary C ∗ algebra and let α be an automorphism of A. Then we set M := A as a linear space. The left action, ρ, of A on M is given, simply, by left multiplication composed with α, i.e., ρ(a)ξ := α(a)ξ . The right action is just right multiplication, σ (a)ξ = ξ a. The A-valued inner product is a, b := a ∗ b. The notion of a C ∗ -correspondence clearly is asymmetric. We don’t require an A-valued inner product for the left action. Perhaps the best way to think of a C ∗ correspondence is as a generalized homomorphism from A into L(M). We shall not pursue here all the details necessary to give clarity to that perspective, referring the reader to [8] instead. However, we do require a bit of the theory that is necessary to give our spaces M π a norm structure. So, fix the correspondence M over A, let π be a C ∗ -representation of A on Hπ , and recall Definition 3.12. To make sense of it in the C ∗ -setting, we must first give M ⊗π Hπ an inner product. The one we want is defined by the formula

ξ ⊗ h, η ⊗ k := h, π( ξ, η M )k Hπ

(5.2)

and extended by linearity. A moment’s reflection reveals that this is a bonafide sesquilinear form that is conjugate linear on the right-hand side and positive in the sense that the inner product of a vector with itself is non-negative. We divide out by the null space of this form, and build the completion of the result. This completion, then, is the Hilbert space we want and we shall continue to denote it by M ⊗π Hπ . Another moment’s reflection reveals that L(M) acts by continuous linear operators on M ⊗π Hπ via the formula X (m ⊗ h) := (X m) ⊗ h,

m ⊗ h ∈ M ⊗π Hπ ,

which is the defining formula for the induced representation π E , (3.8). Since it clearly preserves adjoints, we see that if we cut π E down to L(M) we obtain a C ∗ representation of L(M) on M ⊗π Hπ . which we continue to denote by π E . We also write π E (X ) = X ⊗ I Hπ , 612

X ∈ L(M).

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Since M is a C ∗ -correspondence, the left action of A on M, ρ, is a C ∗ -homomorphism of A into L(M), so π E ◦ ρ is a C ∗ -representation of A on M ⊗π Hπ . Hence, the intertwining space M π = I(π E ◦ ρ, π ) is a closed C-linear subspace of the space of continuous linear operators from M ⊗π Hπ to Hπ , B(M ⊗π Hπ , Hπ ). Thus, we may give M π the norm structure it inherits from B(M ⊗π Hπ , Hπ ). now to the domains on which With the norm structure on the M π in hand, we turn π our functions are to be defined. They are subsets U ⊂  M , and we shall write π∈A  π U = π ∈ A U(π ), where U(π ) := U ∩ M . Unless otherwise stated, U(π ) is to be an open subset of M π . The key feature that the U(π ) will be assumed to enjoy is codified in the following definition. Definition 5.2 We say that U = {U(π )}π ∈ A respects direct sums in case U(π ) ⊕ U(τ ) =

    U(π ) 0 z 0 := | z ∈ U(π ), w ∈ U(τ ) 0 U(τ ) 0w  ⊆ U(π ⊕ τ ), π, τ ∈ A.

(5.3) (5.4)

In this event, we say that U is an (open) matricial set.8 The following supplemental properties of matricial sets play important roles in our work. Definition 5.3 A matricial set is said to respect similarities or to be similarity invariant  every invertible operator s in π(A) and every z ∈ U(π ), in case for every π ∈ A, −1 −1 s · z · s := s z(I M ⊗ s) lies in U(π ). If s is restricted to the unitary group in π(A) , then we’ll say U is unitarily invariant. We shall say U is matricially convex in case for all π and all v ∈ I(π, τ ) satisfying vv∗ ≤ I Hτ , v · U(π ) · v ∗ := vU(π )(I M ⊗ v ∗ ) ⊆ U(π ). One very natural example of a matricial set are “matricial discs”. To define these, we begin by saying that a family of vectors ζ = {ζπ }π ∈ A is an additive field of vectors  in case ζπ ∈ M π and ζπ ⊕τ = ζπ ⊕ ζτ . Given such a field, and a positive real (over A) number R, we define D(ζ, R) := {D(ζπ , R)}π ∈ A, where D(ζπ , R) := {ξ ∈ M π | ξ − ζπ  < R} and the norm is that on M π . Evidently, D(ζ, R) is a matricial set. However, in general, it won’t be matricially convex. It will be matricially convex in case the field ζ is central in the sense that ζτ (I M ⊗ C) = Cζπ  for all C ∈ I(π, τ ) , π, τ ∈ A. For another, examples, let F ∈ T A (M) and define  more comprehensive class of π (z) < 1}.9 Then U(F) is a matricial domain that U(F) := π ∈ A{z ∈ M π |  F is a natural analogue of polynomially convex domains that appear in the theory of functions of several complex variables. 8 Cf. Eq. (4.10) 9 The norm here is the norm on B(H ). π

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Now, observe that the notion of a matricial family of functions makes sense for families of functions on matricial sets. We state the definition for the sake of clarity. Definition 5.4 Let U = {U(π )}π ∈ A be a matricial set, and let { f π }π ∈ A be a family of functions such that f π maps U(π ) into B(Hπ ) for each π . Then { f π }π ∈ A is called a matricial family of functions on U in case C f π (z) = f τ (w)C for every z ∈ U(π ), every w ∈ U(τ ) and every C ∈ I(π, τ ) such that Cz = w(I M ⊗ C); i.e., in case Eq. (4.9) is satisfied for all pairs (z, w) ∈ U(π ) × U(τ ). ∞ ⊗n Recall that the tensor algebra T A (M) is defined to be the direct sum n=0 M  ⊗n (Definition 3.2). Of course, set theoretically, T A (M) is the coproduct, n≥0 M . We  now want to consider the product, n≥0 M ⊗n , which we shall denote by T A ((M)) and call the algebra of formal tensor series. Elements of T A ((A)) are formal sums, θ ∼ k≥0 θk , θk ∈ M ⊗k . These are added component-wise; the multiplication in T A ((M)) is given by the formula θ ∗ η = ζ, where θ ∼



θk , η ∼



ηk , and ζ ∼

ζk :=





ζk , where

θl ⊗ ηm ,

k, l, m ≥ 0.

k=l+m

It is evident, that T A (M) is a subalgebra T A ((M)). Just as in elementary complex analysis, we have an analogue of the CauchyHadamard theorem. Associated to each formal tensor series θ ∼ θk we define R(θ ) to be 1

R(θ ) := (lim sup θk  k )−1 ,

(5.5)

k→∞

obtaining a non-negative real number or +∞.  and each ζπ ∈ M π , Proposition 5.5 Suppose R(θ ) is not zero. Then for each π ∈ A the series ∞ 

Zk (z − ζπ )L θk

(5.6)

k=0

converges in the operator norm on B(Hπ ) for each z ∈ D(ζπ , R(θ )). The convergence of the series is uniform on any subdisc D(ζπ , r ), r < R(θ )), and it is uniform in π .

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Further, the resulting function, f π , is analytic as a map from D(ζπ , R(θ )) to B(Hπ ) and the Frechet derivative of f π is given by the formula D f π (z)[ζ ] =

∞ 

DZk (z − ζπ )[ζ ]L θk ,

z ∈ D(ζπ , R(θ )), ζ ∈ M π .

k=1

Further still, for each r < R(θ )), sup

sup

 z∈D(ζπ ,r ) π∈A

 f π (z) < ∞.

(5.7)

Proof The proof amounts to examining the estimate: Zk (z − ζπ )L θk  ≤ z − ζπ k θk  ≤ (

ρ k ) ρ

that is valid for all z ∈ D(ζπ , ρ) when ρ < ρ  .

 

Corollary 5.6 In the notation of Proposition 5.5, if the ζπ form an additive field of vectors, then not only is the family of discs D(ζ, R(θ )) := {D(ζπ , R(θ ))}π ∈ A a matricial family, but also the family of functions { f π }π ∈ A is a matricial family of functions. Theorem 5.1 of [33] yields the following converse to Proposition 5.5 and its corollary. Theorem 5.7 Suppose D(ζ, R), 0 < R ≤ ∞ is a matricial disc determined by the additive field ζ = {ζπ }π ∈ A and suppose that { f π }π ∈ A is a matricial family of functions defined on D(ζ, R) that is locally uniformly bounded in the sense that the inequality 5.7 is satisfied for all r < R. Then there is a tensorial power series θ ∼ k≥0 θk with R(θ ) ≥ R such f π (z) =



Zk (z − ζπ )L θk ,

z ∈ D(ζπ , R).

k≥0

The proof of Theorem 5.7 follows the basic steps that led to Taylor’s Taylor formula, Theorem 4.6, but pains must be taken to add appropriate estimates on the size of the terms in the formula. The details are spread out through [33, Section 5]. We note, too, in passing that in Section 6 of [33] we consider two C ∗ correspondences M and N and families of maps f := { f π }π ∈ A where f π maps the disc of radius r centered at the zero of M π , D M (0π , r ), to N π . We assume, first, that f respects intertwiners in the following sense that is a variant of Definitions 4.3 and 5.4: For each pair (z, w) ∈ D M (0π , r ) × D M (0τ , r ), I(π × z, τ × w) ⊆ I( f π (z), f τ (w)),

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(5.8)

P. S. Muhly, B. Solel

where I( f π (z), f τ (w)) = {C ∈ B(Hπ , Hτ ) | C f π (z) = f τ (I N ⊗ C). The second assumption we make on the family f is that the maps f π are bounded uniformly in π . This may be expressed succinctly by saying that there is an R ≥ 0 that is independent of π such that f π : D M (0π , r ) → D N (0π , R),

 π ∈ A.

(5.9)

Under these assumptions we prove in [33, Theorem 6.1] that the f π have power series expansions that are similar to those described in Theorem 5.7. We omit the details here, but the ultimate conclusion is: Local uniform boundedness in the sense of equations (5.7) and (5.9) coupled with the notions of Respecting Intertwiners in the senses of Definition 4.3 and Eq. (5.8) are the key properties that underwrite Tensorial Function Theory.

6 Concluding Remarks We close with some telegraphic remarks that we hope someone interested in pursuing the subject we have presented will find useful. In particular, we want to provide a brief survey of the literature that focuses on function-theoretic issues. Because of the broad scope of the subject, we leave aside references to many notable and important contributions to noncommutative function spaces and other related constructs. Much of the current developments in N C-function theory derive from Dan Voiculescu’s work in free probability. A good place to start with this connection is the pair of his papers devoted to questions in free analysis, [53, 54]. We already have mentioned the book by Dima Kaliuzhnyi-Verbovetskyi and Victor Vinnikov [22]. It and its preprints have been our primary source of inspiration. We note that much of the algebraic material we presented here was exposed first in their book, where the algebra A is assumed to be commutative and the bi-module M is such that the left and right actions of A are the same. In addition to treating fully Joe Taylor’s ideas from [52], it has a comprehensive review of N C-function theory literature up to 2014. We call special attention to the voluminous work of Gelu Popescu that will be exposed in his soon-to-be completed book, Noncommutative Multivariable Operator Theory. We mention it here especially because he has a perspective on noncommutative sets and spaces that is different from that which we have described. See in particular [39]. That Memoir inspired us to formulate another notion of noncommutative discs in [34] that is broader than that discussed in this survey. We would be remiss if we did not call attention to the work of Bill Helton and his collaborators, Igor Klep, and Scott McCullough, that is devoted to the structure of noncommutative sets – the matricial sets defined in Definition 5.2. Again, the number of papers in this area is enormous, but we believe the interested reader would find [17–19] very helpful as entrées to the subject. These papers were a major source of inspiration for our paper [33] on which this survey is based. More recently, Jim Agler and John McCarthy have contributed a lot to the subject. We recommend [1–3], for specific connections with what we have presented and to

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the ideas to be discussed below. The same may be said about the very important work of Igor Klep and Špela Špenko [24]. One challenge that we have been facing, and which many may find interesting, is to generalize results from N C-function theory to the tensorial setting. We do not mean this simply as a call to “pick off low hanging fruit”. Rather, in order to have a full bodied function theory - either in the N C-setting or the tensorial setting - one should think in terms of schemes that we mentioned in the introduction. One needs sheaves and ways to glue sheaves together. Such efforts should lead to noncommutative holomorphic spaces and related constructs. We find ourselves trying to channel Herman Weyl, who in the preface of his book on Riemann surfaces [56] famously wrote: I shared his [Felix Klein’s] conviction that Riemann surfaces are not merely a device for visualizing the many-valuedness of analytic functions, but rather an indispensable essential component of the theory; not a supplement, more or less artificially distilled from the functions, but their native land, the only soil in which the functions grow and thrive. So, we ask: What is the natural soil of noncommutative and tensorial functions? As we mentioned earlier, we like to think of matricial sets as analogues of polynomially convex sets in affine spaces. Some efforts to go beyond matricial sets were made very early in the theory. We have in mind the work of Joe Taylor’s student Denis Luminet [28, 29]. The more recent work of Agler and McCarthy and that of Klep and Špenko we cited suggest possibilities for how to proceed further. We add to this list the very important paper of Klep, Vinnikov, and Jurij Volˇciˇc [23]. Our lodestar, what we believe is essential to keep in mind, is this: N C-functions and tensorial functions are fibred structures; they are fibred over the object spaces of categories. The objects of the category parameterize the functions and their domains. The morphisms dictate how to move among the fibres. In the case of N C-functions the category, which we denote here by Mat, is the skeletal category whose objects, Mat0 , are the positive integers and whose morphisms, Mat1 , are the n × m matrices, 1 ≤ n, m < ∞, i.e., HomMat (n, m) is the set of n × m matrices. The composition of morphisms in through matrix multiplication. (We allow m and n to be ∞ from time to time and we treat the corresponding matrices as operators on Hilbert space.) In the  So, when A is a C ∗ -algebra the proper setting to tensorial setting, the category is A. study these function-theoretic constructs may well be C ∗ - and W ∗ -categories [15]. However, we note that the more general category of operator spaces and completely bounded maps may be more appropriate. Gille Pisier already has suggested a theory of tensor algebras in this category [38]. Whatever is the appropriate categorical setting, that categorical structure should be impressed in the “natural soil”. While the N C- and tensorial functions are global constructs, depending upon all the objects in the category, it is interesting and important to study the fibre determined by a single object. In this situation, we have found that one is drawn ineluctably to study the full subcategory generated (in the categorical sense) by the object. This study, in turn, leads directly to interesting issues in geometric invariant theory. Some of our contributions to this perspective are published in [16] which was co-authored with Erin Griesenauer.

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We drew inspiration from several sources. First, there is Claudio Procesi’s fundamental work devoted to the study of the simultaneous similarity classes of d-tuples of n ×n matrices [41]. It, and his two papers with Lieven Le Bruyn [26, 27] made it abundantly clear to us how geometric invariant theory plays a role in N C-function theory. In this connection, we also want to call attention to Le Bruyn’s book [25], which is full of down-to-earth computations that help to reveal the properties of the structures involved. Also, we want to cite Nolan Wallach’s recent text on invariant theory [55] which makes geometric invariant theory easily accessible to the uninitiated. Finally, we want especially to call attention to the paper [43] in which Zinovy Reichstein and Nicolas Vonessen study polynomial identity algebras as rings of functions. It was the principal source of inspiration for [16]. Data availability Data sharing is not applicable. No datasets were generated or analyzed in the research for or writing of this article.

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Complex Analysis and Operator Theory (2022) 16:50 https://doi.org/10.1007/s11785-022-01225-8

Complex Analysis and Operator Theory

Operator Theory on Noncommutative Polydomains, I Gelu Popescu1 Received: 17 January 2022 / Accepted: 6 March 2022 / Published online: 2 April 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract The goal of this paper is to study the noncommutative polydomains and their universal operator models generated by admissible k-tuples of formal power series in several noncommuting indeterminates. Several aspects of the multi-variable operator theory of these polydomains and their universal models are discussed in connection with the noncommutative Hardy algebras they generate. Keywords Multivariable operator theory · Noncommutative polydomains · Universal operator models · Fock spaces · Noncommutative Hardy algebras · C ∗ -algebras Mathematics Subject Classification Primary 47A20, 46L45, 46L52; Secondary 47A60, 47B37

Contents 1 2 3 4 5 6 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formal Power Series, Universal Operator Models, and Noncommutative Polydomains . . . . . . Noncommutative Berezin Kernels, Polydomain Algebras, and Von Neumann Inequality . . . . . Noncommutative Hardy Algebras Associated with Polydomains and Multi-Analytic Operators . . w ∗ -Continuous Functional Calculus for Completely Non-coisometric Elements in Polydomains . Bohr Inequality for the Noncommutative Hardy Algebra F ∞ (g) . . . . . . . . . . . . . . . . . . The Algebra H ol(E rad −1 ) of Free Holomorphic Functions and Weierstrass, Montel, Vitali Theorems g

2 9 32 40 56 63 77

Dedicated to the memory of Jörg Eschmeier. Communicated by Michael Hartz. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz and Mihai Putinar. Research supported in part by NSF Grant DMS 1500922.

B 1

Gelu Popescu [email protected] Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA

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G. Popescu 8 Algebras of Bounded Free Holomorphic Functions on Polydomains and Schwarz Lemma . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90 97

1 Introduction In 1963, Foia¸s [47] (see also de Brange-Rovnyak [42]) proved that the universal model of Rota [93], i.e. the unilateral shift, is in fact a model for all contractions satisfying the condition that T ∗n → 0 strongly as n → ∞. This was one of the stepping stones for the development of the celebrated Sz.-Nagy-Foia¸s theory of contractions on Hilbert spaces [99]. The idea of using weighted shifts as models for operators goes back to Agler [1, 2] in his study of hypercontractions. This was generalized by Müller [59] and reconsidered more recently by Olofsson [61, 62], and by Ball and Bolotnikov [17]. In the multivariable commutative setting, the use of weighted shifts as universal models for n-tuples of operators was initiated by Drury [46], Müller-Vasilescu [60], Athavale [15], Curto-Vasilescu [32, 104] in their attempt to generalize the Sz.-Nagy– Foia¸s dilation theory. This was revitalized by Arveson [13, 14], Davidson-Pitts [38], and by the author [73, 75], and has been extensively studied in recent years by several authors [22, 23, 27, 30, 40, 41, 94, 95, 102], and others. The study of more general commutative domains and their operator models was pursued in [8, 10, 92], and [84]. In several noncommuting variables, the study of the closed operator ball   n ∗ ∗ [B(H)n ]− 1 := (X 1 , . . . , X n ) ∈ B(H) : I − X 1 X 1 − · · · − X n X n ≥ 0 and the associated universal model (S1 , . . . , Sn ) of left creation operators on the full Fock space with n generators has generated a free analogue of Sz.-Nagy–Foia¸s theory (see [12, 16, 29, 36, 37, 39, 48, 51, 52, 55, 56, 67–76, 81, 82], and the references there in). More general noncommutative domains and their operator models were studied by Arias and the author in [11], and by the author in [80, 84, 89], and [91]. This has been accompanied by the development of the theory of free holomorphic functions on these noncommutative sets. Several classical results from complex analysis, hyperbolic geometry, and interpolation theory have free analogues in this noncommutative multivariable setting (see [49–52, 77–79, 82, 83], etc.). We should mention that, in an abstract setting, analytic functions of noncommuting variables originate in the pioneering work of Taylor [100, 101]. This has been revived by Voiculescu in [105, 106] in connection with free probability, and it has been pushed forward in the last decade by Helton, Klep, McCullough, and Slingled [49, 50], Kalyuzhnyi-Verbovetskyi and Vinnikov [53], Agler and McCarthy [3–5], Klep, Vinnikov, and Volˇciˇc [57], and many others. Following Andô’s dilation theorem [9] for two commuting contractions, Brehmer [28] showed that under certain positivity conditions on a family of commuting contractions, one can obtain the so-called regular dilations. Motivated by Agler’s work [2] on weighted shifts, Curto and Vasilescu initiated a theory of standard operator models in the commutative polydisc in [33, 34]. Timotin [102] was able to obtain some of their results from Brehmer’s theorem. More recently, we should mention the work of Jaydeb Sarkar and his collaborators [20, 35, 58, 96, 97], and [21].

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Operator Theory on Noncommutative Polydomains, I

In an attempt to unify the multivariable operator model theory for noncommutative domains and commutative polydiscs, we initiated the study of noncommutative polyballs and polydomains in [73, 85–87], and [88]. The present paper is a continuation of the ongoing program to develop a free analogue of the Sz.-Nagy-Foia¸s theory of contractions for noncommutative polydomains and varieties in several noncommuting variables and to develop the theory of free holomorphic functions on these domains and varieties. The paper is devoted to enhancing the understanding of the structure of large classes of noncommutative polydomains which admit universal models and have rich analytic function theory. This is accompanied by the study of the universal algebras associated with these noncommutative domains in connection with their representation theory and the harmonic analysis on tensor products of full Fock spaces. While the class of these polydomains contains the regular polydomains previously studied in the literature, the main focus of the present paper is on the non-regular noncommutative polydomains with the emphasize on the multi-variable operator theory. Let Z = (Z 1 , . . . , Z k ) be a k-tuple where Z i = (Z i,1 , . . . , Z i,n i ) is an n i -tuple of noncommuting indeterminates and assume that, for each s, t ∈ {1, . . . , k} with s = t, the entries of Z s are commuting with the entries of Z t . Let F+ n i be the unital free semigroup on n i generators g1i , . . . , gni and the identity g0i and set Z i,gi := 1 0

and Z i,α := Z i, j1 · · · Z i, j p if α = g ij1 · · · g ij p ∈ F+ n i . In this setting, the length of α is defined by |α| = p and |g0i | = 0. Denote by S Z the set of all formal power series ζ =



cα1 ,...,αk Z 1,α1 · · · Z k,αk ,

cα1 ,...,αk ∈ C.

αi ∈F+ ni

Let g= (g1 , . . . , gk ) be a k-tuple of formal power series gi ∈ S Z i with gi := 1 + αi ∈F+n ,|αi |≥1 bi,α Z i,α , where bi,α > 0 and gi is a free holomorphic function in i a neighborhood of the origin in B(H)n i , which is equivalent to ⎛ ⎜ lim sup ⎝ k→∞

⎞1/2k



2 ⎟ bi,α ⎠

< ∞.

α∈F+ n i ,|α|=k

We associate with g the Hilbert space F 2 (g) of formal power series in S Z with orthogonal basis {Z 1,α1 · · · Z k,αk : αi ∈ F+ n i } such that Z 1,α1 · · · Z k,αk := √ 1 . Note that b1,α1 ···bk,αk

F 2 (g) =

⎧ ⎪ ⎨ ⎪ ⎩

ζ =



cα1 ,...,αk Z 1,α1 · · · Z k,αk : ζ g :=

αi ∈F+ ni

 αi ∈F+ ni

⎫ ⎪ ⎬ 1 |cα1 ,...,αk |2 < ∞ . ⎪ b1,α1 · · · bk,αk ⎭

We define the left multiplication operators L i, j on F 2 (g) by L i, j ζ = Z i, j ζ , where i ∈ {1, . . . , k} and j ∈ {1, . . . , n i }. It is easy to see that L i, j is a bounded operator if and only if

Reprinted from the journal

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G. Popescu

sup

α∈F+ ni

bi,α < ∞. bi,gi α j

Throughout this paper, we denote by B(H) the algebra of bounded linear operators on a Hilbert space H. Denote by Mg (H) the noncommutative set of all k-tuples X = (X 1 , . . . , X k ) in B(H)n 1 × · · · × B(H)n k with X i = (X i,1 , . . . , X i,n i ) such that the k-tuple L = (L 1 , . . . , L k ) with L i := (L i,1 , . . . , L i,n i ) can play the role of universal operator model for X , i.e. there is a Hilbert space G such that ∗ ∗ X i, j = (L i, j ⊗ IG )|H .

A challenging question is whether the noncommutative sets Mg (H) can be completely described by certain solution sets of operator inequations. We define the defect operator g−1 (X , X ∗ ) := g−1 ,X 1 ◦ · · · ◦ g−1 ,X k (I ), 1

where gi−1 = 1 +

 α∈F+ n i ,|α|≥1

ai,α Z i,α is the inverse of the formal power series gi and

g−1 ,X i (Y ) := i

k

∞ 



∗ ai,α X i,α Y X i,α ,

p=0 α∈F+ n ,|α|= p i

where X i ∈ B(H)n i and Y ∈ B(H) are such that the series converges in the weak pur e operator topology. We introduce the pure noncommutative polydomain Dg−1 (H) ⊂ B(H)n 1 +···+n k as the set of all pure solutions of the operator inequation g−1 (X , X ∗ ) ≥ 0. Our first goal is to find large classes of k-tuples g of formal power series such that the noncommutative sets Mg (H) can be completely described. This goal is achieved pur e by proving that Dg−1 (H) = Mg (H) if and only if         ∗  sup  ai,β Li,β Li,β  < ∞,  m∈N  β∈F+ni |β|≤m 

i ∈ {1, . . . , k}.

In this case, we say that g = (g1 , . . . , gk ) is an admissible k-tuple for operator model pur e theory and Dg−1 (H) is an admissible polydomain. To the benefit of the reader, we distinguish the following examples of admissible k-tuples of formal power series, which we will refer to throughout the paper. Consider any noncommutative polynomials ϕi := α∈F+n ,|α|≥1 di,α Z i,α with nonnegative coefi

ficients and di,α > 0 if |α| = 1. Given any s := (s1 , . . . , sk ) ∈ [1, ∞)k , the k-tuple of formal power series

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ψ s := ((1 − ϕ1 )−s1 , . . . , (1 − ϕk )−sk ) is admissible for operator model theory. The scale {F 2 (ψ s ) : s := (s1 , . . . , sk ) ∈ [1, ∞)k } of generalized noncommutative Bergman spaces and the associated polypur e domains D −1 were never studied in the multivariable noncommutative setting, ψs

as far as we know, except the case when s ∈ Nk which was studied in [84, 85], and [86]. We also remark the scale of weighted noncommutative Bergman spaces {F 2 (γ s ) : s := (s1 , . . . , sk ) ∈ (0, ∞)k }, where the admissible k-tuples of formal power series are ⎛⎛ γ s := ⎝⎝1 −

n1 

⎛

⎞−s1 Z 1, j ⎠

, . . . ⎝1 −

j=1

nk 

⎞−sk ⎞ Z k, j ⎠

⎠.

j=1

Note that if s1 = · · · = sk = 1 we recover the noncommutative polyball. If k = 1, n 1 = n, and s1 = n (resp. s1 = n + 1) the corresponding subspace F 2 (γ s ) can be identified with the the noncommutative Hardy (resp. Bergman) space over the unit ball [B(H)n ]1 . If k = 1, n 1 = n, and s ∈ (0, 1] the Hilbert spaces F 2 (γ s ) are noncommutative generalizations of the classical Besov-Sobolev spaces on the unit 1 ball Bn ⊂ Cn with representing kernel (1−z,w ) s , z, w ∈ Bn . The scale of noncommutative Dirichlet spaces {F 2 (ωs ) : s := (s1 , . . . , sk ) ∈ [0, ∞)k } is another remarkable example, where the admissible k-tuples of formal power series are given by ⎞

⎛ ⎜ ωs := ⎝ α∈F+ n1

1 Z 1,α , . . . , (|α| + 1)s1

 α∈F+ nk

1 ⎟ Z k,α ⎠ . s k (|α| + 1)

When k = 1 and s1 = 1, we get the noncommutative Dirichlet space over the unit ball [B(H)n ]1 . More general classes of admissible k-tuples of formal power series and the associated polydomains are presented in Sect. 2 of this paper. Throughout this paper, unless otherwise specified, we assume that g = (g1 , . . . , gk )  is a k-tuple of free holomorphic functions gi := 1 + αi ∈F+n ,|αi |≥1 bi,α Z i,α in a i neighborhood of the origin in B(H)n i with strictly positive coefficients and such that b supα∈F+n b i,αi < ∞. For convenience, the role of universal operator model associated i

i,g j α

with g will be played by a k-tuple W = (W1 , . . . , Wk ) with Wi := (Wi,1 , . . . , Wi,n i ), where Wi, j are weighted left creation operators acting on the tensor products F 2 (Hn 1 ) ⊗ · · · ⊗ F 2 (Hn k ), where F 2 (Hn i ) is the full Fock space with n i generators. As expected, L is jointly unitarily equivalent to W, i.e. there is a unitary operator such that U Wi, j = L i, j U for all i, j. Our approach to discover larger classes of polydomains that admit universal operator models is different from the one pursued in our previous work [85, 86], where we focused on a particular class of polydomains and then found the coresponding

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universal model. The road map is detailed in Sect. 2. We fix a k-tuple g of free holomorphic functions, as above, and define the associated universal model W. Then we find necessary and sufficient conditions to ensure that a k-tuple X = (X 1 , . . . , X k ) ∈ B(H)n 1 × · · · × B(H)n k with X i = (X i,1 , . . . , X i,n i ) is in the noncommutative set Mg (H), i.e. X admits W as universal model. This leads us to introducing the noncompur e mutative polydomain Dg−1 (H), its pure part Dg−1 (H), and the Cuntz part Dgc−1 (H). We show that Mg (H) contains a neighborhood of the origin in the relative product topology and pur e

Dg−1 (H) ⊂ Mg (H). The main result of this section, Theorem 2.13, provides necessary and sufficient condipur e tions to ensure that Dg−1 (H) = Mg (H). In this case, we say that g is an admissible pur e

k-tuple and Dg−1 (H) is an admissible polydomain. Several classes of admissible noncommutative polydomains, which have not been studied yet, are introduced. In Sect. 3, using Berezin type [19] noncommutative kernels, we provide representations for the pure elements in Dg−1 (H) and show that if X = (X 1 , . . . , X k ) ∈ pur e

Dg−1 (H) with X i = (X i,1 , . . . , X i,n i ), then there is a unital completely contractive homomorphism  X : A(g) → B(H),  X ( p(W)) := p(X ), for any polynomial p in n 1 + · · · + n k noncommutative indeterminates Z i, j , where the noncommutative domain algebra A(g) is defined as the norm-closed non-selfadjoint algebra generated by the weighted shifts Wi, j and the identity. The main result of this section which will be very useful in our investigation is the following. Let f := (f1 , . . . , fk ) and g := (g1 , . . . , gk ) be two admissible k-tuples of free holomorphic functions and let W(f) and W(g) be the universal models associated with the pur e pur e polydomains Df −1 and Dg−1 , respectively. We prove that Dg−1 ⊂ Df −1 if and only if   there is a Hilbert space G and an isometry V : ks=1 F 2 (Hn s ) → ks=1 F 2 (Hn s ) ⊗ G such that (g)∗

(f)∗

V Wi, j = (Wi, j ⊗ IG )V , which turns out to be equivalent to the fact the linear map  : A(f) → A(g) defined by (f)

(f)

(g)

(g)

(W1,α1 · · · Wk,αk ) := W1,α1 · · · Wk,αk ,

αi ∈ F+ ni ,

is completely contractive and has a co-extension to a ∗-representation π : C ∗ (W(f) ) → B(K) which is pure in a certain sense. In Sect. 4, we introduce the noncommutative Hardy algebras F ∞ (g) and provide some basic properties. If g := (g1 , . . . , gk ) is a k-tuple of free holomorphic functions such that

626

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sup

α∈F+ ni

bi,α < ∞, bi,αgi

j ∈ {1, . . . , n i },

j

we show that F ∞ (g) is the sequential SOT-(resp. WOT-, w*-) closure of all polynomials in Wi, j and the identity, and every element in F ∞ (g) has a unique Fourier type representation. We introduce the (right) noncommutative Hardy algebra R ∞ (g) and show that F ∞ (g) = R ∞ (g) , where  stands for the commutant. We also prove the existence of a w ∗ -continuous F ∞ (g)-functional calculus for pure elements in the noncommutative polydomain Dg−1 (H). In Sect. 5, we provide a w ∗ -continuous F ∞ (g)-functional calculus for the completely non-coisometric (c.n.c.) elements in the noncommutative polydomain Dg−1 (H). An element X ∈ Dg−1 (H) is said to be c.n.c. with respect to the polydomain Dg−1 if there is no nonzero joint invariant subspace H0 under the operators ∗ , where i ∈ {1, . . . , k} and j ∈ {1, . . . , n }, such that X i, i j g−1 (X , X ∗ )|H0 = 0. pur e

The main result of this section states that if X ∈ Dg−1 (H) is a c.n.c. element with respect to Dg−1 (H), then the completely contractive linear map  : A(g) → B(H),

( p(W)) := p(X ),

has a unique extension to a w ∗ -continuous homomorphism  X : F ∞ (g) → B(H) which is completely contractive and sequentially WOT-continuous (resp. SOT-continuous). Moreover, if ϕ(W) ∈ F ∞ (g) has the Fourier representation  cβ1 ,...,βk W1,β1 · · · Wk,βk , then + β1 ∈F+ n 1 ,...,βk ∈Fn k

 X (ϕ(W)) = SOT-



lim

N1 →∞...,Nk →∞



×

 1−

(s1 ,...,sk )∈Zk ,|s j |≤N j

   |sk | |s1 | ··· 1 − N1 + 1 Nk + 1

cα1 ,...,αk X 1,α1 · · · X k,αk .

αi ∈F+ n i ,|αi |=si i∈{1,...,k}

 k Bohr’s inequality [25] (see also [65, 98, 103]) asserts that, if f (z) := ∞ k=0 ak z is an analytic function on the open unit disc D := {z ∈ C : |z| < 1} such that

f ∞ ≤ 1, then ∞ 

r k |ak | ≤ 1

for 0 ≤ r ≤

k=0

1 . 3

The fact that 13 is the best possible constant was proved by M. Riesz, Schur, and Weiner, independently. Several attempts have been made (see [6, 7, 18, 24, 43–45, 64, 78, 84]) to find multivariable analogues of Bohr’s inequality in various settings.

Reprinted from the journal

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In Sect. 6, we study the Bohr phenomenon for the noncommutative Hardy algebra F ∞ (g). The multi-homogeneous Bohr radius for the noncommutative Hardy algebra F ∞ (g) is denoted by K mh (F ∞ (g)) and is the largest r > 0 such that             r |β1 |+···+|βk | cβ1 ,...,βk W1,β1 · · · Wk,βk  ≤ ϕ(W)

   +  ( p1 ,..., pk )∈Zk+  β1 ∈F+ n 1 ,...,βk ∈Fn k   |β1 |= p1 ,...,|βk |= pk

for any ϕ(W) ∈ F ∞ (g). The homogeneous Bohr radius for the noncommutative Hardy algebra F ∞ (g), denoted by K h (F ∞ (g)), is defined in a similar manner but using the homogeneous expansions. To give the reader a flavor of our results, we mention the following particular case. Let s := (s1 , . . . , sk ) ∈ [1, ∞)k and ψ s := ((1 − ϕ1 )−s1 , . . . , (1 − ϕk )−sk ),  where each ϕi := α∈F+n ,|α|≥1 di,α Z i,α is a formal power series with nonnegative i coefficients and di,α > 0 if |α| = 1. Then the Bohr radius K mh (F ∞ (ψ s )) satisfies the inequality   1 2 s1 +···+sk < r0 ≤ K mh (F ∞ (ψ s )), 1− 3 where r0 is the nonnegative solution of the equation  ( p1 ,..., pk )∈Zk+ \{0}

r

p1 +···+ pk

1/2 k   1 si + pi − 1 = . pi 2 i=1

We also mention that if the universal model W is radially pure, i.e. r W is pure for any r ∈ [0, 1], then we prove that the homogeneous Bohr radius for the noncommutative Hardy algebra F ∞ (g) satisfies the inequality 13 ≤ K h (F ∞ (g)). In particular, if there  s is there is s ∈ {1, . . . , k}, such that gs−1 = 1 − nj=1 as, j Z s, j , where as, j > 0, then K h (F ∞ (g)) = 13 . In Sect. 7, we provide basic properties for the algebra H ol(Egrad −1 ) of all free holomorphic functions on the radial envelope  pur e Egrad r Dg−1 (H). −1 (H) := 0≤r 0. If 0 < t < √ρn , then the |α|= p |ai,α | i n i -tuple (t IH , . . . , t IH ) is in the ball [B(H)n i ]ρ , which implies M :=

∞ 



|ai,α |t 2 p < ∞.

p=1 α∈F+ n ,|α|= p i

Let 0 < ω < 1 and set δ := ωt. Note that, if X i ∈ [B(H)n i ]δ , then ∞ 



∗ |ai,α | X i,α X i,α

p=1 α∈F+ n ,|α|= p i

≤

∞ 



|ai,α |δ 2 p ≤ ω2

p=1 α∈F+ n ,|α|= p

∞ 



|ai,α |t 2 p = ω2 M.

p=1 α∈F+ n ,|α|= p

i

i

Now, one can easily complete the proof.    Definition 2.6 Let gi = 1+ α∈F+n ,|α|≥1 bi,α Z i,α , i ∈ {1, . . . , k}, be free holomorphic i

−1 functions  in a neighborhood of the origin with bi,α > 0 and with inverse gi = 1 + α∈F+n ,|α|≥1 ai,α Z i,α . The noncommutative polydomain Dg−1 (H) is defined as i the set of all k-tuples X = (X 1 , . . . , X k ) ∈ B(H)n 1 ×c · · · ×c [B(H)n k with X i := (X i,1 , . . . , X i,n i ) ∈ B(H)n i with the property that the defect operator

g−1 (X , X ∗ ) := g−1 ,X 1 ◦ · · · ◦ g−1 ,X k (I ) 1

k

is a well-defined positive operator and g1 ,X 1 ◦ · · · ◦ gk ,X k (g−1 (X , X ∗ )) ≤ I .

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When we want to emphasize the components of g−1 := (g1−1 , . . . , gk−1 ), we use the notation Dg−1 ,...,g−1 (H) for the noncommutative set Dg−1 (H). 1

k

Definition 2.7 Let X ∈ Dg−1 (H). (i) X is said to be a Cuntz type tuple in Dg−1 (H) if g−1 (X , X ∗ ) = 0. (ii) X is said to be a pure element in Dg−1 (H), if g1 ,X 1 ◦ · · · ◦ gk ,X k (g−1 (X , X ∗ )) = I . (iii) The pure part of Dg−1 (H) is defined by setting pur e

Dg−1 (H) := {X ∈ Dg−1 (H) : X is pure}. (iv) The Cuntz part of Dg−1 (H) is defined by setting Dgc−1 (H) := {X ∈ Dg−1 (H) : g−1 (X , X ∗ ) = 0}. pur e

We remark that one can easily show that Dg−1 (H), Dg−1 (H), and Dgc−1 (H) are operator noncommutative sets. Definition 2.8 We say that X = (X 1 , . . . , X k ) ∈ B(H)n 1 +···+n k is δ-radially pure with pur e respect to Dg−1 (H) if there is δ ∈ [0, 1) such that r X := (r X 1 , . . . , r X k ) ∈ Dg−1 (H) for every r ∈ [δ, 1). For simplicity, a 0-radially pure element is called radially pure. A set  ⊂ B(H)n 1 +···+n k is called radially pure with respect to Dg−1 (H) if any X ∈  is radially pure. We say that bounded linear map  : B(H) → B(H) is pure if WOT− lim m (Y ) = 0 for any Y ∈ B(H). m→∞

In particular, if  < 1, then  is pure. Note that if  is a positive linear map, then it is pure if and only if WOT- limm→∞ m (I ) = 0.  Theorem 2.9 Let gi = 1 + α∈F+n ,|α|≥1 bi,α Z i,α , i ∈ {1, . . . , k}, be free holomorphic i

−1 functions  in a neighborhood of the origin with bi,α > 0 nand with inverse gi n = 1 + α∈F+n ,|α|≥1 ai,α Z i,α . Let X = (X 1 , . . . , X k ) ∈ B(H) 1 ×c · · · ×c [B(H) k be i a k-tuple with X i := (X i,1 , . . . , X i,n i ) ∈ B(H)n i such that the following conditions hold:

(i) g−1 (X , X ∗ ) ≥ 0; (ii) There is c ∈ (0, 1) such that 

∗ |ai,α |X 1,α X 1,α < cI .

α∈F+ n i ,|α|≥1

Reprinted from the journal

637

G. Popescu pur e

Then X ∈ Dg−1 (H). Moreover, for any subset {i 1 , . . . , i p } ⊂ {1, . . . , k} with i 1 < i 2 < . . . < i p , the p-tuple (X i1 , . . . , X i p ) is in the pure polydomain D

pur e (H). gi−1 ,...,gi−1 p 1

Proof Define the linear maps ∞ 

g−1 ,X i (Y ) := i



∗ ai,α X i,α Y X i,α ,

Y ∈ B(H),

p=1 α∈F+ n i ,|α|= p

and ˆ −1 (Y ) :=  g ,X i

∞ 

i



∗ |ai,α |X i,α Y X i,α ,

Y ∈ B(H),

p=1 α∈F+ n ,|α|= p i

and note that they are well-defined. It is easy to see that the multi-series  αi ∈F+ n i ,|αi |≤m i

∗ ∗ |a1,α1 | · · · |ak,αk |X 1,α1 · · · X k,αk X k,α · · · X 1,α 1 k

i∈{1,...,k}

ˆ −1 ) ◦ · · · ◦ (id +  ˆ −1 )(I ) = (id +  g ,X 1 g ,X k 1

k

≤ (c + 1)k I is convergent in the weak operator topology. Consequently,           ∗ ∗  sup  |a1,β1 | · · · |ak,βk |X 1,β1 · · · X k,βk X k,βk · · · X 1,β1  < ∞  m 1 ,...,m k ∈N  βi ∈F+ni ,|βi |≤m i    i∈{1,...,k}

and g−1 (X , X ∗ ) := WOT- lim · · · lim m 1 →∞ m k →∞  ∗ ∗ a1,β1 · · · ak,βk X 1,β1 · · · X k,βk X k,β · · · X 1,β 1 k βi ∈F+ n i ,|βi |≤m i i∈{1,...,k}

exists and it does not depend on the order of the iterated limit. Moreover, as above, we can show that, for any i ∈ {1, . . . , k − 1}, g−1 ,...,g−1 (X , X ∗ ) := WOT- lim · · · lim m i →∞ m k →∞ i k  ∗ ∗ ai,βi · · · ak,βk X i,βi · · · X k,βk X k,β · · · X i,β k i βs ∈F+ n s ,|βs |≤m s s∈{i,...,k}

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exists. On the other hand, for each i ∈ {1, . . . , k} and Y = Y ∗ ∈ B(H), we have .& '.  . . ∗ h, h . . |ai,γ1 | · · · |ai,γm |X i,γ1 ···γm Y X i,γ 1 ···γm γ1 ,...,γm ∈F+ ni |γ1 |≥1,...,|γm |≥1

.* +. . m . . ˆ ≤ Y .  −1 (I )h, h .. ≤ cm Y

h 2 gi ,X i

which implies ⎛ ∞ ⎜  ⎜ ⎜ ⎝

m=1

⎞  γ1 ,...,γm ∈F+ ni |γ1 |≥1,...,|γm |≥1

.& '.⎟ c .⎟ . ∗

Y

h 2 . h, h .⎟ ≤ . |ai,γ1 | · · · |ai,γm |X i,γ1 ···γm Y X i,γ ···γ m 1 ⎠ 1−c

Due to the absolute convergence of the series of real numbers ⎛

⎞

∞ ⎜  ⎜ ⎜ ⎝

m=1



γ1 ,...,γm ∈F+ ni |γ1 |≥1,...,|γm |≥1

which is equal to

∞

& '⎟ ⎟ ∗ (−1)m ai,γ1 · · · ai,γm X i,γ1 ···γm Y X i,γ h, h ⎟, ···γ m 1 ⎠

*

+

m m m=1 (−1) g−1 ,X i (Y )h, h i

, we can rearrange it and prove that

it is equal to ∞ 



m=1 β∈F+ n ,|β|=m i

⎛ ( |β| ⎜ ⎜ ⎜ ⎝ j=1

⎞



) ⎟ ⎟ ∗ (−1) ai,γ1 · · · ai,γ j ⎟ X i,γ1 ···γ j Y X i,γ1 ···γ j h, h . ⎠ j

γ1 ···γ j =β |γ1 |≥1,...,|γ j |≥1

On the other hand, if Y = Y ∗ , then .* .* +. . m . m . .  −1 (Y )h, h . ≤ Y .  ˆ −1 . . . gi ,X i

gi

+. . (I )h, h .. ≤ cm Y

h 2 ,X i

* for any h ∈ H and m ∈ N. Consequently, limm→∞  m−1

gi ,X i

+ (Y )h, h = 0 for any

h ∈ H, which shows that g−1 ,X i is a pure linear map. Since gi−1 gi = gi gi−1 = 1, i Lemma 1.1 from [91] shows that bi,α =

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|α| 



j=1

γ1 ···γ j =α |γ1 |≥1,...,|γ j |≥1

639

(−1) j ai,γ1 · · · ai,γ j

G. Popescu

for every α ∈ F+ n i with |α| ≥ 1. Consequently, using the results above, we deduce that ( lim

m→∞

)



∗ bi,β X i,β Y X i,β h, h

β∈F+ n i ,|β|≤m

= Y h, h +

∞ 

& ' ∗ bi,β X i,β Y X i,β h, h



p=1 β∈F+ n ,|β|= p i

= Y h, h + ⎛ ( |β| ⎜ ⎜ ⎜ ⎝ j=1

∞ 



m=1 β∈F+ n ,|β|=m i

⎞

⎟ ⎟ ∗ (−1) ai,γ1 · · · ai,γ j ⎟ X i,γ1 ···γ j Y X i,γ h, h 1 ···γ j ⎠



γ1 ···γ j =β |γ1 |≥1,...,|γ j |≥1

= Y h, h +

∞ * 

)

j

+

(−1)m  m−1

gi ,X i

m=1

(Y )h, h

for any h ∈ H. If i ∈ {1, . . . , k − 1} and take Y = g−1 ,...,g−1 (X , X ∗ ) in the relation i k above, we deduce that ( lim

m→∞

&

)

 β∈F+ n i ,|β|≤m

∗ bi,β X i,β g−1 ,...,g−1 (X , X ∗ )X i,β h, h i

∗

k

*

'

= g−1 ,...,g−1 (X , X )h, h − lim (−1) m→∞ i+1 k ' & = g−1 ,...,g−1 (X , X ∗ )h, h . i+1

m

 m−1 gi ,X i



∗



+

g−1 ,...,g−1 (X , X ) h, h i+1

k

k

The later equality is due to the fact that &

' g−1 ,...,g−1 (X , X ∗ )h, h i k & ' & ' = g−1 ,...,g−1 (X , X ∗ )h, h + g−1 ,X i (g−1 ,...,g−1 )(X , X ∗ )h, h i+1

k

* and limm→∞ (−1)m  m−1

gi ,X i

i

i+1

k

+   g−1 ,...,g−1 (X , X ∗ ) h, h = 0. Note also that, if i+1

k

g−1 ,...,g−1 (X , X ∗ ) ≥ 0, then the relations above show that g−1 ,...,g−1 (X , X ∗ ) ≥ 0. i k i+1 k Since g−1 (X , X ∗ ) = g−1 ,...,g−1 (X , X ∗ ) ≥ 0, 1

k

640

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we deduce that g−1 ,...,g−1 (X , X ∗ ) ≥ 0 for any i ∈ {1, . . . , k}. Moreover, as above, i k one can show that g−1 ,...,g−1 ≥ 0 for any subset {i 1 , . . . , i p } ⊂ {1, . . . , k} with i1

ip

i1 < i2 < . . . < i p . Now, using the relations above when i = 1, . . . , k − 1, we obtain 

lim · · · lim

m k →∞

m 1 →∞



···

αk ∈F+ nk |αk |≤m k

bk,αk

α1 ∈F+ n1 |α1 |≤m 1

∗ ∗ · · · b1,α1 X k,αk · · · X 1,α1 g−1 (X , X ∗ )X 1,α · · · X k,α 1 k  = lim · · · lim bk,αk X k,αk m k →∞

⎛

m 1 →∞

⎛

αk ∈F+ nk |αk |≤m k

⎞

⎞

⎜ ⎜  ⎟ ⎟ ⎜ ⎜ ⎟ ∗ ∗ ⎟ · · · b1,α1 X 1,α1 g−1 ,...,g−1 (X , X ∗ )X 1,α ⎜· · · ⎜ ⎟ ⎟ X k,αk 1 1 k ⎝ ⎝ ⎠ ⎠ + α1 ∈Fn 1 |α1 |≤m 1

= lim · · · lim m k →∞

⎛



m 2 →∞

⎛

bk,αk X k,αk

αk ∈F+ nk |αk |≤m k

⎞

⎞

⎜ ⎜  ⎟ ⎟ ⎜ ⎜ ⎟ ∗ ∗ ⎟ · · · b2,α2 X 2,α2 g−1 ,...,g−1 (X , X ∗ )X 2,α ⎜· · · ⎜ ⎟ ⎟ X k,αk 2 2 k ⎝ ⎝ ⎠ ⎠ + α2 ∈Fn 2 |α2 |≤m 2

= ······ = lim

m k →∞

 αk ∈F+ nk |αk |≤m k

∗ bk,αk X k,αk g−1 (X , X ∗ )X k,α = I − lim (−1)m  m−1 k m→∞

k

gk ,X k

(I ) = I ,

where the limits are in the weak operator topology. This shows that lim · · · lim

m k →∞

m 1 →∞

 αk ∈F+ nk |αk |≤m k

···



bk,αk

α1 ∈F+ n1 |α1 |≤m 1

∗ ∗ · · · b1,α1 X k,αk · · · X 1,α1 g−1 (X , X ∗ )X 1,α · · · X k,α = I. 1 k pur e

Therefore, X ∈ Dg−1 (H). The last part of the theorem can be proved in a similar manner. The proof is complete.   Remark 2.10 The condition (ii) in Theorem 2.9 can be replaced by the weaker condition:

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(ii) the series



α∈F+ ni |α|≥1

∗ is weakly convergent and |ai,α |X 1,α X 1,α

*

ˆ m−1 lim sup 

gi ,X i

m→∞

+1 (I )h, h

m

< 1,

h ∈ H,

where ˆ −1 (Y ) :=  g ,X i i

∞ 



∗ |ai,α |X i,α Y X i,α ,

Y ∈ B(H).

p=1 α∈F+ n i ,|α|= p

Indeed, a close look at the proof of Theorem 2.9 reveals that under the condition (ii) , the defect operator g−1 (X , X ∗ ) is well-defined, the series ⎞

⎛ ∞ ⎜  ⎜ ⎜ ⎝

m=1



γ1 ,...,γm ∈F+ ni |γ1 |≥1,...,|γm |≥1

ˆ m−1 is convergent, and 

gi ,X i

.& '.⎟ . .⎟ ∗ h, h . |ai,γ1 | · · · |ai,γm |X i,γ1 ···γm Y X i,γ .⎟ 1 ···γm ⎠

(I ) → 0 as m → ∞. Now, the proof of Theorem 2.9, when

condition (ii) is replaced by (ii) , can easily be adjusted.  Corollary 2.11 For each i ∈ {1, . . . , k}, let gi = 1 + α∈F+n ,|α|≥1 bi,α Z i,α be a free i holomorphic function in a neighborhood of the origin with bi,α > 0 and with inverse  gi−1 = 1 + α∈F+n ,|α|≥1 ai,α Z i,α . If X ∈ B(H)n 1 ×c · · · ×c B(H)n k is such that i

∞ 



∗ |ai,α |X i,α X i,α ≤ tI,

i ∈ {1, . . . , k},

p=1 α∈F+ n ,|α|= p i

where 0 < t ≤ 21/k − 1, then X is a radially pure element in Dg−1 (H). Moreover, there is  ∈ (0, 1), such that any k-tuple in [B(H)n 1 ] ×c · · ·×c [B(H)n k ] is a radially pure element in the polydomain Dg−1 (H). Proof Let 0 <   ≤ 21/k − 1 and let X = (X 1 , . . . , X k ) ∈ B(H)n 1 ×c · · · ×c [B(H)n k , with X i = (X i,1 , . . . , X i,n i ), be such that, for each i ∈ {1, . . . , k}, ∞ 



∗ |ai,α |X i,α X i,α ≤  I .

(2.5)

p=1 α∈F+ n ,|α|= p i

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Note that g−1 (X , X ∗ ) := I + i

∞ 



∗ ai,α X i,α X i,α ≥I−

p=1 α∈F+ n ,|α|= p

∞ 



∗ |ai,α |X i,α X i,α ≥ 0.

p=1 α∈F+ n ,|α|= p

i

i

  ∗ for Let us define the linear maps g−1 ,X i (Y ) := ∞ a X Y X i,α p=1 α∈F+ n i ,|α|= p i,α i,α i ∞  ˆ −1 (Y ) := + all Y ∈ B(H) and  |ai,α |X i,α Y X ∗ . Note that, p=1 relation (2.5) implies

α∈Fni ,|α|= p

gi ,X i

 + α1 ∈F+ n 1 ,...αk ∈Fn k

i,α

∗ ∗ |a1,α1 | · · · |ak,αk |X 1,α1 · · · X k,αk X k,α · · · X 1,α 1 k

ˆ −1 ) ◦ · · · ◦ (id +  ˆ −1 )(I ) = (id +  g ,X 1 g ,X k 1

k

 k

≤ (1 +  ) I . Consequently,           ∗ ∗  sup  a1,β1 · · · ak,βk X 1,β1 · · · X k,βk X k,βk · · · X 1,β1  < ∞  m 1 ,...,m k ∈N  βi ∈F+ni ,|βi |≤m i    i∈{1,...,k}

and g−1 (X , X ∗ ) := WOT- lim · · · lim m 1 →∞

m k →∞

· · · ak,βk X 1,β1 · · ·



a1,β1

βi ∈F+ n i ,|βi |≤m i

i∈{1,...,k} ∗ ∗ X k,βk X k,βk · · · X 1,β 1

exists. Moreover, we have g−1 (X , X ∗ ) ≥ I − ((1 +   )k I − I ) ≥ 0

for 0 <   ≤ 21/k − 1.

Applying Theorem 2.9, we deduce that any k-tuple X = (X 1 , . . . , X k ) satisfying relapur e tion (2.5) is in Dg−1 (H). Moreover, X is radially pure with respect to the polydomain Dg−1 (H). Now, as in the proof of Proposition 2.5, one can show that there is  ∈ (0, 21/k − 1) such that any k-tuple in [B(H)n 1 ] ×c · · · ×c [B(H)n k ] satisfies relation (2.5). Therefore, [B(H)n 1 ] ×c · · · ×c [B(H)n k ] ⊂ Dg−1 (H) and any k-tuple in [B(H)n 1 ] ×c · · · ×c [B(H)n k ] is radially pure with respect to   Dg−1 (H). The proof is complete.

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 Theorem 2.12 Let g := (g1 , . . . , gk ) with gi = 1 + α∈F+n ,|α|≥1 bi,α Z i,α be a k-tuple i of free holomorphic functions on a neighborhood of the origin such that bi,α > 0 and

sup

α∈F+ ni

bi,α < ∞, bi,gi α

j ∈ {1, . . . , n i }.

j

Then, there is  > 0, such that pur e

[B(H)n 1 ] ×c · · · ×c [B(H)n k ] ⊂ Dg−1 (H) ⊂ Mg (H). Proof Using Theorems 2.1 and 2.9, and Corollary 2.11, the result follows.

 

To insure that the noncommutative set Mg (H) is nontrivial and contains a neighborhood of the origin in B(H)n 1 ×c · · · ×c B(H)n k , we assume throughout this paper, unless otherwise specified, that the g satisfies the hypothesis of Theorem 2.12. Theorem 2.13 Let g := (g1 , . . . , gk ) be a k-tuple of free holomorphic functions in a neighborgood of the origin with 

gi = 1 +

bi,α Z i,α ,

α∈F+ n i ,|αi |≥1

where bi,α > 0 and

sup

α∈F+ ni

bi,α < ∞, bi,gi α

j ∈ {1, . . . , n i }.

j

Then the following statements are equivalent. pur e

(i) Dg−1 (H) = Mg (H).

pur e  (ii) If W is the universal model associated with g, then W ∈ Dg−1 ( ks=1 F 2 (Hn s )).  ai,α Z i,α is the inverse of gi , then the limit (iii) If gi−1 = 1 + α∈F+ n i ,|α|≥1

lim · · · lim

m 1 →∞

m k →∞

 βi ∈F+ n i ,|βi |≤m i i∈{1,...,k}

∗ ∗ a1,β1 · · · ak,βk W1,β1 · · · Wk,βk Wk,β · · · W1,β 1 k

exists in the weak (or strong) operator topology. (iv) For each i ∈ {1, . . . , k},         ∗  sup  ai,β Wi,β Wi,β  < ∞.  m∈N  β∈F+ni |β|≤m 

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(v) For each i ∈ {1, . . . , k},

sup

N ∈N,α∈F+ ni

. . . . . . .  ai,β bi,γ . . . < ∞. . . .0≤|β|≤N bi,α . . βγ =α .

 Proof Since W ∈ Mg ( ks=1 F 2 (Hn s )), the implication (i) ⇒ (ii) is clear. Due to pur e  the definition of the polydomain Dg−1 ( ks=1 F 2 (Hn s )) g−1 (W, W∗ ) := g−1 ,W1 ◦ · · · ◦ g−1 ,Wk (I ) 1

k

is a well-defined positive operator. Consequently, the implication (ii) ⇒ (iii) is true. Now, (iii) ⇒ (iv) is due to the uniform boundedness principle. The equivalence of (iv) with (v) was proved in [91]. We prove that (iv) ⇒ (ii). First, we remark that           ∗ ∗  a1,β1 · · · ak,βk W1,β1 · · · Wk,βk Wk,βk · · · W1,β1      βi ∈F+ni ,|βi |≤m i   i∈{1,...,k}     k      ∗  = ai,β Wi,β Wi,β     i=1 β∈F+  n i |β|≤m i and           ∗ ∗  sup  a1,β1 · · · ak,βk W1,β1 · · · Wk,βk Wk,βk · · · W1,β1  < ∞.  m 1 ,...,m k ∈N  βi ∈F+ni ,|βi |≤m i    i∈{1,...,k}

(2.6) Due the relation (2.2), we have  Wi,β eγi = 

bi,γ

bi,βγ

i eβγ

and

∗ i Wi,β eα =

⎧√ ⎨ √bi,γ ei

if α = βγ

⎩

otherwise

bi,α γ

0

for any α, β ∈ F+ n i . Consequently, ∗ i Wi,β Wi,β eα

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/b =

i,γ i bi,α eα

if α = βγ

0

otherwise.

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(2.7)

G. Popescu

For each m ∈ N and α ∈ F+ n i , we set

(m ) di,αi

:=

⎧ ⎪ ⎨1 + ⎪ ⎩

 βγ =α 1≤|β|≤m

ai,β bi,γ bi,α

,

if |α| ≥ 1 if α = g0i .

1,

Note that lim · · · lim

m 1 →∞



m k →∞

βi ∈F+ n i ,|βi |≤m i

∗ ∗ a1,β1 · · · ak,βk W1,β1 · · · Wk,βk Wk,β · · · W1,β e1 ⊗ · · · ⊗ eαk k 1 α1 k

i∈{1,...,k}

=

(m ) (m ) d1,α11 · · · dk,αkk eα1 1

⊗ · · · ⊗ eαk k .

On the other hand, relation gi−1 gi = 1 implies 

bi,γ +

if γ ∈ F+ n i , |γ | ≥ 1.

ai,β bi,α = 0

βα=γ ,|β|≥1 (m )

(m )

(m )

Consequently, if m i ≥ |αi | ≥ 1, we have di,αii = 0. This implies that d1,α11 · · · dk,αkk =

(m 1 ) (m k ) 0 if there is i ∈ {1, . . . , k} such that m i ≥ |αi | ≥ 1, and d1,α · · · dk,α = 1 if αi = g0i 1 k for any i ∈ {1, . . . , k}. Consequently, we deduce that



lim · · · lim

m 1 →∞

m k →∞

a1,β1 · · · ak,βk W1,β1

βi ∈F+ n i ,|βi |≤m i i∈{1,...,k}

∗ · · · Wk,βk Wk,β k

∗ · · · W1,β e1 ⊗ · · · ⊗ eαk k = 0 1 α1

if |α1 | + · · · + |αk | ≥ 1, and it equal to 1 if αi = g0i for any i ∈ {1, . . . , k}. Therefore, due to relation (2.6), we deduce that the limit defining g−1 (W, W∗ ) exists in the strong operator topology and g−1 (W, W∗ ) = SOT- lim · · · lim m 1 →∞

m k →∞



a1,β1

βi ∈F+ n i ,|βi |≤m i i∈{1,...,k} ∗ ∗ · · · ak,βk W1,β1 · · · Wk,βk Wk,β · · · W1,β 1 k

= PC1 ,  where PC1 is the orthogonal projection of ks=1 F 2 (Hn s ) onto the constants C1. Note that any change of the order in the iterated limit leads to the same result. On the other

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hand, using Proposition 2.4, we deduce that 

lim · · · lim

m 1 →∞

m k →∞

b1,α1 · · · bk,αk W1,α1

αi ∈F+ n i ,|αi |≤m i i∈{1,...,k} ∗ · · · Wk,αk g−1 (W, W∗ )Wk,α k

∗ · · · W1,α = I, 1

where the convergence is in the strong operator topology. Therefore, W ∈ pur e Dg−1 (⊗ks=1 F 2 (Hn s )). It remains to prove that (ii) ⇒ (i). To this end, assume that item (ii) holds. pur e According to Theorem 2.12, we have Dg−1 (H) ⊂ Mg (H). To prove the reverse inclusion, let X = (X 1 , . . . , X k ) ∈ Mg (H), with with X i = (X i,1 , . . . , X i,n i ) ∈ B(H)n i . Then there is a Hilbert spaces D such that ∗ ∗ X i, j = (Wi, j ⊗ ID )|H ,

i ∈ {1, . . . , k}, j ∈ {1, . . . , n i },

where H is identified with a coinvariant subspace for the operators Wi, j ⊗ ID . Then we have  βi ∈F+ n i ,|βi |≤m i i∈{1,...,k}

∗ ∗ a1,β1 · · · ak,βk X 1,β1 · · · X k,βk X k,β · · · X 1,β 1 k

⎛ ⎜ ⎜ = PH ⎜ ⎝

⎞  βi ∈F+ n i ,|βi |≤m i i∈{1,...,k}

⎟ ∗ ∗ ⎟ a1,β1 · · · ak,βk W1,β1 · · · Wk,βk Wk,β · · · W ⎟ |H 1,β 1 k ⎠

which, due to relation (2.6), implies           ∗ ∗  a1,β1 · · · ak,βk X 1,β1 · · · X k,βk X k,β · · · X sup   < ∞. 1,β 1 k m 1 ,...,m k ∈N  βi ∈F+ni ,|βi |≤m i    i∈{1,...,k}

Since item (iii) holds, g−1 (W, W∗ ) = SOT- lim · · · lim m 1 →∞

m k →∞



a1,β1

βi ∈F+ n i ,|βi |≤m i i∈{1,...,k} ∗ ∗ · · · ak,βk W1,β1 · · · Wk,βk Wk,β · · · W1,β 1 k

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converges in the strong operator topology to the orthogonal projection PC and the iterated limit does not depend on the order of the limits. Consequently, 

g−1 (X , X ∗ ) = SOT- lim · · · lim m 1 →∞

a1,β1

m k →∞

· · · ak,βk X 1,β1 · · ·

βi ∈F+ n i ,|βi |≤m i i∈{1,...,k} ∗ ∗ X k,βk X k,β · · · X 1,β 1 k

exists and is a positive operator which does not depend on the order in the iterated limit, and  αk ∈F+ nk



···

∗ ∗ bk,αk · · · b1,α1 X k,αk · · · X 1,α1 g−1 (X , X ∗ )X 1,α · · · X k,α 1 k

α1 ∈F+ n1

⎛

⎜ = PH ⎝



b1,α1 · · · bk,αk W1,α1

+ α1 ∈F+ n 1 ,...αk ∈Fn k

0 ∗ ∗ |H = I · · · Wk,αk g−1 (W, W∗ )Wk,α · · · W1,α 1 k where the convergence is in the strong operator topology. On the other hand, we also have X i, j X s,t = PH Wi, j Ws,t |H = PH Ws,t Wi, j |H = X s,t X i, j for any i, s ∈ {1, . . . , k} with i = s and any j ∈ {1, . . . , n i }, t ∈ {1, . . . , n s }. Therefore, X is a pure element in the noncommutative set Dg−1 (H). This completes the proof.   Remark 2.14 As in the proof of Theorem 2.13, one can show that, for any subset {i 1 , . . . , i p } ⊂ {1, . . . , k} with i 1 < i 2 < · · · < i p ,            sup ai1 ,β1 · · · ai p ,β p Wi1 ,β1 · · · Wi p ,β p Wi∗p ,β p · · · Wi∗1 ,β1  < ∞,    m 1 ,...,m p ∈N  +  βi ∈Fni ,|βi |≤m i  i∈{1,..., p}

and

g−1 ,...,g−1 (W, W∗ ) i1

ip

:= SOT- lim · · · lim m 1 →∞

m p →∞



ai1 ,β1

βi ∈F+ n i ,|βi |≤m i i∈{1,..., p} · · · ai p ,β p Wi1 ,β1 · · · Wi p ,β p Wi∗p ,β p · · · Wi∗1 ,β1

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exists, does not depend on the order of the iterated limit, and is equal to an orthogonal projection. Moreover, we have 

lim · · · lim

m 1 →∞

m p →∞

bi1 ,α1 · · · bi p ,α p Wi1 ,α1

βi ∈F+ n i ,|βi |≤m i i∈{1,..., p}

· · · Wi p ,αk g−1 ,...g−1 (W, W∗ )Wi∗p ,α p · · · Wi∗1 ,α1 = I , i1

ip

where the convergence is in the strong operator topology. Theorem 2.13 leads to the following Definition 2.15 A k-tuple g := (g1 , . . . , gk ) of free holomorphic functions on a neighborhood of the origin such that gi = 1+ α∈Fn ,|αi |≥1 bi,α Z i,α satisfies the conditions i

bi,α > 0

and

sup

α∈F+ ni

bi,α < ∞, bi,gi α

j ∈ {1, . . . , n i },

j

is called admissible for operator model theory if pur e

Dg−1 (H) = Mg (H). In this case, Dg−1 is called admissible polydomain. Now, we present several examples of admissible k-tuples of formal power series g = (g1 , . . . , gk ), the corresponding weighted Hilbert spaces F 2 (g), and the associated noncommutative polydomains Dg−1 . Example 2.16 Let g := (g1 , . . . , gk ) be a k-tuple of  free holomorphic functions in the neighborhood of the origin such that each gi = 1 + α∈F+n ,|α|≥1 bi,α Z i,α satisfies the i following conditions: b

< ∞ for every j ∈ {1, . . . , n j }; (i) bi,α > 0 and supα∈F+n b i,α i gi α j  (ii) If gi−1 = 1 + α∈F+n ,|α|≥1 ai,α Z i,α , there is N0 ≥ 2 such that ai,α ≥ 0 (or ai,α ≤ i

0) for every α ∈ F+ n i with |α| ≥ N0 .

Using Theorem 4.7 from [91], we deduce that g := (g1 , . . . , gk ) is an admissible tuple for operator model theory. When g := (g1 , . . . , gk ) is a k-tuple of free holomorphic functions in the neighborhood of the origin such that ai,α ≤ 0 for any α ∈ F+ n i with |α| ≥ 1 and ai,α > 0 if |α| = 1, then condition (i) is automatically satisfied. In this case, we say that Dg−1 is a regular noncommutative polydomains. These polydomains were extensively studied pur e in [84], if k = 1, and in [85, 86], if k ≥ 2. We remark that the polydomain Dg−1 is radially pure and bi,α bi,β ≤ bi,αβ for any α, β ∈ F+ ni . Reprinted from the journal

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The next proposition provides a large class of admissible k-tuples ψ s of formal power series for operator model theory. The scale of generalized noncommutative Bergman spaces {F 2 (ψ s ) : s := (s1 , . . . , sk ) ∈ [1, ∞)k } and the associated polydomains Dψ −1 have not been studied yet. The only exception is when s ∈ Nk , which was s studied in [84, 86], and [85]. Proposition 2.17 Let ϕi :=



d Z α∈F+ n i ,|α|≥1 i,α i,α

be formal power series with non-

negative coefficients and di,α > 0 if |α| = 1. For each s := (s1 , . . . , sk ) ∈ [1, ∞)k , define the k-tuple of formal power series ψ s := ((1 − ϕ1 )−s1 , . . . , (1 − ϕk )−sk ) 

and assume that (1 − ϕi )−si = 1 + bi,α bi,β ≤

  si + |β| − 1 bi,αβ |β|

b Z . α∈F+ n i ,|α|≥1 i,α i,α 

and bi,α bi,β ≤

Then

 si + |α| − 1 bi,αβ , |α|

α, β ∈ F+ ni ,

and, for any β ∈ F+ n i with |β| = 1, 1 bi,α ≤ bi,βα di,β

and

bi,α 1 ≤ . bi,αβ di,β

If, in addition, ϕi are noncommutative polynomials, then ψ s is an admissible k-tuple for operator model theory. Proof Since, for each i ∈ {1, . . . , k},  ∞   si + k − 1 k ϕ k k=1 ⎛  |α|   ⎜ si + j − 1 ⎜ =1+ ⎝ j

(1 − ϕi )−si = 1 +

α∈F+ n i ,|α|≥1

j=1

⎞  γ1 ···γ j =α |γ1 |≥1,...,|γ j |≥1

⎟ di,γ1 · · · di,γ j ⎟ ⎠ Z i,α ,

we deduce that bg i = 1 0

and

bi,α

 |α|   si + j − 1 = j

 γ1 ···γ j =α |γ1 |≥1,...,|γ j |≥1

j=1

di,γ1 · · · di,γ j

650

if α ∈ F+ n i , |α| ≥ 1.

(2.8)

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 Note that, since si ≥ 1, we have

si + j j +1

 ≥

  si + j − 1 for any j ≥ 1, which j

implies       si + p − 1 s +j+p si + |β| − 1 si + j − 1 ≤ i j p j+p |β| for any j ∈ {1, . . . , |α|} and p ∈ {1, . . . , |β|}. Consequently, using relation (2.8) we obtain bi,α bi,β

  |β|  |α|   si + p − 1 si + j − 1 = j p j=1 p=1



 γ1 ···γ j =α |γ1 |≥1,...,|γ j |≥1

di,γ1 · · · di,γ j di,σ1 · · · di,σ p

σ1 ···σ p =β |σ1 |≥1,...,|σ p |≥1

≥

 |α| |β|    si + |β| − 1   si + j + p |β| j+p j=1 p=1



 γ1 ···γ j =α |γ1 |≥1,...,|γ j |≥1

di,γ1 · · · di,γ j di,σ1 · · · di,σ p

σ1 ···σ p =β |σ1 |≥1,...,|σ p |≥1

≤

  |α|+|β|   si + |β| − 1  si +  − 1 |β|  =1

=

  si + |β| − 1 bi,αβ |β|



di,1 · · · di,

1 ··· =αβ |1 |≥1,...,| |≥1

for any α, β ∈ F+ n i with |α| ≥ 1 and |β| ≥ 1. Note that the equality above holds also if |α| = 0 or |β| = 0. Similarly, on can prove that bi,α bi,β

  si + |α| − 1 ≤ bi,αβ , |α|

α, β ∈ F+ ni .

Since bi,gi = si dgi for every j ∈ {1, . . . , n i }, we can use the inequalities above to j

deduce that

j

bi,α bgi α j

≤

1 dgi

and

j

bi,α bi,αgi

j

≤

1 dgi

for any α ∈ F+ ni .

To prove the last part of the proposition, assume that each ϕi is a noncommutative polynomial. We have (1 − ϕi )si = 1 +

  ∞  s (−1)k i ϕ k = 1 + k k=1

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 α∈F+ n i ,|α|≥1

ai,α Z i,α

G. Popescu

for some coefficient ai,α ∈ R and there is Ns ≥ 2 such that ai,α ≥ 0 (or ai,α ≤ 0) for every α ∈ F+ n i with |α| > Ns . Example 2.16 shows that ψ s is an admissible k-tuple of free holomorphic functions. The proof is complete.   We should mention that it remains an open question whether or not ψ s remains an admissible k-tuple for operator model theory even when ϕi are not polynomials. The answer is positive in the particular case when s ∈ Nk (see [84] and [86]). Another remarkable class of o admissible k-tuples of formal power series is provided by the following Example 2.18 Let {F 2 (γ s ) : s := (s1 , . . . , sk ) ∈ (0, ∞)k } be the scale of weighted noncommutative Bergman spaces, where the admissible k-tuples of formal power series are ⎛⎛ γ s := ⎝⎝1 −

n1 

⎛

⎞−s1 Z 1, j ⎠

, . . . ⎝1 −

j=1

nk 

⎞−sk ⎞ Z k, j ⎠

⎠,

j=1

and let Dγ −1 be the associated polydomains. According to Theorem 4.3 from [91], if s si ∈ (0, 1], then Dψi −1 is a regular domain. Using Proposition 2.16, we conclude that γ s is an admissible k-tuple for any s ∈ (0, ∞)k . Note that, if s1 = · · · = sk = 1, we recover the noncommutative polyball studied in [84] and [86]. If k = 1, n 1 = n, and s1 = n (resp. s1 = n + 1) the corresponding subspace F 2 (γ s ) can be identified with the the noncommutative Hardy (resp. Bergman) space over the unit ball [B(H)n ]1 . If k = 1, n 1 = n, and s ∈ (0, 1] the Hilbert spaces F 2 (γ s ) are noncommutative generalizations of the classical Besov-Sobolev spaces on the unit ball Bn ⊂ Cn with 1 representing kernel (1−z,w ) s , z, w ∈ Bn . Example 2.19 Let {F 2 (ωs ) : s := (s1 , . . . , sk ) ∈ [0, ∞)k } be the scale of noncommutative Dirichlet spaces, where the admissible k-tuples of formal power series are given by ⎞

⎛ ⎜ ωs := ⎝ α∈F+ n1

 1 1 ⎟ . Z 1,α , . . . , Z s sk k,α ⎠ (|α| + 1) 1 (|α| + 1) + α∈Fn k

When k = 1 and s1 = 1, we get the noncommutative Dirichlet space over the unit ball is a regular polydomain. [B(H)n ]1 . Due to Theorem 4.3 from [91], Dω−1 s

3 Noncommutative Berezin Kernels, Polydomain Algebras, and Von Neumann Inequality In this section, we employ noncommutative Berezin kernels to provide canonical pur e representations for the elements of the pure polydomain Dg−1 . We introduce the 652

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polydomain algebra A(g) and describe the class of completely contractive unital reppur e resentations generated by the elements of Dg−1 (H). Several characterizations for the pur e

pur e

inclusion Dg−1 ⊂ Df −1 are also provided. Let g := (g1 , . . . , gk ) be a k-tuple  of free holomorphic functions on a neighborhood of the origin such that gi = 1 + α∈F+n ,|αi |≥1 bi,α Z i,α satisfies the conditions i

bi,α > 0

and

sup

α∈F+ ni

bi,α < ∞, bi,gi α

j ∈ {1, . . . , n i }.

j

Having in mind Theorem 2.1, we define the noncommutative Berezin kernel associated with any element X = (X 1 , . . . , X k ) with X i := (X i,1 , . . . , X i,n i ) in the noncommutative polydomain Dg−1 (H), as the operator K g,X : H →

k 1

F 2 (Hn s ) ⊗ g−1 (X , X ∗ )(H)

s=1

defined by K g,X h :=

 βi ∈F+ ni i∈{1,...,k}



 b1,β1 · · · bk,βk eβ11

∗ ∗ ⊗ · · · ⊗ eβk k ⊗ g−1 (X , X ∗ )1/2 X 1,β · · · X k,β h, 1 k

where the defect operator is defined by g−1 (X , X ∗ ) := g−1 ,X 1 ◦ · · · ◦ g−1 ,X k (I ). 1

k

Theorem 3.1 Let X = (X 1 , . . . , X k ) ∈ Dg−1 (H) with X i = (X i,1 , . . . , X i,n i ) and let W := (W1 , . . . , Wk ) with Wi := (Wi,1 , . . . , Wi,n i ) be the universal model associated with g. Then the following statements hold. (i) The noncommutative Berezin kernel K g,X is a contraction and ∗ ∗ K g,X X i, j = (Wi, j ⊗ ID )K g,X ,

i ∈ {1, . . . , n}, j ∈ {1, . . . , n i }.

pur e

(ii) X ∈ Dg−1 (H) if and only if K g,X is an isometry. pur e

(iii) If X ∈ Dg−1 (H), then there is a Hilbert spaces D such that ∗ ∗ X i, j = (Wi, j ⊗ ID )|H ,

i ∈ {1, . . . , k}, j ∈ {1, . . . , n i },

where H is identified with a co-invariant subspace for the operators Wi, j ⊗ ID . If g is an admissible k-tuple of free holomorphic functions, the converse is also true.

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Proof Using Theorem 2.1, the definition of the polydomain Dg−1 (H), and that of the noncommutative Berezin kernel K g,X , we deduce that K g,X is a contraction and ∗ ∗ K g,X X i, j = (Wi, j ⊗ ID )K g,X ,

i ∈ {1, . . . , n}, j ∈ {1, . . . , n i },

where D := g−1 (X , X ∗ )H. Item (ii) holds due to the fact that ∗ K g,X K g,X =



···

αk ∈F+ nk



bk,αk

α1 ∈F+ n1

∗ ∗ · · · b1,α1 X k,αk · · · X 1,α1 g−1 (X , X ∗ )X 1,α · · · X k,α . 1 k pur e

If X ∈ Dg−1 (H), then using items (i), (ii), and identifying H with K g,X H, we deduce that ∗ ∗ X i, j = (Wi, j ⊗ ID )|H ,

i ∈ {1, . . . , k}, j ∈ {1, . . . , n i }. pur e

If g is an admissible k-tuple, then Dg−1 (H) = Mg (H), due to Theorem 2.13. This completes the proof of item (iii).   pur e

Definition 3.2 Given X ∈ Dg−1 (H) and a Hilbert space G, we say that W ⊗ IG is a dilation (co-extension) of X if ∗ ∗ X i, j = (Wi, j ⊗ IG )|H ,

i ∈ {1, . . . , k}, j ∈ {1, . . . , n i },

where H is identified with a coinvariant subspace under each operators Wi, j ⊗ IG . The dilation of X implemented by the Berezin kernel K g,X (see Theorem 3.1) is called the canonical dilation. Using Theorem 3.1 and the remarks following Theorem 2.13, we deduce the following Corollary 3.3 Let X = (X 1 , . . . , X k ) with X i = (X i,1 , . . . , X i,n i ) ∈ B(H)n i . If g is pur e an admissible k-tuple and X ∈ Dg−1 (H), then (X i1 , . . . , X i p ) ∈ D

pur e (H), gi−1 ,...,gi−1 p 1

for any p, i 1 , . . . , i p ∈ {1, . . . , k} with i 1 < · · · < i p . Moreover, there is a natural imbedding D

pur e (H) gi−1 ,...,gi−1 p 1

pur e

 (X i1 , . . . , X i p ) → (Y1 , . . . , Yk ) ∈ Dg−1 (H,

where Ys := X i j if s = i j and Y = 0, otherwise.

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We remark that, due to Theorems 2.13 and 3.1, if g is admissible and X = pur e (X 1 , . . . , X k ) ∈ Dg−1 (H), then the definition of the defect operator g−1 (X , X ∗ ) does not depend of the order of the iterated limit. Moreover, in this case one can prove that  g−1 (X , X ∗ ) = SOT- lim a1,β1 p→∞

+ β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |≤ p,...,|βk |≤ p ∗ · · · ak,βk X 1,β1 · · · X k,βk X k,β k



= SOT- lim

p→∞

∗ · · · X 1,β 1

a1,β1

+ β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |+···+|βk |≤ p

∗ ∗ · · · ak,βk X 1,β1 · · · X k,βk X k,β · · · X 1,β . 1 k

We refer the reader to [63, 66] for basic facts concerning completely contractive (resp. positive) maps. Theorem 3.4 Let X = (X 1 , . . . , X n ) with X i = (X i,1 , . . . , X i,n i ) be in the polyball pur e Dg−1 (H), where the closure is in the operator norm topology. Then there is a unital completely contractive linear map ∗  X : S := span{W1,α1 · · · Wk,αk Wk,β k ∗ · · · W1,β : αi , βi ∈ F+ n i } → B(H), 1

such that ∗ ∗  X (W1,α1 · · · Wk,αk Wk,β · · · W1,β ) = X 1,α1 1 k ∗ ∗ · · · X k,αk X k,β · · · X 1,β 1 k

for any αi , βi ∈ F+ n i . Moreover, ∗  X (A) = lim K g,Y ( p) (A ⊗ I )K g,Y ( p) , p→∞

A ∈ S,

where the limit exists in the operator norm topology and {Y ( p) }∞ p=1 ⊂ Dg−1 (H) is a convergent sequence to X in the same topology. pur e

Proof Let X be in the polyball Dg−1 (H) and let {Y ( p) }∞ p=1 ⊂ Dg−1 (H) be a convergent sequence to X in the operator norm. According to Theorem 3.1, for any operator p(W, W∗ ) in the linear span, pur e

pur e

∗ ∗ span{W1,α1 · · · Wk,αk Wk,β · · · W1,β : αi , βi ∈ F+ n i }, 1 k

we have ∗

∗ ∗ p(Y ( p) , Y ( p) ) = K g,Y ( p) ( p(W, W ) ⊗ I )K g,Y ( p) .

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G. Popescu ∗

Hence, p(Y ( p) , Y ( p) ) ≤ p(W, W∗ ) . Taking p → ∞, we obtain p(X , X ∗ ) ≤

p(W, W∗ ) . Let A ∈ S and let { pn (W, W∗ )} be a sequence which converges in norm to A. Since

pn (X , X ∗ ) − pm (X , X ∗ ) ≤ pn (W, W∗ ) − pm (W, W∗ ) , the sequence { pn (X , X ∗ )} is convergent in the operator norm. Define  X (A) := limn→∞ pn (X , X ∗ ) and note that  X (A) ≤ A . Similarly, passing to matrices over S, one can prove that  X is a completely contractive linear map. To prove the last part of the theorem, let { pn (W, W∗ )} be a sequence which converges in norm to A and that the considerations above imply

 X (A) − pn (X , X ∗ ) ≤ A − pn (W, W∗ ) .

(3.1)

Let  > 0 and K ∈ N be such that A − p K (W, W∗ ) < 3 . Using again Theorem 3.1 and inequality (3.1), we deduce that ∗

 X (A) − K g,Y ( p) (A ⊗ I )K g,Y ( p)

∗ ∗ ≤  X (A) − K g,Y ( p) ( p K (W, W ) ⊗ I )K g,Y ( p)

∗ ∗ + K g,Y ( p) [(A − p K (W, W )] ⊗ I )K g,Y ( p)

≤  X (A) − p K (Y ( p) , Y ( p)∗ ) + A − p K (W, W∗ )

≤  X (A) − p K (X , X ∗ ) + p K (X , X ∗ ) − p K (Y ( p) , Y ( p)∗ ) + A − p K (W, W∗ )

≤ 2 A − p K (W, W∗ ) + p K (X , X ∗ ) − p K (Y ( p) , Y ( p)∗ ) .

Since Y ( p) → X in the operator norm, there is N ∈ N such that p K (X , X ∗ ) − p K (Y ( p) , Y ( p)∗ ) < 3 for any p ≥ N . Using the estimations above, we conclude that ∗  

 X (A) − K g,Y ( p) (A ⊗ I )K g,Y ( p) <  for any p ≥ N . The proof is complete. We remark the map  X in Theorem 3.4 is also completely positive due to the fact that S is an operator system. We introduce the noncommutative domain algebra A(g) as the norm-closed nonself-adjoint algebra generated by the weighted shifts Wi, j and the identity. The following result provides a von Neumann [107] type inequality for the noncommutapur e tive domain Dg−1 (H). pur e

Corollary 3.5 If X = (X 1 , . . . , X k ) ∈ Dg−1 (H) with X i = (X i,1 , . . . , X i,n i ), then           ∗ ∗  dα,β X 1,α1 · · · X k,αk X k,βk · · · X 1,β1      αi ,βi ∈F+ni    |αi |,|βi |≤m

656

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          ∗ ∗  ≤ dα,β W1,α1 · · · Wk,αk Wk,β · · · W 1,β1  k     αi ,βi ∈F+ni   |αi |,|βi |≤m

for any m ∈ N, dα,β ∈ C. Moreover, there is a unital completely contractive homomorphism  X : A(g) → B(H),  X ( p(W)) := p(X ), for any polynomial p in n 1 + · · · + n k noncommutative indeterminates Z i, j . (g)

(g)

(g)

(g)

(g)

Let W(g) := (W1 , . . . , Wk ) with Wi := (Wi,1 , . . . , Wi,n i ) be the universal model of the noncommutative polydomain Dg−1 . The C ∗ -algebra generated by the (g)

operators Wi, j and the identity will be denoted by C ∗ (W(g) ). Given two polydomains pur e pur e pur e pur e Dg−1 and Df −1 , we say that Dg−1 ⊂ Df −1 if Dg−1 (H) ⊂ Df −1 (H) for any Hilbert space H. Theorem 3.6 Let f := (f1 , . . . , fk ) and g := (g1 , . . . , gk ) be two admissible k-tuples of free holomorphic functions and let W(f) and W(g) be the universal models associated with the polydomains Df −1 and Dg−1 , respectively. Then the following statements are equivalent. pur e

pur e

(i) Dg−1 ⊂ Df −1 .

 (ii) W(g) is a pure element in Df −1 ( ks=1 F 2 (Hn s )). (iii) There is a Hilbert space G and an isometry V k 2 s=1 F (Hn s ) ⊗ G such that (g)∗

(f)∗

V Wi, j = (Wi, j ⊗ IG )V ,

:

k s=1

F 2 (Hn s ) →

i ∈ {1, . . . , k}, j ∈ {1, . . . , n i }.

(iv) The linear map  : A(f) → A(g) defined by (g)

(g)

(f) (f) (W1,α · · · Wk,α ) := W1,α1 · · · Wk,αk , 1 k

αi ∈ F+ ni ,

is completely contractive and has a co-extension to a ∗-representation π : C ∗ (W(f) ) → B(K) which is pure, i.e. (f)

(f)

(f)

(f)

(W1,α1 · · · Wk,αk )∗ = π(W1,α1 · · · Wk,αk )∗ |k

s=1

F 2 (Hn s ) ,

αi ∈ F+ ni ,

and π is unitarily equivalent to a multiple of the identity representation of C ∗ (W(f) ). Proof  Since g is admissible, Theorem 2.13 shows that W(g) is a pure tuple in Dg−1 ( ks=1 F 2 (Hn s )). Consequently, the implication (i) ⇒ (ii) is clear. To prove Reprinted from the journal

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that (ii) ⇒ (iii), suppose that W(g) is a pure tuple in Df −1 ( Theorem 3.1, we have (g)∗

(f)∗

K f,W(g) Wi, j = (Wi, j ⊗ IE )K f,W(g) ,

k s=1

F 2 (Hn s )). Due to

i ∈ {1, . . . , k}, j ∈ {1, . . . , n i },

and the noncommutative Berezin kernel K f,W(g) is an isometry. Taking V := K f,W(g) , we conclude that item (iii) holds. Now, we prove  that (iii) ⇒ (ii) ⇒ (i). Suppose that (iii) holds. Assume that fi = 1 + α∈F+n ,|α|≥1 bi,α Z i,α and fi−1 = i  1 + α∈F+n ,|α|≥1 ai,α Z i,α . Due to Theorem 2.13, i



f −1 (W(f) , W(f)∗ ) = SOT- lim · · · lim m 1 →∞

m k →∞

(f)

a1,β1

+ β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |≤m 1 ,...,|βk |≤m k

(f)

(f)∗

(f)∗

· · · ak,βk W1,β1 · · · Wk,βk Wk,βk · · · W1,β1 exists in the strong operator topology and f −1 (W(f) , W(f)∗ ) = PC1 , the orthogonal  projection of ks=1 F 2 (Hn s ) onto the constants C1. Consequently, f −1 (W(g) , W(g)∗ )



= SOT- lim · · · lim m 1 →∞

m k →∞

+ β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |≤m 1 ,...,|βk |≤m k

(g)

(g)

(g)∗

(g)∗

a1,β1 · · · ak,βk W1,β1 · · · Wk,βk Wk,βk · · · W1,β1

= V ∗ f −1 (W(f) , W(f)∗ )V ≥ 0

where the series converge in the strong operator topology. On the other hand, using again Theorem 2.13, we have 

(g)

+ α1 ∈F+ n 1 ,...αk ∈Fn k

b1,α1 · · · bk,αk W1,α1

(g)

(g)∗

(g)∗

· · · Wk,αk f −1 (W(g) , W(g)∗ )Wk,αk · · · W1,α1 ⎛  ⎜ (f) = V∗ ⎝ b1,α1 · · · bk,αk W1,α 1 + α1 ∈F+ n 1 ,...αk ∈Fn k

⎞

(f) (f)∗ (f)∗ ⎟ · · · Wk,α f −1 (W(f) , W(f)∗ )Wk,α · · · W1,α ⎠V 1 k k

= V V∗ = I, 658

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where the is in the strong operator topology. Hence, W(g) is a pure tuple convergence k 2 in Df −1 ( s=1 F (Hn s )). Consequently, we have f −1 (W(g) , W(g)∗ ) ≥ 0 and 

(g)

+ α1 ∈F+ n 1 ,...αk ∈Fn k

b1,α1 · · · bk,αk W1,α1

(g)

(g)∗

(g)∗

· · · Wk,αk f −1 (W(g) , W(g)∗ )Wk,αk · · · W1,α1 = I . pur e

According to Theorem 3.1, if X ∈ Dg−1 (H), then there is a Hilbert spaces D such that ∗ ∗ X i, j = (Wi, j ⊗ ID )|H ,

i ∈ {1, . . . , k}, j ∈ {1, . . . , n i },

where H is identified with a co-invariant subspace for the operators Wi, j ⊗ ID . Now, we take the tensor product of the relations above with ID and compress them to the pur e Hilbert space H, and conclude that X ∈ Df −1 (H). Therefore, item (i) holds. To prove the implication (iii) ⇒ (iv), suppose that (iii) holds. Since W(g) is a pure tuple in Df −1 ( ks=1 F 2 (Hn s )), Corollary 3.5 shows that the linear map  : A(f) → A(g) defined by (g)

(g)

(f) (f) (W1,α · · · Wk,α ) := W1,α1 · · · Wk,αk , 1 k

αi ∈ F+ ni ,

is a completely contractive homomorphism. Consider the ∗-representation 2 ∗

(f)

π : C (W ) → B

k 1

3 F (Hn s ) ⊗ G , π(A) := A ⊗ IG . 2

s=1

Since (g)∗

(f)∗

V Wi, j = (Wi, j ⊗ IG )V ,

i ∈ {1, . . . , k}, j ∈ {1, . . . , n i }.

  and V is an isometry, we can identify ks=1 F 2 (Hn s ) with V ( ks=1 F 2 (Hn s )). Con (f)∗ sequently, ks=1 F 2 (Hn s ) is invariant under each operator Wi, j ⊗ IG and (g)∗

(g)∗

(f) (f) ∗  · · · Wk,α ) | k W1,α1 · · · Wk,αk = π(W1,α 1 k

s=1

F 2 (Hn s ) ,

αi ∈ F+ ni .

To complete the proof, we show that (iv) ⇒ (ii). To this end, suppose that item (iv) holds. Then there is a unitary operator U : K → ks=1 F 2 (Hn s ) ⊗ G such that (f)

(f)

(f)

(f)

π(W1,α1 · · · Wk,αk ) = U ∗ (W1,α1 · · · Wk,αk ⊗ IG )U , Reprinted from the journal

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αi ∈ F+ ni .

G. Popescu

Since f is an admissible tuple, Theorem 3.1 implies f −1 (W(g) , W(g)∗ ) = Pk

s=1

F 2 (Hn s ) U

∗

(f −1 (W(f) , W(f)∗ ) ⊗ IG )U |k

s=1

F 2 (Hn s )

≥0

and  + α1 ∈F+ n 1 ,...αk ∈Fn k

(f) (f) (f)∗ (f)∗ b1,α1 · · · bk,αk W1,α · · · Wk,α f −1 (W(f) , W(f)∗ )Wk,α · · · W1,α =I 1 1 k k

where the convergence is in the strong topology. Hence, we deduce that 

b1,α1 · · · bk,αk U ∗

+ α1 ∈F+ n 1 ,...αk ∈Fn k

4  5 (f) (f) (f)∗ (f)∗ W1,α1 · · · Wk,αk f −1 (W(f) , W(f)∗ )Wk,αk · · · W1,α1 ⊗ IG U = I .

Taking the compression to (g)

k s=1

F 2 (Hn s ) and using the fact that

(g)

(f) (f) ∗ (W1,α1 · · · Wk,αk )∗ = (W1,α · · · Wk,α ) 1 k (f)

(f)

= π(W1,α1 · · · Wk,αk )∗ |k

s=1

=U

∗

(f) ((W1,α 1

(f) ∗ · · · Wk,α ) k

F 2 (Hn s )

⊗ IG )U |k

s=1

F 2 (Hn s )

for any αi ∈ F+ n i , we obtain  + α1 ∈F+ n 1 ,...αk ∈Fn k

(g)

b1,α1 · · · bk,αk W1,α1

(g)

(g)∗

(g)∗

· · · Wk,αk f −1 (W(g) , W(g)∗ )Wk,αk · · · W1,α1 = I which shows that W(g) is a pure element in Df −1 ( holds. The proof is complete.

k s=1

F 2 (Hn s )). Therefore item (ii)  

4 Noncommutative Hardy Algebras Associated with Polydomains and Multi-Analytic Operators In this section, we introduce the noncommutative Hardy algebras F ∞ (g) and R ∞ (g) and present basic results concerning multi-analytic operators with respect to universal models. We provide a w∗ -continuous F ∞ (g)-functional calculus for pure elements in the noncommutative polydomain Dg−1 (H).

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 For each i ∈ {1, . . . , k}, let gi = 1 + α∈F+n ,|α|≥1 bi,α Z i,α be a formal power i series such that the coefficients {bi,α }α∈F+n are strictly positive real numbers such that i

sup

α∈F+ ni

bi,α < ∞, bi,gi α

j ∈ {1, . . . , n i },

j

and let W = (W1 , . . . , Wk ) with Wi = (W i,1 , . . . , Wi,n i ) be the associated weighted cβ1 ,...,βk W1,β1 · · · Wk,βk be a forleft creation operators. Let ϕ(W) = mal series such that, for each γi ∈

+ β1 ∈F+ n 1 ,...,βk ∈Fn k + Fn i ,



cβ1 ,...,βk W1,β1

+ β1 ∈F+ n 1 ,...,βk ∈Fn k

· · · Wk,βk (eγ11 ⊗ · · · ⊗ eγkk ) ∈

k 1

F 2 (Hn s )

s=1

and          sup cβ1 ,...,βk W1,β1 · · · Wk,βk ( p) < ∞,   p∈P , p ≤1   βi ∈F+ni where P is the linear span of {eγ11 ⊗ · · · ⊗ eγkk }γi ∈F+n . In this case, there is a unique i bounded linear operator acting on the Fock space ks=1 F 2 (Hn s ), which we denote by ϕ(W), such that 

ϕ(W) p =

cβ1 ,...,βk W1,β1 · · · Wk,βk ( p)

for any p ∈ P.

+ β1 ∈F+ n 1 ,...,βk ∈Fn k

 The set of all operators ϕ(W) ∈ B( ks=1 F 2 (Hn s )) satisfying the above-mentioned ∞ properties is denoted by F (g). Using standard arguments, one can prove that F ∞ (g) is a Banach algebra, which we call (left) noncommutative Hardy algebra associated with k-tuple g = (g1 , . . . , gk ).  Remark 4.1 Each ϕ(W) ∈ F ∞ (g) has a unique Fourier representation βi ∈F+ n i

i∈{1,...,k}

cβ1 ,...,βk W1,β1 · · · Wk,βk . Now, we assume that, for each i ∈ {1, . . . , k}, sup

α∈F+ ni

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bi,α < ∞, bi,αgi

j ∈ {1, . . . , n i }.

j

661

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Define the weighted right creation operators i, j : F 2 (Hn i ) → F 2 (Hn i ) by setting i, j := Ri, j G i, j , where Ri,1 , . . . , Ri,n i are the right creation operators on the full Fock space F 2 (Hn i ) and each diagonal operator G i, j is defined by 6 G i, j eαi

=

bi,α i e , α ∈ F+ ni . bi,αgi α j

In this case, we have  i,β eγi = 

bi,γ

bi,γ β˜

eγi β˜

and

∗ i i,β eα =

⎧√ ⎨ √bi,γ e

if α = γ β˜

⎩

otherwise

bi,α

γ

0

(4.1)

i i + ˜ for any α, β ∈ F+ n i , where β denotes the reverse of β = g j1 · · · g j p ∈ Fn i , i.e., β˜ = g ij p · · · g ij1 . Note that i, j Wi, p = Wi, p i, j for any i ∈ {1, . . . , k} and j, p ∈  {1, . . . , n i }. We introduce the operator i, j acting on ks=1 F 2 (Hn s ) and given by

i, j := I ⊗ ·· · ⊗ I! ⊗i, j ⊗ I ⊗ ·· · ⊗ I! . i−1 times

k−i times

Set i := (i,1 , . . . , i,n i ) for each i ∈ {1, . . . , k}, and  := (1 , . . . , k ). When necessary, we also denote by i the row operator [i,1 · · · i,n i ] acting on the direct   sum ( ks=1 F 2 (Hn s )(n i ) of n i copies of ks=1 F 2 (Hn s ) . We remark that  is jointly unitarily equivalent to the k-tuple R = (R1 , . . . , Rk ) with Ri := (Ri,1 , . . . , Ri,n i ) of right multiplication operators on the Hilbert space F 2 (g) defined by Ri, j ζ = ζ Z i, j .  Indeed, the operator U : ks=1 F 2 (Hs ) → F 2 (g) defined by U (eα1 1 ⊗ · · · ⊗ eαk k ) :=

 b1,α1 · · · bk,αk Z 1,α1 · · · Z k,αk ,

αi ∈ F+ ni ,

is unitary and U  i, j = Ri, j U for any i ∈ {1, . . . , k} and j ∈ {1, . . . , n i }. cβ˜1 ,...,β˜k 1,β1 · · · k,βk be a formal series such that, for Let ϕ() = βi ∈F+ n i

each γi ∈

i∈{1,...,k}

F+ ni ,  βi ∈F+ ni

cβ˜1 ,...,β˜k 1,β1 · · · k,βk (eγ11 ⊗ · · · ⊗ eγkk ) ∈

k 1

F 2 (Hn s )

s=1

i∈{1,...,k}

and            sup cβ˜1 ,...,β˜k 1,β1 · · · k,βk ( p) < ∞,    p∈P , p ≤1  +   βi ∈Fni  i∈{1,...,k}

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where P is the linear span of {eγ11 ⊗ · · · ⊗ eγkk }γi ∈F+n . In this case, there is a unique i bounded linear operator acting on the Fock space ks=1 F 2 (Hn s ), which we denote by ϕ(), such that 

ϕ() p =

βi ∈F+ ni i∈{1,...,k}

cβ˜1 ,...,β˜k 1,β1 · · · k,βk ( p)

for any p ∈ P.

 The set of all operators ϕ() ∈ B( ks=1 F 2 (Hn s )) satisfying the above-mentioned properties is denoted by R ∞ (g). One can prove that R ∞ (g) is a Banach algebra, which we call (right) noncommutative Hardy algebra associated with k-tuple g = (g1 , . . . , gk ). Proposition 4.2 Let g := (g1 , . . . , gk ) be a k-tuple of formal power series gi =  1 + α∈Fn ,|α|≥1 bi,α Z i,α such that the coefficients {bi,α }α∈F+n are strictly positive i i real numbers and such that sup

α∈F+ ni

bi,α < ∞, bi,gi α

bi,α < ∞, bi,αgi

sup

α∈F+ ni

j

j ∈ {1, . . . , n i },

j

and let  := (1 , . . . , k ) with i := (i,1 , . . . , i,n i ) be the associated weighted right creation operators. Then   (i) F ∞ (g) = s,t s∈{1,...,k} = R ∞ (g) , where  stands for the commutant, t∈{1,...,n s }

(ii) F ∞ (g) = F ∞ (g) and R ∞ (g) = R ∞ (g). In particular, the noncommutative Hardy algebra F ∞ (g) is WOT-(resp. SOT-, w*-) closed. Proof The relation F ∞ (g) ⊂ {s,t }s,t holds due to the the definition of F ∞ (g) and the fact that Wi, j s,t = s,t Wi, j for any i, s ∈ {1, . . . , k}, j ∈ {1, . . . , n i }, and  t ∈ {1, . . . , n s }. Let A ∈ {s,t }s,t ⊂ B( ks=1 F 2 (Hn s )). Then, using relation (2.7), we have 

A(1) =

cβ1 ,...,βk 

βi ∈F+ ni i∈{1,...,k}

for some coefficients {cβ1 ,...,βk }

1 1 ···  eβ1 ⊗ · · · ⊗ eβk k b1,β1 bk,βk 1

βi ∈F+ ni i∈{1,...,k}

⊂ C with

< ∞. Since As,t = s,t A for any s ∈ (2.7) and (4.1) imply

Reprinted from the journal



1 1 |cβ1 ,...,βk |2 b1,β · · · bk,β βi ∈F+ ni 1 k i∈{1,...,k} {1, . . . , k} and t ∈ {1, . . . , n s }, relations

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 b1,α1 · · · bk,αk A1,α˜ 1 · · · k,α˜ k (1)   = b1,α1 · · · bk,αk 1,α˜ 1 · · · k,α˜ k A(1)    bk,αk b1,α1 cβ1 ,...βk  ···  1,α˜ 1 · · · k,α˜ k (eβ11 ⊗ · · · ⊗ eβk k ) = b1,β1 bk,βk +

A(eα1 1 ⊗ · · · ⊗ eαk k ) =

βi ∈Fn i i∈{1,...,k}



=

βi ∈F+ ni i∈{1,...,k}



=

βi ∈F+ ni i∈{1,...,k}

  bk,αk 1 b1,α1 cβ1 ,...βk  ···  eβ α ⊗ · · · ⊗ eβk k αk b1,β1 α1 bk,βk αk 1 1 cβ1 ,...βk W1,β1 · · · Wk,βk (eα1 1 ⊗ · · · ⊗ eαk k ).

+ Therefore, for each (α1 , . . . , αk ) ∈ F+ n 1 ×· · ·×Fn k ,

< ∞. and A(q) =



cβ1 ,...βk W1,β1 βi ∈F+ ni i∈{1,...,k}



1 |cβ1 ,...,βk |2 b1,β βi ∈F+ ni 1 i∈{1,...,k}

1 · · · bk,β

k

· · · Wk,βk (q) for any q in the linear span

. Since A is a bounded operator, we deduce that A ∈ F ∞ (g) γi ∈F+ ni i∈{1,...,k} F ∞ (g) = {s,t }s,t . Similarly, one can show that R ∞ (g) = {Ws,t }s,t .

of {eγ11 ⊗· · ·⊗eγkk }

and, therefore, The rest of the proof is straightforward.    Let : Tk → B( ks=1 F 2 (Hn s )) be the strongly continuous unitary representation of the k-dimensional torus, defined by 

(eiθ1 , . . . , eiθk ) f :=

αs ∈F+ ns s∈{1,...,k}

for any f =



aα1 ,...,αk eα1 1 αs ∈F+ ns s∈{1,...,k}

eiθ1 |α1 | · · · eiθk |αk | aα1 ,...,αk eα1 1 ⊗ · · · eαk k .

⊗ · · · eαk k ∈

k s=1

F 2 (Hn s ). We have the orthog-

onal decomposition k 1

F 2 (Hn s ) =



E p1 ,..., pk ,

( p1 ,..., pk )∈Zk

s=1

where the spectral subspace E p1 ,..., pk is the image of the orthogonal projection  P p1 ,..., pk ∈ B( ks=1 F 2 (Hn s )) defined by  P p1 ,..., pk :=

1 2π

k 7

2π 0

7

2π

···

e−i p1 θ1 · · · e−i pk θk (eiθ1 , . . . , eiθk )dθ1 . . . dθk ,

0

where the integral is defined as a weak integral and the integrant is a continuous function in the strong operator topology. We remark that if ps < 0 for some s ∈

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{1, . . . , k}, then P p1 ,..., pk = 0 and, therefore, E p1 ,..., pk = {0}. Note that the spectral subspaces of are / E p1 ,..., pk :=

f ∈

k 1

8 F (Hn s ) : (e 2

iθ1

,...,e

iθk

)f =e

iθ1 p1

···e

iθk pk

f

s=1

for ( p1 , . . . , pk ) ∈ Zk . From now on, we use the notation (eiθ ) := (eiθ1 , . . . , eiθk ).  Definition 4.3 If T ∈ B( ks=1 F 2 (Hn s ) ⊗ K) and (s1 , . . . , sk ) ∈ Zk we define the (s1 , . . . , sk )-multi-homogeneous part of T to be the operator Ts1 ,...,sk ∈  B( ks=1 F 2 (Hn s ) ⊗ K) defined by  Ts1 ,...,sk :=

1 2π

k 7

2π

7

2π

e−is1 θ1    ∗

(eiθ ) ⊗ IK T (eiθ ) ⊗ IK dθ1 . . . dθk . ···

0

· · · e−isk θk

0

It is easy to see that Ts1 ,...,sk ≤ T , (T ∗ )s1 ,...,sk = (T−s1 ,...,−sk )∗ , and

(eiθ )∗ |E p1 ,..., pk = e−i p1 θ1 · · · e−i pk θk IE p1 ,..., pk for any ( p1 , . . . , pk ) ∈ Zk . Fix (s1 , . . . , sk ) ∈ Zk and note that, for any f ∈ E p1 ,..., pk ⊗ K,  Ts1 ,...,sk f =

1 2π

k 7

2π 0

7

2π

··· 0

  e−i(s1 + p1 )θ1 · · · e−i(sk + pk )θk (eiθ ) ⊗ IK T f dθ1 . . . dθk

= (Ps1 + p1 ,...,sk + pk ⊗ IK )T f .

Consequently, 9 0 Ts1 ,...,sk E p1 ,..., pk ⊗ K ⊂ Es1 + p1 ,...,sk + pk ⊗ K for any ( p1 , . . . , pk ) ∈ Zk and (s1 , . . . , sk ) ∈ Zk . An operator A ∈ B( ⊗ K) is called multi-homogeneous of degree (s1 , . . . , sk ) ∈ Zk if

k s=1

F 2 (Hn s )

9 0 A E p1 ,..., pk ⊗ K ⊂ Es1 + p1 ,...,sk + pk ⊗ K for any ( p1 , . . . , pk ) ∈ Zk . We recall from [54] that if X is a Banach space, ϕ is a continuous X -valued function on T, and κn is a summability kernel, then 1 n→∞ 2π

7

ϕ(0) = lim

Reprinted from the journal

2π

κn (eiθ )ϕ(eiθ )dθ.

0

665

G. Popescu

 Using this result, we proved in [90] that if T ∈ B( ks=1 F 2 (Hn s ) ⊗ K) and ∈ {Ts ,...,s } k are the multi-homogeneous parts of T , then, for any f k1 k 2(s1 ,...,sk )∈Z s=1 F (Hn s ) ⊗ K, we have 

T f = lim . . . lim N1 →∞

Nk →∞

 ··· 1 −

 |sk | Ts1 ,...,sk f Nk + 1

(s1 ,...,sk )∈Zk ,|s j |≤N j



|s1 | 1− N1 + 1



and 

T g = lim . . . lim N1 →∞

Nk →∞

Ts1 ,...,sk g

(s1 ,...,sk )∈Zk ,|s j |≤N j

for any g ∈ E p1 ,..., pk ⊗ K any ( p1 , . . . , pk ) ∈ Zk .  Proposition 4.4 If T ∈ B( ks=1 F 2 (Hn s )⊗K) and {Ts1 ,...,sk }(s1 ,...,sk )∈Zk are the multihomogeneous parts of T , then      (s

 1−



k 1 ,...,sk )∈Z ,|s j |≤N j

      |s1 | |sk | ··· 1 − Ts1 ,...,sk   ≤ T

N1 + 1 Nk + 1 

for any N1 , . . . , Nk ) ∈ Nk . Proof For each j ∈ {1, . . . , k}, we consider the Fejér kernel K N j (eiθ j ) :=    |s j | is j θ j . Note that 1 − |s j |≤N j N j +1 e 

k 7

7

  ∗  K N1 (eiθ1 ) · · · K Nk (eiθk ) (eiθ ) ⊗ IK T (eiθ ) ⊗ IK f dθ1 . . . dθk 0 0 ⎛ ⎞  k    |s j | ⎝ ⎠ 1− = Nj + 1

1 2π

2π

···

j=1 |s j |≤N j

 × =

1 2π

k 7 

0

2π

2π

7 2π    ∗ ··· e−is1 θ1 · · · e−isk θk (eiθ ) ⊗ IK T (eiθ ) ⊗ IK f dθ1 . . . dθk 0     |sk | |s1 | ··· 1 − Ts1 ,...,sk f 1− N1 + 1 Nk + 1

(s1 ,...,sk )∈Zk ,|s j |≤N j

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for any f ∈

k s=1

     (s

F 2 (Hn s ) ⊗ K. Hence, we deduce that  1−



k 1 ,...,sk )∈Z ,|s j |≤N j



1 2π

k 7

2π

7







|sk | |s1 | ··· 1 − Ts1 ,...,sk N1 + 1 Nk + 1

   f  

2π

≤ ··· K N1 (eiθ1 ) · · · K Nk (eiθk ) 0 0    ∗     (eiθ ) ⊗ IK T (eiθ ) ⊗ IK f  dθ1 . . . dθk ≤ T

f

for any N j ∈ N and f ∈

k s=1

F 2 (Hn s ) ⊗ K. This completes the proof.

 

We remark that, as in the proof of Proposition 4.2, one can show that any operator in the commutant of the set { i, j ⊗ IK : i ∈ {1, . . . , k}, j ∈ {1, . . . , n i }} W1,β1 · · · Wk,βk ⊗ Cβ1 ,...,βk for some has a unique Fourier representation βi ∈F+ n i

i∈{1,...,k}

operators {Cβ1 ,...,βk } in B(K).

Theorem 4.5 Let g := (g1 , . . . , gk ) be a k-tuple of formal power series gi = 1 +  + are strictly positive real α∈Fni ,|α|≥1 bi,α Z i,α such that the coefficients {bi,α }α∈F+ ni numbers and such that sup

α∈F+ ni

bi,α < ∞, bi,gi α j

sup

α∈F+ ni

bi,α < ∞, bi,αgi

j ∈ {1, . . . , n i }.

j

Let W := (W1 , . . . , Wk ) with Wi := (Wi,1 , . . . , Wi,n i ) (resp.  := (1 , . . . , k ) with i := (i,1 , . . . , i,n i )) be the associated weighted left (resp. right) creation  operators. Let A ∈ B( ks=1 F 2 (Hn s ) ⊗ K) be such that A(i, j ⊗ IK ) = (i, j ⊗ IK )A, If A has the Fourier representation

A = SOT×

lim

βi ∈F+ ni

W1,β1 · · · Wk,βk ⊗ Cβ1 ,...,βk , then 



N1 →∞...,Nk →∞





i ∈ {1, . . . , k}, j ∈ {1, . . . , n k }.

(s1 ,...,sk )∈Zk ,|s j |≤N j

W1,α1 · · · Wk,αk ⊗ Cα1 ,...,αk ,

αi ∈F+ n i ,|αi |=si

Reprinted from the journal

   |s1 | |sk | 1− ··· 1 − N1 + 1 Nk + 1

667

G. Popescu

           W1,α1 · · · Wk,αk ⊗ Cα1 ,...,αk  ≤ A

   α1 ∈F+n1 ,...,αk ∈F+nk   

for any (s1 , . . . , sk ) ∈ Zk+ ,

|α1 |=s1 ,...,|αk |=sk

and          |s1 | |sk |  1− ··· 1 −   N1 + 1 Nk + 1  (s1 ,...,sk )∈Zk ,|s j |≤N j        W1,α1 · · · Wk,αk ⊗ Cα1 ,...,αk  ≤ A

 +  α1 ∈F+ n 1 ,...,αk ∈Fn k  |α1 |=s1 ,...,|αk |=sk

for any N1 , . . . , Nk ∈ N. Proof Let us prove that the (s1 , . . . , sk )-multi-homogeneous part of A is 

As1 ,...,sk =

W1,α1 · · · Wk,αk ⊗ Cα1 ,...,αk

+ α1 ∈F+ n 1 ,...,αk ∈Fn k |α1 |=s1 ,...,|αk |=sk

if s1 , . . . , sk ≥ 0, and As1 ,...,sk = 0 if there is si < 0 for some i ∈ {1, . . . , k}. Fix x ∈ E p1 ,..., pk ⊗ K, pi ∈ Z, and note that  As1 ,...,sk x =

1 2π

k 7

2π 0

7 ··· 0

2π

  e−i(s1 + p1 )θ1 · · · e−i(sk + pk )θk (eiθ ) ⊗ IK Axdθ1 . . . dθk

= (Ps1 + p1 ,...,sk + pk ⊗ IK )Ax ⎛ ⎜ = (Ps1 + p1 ,...,sk + pk ⊗ IK ) ⎝



⎞ 9 0 ⎟ W1,α1 · · · Wk,αk ⊗ Cα1 ,...,αk x ⎠ .

+ α1 ∈F+ n 1 ,...,αk ∈Fn k

If x = eβ11 ⊗· · ·⊗eβk k ⊗h with |βi | = pi ≥ 0 for any i ∈ {1, . . . , k} and s1 , . . . , sk ≥ 0, then As1 ,...,sk (eβ11 ⊗ · · · ⊗ eβk k ⊗ h) = (Ps1 + p1 ,...,sk + pk ⊗ IK ) ⎛ ⎞ ⎜  ⎟ W1,α1 · · · Wk,αk (eβ11 ⊗ · · · ⊗ eβk k ) ⊗ Cα1 ,...,αk h ⎠ ⎝ αi ∈F+ ni

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=

+ α1 ∈F+ n 1 ,...,αk ∈Fn k |α1 |=s1 ,...,|αk |=sk

W1,α1 · · · Wk,αk (eβ11 ⊗ · · · ⊗ eβk k ) ⊗ Cα1 ,...,αk h.

On the other hand, note that if si < 0 for some i ∈ {1, . . . , k}, then si + pi < ki + pi for any ki ≥ 0 and, consequently, As1 ,...,sk (eβ11 ⊗ · · · ⊗ eβk k ⊗ h) = 0. This proves our assertion. Now, we can use Proposition 4.4 to complete the proof.   k An operator A ∈ B( s=1 F 2 (Hn s ) ⊗ K) is called multi-analytic with respect to  ⊗ IK , where  ⊗ IK := (1 ⊗ IK , . . . , k ⊗ IK ) and i ⊗ IK := (i,1 ⊗ IK , . . . , i,n i ⊗ IK ), if A(i, j ⊗ IK ) = (i, j ⊗ IK )A,

i ∈ {1, . . . , k}, j ∈ {1, . . . , n i }.

¯ min B(K), the WOT-closure of the spatial tensor product We denote by F ∞ (g)⊗ F ∞ (g)⊗min B(K). In addition, we denote by P(W) the algebra of all polynomials in Wi, j and the identity. Theorem 4.6 Let g := (g1 , . . . , gk ) be a k-tuple of formal power series gi = 1 +  + are strictly positive real α∈Fni ,|α|≥1 bi,α Z i,α such that the coefficients {bi,α }α∈F+ ni numbers and such that sup

α∈F+ ni

bi,α < ∞, bi,gi α j

sup

α∈F+ ni

bi,α < ∞, bi,αgi

j ∈ {1, . . . , n i },

j

and let W := (W1 , . . . , Wk ) with Wi := (Wi,1 , . . . , Wi,n i ) (resp.  := (1 , . . . , k ) with i := (i,1 , . . . , i,n i )) be the associated weighted left (resp. right) creation operators. Then the set of all multi-analytic operators with respect to ¯ min B(K). Moreover,  ⊗ IK coincides with F ∞ (g)⊗ ¯ min B(K) = P(W)⊗min B(K) F ∞ (g)⊗

SOT

= P(W)⊗min B(K)

w∗

¯ min B(K) is the sequential SOT-(resp. WOT-, w*-) cloand each element in F ∞ (g)⊗  sure of operator-valued polynomials of the form βi ∈F+n ,|βi |≤m W1,β1 · · · Wk,βk ⊗ i

i∈{1,...,k}

Cβ1 ,...,βk , where Cβ1 ,...,βk ∈ B(K).

Proof Let M be the set of all multi-analytic operators with respect to  ⊗ IK and note that M is WOT-(resp. SOT-, w*-) closed. Using Theorem 4.5, one can easily prove that M = P(W)⊗min B(K)

SOT

= P(W)⊗min B(K)

W OT

= P(W)⊗min B(K)

w∗

and that each element in M is the sequential SOT-(resp. WOT-, w*-) limit of operator valued polynomials of the form βi ∈F+n ,|βi |≤m W1,β1 · · · Wk,βk ⊗ Cβ1 ,...,βk , where i

i∈{1,...,k}

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G. Popescu

Cβ1 ,...,βk ∈ B(K). On the other hand, we have W OT

M = P(W)⊗min B(K) ¯ min B(K). = F ∞ (g)⊗

⊂ F ∞ (g)⊗min B(K)

W OT

Since each element in F ∞ (g) ⊗min B(K) commutes with i, j ⊗ IK for any i ∈ ¯ min B(K) ⊂ M, which com{1, . . . , k} and j ∈ {1, . . . , n i }, we deduce that F ∞ (g)⊗ pletes the proof.   As a particular case of Theorem 4.6, we deduce that the noncommutative Hardy space F ∞ (g) satisfies the relation F ∞ (g) = P(W)

SOT

= P(W)

W OT

= P(W)

w∗

.

Moreover, F ∞ (g) is the sequential SOT-(resp. WOT-, w*-) closure of P(W). We also remark that similar results hold for the Hardy algebra R ∞ (g). In what follows, we prove the existence of a w∗ -continuous F ∞ (g)-functional calculus for the pure elements in the noncommutative polydomain Dg−1 (H) and several other properties needed later on. Theorem 4.7 Let g := (g1 , . . . , gk ) be a k-tuple of free holomorphic functions such that sup

α∈F+ ni

bi,α < ∞, bi,αgi

j ∈ {1, . . . , n i },

j

and let W := (W1 , . . . , Wk ) with Wi := (Wi,1 , . . . , Wi,n i ) be the universal operator model of the noncommutative polydomain Dg−1 . Then the following statements hold. (i) If T = (T1 , . . . , Tn ) ∈ Dg−1 (H), then, for any ϕ(W) ∈ F ∞ (g) with Fourier  cβ1 ,...,βk W1,β1 · · · Wk,βk , representation βi ∈F+ n pur e

i

i∈{1,...,k}

ϕ(r T ) :=

∞ 



r q cβ1 ,...,βk T1,β1 · · · Tk,βk

+ q=0 β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |+···+|βk |=q

converges in the operator norm topology for any r ∈ [0, 1) and

ϕ(r T ) ≤ ϕ(r W) ,

r ∈ [0, 1).

(ii) If T ∈ Dg−1 (H), then the map T : F ∞ (g) → B(H) defined by pur e

∗ T (ϕ(W)) := K g,T (ϕ(W) ⊗ IH )K g,T ,

670

ϕ(W) ∈ F ∞ (g),

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Operator Theory on Noncommutative Polydomains, I

is a w ∗ -continuous linear map such that ⎛ ⎜ ⎜ T ⎜ ⎝

⎞  αi ∈F+ n i ,|αi |≤m i∈{1,...,k}



=

⎟ ⎟ cα1 ,...,αk W1,α1 · · · Wk,αk ⎟ ⎠

cα1 ,...,αk T1,α1 · · · Tk,αk

αi ∈F+ n i ,|αi |≤m i∈{1,...,k}

for any m ∈ N, cα1 ,...,αk ∈ C. Moreover, T is a unital completely contractive homomorphism which is sequentially WOT-continuous (resp. SOT-continuous). (iii) If r ∈ [0, 1) and r W is a pure element with respect to Dg−1 , then

ϕ(r W) ≤ ϕ(W)

for any ϕ ∈ F ∞ (g). (iv) If W is δ-radially pure with respect to Dg−1 , then ϕ(W) = SOT- lim ϕ(r W). r →1

pur e

(v) If T ∈ Dg−1 (H) and W is δ-radially pure with respect to Dg−1 , then T (ϕ(W)) = SOT- lim ϕ(r T ). r →1

pur e

(vi) If T ∈ Dg−1 (H) and t0 := sup{t ∈ [0, 1] : r W is pure for any r ∈ [0, t)}, then, t0 > 0 and T (ϕ(t0 W)) = SOT- lim ϕ(r T ), r t0

ϕ(W) ∈ F ∞ (g).

Moreover, sup X ϕ(X ) < ∞, where the supremum is taken over all X ∈ : pur e r ∈[0,t0 ) r Dg−1 (H).  Proof Let ϕ(W) ∈ F ∞ (g) have Fourier representation βi ∈F+n cβ1 ,...,βk W1,β1 · · · i Wk,βk . According to Theorem 4.5, we have            cβ1 ,...,βk W1,β1 · · · Wk,βk  ≤ ϕ(W) ,      β1 ∈F+n1 ,...,βk ∈F+nk   |β1 |= p1 ,...,|βk |= pk

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Consequently, if r ∈ [0, 1), then we have     ∞        p r cβ1 ,...,βk W1,β1 · · · Wk,βk      p=0 p1 ,..., pk ∈N∪{0} β1 ∈F+n1 ,...,βk ∈F+nk    p1 +···+ pk = p |β1 |= p1 ,...,|βk |= pk

≤

∞ 

rp

≤

ϕ(W)

p1 ,..., pk ∈N∪{0} p1 +···+ pk = p

p=0 ∞ 



 r

p

p=0

 p+k−1

ϕ(W) . k−1

This shows that ϕ(r W) :=

∞ 



r q cβ1 ,...,βk W1,β1 · · · Wk,βk

+ q=0 β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |+···+|βk |=q

pur e

converges in the operator norm topology for any r ∈ [0, 1). Since T ∈ Dg−1 (H), Theorem 3.4 implies       m      q r cβ1 ,...,βk T1,β1 · · · Tk,βk      q=0 β1 ∈F+n1 ,...,βk ∈F+nk   |β1 |+···+|βk |=q      m       q ≤ r cβ1 ,...,βk W1,β1 · · · Wk,βk    q=0 β1 ∈F+n1 ,...,βk ∈F+nk    |β1 |+···+|βk |=q

for any m ∈ N. Consequently, using the results above, we deduce that ϕ(r T ) :=

∞ 



r q cβ1 ,...,βk T1,β1 · · · Tk,βk

+ q=0 β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |+···+|βk |=q

converges in the operator norm topology for any r ∈ [0, 1) and ϕ(r T ) ≤ ϕ(r W) . Therefore item (i) holds. pur e To prove item (ii), assume that T ∈ Dg−1 (H). Note that if {ϕι (W)}ι is a net in F ∞ (g) such that w ∗ -limι ϕι (W) = ϕ(W) ∈ F ∞ (g), then w ∗ -limι ϕι (W) ⊗ IH = ϕ(W) ⊗ IH . Since

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ϕ(W) ∈ F ∞ (g),

we deduce that w ∗ -limι T (ϕι (W)) = T (ϕ(W)). Therefore T is w ∗ -continuous. Let {ϕk (W)}k ⊂ F ∞ (g) be a sequence such that WOT-limk→∞ ϕk (W) = ϕ(W). Since the WOT and w∗ -topologies concide on bounded sets, we have w∗ limk→∞ ϕk (W) = ϕ(W) which implies w∗ -limk→∞ (ϕk (W) ⊗ IK ) = ϕ(W) ⊗ IH . Hence, WOT-limk→∞ (ϕk (W)⊗ IK ) = ϕ(W)⊗ IH and WOT-limk→∞ T (ϕk (W)) = T (ϕ(W)). Similarly, if SOT-limk→∞ ϕk (W) = ϕ(W), then we also have SOTlimk→∞ (ϕk (W)⊗ IK ) = ϕ(W)⊗ IH which implies that SOT-limk→∞ T (ϕk (W)) = T (ϕ(W)). According to Theorem 3.1, the Berezin kernel K g,T is an isometry and ∗ ⊗ I )K K g,T Ti,∗ j = (Wi, g,T for any i ∈ {1, . . . , k} and j ∈ {1, . . . , n i }. ConseD j quently, ⎛ ⎜ ⎜ T ⎜ ⎝

⎞ 

αi ∈F+ n i ,|αi |≤m i∈{1,...,k}



=

⎟ ⎟ cα1 ,...,αk W1,α1 · · · Wk,αk ⎟ ⎠

cα1 ,...,αk T1,α1 · · · Tk,αk

αi ∈F+ n i ,|αi |≤m i∈{1,...,k}

for any m ∈ N, cα1 ,...,αk ∈ C.. Since T is a homomorphism on the algebra P(W) and W OT

, one can easily show that T is a homomorphism on F ∞ (g). F ∞ (g) = P(W) On the other hand, since   9 p 0 p ∗ [ϕi j (W) ⊗ IH ] p× p ⊕1 K g,T , [T (ϕi j )] p× p = ⊕1 K g,T

p ∈ N,

and K g,T is an isometry, it is clear that T is a completely contractive homomorphism. This completes the proof of item (ii). Now, to prove item (iii), fix r ∈ [0, 1) and assume that r W is a pure element with respect to Dg−1 . In what follows, we show that K g,r W ϕ(r W)∗ = [ϕ(W)∗ ⊗ Ik

i=1

F 2 (Hni ) ]K g,r W .

Due to the proof of item (i), we have ϕ(r W) = lim pn (r W), n→∞

where pn (W) :=

n q=0



+ β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |+···+|βk |=q

cβ1 ,...,βk W1,β1 · · · Wk,βk and the conver-

gence is in the operator norm topology. For each i ∈ {1, . . . , k}, let γi , σi , i ∈ F+ ni ∗ ∗ (e1 ⊗ · · · ⊗ ek ) = 0 for any and set n := |γ1 | + · · · + |γk |. Since W1,β · · · W γk k,βk γ1 1 βi ∈ F+ n i with |β1 | + · · · + |βk | > n, we have Reprinted from the journal

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ϕ(r W)∗ (eα1 1 ⊗ · · · ⊗ eαk k ) = pn (r W)∗ (eα1 1 ⊗ · · · ⊗ eαk k ) for any αi ∈ F+ n i with |α1 | + · · · + |αk | ≤ n. Since r W is a pure element in the k F 2 (Hn i )), we can apply Theorem 3.1 to noncommutative polydomain Dg−1 ( i=1 obtain K g,r W pn (r W)∗ = [ pn (W)∗ ⊗ Ik

i=1

F 2 (Hni ) ]K g,r W .

Using the definition of the noncommutative Berezin kernel, careful calculations reveal that & ' K g,r W ϕ(r W)∗ (eγ11 ⊗ · · · ⊗ eγkk ), (eσ11 ⊗ · · · ⊗ eσk k ) ⊗ (e11 ⊗ · · · ⊗ ekk ) & ' = K g,r W pn (r W)∗ (eγ11 ⊗ · · · ⊗ eγkk ), (eσ11 ⊗ · · · ⊗ eσk k ) ⊗ (e11 ⊗ · · · ⊗ ekk ) & = [( pn (W)∗ ⊗ Ik F 2 (Hn ) )]K g,r W (eγ11 ⊗ · · · ⊗ eγkk ), i=1 i ' 1 k 1 (eσ1 ⊗ · · · ⊗ eσk ) ⊗ (e1 ⊗ · · · ⊗ ekk )    ; r |β1 |+···|βk | b1,β1 · · · bk,βk pn (W)∗ = βi ∈F+ n i ,i=1,...,k

' (eβ11 ⊗ · · · ⊗ eβk k ), eσ11 ⊗ · · · ⊗ eσk k & ∗ ∗ × W1,β · · · Wk,β (eγ11 ⊗ · · · ⊗ eγkk ), g−1 (r W, r W∗ )1/2 1 k ' (e11 ⊗ · · · ⊗ ekk )   r |β1 |+···|βk | b1,β1 = βi ∈F+ n i ,i=1,...,k

& '  · · · bk,βk ϕ(W)∗ (eβ11 ⊗ · · · ⊗ eβk k ), eσ11 ⊗ · · · ⊗ eσk k & ' ∗ ∗ 1 k ∗ 1/2 1 k × W1,β · · · W (e ⊗ · · · ⊗ e ),  (r W, r W ) (e ⊗ · · · ⊗ e ) −1 g γk 1 k k,βk γ1 1 & ∗ 1 k = [(ϕ(W) ⊗ Ik F 2 (Hn ) )]K g,r W (eγ1 ⊗ · · · ⊗ eγk ), i=1 i ' 1 k 1 (eσ1 ⊗ · · · ⊗ eσk ) ⊗ (e1 ⊗ · · · ⊗ ekk ) for any σi , i ∈ F+ , i ∈ {1, . . . , k}. Hence, since ϕ(r W) and ϕ(W) are bounded k n i 2 operators on i=1 F (Hn i ), we deduce that K g,r W ϕ(r W)∗ = [ϕ(W)∗ ⊗ Ik

i=1

F 2 (Hni ) ]K g,r W .

k F 2 (Hn i )), Since r W is a pure element in the noncommutative polydomain Dg−1 ( i=1 Theorem 3.1 shows that the Berezin kernel K g,r W is an isometry and, therefore, the

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equality above implies

ϕ(r W)) ≤ ϕ(W)) , which completes the proof of item(iii). To prove item (iv), assume that W is a δ-radially pure element with respect to Dg−1 pur e for some δ ∈ (0, 1). Then r W ∈ Dg−1 (⊗ks=1 F 2 (Hn s )) for all r ∈ [δ, 1). Due to part (iii), we have ϕ(r W) ≤ ϕ(W) for r ∈ [δ, 1). Hence, and due to the fact that ϕ(W)(eα1 1 ⊗ · · · ⊗ eαk k ) = lim ϕ(r W)(eα1 1 ⊗ · · · ⊗ eαk k ) for any αi ∈ F+ n i , an approximation argument implies

r →1

ϕ(W) = SOT- lim ϕ(r W). r →1

pur e

To prove item (v), assume that T ∈ Dg−1 (H) and W is δ-radially pure with respect to Dg−1 . Then, the items (i) and (ii) imply ∗ ϕ(r T ) = K g,T (ϕ(r W) ⊗ IH )K g,T ,

r ∈ (0, 1).

(4.2)

Taking into account that the map Y → Y ⊗ I is SOT-continuous on bounded sets and using item (iv), we can pass to the limit in relation (4.2) and deduce that ∗ (ϕ(W) ⊗ IH )K g,T = T (ϕ(W)). SOT- lim ϕ(r T ) = K g,T r →1

Therefore, item (v) holds. It remains to prove item (vi). Note that, due to Corollary 2.11, there is  ∈ (0, 1) such that any k-tuple in [B(H)n 1 ] ×c · · · ×c [B(H)n k ] is a radially pure element in the polydomain Dg−1 (H). Hence, t0 > 0. Due to item (i), if ϕ(W) ∈ F ∞ (g), then ϕ(t0 W) ∈ A(g), ϕ(t0 T ) ≤ ϕ(t0 W) , and ∗ T (ϕ(t0 W)) := K g,T (ϕ(t0 W) ⊗ IH )K g,T = ϕ(t0 T ).

Applying item (ii), we have ϕ(r W) ≤ ϕ(W) for any r ∈ [0, t0 ). On the other hand, part (i) shows that ϕ(r T ) ≤ ϕ(r W) for any r ∈ [0, 1). Combining these inequalities, we get

ϕ(r T ) ≤ ϕ(r W) ≤] ϕ(W) ,

r ∈ [0, t0 ).

Consequently, since ϕ(t0 W)(eα1 1 ⊗ · · · ⊗ eαk k ) = lim ϕ(r W)(eα1 1 ⊗ · · · ⊗ eαk k ) for any αi ∈ F+ n i , an approximation argument implies

r t0

ϕ(t0 W) = SOT- lim ϕ(r W). r t0

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Hence, and using again the fact that the map Y → Y ⊗ I is SOT-continuous on bounded sets, we deduce that ∗ ϕ(t0 T ) = K g,T (ϕ(t0 W) ⊗ IH )K g,T ∗ = SOT- lim K g,T (ϕ(r W) ⊗ IH )K g,T r t0

= SOT- lim ϕ(r T ) r t0

 

This completes the proof.

5 w ∗ -Continuous Functional Calculus for Completely Non-coisometric Elements in Polydomains In this section we provide a w ∗ -continuous F ∞ (g)-functional calculus for the completely non-coisometric (c.n.c.) elements in the closed noncommutative polydomain pur e Dg−1 (H). Definition 5.1 A tuple T = (T1 , . . . , Tk ) ∈ Dg−1 (H) is said to be completely noncoisometric (c.n.c.) with respect to the polydomain Dg−1 if there is no nonzero joint invariant subspace H0 under the operators Ti,∗ j , where i ∈ {1, . . . , k} and j ∈ {1, . . . , n i }, such that g−1 (T , T ∗ )|H0 = 0, where the defect operator is defined by g−1 (T , T ∗ ) := g−1 ,T1 ◦ · · · ◦ g−1 ,Tk (I ). 1

k

Lemma 5.2 If T ∈ Dg−1 (H), then the following statements are equivalent. (i) T is completely non-coisometric with respect to Dg−1 . (ii) There is no h ∈ H, h = 0 such that ∗ ∗ g−1 (T , T ∗ )T1,α · · · Tk,α h = 0, 1 k

+ (α1 , . . . , αk ) ∈ F+ n 1 × · · · × Fn k .

(iii) The noncommutative Berezin kernel K g,T is a one-to-one operator. Proof First, we remark that T is completely non-coisometric with respect to Dg−1 if and only if < + (α1 ,...,αk )∈F+ n 1 ×···×Fn k

∗ ∗ ker g−1 (T , T ∗ )T1,α · · · Tk,α = {0}, 1 k

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which shows that (i) is equivalent to (ii). Since the noncommutative Berezin kernel associated with T ∈ Dg−1 (H) is the operator K g,T : H →

k 1

F 2 (Hn s ) ⊗ g−1 (T , T ∗ )(H)

s=1

defined by K g,T h :=





βi ∈F+ n i ,i=1,...,k

 b1,β1 · · · bk,βk eβ11

∗ ∗ ⊗ · · · ⊗ eβk k ⊗ g−1 (T , T ∗ )1/2 T1,β · · · Tk,β h, 1 k

we have ' & ∗ K g,T h, h = K g,T



b1,α1

+ α1 ∈F+ n 1 ,...αk ∈Fn k

; = ∗ ∗ · · · bk,αk T1,α1 · · · Tk,αk g−1 (T , T ∗ )Tk,α · · · T1,α h, h 1 k for any h ∈ H. Consequently, K g,T h ∗ · · · T ∗ h = 0 for any (α , . . . , α ) ∈ = 0 if and only if g−1 (T , T ∗ )1/2 Tk,α 1 k 1,α1 k + F+ n 1 × · · · × Fn k , if and only if
0 and let N ∈ N be such that x N − h < 2 ϕ(W)

. Using the inequality (5.3), we obtain

Ah − p N1 ,...,Nk (T (m) )h

≤ A

h − x N + p N1 ,...,Nk (T (m) )

x N − h + Ay N − p N1 ,...,Nk (T (m) )x N

≤ 2 ϕ(W)

x N − h + Ax N − p N1 ,...,Nk (T (m) )x N

≤  + Ax N − p N1 ,...,Nk (T (m) )x N . 680

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Taking m → ∞, we deduce that

Ah − p N1 ,...,Nk (T )h ≤  + Ay N − p N1 ,...,Nk (T )x N . ∗ , we can use relation (5.2) to deduce that Consequently, since x N ∈ range K g,T

lim sup

N1 →∞,...,Nk →∞

Ah − p N1 ,...,Nk (T )h ≤ 

for any  > 0. This implies lim

N1 →∞,...,Nk →∞

Ah − p N1 ,...,Nk (T )h = 0,

h ∈ H,

which proves our assertion. Define the map T : F ∞ (g) → B(H) by setting ψT (ϕ(W)) = ϕ(T ) := A and let {ϕι (W)}ι ⊂ F ∞ (g) be a bounded net such that ϕι (W) → ϕ(W) ∈ F ∞ (g) in the strong operator topology. Then ϕι (W) ⊗ IH → ϕ(W) ⊗ IH in the same topology. According to relation (5.2), we have ∗ ∗ ∗ ∗ ϕι (T )K g,T ξ = K g,T (ϕι (W) ⊗ IH )ξ → K g,T (ϕ(W) ⊗ IH )ξ = ϕ(T )K g,T ξ

 for any ξ ∈ ks=1 F 2 (Hn s ) ⊗ H. Since ϕι (T ) ≤ ϕι (W) , the net {ϕι (T )}ι is also ∗ is dense in H, a standard approximation bounded. Taking into account that range K g,T argument. shows that SOT- limι ϕι (T ) = ϕ(T ). One can prove a similar result when SOT is replaced by WOT or w∗ -topology. Therefore, since the map T : F ∞ (g) → B(H) is sequentially w∗ -continuous, we can use Lemma 5.3 to conclude that T is w ∗ -continuous. In what follows, we show that T is a completely contractive homomorphism. Let [ϕi j (W)]q ∈ Mq×q (C) ⊗ F ∞ (g) and note that, due to Theorem 4.5, there ij a multi-sequence of matrices [ p N1 ,...,Nk (W)]q ∈ Mq×q (C) ⊗ P(W) such that ij

ij

[ p N1 ,...,Nk (W)]q → [ϕi j (W)]q strongly as k → ∞ and [ p N1 ,...,Nk (W)]q ≤  q q

[ϕ i j (W)]q . Let h = ⊕i=1 h i ∈ ⊕i=1 ( ks=1 F 2 (Hn s )) and let {T (m) }m∈N ⊂ pur e Dg−1 (H) be a sequence such that T − T (m) → 0 as m → ∞. According to ij

the results above, we have [ϕi j (T )]q h = lim N1 →∞,...Nk →∞ [ p N1 ,...,Nk (T )]q h and

[ϕi j (T )]q h ≤

sup

N1 ,...Nk ∈N

ij

[ p N1 ,...,Nk (T )]q

h ≤

sup

N1 ,...,Nk ,m∈N

[ p N1 ,...,Nk (T (m) )]q

h

ij

≤ [ϕi j (W)]q

h .

Hence, [ϕi j (T )]q ≤ [ϕi j (W)]q , which proves that T is a completely contractive linear map. Since T is a homomorphism on the algebra of polynomials P(W) and T is sequentially WOT-continuous, one can use the WOT-density of P(W) in F ∞ (g) to show that T is a homomorphism on F ∞ (g).

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Now we prove the last part of the theorem. To this end, assume that W is a δradially pure element with respect to Dg−1 . Then there exists δ ∈ (0, 1) such that r W is  pure for any r ∈ [δ, 1). Let ϕ(W) ∈ F ∞ (g) have the Fourier representation cβ1 ,...,βk W1,β1 · · · Wk,βk . Due to Theorem 4.7, + β1 ∈F+ n 1 ,...,βk ∈Fn k

ϕ(r W) :=

∞ 



r q cβ1 ,...,βk W1,β1 · · · Wk,βk

+ q=0 β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |+···+|βk |=q

converges in the operator norm topology for any r ∈ [0, 1), SOT- lim ϕ(r W) = ϕ(W), r →1

and

ϕ(r W) ≤ ϕ(W)

for any r ∈ [δ, 1). Moreover, ϕ(r T ) :=

∞ 



r q cβ1 ,...,βk T1,β1 · · · Tk,βk

+ q=0 β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |+···+|βk |=q

converges in the operator norm topology for any r ∈ [0, 1), and

ϕ(r T ) ≤ ϕ(r W . ∗ = K ∗ (W ⊗ I ) for any i ∈ {1, . . . , k}, j ∈ {1, . . . , n }, and Now, since Ti. j K g,T i, j i H g,T using the convergence in norm of the series defining ϕ(r T ) and ϕ(r W), we deduce that ∗ ∗ = K g,T (ϕ(r W) ⊗ IH ) (5.4) ϕ(r T )K g,T

for any r ∈ [δ, 1). Taking r → 1 in this relation, we deduce that the map A˜ : ∗ → H given by Ay ˜ := limr →1 ϕ(r T )y for y ∈ rangeK ∗ , is well-defined, range K g,T g,T linear, and ∗ ∗ ∗ ˜ g,T A(K x) ≤ lim sup ϕ(r T )

K g,T x ≤ ϕ(W)

K g,T x . r →1

˜ ≤ ϕ(W) . Using a Consequently, A˜ has a unique continuous extension to H and A

˜ standard approximation argument, one can show that Ah = limr →1 ϕ(r T )h for every ˜ ∗ = h ∈ H. Taking into account the definition of A˜ and relation (5.4), we have AK g,T ∗ ∗ ∗ K g,T (ϕ(W) ⊗ IH ). Due to relation (5.2), we have AK g,T = K g,T (ϕ(W) ⊗ IH ). Since ∗ is dense in H, we deduce that A = A. ˜ A and A˜ are bounded operators and range K g,T This completes the proof.  

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6 Bohr Inequality for the Noncommutative Hardy Algebra F ∞ (g) In this section we study the Bohr phenomenon for the noncommutative Hardy algebra F ∞ (g). Using multi-homogeneous and homogeneous expansions for the elements of F ∞ (g), we introduce the multi-homogeneous Bohr radius K mh (F ∞ (g)) and the homogeneous Bohr radius K h (F ∞ (g)), respectively, and provide several estimates. For each q, p1 . . . , pk ∈ Z+ , we set   + q := (α1 , . . . , αk ) ∈ F+ n 1 × · · · × Fn k : |α1 | + · · · + |αk | = q and   +  p1 ,..., pk := (α1 , . . . , αk ) ∈ F+ n 1 × · · · × Fn k : |αi | = pi . Note that + F+ n 1 × · · · × Fn k =

∞ 

q =



 p1 ,..., pk .

( p1 ,..., pk )∈Zk+

q=0

 Let g := (g1 , . . . , gk ) be such that gi = 1+ α∈F+n ,|α|≥1 bi,α Z i,α is a free holomorphic i function in a neighborhood of the origin with strictly positive coefficients and such that sup

α∈F+ ni

bi,α < ∞, bi,gi α

j ∈ {1, . . . , n i },

j

and let W = (W1 , . . . , Wk ) with Wi = (Wi,1 , . . . , Wi,n i ) be the associated weighted left creation operators. The next result is an extension of Theorem 2.2 from [88]. We include a proof for completeness. Theorem 6.1 Let  be any of the sets q or  p1 ,..., pk , where q ∈ Z+ \{0} and Zk+ \{0}, and let ϕ(W) ∈ F ∞ (g) with ϕ(W) ≤ 1 have the Fourier ( p1 . . . , pk ) ∈  cα1 ,...,αk W1,α1 · · · Wk,αk . Then representation αi ∈F+ n i

i∈{1,...,k}

 (α1 ,...,αk )∈

|cα1 ,...,αk |2 ≤ (1 − |c0 |2 )2 b1,α1 · · · bk,αk

and  αi ∈F+ n i ,i∈{1,...,k}

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Proof Let M be the closed linear span of the vectors 1, eβ11 ⊗ · · · ⊗ eβk k , where (β1 , . . . , βk ) ∈ . Note that PM ϕ(W)|M 1, 1 = ϕ(W)1, 1 = c0 , ' & ' & PM ϕ(W)|M (eβ11 ⊗ · · · ⊗ eβk k ), 1 = ϕ(W))(eβ11 ⊗ · · · ⊗ eβk k ), 1 = 0, and &

' & ' PM ϕ(W)|M 1, eβ11 ⊗ · · · ⊗ eβk k = ϕ(W)1, eβ11 ⊗ · · · ⊗ eβk k = 

1 cβ1 ,...,βk , b1,β1 · · · bk,βk

for any (β1 , . . . , βk ) ∈ . Since  is either q or  p1 ,..., pk , we deduce that &

PM ϕ(W)|M (eβ11 ⊗ · · · ⊗ eβk k ), eγ11 ⊗ · · · ⊗ eγkk & ' = ϕ(W)(eβ11 ⊗ · · · ⊗ eβk k ), eγ11 ⊗ · · · ⊗ eγkk    b1,α1 · · · bk,αk cα1 ,...,αk =

'

p∈Zk+ (α1 ,...,αk )∈

& ' eα1 1 β1 ⊗ · · · ⊗ eαk k βk , eγ11 ⊗ · · · ⊗ eγkk = c0 δβ1 ,γ1 · · · δβk ,γk for any (β1 , . . . , βk ), (γ1 , . . . , γk ) ∈ . One can easily see that the matrix representation of the contraction PM ϕ(W)|M with respect to the decomposition 

M = C1 ⊕

(β1 ,...,βk )∈

C(eγ11 ⊗ · · · ⊗ eγkk )

is ⎛

⎤ ⎡[0 ⎜ √ 1 c α ,...,α c0 1 k ⎜ ⎜ ⎢ b1,α1 ···bk,αk ⎥⎢ . ⎜⎢ ⎥⎣ . .. ⎝⎣ ⎦ . . 0 (α1 , . . . , αk ) ∈  c0

⎡

⎞ 0] ⎤⎟ 0 ⎟ .. ⎥ ⎟ ⎟. . ⎦⎠ · · · c0 ··· ··· .. .

Lemma 3.4 from [64] states that if A, Bare Hilbert  spaces, B is a bounded linear λIA 0 operator from A to B, and λ ∈ C, then is a contraction if and only if B λIB

B ≤ 1 − |λ|2 . Using this result in our setting, we obtain  (α1 ,...,αk )∈

|cα1 ,...,αk |2 ≤ (1 − |c0 |2 )2 . b1,α1 · · · bk,αk

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To obtain the second inequality in the theorem, note that 

ϕ(W)1 =



αi ∈F+ n i ,i∈{1,...,k}

1 cα1 ,...,αk eα1 1 ⊗ · · · ⊗ eαk k . b1,α1 · · · bk,αk

Since ϕ(W)1 ≤ ϕ(W) , we complete the proof.

 

A close look at the  proof of Theorem 4.7 reveals that, if ϕ(W) ∈ F ∞ (g) has the cβ1 ,...,βk W1,β1 · · · Wk,βk , then Fourier representation βi ∈F+ n i

i∈{1,...,k}

Dr (ϕ) :=

 ( p1 ,..., pk )∈Zk+

      (β1 ,...,βk )∈ p

1 ,..., pk

   |β1 | |βk | r1 · · · rk cβ1 ,...,βk W1,β1 · · · Wk,βk   0 such that Dr (ϕ) :=

 ( p1 ,..., pk )∈Zk+

      (β1 ,...,βk )∈ p

r

|β1 |+···+|βk |

1 ,..., pk

   cβ1 ,...,βk W1,β1 · · · Wk,βk   ≤ ϕ(W)



for any ϕ(W) ∈ F ∞ (g). If we replace the Hardy space F ∞ (g) with the subspace F0∞ (g) := {ϕ ∈ F ∞ (g) : ϕ(0) = 0}, the corresponding Bohr radius is denoted by K mh (F0∞ (g)). Theorem 6.2 Let ϕ(W) ∈ F ∞ (g) have the Fourier representation

 βi ∈F+ ni

cβ1 ,...,βk W1,β1

· · · Wk,βk and assume that ϕ(W) ≤ 1. Let r := (r1 , . . . , rk ) ∈ [0, 1)k and set 

r :=

p

( p1 ,..., pk )∈Zk+ \{0}

,

p

r1 1 · · · rk k

max

βi ∈F+ n i |βi |= pi

b1,β1 · · · bk,βk W1,β1 · · · Wk,βk .

If r ≤ 21 , then Dr (ϕ) :=

 ( p1 ,..., pk )∈Zk+

      (β1 ,...,βk )∈ p

1 ,..., pk

   |β | |β | r1 1 · · · rk k cβ1 ,...,βk W1,β1 · · · Wk,βk   ≤ 1. 

If, in addition, ϕ(W) ∈ F0∞ (g), then Dr (ϕ) ≤ r . Moreover, the following statements hold.

Reprinted from the journal

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(i) If r0 is the nonnegative solution of the equation r = 21 , where r = (r ,...,r ) , then the multi-homogeneous Bohr radius satisfies the inequality r0 ≤ K mh (F ∞ (g)). (ii) If t0 is the nonnegative solution of the equation r = 1, then t0 ≤ K mh (F0∞ (g)).  Proof Since gi = 1+ |α|≥1 bi,α Z α is a free holomorphic function on a neighborhood of the origin, we have ⎛ ⎜ lim sup ⎝ k→∞

⎞1/2k



2 ⎟ bi,α ⎠

< ∞.

α∈F+ n i ,|α|=k

  Hence, we deduce that ωi := 1 + α∈F+n ,|α|≥1 bi,α Z i,α is also a free holoi morphic function on a neighborhood of the origin. each i ∈ {1, . . . , k}, let  For  Mi := sup j∈{1,...,n i } Wi, j . Note that each series ∞ b (r M) pi pi =0 α∈F+ n i ,|α|= pi i,α i is convergent for ri > 0 small enough. Now, it is easy yo see that r ≤







b1,β1 · · · bk,βk (r1 M1 )|β1 | · · · (rk Mk )|βk |

( p1 ,..., pk )∈Zk+ (β1 ,...,βk )∈ p1 ,..., pk

and the later series is convergent for some ri > 0. Assume that ( p1 , . . . , pk ) ∈ Zk+ \{0}. Using Theorem 6.1, we deduce that        |β1 | |βk |   r · · · r c W · · · W β ,...,β 1,β k,β 1 1 k k 1 k   (β1 ,...,βk )∈ p ,..., p  1 k   4 5  |β | |β |  =  r1 1 · · · rk k b1,β1 · · · bk,βk W1,β1 · · · Wk,βk : (β1 , . . . , βk ) ∈  p1 ,..., pk   ⎡ ⎤ c  √ β1 ,...,βk  b ···b 1,β k,β 1 k ⎢ ⎥ ⎣ ⎦ :  (β1 , . . . , βk ) ∈  p1 ,..., pk  1/2       2|β | 2|β | ∗  ≤ r1 1 · · · rk k b1,β1 · · · bk,βk W1,β1 · · · Wk,βk Wk,β · · · W 1,β 1   k  (β1 ,...,βk )∈ p ,..., p 1 k  1/2     |cβ1 ,...,βk |2   ×  b1,β1 · · · bk,βk    (β1 ,...,βk )∈ p1 ,..., pk

686

Reprinted from the journal

Operator Theory on Noncommutative Polydomains, I

≤ (1 − |c0 |

2

p )r1 1

p · · · rk k

      (β1 ,...,βk )∈ p

∗ b1,β1 · · · bk,βk W1,β1 · · · Wk,βk Wk,β k

1/2  

∗  · · · W1,β 1

1 ,..., pk



.

Since the operators {Wi,βi }βi ∈F+n ,|βi |= pi have orthogonal ranges, it is easy to see that i

1/2        ∗  bi,βi Wi,βi Wi,β   i   βi ∈F+ni ,|βi |= pi  = max bi,βi Wi,βi

βi ∈F+ n i ,|βi |= pi

for each i ∈ {1, . . . , k}. Consequently, using the relations above and that r ≤ 21 , we obtain  ( p1 ,..., pk )∈Zk+

      (β1 ,...,βk )∈ p

≤ |c0 | + (1 − |c0 |2 )

   |β1 | |βk | r1 · · · rk cβ1 ,...,βk W1,β1 · · · Wk,βk    1 ,..., pk  p p r1 1 · · · rk k

( p1 ,..., pk )∈Zk+ \{0}

      (β1 ,...,βk )∈ p

∗ b1,β1 · · · bk,βk W1,β1 · · · Wk,βk Wk,β k

1 ,..., pk



≤ |c0 | + (1 − |c0 |2 ) max

,

βi ∈F+ n i |βi |= pi

≤ |c0 | +

p

1/2   ∗  · · · W1,β 1 

p

r1 1 · · · rk k

( p1 ,..., pk )∈Zk+ \{0}

b1,β1 · · · bk,βk W1,β1 · · · Wk,βk

1 − |c0 |2 ≤ 1. 2

To prove the last part of the theorem, take r1 = · · · = rk = r and let r0 be the nonnegative solution of the equation r = 21 . Since the function r → r is strictly increasing on the interval [0, 1), we can use the first part of the theorem to deduce that r0 ≤ K mh (F ∞ (g)). To prove part (ii), note that, in the particular case when the coefficient c0 = 0, the relations above imply Dr (ϕ) ≤ r . Consequently, if t0 is the nonnegative solution of   the equation r = 1, then t0 ≤ K mh (F0∞ (g)). The proof is complete. Proposition 6.3 Let s := (s1 , . . . , sk ) ∈ [1, ∞)k and ψ s := ((1 − ϕ1 )−s1 , . . . , (1 − ϕk )−sk ),

Reprinted from the journal

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 where each ϕi := d Z is a formal power series with nonnegaα∈F+ n i ,|α|≥1 i,α i,α tive coefficients and di,α > 0 if |α| = 1. If W := (W1 , . . . , Wk ) with Wi := (Wi,1 , . . . , Wi,n i ) is the universal model associated with ψ s , then 

b1,β1 · · · bk,βk W1,β1

(β1 ,...,βk )∈ p1 ,..., pk ∗ ∗ · · · Wk,βk Wk,β · · · W1,β ≤ 1 k

 k   si + pi − 1 I. pi i=1

Proof First, note that ∗ i Wi,β Wi,β eα

/b =

i,γ i bi,α eα

if α = βγ

0

otherwise

and ∗

= sup

Wi,β Wi,β

γ ∈F+ ni

bi,γ . bi,βγ

According to Proposition 2.17, we have  bi,β bi,γ ≤

 si + |β| − 1 bi,βγ , |β|

β, γ ∈ F+ ni .

Consequently, we deduce that 1

Wi,β ≤  bi,β

6  si + |β| − 1 , |β|

β ∈ F+ ni .

Since the operators {Wi,βi }βi ∈F+n have orthogonal ranges, we have i

 βi ∈F+ n i ,|βi |= pi

∗ bi,βi Wi,βi Wi,β i

  si + pi − 1 I. ≤ pi  

Now, it is easy to complete the proof. Theorem 6.4 Let s := (s1 , . . . , sk ) ∈ [1, ∞)k and ψ s := ((1 − ϕ1 )−s1 , . . . , (1 − ϕk )−sk ),

 where each ϕi := α∈F+n ,|α|≥1 di,α Z i,α is a formal power series with nonnegative i coefficients and di,α > 0 if |α| = 1. Then the following statements hold.

688

Reprinted from the journal

Operator Theory on Noncommutative Polydomains, I

(i) The Bohr radius K mh (F ∞ (ψ s )) satisfies the inequality   1 2 s1 +···+sk 1− < r0 ≤ K mh (F ∞ (ψ s )), 3 where r0 is the nonnegative solution of the equation 

ωr :=

r

p1 +···+ pk

( p1 ,..., pk )∈Zk+ \{0}

1/2 k   1 si + pi − 1 = . pi 2 i=1

(ii) The Bohr radius K mh (F0∞ (ψ s )) satisfies the inequality t0 ≤ K mh (F0∞ (ψ s )), where t0 is the nonnegative solution of the equation ωr = 1. Proof Let ϕ(W) ∈ F ∞ (ψ s ) have the Fourier representation



cβ1 ,...,βk W1,β1 βi ∈F+ ni i∈{1,...,k}

· · · Wk,βk . As in the proof of Theorem 6.2, we can use Proposition 6.3 to deduce that  ( p1 ,..., pk )∈Zk+

      (β1 ,...,βk )∈ p

1 ,..., pk

   |β | |β | r1 1 · · · rk k cβ1 ,...,βk W1,β1 · · · Wk,βk   



≤ |a0 | + (1 − |a0 |2 )

p

p

r1 1 · · · rk k

( p1 ,..., pk )∈Zk+ \{0}

1/2 k   si + pi − 1 pi i=1

and, consequently, taking r1 = · · · = rk = r , we deduce that r0 ≤ K mh (F ∞ (ψ s )), where r0 is the nonnegative solution of the equation ωr = 21 . On the other hand, since  ( p1 ,..., pk )∈Zk+ \{0}

r

p1 +··· pk

E 1/2 D k  k  si + pi − 1 −si < (1 − r ) − 1 pi i=1

i=1

for any r ∈ (0, 1) and D k 

E (1 − γ0 )

−si

−1 =

i=1

Reprinted from the journal

689

1 , 2

G. Popescu

9 0 1 where γ0 := 1 − 23 s1 +···+sk , we must have γ0 < r0 . To prove part (ii), note that, in the particular case when c0 = 0, the inequalities above imply         |β1 |+···+|βk |  r cβ1 ,...,βk W1,β1 · · · Wk,βk    ≤ ωr .   ( p ,..., p )∈Zk \{0} (β1 ,...,βk )∈ p ,..., p 1

+

k

1

k

Consequently, if t0 is the nonnegative solution of the equation ωr = 1, then t0 ≤   K mh (F0∞ (ψ s )). The proof is complete.  ∞ Let ϕ ∈ F (g) have the Fourier representation cβ1 ,...,βk W1,β1 · · · Wk,βk and βi ∈F+ ni

let Dr (ϕ) :=

      (β1 ,...,βk )∈ p

 ( p1 ,..., pk )∈Zk+

1 ,..., pk

   |β | |β | r1 1 · · · rk k cβ1 ,...,βk W1,β1 · · · Wk,βk  , 

be the associated majorant series, where W is the universal model associated with g and r := (r1 , . . . , rk ) ∈ [0, 1)k . Define dr (F ∞ (g)) := sup

Dr (ϕ) ,

ϕ

where the supremum is taken over all ϕ ∈ F ∞ (g) with ϕ not identically 0. When r1 = · · · = rk = r ∈ [0, 1), we use the notation Dr (ϕ) and dr (F ∞ (g)), respectively. Given r := (r1 , . . . , rk ) ∈ [0, 1)k , we define Mr := min {Ar , Br } , where / 1 if r ≤ 21 Ar := 1 r + 4r if r > 21 ,  p p r := r1 1 · · · rk k ( p1 ,..., pk )∈Zk+ \{0}

      (β1 ,...,βk )∈ p

∗ b1,β1 · · · bk,βk W1,β1 · · · Wk,βk Wk,β k

1 ,..., pk

1/2   ∗  · · · W1,β , 1 

and ⎛ ⎜ Br := ⎝



2 p1

r1

2 pk

· · · rk

( p1 ,..., pk )∈Zk+

      (β1 ,...,βk )∈ p

b1,β1

1 ,..., pk

∗ · · · bk,βk W1,β1 · · · Wk,βk Wk,β k

690

⎞1/2   ∗ ⎟ · · · W1,β1 ⎠ .  

Reprinted from the journal

Operator Theory on Noncommutative Polydomains, I

We also set ⎛ 

⎜ Br0 := ⎝

2 p1

r1

2 pk

· · · rk

( p1 ,..., pk )∈Zk+ \{0}

      (β1 ,...,βk )∈ p

b1,β1

1 ,..., pk

∗ · · · bk,βk W1,β1 · · · Wk,βk Wk,β k

⎞1/2   ∗ ⎟ · · · W1,β1 ⎠ .  

Theorem 6.5 If r := (r1 , . . . , rk ) ∈ [0, 1)k , then 1 ≤ dr (F ∞ (g)) ≤ Mr . where Mr := min {Ar , Br }, and dr (F0∞ (g)) ≤ K r ,   where K r := min r , Br0 . Proof If we take ϕ = 1 we deduce that 1 ≤ dr (F ∞ (g)). As in the proof of Theorem 6.2 and using Cauchy-Schwarz inequality, we obtain        |β1 | |βk |  r1 · · · rk cβ1 ,...,βk W1,β1 · · · Wk,βk  Dr (ϕ) :=     ( p1 ,..., pk )∈Zk+ (β1 ,...,βk )∈ p1 ,..., pk      2|β | 2|β |  ≤ r1 1 · · · rk k b1,β1   ( p1 ,..., pk )∈Zk+ (β1 ,...,βk )∈ p1 ,..., pk 1/2    ∗ · · · bk,βk W1,β1 · · · Wk,βk Wk,βk · · · W1,β1     1/2     |cβ1 ,...,βk |2   ×   (β1 ,...,βk )∈ p ,..., p b1,β1 · · · bk,βk  1 k ⎛      ⎜ 2|β | 2|β |  ≤ ⎝ r1 1 · · · rk k b1,β1   ( p ,..., p )∈Zk (β1 ,...,βk )∈ p ,..., p 

1

k

+

1

k

∗ · · · bk,βk W1,β1 · · · Wk,βk Wk,β k

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691

⎞1/2   ⎟ · · · W1,β1 ⎠  

G. Popescu

⎛ 

⎜ ×⎝

( p1 ,..., pk )∈Zk+

      (β1 ,...,βk )∈ p

⎞1/2  ⎟ |cβ1 ,...,βk ⎠ b1,β1 · · · bk,βk   |2

1 ,..., pk

≤ Br ϕ(1) ≤ Br ϕ

for any ϕ ∈ F ∞ (g). Now, we prove that Dr (ϕ) ≤ Ar . We may assume that ϕ ∞ ≤ 1 and, consequently |c0 | ≤ 1. It is easy to see that, if 0 ≤ r ≤ 21 , then sup{x +(1− x 2 )r : 0 ≤ x ≤ 1} ≤ 1 1. If r > 21 , we have sup{x + (1 − x 2 )r : 0 ≤ x ≤ 1} = r + 4 . Consequently, r using Theorem 6.2 (see the proof), we deduce that Dr (ϕ) ≤ |c0 | + (1 − |c0 |2 )r ≤ Ar . Therefore, Dr (ϕ) ≤ Mr ϕ for any ϕ ∈ F ∞ (g). Now, we prove the second part of the theorem for the algebra F0∞ (g). Note that, as above, one can show that Dr (ϕ) ≤ Br0 ϕ for any ϕ ∈ F0∞ (g). According to the   proof of Theorem 6.2, we have Dr (ϕ) ≤ r . This completes the proof. Theorem 6.6 Let s := (s1 , . . . , sk ) ∈ [1, ∞)k and ψ s := ((1 − ϕ1 )−s1 , . . . , (1 − ϕk )−sk ),  where each ϕi := α∈F+n ,|α|≥1 di,α Z i,α is a formal power series with nonnegative i coefficients and di,α > 0 if |α| = 1. Then the following statements hold. (i) For any r ∈ [0, 1), /



∞

1 ≤ dr (F (ψ s )) ≤ min ar , √

1

s1 +···+sk 8

1 − r2

,

where / ar := ωr :=

1 ωr +

if 0 ≤ r ≤ r0 if r0 < r < 1,

1 4ωr



r

p1 +···+ pk

( p1 ,..., pk )∈Zk+ \{0}

1/2 k   si + pi − 1 , pi i=1

and r0 is the nonnegative solution of the equation ωr = 21 . (ii) For any r ∈ [0, 1), ⎧ ⎨

dr (F0∞ (ψ s )) ≤ min ωr , ⎩

2

1 1 − r2

692

s1 +···+sk

31/2 ⎫ ⎬ −1 . ⎭

Reprinted from the journal

Operator Theory on Noncommutative Polydomains, I

Proof According to Proposition 6.3 we have 

b1,β1 · · · bk,βk W1,β1

(β1 ,...,βk )∈ p1 ,..., pk ∗ · · · Wk,βk Wk,β k

∗ · · · W1,β 1

 k   si + pi − 1 I. ≤ pi i=1

Hence, we deduce that 

Dr (ϕ) :=

( p1 ,..., pk )∈Zk+

      (β1 ,...,βk )∈ p

1 ,..., pk

≤ |a0 | + (1 − |a0 | ) 2

   |β1 |+···+|βk | r cβ1 ,...,βk W1,β1 · · · Wk,βk   



r

( p1 ,..., pk )∈Zk+ \{0}

p1 +···+ pk

1/2 k   si + pi − 1 . pi i=1

Now, following the lines of the proof of Theorem 6.5, one can complete the proof of part (i). Similarly, one can prove part (ii).   For the Hardy algebra H ∞ (D), Bombieri and Bourgain (see [26]) proved that dr (H ∞ (D)) ∼ √ 1 2 as r → 1. It remains an open problem if, in our setting, limr →1



1−r dr (F ∞ (ψ s )) s1 +···+sk √1

= 1. We know that the result is true for the polyball Bn (see

1−r 2

[87]).  Let ϕ(W) ∈ F ∞ (g) have the Fourier representation βi ∈F+n cβ1 ,...,βk W1,β1 · · · i Wk,βk and recall that   + q := (α1 , . . . , αk ) ∈ F+ n 1 × · · · × Fn k : |α1 | + · · · + |αk | = q ,

q ∈ Z+ .

The homogeneous Bohr radius for the noncommutative Hardy algebra F ∞ (g) is denoted by K h (F ∞ (g)) and is the largest r > 0 such that    ∞      |β1 |+···+|βk |  Mr (ϕ) := r cβ1 ,...,βk W1,β1 · · · Wk,βk    ≤ ϕ(W)

 q=0 ( p1 ,..., pk )∈q for any ϕ(W) ∈ F ∞ (g). Note that K h (F ∞ (g)) ≥ K mh (F ∞ (g)) and K h (F0∞ (g)) ≥ K mh (F0∞ (g)).

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Theorem 6.7 Let ϕ(W) ∈ F ∞ (g) have the Fourier representation · · · Wk,βk and assume that ϕ(W) ≤ 1. Let r ∈ [0, 1) and set

 βi ∈F+ ni

cβ1 ,...,βk W1,β1

 1/2      ∗ ∗  rq  b · · · b W · · · W W · · · W . Er := 1,β1 k,βk 1,β1 k,βk k,βk 1,β1   ( p1 ,..., pk )∈q  q=1 ∞ 

If Er ≤ 21 , then Mr (ϕ) ≤ 1. If, in addition, ϕ(W) ∈ F0∞ (g), then Mr (ϕ) ≤ Er . Moreover, the following statements hold. (i) If γ0 is the nonnegative solution of the equation Er = 21 , then the homogeneous Bohr radius satisfies the inequality γ0 ≤ K h (F ∞ (g)). (ii) If σ0 is the nonnegative solution of the equation Er = 1, then σ0 ≤ K h (F0∞ (g)). Proof The proof is similar to that of Theorem 6.2.

 

For each r ∈ [0, 1), define m r (F ∞ (g)) := sup

Mr (ϕ) ,

ϕ

where the supremum is taken over all ϕ ∈ F ∞ (g) with ϕ not identically 0. Set Nr := min {Cr , Dr } , where / 1 if Er ≤ 21 Cr := Er + 4E1 r if Er > 21 ,  1/2   ∞     q ∗ ∗  Er := r  b1,β1 · · · bk,βk W1,β1 · · · Wk,βk Wk,βk · · · W1,β1  , (β1 ,...,βk )∈q  q=1

694

Reprinted from the journal

Operator Theory on Noncommutative Polydomains, I

and  ⎞1/2      2q  ∗ ∗ ⎠ ⎝ Dr := r  b1,β1 · · · bk,βk W1,β1 · · · Wk,βk Wk,βk · · · W1,β1  . (β1 ,...,βk )∈q  q=0 ⎛

∞ 

We also define  ⎞1/2      0 2q ∗ ∗  ⎠ . Dr := ⎝ r  b1,β1 · · · bk,βk W1,β1 · · · Wk,βk Wk,βk · · · W1,β1   (β1 ,...,βk )∈q  q=1 ⎛

∞ 

Theorem 6.8 If r ∈ [0, 1), then 1 ≤ m r (F ∞ (g)) ≤ Nr . where Nr := min {Cr , Dr }, and m r (F0∞ (g)) ≤ Q r ,   where Q r := min Er , Dr0 . Proof The proof is similar to that of Theorem 6.5.

 

For the rest of this section, we assume that g := (g1 , . . . , gk ) is a k-tuple of free holomorphic functions such that sup

α∈F+ ni

bi,α < ∞, bi,αgi

j ∈ {1, . . . , n i },

j

and the associated universal operator model W := (W1 , . . . , Wk ) with Wi := (Wi,1 , . . . , Wi,n i ) is radially pure. Theorem 6.9 Let g := (g1 , . . . , gk ) be a k-tuple of free holomorphic functions such that sup

α∈F+ ni

bi,α < ∞, bi,αgi

j ∈ {1, . . . , n i },

j

and let W := (W1 , . . . , Wk ) with Wi := (Wi,1 , . . . , Wi,n i ) be the universal operator model of the noncommutative polydomain Dg−1 . If W is radially pure, then the homogeneous Bohr radius for the noncommutative Hardy algebra F ∞ (g) satisfies the inequality 1 ≤ K h (F ∞ (g)). 3 Reprinted from the journal

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G. Popescu

In particular, if there is there is s ∈ {1, . . . , k}, such that gs−1 = 1 − where as, j > 0, then K h (F ∞ (g)) =

n s

j=1 as, j Z s, j ,

1 . 3

Proof Let ϕ(W) ∈ F ∞ (g) have the Fourier representation



cβ1 ,...,βk W1,β1 βi ∈F+ ni i∈{1,...,k}

· · · Wk,βk and assume that ϕ(W) =1. Since W is radially pure, we have zW ∈  Dg−1 ( ks=1 F 2 (Hn s ) for any z ∈ D, and, due to Theorem 4.7, we have        q r  cβ1 ,...,βk W1,β1 · · · Wk,βk   < ∞, (β1 ,...,βk )∈q  q=0 ∞ 

r ∈ [0, 1),

and supz∈D ϕ(zW) ≤ ϕ(W) = 1 for any z ∈ D. This shows that g(z) := ϕ(zW) =

∞  q=0

⎛ ⎝



⎞ cβ1 ,...,βk W1,β1 · · · Wk,βk ⎠ z q ,

z ∈ D,

(β1 ,...,βk )∈q

is an operator-valued analytic function on D with g ∞ ≤ 1 and g(0) = c0 I . Applying Theorem 2.9 from [87], when k = 1 and g −1 = Z , and taking into account that 0 ≤ |c0 | ≤ 1, we obtain     ∞     r 2  ≤1 rq  c W · · · W β1 ,...,βk 1,β1 k,βk  ≤ |c0 | + (1 − |c0 | )  1 − r   q=0

(β1 ,...,βk )∈q

for any r ∈ [0, 13 ]. This implies that 13 ≤ K h (F ∞ (g)). To prove the secondpart of the theorem, assume that there is there is s ∈ {1, . . . , k}, s as, j Z s, j , where as, j > 0. Note that the map X → " X := such that gs−1 = 1 − nj=1 I ⊗ · · · ⊗ I ⊗ X ⊗ I ⊗ · · · ⊗ I , where X is on the s th position, is a a complete isometry from F ∞ (gs ) to F ∞ (g), and consequently K h (F ∞ (g)) ≤ K h (F ∞ (gs )). On the other hand, according to Proposition 3.9 from [88], F ∞ (gs ) is completely isometric isomorphic with F ∞ (ψ), where ψ −1 = 1 − (Z 1 + · · · + Z n s ), via the mapping Wi,α → Sα , α ∈ F+ n s , where S1 , . . . , Sn s are the left creation operators on the full Fock space F 2 (Hn s ). Consequently, K h (F ∞ (gs )) = K h (F ∞ (ψ)). According to [64], we have K h (F ∞ (ψ)) = 13 . Combining the results above, we   conclude that K h (F ∞ (g)) = 13 . The proof is complete. 696

Reprinted from the journal

Operator Theory on Noncommutative Polydomains, I

Corollary 6.10 Under the hypothesis of Theorem 6.9, we have 1 ≤ K h (F0∞ (g)). 2 In particular, if there is there is s ∈ {1, . . . , k}, such that gs−1 = 1 − where as, j > 0, then

n s

j=1 as, j Z s, j ,

1 K h (F0∞ (g)) ≤ √ . 2 Proof As in the proof of Theorem 6.9, one can show that     ∞     r  ≤1 rq  c W · · · W β1 ,...,βk 1,β1 k,βk  ≤  1 − r (β1 ,...,βk )∈q  q=1 for any r ∈ [0, 21 ]. This implies that 21 ≤ K h (F0∞ (g)). Now, assume that there is there  s is s ∈ {1, . . . , k}, such that gs−1 = 1 − nj=1 as, j Z s, j , where as, j > 0. As in the proof of Theorem 6.9, one can use the results from [87, 88], and [64] to show that 1 K h (F0∞ (g)) ≤ K h (F0∞ (gs )) = K h (F0∞ (ψ)) ≤ √ . 2  

This completes the proof.

7 The Algebra Hol(Egrad −1 ) of Free Holomorphic Functions and Weierstrass, Montel, Vitali Theorems We provide basic properties for the algebra H ol(Egrad −1 ) of all free holomorphic funcpur e

tions on the radial envelope Egrad −1 )(H) associated with the closed polyball Dg−1 (H). We provide Cauchy type inequalities for the coefficients of free holomorphic functions on radial envelopes and an analogue of Weierstrass’ theorem. This is used to obtain noncommutative versions of Montel’s and Vitali’s theorems from complex analysis in our setting. Let g := (g1 , . . . , gk ) be an admissible k-tuple of free holomorphic functions and let W be the universal operator model of the noncommutative polydomain Dg−1 . We pur e

introduce the radial envelope of Dg−1 (H) by setting Egrad −1 (H) :=

 0≤r γ , then there is a Hilbert space H and pur e Y ∈ sDg−1 (H) such that the series 



cα1 ,...,αk Y1,α1 · · · Yk,αk

( p1 ,..., pk )∈Zk+ ) αi ∈F+ n i ,|αi |= pi i∈{1,...,k}

is divergent in the operator norm topology. Proof Let γ ∈ (0, ∞) and 0 ≤ r < ρ < γ . This implies   1   p1 +···+ pk      1   cα1 ,...,αk W1,α1 · · · Wk,αk  <    ρ αi ∈F+ni ,|αi |= pi    i∈{1,...,k}

for all but finitely many k-tuples ( p1 , . . . , pk ) ∈ Zk+ . Theorem 3.4 shows that, for any pur e

X ∈ r Dg−1 (H), 700

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Operator Theory on Noncommutative Polydomains, I

           cα1 ,...,αk X 1,α1 · · · X k,αk     αi ∈F+ni ,|αi |= pi    i∈{1,...,k}            r p1 +···+ pk   ≤ cα1 ,...,αk r p1 +···+ pk W1,α1 · · · Wk,αk  ≤   ρ αi ∈F+ni ,|αi |= pi    i∈{1,...,k}

for all but finitely many k-tuples ( p1 , . . . , pk ) ∈ Zk+ . Therefore, the series             cα1 ,...,αk X 1,α1 · · · X k,αk      ( p1 ,..., pk )∈Zk+ αi ∈F+ n i ,|αi |= pi   i∈{1,...,k}

pur e

is convergent for any X ∈ γ Egrad −1 (H) and uniform convergent on r Dg−1 (H). One can similarly prove the result when γ = ∞. To prove the last part of the theorem, assume that γ ∈ [0, ∞) and γ < ρ < s. Set Y := sW and note that, since ρ1 < γ1 , there are infinitely many k-tuples ( p1 , . . . , pk ) ∈ Zk+ such that

           cα1 ,...,αk Y1,α1 · · · Yk,αk      αi ∈F+ni ,|αi |= pi   i∈{1,...,k}            s p1 +···+ pk   = s p1 +···+ pk  cα1 ,...,αk W1,α1 · · · Wk,αk  > .   ρ αi ∈F+ni ,|αi |= pi    i∈{1,...,k}

Consequently, the series             cα1 ,...,αk Y1,α1 · · · Yk,αk      ( p1 ,..., pk )∈Zk+ ) αi ∈F+ n i ,|αi |= pi   i∈{1,...,k}

 

is divergent. The proof is complete.

Reprinted from the journal

701

G. Popescu

Theorem 7.3 Let ϕ =



cβ1 ,...,βk Z 1,β1 βi ∈F+ ni i∈{1,...,k}

· · · Z k,βk be a formal power series

and let γ ∈ [0, ∞] be the polydomain radius of convergence. Then the following statements hold. (i) The series       ∞       cα1 ,...,αk X 1,α1 · · · X k,αk    +  p=0 α∈Fn ,...,αk ∈F+  nk 1   |α1 |+···+|αk |= p

pur e

is convergent for any X ∈ γ Egrad −1 (H) and uniform convergent on r Dg−1 (H) if 0 ≤ r < γ. pur e (ii) If γ ∈ [0, ∞) and s > γ , there is a Hilbert space H and Y ∈ sDg−1 (H) such that the series ∞ 



cα1 ,...,αk Y1,α1 · · · Yk,αk

+ p=0 α∈F+ n 1 ,...,αk ∈Fn k |α1 |+···+|αk |= p

is divergent in the operator norm topology. (iii) The polydomain radius of convergence satisfies relation  1  p      1   = lim sup  cα1 ,...,αk W1,α1 · · · Wk,αk  .  γ p→∞  α∈F+n1 ,...,αk ∈F+nk    |α1 |+···+|αk |= p

Proof Since       ∞      cα1 ,...,αk X 1,α1 · · · X k,αk    +  p=0 α∈Fn ,...,αk ∈F+  nk 1   |α1 |+···+|αk |= p             ≤ cα1 ,...,αk Y1,α1 · · · Yk,αk  ,     ( p1 ,..., pk )∈Zk+ ) αi ∈F+ n i ,|αi |= pi   i∈{1,...,k}

702

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Theorem 7.2 implies item (i). To prove item (ii), it is enough to show that, if s > γ , then ϕs (W) :=

∞ 



cα1 ,...,αk s | p1 |+···+ pk W1,α1 · · · Wk,αk

+ p=0 α∈F+ n 1 ,...,αk ∈Fn k |α1 |+···+|αk |= p

is divergent in the operator norm topology. Assume, by contradiction, that the series above is convergent. Then ϕs (W) ∈ F ∞ (g) and, for any r ∈ [0, 1), ∞  p=0



rp

( p1 ,..., pk )∈Zk+ ) p1 +···+ pk = p

           |α1 |+···+|αk | cα1 ,...,αk s W1,α1 · · · Wk,αk     αi ∈F+ni ,|αi |= pi    i∈{1,...,k}   ∞ ∞    p p p+k−1 ≤ r

ϕs (W) ≤ r

ϕs (W) < ∞ k−1 p=0

( p1 ,..., pk )∈Zk+ ) p1 +···+ pk = p

p=0

for any r ∈ [0, 1). Hence, we deduce that       ∞        cα1 ,...,αk X 1,α1 · · · X k,αk  < ∞    p=0 ( p1 ,..., pk )∈Zk ) αi ∈F+  n i ,|αi |= pi +   p1 +···+ pk = p i∈{1,...,k}

for any X ∈ ρDgrad −1 (H), where ρ ∈ (γ , s), which contradicts the result of Theorem 7.2. Therefore, item (ii) holds. Now, item (iii) is due to the results above and Theorem 7.2. This completes the proof.   The next result provides Cauchy type inequalities for the coefficients of free holo+ morphic functions on the radial envelope Egrad −1 . If α, β, γ ∈ Fn i are such that α = βγ , + we set α\β := γ . If α = βγ for any γ ∈ Fn i , we set bi,α\β := 0 Theorem 7.4 Let F be a free holomorphic function on the radial envelope Egrad −1 with representation F(X ) =





( p1 ,..., pk )∈Zk+ ) αi ∈F+ n i ,|αi |= pi i∈{1,...,k}

Reprinted from the journal

703

cα1 ,...,αk X 1,α1

G. Popescu

· · · X k,αk ,

X ∈ Egrad −1 (H).

If 0 < r < 1 and M(r ) := F(r W) , then   |cα1 ,...,αk | W1,α1 · · · Wk,αk           1   ≤ cα1 ,...,αk W1,α1 · · · Wk,αk  ≤ p +···+ p M(r ), 1 k   r  αi ∈F+ni ,|αi |= pi   i∈{1,...,k}

for any αi ∈ F+ n i such that |αi | = pi , and k    W1,α · · · Wk,α  = sup 1 k

+ i=1 α∈Fni

6 bi,α\β . bi,α

Proof Due to Theorem 7.2, we deduce that F(r W) ∈ F ∞ (g) for any r ∈ (0, 1). Applying Theorem 4.5, we obtain            p1 +···+ pk cα1 ,...,αk r W1,α1 · · · Wk,αk  ≤ M(r ).     αi ∈F+ni ,|αi |= pi  

(7.1)

i∈{1,...,k}

  Since the set W1,α1 · · · Wk,αk : αi ∈ F+ n i , |αi | = pi , i ∈ {1, . . . , k} consists of operators with orthogonal ranges, it is easy to see that            cα1 ,...,αk W1,α1 · · · Wk,αk     αi ∈F+ni ,|αi |= pi    i∈{1,...,k}   = max |cα1 ,...,αk | W1,α1 · · · Wk,αk  . αi ∈F+ n i ,|αi |= pi i∈{1,...,k}

Using now relation (7.1), we deduce the first part of the theorem. To prove the last part, note that ∗ i Wi,β Wi,β eα

/b =

i,γ i bi,α eα

if α = βγ

0

otherwise

704

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Operator Theory on Noncommutative Polydomains, I ∗ for every i ∈ {1, . . . , k} and α, β ∈ F+ n i . Since Wi,β Wi,β is a diagonal oper   bi,α\β W1,α · · · Wk,α  = ator, it is clear that Wi,β = supα∈F+n . Since 1 k b i,α i     W1,α  · · · Wk,α , we can complete the proof.   1

k

Here is an analogue of Weierstrass theorem for free holomorphic functions on radial envelopes. Theorem 7.5 Let {Fm }∞ m=1 be a sequence of free holomorphic functions on the radial envelope Egrad with the property that, for each r ∈ [0, 1), Fm converges uniformly on −1 pur e

r Dg−1 (H). Then there is F ∈ H ol(Egrad −1 ) such that Fm converges to F uniformly on pur e

r Dg−1 (H) for any r ∈ [0, 1). Proof According to Theorem 7.1, we have

Fm (X ) − Fn (X ) = Fm (r W) − Fn (r W) ,

sup X ∈r D

pur e (H ) g−1

where the supremum is taken over all Hilbert spaces. Since Fm converges uniformly pur e on r Dg−1 (H), we deduce that Fm (r W) converges in the operator norm topology to

an operator F (r ) (W) in the noncommutative polydomain algebra A(g) ⊂ F ∞ (g). Assume that Fm and F (r ) have the Fourier representations 

Fm =



( p1 ,..., pk )∈Zk+ ) αi ∈F+ n i ,|αi |= pi i∈{1,...,k}

cα(m) Z · · · Z k,αk 1 ,...,αk 1,α1

and 

F (r ) =



( p1 ,..., pk )∈Zk+ ) αi ∈F+ n i ,|αi |= pi

cα(r1),...,αk Z 1,α1 · · · Z k,αk ,

i∈{1,...,k}

respectively. Consequently, Fm (r W) − F (r ) (W) =





( p1 ,..., pk )∈Zk+ ) αi ∈F+ n i ,|αi |= pi i∈{1,...,k}

  (r ) r p1 +···+ pk cα(m) − c α1 ,...,αk W1,α1 · · · Wk,αk 1 ,...,αk

Reprinted from the journal

705

G. Popescu

which, due to Theorem 4.5, implies              p1 +···+ pk (m) (r ) r cα1 ,...,αk − cα1 ,...,αk W1,α1 · · · Wk,αk      αi ∈F+ni ,|αi |= pi   i∈{1,...,k}

≤ Fm (r W) − F (r ) (W) .

(7.2)

  Consequently, since the set W1,α1 · · · Wk,αk : αi ∈ F+ n i , |αi | = pi , i ∈ {1, . . . , k} consists of operators with orthogonal ranges, we obtain − cα(r1),...,αk | W1,α1 · · · Wk,αk ≤ Fm (r W) − F (r ) (W) . |r p1 +···+ pk cα(m) 1 ,...,αk (m)

(r )

Taking m → ∞, we conclude that cα1 ,...,αk := limm→∞ cα1 ,...,αk exists and cα1 ,...,αk = r p1 +···+ pk cα1 ,...,αk . We consider the formal power series F :=





cα1 ,...,αk Z 1,α1 · · · Z k,αk

( p1 ,..., pk )∈Zk+ ) αi ∈F+ n i ,|αi |= pi i∈{1,...,k}

and prove that F ∈ H ol(Egrad −1 ). Due to relation (7.2), we have

r p1 +···+ pk

           (m) cα1 ,...,αk W1,α1 · · · Wk,αk − cα1 ,...,αk W1,α1 · · · Wk,αk     αi ∈F+ni ,|αi |= pi    i∈{1,...,k}

≤ Fm (r W) − F (r ) (W)

which implies r p1 +···+ pk



cα(m) W1,α1 1 ,...,αk

αi ∈F+ n i ,|αi |= pi i∈{1,...,k} · · · Wk,αk → r p1 +···+ pk cα1 ,...,αk W1,α1

· · · Wk,αk ,

(7.3)

as m → ∞, uniformly with respect to p1 , . . . , pk ∈ Z+ . Assume that   1   p1 +···+ pk        lim sup  cα1 ,...,αk W1,α1 · · · Wk,αk  > γ > 1.   k ( p1 ,..., pk )∈Z+ α ∈F+ ,|α |= p   i ni i i  i∈{1,...,k}

706

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Then we have

r p1 +···+ pk

           cα1 ,...,αk W1,α1 · · · Wk,αk  > (γ r ) p1 +···+ pk    αi ∈F+ni ,|αi |= pi    i∈{1,...,k}

for infinitely many k-tuples ( p1 , . . . , pk ) ∈ Zk+ . Due to relation (7.3), there is K ∈ N such that           p1 +···+ pk  (m) cα1 ,...,αk W1,α1 · · · Wk,αk  > (γ r ) p1 +···+ pk r    αi ∈F+ni ,|αi |= pi    i∈{1,...,k}

for any m ≥ K and infinitely many k-tuples ( p1 , . . . , pk ) ∈ Zk+ . Hence, we deduce that   1   p1 +···+ pk        lim sup  cα(m) W · · · W ≥ γ > 1.  1,α k,α 1 ,...,α k 1 k   ( p1 ,..., pk )∈Zk+ α ∈F+ ,|α |= p  i i i ni   i∈{1,...,k}

Due to Theorem 7.2, we conclude that Fm ∈ / H ol(Egrad −1 ), which is a contradiction. Therefore, F ∈ H ol(Egrad −1 ) and, consequently, 

F(r W) =



r p1 +··· pk cα1 ,...,αk W1,α1 · · · Wk,αk

( p1 ,..., pk )∈Zk+ ) αi ∈F+ n i ,|αi |= pi i∈{1,...,k}

is convergent in the operator norm. Now, it clear that F(r W) = F (r ) (W) and Fm (r W) converges in the operator norm topology to an operator F(r W) for every r ∈ (0, 1). This completes the proof.   Definition 7.6 We say that a set N ⊂ H ol(Egrad −1 ) is normal if each sequence ⊂ N has a subsequence {F } which converges to an element F ∈ N {Fm }∞ m k m=1 pur e

uniformly on r Dg−1 (H) for any r ∈ [0, 1). A set G ⊂ H ol(Egrad −1 ) is called locally bounded if, for each r ∈ [0, 1), there exists Mr > 0 such that sup X ∈r D

Reprinted from the journal

F(X ) ≤ Mr ,

pur e (H ) g−1

707

for any f ∈ G.

G. Popescu

The following result is a noncommutative version of Montel’s theorem from complex analysis. Theorem 7.7 A subset G ⊂ H ol(Egrad −1 ) is locally bounded if and only if it is a normal set. Proof Assume that G is locally bounded. Then, for each r ∈ [0, 1), there is Mr > 0 such that F(r W) ≤ Mr for any F ∈ G. Let {Fm }∞ m=1 ⊂ G have the representations 



Fm =

( p1 ,..., pk )∈Zk+ ) αi ∈F+ n i ,|αi |= pi i∈{1,...,k}

cα(m) Z · · · Z k,αk . 1 ,...,αk 1,α1

Note that |cg1 ,...gk | = |Fm (0)| ≤ M0 and, according to Theorem 7.4, we have 0

0

|cα(m) |≤ 1 ,...,αk

Mr 1 r p1 +···+ pk W1,α1 · · · Wk,αk

for any r ∈ [0, 1), m ∈ N, and αi ∈ F+ n i with |αi | = pi . Applying Bolzano-Weierstrass theorem, a standard inductive argument and the diagonal process proves the exis(m) ∞ ∞ tence of a subsequence {Fm s }∞ s=1 of {Fm }m=1 such that {cα1 ,...,αk }s=1 is a convergent + + sequence in C, for each (α1 , . . . , αk ) ∈ Fn 1 × · · · × Fn k . In what follows, we prove that, for any r , t ∈ [0, 1), the sequence {Fm s (r tW)}∞ s=1 is convergent in the operator norm topology. Applying Theorem 7.4, we deduce that

Fm s (r tW) − Fm  (r tW)

≤

K 



(r t) p

( p1 ,..., pk )∈Zk+ p1 +···+ pk = p

p=0

           (m s ) (m  ) (cα1 ,...,αk − cα1 ,...,αk )W1,α1 · · · Wk,αk     αi ∈F+ni ,|αi |= pi    +

i∈{1,...,k} ∞ 

p=K +1

≤



(r t) p

K  p=0

(r t) p

( p1 ,..., pk )∈Zk+ p1 +···+ pk = p

1 r p1 +···+ pk

Fm s (r W − Fm  (r W)

 ( p1 ,..., pk )∈Zk+ p1 +···+ pk = p

         (m s ) (m  ) (cα1 ,...,αk − cα1 ,...,αk )W1,α1 · · · Wk,αk     αi ∈F+ni ,|αi |= pi  708

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+

∞ 

 t

p

p=K +1

 p+k−1 2Mr . k−1 

 p+k−1 ∞ → 0 as K → ∞, and {cα(m) 1 ,...,αk }s=1 k−1 is a convergent sequence in C, it is easy to see that Fm s (r tW) − Fm  (r tW) → 0, as m s → ∞ and m  → ∞, for any r , t ∈ [0, 1). Applying Theorem 7.5, we find F ∈ H ol(Egrad −1 ) with the property that F(r W) = lim s→∞ Fm s (r W) in the operator norm, for any r ∈ [0, 1). Therefore, G is a normal set. Conversely, assume that G is a normal set and that there is r0 ∈ [0, 1) such that sup F∈G F(r0 W) = ∞. Then there is a sequence {Fm } ⊂ G such that Fm (r0 W) → ∞ as m → ∞. Since G is normal, there is F ∈ H ol(Egrad −1 ) with the property that F(r W) = lims→∞ Fm s (r W) in the operator norm, for any r ∈ [0, 1). The later   relation contradicts the fact that Fm (r0 W) → ∞. This completes the proof. Consequently, since

∞

p p=K +1 t

Given F, G ∈ H ol(Egrad −1 ) and r ∈ [0, 1), we define dr (F, G) :=

F(X ) − G(X ) ,

sup X ∈r D

pur e (H ) g−1

where the supremum is taken over all Hilbert spaces H. Let {rm } ⊂ (0, 1) be a sequence such that rm → 1 and define d(F, G) :=

∞  1 drm (F, G) . 2m 1 + drm (F, G)

m=1

Using standard arguments, one can show that d is a metric on H ol(Egrad −1 ).

  Theorem 7.8 H ol(Egrad is a complete metric space. A subset G ⊂ H ol(Egrad −1 ), d −1 ) is compact if and only if G is closed and locally bounded.

rad Proof Let {F p }∞ p=1 ⊂ H ol(Eg−1 ) be a Cauchy sequence with respect to the metric d. Then, for each m ∈ N, {F p (rm W}∞ p=1 is a Cauchy sequence in the operator norm and, consequently, it is convergent. Applying Theorem 7.5, we find F ∈ H ol(Egrad −1 ) such that F p (r W) → F(r W), as p → ∞, for eachr ∈ [0, 1). Now,  it is easy to see that

d(F p , F) → 0 as p → ∞. This proves that H ol(Egrad is a complete metric −1 ), d space. Applying now Theorem 7.7, we complete the proof.   As an application of Theorems 7.7 and 7.5, we can obtain the following analogue of Vitali’s theorem in our noncommutative multivariable setting. Since the proof is very similar to the proof of Theorem 5.3 from [77], we leave it to the reader. Theorem 7.9 Let {Fm }∞ m=1 be a sequence of free holomorphic functions on the radial envelope Egrad with the property that, for each r ∈ [0, 1), {Fm }∞ −1 m=1 is uniformly Reprinted from the journal

709

G. Popescu pur e

bounded on r Dg−1 (H), and there is γ ∈ (0, 1) such that Fm is uniformly convergent pur e

on γ Eg−1 (H). Then there is F ∈ H ol(Dgrad −1 ) such that Fm is uniformly convergent pur e

to F on r Dg−1 (H) for any r ∈ [0, 1).

8 Algebras of Bounded Free Holomorphic Functions on Polydomains and Schwarz Lemma In this section, we assume that g is an admissible k-tuple of free holomorphic functions and the associated universal model W is radially pure. We introduce the algebra rad H ∞ (Egrad −1 ) of all bounded free holomorphic functions on Eg−1 and proved that it is completely isometric isomorphic to noncommutative Hardy algebra F ∞ (g). We also rad introduce the algebra A(Egrad −1 ) of all free holomorphic functions on Eg−1 which have continuous extension to Egrad −1 (H) for any Hilbert space H, and show that it is completely isometric isomorphic to the noncommutative polydomain algebra A(g). A Schwarz type lemma is also provided. Let g := (g1 , . . . , gk ) be an admissible k-tuple of free holomorphic functions such that sup

α∈F+ ni

bi,α < ∞, bi,αgi

j ∈ {1, . . . , n i },

j

and let W := (W1 , . . . , Wk ) with Wi := (Wi,1 , . . . , Wi,n i ) be the universal operator model of the noncommutative polydomain Dg−1 . Throughout this section we assume pur e  that W is radially pure, i.e. r W ∈ Dg−1 ( ks=1 F 2 (Hn i )) for any r ∈ [0, 1). Lemma 8.1 Let W := (W1 , . . . , Wk ) be the universal operator model of the noncompur e mutative polydomain Dg−1 . If W is radially pure and X = (X 1 , . . . , X k ) ∈ Dg−1 (H), pur e

then (z 1 X 1 , . . . , z k X k ) ∈ Dg−1 (H) for any z 1 , . . . , z k ∈ D. Proof Since r W is pure for any r ∈ [0, 1), Corollary 3.3 shows that r Wi is pure with respect to the domain Dg−1 for each i ∈ {1, . . . , k}. Therefore, i

g−1 (r Wi , r Wi∗ ) = i



∗ r 2|αi | ai,αi Wi,αi Wi,α ≥ 0, i

αi ∈F+ ni

where the convergence is in the strong operator topology, and  αi ∈F+ ni

∗ r 2|αi | bi,αi Wi,αi g−1 (r Wi , r Wi∗ )Wi,α =I i i

710

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for any r ∈ [0, 1). Using these relations and setting r = (r1 , . . . , rk ) ∈ [0, 1)k and rW := (r1 W1 , . . . , rk Wk ), we obtain g−1 (rW, rW∗ ) = SOT-



lim

2|β1 |

r1

m 1 →∞,...,m k →∞

2|βk |

· · · rk

a1,β1

+ β1 ∈F+ n 1 ,...,βk ∈Fn k |β1 |≤m 1 ,...,|βk |≤m k ∗ ∗ · · · ak,βk W1,β1 · · · Wk,βk Wk,β · · · W1,β 1 k

=

k 

g−1 (ri Wi , ri Wi∗ ) ≥ 0.

i=1

i

Consequently, we have 

2|α1 |

r1

2|αk |

· · · rk

b1,α1

+ α1 ∈F+ n 1 ,...αk ∈Fn k

∗ ∗ · · · bk,αk W1,α1 · · · Wk,αk g−1 (rW, rW∗ )Wk,α · · · W1,α k ⎛ ⎞ 1 k  ⎜  2|αi | ∗ ⎟ = ri bi,αi Wi,αi g−1 (r Wi , r Wi∗ )Wi,α ⎝ ⎠ = I. i i=1

i

αi ∈F+ ni

pur e

Therefore, rW is pure. Now, assume that X = (X 1 , . . . , X k ) ∈ Dg−1 (H). According to Theorem 3.1, we have ∗ ∗ X i, j = (Wi, j ⊗ ID )|H ,

i ∈ {1, . . . , k}, j ∈ {1, . . . , n i },

where H is identified with a coinvariant subspace for the operators Wi, j ⊗ ID . ∗ = (r W∗ ⊗ I )| and using the fact that rW := Hence, we deduce that ri X i, i i, j D H j (r1 W1 , . . . , rk Wk ) is pure, it is easy to see that (r1 X 1 , . . . , rk X k ) is a pure element in pur e Dg−1 (H). Now, it is clear that (z 1 X 1 , . . . , z k X k ) ∈ Dg−1 (H) for any z 1 , . . . , z k ∈ D. The proof is complete.   pur e

Due to the result above, we say that Dg−1 is a complete Reinhardt set centered at 0. pur e

Recall that the radial envelope of Dg−1 (H) is defined by Egrad −1 (H) :=

 0≤r a and X ∈ / |λ| Eg−1 (H) if 0 < c < a. Consequently, X ∈ |λ| a   m rad (X ) = |λ| . Therefore, item (ii) follows.

Here is an analogue of Schwarz lemma in our noncommutative multivariable setting. Theorem 8.5 Let G : Egrad −1 (H) → B(H) be a bounded free holomorphic function with G ∞ ≤ 1 and G(0) = 0. Then

G(X ) ≤ m rad (X ),

X ∈ Egrad −1 (H).

Moreover, if G has the representation G(X ) = · · · X k,αk then



∞

p=1

|α1 |+···+|αk |= p

           cα1 ,...,αk X 1,...,α1 · · · X k,αk  ≤ m rad (X ),    α1 ∈F+n1 ,...,αk ∈F+nk   

cα1 ,...,αk X 1,...,α1

X ∈ Egrad −1 (H).

|α1 |+···+|αk |=1

Proof Let X ∈ Egrad −1 (H) and fix an arbitrary t ∈ (0, 1) such that m rad (X ) < t < 1. Due to Lemma 8.4, we deduce that m rad ( 1t X ) < 1 which implies

1 t X

∈ Egrad −1 (H).

z rad rad Since zEgrad −1 (H) ⊂ Eg−1 (H) for any z ∈ D, we have t X ∈ Eg−1 (H) for any z ∈ D. For every u, v ∈ H with u ≤ 1 and v ≤ 1, we define the map ϕu,v : D → C by setting

ϕu,v (z) :=

∞ 



cα1 ,...,αk

+ p=1 α1 ∈F+ n 1 ,...,αk ∈Fn k |α1 |+···+|αk |= p

 z |α1 |+···+|αk | ; t

= X 1,...,α1 · · · X k,αk u, v .

Since G is a bounded free holomorphic function on Egrad −1 (H) with G ∞ ≤ 1, it is clear that ϕu,v is an analytic function on the open unit disc D and |ϕu,v (z)| ≤ 1. Since ϕu,v (0) = 0, we can apply the classical Schwarz lemma and obtain |ϕu,v (z)| ≤ |z| for any z ∈ D. Since m rad (X ) < 1, we set z = m rad (X ) in the later inequality and obtain .  *  . m rad (X ) X u, v .. ≤ m rad (X ) G t 716

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Operator Theory on Noncommutative Polydomains, I

for any t ∈ (0, 1) such that m rad (X ) < t < 1. Since G is a continuous function on Egrad −1 (H) (see Theorem 7.1), and taking t → m rad (X ), we deduce that |G(X )u, v | ≤ m rad (X ) for any X ∈ Egrad −1 (H) and any u, v ∈ H with u ≤ 1 and v ≤ 1. Consequently,

G(X ) ≤ m rad (X ),

X ∈ Egrad −1 (H).

 (0)| ≤ 1. On the other hand, due to the classical Schwarz lemma, we also have |ϕu,v Using the definition of ϕu,v , we deduce that



 (0) = ϕu,v

cα1 ,...,αk

+ α1 ∈F+ n 1 ,...,αk ∈Fn k |α1 |+···+|αk |=1

  = 1 ; X 1,...,α1 · · · X k,αk u, v t

for any u, v ∈ H with u ≤ 1 and v ≤ 1. Consequently,            cα1 ,...,αk X 1,...,α1 · · · X k,αk  ≤ t.    α1 ∈F+n1 ,...,αk ∈F+nk    |α1 |+···+|αk |=1

Taking t → m rad (X ), we complete the proof.

 

The proof of the following result is similar to that of the theorem above. p Remark 8.6 If G : Egrad −1 (H) → B(H) , p ∈ N, is a bounded free holomorphic function with G ∞ ≤ 1 and G(0) = 0, then

G(X ) ≤ m rad (X ),

X ∈ Egrad −1 (H).

Declarations Conflict of interest The author states that there is no conflict of interest. No datasets were generated or analyzed during the current study.

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Operator Theory on Noncommutative Polydomains, I 95. Salomon, G., Shalit, O.M., Shamovich, E.: Algebras of noncommutative functions on subvarieties of the noncommutative ball: the bounded and completely bounded isomorphism problem. J. Funct. Anal. 278(7), 108427 (2020) 96. Sarkar, J.: An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces-I. J. Oper. Theory 73, 433–441 (2015) 97. Sarkar, J.: An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces-II. Complex Anal. Oper. Theory 10(4), 769–782 (2016) 98. Sidon, S.: Uber einen Satz von Herrn Bohr. Math. Z. 26, 731–732 (1927) 99. B. Sz.-Nagy, Foia¸s, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space, Second edition. Revised and enlarged edition. pp. xiv+474 Universitext. Springer, New York (2010) 100. Taylor, J.L.: A general framework for a multi-operator functional calculus. Adv. Math. 9, 183–252 (1972) 101. Taylor, J.L.: Functions of several noncommuting variables. Bull. Am. Math. Soc. 79, 134 (1973) 102. Timotin, D.: Regular dilations and models for multicontractions. Indiana Univ. Math. J. 47(2), 671– 684 (1998) 103. Tomic, M.: Sur un theoreme de H. Bohr. Math. Scand. 1, 103–106 (1962) 104. Vasilescu, F.-H.: An operator-valued Poisson kernel. J. Funct. Anal. 110(1), 47–72 (1992) 105. Voiculescu, D.V.: Free analysis questions I. Duality transform for the coalgebra of ∂ X : B . Int. Math. Res. Not. 2004(16), 793–822 106. Voiculescu, D.V.: Free analysis questions II: The Grassmannian completion and the series expansions at the origin. J. Reine Angew. Math. 645, 155–236 (2010) 107. von Neumann, J.: Eine Spectraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr. 4, 258–281 (1951) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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(2022) 16:75 Complex Analysis and Operator Theory https://doi.org/10.1007/s11785-022-01258-z

Complex Analysis and Operator Theory

Radial-like Toeplitz Operators on Cartan Domains of Type I Raúl Quiroga-Barranco1 Received: 30 April 2022 / Accepted: 17 June 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract Let DIn×n be the Cartan domain of type I which consists of the complex n × n matrices Z that satisfy Z ∗ Z < In . For a symbol a ∈ L ∞ (DIn×n ) we consider three radiallike type conditions: 1) left (right) U(n)-invariant symbols, which can be defined   1 1 by the condition a(Z ) = a (Z ∗ Z ) 2 (a(Z ) = a (Z Z ∗ ) 2 , respectively), and 2) U(n) × U(n)-invariant symbols, which are defined by the condition a(A−1 Z B) = a(Z ) for every A, B ∈ U(n). We prove that, for n ≥ 2, these yield different sets of symbols. If a satisfies 1), either left or right, and b satisfies 2), then we prove that the corresponding Toeplitz operators Ta and Tb commute on every weighted Bergman space. Furthermore, among those satisfying condition 1), either left or right, there exist, for n ≥ 2, symbols a whose corresponding Toeplitz operators Ta are non-normal. We use these facts to prove the existence, for n ≥ 2, of commutative Banach non-C ∗ algebras generated by Toeplitz operators. Keywords Toeplitz operators · Representation theory · Radial symbols Mathematics Subject Classification Primary 47B35; Secondary 22D10

1 Introduction The theory of Toeplitz operators acting on Bergman spaces has proven to be a very interesting and active line of research. A very general, and at the same time accessible, setup is given by choosing a circular bounded symmetric domain D where we can

Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

B 1

Raúl Quiroga-Barranco [email protected] Centro de Investigación en Matemáticas, Guanajuato, México

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consider Bergman spaces A2λ (D) and Toeplitz operators acting on them. This can be done for every λ > p − 1, where p is the genus of D (see [33]). An approach to the study of Toeplitz operators is achieved by considering the algebras that they generate when the symbols are restricted to some particular sets. For example, for some subset S ⊂ L ∞ (D), one studies the features of the unital Banach algebra generated by the Toeplitz operators, acting on A2λ (D), whose symbols belong to S. From now on, we will denote by T (λ) (S) such unital Banach algebra. It turns out that, in many cases, the right choice of set of symbols S yields commutative algebras, or at least some setup where studying the commutativity of Toeplitz operators is quite interesting. We refer to [1–3, 6, 8, 9, 23, 29, 36] for just a few examples of this fact. For a circular bounded symmetric domain D, the group of biholomorphism G acts transitively and provides a tool to select special sets of symbols. More precisely, for any closed subgroup H of G, we can consider the set of essentially bounded symbols that are H -invariant. We will denote by L ∞ (D) H the space of such H -invariant symbols. This has allowed to prove the existence of a variety of commutative C ∗ -algebras of the form T (λ) (L ∞ (D) H ). Such algebras have shown to be very interesting and complicated in some cases. We refer to [10, 11, 27, 28] for some general constructions of commutative C ∗ -algebras obtained with these techniques for the n-dimensional unit ball Bn as well as general irreducible bounded symmetric domains. Although the general setup of arbitrary bounded symmetric domains is quite important, the case of radial symbols on the unit disk D can still be considered as particularly interesting. We recall that a ∈ L ∞ (D) is called radial if and only if a(z) = a(|z|), for every z ∈ D. Besides the references mentioned above, we can also refer to [5, 13, 15, 17, 22, 25, 26, 30, 34, 35] for examples of research on Toeplitz operators with radial symbols and some generalizations to the unit ball Bn . From these references we would like to highlight [22] where it was discovered the importance of Toeplitz operators with radial symbols. It was first proved there that for any radial symbol a on the unit disk D, the Toeplitz operator Ta acting on the (weightless) Bergman space A2 (D) can be diagonalized with respect to the natural monomial basis. In other words, such Toeplitz operator Ta with radial symbol a preserves the Hilbert direct sum A2 (D) =



Cz n ,

(1.1)

n∈N

thus providing its diagonal form. In particular, any Toeplitz operator with radial symbol is normal. Furthermore, it follows that the C ∗ -algebra generated by Toeplitz operators with radial symbols is commutative (see [16, 18]). We note that a symbol a ∈ L ∞ (D) is radial if and only if a(t z) = a(z) for every t ∈ T and z ∈ D. In other words, the space of radial symbols on the unit disk is given by L ∞ (D)T , for the T-action on D by rotations around the origin. This action realizes the isotropy at the origin of the biholomorphisms of D. A natural problem is to generalize to higher dimensions the notion of radial symbols and to study the features of the corresponding Toeplitz operators. This has already been considered for the n-dimensional unit ball Bn . In particular, radial symbols were studied in [15], separately radial symbols (also known as quasi-elliptic symbols) have

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Radial-like Toeplitz Operators on Cartan…

been studied in [25, 27], and in other references several variations of these have been considered, for example, [13, 26, 34, 35]. In all these previous works, the corresponding Toeplitz operators have also been studied and used to construct commutative Banach and C ∗ -algebras. Our main goal here is to consider an alternative generalization, to higher dimensions, of the notion of radial symbol from the unit disk D. More specifically, for the Cartan domains of type I we introduce three possible generalizations of the notion of radial symbols in D. We study the corresponding Toeplitz operators and obtain some commutative Banach algebras that they generate. We recall that, for every n ≥ 1, the Cartan domain DIn×n of type I is the domain of matrices Z ∈ Mn×n (C) that satisfy Z ∗ Z < In . We note that this defining condition is equivalent to Z Z ∗ < In . Also observe that for n = 1 we recover the unit disk D. Given the obvious similarity in the definitions of D and DIn×n , one might guess that a natural generalization of a radial symbol a for DIn×n can be given by the condition  1 a(Z ) = a (Z ∗ Z ) 2

(1.2)

for every Z ∈ DIn×n . Alternatively, we can also consider the condition  1 a(Z ) = a (Z Z ∗ ) 2

(1.3)

for every Z ∈ DIn×n . On the other hand, since the radial symbols in D are those invariant by the biholomorphism subgroup fixing the origin, we can consider the corresponding condition for DIn×n . As noted in Sect. 2.1, the biholomorphisms fixing the origin in DIn×n are realized by the linear action of U(n) × U(n) given by (A, B) · Z = AZ B −1 where A, B ∈ U(n) and Z ∈ DIn×n . Hence, we can consider a third condition for a symbol a in DIn×n , which is a(A−1 Z B) = a(Z )

(1.4)

for every A, B ∈ U(n) and Z ∈ DIn×n . In some sense, we can consider these three conditions as yielding radial-like symbols on the domain DIn×n . Note that, for n = 1, these three conditions are clearly equivalent and correspond to the actual radial symbols on D. However, it turns out that, for n ≥ 2, the three conditions (1.2), (1.3) and (1.4) are non-equivalent by pairs. To prove this, we introduce in Proposition 3.1 and Corollary 3.3 a sort of polar coordinates similar (but not the same) to those considered in Section 3.4 from [19] (see Remark 3.4). Then, Proposition 3.5 provides a polar coordinates description of the U(n) × U(n)-invariant symbols, i.e. those satisfying (1.4). On the other hand, we prove in Proposition 3.8 and Corollary 3.11 (see also Remark 3.10) that conditions (1.2) and (1.3) are equivalent to requiring the symbol a to be invariant under the action of the factors U(n) L = U(n) × {In } and U(n) R = {In } × U(n), respectively. For this reason, we call the symbols satisfying (1.2) left U(n)-invariant,

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and we call those satisfying (1.3) right U(n)-invariant (see Sect. 3.2). We also provide a polar coordinates description of both of them in Proposition 3.12. These results allow us to show that, for n ≥ 2, all three of the above conditions on a symbol a are different (see Remarks 3.13 and 3.14). This will actually bring some richness that allows to construct special Toeplitz operators and Banach algebras. Hence, our next step is to study the Toeplitz operators whose symbols satisfy one of the three conditions mentioned above. Again, we take as model to generalize the case of the unit disk D. More precisely, we proceed to generalize the Hilbert direct sum decomposition (1.1). For this, we note that the latter is precisely the decomposition into irreducible subspaces for a natural T-action on the Bergman space A2 (D). We consider in Sect. 2.3 a natural unitary representation πλ of U(n) × U(n) on the weighted Bergman space A2λ (DIn×n ). Then, we obtain in Theorem 4.10 a Hilbert direct sum which is U(n) × U(n)-invariant and that reduces to (1.1) for n = 1 in the weightless case. There is a corresponding result for the actions of U(n) L and of U(n) R which is obtained in Theorem 4.13. In all these cases we provide a precise description of the terms in the Hilbert direct sums as modules over the corresponding groups. This turns out to be fundamental since for all three cases the Hilbert direct sum is exactly the same, and can only be distinguished by the unitary actions on the terms of the corresponding groups. In fact, it is the notion of isotypic decomposition from representation theory that allows us to distinguish between the three actions (see Sect. 4 for further details and definitions). Furthermore, we prove in Proposition 4.2 a criterion for a C ∗ -algebra of the form (λ) T (L ∞ (DIn×n ) H ), where H ⊂ U(n) × U(n) is a closed subgroup, to be commutative. This result is applied by testing whether or not an isotypic decomposition is multiplicity-free (see Sect. 4 for definitions and further details). This allows us to prove in Theorem 4.11 that the C ∗ -algebra T (λ) (L ∞ (DIn×n )U(n)×U(n) ) is commutative. Such result was already obtained in [10, 11]. However, as noted in Remark 4.12 our proof provides more information that can be used to find explicit diagonal forms of the corresponding Toeplitz operators. In contrast with the case of U(n) × U(n)-invariance, the left and right U(n)invariant symbols yield non-commutative C ∗ -algebras. This is proved in Theorem 4.14, where we show that T (λ) (L ∞ (DIn×n )U(n) L ) and T (λ) (L ∞ (DIn×n )U(n) R ) are not commutative. Nevertheless, we are able to prove in Theorem 4.19 that these algebras centralize each other (see also Corollary 4.20). This is a very interesting phenomenon: conditions (1.2) and (1.3), which generalize radial symbols on D, yield symbols whose Toeplitz operators respectively generate two non-commutative (for n ≥ 2) C ∗ -algebras that centralize each other. In particular, some degree of commutativity is still present for n ≥ 2. These results are possible due to the detailed information provided by the isotypic decompositions obtained for the three groups considered. It also allow us to prove in Theorem 4.15, for n ≥ 2, the existence of symbols a satisfying either (1.2) or (1.3) (λ) for which the corresponding Toeplitz operators Ta are not normal. This is again in contrast with the behavior observed in the case of the unit disk, where every radial symbol always yields normal Toeplitz operators.

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This variety of symbols whose Toeplitz operators are non-normal lead to the existence of some interesting Banach algebras generated by Toeplitz operators. The main results in this regard are Theorems 4.17 and 4.22, and Corollary 4.21 (see also Remark 4.23). In all cases by choosing suitable left and right U(n)-invariant symbols, one obtains, in the case n ≥ 2, Banach non-C ∗ algebras generated by Toeplitz operators. The main tool from representation theory that we used is based on the relationship that exist between the representations of U(n) and GL(n, C). Such relationship comes from the fact that the latter is the complexification of the former (see Lemma 4.3 and the remarks that follow). From this, a detailed analysis of the well known representation of GL(n, C) × GL(n, C) on polynomials over Mn×n (C), reviewed in Sect. 4, provides the means to compute the required isotypic decompositions of Bergman spaces. The author would like to respectfully dedicate this work to the memory of Jörg Eschmeier, with whom he had the fortune to share an interest on Toeplitz operators acting on higher dimensional domains.

2 Cartan Domains of Type I As observed in the introduction, we will denote by DIn×n the domain of complex n × n matrices Z that satisfy Z ∗ Z < In . We recall that this condition is equivalent to Z Z ∗ < In (see [19]). It is well known that DIn×n is an irreducible circled bounded symmetric domain. The most elementary case is given by DI1×1 = D, the unit disk in the complex plane C. The domain DIn×n is clearly n 2 -dimensional. Furthermore, this domain has rank n, genus 2n and its characteristic multiplicities are a = 2 and b = 0. In particular, it has an unbounded tube-type realization. We refer to [33] for the definitions and proofs of these claims. Let us consider the pseudo-unitary group of matrices M ∈ GL(2n, C) that satisfy the condition M  In,n M = In,n where In,n is the block diagonal 2n × 2n matrix In,n = diag(In , −In ). The group of such matrices M will be denoted by U(n, n), and it yields the linear isometries of the Hermitian form on C2n defined by the matrix In,n . 2.1 The Biholomorphism Group of DIn×n It is well known that the biholomorphism group of DIn×n is realized by the following action of U(n, n) U(n, n) × DIn×n → DIn×n   A B · Z = (AZ + B)(C Z + D)−1 , C D where A, B, C, D have size n × n. We refer to [20, 24] for the details of this fact. Furthermore, this action is transitive and the isotropy subgroup that fixes the origin is the subgroup U(n) × U(n) block diagonally embedded in U(n, n). In other words,

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we have the injective homomorphism of Lie groups U(n) × U(n) → U(n, n)   A 0 (A, B) → , 0 B that we will consider from now on. In particular, the group of biholomorphisms of DIn×n that fix the origin is given by the linear action U(n) × U(n) × DIn×n → DIn×n (A, B) · Z = AZ B −1 . Hence, we have a representation of DIn×n as a homogeneous space given by DIn×n U(n, n)/U(n) × U(n). However, we note that this representation comes from an action that is not effective. We recall that an action is called effective if the identity element is the only one acting trivially. In this case, it is easy to see that, for every t ∈ T, the element (t In , t In ) ∈ U(n) × U(n) acts trivially on DIn×n . One way to avoid this situation is by considering matrices with determinant 1. More precisely, we consider the groups SU(n, n) = U(n, n) ∩ SL(2n, C) S(U(n) × U(n)) = {(A, B) | det(A) det(B) = 1}, with the corresponding actions on DIn×n . These new groups continue to realize the whole biholomorphism group and the isotropy subgroup at the origin, respectively, and thus yield the representation DIn×n SU(n, n)/S(U(n) × U(n)). Even in this case there are elements of SU(n, n) that act trivially, but now they form a finite group, which is enough to deal with most situations. We will be mainly interested in the biholomorphisms of DIn×n that fix the origin, which we just saw that come from the actions of either of the groups U(n) × U(n) or S(U(n) × U(n)). For our purposes, it will be more useful to consider the former subgroup rather than the latter. One reason is that this will make more natural to work with two actions of the group U(n). More precisely, from now on we will refer to the left and the right U(n)-actions which are given respectively by Z → AZ ,

Z → Z B −1 ,

(2.1)

for A, B ∈ U(n) and Z ∈ DIn×n . To simplify our notation we will denote U(n) L = U(n) × {In }, U(n) R = {In } × U(n),

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and so the left and right U(n)-actions are given by the actions of U(n) L and U(n) R , respectively. Note the actions of the three groups U(n) × U(n), U(n) L and U(n) R extend naturally to linear actions on Mn×n (C), which we will use latter on. For n = 1, we have U(1) = T, and the left and right U(1)-actions yield the same collection of biholomorphisms on the unit disk D: the rotations around the origin. However, for n ≥ 2, the group U(n) is reductive with 1-dimensional center and its semisimple part is SU(n) which is in fact simple. The left and right U(n)-actions have in common only the transformations of the form Z → t Z where t ∈ T. In other words, if A, B ∈ U(n) satisfy AZ = Z B, for all Z ∈ DIn×n , then A = B = t In for some t ∈ T. Since DIn×n is open in Mn×n (C), this claim is an easy linear algebra exercise. In particular, for n ≥ 2, the left and right U(n)-actions on DIn×n are not the same. In the rest of this work, the actions of either U(n) × U(n), U(n) L or U(n) R will be considered both on DIn×n and Mn×n (C). As noted above, such actions are linear on Mn×n (C). 2.2 Bergman Spaces and Toeplitz Operators on DIn×n Let us denote by dv(Z ) the Lebesgue measure on Mn×n (C) normalized by the condition v(DIn×n ) = 1. The (weightless) Bergman space on DIn×n is the subspace of holomorphic functions on DIn×n that belong to L 2 (Mn×n (C), v), and such subspace will be denoted by A2 (DIn×n ). It is very well known that A2 (DIn×n ) is a closed subspace and a reproducing kernel Hilbert space whose kernel is the function K : DIn×n × DIn×n → C given by K (Z , W ) = det(In − Z W ∗ )−2n . We refer to [19, 33] for the details of this fact as well as for the claims made in this subsection. For every λ > 2n − 1, we consider the measure given by det(In − Z Z ∗ )λ−2n dv(Z ), which is in fact finite on DIn×n . We let cλ > 0 be the constant such that the measure dvλ (Z ) = cλ det(In − Z Z ∗ )λ−2n dv(Z ), satisfies vλ (DIn×n ) = 1. Then, the weighted Bergman space corresponding to a given weight λ > 2n − 1 is the subspace of holomorphic functions that belong to L 2 (DIn×n , vλ ). We denote this subspace with A2λ (DIn×n ) which is, as before, Reprinted from the journal

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closed and a reproducing kernel Hilbert space. In this case the kernel is the function K λ : DIn×n × DIn×n → C given by K λ (Z , W ) = det(In − Z W ∗ )−λ . Note that for λ = 2n we recover the weightless Bergman space. The Bergman projection associated to the Bergman space A2λ (DIn×n ) is the projection Bλ : L 2 (DIn×n , vλ ) → A2λ (DIn×n ) given by  Bλ ( f )(Z ) =

DIn×n

f (W )K λ (Z , W )dvλ (W ),

for every Z ∈ DIn×n . The above remarks lead to the so-called Toeplitz operators. More precisely, for a function a ∈ L ∞ (DIn×n ), called a symbol, the Toeplitz operator with symbol a is the (λ) bounded operator Ta = Ta ∈ B(A2λ (DIn×n )) given by Ta(λ) ( f )(Z )

 = Bλ (a f )(Z ) =

DIn×n

a(W ) f (W )K λ (Z , W )dvλ (W )



= cλ

DIn×n

a(W ) f (W ) det(In − W W ∗ )λ−2n dv(W ) , det(In − Z W ∗ )λ

for every Z ∈ DIn×n . 2.3 A Unitary Action on Bergman Spaces The normalized Lebesgue measure dv(Z ) introduced before is obtained from a corresponding Hermitian inner product on Mn×n (C). Up to our normalization, such inner product is given by (Z , W ) → tr(Z W ∗ ), which is clearly invariant under the action of U(n) × U(n) on Mn×n (C). It follows that the U(n) × U(n)-action preserves the normalized Lebesgue measure dv(Z ). Furthermore, since the function Z → det(In − Z Z ∗ )λ−2n , defined on DIn×n , is clearly U(n) × U(n)-invariant, it follows that the measure dvλ (Z ) is U(n) × U(n)-invariant as well. From the previous discussion it follows that, for every λ > 2n − 1, we have a unitary representation given by U(n) × U(n) × L 2 (DIn×n , vλ ) → L 2 (DIn×n , vλ ) ((A, B) · f )(Z ) = f (A−1 Z B),

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which is continuous in the strong operator topology. Furthermore, this action clearly leaves invariant the Bergman space A2λ (DIn×n ). We will denote by πλ this unitary representation of U(n) × U(n) on A2λ (DIn×n ). We note that corresponding properties hold for the subgroups U(n) L and U(n) R . More precisely, the restrictions πλ |U(n) L and πλ |U(n) R are unitary representations as well. In words, the left and right U(n)-actions define unitary representations on each weighted Bergman space.

3 Invariant Symbols Given the unitary action of U(n) × U(n) on A2λ (DIn×n ) introduced in Sect. 2.3 it is useful to consider symbols that are invariant under this group and its subgroups U(n) L and U(n) R . This will yield special types of Toeplitz operators. 3.1 Symbols Invariant Under U(n) × U(n) We note that there is a natural action of the group U(n) × U(n) on the symbols a ∈ L ∞ (DIn×n ) given by ((A, B) · a)(Z ) = a(A−1 Z B) for every (A, B) ∈ U(n) × U(n) and for every Z ∈ DIn×n . A symbol a will be called U(n) × U(n)-invariant if it is invariant under this action, in other words if it satisfies (A, B) · a = a for every (A, B) ∈ U(n) × U(n). We will denote by L ∞ (DIn×n )U(n)×U(n) the space of all U(n) × U(n)-invariant symbols. More generally, for every closed subgroup H ⊂ U(n) × U(n), we will denote by L ∞ (DIn×n ) H the space of symbols a ∈ L ∞ (DIn×n ) that satisfy h · a = a, for every h ∈ H . These are also called H -invariant symbols. For every z ∈ Cn we will denote by D(z) the diagonal matrix whose diagonal entries are given by the components of z. In particular, the collection of matrices D(t) with t ∈ Tn is precisely the subgroup of diagonal matrices of U(n). The assignment t → D(t) yields an isomorphism between Tn and the diagonal subgroup of U(n). From now on we will identify these groups through such assignment. From linear algebra (see also [19]) it follows that for every Z ∈ Mn×n (C) there exist U , V ∈ U(n) and x ∈ Rn satisfying x1 ≥ · · · ≥ xn ≥ 0 such that Z = U D(x)V .

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This representation is not unique, since for every t ∈ Tn we can rewrite it as Z = U D(x)V = U D(t)D(x)D(t)V ,

(3.2)

where we still have U D(t), D(t)V ∈ U(n). In the rest of this work we will denote by Mn×n (C)× the subset of matrices Z ∈ Mn×n (C) such that Z Z ∗ is positive definite with all of its eigenvalues of multiplicity one. We recall that the eigenvalues of Z Z ∗ and Z ∗ Z are exactly the same, so we can use Z ∗ Z instead of Z Z ∗ in the definition of Mn×n (C)× . We also note that Mn×n (C)× ⊂ GL(n, C), and that both of these sets are open conull dense in Mn×n (C). For n = 1, we have M1×1 (C)× = GL(1, C) = C× = C \ {0}, the multiplicative group of non-zero complex numbers, hence our notation. It is easy to prove that for every Z ∈ Mn×n (C)× the representation (3.1) is unique up to the alternative representations given by (3.2). This follows, for example, from the computations found in Section 3.4 of [19]. We can use these remarks to obtain coordinates on Mn×n (C) almost everywhere that are well-defined up to the ambiguity given by (3.2). On the product U(n) × U(n) we consider the Tn -action given by t · (U , V ) = (U D(t), D(t)V ), − → where U , V ∈ U(n) and t ∈ Tn . Let us denote by R n+ the subset of elements x belonging to Rn that satisfy x1 > · · · > xn > 0. We define the smooth map − → ϕ : U(n) × U(n) × R n+ → Mn×n (C)× ϕ(U , V , x) = U D(x)V .

(3.3)

The next result shows that ϕ parameterizes the open set Mn×n (C)× as the manifold − → U(n) × U(n) × R n+ up to the Tn -action given above, thus providing coordinates for Mn×n (C) that are defined almost everywhere and up to such action. Furthermore, it yields an alternative description, up to diffeomorphism, of the set Mn×n (C)× in terms of the groups under consideration. We recall that a submersion is a surjective smooth map whose differential at every point is also surjective. We refer to [21] for the notion of principal fiber bundle, but we note that the properties established below provide one possible definition. − → Proposition 3.1 The smooth map ϕ : U(n) × U(n) × R n+ → Mn×n (C)× given by (3.3) satisfies the following properties. (1) ϕ is a smooth submersion. − → (2) The Tn -action on U(n) × U(n) × R n+ is proper and free. (3) For every Z ∈ Mn×n (C)× , the fiber ϕ −1 (Z ) is a Tn -orbit. − → In other words, U(n) × U(n) × R n+ is a principal fiber bundle over Mn×n (C)× with projection ϕ and structure group Tn . In particular, we have an induced diffeomorphism − →  ϕ : (U(n) × U(n))/Tn × R n+ → Mn×n (C)× 732

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 ϕ ([U , V ], x) = U D(x)V , where (U(n) × U(n))/Tn denotes the quotient by the Tn -action on U(n) × U(n) given above, and [U , V ] denotes the class of (U , V ) ∈ U(n) × U(n) in such quotient. Proof The properness of the Tn -action is clear from the compactness of this group, and the freeness of the action is a consequence of the freeness of left and right translation actions on groups. This proves (2). The uniqueness of the representation (3.1) up to the alternatives given by (3.2) prove that, for every Z ∈ Mn×n (C)× , the fiber ϕ −1 (Z ) is precisely a Tn -orbit. This proves (3). Clearly the map ϕ is smooth, so it remains to show that its differential at every point is surjective. We note that both the domain and range of ϕ admit natural left and right U(n)-actions which thus define diffeomorphisms. For the domain, these actions are given, respectively, by U · (A, B, x) = (U A, B, x), (A, B, x) · V = (A, BV , x). Since ϕ is clearly equivariant for both of these actions, it is enough to prove that − → dϕ(In ,In ,x) is surjective for every x ∈ R n+ . A straightforward computation using the multi-linearity of the product of matrices implies that dϕ(In ,In ,x) (A, B, v) = AD(x) + D(x)B + D(v), for every (A, B, v) ∈ u(n) × u(n) × Rn . We will prove that the image of such linear map has real dimension 2n 2 = dimR (Mn×n (C)). If there exist A, B ∈ u(n) such that AD(x)+ D(x)B = 0, then applying the adjoint we also have D(x)A + B D(x) = 0, because x ∈ Rn . Hence, for every j, k we have a jk xk = −x j b jk , x j a jk = −b jk xk , and this clearly yields (x 2j − xk2 )a jk b jk = 0. Since x j > xk > 0 for every j > k, we conclude that both A, B are diagonal and A = −B. It follows that the linear map u(n) × u(n) → Mn×n (C) given by (A, B) → AD(x) + D(x)B, has real n-dimensional kernel, and so its image has real dimension 2n 2 − n. Now let us assume that there exist A, B ∈ u(n) and v ∈ Rn such that AD(x) + D(x)B = D(v).

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Since both x, v ∈ Rn , we also have D(x)A + B D(x) = −D(v), and these yield the identities a jk xk + x j b jk = δ jk v j , x j a jk + b jk xk = −δ jk v j , which for j = k give the same equations obtained above. We conclude again that A = −B and that these are diagonal matrices. Hence, for some y ∈ Rn we have A = D(i y) and the previous identities reduce to 0 = D(i y)D(x) − D(x)D(i y) = D(v), which implies that v = 0. Hence, the n-dimensional image of the map v → D(v) is complementary to the image of the map (3.4). This proves that the image of the map dϕ(In ,In ,x) has real dimension 2n 2 and so it is surjective. The rest of the claims now follow from the definition and properties of principal fiber bundles. 

We will use the previous result to introduce coordinates in DIn×n which, as before, will be defined almost everywhere and up to a Tn -action. First we state the next result which is an immediate consequence of the condition Z ∗ Z < In that defines DIn×n . Lemma 3.2 For a given Z ∈ Mn×n (C), and with respect to the representation Z = U D(x)V from (3.1) we have: Z ∈ DIn×n if and only if 1 > x1 ≥ · · · ≥ xn ≥ 0. −−−→ We will denote by (0, 1)n the set of all x ∈ Rn such that 1 > x1 > · · · > xn > 0. The next result is a consequence of Proposition 3.1 and Lemma 3.2. We note that the set DIn×n ∩ Mn×n (C)× is an open conull dense subset of the domain DIn×n . For simplicity we will use the same notation  ϕ for the map introduced in Proposition 3.1 that now considers the domain DIn×n . Corollary 3.3 In the notation of Proposition 3.1 the map given by −−−→  ϕ : (U(n) × U(n))/Tn × (0, 1)n → DIn×n ∩ Mn×n (C)×  ϕ ([U , V ], x) = U D(x)V , is a diffeomorphism. Remark 3.4 It is an easy exercise to verify that the diffeomorphism  ϕ considered in Proposition 3.1 and Corollary 3.3 yields, for n = 1, the usual polar coordinates on M1×1 (C)× = C× and DI1×1 ∩ M1×1 (C)× = D \ {0}, respectively. To see this, we note that there is natural isomorphism (U(1) × U(1))/T T,

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of Lie groups. Hence, the diffeomorphism  ϕ gives, in both cases, a natural generalization of polar coordinates for arbitrary n ≥ 2. These coordinates are basically equivalent to those considered in Section 3.4 from [19]. However, both polar coordinates, ours and the ones found in [19], use different parameterizing spaces. We now use the coordinates provided by the previous discussion to obtain the next characterization of the U(n) × U(n)-invariant symbols. Proposition 3.5 Let a ∈ L ∞ (DIn×n ) be given. Then, the following conditions are equivalent. (1) The symbol a satisfies a(U −1 Z V ) = a(Z ) for every U , V ∈ U(n) and Z ∈ DIn×n ∩ Mn×n (C)× . −−−→ (2) For every U , V ∈ U(n) and x ∈ (0, 1)n we have a(U D(x)V ) = a(D(x)). −−−→ (3) The function a ◦ ϕ defined on U(n) × U(n) × (0, 1)n depends only on the factor −−−→n (0, 1) −−−→ (4) The function a ◦  ϕ defined on (U(n) × U(n))/Tn × (0, 1)n depends only on the −−−→ factor (0, 1)n . Proof From the definitions of ϕ and  ϕ , we clearly have that (2), (3) and (4) are equivalent. Also, it is immediate that (1) implies (2). Let us assume that (2) holds. Let Z ∈ DIn×n ∩ Mn×n (C)× be given and consider its decomposition Z = U D(x)V as given in (3.1), so that in particular we have −−−→ x ∈ (0, 1)n . Then, for every U1 , V1 ∈ U(n) we have a(U1−1 Z V1 ) = a(U1−1 U D(x)V V1 ) = a(D(x)) = a(U D(x)V ) = a(Z ), 

and this implies (1). Remark 3.6 We recall that a symbol a ∈ L ∞ (D) is radial if and only if we have a(z) = a(|z|) = a(|t1 zt2 |) = a(t1 zt2 )

for all z ∈ D and t1 , t2 ∈ T, which trivially recovers condition (2) from Proposition 3.5. Furthermore, using Remark 3.4 on polar coordinates, the first identity above is also trivially equivalent to (4) of the same proposition. As noted before, in this case M1×1 (C)× = C× . 3.2 Left and Right U(n)-invariant Symbols We recall the left and right U(n)-actions on the domain DIn×n given by (2.1), which correspond to the actions of the subgroups U(n) L and U(n) R , defined in (2.2), respectively. These allow us to consider two additional families of invariant symbols. A

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symbol a ∈ L ∞ (DIn×n ) will be called left U(n)-invariant if it is invariant under the U(n) L -action, in other words if we have a(U Z ) = a(Z ), for every U ∈ U(n) and for every Z ∈ DIn×n . Similarly, the symbol a ∈ L ∞ (DIn×n ) will be called right U(n)-invariant if it is invariant under the U(n) R -action, which is now equivalent to a(Z V ) = a(Z ), for every V ∈ U(n) and for every Z ∈ DIn×n . In particular, we can alternatively speak of U(n) L -invariant symbols and U(n) R -invariant symbols, respectively. The space consisting of the former will be denote by L ∞ (DIn×n )U(n) L , while the space consisting of the latter will be denoted by L ∞ (DIn×n )U(n) R . Let us denote by Pos(n, C) the cone of positive definite Hermitian elements of Mn×n (C). We recall the following elementary fact from linear algebra. Lemma 3.7 For every Z ∈ GL(n, C) there exist unique elements U , V ∈ U(n) and P, Q ∈ Pos(n, C) such that Z = U P = QV . 1

1

Furthermore, we have P = (Z ∗ Z ) 2 and Q = (Z Z ∗ ) 2 . In the notation of Lemma 3.7 and from now on, we will refer to the expressions Z = U P and Z = QV as the left and right polar decompositions, respectively. The next result provide a characterization of left and right U(n)-invariant symbols in terms of these decompositions. We recall that GL(n, C) is open conull dense in Mn×n (C) and so DIn×n ∩ GL(n, C) has the same properties with respect to DIn×n . Proposition 3.8 For any symbol a ∈ L ∞ (DIn×n ), the following properties hold. (1) If a satisfies  1 a(Z ) = a (Z ∗ Z ) 2 for every Z ∈ DIn×n , then a is U(n) L -invariant. Conversely, if a is U(n) L -invariant, then we have  1 a(Z ) = a (Z ∗ Z ) 2 for every Z ∈ DIn×n ∩ GL(n, C). (2) If a satisfies  1 a(Z ) = a (Z Z ∗ ) 2

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for every Z ∈ DIn×n , then a is U(n) R -invariant. Conversely, if a is U(n) R -invariant, then we have  1 a(Z ) = a (Z Z ∗ ) 2 for every Z ∈ DIn×n ∩ GL(n, C). Proof If a satisfies the displayed identity in (1) of the statement, then for any given U ∈ U(n) we have    1 1 1 a(U Z ) = a ((U Z )∗ (U Z )) 2 = a (Z ∗ U ∗ U Z ) 2 = a (Z Z ∗ ) 2 = a(Z ) for every Z ∈ DIn×n , which proves the U(n) L -invariance of a. If a is U(n) L -invariant, and we choose Z ∈ DIn×n ∩ GL(n, C) with left polar 1

decomposition Z = U (Z ∗ Z ) 2 , then we have   1 1 a(Z ) = a U (Z ∗ Z ) 2 = a (Z ∗ Z ) 2 , thus completing the proof of (1). The proof of (2) is similar. 

√ √ Remark 3.9 For n = 1, we have |z| = zz = zz for every z ∈ DI1×1 = D. This yields the trivial fact that, for a symbol a, left and right T-invariance are mutually equivalent, and also equivalent to the condition a(z) = a(|z|) for every z ∈ D. Note than in this case we have GL(n, C) = C× . Remark 3.10 For a given symbol a ∈ L ∞ (DIn×n ) let us consider the symbol  a = χ E a. where we take E = DIn×n ∩ GL(n, C). In other words,  a redefines a as 0 on the closed null subset of singular matrices in DIn×n . If Z ∈ DIn×n is singular, then the  1 1 a (Z ) = 0 =  a (Z ∗ Z ) 2 . Since matrix (Z ∗ Z ) 2 is singular as well and so we have  the U(n) L -action leaves invariant the subset E, it follows that if a is U(n) L -invariant, then  a is U(n) L -invariant as well. We conclude from these observations and Proposition 3.8 that a is U(n) L -invariant if and only if  1  a (Z ) =  a (Z ∗ Z ) 2 for every Z ∈ DIn×n , and not just for Z ∈ DIn×n ∩GL(n, C). In other words, any symbol a ∈ L ∞ (DIn×n ) can be redefined outside the U(n) L -invariant open conull dense subset DIn×n ∩ GL(n, C) so that its U(n) L -invariance is equivalent to the condition  1 a(Z ) = a (Z ∗ Z ) 2

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to hold for every Z ∈ DIn×n . On the other hand, we note that for continuous symbols this equivalence is immediate from the density of DIn×n ∩ GL(n, C) in DIn×n , in which case there is no need to redefine any of the values of a. We also note that similar remarks hold for U(n) R -invariant symbols. As a consequence of Remark 3.10 we obtain the next result. Corollary 3.11 Up to the identification of symbols that are equal almost everywhere, the left and right U(n)-invariant symbols are given as follows.  1 L ∞ (DIn×n )U(n) L = {a ∈ L ∞ (DIn×n ) | a(Z ) = a (Z ∗ Z ) 2 for every Z ∈ DIn×n },  1 L ∞ (DIn×n )U(n) R = {a ∈ L ∞ (DIn×n ) | a(Z ) = a (Z Z ∗ ) 2 for every Z ∈ DIn×n }. Straight from the definition of invariance we have L ∞ (DIn×n )U(n)×U(n) = L ∞ (DIn×n )U(n) L ∩ L ∞ (DIn×n )U(n) R . In particular, the space L ∞ (DIn×n )U(n)×U(n) is a subspace of both L ∞ (DIn×n )U(n) L and L ∞ (DIn×n )U(n) R . As noted in Remark 3.9, these three spaces are all the same for n = 1. However, we will show that for n ≥ 2 there are plenty of both left and right U(n)-invariant symbols that are not U(n) × U(n)-invariant. We achieve this through the next result which corresponds to Proposition 3.5. As noted above, we will consider that two given functions are the same if they agree almost everywhere. Proposition 3.12 Let a ∈ L ∞ (DIn×n ) be a given symbol. Then, the following conditions are equivalent. (1) The symbol a is left (right, respectively) U(n)-invariant. (2) The function ([U , V ], x) → a ◦  ϕ ([U , V ], x) = a(U D(x)V ) defined on the set −−−→n n (U(n) × U(n))/T × (0, 1) is independent of U (independent of V , respectively). −−−→ (3) There is a measurable function f defined on (Tn \U(n)) × (0, 1)n (defined on − − − → (U(n)/Tn ) × (0, 1)n , respectively) such that a◦ ϕ = f ◦ρ where ρ is given by ([U , V ], x) → ([V ], x) (given by ([U , V ], x) → ([U ], x), respectively). In particular, the assignment f → f ◦ ρ ◦  ϕ −1 −−−→ establishes a one-to-one correspondence between either L ∞ ((Tn \U(n)) × (0, 1)n ) − − − → or L ∞ ((U(n)/T n ) × (0, 1)n ), and the U(n) L -invariant or U(n) R -invariant symbols, respectively, where ρ is either of the assignments ([U , V ], x) → ([V ], x) or ([U , V ], x) → ([U ], x), respectively.

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Proof We will only consider the case of left U(n)-invariance since the other case is proved similarly. If a is U(n) L -invariant, then we have a◦ ϕ ([U , V ], x) = a(U D(x)V ) = a(D(x)V ) = a ◦  ϕ ([In , V ], x), and so it follows immediately that (1) implies (2). Let us assume that (2) holds. For every V ∈ U(n) and its corresponding class [V ] ∈ U(n) we define f ([V ], x) = a ◦  ϕ ([In , V ], x) = a(D(x)V ), −−−→ for every x ∈ (0, 1)n . From the definition of the corresponding quotients it is clear that [In , V ] = [In , V1 ] implies [V ] = [V1 ]. Furthermore, if [V ] = [V1 ], then there exist t ∈ Tn such that V = D(t)V1 and we have a(D(x)V ) = a(D(x)D(t)V1 ) = a(D(t)D(x)V1 ) = a(D(x)V1 ), where we have used (2) in the last identity. It follows that f is a well defined function −−−→ on Tn \U(n) × (0, 1)n . Also, using any measurable section of ρ it is easy to see that f is measurable. This proves (3). Let us now assume that (3) holds. Hence, we have a(U D(x)V ) = a ◦  ϕ ([U , V ], x) = f ([V ], x) = a ◦  ϕ ([In , V ], x) = a(D(x)V ) −−−→ for every U , V ∈ U(n) and x ∈ (0, 1)n . Hence, for every Z ∈ DIn×n ∩ Mn×n (C)× with decomposition Z = U D(x)V as in (3.1) and U1 ∈ U(n) we have a(U1 Z ) = a(U1 U D(x)V ) = a(D(x)V ) = a(U D(x)V ) = a(Z ). This proves the U(n) L -invariance of a on a conull subset of DIn×n which, according to our convention, is enough to conclude (1). For the last claim it is enough to recall from Corollary 3.3 that  ϕ is a diffeomorphism −−−→n n from (U(n) × U(n))/T ×(0, 1) onto the open conull dense subset DIn×n ∩Mn×n (C)× of DIn×n , and so it has a smooth inverse. 

Remark 3.13 We note that condition (3) from Proposition 3.12 for the left U(n)invariant case yields the commutative diagram a

DIn×n  ϕ

f

−−−→ (U(n) × U(n))/Tn × (0, 1)n

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where the lower horizontal arrow is given by the assignment ([U , V ], x) → ([V ], x). And for the right U(n)-invariant case it yields the commutative diagram a

DIn×n

C

 ϕ

f

−−−→ (U(n) × U(n))/Tn × (0, 1)n

−−−→ (U(n)/Tn ) × (0, 1)n

where the lower horizontal arrow is given by the assignment ([U , V ], x) → ([U ], x). Remark 3.14 By comparing Propositions 3.5 and 3.12, we observe that the U(n) × U(n)−−−→ invariant symbols correspond to functions defined on (0, 1)n , while the left and right −−−→ U(n)-invariant symbols correspond to functions defined on either (Tn \U(n))× (0, 1)n − − − → or (U(n)/Tn ) × (0, 1)n , respectively. The first factor is in both cases a manifold with real dimension n(n − 1). In fact, both manifolds are easily seen to be diffeomorphic through the assignment [U ] → [U −1 ], so one can work with either once this identification has been considered. For n = 1, this manifold is just a point, which corresponds to the equivalence of the invariance with respect to the three groups involved. However, for n ≥ 2, the manifold has positive dimension and Proposition 3.12 proves the existence of plenty of either left or right U(n)-invariant symbols that are not U(n) × U(n)-invariant. More precisely, there are as many of such symbols as elements in L ∞ (Tn \U(n)). In particular, there are plenty of symbols that are either left or right U(n)-invariant but not both.

4 Toeplitz Operators with Invariant Symbols The principal tool to obtain our results will be representation theory. Hence, we recall some notions and refer to [7, 14, 20] for further details and proofs. If H is a compact Lie group, then a unitary representation of H on a Hilbert space H is a strong topology continuous linear action π : H × H → H. In this case, we also say that H is an H -module. For K a closed subspace of H we will say that K is H invariant, or an H -submodule, if π(g)(K) = K for every g ∈ H . If K is H -invariant and its only H -invariant subspaces are 0 and K itself, then K is called irreducible. It is known that, since H is compact, all the irreducible H -modules are finite dimensional and every H -module, finite dimensional or not, is a Hilbert direct sum of irreducible H -modules. If H1 and H2 are two given H -modules, with corresponding representations π1 and π2 , then a bounded operator T : H1 → H2 such that T ◦ π1 (h) = π2 (h) ◦ T , for every

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h ∈ H , is called an intertwining operator. If there is a unitary intertwining operator H1 → H2 , then we say that the H -modules are isomorphic. As usual this yields an equivalence relation. For H = H1 = H2 , the algebra of all intertwining operators is denoted by End H (H), and it is a von Neumann algebra.  the family of all equivalence classes of irreducible H -modules. We will denote by H We now state our main abstract tool from representation theory of compact groups. Proposition 4.1 Let H be a compact group with a unitary representation on a Hilbert  let us denote by space H. For every equivalence class [K] ∈ H H[K] =



{W ⊂ H | W is an H -submodule in the class of [K]}.

Then, the following properties hold. , the subspace H[K] is a (closed) H -submodule of H. This is (1) For every [K] ∈ H called the isotypic component of H associated to the class [K]. (2) There is a Hilbert direct sum H=



H[K] .

 [K]∈ H

This is called the isotypic decomposition of H as an H -module. If T belongs to End H (H), then T preserves this Hilbert direct sum. (3) The von Neumann algebra End H (H) is commutative if and only if for every [K] ∈  the isotypic component H[K] is either 0 or irreducible. If this is the case, then we H say that the isotypic decomposition is multiplicity-free, and every T belonging to End H (H) acts by a constant multiple of the identity on each isotypic component. 4.1 Invariance with Respect to Closed Subgroups We are interested on the unitary representations πλ (for λ > 2n − 1) of the group U(n) × U(n) on the weighted Bergman spaces A2λ (DIn×n ), as well as on the restrictions πλ |U(n) L and πλ |U(n) R (see Sect. 2.3). However, for simplicity, in this subsection we will consider any closed subgroup H ⊂ U(n) × U(n). Then there is a correspondence between H -invariant symbols and intertwining Toeplitz operators. We recall from Sect. 3 that a symbol a ∈ L ∞ (DIn×n ) is H -invariant if and only if a ◦ h = a for every h ∈ H , and that the family of all such symbols is denoted by L ∞ (DIn×n ) H . We also denote by πλ | H the restriction of the representation πλ to H . We recall that for a set S ⊂ L ∞ (DIn×n ) of symbols we denote by T (λ) (S) the unital Banach algebra generated by Toeplitz operators with symbols in S and acting on A2λ (DIn×n ), where the weight λ > 2n − 1. In particular, for any closed subgroup H ⊂ U(n) × U(n), the algebra T (λ) (L ∞ (DIn×n ) H ) is a C ∗ -algebra. This is a well known consequence of the fact that L ∞ (DIn×n ) H is conjugation invariant. The next proposition is the main tool from representation theory that we will use in the setup of Toeplitz operators. Similar results can be found in [10], but we will

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use the one below that requires a simplified proof which is enough for our purposes. Furthermore, our particular statement will provide some useful tools. Proposition 4.2 If H ⊂ U(n) × U(n) is a closed subgroup, then the following holds for every λ > 2n − 1. (1) A symbol a ∈ L ∞ (DIn×n ) is H -invariant if and only if Ta(λ) is intertwining for πλ | H . In other words, we have T (λ) (L ∞ (DIn×n ) H ) ⊂ End H (A2λ (DIn×n )). (2) For every T ∈ End H (A2λ (DIn×n )) and any finite dimensional πλ | H -invariant subspace W ⊂ A2λ (DIn×n ), there exists a symbol a ∈ L ∞ (DIn×n ) H such that T f , gλ = Ta(λ) f , gλ for every f , g ∈ W . (3) If all the isotypic components associated to πλ | H are finite dimensional, then the C ∗ -algebra T (λ) (L ∞ (DIn×n ) H ) is commutative if and only if the isotypic decomposition of the restriction πλ | H is multiplicity-free. Proof The proof of (1) is an easy exercise that uses the formula Ta(λ) f , gλ = a f , gλ , for every f , g ∈ A2λ (DIn×n ), together with the fact that πλ is unitary. To prove (2), let T and W be given as in its statement. Theorem 2 from [12] proves the existence of a symbol a ∈ L ∞ (DIn×n ) such that T f , gλ = Ta(λ) f , gλ for every f , g ∈ W . Let us consider the symbol given by 

a(h −1 · Z )dh,

 a (Z ) = H

where dh is the probability Haar measure of H . Clearly  a belongs to L ∞ (DIn×n ) H , as a consequence of the left and right invariance of the Haar measure for compact groups. Then, we can compute as follows for every f , g ∈ W (λ)

a f , gλ Ta f , gλ =    = a(h −1 · Z ) f (Z )g(Z )dhdvλ (Z ) DI

H

H

DIn×n

 n×n  =

a(Z ) f (h · Z )g(h · Z )dvλ (Z )dh

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 = = =



H

H 

H

= H

aπλ (h)−1 f , πλ (h)−1 gλ dh Ta(λ) πλ (h)−1 f , πλ (h)−1 gλ dh T πλ (h)−1 f , πλ (h)−1 gλ dh T f , gλ dh = T f , gλ ,

where we have used the properties of T , W and a. This proves (2) for the symbol  a. For (3) we first observe that (1) implies that the commutativity of the C ∗ -algebra T (λ) (L ∞ (DIn×n ) H ) follows from that of End H (A2λ (DIn×n )), which occurs when πλ is multiplicity-free. Conversely, let us assume that πλ is not multiplicity-free. Let us choose a nonirreducible isotypic component W of πλ which, by assumption, is finite dimensional. Then, there exist T1 , T2 ∈ End H (A2λ (DIn×n )) that necessarily preserve W and whose restrictions to W do not commute with each other. By (2), there exist symbols a1 , a2 ∈ L ∞ (DIn×n ) H such that T j f , gλ = Ta(λ) f , gλ j for every f , g ∈ W , and j = 1, 2. By (1) and Proposition 4.1(2), the Toeplitz operators (λ) (λ) Ta1 and Ta1 preserve W as well, because the latter is an isotypic component. It follows that T1 |W = Ta(λ) | , T2 |W = Ta(λ) | , 1 W 2 W (λ)

(λ)

which implies that Ta1 and Ta1 do not commute with each other and so the C ∗ -algebra 

T (λ) (L ∞ (DIn×n ) H ) is not commutative. Propositions 4.1 and 4.2 lead us to find the isotypic decompositions of the U(n) × U(n)-action and the left and right U(n)-actions to determine the commutativity of the algebras generated by Toeplitz operators whose symbols are invariant under such actions. 4.2 Toeplitz Operators with U(n) × U(n)-invariant Symbols Let us denote by P(Mn×n (C)) the vector space of polynomials on Mn×n (C), and by P d (Mn×n (C)) the subspace of those that are homogeneous of degree d. Then, we have an algebraic direct sum P(Mn×n (C)) =

∞  d=0

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which is invariant with respect to transformations of the form Z → t Z for every t ∈ T. Since these transformations belong to the U(n) × U(n)-action, they induce a unitary representations on the weighted Bergman spaces. Furthermore, it is well known that P(Mn×n (C)) is dense in every weighted Bergman space A2λ (DIn×n ) (see [33]). Hence, for every λ > 2n − 1, we have a Hilbert direct sum decomposition A2λ (DIn×n ) =

∞ 

P d (Mn×n (C))

(4.1)

d=0

that corresponds to the isotypic decomposition of the T-action just described. More precisely, every subspace P d (Mn×n (C)) is the isotypic component associated to the character χ−d (t) = t −d of T. Since the U(n) × U(n)-action is linear, it follows that it preserves the Hilbert direct sum (4.1). Hence, to obtain a decomposition of A2λ (DIn×n ) into irreducible U(n) × U(n)-submodules it is enough to do so for each term of (4.1). We will achieve this by using representation theory for the group GL(n, C). The next result establishes an equivalence between representations of suitable complex and compact Lie groups, which justify our passage from U(n) to GL(n, C). It is well known from representation theory as part of the so-called Weyl’s unitary trick. Hence, we provide a sketch of the proof and refer to [14, 20] for further details. Lemma 4.3 Let G be a connected complex Lie group and H a closed subgroup. Assume that the Lie algebra of G satisfies g = h ⊕ ih, where h is the Lie algebra of H . If π : G → GL(W ) is a finite dimensional complex representation, then the following properties hold. (1) If W0 ⊂ W is a (complex) subspace, then W0 is G-invariant if and only if it is H -invariant. (2) The representation π is irreducible if and only if π | H is irreducible. Furthermore, if π j : G → GL(W j ), for j = 1, 2, are two given finite dimensional representations, then W1 W2 as G-modules if and only if W1 W2 as H -modules. Proof Clearly (1) implies (2), so we will prove the former. It is obvious that G-invariance of a subspace implies its H -invariance. So we consider an H -invariant subspace W0 and show that it is G-invariant. Let dπ : g → gl(W ) be the induced representation on Lie algebras obtained by differentiation. In particular, we have a commutative diagram g

G

dπ

gl(W )

π

GL(W )

where the vertical arrows are the corresponding exponential maps. Since π(h)(W0 ) = W0 for every h ∈ H , it follows that π(exp(r X ))(w) ∈ W0

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for every X ∈ h, w ∈ W0 and r ∈ R. Hence, differentiation with respect to r yields dπ(X )(W0 ) ⊂ W0 for every X ∈ h. Since W0 is complex, dπ is complex linear and g = h ⊕ ih this implies that dπ(X )(W0 ) ⊂ W0 for every X ∈ g. We conclude that exp(dπ(X ))(w) =

∞ dπ(X ) j j=0

j!

(w) ∈ W0

for every X ∈ g and w ∈ W0 . The above commutative diagram now implies that π(exp(X ))(W0 ) = exp(dπ(X ))(W0 ) = W0 for every X ∈ g. Since G is connected it is generated by exp(g) and this yields the G-invariance of W0 . To prove the last claim, note that the non-trivial part is showing that W1 W2 as H -modules implies W1 W2 as G-modules. Let T : W1 → W2 be an isomorphism of H -modules. In particular, we have T ◦ π1 (h) = π2 (h) ◦ T , for every h ∈ H . As before, taking h = exp(r X ) and differentiating with respect to r we obtain T ◦ dπ1 (X ) = dπ2 (X ) ◦ T , for every X ∈ h. But this implies that the same identity holds for every X ∈ g since T , dπ1 and dπ2 are complex linear. Using the series expansion of the exponential of linear maps we conclude that T ◦ exp(dπ1 (X )) = exp(dπ2 (X )) ◦ T , for every X ∈ g. Commutative diagrams similar to the one used above imply that T ◦ π1 (exp(X )) = π2 (exp(X )) ◦ T , for every X ∈ g. Since G is connected it is generated by exp(g) and we conclude that T is an isomorphism of G-modules. 

Lemma 4.3 clearly applies to G = GL(n, C) and H = U(n). This will allow us to describe isotypic decompositions involving U(n) in terms of the irreducible representations of GL(n, C). For this reason, we will recall some of the properties of the irreducible rational representations of GL(n, C). This will be enough for our purposes. We refer to [14] for further details, definitions and proofs of the results and claims found in the rest of this subsection. Let us denote by C×n the subgroup of diagonal matrices in GL(n, C). In particular, we have Tn = U(n) ∩ C×n , the subgroup of diagonal matrices in U(n). Then, the Lie

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algebra of C×n is Cn and the Lie algebra of Tn is iRn , both considered as spaces of diagonal matrices. In other words, we will use from now on the embeddings Cn → gl(n, C), C×n → GL(n, C), of Lie algebras and Lie groups, respectively, both given by the assignment z → D(z). Note that a corresponding remark applies for the Lie group U(n). Let π : GL(n, C) → GL(W ) be a rational representation of GL(n, C), where W is a finite dimensional complex vector space. Then, for every μ ∈ Cn∗ , the complex dual space of Cn , we will denote W (μ) = {w ∈ W | dπ(X )(w) = μ(X )w, for every X ∈ Cn }. If W (μ) = 0, then this subspace is called a weight space and the functional μ is called a weight, both associated to W as GL(n, C)-module. If we denote by X(W ) the set of weights for the GL(n, C)-module W , then it is well known (see Section 3.1.3 from [14]) that W is the direct sum of its weight spaces W =



W (μ).

μ∈X(W )

A particular, case is given by the adjoint representation of GL(n, C) Ad : GL(n, C) → GL(gl(n, C)) Ad(g)(X ) = g Xg −1 , whose differential is well known to be the adjoint representation of gl(n, C) ad : gl(n, C) → gl(gl(n, C)) ad(X )(Y ) = [X , Y ]. The collection of non-zero weights for the adjoint representation of GL(n, C) is called the set of roots and it will be denoted by . Let {e j }nj=1 be the canonical basis of the dual space Cn∗ of Cn . In other words, we define e j (z) = z j , for every z ∈ Cn . Recall that Cn is being identified with the subspace of diagonal matrices in Mn×n (C). Then, it is easily seen that the set of roots for GL(n, C) is given by = {e j − ek | j, k = 1, . . . , n, j = k}. We note that the functionals {e j }nj=1 are real valued on Rn . Hence, we can identify its real span with the dual space Rn∗ of Rn . In particular, we will consider from now on ⊂ Rn∗ , and also that e j ∈ Rn∗ for every j = 1, . . . , n. The lexicographic order with respect to the ordered basis e1 , . . . , en yields a partial order on Rn∗ . We will

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denote by + the set of positive roots with respect to this order. In particular, we now have + = {e j − ek | 1 ≤ j < k ≤ n}. As before, let us consider any given finite dimensional rational representation π : GL(n, C) → GL(W ). Then, we define the root order in X(W ) by μ1 ≺ μ2 ⇐⇒ μ2 − μ1 = α1 + · · · + αm for some α1 , . . . , αm ∈ + . We note that the same constructions and properties considered so far apply to the Lie group SL(n, C) and its Lie algebra sl(n, C) without any essential change. The only modification that has to be done is to replace the diagonal subgroup C×n of GL(n, C) ×n such that z · · · · · z = 1. Correspondingly, the by the subgroup C×n 1 n 0 of z ∈ C × Lie algebra of C0 is the subspace Cn0 of z ∈ Cn such that z 1 + · · · + z n = 0. One advantage of considering the subgroup SL(n, C) is that it is semisimple (for n ≥ 2) and we have at our disposal the Theorem of the Highest Weight for Lie algebras (see Section 3.2.1 from [14]). This allows us to obtain the next result. We sketch the additional arguments required to obtain the needed statement for the Lie group SL(n, C) from the results found in [14]. Note that for a finite dimensional rational representation π of either GL(n, C) or SL(n, C) (the cases under consideration) the theory of weights is obtained from the representation dπ of either gl(n, C) or sl(n, C), respectively. Proposition 4.4 The finite dimensional rational representations of SL(n, C) satisfy the following properties. (1) If π : SL(n, C) → GL(W ) is an irreducible finite dimensional rational representation, then there exists a unique weight μW ∈ X(W ) such that μ ≺ μW for every μ ∈ X(W ) \ {μW }. The weight μW is called the highest weight of W . (2) If π j : SL(n, C) → GL(W j ), for j = 1, 2, are two irreducible finite dimensional rational representations, then W1 W2 as SL(n, C)-modules if and only if μW1 = μW2 . In other words, two such irreducible representations are equivalent if and only if they have the same highest weight. Proof The existence of highest weights for the noted irreducible representations is a consequence of Corollary 3.2.3 from [14], thus proving (1). Let us consider two representations π1 and π2 as in (2). If W1 and W2 are isomorphic as SL(n, C)-modules, then differentiation proves that they are isomorphic as sl(n, C)modules. From the definitions it is easy to see that this implies X(W1 ) = X(W2 ), and so we conclude that μW1 = μW2 . We now assume that μW1 = μW2 . Then, Theorem 3.2.5 from [14] implies that W1 W2 as sl(n, C)-modules. Let T : W1 → W2 be an isomorphism of sl(n, C)modules. Then, we have T ◦ dπ1 (X ) = dπ2 (X ) ◦ T ,

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for every X ∈ sl(n, C). At this point we can repeat the arguments found at the end of the proof of Lemma 4.3 to show that T is an isomorphism of SL(n, C)-modules.  The fundamental dominant weights (see [14]) for the group SL(n, C) are the linear functionals given by the restrictions ω j = (e1 + · · · + e j )|Cn0 n for every j = 1, . . . , n − 1, which belong to Cn∗ 0 , the dual space of C0 ⊂ sl(n, C). n Note that the restriction of e1 + · · · + en to C0 is 0. The set of dominant weights for the group SL(n, C) is given by (see Section 3.1.4 from [14])

P++ (SL(n, C)) =

n−1 

Nω j .

j=1

With this notation, the next result is a consequence of Section 3.1.4 and Theorem 5.5.21 from [14]. Corollary 4.5 The set P++ (SL(n, C)) is precisely the collection of the highest weights of the irreducible rational representations of the group SL(n, C). In particular, there is a one-to-one correspondence between the family of equivalence classes of irreducible rational representations of SL(n, C) and the set P++ (SL(n, C)). Such correspondence assigns to every equivalence class the highest weight of any of its elements. Remark 4.6 For every μ ∈ P++ (SL(n, C)) we will denote by W μ a SL(n, C)-module with highest weight μ. In particular, W μ is well defined up to an isomorphism of SL(n, C)-modules. Thus, the inverse of the correspondence stated in Corollary 4.5 is given by μ → [W μ ]. We will now use the previous constructions and follow Section 5.5.4 from [14] to describe the irreducible rational representations of GL(n, C). The set of dominant weights of the group GL(n, C) is the set P++ (GL(n, C)) of elements μ ∈ Cn∗ , the dual space of Cn ⊂ gl(n, C), that can be written as μ = m 1 e1 + · · · + m n en ,

(4.2)

where m 1 ≥ · · · ≥ m n and m j ∈ Z for every j = 1, . . . , n. For every μ ∈ P++ (GL(n, C)) given by the expression (4.2) we will consider the element of P++ (SL(n, C)) given by μ0 = (m 1 − m 2 )ω1 + · · · + (m n−1 − m n )ωn−1 .

(4.3)

With the previous notation, the next result describes the irreducible rational representations of GL(n, C). This is basically a restatement of Theorem 5.5.22 from [14].

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Proposition 4.7 There is a one-to-one correspondence between the set of dominant weights P++ (GL(n, C)) and the equivalence classes of irreducible rational representations of GL(n, C). This correspondence assigns to every μ ∈ P++ (GL(n, C)) an irreducible rational representation π μ : GL(n, C) → GL(W μ ) that satisfies the following properties, where μ has the representation given by (4.2). (1) The restriction π μ |SL(n,C) is an irreducible representation of SL(n, C) with highest weight μ0 given by (4.3). (2) The restriction π μ |C× In yields the representation of the diagonal subgroup C× C× In ⊂ GL(n, C) given by the action on W μ through the character z → z m 1 +···+m n In . We will use the previous constructions to study suitable representations of U(n). Hence, the following result will be very useful. Lemma 4.8 With the notation of Proposition 4.7, for every μ ∈ P++ (GL(n, C)) the irreducible representation π μ : GL(n, C) → GL(W μ ) restricted to U(n) is irreducible as well. In other words, W μ is an irreducible U(n)-module for every μ ∈ P++ (GL(n, C)). Furthermore, if μ1 , μ2 ∈ P++ (GL(n, C)) then W μ1 W μ2 as U(n)-modules if and only if μ1 = μ2 . Proof Lemma 4.3(2) implies the first part of the statement. The second part of the statement follows from Proposition 4.7 and the last claim of Lemma 4.3. In both cases, to apply Lemma 4.3 we take G = GL(n, C) and H = U(n). 

The representation πλ of U(n) × U(n) on P(Mn×n (C)) has a natural extension to the representation GL(n, C) × GL(n, C) × P(Mn×n (C)) → P(Mn×n (C)) (A, B) · p(Z ) = p(A−1 Z B), which clearly preserves P d (Mn×n (C)) for every d ∈ N. Furthermore, the induced representation of GL(n, C) × GL(n, C) on each of these subspaces is rational (see [14]). We will now describe its decomposition into irreducible submodules. First, we recall some properties of representations of a product of groups and tensor products that we will use freely. We refer to [7, 14] for further details and proofs. Recall that for two given Lie groups H1 , H2 and corresponding finite dimensional modules W1 , W2 , the tensor product W1 ⊗ W2 admits a natural representation of H1 × H2 . If both W1 and W2 are irreducible, then W1 ⊗ W2 is irreducible as well. Furthermore, if W1 , W1 and W2 , W2 are finite dimensional irreducible modules over H1 and H2 , respectively, then W1 ⊗ W2 and W1 ⊗ W2 are isomorphic over H1 × H2 if and only if W1 W1 and W2 W2 over H1 and H2 , respectively. We now recall the definition of a special type of dominant weight that appears in the representation of GL(n, C) × GL(n, C) on the space of polynomials given above. A dominant weight μ will be called non-negative if m 1 ≥ · · · ≥ m n ≥ 0, where μ is given by (4.2). In this case we will write |μ| = m 1 + · · · + m n ,

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and we call |μ| the size of μ. Let us denote by P(n) the set of all non-negative dominant weights of GL(n, C), and by Pd (n) the subset of those with size d ∈ N. In particular, we have Pd (n) ⊂ P(n) ⊂ P++ (GL(n, C)) for every d ∈ N. Then, the next result is obtained using Theorem 5.6.7 from [14]. Proposition 4.9 For every μ ∈ P(n) there is a GL(n, C) × GL(n, C)-submodule P μ (Mn×n (C)) of P(Mn×n (C)) such that the following properties are satisfied. (1) For every d ∈ N we have a direct sum decomposition 

P d (Mn×n (C)) =

P μ (Mn×n (C))

μ∈Pd (n)

which is GL(n, C) × GL(n, C)-invariant. (2) For every μ ∈ P(n), the spaces P μ (Mn×n (C)) and W μ∗ ⊗ W μ are isomorphic as GL(n, C) × GL(n, C)-modules, where W μ is the GL(n, C)-module given by Proposition 4.7 and W μ∗ is its dual GL(n, C)-module. In particular, the direct sum in (1) is the isotypic decomposition for the GL(n, C) × GL(n, C)-action and it is multiplicity-free. (3) The algebraic direct sum P(Mn×n (C)) =



P μ (Mn×n (C))

μ∈P(n)

is the isotypic decomposition for the GL(n, C) × GL(n, C)-action, and it is multiplicity-free. Proof Claims (1) and (2) follow directly from Theorem 5.6.7 from [14]. The direct sum in (3) and its invariance is clear. On the other hand, for any two given summands P μ1 (Mn×n (C)) and P μ2 (Mn×n (C)) the action of the subgroup {(z In , z −1 In ) | z ∈ C× } C× of GL(n, C) × GL(n, C) on them is given by the characters χ−2|μ1 | (z) = z −2|μ1 | , χ−2|μ2 | (z) = z −2|μ2 | , since they are subspaces of P |μ1 | (Mn×n (C)) and P |μ2 | (Mn×n (C)), respectively. It follows from this together with (1) and (2) that the direct sum in (3) consists of mutually non-isomorphic irreducible GL(n, C) × GL(n, C)-submodules. This completes the proof of (3). 

As a consequence, we obtain the next result for Bergman spaces. Theorem 4.10 For every λ > 2n − 1, we have a Hilbert direct sum A2λ (DIn×n ) =



P μ (Mn×n (C)).

μ∈P(n)

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which is the isotypic decomposition for the U(n) × U(n)-action. Furthermore, for every μ ∈ P(n) the subspace P μ (Mn×n (C)) is isomorphic to W μ∗ ⊗ W μ as a module over U(n) × U(n). In particular, this U(n) × U(n)-action is multiplicity-free. Proof Lemma 4.3(2) and Proposition 4.9(2) imply that for every μ ∈ P(n) the subspace P μ (Mn×n (C)) is irreducible and isomorphic to W μ∗ ⊗ W μ , both over U(n) × U(n). The last claim of Lemma 4.3 and Proposition 4.9(3) imply that all such subspaces are mutually non-isomorphic over U(n) × U(n). Note that we applied Lemma 4.3 to the case G = GL(n, C) × GL(n, C) and H = U(n) × U(n). It follows that, for every d ∈ N, the direct sum P d (Mn×n (C)) =



P μ (Mn×n (C))

μ∈Pd (n)

from Proposition 4.9(1), is the isotypic decomposition for the U(n) × U(n)-action. In particular, this sum is orthogonal with respect to any U(n) × U(n)-invariant inner product. The previous arguments and (4.1) imply that the sum in the statement is indeed a Hilbert direct sum. The rest of the claims of the statement follow as well from the previous remarks. 

Theorem 4.10 yields the following result. It provides an alternative proof to some of the results from [11]. Theorem 4.11 For any λ > 2n − 1, the C ∗ -algebra T (λ) (L ∞ (DIn×n )U(n)×U(n) ) acting on A2λ (DIn×n ) is commutative. Furthermore, every operator belonging to T (λ) (L ∞ (DIn×n )U(n)×U(n) ) preserves the Hilbert direct sum from Theorem 4.10 and acts by a constant multiple of the identity on each of its summands. Proof By Theorem 4.10, the restriction πλ |U(n)×U(n) is multiplicity-free and so Proposition 4.1(3) implies that EndU(n)×U(n) (A2λ (DIn×n )) is commutative. Hence, the first claim follows from Proposition 4.2(1) applied to H = U(n) × U(n). The second claim follows from Proposition 4.1(3) and Proposition 4.2(1). 

Remark 4.12 The commutativity claim from Theorem 4.11 was proved in [11] for the domain DIn×n among others. However, the corresponding isotypic decomposition was not explicitly computed in [11]. Hence, Theorems 4.10 and 4.11 together add information that can be used to study with more detail the operators belonging to the C ∗ -algebra T (λ) (L ∞ (DIn×n )U(n)×U(n) ). 4.3 Toeplitz Operators with Left and Right U(n)-invariant Symbols Let us now consider the left and right U(n)-actions on the weighted Bergman spaces A2λ (DIn×n ) for every λ > 2n − 1. These actions correspond to the subgroups U(n) L and U(n) R , respectively, of U(n) × U(n). In particular, the Hilbert direct sum from Theorem 4.10 is invariant under both the left and right U(n)-actions. However, as we will show, the terms of such direct sum are no longer irreducible for them.

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The next result yields the previous claims by describing the isotypic decompositions for the restrictions πλ |U(n) L and πλ |U(n) R . Theorem 4.13 For every λ > 2n − 1, the Hilbert direct sum A2λ (DIn×n ) =



P μ (Mn×n (C)),

μ∈P(n)

is the isotypic decomposition for both the left and right U(n)-actions. More precisely, for every μ ∈ P(n), the subspace P μ (Mn×n (C)) is an isotypic component for both the left and right U(n)-actions. Furthermore, these decompositions satisfy the following properties. (1) For the representation πλ |U(n) L on A2λ (DIn×n ) we have an isomorphism of U(n)modules A2λ (DIn×n )

=



μ

P (Mn×n (C))

μ∈P(n)



dim Wμ 

μ∈P(n)

j=1

W μ∗ ,

obtained from the fact that, for every μ ∈ P(n), the subspace P μ (Mn×n (C)) is isomorphic, as U(n) L -module, to the sum of dim W μ copies of the U(n)-module W μ∗ . (2) For the representation πλ |U(n) R on A2λ (DIn×n ) we have an isomorphism of U(n)modules A2λ (DIn×n )

=



μ

P (Mn×n (C))

μ∈P(n)



dim Wμ 

μ∈P(n)

j=1

W μ,

obtained from the fact that, for every μ ∈ P(n), the subspace P μ (Mn×n (C)) is isomorphic, as U(n) R -module, to the sum of dim W μ copies of the U(n)-module W μ . In particular, for n ≥ 2, the restricted representations πλ |U(n) L and πλ |U(n) R are not multiplicity-free. Proof The Hilbert direct sum holds by Theorem 4.10 and it is left and right U(n)invariant as noted above. To prove the rest of the statement we will only consider the left U(n)-action since the case of the right U(n)-action can be handled similarly. Let μ ∈ P(n) be given. By Theorem 4.10 there is an isomorphism T : P μ (Mn×n (C)) → W μ∗ ⊗ W μ of U(n) × U(n)-modules. In particular, T is an isomorphism of U(n) L -modules. Note that the U(n) L -module structure on the target is given by U · (w ∗ ⊗ w) = (U · w ∗ ) ⊗ w,

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for every U ∈ U(n), w ∗ ∈ W μ∗ and w ∈ W μ . Hence, for every w ∈ W μ \ {0} the subspace W μ∗ ⊗ w is a U(n) L -submodule naturally isomorphic to the U(n)-module W μ∗ . If we choose a basis w1 , . . . , wdim W μ of W μ , then we conclude that W

μ∗

μ

⊗W =

dim Wμ 

W

μ∗

⊗ wj

j=1

dim Wμ 

W μ∗

j=1

is a decomposition into irreducible U(n) L -submodules, where the indicated isomorphism is over U(n). These remarks and T yield the isomorphism of U(n)-modules μ

P (Mn×n (C))

dim Wμ 

W μ∗ .

j=1

On the other hand, for every μ1 , μ2 ∈ P(n), we have W μ1 ∗ W μ2 ∗ if and only W μ1 W μ2 , where both isomorphisms are considered over U(n). By Lemma 4.3 (for G = GL(n, C) and H = U(n)) and Proposition 4.7 such conditions are also equivalent to μ1 = μ2 . The previous arguments show that the subspace P μ (Mn×n (C)), where μ ∈ P(n), is an isotypic component for the U(n) L -action. They also prove the claims in (1). Finally, let us assume that n ≥ 2. Then, it is well known that the group U(n) has infinitely many irreducible representations of the form W μ , for some μ ∈ P(n), and so that dim W μ ≥ 2. For example, if d ∈ N, then we have de1 ∈ P(n) and W de1 P d (Cn )∗ as U(n)-modules, and so dim W de1 ≥ 2 for d ≥ 1. Since P μ (Mn×n (C)) is the sum of dim W μ irreducible U(n)-modules, we conclude that the 

isotypic decomposition for πλ |U(n) L is not multiplicity-free. We now obtain the next result for Toeplitz operators whose symbols are left or right U(n)-invariant but not necessarily both. Theorem 4.14 For every n ≥ 2 and λ > 2n − 1, both of the unital C ∗ -algebras T (λ) (L ∞ (DIn×n )U(n) L ) and T (λ) (L ∞ (DIn×n )U(n) R ) are not commutative. However, every operator belonging to either of these algebras preserve the Hilbert direct sum from Theorem 4.13. Proof We only consider T (λ) (L ∞ (DIn×n )U(n) L ), since the other algebra can be treated similarly. Note that we are assuming that n ≥ 2 and λ > 2n − 1. Theorem 4.13 shows that the restriction πλ |U(n) L is not multiplicity-free and that its isotypic components are finite dimensional. Thus, it follows from Proposition 4.2(3) that the C ∗ -algebra T (λ) (L ∞ (DIn×n )U(n) L ) is not commutative. The second claim follows from Proposition 4.2(1) and Proposition 4.1(2) applied to the subgroup H = 

U(n) L . We now show that the non-commutativity of the C ∗ -algebras from Theorem 4.14 is quite strong. It will also be useful in constructing commutative Banach algebras generated by Toeplitz operators that are non-C ∗ .

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Theorem 4.15 Let us assume that n ≥ 2. Then, for every λ > 2n − 1, there exist a ∈ L ∞ (DIn×n )U(n) L and b ∈ L ∞ (DIn×n )U(n) R such that the Toeplitz operators Ta(λ) and Tb(λ) are not normal. Proof We will consider only the case of the subgroup U(n) L , since the case of U(n) R is proved similarly. By Theorem 4.13, the isotypic decomposition for πλ |U(n) L is not multiplicity-free. Hence, there exist μ ∈ P(n) such that the subspace P μ (Mn×n (C)) is not irreducible over U(n) L . Furthermore, by the claims in Theorem 4.13(1) it follows that EndU(n) L (P μ (Mn×n (C))) Mm×m (C) as C ∗ -algebras, where m = dim W μ ≥ 2. Let A ∈ Mm×m (C) be some matrix which is not normal. This yields an operator T A ∈ EndU(n) L (P μ (Mn×n (C))) which is not normal either. Extend T A by zero in the rest of the terms of the Hilbert direct sum from Theorem 4.13 to obtain an operator T ∈ EndU(n) L (A2 (DIn×n )) such that T |P μ (Mn×n (C)) = T A . By Proposition 4.2(2) there exist a ∈ L ∞ (DIn×n )U(n) L such that T f , gλ = Ta(λ) f , gλ for every f , g ∈ P μ (Mn×n (C)). As we have done previously, Propositions 4.1 and (λ) (λ) (λ) 4.2 imply that the Toeplitz operators Ta and (Ta )∗ = Ta preserve the subspace P μ (Mn×n (C)). From the previous remarks we conclude that Ta(λ) |P μ (Mn×n (C)) = T |P μ (Mn×n (C)) = T A (Ta(λ) )∗ |P μ (Mn×n (C)) = T ∗ |P μ (Mn×n (C)) = T A∗ , (λ)

and so that Ta



is not normal.

Remark 4.16 The arguments used in the proof of Theorem 4.15 are easily seen to imply some stronger properties. More precisely, if we consider the subspace of A2λ (DIn×n ) given by W =

k 

P μ j (Mn×n (C))

j=1

for a finite family of elements μ1 , . . . , μk ∈ P(n), then for every operator T ∈ EndU(n) L (W ) ⊂ EndU(n) L (A2λ (DIn×n )) there exist symbols a ∈ L ∞ (DIn×n )U(n) L and b ∈ L ∞ (DIn×n )U(n) R such that (λ)

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In particular, for every μ ∈ P(n), the homomorphism of C ∗ -algebras given by T (λ) (L ∞ (DIn×n ) H ) → End H (P μ (Mn×n (C))) T → T |P μ (Mn×n (C)) , is surjective when H is taken to be either U(n) L or U(n) R . 4.4 Commutative Banach Algebras Generated by Toeplitz Operators In this subsection we will use both left and right U(n)-invariant symbols to obtain commutative Banach algebras generated by Toeplitz operators that are not C ∗ -algebras. Our first examples are stated in the next result, which is a consequence of the existence of some special non-normal Toeplitz operators proved in Theorem 4.15 (see also Proposition 3.8 and Remark 3.10). Theorem 4.17 Let us assume that n ≥ 2. Then, for every λ > 2n − 1 the following holds.  1 (1) There exists a ∈ L ∞ (DIn×n ) that satisfies a(Z ) = a (Z ∗ Z ) 2 , for almost every Z ∈ DIn×n , such that the unital Banach algebra generated by Ta(λ) is commutative and non-C ∗ . In particular, the unital C ∗ -algebra generated by Ta(λ) is not commutative.  1 (2) There exists a ∈ L ∞ (DIn×n ) that satisfies a(Z ) = a (Z Z ∗ ) 2 , for almost every (λ) Z ∈ DIn×n , such that the unital Banach algebra generated by Ta is commuta(λ) tive and non-C ∗ . In particular, the unital C ∗ -algebra generated by Ta is not commutative. We will also obtain commutative Banach algebras that contain the C ∗ -algebra as well as Toeplitz operators with left and right U(n)T invariant symbols. For this, we will use the next elementary auxiliary result. (λ) (L ∞ (DI U(n)×U(n) ) n×n )

Lemma 4.18 Let H be a Lie group with an irreducible unitary representation π on a Hilbert space H. Then, the following properties are satisfied. (1) The von Neumann algebra generated by π(H ) is the algebra B(H) of all bounded operators on H. (2) If K is a Hilbert space, and π L is the unitary representation of H on H ⊗ K given by π L (h)(u ⊗ v) = (π(h)(u)) ⊗ v for every h ∈ H , u ∈ H and v ∈ K, then the von Neumann algebra generated by π L (H ) consists of all the operators of the form T ⊗ IK where T ∈ B(H). Proof By Schur’s Lemma, the commutant π(H ) is the algebra CIH . Hence, the von Neumann algebra generated by π(H ) is given by π(H ) = B(H). This proves (1). Hence, (2) follows from (1) and the elementary properties of tensor products (see Corollary 1.5 in Chapter IV of [32]). 

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We now obtain a centralizing result for the C ∗ -algebras considered in Theorem 4.14. It will be used to prove the existence of commutative Banach algebras as noted before. Nevertheless, this result is interesting by itself. Theorem 4.19 For every λ > 2n − 1, the C ∗ -algebras T (λ) (L ∞ (DIn×n )U(n) L ) and T (λ) (L ∞ (DIn×n )U(n) R ) centralize each other. More precisely, we have ST = T S for every S ∈ T (λ) (L ∞ (DIn×n )U(n) L ) and T ∈ T (λ) (L ∞ (DIn×n )U(n) R ). Proof By Proposition 4.2(1) it is enough to prove the result for the von Neumann algebras EndU(n) L (A2λ (DIn×n )) and EndU(n) R (A2λ (DIn×n )). Since both of these algebras preserve the Hilbert direct sum from Theorem 4.13, it suffices to prove the result for the algebras EndU(n) L (P μ (Mn×n (C))) and EndU(n) R (P μ (Mn×n (C))), for every μ ∈ P(n). By Theorem 4.10, for every μ ∈ P(n) we have P μ (Mn×n (C)) W μ∗ ⊗ W μ , as U(n) × U(n)-modules. So the result finally reduces to showing that for every μ ∈ P(n), the von Neumann algebras EndU(n) L (W μ∗ ⊗ W μ ) and EndU(n) R (W μ∗ ⊗ W μ ) centralize each other. Since, both W μ and W μ∗ are irreducible U(n)-modules, Lemma 4.18 shows that EndU(n) L (W μ∗ ⊗ W μ ) and EndU(n) R (W μ∗ ⊗ W μ ) consist precisely of maps of the form T ⊗ IW μ and IW μ∗ ⊗ S, respectively, where T ∈ End(W μ∗ ) and S ∈ End(W μ ). This clearly completes the proof. 

From the proof of Theorem 4.19 we obtain the following immediate consequence. Corollary 4.20 For every λ > 2n − 1, the von Neumann algebras of intertwining operators EndU(n) L (A2λ (DIn×n )) and EndU(n) R (A2λ (DIn×n )) are the commutant of each other. As a consequence of Theorems 4.17 and 4.19 we obtain the following example of a commutative Banach algebra generated by Toeplitz operators. Corollary 4.21 Let us assume that n ≥ 2. For any given λ > 2n − 1, let a, b ∈ L ∞ (DIn×n ) be symbols such that a and b satisfy (1) and (2) from Theorem 4.17, (λ) (λ) respectively. Then, the unital Banach algebra generated by Ta and Tb is commuta∗ ∗ tive and non-C . In particular, the unital C -algebra generated by these two Toeplitz operators is not commutative. Finally, we prove the existence of a commutative Banach algebra with a quite large set of generators which are Toeplitz operators. As before, for n ≥ 2, the existence of the non-normal Toeplitz operators from the statement is guaranteed by Theorem 4.15. We recall that for a set of symbols S, we denote by T (λ) (S) the unital Banach algebra generated by the Toeplitz operators with symbols in S.

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Theorem 4.22 Assume that n ≥ 2. For a given λ > 2n − 1, choose symbols a ∈ L ∞ (DIn×n )U(n) L and b ∈ L ∞ (DIn×n )U(n) R such that the Toeplitz operators Ta(λ) and Tb(λ) are not normal, and let S = L ∞ (DIn×n )U(n)×U(n) ∪ {a, b}. Then, the Banach algebra T (λ) (S) is commutative and non-C ∗ . In particular, the unital C ∗ -algebra generated by S is not commutative. Proof To prove the commutativity of the given Banach algebra T (λ) (S), it is enough (λ) (λ) to show that for every pair of symbols ϕ, ψ ∈ S the Toeplitz operators Tϕ and Tψ commute with each other. If ϕ, ψ both belong to L ∞ (DIn×n )U(n)×U(n) , then this follows from Theorem 4.11. If {ϕ, ψ} = {a, b}, then the claim follows from Theorem 4.19. Let us now assume that ϕ ∈ L ∞ (DIn×n )U(n)×U(n) and that ψ is either a or b. In particular, ψ is either left or right U(n)-invariant. By Theorems 4.11 and 4.14, both (λ) (λ) Toeplitz operators Tϕ and Tψ preserve the Hilbert direct sum A2λ (DIn×n ) =



P μ (Mn×n (C)),

μ∈P(n)

(λ)

and the operator Tϕ acts by a constant multiple of the identity on each term. Hence, the Toeplitz operators Tϕ(λ) and Tψ(λ) commute with each other. This proves that T (λ) (S) is commutative. (λ) (λ) Since Ta and Tb are not normal their adjoints do not belong to T (λ) (S). Hence, T (λ) (S) is not C ∗ and so the unital C ∗ -algebra generated by S is not commutative.  Remark 4.23 For n ≥ 2, and with our current notation, if we choose non-normal operators S ∈ T (λ) (L ∞ (DIn×n )U(n) L ) and T ∈ T (λ) (L ∞ (DIn×n )U(n) R ), then the unital Banach algebra generated by T (λ) (L ∞ (DIn×n )U(n)×U(n) ) together with S, T is commutative, it is not C ∗ , and the unital C ∗ -algebra generated by such operators is not commutative. The proof is the same one used to obtain Theorem 4.22. This provides some more general commutative Banach non-C ∗ algebras generated by Toeplitz operators. We also note that, if we take S = L ∞ (DIn×n )U(n)×U(n) ∪ {ψ}, where ψ ∈ {a, b} and a, b are as in Theorem 4.22, then the Banach algebra T (λ) (S) satisfies the same conclusions from such theorem. This yields additional examples of commutative Banach non-C ∗ algebras generated by Toeplitz operators. Of course, we can as well consider an even more general example given by the Banach algebra generated by T (λ) (L ∞ (DIn×n )U(n)×U(n) ) together with some non-normal operator in either T (λ) (L ∞ (DIn×n )U(n) L ) or T (λ) (L ∞ (DIn×n )U(n) R ).

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R. Quiroga-Barranco Acknowledgements This research was partially supported by SNI-Conacyt and Conacyt Grants 280732 and 61517. Data availability statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References 1. Appuhamy, Amila, Le, Trieu: Commutants of Toeplitz operators with separately radial polynomial symbols. Complex Anal. Oper. Theory 10(1), 1–12 (2016) ˇ ckovi´c, Ž, Rao, N.V.: Commutants of analytic Toeplitz operators on the Bergman space. 2. Axler, S., Cuˇ Proc. Amer. Math. Soc. 128(7), 1951–1953 (2000) 3. Bauer, Wolfram, Choe, Boo Rim, Koo, Hyungwoon: Commuting Toeplitz operators with pluriharmonic symbols on the Fock space. J. Funct. Anal. 268(10), 3017–3060 (2015) 4. Bauer, Wolfram, Yañez, C.H., Vasilevski, N.: Eigenvalue characterization of radial operators on weighted Bergman spaces over the unit ball. Integral Equations Operator Theory 78(2), 271–300 (2014) 5. Bauer, Wolfram, Vasilevski, Nikolai: On the structure of a commutative Banach algebra generated by Toeplitz operators with quasi-radial quasi-homogeneous symbols. Integral Equations Operator Theory 74(2), 199–231 (2012) 6. Bauer, Wolfram, Vasilevski, Nikolai: On the structure of commutative Banach algebras generated by Toeplitz operators on the unit ball. Quasi-elliptic case. I: Generating subalgebras. J. Funct. Anal. 265(11), 2956–2990 (2013) 7. Bröcker, Theodor, tom Dieck, Tammo: Representations of compact Lie groups. Translated from the German manuscript. Corrected reprint of the 1985 translation. Graduate Texts in Mathematics, 98. Springer-Verlag, New York (1995) 8. Choe, B.R., Koo, H., Lee, Y.J.: Commuting Toeplitz operators on the polydisk. Trans. Amer. Math. Soc. 356(5), 1727–1749 (2004) ˇ ckovi´c, Željko, Louhichi, Issam: Finite rank commutators and semicommutators of quasihomoge9. Cuˇ neous Toeplitz operators. Complex Anal. Oper. Theory 2(3), 429–439 (2008) 10. Dawson, Matthew, Ólafsson, Gestur, Quiroga-Barranco, Raul: Commuting Toeplitz operators on bounded symmetric domains and multiplicity-free restrictions of holomorphic discrete series. J. Funct. Anal. 268(7), 1711–1732 (2015) 11. Dawson, Matthew, Quiroga-Barranco, Raul: Radial Toeplitz operators on the weighted Bergman spaces of Cartan domains. Representation theory and harmonic analysis on symmetric spaces, 97-114, Contemp. Math., 714, Amer. Math. Soc., [Providence], RI (2018) 12. Engliš, M.: Density of algebras generated by Toeplitz operators on Bergman spaces. Arkiv förMatematik 30(2), 227–243 (1992) 13. Esmeral, Kevin, Maximenko, Egor A.: Radial Toeplitz operators on the Fock space and square-rootslowly oscillating sequences. Complex Anal. Oper. Theory 10(7), 1655–1677 (2016) 14. Goodman, Roe, Wallach, Nolan R.: Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255. Springer, Dordrecht (2009) 15. Grudsky, S., Karapetyants, A., Vasilevski, N.: Toeplitz operators on the unit ball in Cn with radial symbols. J. Operator Theory 49(2), 325–346 (2003) 16. Grudsky, S., Karapetyants, A., Vasilevski, N.: Dynamics of properties of Toeplitz operators with radial symbols. Integral Equations Operator Theory 50(2), 217–253 (2004) 17. Grudsky, Sergei M., Maximenko, Egor A., Vasilevski, Nikolai L.: Radial Toeplitz operators on the unit ball and slowly oscillating sequences. Commun. Math. Anal. 14(2), 77–94 (2013) 18. Grudsky, S., Quiroga-Barranco, R., Vasilevski, N.: Commutative C ∗ -algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal. 234(1), 1–44 (2006) 19. Hua, L.K.: Harmonic analysis of functions of several complex variables in the classical domains. Translated from the Russian, which was a translation of the Chinese original, by Leo Ebner and Adam Korányi. With a foreword by M. I. Graev. Reprint of the 1963 edition. Translations of Mathematical Monographs, 6. American Mathematical Society, Providence, R.I. (1979) 20. Knapp, Anthony W.: Lie groups beyond an introduction. Second edition. Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA (2002)

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Radial-like Toeplitz Operators on Cartan… 21. Kobayashi, Shoshichi; Nomizu, Katsumi: Foundations of differential geometry. Vol. I. Reprint of the 1963 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York (1996) 22. Korenblum, Boris, Zhu, Ke He.: An application of Tauberian theorems to Toeplitz operators. J. Operator Theory 33(2), 353–361 (1995) 23. Le, Trieu: Commutants of separately radial Toeplitz operators in several variables. J. Math. Anal. Appl. 453(1), 48–63 (2017) 24. Mok, Ngaiming: Metric rigidity theorems on Hermitian locally symmetric manifolds. Series in Pure Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ (1989) 25. Quiroga-Barranco, Raul: Separately radial and radial Toeplitz operators on the unit ball and representation theory. Bol. Soc. Mat. Mex. (3) 22(2), 605–623 (2016) 26. Quiroga-Barranco, Raul, Sanchez-Nungaray, Armando: Toeplitz operators with quasi-homogeneous quasi-radial symbols on some weakly pseudoconvex domains. Complex Anal. Oper. Theory 9(5), 1111–1134 (2015) 27. Quiroga-Barranco, Raul, Vasilevski, Nikolai: Commutative C ∗ -algebras of Toeplitz operators on the unit ball. I. Bargmann-type transforms and spectral representations of Toeplitz operators. Integral Equations Operator Theory 59(3), 379–419 (2007) 28. Quiroga-Barranco, Raul, Vasilevski, Nikolai: Commutative C ∗ -algebras of Toeplitz operators on the unit ball. II. Geometry of the level sets of symbols. Integral Equations Operator Theory 60(1), 89–132 (2008) 29. Rodriguez, M.A.R.: Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasihomogeneous symbols. Bol. Soc. Mat. Mex. (3) 26(3), 1243–1271 (2020) 30. Suárez, Daniel: The eigenvalues of limits of radial Toeplitz operators. Bull. Lond. Math. Soc. 40(4), 631–641 (2008) 31. Schmidt, Robert: Über divergente Folgen and lineare Mittelbildungen. Math. Z. 22, 89–152 (1925) 32. Takesaki, Masamichi: Theory of operator algebras. I. Springer-Verlag, New York-Heidelberg (1979) 33. Upmeier, Harald: Toeplitz operators and index theory in several complex variables. Operator Theory: Advances and Applications, 81. Birkhäuser Verlag, Basel (1996) 34. Vasilevski, Nikolai: Quasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators. Integral Equations Operator Theory 66(1), 141–152 (2010) 35. Vasilevski, Nikolai: On Toeplitz operators with quasi-radial and pseudo-homogeneous symbols. In: Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory, vol. 2, pp. 401–417, Springer Verlag, Heidelberg (2017) 36. Zorboska, Nina: The Berezin transform and radial operators. Proc. Amer. Math. Soc. 131(3), 793–800 (2003) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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Complex Analysis and Operator Theory (2022) 16: 81 https://doi.org/10.1007/s11785-022-01259-y

Complex Analysis and Operator Theory

A Short Proof for the Twisted Multiplicativity Property of the Operator-Valued S-transform Roland Speicher1 Received: 16 April 2022 / Accepted: 18 June 2022 / Published online: 11 July 2022 © The Author(s) 2022

Abstract We provide a short proof for the twisted multiplicativity property of the operator-valued S-transform. Keywords Operator-valued free probability · S-transform · Cumulants · Multilinear function series Mathematics Subject Classification 46L54

1 Introduction In free probability the most basic operations are the free addititive and multiplicative convolutions, given by the sum and the product, respectively, of two freely independent random variables x and y. Voiculescu provided with the R-transform [1] and the S-transform [2] analytic functions which describe these operations via R x+y (z) = R x (z)+ R y (z) and Sx y (z) = Sx (z)· S y (z); Haagerup provided in [3] different proofs of Voiculescu’s results, relying on Fock space and elementary Banach algebra techniques. Those functions can also be considered as formal power series; their coefficients are then determined in terms of the moments of the considered variables and the above Dedicated to the memory of Jörg Eschmeier. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht. The author was partially supported by the SFB-TRR 195 of the German Research Foundation DFG.

A previous version of this chapter was published Open Access under a Creative Commons Attribution 4.0 International License at http://link.springer.com/10.1007/s11785-022-01259-y.

B 1

Roland Speicher [email protected] Universität des Saarlandes, Fachrichtung Mathematik, Postfach 151150, 66041 Saarbrücken, Germany

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relations are translations into generating series of how moments of the sum or the product of free variables are determined in terms of the moments of the individual variables. In the case of the additive convolution, this is quite straightforward, the coefficients of the R-transform are then the free cumulants. For the multiplicative case the situation is a bit more complicated, but still one can get the above mentioned multiplicativity from the basic properties of free cumulants and playing around with formal power series; for this we refer in particular to the book [4], which covers the combinatorial facet of free probability. Other general introductions to the basics of free probability can be found in [5–7]. There exists a (very powerful) operator-valued extension of free probability [8] with its operator-valued versions of the additive and multiplicative free convolutions and of the R-transform and the S-transform. Whereas for the additive case and the Rtransform, the statements and proofs can easily be extended to the operator-valued case, the multiplicative situation is more complicated. It was actually discovered by Dykema in [9] that in this case the formula for the S-transform of a product of free variables involves a twist, due to the non-commutativity of the underlying algebra of “scalars". Though Dykema uses the language of formal power series adapted to this operatorvalued setting (formal multilinear function series), both his proofs in [9] and [10] use quite involved Fock space realizations, modeled according to Haagerup’s approach in the scalar-valued case [3]. Our goal here is to give a more direct proof of the twisted multiplicativity of the S-transform, just using the basic definitions and properties of free cumulants and of the S-transform, as well as easy formal manipulations with power series. This is in principle just an operator-valued adaptation of the same kind of arguments from [11]. However, since the order matters now, finding the right way of writing and manipulating the formulas was not as straighforward as it might appear from the polished final write-up. So it could be beneficial for future use by others to record those calculations here. Reconsidering the operator-valued S-transform was initiated by discussions with Kurusch Ebrahimi-Fard and Nicolas Gilliers on their preprint [12]. There they provide an “understanding” of the twist from an higher operadic point of view. Our calculations here can also be seen as a more pedestrian version of parts of their work. One should note that in their setting (as well as in Dykema’s combinatorial interpretation via linked non-crossing partitions) the T -transform, which is the multiplicative inverse of the Stransform, seems to be the more appropriate object. For our formal manipulations with power series there is no such distinction.

2 Basic Definitions We are working in an operator-valued probability space (A, B, E); this means that A is a unital algebra with a unital subalgebra B ⊂ A and a conditional expectation E : A → A, i.e., a linear map with the additional properties E[b] = b,

E[b1 ab2 ] = b1 E[a]b2

for all b ∈ B, a1 , a2 ∈ A.

Elements in A will be called (random) variables and the basic information about them is encoded in their operator-valued moments

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E[xb1 xb2 · · · bn x]

(n ∈ N; b1 , . . . , bn ∈ B)

or, equivalently, in their operator-valued cumulants κ(xb1 , xb2 , . . . , xbn , x)

(n ∈ N; b1 , . . . , bn ∈ B).

The latter can be defined in a recursive way via E[xb1 xb2 · · · bn x] =

n 



k=0 1≤q1 0 and δ > 0 such that for every z ∈ C with |z| < δ we have |x1i (z x) ˆ − x1 j (z x| ˆ ≥ a|z|, i = j.

(2.2)

The constant a continuously depends on  n     ∂ x ∂ x1 j 1j   min  (0) − (0) xk  ,   i, j ∂ xk ∂ xk k=2

and, therefore, may be chosen such that (2.2) holds uniformly in a neighborhood of x. ˆ Recall that for each j ∈ Iλ the point y j,λ (x) ˆ in the component { R˜ j = 0} was ˆ we will write y j (x). ˆ For defined by (1.8). In our case λ = 1, and instead of y j,1 (x) each i ∈ I1 and z ∈ C close to 0 define τi j (z) ∈ C by the condition that it is the closest to 1 root of the following equation in τ ˆ = 0. R˜ i (τ y j (z x))

(2.3)

As mentioned above, conditions (a) and (b) imply that every line in Cn passing through the origin and close to the x1 -axis has only one point of intersection with the hypersurface { R˜ i = 0} that is close to (1, 0, ..., 0). When this line is the line passing through ˆ this point determines τi j (z) in the following way. Write the the origin and y j (z x), homogeneous decomposition of R˜ j : R˜ i (x) = R˜ i (x1 , x) ˆ =

ti

Sil (x),

l=0

where Sil (x) is a homogeneous polynomial of degree l in x1 , ..., xn and ti is the degree of R˜ i . We always assume that R˜ 0 = −1. Then τi j (z) is the only root of the equation (in τ ) ti

τ l Sil (y j (z x)) ˆ =0

l=0

which is close to 1. Since this root has multiplicity 1, τi j (z) is an analytic functiion of z in a small neighborhood of the origin in C, which we denote by U . Of course, τ j j (z) = 1, τi j (z) = 1 for i = j, z = 0, and, of courses, ˆ = 0 for all z ∈ U . It is easy to see that (2.2) implies that there R˜ i (τi j (z)y j (z x))

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exist positive constants c and C such that for all j = i c|z| ≤ |τi j (z) − 1| ≤ C|z|.

(2.4)

Using the notation A(x) = x1 A1 + · + xn An , it follows from the definition of the joint ˆ Now, (2.4) spectrum that the reciprocals μi j (z) = τi j1(z) are eigenvalues of A(y j (z x)). implies that a similar estimate (with different constants) holds for the eigenvalues μi j : c1 |z| ≤ |μi j (z) − 1| ≤ C1 |z|, j = i.

(2.5)

This shows that δ j in (1.9) should be chosen smaller than c1 |z| (of course, the integral (1.9) is the same for all δ j < c1 |z|). We call it δ j (z). Let e1 , ..., e N be an eigenbasis for A1 , where e1 , ..., e M1 are eigenvectors with eigenvalue 1, and e M1 +1 , ..., e N correspond to eigenvalues different from 1, so that in the basis e1 , ..., e N the  matrix A1 is diagonal with the first M1 diagonal entries equal to 1 (recall, M1 = j∈I1 m j ), and the others, which we denote by α M1 +1 , ..., α N , being different from 1. Write A2 , ..., An in the basis e1 , ..., e N :  N j A j = alm

l,m=1

.

For t ∈ C and j ∈ I1 let M j (t) be the M1 × M1 block of the matrix 



 n ∂ x1 j  t− ∂x  l=2

l

xl

I−

(0,...,0)

n

x l Al

l=2

formed by the first M1 rows and the first M1 columns. Here xˆ = (x2 , . . . , xn ) is the point that appeared in Theorem 1.11. Consider the following polynomials in t S j (t) = det(M j (t)).  For each j this is a non-trivial polynomial of degree M1 S j (t) → ∞ as t → ∞ . Thus, there exists some positive number b < c1 (the constant from (2.5)) such that these polynomials do not vanish in the punctured disk of radius b centered at the origin: S j (t) = 0 for t ∈ C, 0 < |t| < b, j = 1, ..., k.

(2.6)

Choose some t0 satisfying 0 < t0 < b. Next, we will show that there exists a positive constant M such that for |w − 1| = t0 |z|, where z is close to zero, the following norm estimate holds:  −1  n  M  ≤  w I − x1 j (z x)A . ˆ − zx A 1 s s   |z|

(2.7)

s=2

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The norm here is understood as the norm of an operator acting on C N . Let η ∈ C N , η = 1, and ζ ∈ C N satisfies   n w I − x1 j (z x)A ˆ 1− zxl Al ζ = η,

(2.8)

l=2

which is a system of N equations in variables ζ1 , ...,ζ N , coordinates of ζ in the basis n ∂ x1 j 2 ˆ =z e1 , ..., e N . Since w−1 = t0 zeiθ and 1−x1 j (z x) s=2 ∂ xs |(0,...,0) x s +O(|z| ), the first M1 equation of this system can be written as   (2.9) z M j (t0 eiθ ) + O(z)I M1 )( M1 ζ ) = ( M1 η) − zN (ζ N −M1 ), where I M1 is the M1 × M1 identity matrix, ( M1 ζ ) = (ζ1 , ..., ζ M1 ) and ( M1 η) = (η1 , ..., η M1 ), (ζ N −M1 ) = (ζ M1 +1 , ..., ζ N ), and N is the M1 × (N − M1 ) part of the matrix x2 A2 + · · · + xn An consisting of the entrees in the first M1 rows and the last N − M1 columns. Consider (2.9) as a system of equations in ζ1 , ..., ζ M1 . Then (2.6) implies that for sufficiently small z the main determinant of this system does not vanish (it is equal to z M1 S j (t0 eiθ )+ O(|z| M1 +1 )). Now, Cramer’s rule implies that ζ1 , ..., ζ M1 are expressed in terms of ζ M1 +1 , ..., ζ N and η1 , ..., η M1 in the following way: ! ζi =

" M1 N 1 Dil (z) ψil (z) η ζl , i = 1, ..., k1 . + l iθ z S j (t0 e ) S j (t0 eiθ ) l=1

(2.10)

l=M+1

where Dil (z) and ψil (z) are uniformly bounded analytic functions in a small punctured neighborhood of the origin (and, therefore, analytically extendable to the origin). Substitute these expressions of ζi , i = 1, ..., M1 in the remaining N − M1 equations of the system (2.8). We obtain a system in the following form:   (1 + t0 zeiθ )I N −M1 − x1 j (z x) ˆ − L(z) (ζ N −M1 ) = (η N −M1 ) − T (z)( M1 η),

(2.11)

where I N −M1 is the (N −M1 )×(N −M1 ) identity matrix,  is the (N −M1 )×(N −M1 ) diagonal matrix with α M1 +1 , ..., α N on the main diagonal (they are coming from A1 ), L(z) is an (N − M1 ) × (N − M1 ) matrix whose all entries are of the order of z, and T (z) is the following (N − M1 ) × M1 matrix with entrees (T (z))im =

1 1 x ailr Dlm (z), r S j (t0 eiθ )

n

M

r =2

l=1

that is the product of the lower left  (N − M1 ) × M1 block of the matrix

and the matrix D(z) =

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Dlm (z) S j (t0 eiθ )

.

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Since for z close to zero, x1 j (z x) ˆ is close to one, and since α M+1 , ..., α N are not equal to one, it follows that the main determinant of (2.11) considered as a system in ζ M1 +1 , ..., ζ N stays away from 0 as z → 0. Since for all j = 1, ..., N , |η J | ≤ 1, this implies that ζ M1 +1 , ..., ζ N stay bounded as z → 0. In summary: there exists a constant M˜ such that |ζ j | ≤

M˜ j = 1, ..., M1 , |ζ j | ≤ M˜, j = M1 + 1, ..., N . |z|

(2.12)

M1 |z|

(2.13)

This yeilds

ζ ≤

for some constant M1 independent of z and η, which proves (2.7). Now, (2.7) implies     −1   n    1    P j (y( z x))  w I − x1 j (z x)A ˆ = ˆ 1− zxl Al dw    2πi |w−1|=t0 |z| l=2    −1  n   1  2π t0 |z| ≤ M1 t0 . w I − x ≤ (z x)A ˆ − zx A max  1j 1 l l   2π |w−1|=t0 |z| l=2

(2.14) Since, the integral  |w−1|=t0 |z|

 −1 n w I − x1 j (z x)A ˆ 1− zxl Al dw l=2

is an analytic function in z in a small punctured disk centered at the origin, Riemann’s theorem implies that it is analytically extendable to z = 0. This completes the proof of Theorem 1.11 for λ = 0. 2. λ = 0. The details of the proof in this case are very similar to those when λ = 0. The only differences are the following. If xˆ is sufficiently close to 0, then close to 0 spectral points ˆ are xn+1,i (x) ˆ − xn+1, j (x), ˆ i ∈ I0 of the matrix A1 + x2 A2 + · · · + xn An − xn+1, j (x)I ˜ implies that there (here we again write xn+1, j instead xn+1 ,0, j ). Now, condition b) exists c > 0 such that for sufficiently small z |xn+1,i (z x) ˆ − xn+1, j (z x)| ˆ > c|z|, i, j ∈ I0 , i = j. Further, we choose a basis e1 , . . . , e N which is an eigenbasis for A1 so that eigenvectors e1 , . . . , e M0 are 0-eigenvectors, and e M0 +1 , . . . , e N are eigenvectors with non-trivial eigenvalues, and write A2 , . . . , An in this basis. The matrix M j (t) in

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this case is defined as M0 × M0 block of the matrix   n n ∂ xn+1, j t+ xl I − x l Al ∂ xl l+2

l=2

of the first M0 rows and the first M0 columns. Again, there exists b > 0 such that the polynomial S j (t) = det(M j (t)) does not vanish in the punctured disk of radius b centered at the origin and we pick 0 < t0 < b. The remaining details of the proof are practically identical to those in the case λ = 0 (of course, the integral (1.9) in (2.14) is replaced with the integral (1.10)). The proof is complete. Remark 1 We would like to note that the above proof holds when A1 is just a diagonal matrix, not necessarily normal. Remark 2 In general, limit projections depend on the choice of x. ˆ Remark 3 It is easily seen that a similar proof holds when a point approaches (1, 0, ..., 0) not along a line, but along a smooth curve γ (z) ∈ Cn−1 which is not tangent to any component of Uˆ at the origin.

3 Relations Between Component Projections Pj, : Proofs of Theorems 1.15 and 1.20 Each projection Pλ in the spectral resolution (1.13) is, of course, represented by the integral Pλ =

1 2πi

 γλ

(w − A1 )−1 dw,

where γλ is a contour which separates λ from the rest of the spectrum of A1 . Proposition 3.1 The limit projections P j,λ from Theorem 1.11 satisfy the following relations: Pi,λ P j,λ = 0 if i = j, i, j ∈ Iλ .  j∈Iλ P j,λ = Pλ

(3.2) (3.3)

Proof The proofs for λ = 0 and λ = 0 are very similar, so we will give it for λ = 0. Fix λ ∈ σ (A1 ), λ = 0 and j ∈ Iλ . Let xˆ ∈ O - the neighborhood from the proof ˆ i ∈ Iλ , the eigenvalues of Theorem 1.11. Equation (2.5) gives an estimate for μi j (x), ˆ = x1,λ, j (x)A ˆ 1 + · · · + xn An . Recall that μi j (x) ˆ close to 1 of the operator A(y j,λ (x)) are the reciprocals of the roots of Eq. (2.3) which in the proof of Theorem 1.11 were denoted by τi, j (1). In the proof of Theorem 1.11 we supresed the dependance of these

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roots on λ, assuming that λ = 1, but here it is appropriate to call them τi, j,λ (z). Recall ˆ was defined by (1.8). Write that y j,λ (x) ˆ = τi, j (1), y˜i, j,λ (x) ˆ = τi, j,λ (x)y ˆ j,λ (x). ˆ τi, j,λ (x) Of course, ˆ = y j,λ (x). ˆ y˜ j, j,λ (x) It was mentioned above that since R˜ i is an irreducible polynomial, every regular point ˆ is a bounded analytic function of xˆ in a of { R˜ i = 0} has multiplicity 1, so τi, j,λ (x) punctured neighborhood of the origin, and, therefore, can be extended to the origin to become an analytic function in the whole neighborhood satisfying τi, j,λ (0) = 1. It is also easy to compute that for m = 2, . . . , n  ∂x ∂x j ˆ ∂ 1,λ,i ˆ − ∂1,λ, ˆ τi, j,λ (x) ∂τi, j,λ xm ( x) xm ( x) (x) ˆ = , ∂x ∂ xm x1,λ, j (x) ˆ − xm 1,λ,i (x) ˆ

(3.4)

∂ xm

and, therefore, (3.4)implies ∂τ j,λ,i (0) = ∂ xm

∂ x1i ∂ xm

(0) −

∂ x1 j ∂ xm

(0)

λ

.

(3.5)

Further, the projections Pi,λ ( y˜ j, j,λ (x)) ˆ and P j,λ ( y˜i, j,λ (x)) ˆ (defined by the integral (1.9) ) satisfy ˆ ˆ = 0, i = j. Pi,λ ( y˜i, j,λ (x))P j,λ ( y˜ j, j,λ ( x)) ˆ xˆ → 0 as xˆ → 0, Theorem 1.11 implies P j,λ ( y˜ j, j,λ (x)) ˆ → P j,λ Since τi, j,λ (x) and Pi,λ ( y˜i, j,λ (x)) ˆ → Pi,λ as xˆ → 0. This proves (3.2). To prove (3.3) we fix a contour γλ that separates λ from the rest of the spectrum of A1 . Suppose that x is close to (1/λ, 0, . . . , 0). Consider the matrix A(x) = x1 A1 + · · · + xn An . Then 1 ∈ σ (A1 /λ) and Pλ is the projection on the 1-eigenspace of A1 /λ. Also, γλ /λ separates 1 from the rest of the spectrum of A1 /λ. Of course, A(x) → A1 /λ as xˆ → 0, and x1 → 1/λ. If |x1 −1/λ|+|x2 |+· · ·+|xn | is small enough, there are no spectral points of A1 /λ on γλ /λ, and, therefore, 1 2πi

 γλ /λ

(w − A(x))−1 dw →

1 2πi

 −1 1 w − A1 dw = Pλ , λ γλ /λ



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as x → (1/λ, 0, . . . , 0). ˆ then the poles of (w − A(x))−1 which lie inside γλ /λ are the If x1 = x1,λ, j (x), ˆ i ∈ Iλ . Now, by the residue theorem reciprocals of τi, j,λ (x), 1 2πi

 γλ

(w − A(x))−1 dw =



Pi,λ ( y˜i, j,λ (x)) ˆ →

i∈Iλ



Pi,λ ,

iıIλ

and we are done.   We now concentrate on the case when n = 2. Of course, in this case Uˆ = {0}, and the limit projections are the same for all x2 (which is xˆ in this situation). ˜ and Proposition 3.6 Let n = 2, λ ∈ σ (A1 ). Assume that conditions (a) and (b), or a) ˜ are satisfied at λ when λ = 0 or λ = 0 respectively. For i, j ∈ Iλ we have b) P j,λ A2 Pi,λ = 0, i = j

(3.7)

If there exists a neighborhood O of the origin, such that at every regular point of { R˜ j = 0} ∩ O, the range of P j,λ (x2 ) consists of 1-eigenvectors for A(y j,λ ((x2 )) = x1,λ, j (x2 )A1 + x2 A2 , if λ = 0, and , 0-eigenvectors of A(y j,0 ((x2 )) = A1 + x2 A2 − x3,0, j (x2 )I when λ = 0, or equivalently, 

   A(y j,λ (x2 )) − I P j,λ (x2 ) = P j,λ (x2 ) A(y j,λ (x2 )) − I = 0, λ = 0, (3.8) A(y j,0 ((x2 ))P j,0 (x2 ) = P j,0 (x2 )A(y j,0 (x2 )) = 0, λ = 0,

(3.9)

then        P x (0)A + A = P (0)A + A P j,λ x1,λ, 1 2 j,λ j,λ 1 2 P j,λ j 1,λ, j

P j,0

P j,λ





=−

 x1,λ, j (0) P j,λ , 2

(3.10)

 x3,0, j (0) P j,0 , 2

(3.11)

λ = 0,      A2 − x3,,0, j (0)I P j,0 = P j,0 A2 − x 3,0, j (0)I

λ=0       x1,λ, j (0)A1 + A2 Pi,λ = P j,λ x1,λ, j (0)A1 + A2 Pi,λ

P j,0

=



A2 −

where P j,λ = lim x2 →0

Reprinted from the journal

= 0, i = j, λ = 0,      Pi,0 = P j,0 A2 − x3,0, j (0)I Pi,0

(3.12)

= 0, i = j, λ = 0.

(3.13)

 x3,0, j (0)I

d P j,λ (x2 ) , d x2

j ∈ Iλ .

783

M. I. Stessin

Proof I. λ = 0. We have for every x2 close to 0: P j,λ (x2 )A(x1,λ, j (x2 ), x2 ) = A(x1,λ, j (x2 ), x2 )P j,λ (x2 ).

(3.14)

By Theorem 1.11 both sides of this equality are analytic functions of x2 in a neighborhood of the origin. Differentiate with respect to x2 : d P j,λ (x2 )  A(x1,λ, j (x2 ), x2 ) + P j,λ (x2 )(x1,λ, j (x 2 )A1 + A2 ) d x2 d P j,λ (x2 )  = A(x1,λ, j (x2 ), x2 ) + (x1,λ, j (x 2 )A1 + A2 )P j,λ (x 2 ). d x2 Passing to the limit as x2 → 0 and using P j,λ A1 = A1 P j,λ = λP j,λ , we obtain 1 1    P j,λ A1 + λx1,λ, A1 P j,λ + λx1,λ, j (0)P j,λ + P j,λ A2 = j (0)P j,λ + A2 P j,λ , λ λ so that 1  1 P j,λ A1 + P j,λ A2 = A1 P j,λ + A2 P j,λ . λ λ Multiply the last equality from the left by Pi,λ with i = j, i ∈ Iλ . Since Pi,λ A1 = A1 Pi,λ = λPi,λ , we obtain 1 Pi,λ P j,λ A1 = Pi,λ P j,λ + Pi,λ A2 P j,λ λ yeilding Pi,λ P j,λ



1 A1 − I λ

 = Pi,λ A2 P j,λ .

Multiply the last equality by P j,λ from the right. Since 

 1 A1 − I P j,λ = 0, λ

and P 2j,λ = P j,λ , we conclude that Pi,λ A2 P j,λ = 0, and (3.7) is proved for λ = 0..

784

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If (3.8) holds, then P j,λ (x2 )A(x1,λ, j (x2 ), x2 ) = P j,λ (x2 )

(3.15)

A(x1,λ, j (x2 ), x2 )P j,λ (x2 ) = P j,λ (x2 ).

(3.16)

Differentiating twice (3.15) and (3.16) with respect to x2 and passing to the limit as x2 → 0 yields 1      λ P j,λ A1 + 2P j,λ (x 1,λ, j (0)A1 + A2 ) + x 1,λ, j (0)P j,λ A1 = P j,λ , 1      x1,λ, j (0)A1 P j,λ + 2(x 1,λ, j (0)A1 + A2 )P j,λ + λ A1 P j = P j,λ ,

where P j,λ = lim x2 →0 to the left gives

d 2 P j,λ (x2 ) . Moving the righthand side of the last two equations d x22

  P j,λ ( λ1 A1 − I ) + 2P j,λ (x1,λ, j (0)A1 + A2 ) + x 1,λ, j (0)P j,λ A1 = 0, (3.17) 1     x1,λ, j (0)A1 P j,λ + 2(x 1,λ, j j (0)A1 + A2 )P j,λ + ( λ A1 − I )P j,λ = 0. (3.18)

Multiply (3.17) from the right by P j,λ and by Pi,λ , i = j. Again, since 

 1 λ A1

− I Pl,λ = Pl,λ



 1 λ A1

−I

= 0, l ∈ Iλ ,

Pl,λ A1 = A1 Pl,λ = λPl,λ , l ∈ Iλ Pi,λ P j,λ = P j,λ Pi,λ = 0, i, j ∈ Iλ , i = j, we obtain    x1,λ, j (0)  P j,λ x1,λ, P j,λ j (0)A1 + A2 P j,λ = − 2    P j,λ x1,λ, j (0)A1 + A2 Pi,λ = 0, i, j ∈ Iλ , i  = j, Similarly, multiplication of (3.18) from the left by P j,λ and by Pi,λ results in    x1,λ, j (0)   P j,λ P j,λ x1,λ, j (0)A1 + A2 P j,λ = − 2     Pi,λ x1,λ, j (0)A1 + A2 P j,λ = 0, i, j ∈ Iλ , i  = j, which finishes the proof for λ = 0. II λ = 0.

Reprinted from the journal

785

M. I. Stessin

The proof in this case is similar and goes along the following lines. 

   A1 + x2 A2 − x3,o, j (x2 )I P j,0 (x2 ) = P j,0 (x2 ) A1 + x2 A2 − x3,0, j (x2 )I ,       = P A + P  A A2 − x3,0, (0)I P + A P − x (0)I . j,0 1 j,0 j,0 2 j j,0 1 3,0, j

Multiply the last relation from the right by Pi,0 and use P j,0 Pi,0 = Pi,0 P j,0 = 0, i = j, A1 Pl,0 = Pl,0 A1 = 0, l ∈ I0 . Since  P j,0 (x2 )Pi,0 (x2 ) ≡ 0, i = j, P j,0 Pi,0 = −P j,0 Pi,0 ,

we obtain  0 = −A1 P j,0 Pi,0 = A1 P j,0 Pi,0 = P j,0 A2 Pi,0 ,

which is (3.7) for λ = 0. Finally, under the assumption of (3.9), the proof of (3.11) and (3.13) is obtained the same way as the one of (3.10) and (3.12) by twice differentiating   P j,0 (x2 ) A1 + x2 A2 = x3,,0, j (x2 )P j,0 (x2 )   A1 + x2 A2 P j,0 (x2 ) = x3,0, j (x2 )P j,0 (x2 ) with respect to x2 , passing to the limit as x2 → 0, and multiplying by P j,0 and Pi,0 . We are done.   Obviously, (3.8) and (3.9) hold when every spectral component passing through (1/λ, 0) has multiplicity 1. Proof of Theorem 1.15 I. λ = 0 Fix j ∈ Iλ . In the results of the previous section we were interested only in the roots of (2.3) (3.19) R˜ i (τ x1,λ, j (x2 ), τ x2 ) = 0. which were close to 1. Now we will consider all the roots of this equation. Observe that if x2 is small enough, all these roots are close to the ratios λ/μ where μ ∈ σ (A1 ). Moreover, since condition b) holds at every spectral point of A1 , the multiplicity of each of these roots is 1 (when x2 is close to 0). We denote them by τ j,λ,i,μ (x2 ), 1 ≤ i ≤ s, , where μ is defined by λ (3.20) τ j,λ,i,μ (x2 ) → , as x2 → 0, μ which, of course, implies i ∈ Iμ . It was mentioned above that τ j,λ,i,μ (x2 ) are analytic functions of x2 in a neghborhood of the origin. Obviously, τ j,λ, j,λ (x2 ) = 1.

786

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Spectral Analysis Near Regular Point of...

Further, (1.9) defined projections P j,λ (x2 ). We define P j,λ,i,μ (x2 ) = Pi,μ (τ j,λ,i,μ x2 ).

(3.21)

It follows from Theorem 1.11 and (3.21) that P j,λ,i,μ (x2 ) are analytic operator-valued functions of x2 in a neighborhood of the origin, and lim P j,λ,i,μ (x2 ) = Pi,μ .

x2 →0

(3.22)

Also   d Pi,λ (τ j,λ,i,μ x2 ) dτ j,λ,i,μ (x2 ) d P j,λ,i,μ τ j,λ,i,μ (x2 ) + x2 , (x2 ) = d x2 d x2 d x2 so that (3.5) and (3.20) imply that for λ = μ we have d P j,λ,i,λ d P j,λ,i,λ  (0) = lim (x2 ) = Pi,λ (0). x2 →0 d x2 d x2

(3.23)

Recall that projections (3.21) appeared in Proposition 3.1 for λ = μ. We renamed them here, since we need these projections corresponding to all roots of (3.19) and want to indicate to which reciprocal of a point in σ (A1 ) the corresponding point of the proper joint spectrum converges. Note that τ j,λ, j,λ (x2 ) = 1 implies P j,λ, j,λ (x2 ) = P j,λ (x2 ). Since for i ∈ Iλ components { R˜ i (x1 , x2 ) = 0} have multiplicity 1 in the projective joint spectrum , the rank of P j,λ,i,μ (x2 ) is equal to 1 for every i ∈ Iλ ∩ Iμ . For other i∈ / Iλ the rank of P j,λ,i,μ is equal to m i - the multiplicity of the spectral component { R˜ i = 0}. As it was mentioned above, for a rank 1 projection (3.8) holds, and, therefore, if  is small, we have 

 x1,λ, j (x2 + )A1 + (x2 + )A2 − I P j,λ (x2 + ) = 0.

We write this relation in the form 1 2πi

 γ

−1   dw = 0, w − 1 w − x1,λ, j (x2 + )A1 − (x2 + )A2

(3.24)

where, as above, γ is a circle centered at 1 which separates 1 from all other eigenvalues of x1,λ, j (x2 )A1 + x2 A2 , and, therefore, the same is true for all x1,λ, j (x2 + )A1 + (x2 + )A2 for all sufficiently small .

Reprinted from the journal

787

M. I. Stessin

Let us write (3.24) as 1 0= 2πi !

 γ

−1   w − 1 w − x1,λ, j (x2 )A1 − x2 A2  

 ∞ (k)   x1,λ, j (x2 ) k  A1 × I− + A2  + k! k=2  −1 −1 × w − x1,λ, j (x2 )A1 − x2 A2 dw  −1   1 = w − 1 w − x1,λ, j (x2 )A1 − x2 A2 2πi γ  ∞ (k) !   ∞  x1,λ, j (x2 )   k x1,λ, j (x2 )A1 + A2  + ×  A1 k! l=0 k=2  −1 l × w − x1,λ, j (x2 )A1 − x2 A2 dw.  x1,λ, j (x 2 )A1

(3.25)

The right hand side of the last expression is an analytic function of  in a small neighborhood of 0, so that all the derivatives at  = 0 vanish. We will need only the first and the second ones. Here are the corresponding relations.  −1   1  w − 1 w − x1,λ, j (x2 )A1 − x2 A2 (x1,λ, j (x 2 )A1 + A2 ) 2πi γ −1  × w − x1,λ, j (x2 )A1 − x2 A2 dw = 0. (3.26)     −1  1  (x1,λ, w − 1 w − x1,λ, j (x2 )A1 − x2 A2 j (x 2 )A1 + A2 ) 2πi γ −1 −1     × w−x1,λ, j (x2 )A1 −x2 A2 (x1,λ, (x )A +A ) w−x (x )A −x A 1 2 1,λ, j 2 1 2 2 j 2 +

 x1,λ, j (x 2 )

2

 −1

A1 w − x1,λ, j (x2 )A1 − x2 A2 dw = 0.

(3.27)

We will now express relations (3.26) and (3.27) in terms of projections P j,λ,i,μ (x2 ). The spectrum of the matrix x1,λ, j (x2 )A1 + x2 A2 consists of complex numbers μ j,λ,i,ν (x2 ) which are reciprocals of τ j,λ,i,ν (x2 ), ν ∈ σ (A1 ) and, possibly, 0. If x2 is close to 0, the latter might occur only when 0 ∈ σ (A1 ). If 0 is an eigenvalue of x1,λ, j (x2 )A1 + x2 A2 , we include it in the formulas below as μ j,λ,i,0 (x2 ). As mentiioned above, each eigenvalue μ j,λ,i,ν (x2 ), i ∈ Iλ , ν = λ is close to 1. These eigenvalues tend to 1 as x2 → 0 (and, of course, μ j,λ, j,λ (x2 ) = 1 for all x2 ). All other μ j,λ,i,ν (x2 ) stay away from 1 as x2 → 0, that is for some positive constant a / Iλ . (3.28) |μ j,λ,i,ν (x2 ) − 1| > a, if i ∈ Iλ and ν = λ, or i ∈

788

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Spectral Analysis Near Regular Point of...

The Jordan decomposition of the matrix A(x2 ) = x1,λ, j (x2 )A1 + x2 A2 consists of Jordan blocks which have sizes between 1 and m i corresponding to eigenvalue μ j,λ,i,ν (we consider an eigenvector which is not in a Jordan cell of dimension higher than 1 as a Jordan cell of size 1), so for w close to 1 we have  −1 w − x1 j (x2 )A1 − x2 A2 P j,λ, j,λ (x2 ) + w−1



P j,λ,i,λ (x2 ) w − μ j,λ,i,λ (x2 ) i∈Iλ ,i= j  r i −1 μ j,λ,i,ν (x 2 )I − x 1,λ, j (x 2 )A 1 − x 2 A 2 P j,λ,i,ν (x2 ) m + ,  r +1 ν∈σ (A1 ),ν=λ i∈Iν r =0 w − μ j,λ,i,ν (x2 )

=

(3.29) The integrands of (3.26) and (3.27) are meromorphic functions in w in a neighborhood of w = 1. We use (3.29) to compute their residues at 1. A simple but tedious computation shows that    −1   Res  (w − 1) w − x1,λ, j (x2 )A1 − x2 A2 (x1,λ, j (x 2 )A1 + A2 ) w=1  −1  × w − x1,λ, j (x2 )A1 − x2 A2  = P j,λ, j,λ (x2 )(x1,λ, j (x 2 )A1 + A2 )P j,λ, j,λ (x 2 ) = 0

(3.30)

Passing to the limit as x2 → 0 we obtain  P j,λ (x1,λ, j (0)A1 + A2 )P j,λ = 0,

(3.31)

which is the first relation claimed in Theorem 1.15. To simplify the computation of the residue of (3.27) at w = 1 let us introduce the following operators  R (x2 ) = x1,λ, j (x 2 )A1 + A2 , ⎛

S1 (x2 ) = P j,λ, j,λ (x2 )R (x2 ) ⎝



i∈Iλ ,i= j

⎞ P j,λ,i,λ (x2 ) ⎠ R (x2 )P j,λ, j,λ (x2 ), 1 − μ j,λ,i,λ (x2 )

S2 (x2 ) = P j,λ, j,λ (x2 )R (x2 )

 r i −1 μ j,λ,i,ν (x 2 )I − x 1,λ, j (x 2 )A 1 − x 2 A 2 P j,λ,i,ν (x2 )  m × r +1  ν∈σ (A1 ),ν=λ i∈Iν r =0 1 − μ j,λ,i,ν (x2 ) 



×R (x2 )P j,λ, j,λ (x2 ).

Reprinted from the journal

789

M. I. Stessin

A straightforward computation using (3.29) and (3.30) shows that (3.27) turns into S1 (x2 ) + S2 (x2 ) +

 x1,λ, j (x 2 )

2

P j,λ, j,λ (x2 )A1 P j,λ, j,λ (x2 ) = 0.

(3.32)

Our next step is finding the limits of each of terms of (3.32) as x2 → 0. If i ∈ Iλ , μ j,λ,i,λ (x2 ) → 1 as x2 → 0, and condition (b) implies that μj,λ,i,λ (0) = 0, so we have ⎛

⎞ P j,λ,i,λ (x2 ) ⎠ R(x2 )P j,λ, j,λ (x2 ) S1 (x2 ) = P j,λ, j,λ (x2 )R(x2 ) ⎝ 1 − μ j,λ,i,λ (x2 ) i∈Iλ ,i= j ⎛ ⎞ P j,λ,i,λ (x2 ) ⎠R(x2 )P j,λ, j,λ (x2 ) = P j,λ, j,λ (x2 )R(x2 )⎝ −μj,λ,i,λ (0)x2 +O(|x2 |2 ) i∈Iλ ,i= j ⎛ ⎞ P j,λ, j,λ (x2 )R(x2 )P j,λ,i,λ (x2 ) ⎠ R(x2 )P j,λ, j,λ (x2 ) =⎝ x2 (−μj,λ,i,λ (0) + o(1)) i∈Iλ ,i= j  ⎛ ⎞  2 ))A P (0)x + O(|x | (x ) P j,λ, j,λ (x2 ) R(0) + (x1,λ, 2 2 1 j,λ,i,λ 2 j ⎠ =⎝ x2 (−μj,λ,i,λ (0) + o(1))

i∈Iλ ,i= j

×R(x2 )P j,λ, j,λ (x2 ). Write (x2 ) = P j,λ, j,λ (x2 )R(0)P j,λ,i,λ (x2 ). Since P j,λ A1 Pi,λ = λP j,λ Pi,λ = 0 for i, j ∈ Iλ , i = j, relation (3.7) in Proposition 3.1 implies (0) = 0. Also, (3.23) and (3.12) yield  (0) = 0, and, hence, (x2 ) = O(|x2 |2 ), as x2 → 0, resulting in lim S1 (x2 ) = 0.

x2 →0

790

(3.33)

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Spectral Analysis Near Regular Point of...

To evaluate the limit of S2 (x2 ) we observe that since A1 is normal, Theorem 1.11 amd (3.20) impliy that for λ = ν, i ∈ Iν and ((μ j,λ,i,ν (x2 )I − x1,λ, j (x2 )A1 − x2 A2 )r P j,λ,i,ν (x2 ) → 0 for r > 0 as x2 → 0. Since by (3.28) we have |1 − μ j,λ,i,λ (x2 )| > a > 0, and, since by Theorem 1.11 all projections P j,λ,i,ν (x2 ) are bounded, all terms in the expression of S2 (x2 ) with r > 0 tend to 0 as x2 → 0. This yields ⎛ lim S2 (x2 ) = P j,λ A2 ⎝

x2 →0





ν∈σ (A1 ),ν=λ i∈Iν

⎞ Pi,ν ⎠ A2 P j,λ . 1 − μ j,λ,i,ν (0)

(3.34)

Equations (3.3) in Proposition 3.1, (3.20), and (3.34) now give ⎛ lim S2 (x2 ) = P j,λ A2 ⎝

x2 →0

⎞



λPν ⎠ A2 P j,λ = λP j,λ A2 Tλ A2 P j,λ . λ−ν

ν∈σ (A1 ),ν=λ

(3.35) Finally, passing to the limit in (3.32) as x2 → 0, (3.33), and (3.35) along with P j,λ A1 = A1 P j,λ = λP j,λ , P 2j,λ = P j,λ , result in P j,λ A2 Tλ A2 P j,λ +

 x1,λ, j (0)

2

P j,λ = 0,

which finishes the proof of Theorem 1.15 for λ = 0. II. λ = 0 In this case (1.10) gives the expression for P j,0 (x2 ). The spectrum of A1 + x2 A2 consists of the roots of Ri (1, x2 , z) = R˜ j (x2 , z) = 0. If x2 is close to 0, we denote these roots by x3,ν,i (x2 ), where ν ∈ σ (A1 ) is the eigenvalue of A1 to which x3,ν,i (x2 ) converges as x2 → 0, and i ∈ Iν . Each x3,ν,i has multiplicity m i , and, in particular, this multiplicity is equal to 1 for i ∈ I0 . Fix j ∈ I0 and x2 close to 0. Similar to what we had in the case λ = 0, here 

 A1 + (x2 + )A2 − x3,0, j (x2 + )I P j,0 (x2 ) = 0,

if  is small enough. An analog of (3.24) in this setting is 1 2πi

 γ

−1  w (w + x3,0, j (x2 + ))I − A1 − (x2 + )A2 dw = 0,

Reprinted from the journal

791

(3.36)

M. I. Stessin

and, as above, γ separates 0 from the rest of the spectrum of A1 + x2 A2 − x3,0, j (x2 )I . ˜ implies that there is a > 0 independent of x2 such that γ can be taken Condition b) to be circle centered at the origin of radius a|x2 |. Following a similar line of presentation as for λ = 0, we write (3.36) as a power series in :   −1 1 w (w + x3,0, j (x2 )I − A1 − x2 A2 0= 2πi γ !   ∞ (k)   x3,0, j (x2 )   k  I × I − A2 − x3,0, j (x2 )I  − k! k=2  −1 −1 × (w + x3,0, j (x2 )I − A1 − x2 A2 dw   −1 1 = w (w + x3,0, j (x2 )I − A1 − x2 A2 2πi γ  ∞ (k) !   ∞ x3,0, j (x2 )   k  I × A2 − x3,0, j (x2 )I  − k! l=0 k=2  −1 l × (w + x3,0, j (x2 )I − A1 − x2 A2 dw. (3.37) and equate the coefficients for powers of  to 0. Again, we are interested in the the coefficients for  and 2 . In this case the inverse is given by: 

(w + x3,0, j (x2 )I − A1 − x2 A2 =

P j,0 (x2 ) + w

i∈I0 ,i= j

−1

Pi,0 (x2 ) w + x3,0, j (x2 ) − x3,0,i (x2 )

l  i −1 m A1 + x2 A2 − x3,ν,i (x2 )I Pi,0 (x2 ) +  l+1 . w + x3,0, j (x2 ) − x3,ν,i (x2 ) ν∈σ (A1 ),ν=0 i∈Iν l=0

(3.38)

We substitute (3.38) into (3.37) and evaluate the residues of the coefficients for  and 2 . The details of this evaluation are similar to the ones in the case λ = 0 and are omitted. The proof of Theorem 1.15 is complete. Our next result shows that under the assumptions of Theorem 1.15 the limit component projections coming out of the joint spectrum σ p (A1 , A1 A2 ) coincide with P j,λ . In a similar way as in our previous consideration, it follows from the implicit function theorem that if σ (A1 , A1 A2 ) satisfies conditions (a) and (b), then for each spectral component of σ p (A1 , A1 A2 ) passing through (1/λ, 0) the first coordinate is expressed as an analytic function of the second one. To distinguish from the previous

792

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case when we considered the pair (A1 , A2 ), we denote the coordinates of a spectral point in σ p (A1 , A1 A2 ) by (z 1 , z 2 ), so that the in the jth component passing through (1/λ, 0), z 1 = z 1,λ, j (z 2 ). Similarly, we denote by Q j,λ (z 2 ) the projection 1 Q j,λ (z 2 ) = 2πi

 |w−1|=δ˜ j

 −1 w − z 1,λ, j (z 2 )A1 − z 2 A1 A2 dw,

˜ where the contours {w : |w − 1| = δ j } separates 1 from the rest of the spectrum of z 1,λ, j (z 2 )A1 + z 2 A1 A2 , as it was in (1.9). Since A1 is normal, Theorem 1.11 implies that there are limits Q j,λ of Q j,λ (x2 ) as x2 → 0. Lemma 3.39 Suppose that the pairs of matrices (A1 , A2 ) and (A1 , A1 A2 ) satisfy the conditions of Theorem 1.15, and let λ ∈ σ (A1 ), λ = 0. Then P j,λ = Q j,λ . Proof First we observe that relation (3.31) (the first relation in Theorem 1.15) implies  that for j ∈ Iλ the compression of A2 to the range of P j,λ is −x1,λ, j (0)IR(P j,λ ) , where  IR(P j,λ ) is the identity matrix on the range of P j,λ . This implies that (−x1,λ, j (0)), j ∈ Iλ are the eigenvalues of the compression of A2 to the λ-eigenspace of A1 , which by (3.3) is the sum of the ranges of P j,λ . Each of these ranges is of dimension 1, and by  condition b) all numbers (x1,λ, j (0)) are different. Let e j , j ∈ Iλ be an eigenvector for P j,λ A2 P j,λ . Then these eigenvectors form a basis of the λ-eigenspace of A1 . Similarly, applying Theorem 1.15 to the pair (A1 , A1 A2 ) we obtain that the com pression of A1 A2 to the λ-eigenspace of A1 has (−z 1,λ. j (0)) as eigenvalues of multiplicity one each. Propositions 3.1 and 3.6 imply ⎞ ⎛ ⎞ ⎛ kλ kλ kλ P j,λ ⎠ A1 A2 ⎝ P j,λ ⎠ = A1 P j,λ A2 P j,λ , Pλ A1 A2 Pλ = ⎝ j=1

j=1

j=1

so that for j ∈ Iλ   Pλ A1 A2 Pλ e j = A1 P j,λ A2 P j,λ e j = −x1,λ, j (0)A1 e j = −λx 1,λ, j (0)e j .  This shows that −λx1,λ, j (0), j ∈ Iλ form the spectrum of the compression of A1 A2 to the λ-eigenspace of A1 and that e j , j ∈ Iλ form the corresponding eigenbasis. Of course, this implies the statement we are proving.  

Proof of Theorem 1.20 Let λ = 0. Apply Theorem 1.15 to each of the pairs (A1 , A2 ) and (A1 , A1 A2 ) and use Lemma 3.39. We obtain  x1,λ, j (0) P j,λ , 2  z (0) P j,λ A1 A2 Tλ A1 A2 P j,λ = − 1,λ,2j P j,λ .

P j,λ A2 Tλ A2 P j,λ = −

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It follows from the definition of the operator Tλ , (1.14), that  Tλ A1 = A1 Tλ = −



 Pμ + λTλ = −I + Pλ + λTλ .

μ∈σ (A1 ),μ=λ

Since P j,λ A1 = λP j,λ , this implies −λP j,λ A22 P j,λ + λP j,λ A2 Pλ A2 P j,λ + λ2 P j,λ A2 T A2 P j,λ  z 1,λ, j (0) P j,λ , =− 2     λ2 x1,λ, j (0) − z 1,λ, j (0) P j,λ , −λP j,λ A22 P j,λ + λP j,λ A2 Pl,λ A2 P j,λ = 2 l∈Iλ

−λP j,λ A22 P j,λ + λP j,λ A2 P j,λ A2 P j,λ =

  λ2 x1,λ, j (0) − z 1,λ, j (0)

2

P j,λ .

Now, P 2j,λ = P j,λ results in 2  P j,λ A2 P j,λ A2 P j,λ = P j,λ A2 P j,λ , and (3.31) yields P j,λ A22 P j,λ

=

 2 z 1 j (0) + 2λ x1 j (0) − λ2 x1 j (0) 2λ

P j,λ .  

The proof is complete.

4 Application to Representations of Coxeter Groups: Proof of Theorem 1.22 Recall that a Coxeter group is a finitely generated group G with generators g1 , . . . , gn satisfy the following relations: (gi g j )m i j = 1, i, j = 1, . . . , n, where m ii = 1 and m i j ∈ N ∪ {∞}, m i j ≥ 2 when i = j. It is easy to see that to avoid redundancies we must have m i j = m ji , and that m i j = 2 means gi and g j commute. Coxeter set of generators, and m i j are The set of generators {g1 , . . . , gn } is called  a called the Coxeter exponents. The matrix m i j  is called a Coxeter matrix. A Coxeter group with 2 generators is called a Dihedral group. The monographs [1, 20, 27] are good sources for information on Coxeter groups. Let G be a Coxeter group with Coxter generators g1 , ..., gn , and ρ : G → V be a finite dimensional unitary representation of G. Suppose that A1 , ...., An is a

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tuple of N × N matrices. In this section we investigate what information about relations between A1 , . . . , An can be obtained from the fact that the joint spectrum σ p (ρ(g1 ), . . . , ρ(gn )) is contained in σ p (A1 , ..., An ). Of course, for every pair of generators gi , g j the representation ρ generates a unitary representation of the Dihedral group generated by gi and g j . It is well-known that every irreducible unitary representation of a Dihedral group is either 1- or 2-dimensional, and by Maschke’s Theorem ([31, 32]) that every representation is a sum of irreducible ones. The one dimensional representations of a Dihedral group are: ρ(g1 ) = ρ(g2 ) = I , or ρ(g1 ) = ρ(g2 ) = −I for an odd order group. For an even order group there is an additioal one-dimensional representation ρ(g1 ) = I , ρ(g2 ) = −I . Two-dimensional irreducible representations are equivalent to those generated by 

1 ρ(g1 ) = 0

  0 cos α , ρ(g2 ) = −1 sin α

 sin α , − cos α

where 0 < α < π . If the group is finite, α is a rational multiple of π . It is easy to see (cf [8]) that the proper joint spectrum of images of the Coxeter generators of a Dihedral group under an irreducible representation could be either a line of the form (one-dimensional) {(x1 , x2 ) : x1 ± x2 = ±1}, or a “complex ellips" (two-dimensional) {(x1 , x2 ) : x12 + 2(cos α)x1 x2 + x22 = 1}, and the joint spectrum of (ρ(g1 ) and ρ(g1 )ρ(g2 )) could be {x1 ± x2 = ±1} for one-dimensional representations, and {x12 − x22 + 2(cos α)x2 = 1} for two-dimensional. Proof of Theorem 1.22 Consider 2 ≤ i ≤ n. Condition (∗) in the statement of Theorem 1.22 implies that every line and ellipse in the joint spectrum of ρ(g1 ) and ρ(gi ) has multiplicity one, and, therefore, by condition (II) in Theorem 1.22, σ p (A1 , Ai ) and σ p (A1 , A1 Ai ) satisfy Theorems 1.15 and 1.20 at (±1, 0). Details of the local analysis

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near each of them are similar, so we concentrate on (1, 0). Following the notations of the previous section we denote by P j,1 the component projections from Theorem 1.11. We will apply Theorem 1.20. First suppose that the jth component of σ p (A1 , Ai ) which passes through (1, 0) is the line x1 ± x2 = 1. Then the corresponding component of the joint spectrum of A1 and A1 A2 is also a line, and x1 j (x2 ) = 1 ± x2 , z 1 j (x2 ) = 1 ± z 2 , so that x1 j = z 1 j ≡ 0, (x1 j )2 ≡ 1. Theorem 1.20 now implies (4.1) P j,1 A22 P j,1 = P j,1 . Let the jth component is the ellipse   x12 + 2 cos α x1 x2 + x22 = 1. Then the corresponding component of σ p (A1 , A1 A2 ) is given by   z 12 − z 22 + 2 cos α z 2 = 1. In this case x1 j (0) = z 1 j (0) = − cos α, x1 j (0) = −1 + cos2 α, z 1 j (0) = 1 − cos2 α, and Theorem 1.20 for λ = 1 shows that (4.1) holds in this case too. Thus, (3.3) shows that the compression of A22 to the 1-eigenspace of A1 is the identity. Since A1 is normal, the projection P1 , onto this subspace is orthogonal. Since the norm of A2 is equal to 1, this implies that every 1-eigenvector of A1 is a 1-eigenvector of A22 . A similar proof shows that every (−1)-eigenvector of A1 is a 1-eigenvector of A22 . Further, every component of the joint spectrum of Coxeter generators under a representation of a Dihedral group, either an ellipse, or a straight line, passes the same number of times through (±1, 0) and through (0, ±1). That is, if t1 and t2 are the multiplicities of 1 and -1 as eigenvalues of ρ(g1 ), and u 1 and u 2 are the same numbers for ρ(g2 ), then t 1 + t2 = u 1 + u 2 . Now, condition II) in the statement of Theorem 1.22 implies that the sum of multiplicities of eigenvalues 1 and − 1 of A1 and Ai are the same. This sum is equal of the sum of dimensions of all Jordan cells in the Jordan representation of Ai corresponding to eigenvalues ±1 and, of course, to the multiplicity of eigenvalue 1 for Ai2 . Thus, the sum of 1- and (−1)-eigenspaces of A1 is exactly the invariant subspace for

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Ai2 corresponding to eigenvalue 1, and the restriction of Ai2 to this subspace is the identity. Since the square of a Jordan cell which corresponds to a non-zero eigenvalue and has dimension higher than 1 is never diagonal, we see that ±1-eigenvectors for A1 and ±1-eigenvectors of Ai span the same subspace, which we call L, that is invariant under the action of both A1 and Ai . As mentioned above, every component of the joint spectrum of ρ(g1 ) and ρ)g2 passes through (±1, 0), so that the dimension of L is equal the dimension of ρ. Of course, L being spanned by ±1-eigenvectors of A1 is independent of i = 2, ..., n, and (1) is proved.  The fact that Ai  are unitary and self-adjoint is straightforward. Indeed, it follows L from L being spanned by ±1-eigenvectors of Ai and from a simple fact that for an operator of norm 1 any two eigenvectors corresponding to different unimodular eigenvalues are orthogonal. Since every component of the joint spectrum of Coxeter generators of a representation of a Coxeter group passes through (±1, 0, ..., 0), we   see that (1.23) holds. The fact that restrictions of Ai  generate a representation of G L follows from the Theorem 1.1 in [8] stating that for a representation of a pair of unitary matrices U1 , U2 their joint spectrum determines a number m such that (U! U2 )m = 1 (m is infinite, if there is no such relation). Thus, the Coxeter matrices of ρ and ρ˜ are the same, and (2) is proved. Finally, (3) follows from (1.23) and Theorem 1.2 in [8]. We are done.

References 1. Bjorner, A., Brenti, F.: Combinatorics of Coxeter Groups. Springer, New York (2005) 2. Bannon, J.P., Cade, P., Yang, R.: On the spectrum of Banach algebra-valued entire functions. Ill. J. Math. 55(4), 1455–1465 (2011) 3. Cade, P., Yang, R.: Projective spectrum and cyclic cohomology. J. Funct. Anal. 265(9), 1916–1933 (2013) 4. Catanese, F.: Babbage’s conjecture, contact surfaces, symmetric determinantal varieties and applications. Invent. Math. 63, 433–465 (1981) 5. Chagouel, I., Stessin, M., Zhu, K.: Geometric spectral theory for compact operators. Trans. Am. Math. Soc. 368(3), 1559–1582 (2016) 6. Coxeter, H.M.S.: Discrete groups generated by reflections. Ann. Math. (2) 35(3), 588–621 (1934) 7. Coxeter, H.M.S.: The complete enumeration of finite groups of the form ri2 = (ri r j )ki j = 1. J. Lond. Math. Soc. 10(1), 21–25 (1935) ˇ ckoviˇc, Z., Stessin, M., Tchernev, A.: Determinantal hypersurfaces and representations of Coxeter 8. Cuˇ groups. Pac. J. Math. 113(1), 103–135 (2021) 9. Dickson, L.E., An elementary exposition of Frobenius’ theory of group characters and group determinants, Ann. of Math. 2(4),: pp. 25–49. also in Mathematical papers, Vol. II, Chelsea, New York 1975, 737–761 (1902) 10. Dickson, L.E.: On the group defined for any given field by the multiplication table of any given finite group, Trans. Am. Math. Soc. 3, 377–382; also in Mathematical Papers, Vol. II, Chelsea, New York 1975, 75–91 (1902) 11. Dickson, L.E.: Modular theory of group-matrices. Trans. Am. Math. Soc. 8, 389–398; also in Mathematical Papers, Vol. II, Chelsea, New York 1975, 251–260 (1907) 12. Dickson, L.E.: Modular theory of group-characters. Bull. Am. Math. Soc. 13, 477–488; also in Mathmatical Papers, Vol. IV, Chelsea, New York 1975, 535–546 (1907) 13. Dickson, L.E.: Determination of all general homogeneous polynomials expressible as determinants with linear elements. Trans. Am. Math. Soc. 22, 167–179 (1921)

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M. I. Stessin 14. Dolgachev, I.: Classical Algebraic Geometry: A Modern View. Cambridge University Press, Cambridge (2012) 15. Douglas, R. G., Yang, R.: Hermitian geometry on reslovent set (I). In: Proceedings of the International Workshop on Operator Theory (IWOTA 2016) (to appear) 16. Douglas, R. G., Yang, R.: Hermitian geometry on resolvent set (II), preprint 17. Frobenius, F. G.: Über vertauschbare Matrizen. Sitz. Kön. Preuss. Akad. Wiss. Berlin 601–614 (1896); also in Gesammelte Abhandlungen, Band II, Springer-Verlag, New York, 1968, 705–718 18. Frobenius, F. G.: Über Gruppencharaktere. Sitz. Kön. Preuss. Akad. Wiss. Berlin (1896) 985–1021; also in Gesammelte Abhandlungen, Band III, Springer-Verlag, New York, 1968, 1–37 19. Frobenius, F. G.: Über die Primfactoren der Gruppendeterminante. Sitz. Kön. Preuss. Akad. Wiss. Berlin 1343–1382 (1896); also in Gesammelte Abhandlungen, Band III, Springer-Verlag, New York, 1968, 38–77 20. Geck, M., Pfeiffer, G.: Characters of Finite Coxeter Groups and Iwahory–Hecke Algebras. Calderon Press, Oxford (2000) 21. Goldberg, B., Yang, R.: Self-similiarity and spectral dynamics. arXiv:2002.09791 (Math FA) 22. Gonzalez-Vera, P., Stessin, M.I.: Joint spectra of Toeplitz operators and optimal recovery of analytic functions. Constr. Approx. 36(1), 53–82 (2012) 23. Grigorchuk, R., Yang, R.: Joint spectrum and infinite dihedral group. Proc. Steklov Inst. Math. 297, 145–178 (2017) 24. Helton, J.W., McCullogh, S.A., Vinnikov, V.: Non-commutative convexity arises from linear matrix inequalities. J. Funct. Anal. 240(1), 105–191 (2006) 25. Helton, J.W., Vinnikov, V.: Linear matrix inequality representation set. Commun. Pure Appl. Math. 60, 654–674 (2007) 26. He, W., Wang, X., Yang, R.: Projective spectrum and kernel bundle (II). J. Oper. Theory 297, 417–433 (2017) 27. Humphreys, J.E.: Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, vol. 10. Cambridge University Press, Cambridge (1992) 28. Kerner, D., Vinnikov, V.: Determinantal representations of singular hypersurfaces in Pn . Adv. Math. 231, 1619–1654 (2012) 29. Klep, I., Volˇciˇc, J.: A note on group representations, determinantal hypersurfaces and their quantizations. In: Proceedings IWOTA (2019) (to appear) 30. Mao, T., Qiao, Y., Wang, P.: Commutativity of normal compact operator via projective spectrum. Proc. AMS 146(3), 1165–1172 (2017) 31. Maschke, H.: Ueber den arithmetischen Character der Coefficienten der Substitutionen endlicher Substitutionsgruppen. Math. Ann. 50(2), 253–298 (1898) 32. Maschke, H.: Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten intransitiv sind. Math. Ann. 52(2–3), 363–368 (1899) 33. Motzkin, T.S., Taussky, O.: Pairs of matrices with property L. Trans. Am. Math. Soc. 73, 108–114 (1952) 34. Peebles, T., Stessin, M.: Spectral surfaces for operator pairs and Hadamard matrices of F type. Adv. Oper. Theory 6, 13 (2021) 35. Stessin, M., Tchernev, A.: Geometry of joint spectra and decomposable operator tuples. J. Oper. Theory 82(1), 79–113 (2019) 36. Stessin, M., Yang, R., Zhu, K.: Analyticity of a joint spectrum and a multivariable analytic Fredhom theorem. N. Y. J. Math. 17A, 39–44 (2011) 37. Vinnikov, V.: Complete description of determinantal representations of smooth irreducible curves. Lin. Albebra Appl. 125, 103–140 (1989) 38. Yang, R.: Projective spectrum in Banach algebras. J. Topol. Anal. 1, 289–306 (2009) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Complex Analysis and Operator Theory (2023) 17:74 https://doi.org/10.1007/s11785-023-01377-1

Complex Analysis and Operator Theory

Stratified Hilbert Modules on Bounded Symmetric Domains Harald Upmeier1 Received: 7 May 2022 / Accepted: 4 June 2023 / Published online: 5 July 2023 © The Author(s) 2023

Abstract We analyze the “eigenbundle” (localization bundle) of certain Hilbert modules over bounded symmetric domains of rank r , giving rise to complex-analytic fibre spaces which are stratified of length r +1. The fibres are described in terms of Kähler geometry as line bundle sections over flag manifolds, and the metric embedding is determined by taking derivatives of reproducing kernel functions. Important examples are the determinantal ideals defined by vanishing conditions along the various strata of the stratification. Keywords Bounded symmetric domain · Hilbert module · Complex-analytic fibre space · Flag manifold · Jordan triple Mathematics Subject Classification Primary 32M15 · 46E22; Secondary 14M12 · 17C36 · 47B35

0 Introduction Jörg Eschmeier was a master in the interplay between operator theory and multivariable complex analysis, in particular in its modern sheaf-theoretic form. An interesting concept in this respect is the “eigenbundle” of a Hilbert module arising from a commuting tuple of non-selfadjoint operators T1 , . . . , Td . Just as the spectral theorem is the basic tool for the analysis of self-adjoint operators, the eigenbundle plays Dedicated to the Memory of Jörg Eschmeier. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

A previous version of this chapter was published Open Access under a Creative Commons Attribution 4.0 International License at http://link.springer.com/10.1007/s11785-023-01377-1.

B 1

Harald Upmeier [email protected] Fachbereich Mathematik, Universität Marburg, 35032 Marburg, Germany

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a similar role in the non-selfadjoint case, naturally involving multi-variable complex analysis instead of “real” measure theory. In the original approach by Cowen-Douglas [6] for Hilbert modules H of holomorphic functions on a bounded domain D ⊂ Cd , the eigenbundle, denoted by H ,  is a genuine holomorphic vector bundle on D whose hermitian geometry determines the underlying operator tuple. In more general situations [8, 9] the eigenbundle will have singularities along certain subvarieties of D, making the geometric aspects more complicated. For example, if H = I is the Hilbert closure of a prime ideal I of polynomials, whose vanishing locus X is smooth, then by a result of Duan-Guo [8] the eigenbundle H has rank 1 on the regular set D\X , whereas H | X is isomorphic to the  (dual) normalbundle of X . Thus we have a “stratification” of length 2. In this paper, we study K -invariant Hilbert modules H over bounded symmetric domains D = G/K of rank r , leading to singular vector bundles which are stratified of length r + 1. This study was initiated in [23] where the eigenbundle of certain polynomial ideals J λ , for a given partition λ of length  r , was determined explicitly. These ideals are not prime ideals except for the “fundamental” partitions λ = (1, . . . , 1, 0, . . . , 0). The main result of [23] describes the fibres of the eigenbundle using representation theory of the compact Lie group K . The current paper extends and generalizes this analysis. For the partition ideals λ J λ and Hilbert closures H = J we construct a Hilbert space embedding of the eigenbundle by taking certain derivatives of the reproducing kernel function of H . This is important to study the hermitian structure and is related to the “jet construction” introduced in [9]. Moreover, we give a holomorphic characterization of the eigenbundle in terms of holomorphic sections of line bundles over a flag manifold. Such a geometric characterization may also hold in more general situations. Beyond the setting of the partition ideals we consider arbitrary K -invariant polynomial ideals, in particular the so-called “determinantal” ideals which have a direct geometric meaning.

1 Hilbert Modules and Their Eigenbundle Let D be a bounded domain in a finite dimensional complex vector space E ≈ Cd . Denote by P E ≈ C[z 1 , . . . , z d ] the algebra of all polynomials on E. A Hilbert space H of holomorphic functions f on D (supposed to be scalar-valued) is called a Hilbert module if for any polynomial p ∈ P E the multiplication operator T p f := p f leaves H invariant and is bounded. Using the adjoint operators T p∗ , the closed linear subspace H ζ := { f ∈ H : T p∗ f = p(ζ ) f ∀ p ∈ P E } 

(1.1)

is called the joint eigenspace at ζ ∈ D. Since T p Tq = T pq for polynomials p, q it suffices to consider linear functionals or just the coordinate functions. The disjoint union  H H=  ζ ∈D  ζ 800

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becomes a subbundle of the trivial vector bundle D × H , which is called the eigenbundle of H , although it is not locally trivial in general. One also requires that the fibres have finite dimension and their union is total in H . The map φ → [φ] from H ζ  to H /Mζ H is a Hilbert space isomorphism, with inverse map H /Mζ H → H ζ , 

f + Mζ H → πζ f ,

where πζ : H → H ζ is the orthogonal projection. Thus H ζ becomes the “quotient   module” for the submodule Mζ H . Classical examples of Hilbert modules are the Bergman space H 2 (D) of squareintegrable holomorphic functions, whose reproducing kernel is called the Bergman kernel, and the Hardy space H 2 (∂ D) if D has a smooth boundary ∂ D. For general Hilbert modules H , a reproducing kernel function is a sesqui-holomorphic function K(z, ζ ) on D × D such that for each ζ ∈ D the holomorphic function Kζ (z) := K(z, ζ ) belongs to H , and we have ψ(z) = (Kz |ψ) H for all ψ ∈ H and z ∈ D. Here (φ|ψ) H is the inner product, anti-linear in φ. Thus H is the closed linear span of the holomorphic functions Kζ , where ζ ∈ D is arbitrary. If φα is any orthonormal basis of H then K(z, ζ ) =



φα (z)φα (ζ ).

α

For each ζ ∈ D we have Kζ ∈ H ζ , as follows from the identity  (T p∗ Kζ |ψ) H = (Kζ | pψ) H = p(ζ )ψ(ζ ) = p(ζ )(Kζ |ψ) H = ( p(ζ )Kζ |ψ) H for p ∈ P E and ψ ∈ H . If the reproducing kernel K has no zeros (e.g., the Bergman kernel of a strongly pseudo-convex domain) then the eigenbundle H is spanned by the functions Kζ and  hence becomes a hermitian holomorphic line bundle. In more general cases the kernel function vanishes along certain analytic subvarieties of D and the eigenbundle is not locally trivial, its fibre dimension can jump along the varieties and we obtain a singular vector bundle on D, also called a “linearly fibered complex analytic space” [13]. Such singular vector bundles are important in Several Complex Variables since they are in duality with the category of coherent analytic module sheaves, whereas (regular) vector bundles correspond to locally free sheaves. In [4] the connection to coherent analytic module sheaves associated with H is made explicit. An important class of Hilbert modules is given by the Hilbert closure H = I of a polynomial ideal I ⊂ P E . In this case the fibres (1.1) of the eigenbundle have finite

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dimension. More precisely, by [8] we have H ≈ I | D for the “localization bundle” I    over E, with fibre I := I /Mζ I ζ at ζ ∈ E. For any set of generators p1 , . . . , pt of I the linear map t  Ct → I ζ , (a1 , . . . , at ) → Mζ I + ai pi  i=1

is surjective, showing that dim I ζ  t. In [4, Lemma 2.3] it is shown that f ∈ H ζ   satisfies pi (ζ ) f = ( pi | f ) H Kζ for all i. This implies that the eigenbundle H restricted to the open dense subset  Dˇ :=

t 

{ζ ∈ D : p j (ζ ) = 0}

j=1

ˇ of D is a holomorphic line bundle spanned by the reproducing kernel Kζ , ζ ∈ D. ˇ The behavior of H on the singular set D\ D is more complicated and has so far been studied mostlywhen the vanishing locus of the reproducing kernel is a smooth subvariety of D, for example given as a complete intersection of a regular sequence of polynomials. The case where I is a prime ideal whose vanishing locus X consists of smooth points has been studied by Duan-Guo [8]. They showed that for ζ ∈ D\X H ζ = Kζ   is the 1-dimensional span of the reproducing kernel vector, whereas for ζ ∈ X H ζ ≈ Tζ⊥ (X )  is isomorphic to the normal space (more precisely its linear dual.) Thus we have a stratification of length 2. We consider a more complicated situation for bounded symmetric domains D of arbitrary rank r , where we have a stratification of length r + 1, the relevant varieties are not smooth and the ideal I is not prime in general.

2 K -invariant Ideals on Bounded Symmetric Domains Let D = G/K be an irreducible bounded symmetric domain of rank r , realized as the (spectral) unit ball of a hermitian Jordan triple (J ∗ -triple) E. Let {uv ∗ w} ∈ E denote the Jordan triple product of u, v, w ∈ E. The compact Lie group K acts by

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linear transformations on E preserving the Jordan triple product. For background on the Jordan theoretic approach towards symmetric domains, see [2, 5, 12, 17, 21]. Let P E denote the algebra of all polynomials on E. Under the natural action (k · f )(z) := f (k −1 z) of k ∈ K on functions f on E the polynomial algebra P E has a Peter–Weyl decomposition [11, 20] 

PE =

P Eλ

λ∈Nr+

into pairwise inequivalent irreducible K -modules P Eλ . Here Nr+ denotes the set of all partitions λ = (λ1 , . . . , λr ) of integers λ1  . . .  λr  0. The polynomials in P Eλ are homogeneous of degree |λ| := λ1 + . . . + λr . We often identify a partition λ with its Young diagram [λ] = {(i, j) : 1  i  r , 1  j  λi }. For fixed μ ∈ Nr+ denote by μ

π μ : PE → PE ,

f → π μ f =: f μ

μ

the K -invariant projection onto P E . As a consequence of Schur orthogonality we have [23, Lemma 3.1]  f μ (z) =

dk χμ (k) f (kz),

(2.1)

K μ

where χμ denotes the character of the K -representation on P E . An ideal I ⊂ P E is called K -invariant if k · f ∈ I for all k ∈ K and f ∈ I . A similar definition applies to Hilbert modules of holomorphic funtions on a K -invariant domain. The formula (2.1) implies that a K -invariant ideal (resp., Hilbert module) is μ a direct sum (resp., Hilbert sum) of its Peter-Weyl subspaces P E . λ For a given partition λ denote by J ⊂ P E the K -invariant ideal generated by P Eλ . The first main result of [23] asserts that J λ has the Peter-Weyl decomposition Jλ =



μ

PE ,

(2.2)

μλ

where λ  μ means λi  μi for all i. This is equivalent to the inclusion [λ] ⊂ [μ] for the corresponding Young diagrams. As a consequence, J μ ⊂ J λ if and only if μ  λ. In this section we show that these “partition” ideals J λ are fundamental for the study

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of general K -invariant ideals. Given a K -invariant ideal I , define I # := {λ ∈ Nr+ : J λ ⊂ I }. Proposition 2.1 Let I ⊂ P E be a K -invariant ideal. Then there is a finite set ⊂ I # of partitions such that I =



J λ.

λ∈

Proof If f ∈ I then (2.1) shows that f μ also belongs to I for all μ ∈ Nr+ . Let f 1 , . . . , f t be a finite set of generators of I . Then their non-zero K -homogeneous parts belong to I and form a finite set of generators. Thus we may assume that each f s ∈ P Eλs for some partition λs . We claim that I =

t 

J λs =: J .

s=1

Since f s ∈ I ∩ P Eλs is non-zero, I is K -invariant and P Eλs is irreducible, it follows that P Eλs ⊂ I and hence J λs ⊂ I . Thus J ⊂ I . Conversely, each generator f s of I belongs to J . Therefore I ⊂ J .   A subset ⊂ I # is called “full” if I =



J λ.

λ∈

/ . A subset A ⊂ I # is called “minimal” if α ∈ A and λ < α implies λ ∈ / I # , i.e., J λ ⊂I Lemma 2.2 Let A ⊂ I # be minimal and ⊂ I # be full. Then A ⊂ . Proof / . Let p ∈ P Eα . Then p ∈ J α ⊂ I =  λSuppose there exists α ∈ A such that α ∈ J and therefore

λ∈

p=



f λ (finite sum)

λ∈

for some f λ ∈ J λ . By (2.2) we have f λα = 0 unless α  λ. Since α ∈ / this implies p = pα =

 λ∈

f λα =

 λ∈ , αλ

f λα =

since {λ ∈ : λ < α} = ∅. This is a contradiction.

 λ∈ , α>λ

f λα = 0  

Corollary 2.3 Every minimal set A ⊂ I # is finite.

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Proof By Proposition 2.1 there exists a full set ⊂ I # which is finite. By Lemma 2.2 we have A ⊂ . Hence A is finite.   Proposition 2.4 There exists a (finite) set A ⊂ I # which is both full and minimal. Proof For any finite subset ⊂ I # we put | | :=



|λ|,

λ∈

where |λ| := λ1 + . . . + λr . By Proposition 2.1 there exists a finite subset ⊂ I # which is full. Put k := min{| | : ⊂ I # full and finite}. Then there exists a full and finite set A ⊂ I # such that |A| = k. We claim that A is minimal. Suppose there exist α ∈ A and λ ∈ I # with λ < α. Then J α ⊂ J λ ⊂ I which shows that the finite set = (A\{α}) ∪ {λ} is still full. On the other hand, we have |λ| < |α| and hence | | = |A\{α}| + |λ| = |A| − |α| + |λ| < |A| = k.  

This contradiction shows that A is minimal.

# ⊂ I # which is both The arguments above show that there is a unique finite set Imin full and minimal. We formulate this as

Proposition 2.5 Let I ⊂ P E be a K -invariant ideal. There exists a unique finite set

⊂ Nr+ such that  I = Jλ (2.3) λ∈

and

/ J μ ⊂I

if μ < λ for some λ ∈ .

For example, the n-th power of the maximal ideal M0 has the form Mn0 =



Jλ

|λ|=n

since the polynomials in P Eλ are homogeneous of degree |λ|. For n = 1 we have M0 = I 1,0,...,0 since P E1,0,...,0 = E ∗ is the linear dual space of E. More interesting examples will be studied in the next section. As an application of Proposition 2.5 we determine for each K -invariant ideal I the “maximal fibre” I 0 = H 0 of the eigenbundle.  

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Theorem 2.6 Let I be a K -invariant ideal, written in the “minimal” form (2.3). Then any Hilbert module closure H = I has the maximal fibre  P Eλ (direct sum) H0 =  λ∈

at the origin. Proof We first show that P Eλ ⊂ H 0 for λ ∈ . Let p ∈ P Eλ and  ∈ E ∗ a linear form  on E. By [22] we have T∗ p =

r  (T∗ p)λ−ε j . j=1

If T∗ p = 0 then (T∗ p|q) = 0 for some q ∈ J μ with μ ∈ . By (2.2) we have q=



qν.

νμ

Therefore (T∗ p|q) =

r r    ((T∗ p)λ−ε j |q) = ((T∗ p)λ−ε j |q ν ). j=1 νμ

j=1

It follows that there exists j such that λ − ε j = ν. Therefore λ > ν  μ. Since both λ, μ ∈ this contradicts the fact that is minimal.  Conversely, suppose there exists φ ∈ H 0 which is orthogonal to λ∈ P Eλ . By  μ averaging over K we may assume that φ ∈ P E for some μ ∈ / . We can write φ ∈ H = I as φ=



φλ

λ 

(n)

M E

j−1

πc

(n)

− → MW 

j−−1

.

(3.7)

λ Proof By Theorem 3.1 we may assume that f ∈ M(n) E belongs to J for a partition λ = (λ1 , . . . , λr ) satisfying (3.8) λ j + . . . + λr  n j

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for all 1  j  r . Now fix  and consider the partition λ∗ := (λ+1 , . . . , λr ) of length r − . By (2.7) we have πc f ∈



α JW

αλ∗

where α  λ∗ is the (partial) containment order, i.e. αi  λi for all  < i  r . For  < j  r we have α j + . . . + αr  λ j + . . . + λr  n j . Therefore α satisfies the analogue of (3.8) relative to W . Applying Theorem 3.1 to W and the sequence n +1  · · ·  nr of length r −  = rank(W ), it follows that ∗) πc J λ ⊂ M(n W . The special case (3.7) corresponds to n 1 = · · · = n j = n, n j+1 = · · · = nr = 0.   We now consider the special case of “step 1” partitions. Let n ∈ N and consider the ideal (n)  }, (3.9) M E = { p ∈ P E : ordζ ( p)  n ∀ ζ ∈ E 

where 0    r is fixed. This corresponds to n = (n (+1) , 0(r −−1) ). In this case Theorem 3.1 yields 

(n)

M E = 

J λ.

λ+1 +...+λr n

For example 

(n)

M E = Mn0 = 0

Jλ

|λ|=λ1 +...+λr n

and (n)

M E

r −1

=



J λ.

λr n

For the K -invariant ideals (3.9) it is easy to find a minimal decomposition: Theorem 3.3 For 0   < r the ideal M(n)  is the minimal and finite sum E 

(n)

M E = 



α J

α

taken over the (finitely many) integer tuples α+1  . . .  αr  0 satisfying |α| := α+1 + . . . + αr = n.

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Here we put ()  α := (α+1 , α+1 , . . . , αr ) ∈ Nr+ .

Proof We first show that the sum is minimal. If α, β ∈ Nr+− satisfy |α| = n = |β| , then α  β and hence α = β, showing that  . Of course there are and  αβ α=β only finitely many partitions α = (α+1 , . . . , αr ) satisfying |α| = n. By Theorem 3.1 r M(n)  corresponds to partitions λ ∈ N+ such that the single inequality E 

λ+1 + . . . + λr  n

(3.10)

holds. Clearly, λ =  α satisfies (3.10), since λi  αi for  < i  r . Thus it remains to show that (3.10) implies λ   α for some α ∈ Nr+− with |α| = n. If λ+1 +. . .+λr  n then there exists α+1  . . .  αr such that |α| = n and λi  αi for  < i  r . To see this, we may assume (by induction) that λ+1 + . . . + λr = n + 1. Let λ+1  . . .  λm > 0 = λm+1 = · · · = λr , where  < m  r . Then α := (λ+1 , . . . , λm−1 , λm − 1, 0,

. . . , 0) is decreasing, λ+1 m >+1 satisfies |α| = n and λ   α since for 1  i   we have  αi = λ+1 − 1 m =  + 1   and therefore  αi  λ+1  λi . In the simplest case n = 1 there is only a single choice α+1 = 1, α+2 = = αr = 0 yielding the “fundamental partition”  α = (1(+1) , 0(r −−1) ) ≡ 1(+1) . Thus the prime (1) ideal M E = M E has the form 

(1)

α M E = J  = J1 

(+1)

.

(n)

For n > 1 we cannot represent M E by a single partition. 

4 Reproducing Kernels and Hermitian Structure The main result of [23], formulated as the isomorphism (2.9), determines the localization bundle J λ as an abstract (singular) holomorphic vector bundle, without reference  to a hermitian metric. If H = I is a Hilbert module completion of a K -invariant ideal I , with reproducing kernel function K(z, ζ ), the corresponding eigenbundle H ≈ I | D carries the all-important hermitian structure as a subbundle of D × H . To  makethe connection one needs an explicit embedding I |D − → H ⊂ D × H.   Reprinted from the journal

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As suggested by the “jet construction” developed in [9] for smooth submanifolds, such a map should involve certain derivatives of K(z, ζ ) in the normal direction. In this section we carry out this program in the more complicated geometric situation related to an arbitrary partition ideal I = J λ . Any polynomial p induces a constant coefficient differential operator p(∂z ) on E, depending in a conjugate-linear way on p. Let ( p|q) denote the Fischer-Fock inner product of polynomials p, q ∈ P E (anti-linear in p) and let E λ (z, ζ ) = Eζλ (z) denote the reproducing kernel of P Eλ . Every irreducible J ∗ -triple E has two “characteristic multiplicities” a, b [2, 16, 17, 21] such that a d = 1 + (r − 1) + b. r 2 Lemma 4.1 For a Jordan triple E, the determinant function Ne at a maximal tripotent e ∈ Sr satisfies (r ) n (4.1) E λ (z, ζ ) = Crn (λ) Nen (z)Ne (ζ ) E λ−n (z, ζ ) for λ  n (r ) and ζ ∈ E e , where Crn (λ) =

r  j=1

Moreover,

1 . (λ j − n + 1 + a2 (r − j))n n

(r )

N e (∂z )Eζλ = Ne (ζ ) Eζλ−n . n

(4.2)

Proof We use the Jordan theoretic Pochhammer symbols (s)λ and the “Faraut-Korányi formula” [11, 12]. Suppose first that E = E e is unital, with unit element e and determinant e . The parameter s = d/r in the continuous Wallach set corresponds to the Hardy space H 2 (S) over the Shilov boundary S of D. Since |e | = 1 on S we obtain for p, q ∈ P Eλ n

(e (∂z )(ne p)|q) = (ne p|ne q) = (d/r )λ+n (r ) (ne p|ne q) S (d/r )λ+n (r ) = (d/r )λ+n (r ) ( p|q) S = ( p|q). (d/r )λ Since q is arbitrary, it follows that n

e (∂z )(ne p) =

(d/r )λ+n (r ) p (d/r )λ

for all partitions λ and p ∈ P Eλ . An application of Schur orthogonality [12, Proposition XI.4.1] shows that E λ (e, e) =

dλ , (d/r )λ

816

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where dλ := dim P Eλ . If λ  n (r ) we can write (r )

n

E λ (z, ζ ) = aλ ne (z)e (ζ ) E λ−n (z, ζ ) for some coefficient aλ . It follows that dλ (r ) n = E λ (e, e) = aλ ne (e)e (e) E λ−n (e, e) (d/r )λ dλ−n (r ) (r ) . = aλ E λ−n (e, e) = aλ (d/r )λ−n (r ) Since dλ = dλ−n (r ) in the unital case it follows that aλ =

dλ dλ−n (r )

(d/r )λ−n (r ) (d/r )λ−n (r ) = = Crn (λ). (d/r )λ (d/r )λ

This proves (4.1). Moreover, 



(r ) 1 n n n  (∂z )Eζλ = e (∂z ) ne e (ζ ) Eζλ−n Crn (λ) e (r ) (r ) (r ) 1 1 n n n n e (ζ ) Eζλ−n . = e (ζ ) e (∂z )(ne Eζλ−n ) = e (ζ ) E λ−n = n Crn (λ) ζ Cr (λ)

In the non-unital case, let Pe be the Peirce 2-projection onto E e . Since Ne (z) = e (Pe z) by definition, applying the unital case to E e and using Pe ζ = ζ we obtain (r )

n

E λ (z, ζ ) = E λ (Pe z, ζ ) = Crn (λ) ne (Pe z)e (ζ ) E λ−n (Pe z, ζ ) n

(r )

= Crn (λ) Nen (z)Ne (ζ ) E λ−n (z, ζ ). n

n

The second assertion follows with e (∂z ) f = N e (∂z )( f ◦ Pe ).

 

The identity (4.2), written as n

n

(r )

N e (∂z )E λ (z, ζ ) = Ne (ζ ) E λ−n (z, ζ ), implies (r )

Nen (∂ ζ )E λ (z, ζ ) = Nen (z) E λ−n (z, ζ ), since E λ (z, ζ ) = E λ (ζ, z). Thus for ζ ∈ E e we have Nen (∂ ζ )Eζλ = Nen Eζλ−n as holomorphic polynomials in z.

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(4.3)

H. Upmeier

Lemma 4.2 Let λ ∈ Nr+ be a partition and ζ ∈ E have rank . Then Eζλ = 0 if and only if λ has length  .  . Therefore Proof If λ has length >  then all p ∈ P Eλ vanish on E (Eζλ | p) = p(ζ ) = 0 and hence Eζλ = 0 since p is arbitrary. Now suppose λ has length  . Consider the spectral decomposition ζ =

 

ζi ei

i=1

for a frame (ei ). The conical polynomial N λ defined as in (3.5) satisfies N λ (ζ ) = ζ1λ1 · · · ζλ = 0. Since N λ (ζ ) = (Eζλ |N λ ) it follows that Eζλ = 0.

 

For n ∈ N, define n (m) = (n, . . . , n, 0, . . . , 0), with n repeated m times. Thus the Young diagram [n (m) ] = [1, m] × [1, n]. Any partition λ can be written as ( )

( −1 )

λ = (n 1 1 , n 2 2

(t −t−1 )

, . . . , nt

, 0(r −t ) ),

(4.4)

where 1  1 < · · · < t  r and n 1 > n 2 > · · · > n t > 0. Thus t is the number of “steps” in the partition. In other words, λ1 = · · · = λ1 = n 1 > λ1+1 = · · · = λ2 = n 2 > λ1+2 . Define for 1  s  t s := {ζ ∈ D : s−1  rank(ζ ) < s } = D ∩

 s −1

Eˇ  .

i=s−1

We also consider the “regular” points Dˇ := {ζ ∈ D : rank(ζ )  t }. Since λ has length t , Lemma 4.2 implies Eζλ = 0 if rank(ζ )  t . Hence Kζ does ˇ For the “singular” points ζ ∈ D there exists a unique s  t such not vanish at ζ ∈ D. that ζ ∈ s . For 1  h  k  t define ( −h−1 )

λkh := (n h h

(

h+1 , n h+1

−h )

818

(k −k−1 )

, . . . , nk

)

Reprinted from the journal

Stratified Hilbert Modules on Bounded Symmetric Domains s as a partition of length k − h−1 . Let μ ∈ N+ satisfy μ  λs1 . For each s  k  t consider the partition (

s+1 (μ, λks+1 ) = (μ, n s+1

−s )

(

s+2 , n s+2

−s+1 )

(k −k−1 )

, . . . , nk

)

of length k . Here λks+1 is empty for s = k. Starting with the Peter-Weyl expansion K(z, ζ ) =



aμ E μ (z, ζ )

μλ λ

of the reproducing kernel K(z, ζ ) of H = J , with coefficients aμ > 0, we define K invariant sesqui-holomorphic functions Ks (z, ζ ) = Kζs (z) on D× D by the Peter-Weyl expansion Kζs =



a(μ,λt

s+1 )

s N+ μλs1

(s )

μ−n s

Eζ

t 

n −n k+1

Ckk

( )

k ((μ, λks+1 ) − n k+1 ).

(4.5)

k=s

More explicitly, the constant is given by n

−n t

((μ, λt−1 s+1 ) − n t

n

−n s+2

t−1 Cntt (μ, λts+1 ) · Ct−1

s+1 · · · Cs+1

(t−1 )

(

s+1 ((μ, n s+1

−s )

−n t−1

n

t−2 ) · Ct−2

(

(

)

t−2 ((μ, λt−2 s+1 ) − n t−1 )

)

n −n s+1

s+1 ) − n s+2 ) · Css

(s ) (μ − n s+1 ).

Lemma 4.3 The kernel Kζs does not vanish if rank(ζ )  s−1 and vanishes if  rank(ζ ) < s−1 . Thus Ks vanishes precisely on s−1 = 0k 0 repeated  times. Here t = 1 and one obtains an isomorphism (cf. [3]) ()

P En

≈ (M , Ln ),

where Ln is the n-th tensor power of the line bundle L defined by (5.1). If  = r and E (r )

is unital with unit element e, then Mr = {E} is a singleton and P En is 1-dimensional, spanned by Nen . If  is arbitrary and n = 1 we have the “fundamental” partitions and ()

P E1

≈ (M , L ).

In the simplest case  = 1 we obtain a kind of projective space M1 consisting of all Peirce 2-spaces of rank 1, or equivalently, all lines spanned by tripotents of rank 1. The associated “tautological” line bundle T =



U

U ∈M1

has no holomorphic sections. On the other hand each f ∈ E ∗ yields a holomorphic section σ f : M → T ∗ of the dual line bundle T∗=



U∗

U ∈M1

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via M1  U → σU := f |U ∈ TU∗ . f

The calculation f ◦h −1

σhU

= ( f ◦ h −1 )|hU = ( f |U ) ◦ h −1 = σU ◦ h −1 = (h · σ f )hU f

−1  → C is given by shows that h · σ f = σ f ◦h . The homogeneous lift  σf :K

 σ f (h) := f (he1 ). E e1 then γ e1 = χ1 (γ )e1 for the holomorphic character χ1 : K E e1 → C× If γ ∈ K given by χ1 (γ ) = (γ e1 |e1 ) = Ne1 (γ e1 ). This is covered by the general formula (5.4): Lemma 5.3 For each f ∈ E ∗ = P E1,0,...,0 we have  σ f (h) = (Ne1 ◦ h ∗ | f ). Proof Putting f z := (z|v) the relation (Ne1 ◦ h ∗ )(z) = Ne1 (h ∗ z) = (h ∗ z|e1 ) = (z|he1 ) implies (Ne1 ◦ h ∗ | f ) = ((·|he1 )|(·|v)) = (he1 |v) = f (he1 ).   In order to describe the localization J λζ at non-zero points ζ of rank , where  s−1   < s , we pass to certain submanifolds of the Peirce flag manifold. Consider the “incidence space” ζ

Ms ,...,t := {(Us , . . . , Ut ) ∈ Ms ,...,t : ζ ∈ Us } and the “polar space” ζ∗

Ms −,...,t − := {(Ws , . . . , Wt ) ∈ Ms −,...,t − : Wt ζ ∗ = 0}. For a tripotent ζ = c this means that the flag is contained in W = E c . These compact ζ submanifolds are closely related: If (Us , . . . , Ut ) ∈ Ms ,··· ,t then the Peirce 0-spaces ζ∗

Wk := Ukc form a flag (Ws , · · · , Wt ) ∈ Ms −,··· ,t − . Conversely, if (Ws , · · · , Wt ) ∈ ζ∗

Ms −,··· ,t − and we put Wk := Wc2k for a tripotent chain cs < · · · < ct of rank s −  < · · · < t −  in W , then c + cs < · · · < c + ct is a tripotent chain of ζ 2 defines a flag (Us , . . . , Ut ) ∈ Ms ,··· ,t . rank s < · · · < t in E and Uk := E c+c k Reprinted from the journal

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H. Upmeier

Moreover, if ck ∼ ck are Peirce equivalent in W , then c + ck ∼ c + ck are Peirce equivalent in E. The closed complex subgroup ζ := {κ ∈ K  : κζ = ζ } K ζ

acts transitively on the incidence space Ms ,...,t and, similarly, the closed complex subgroup ζ ∗ := {κ ∈ K  : κ∗ ∈ K ζ } = {κ ∈ K  : κ ∗ζ = ζ } K ζ∗

acts transitively on the polar space Ms −,...,t − since h(z ζ ∗ )h −1 = (hz)(h −∗ ζ )∗ . Therefore for all h ∈ K ζ∗ ζ ∗ /( K ζ ∗ )Wcs ,...,Wct Ms −,...,t − = K

where ζ ∗ )Wcs ,...,Wct = {γ ∈ K ζ ∗ : γ Wc j = Wc j ∀ s  j  t} (K and Wcs ⊂ . . . ⊂ Wct ⊂ W is the Peirce 2-flag for a given chain of tripotents cs < . . . < ct in W of type s −  < . . . < t − . We may choose c j := e+1 + e j . The restriction homomorphism ζ ∗ → K W , κ → κ|W K defines a biholomorphic map  ζ∗ ζ ∗ /( K ζ ∗ )Wcs ,...,Wct − Wcs W / K Ms −,...,t − = K →K W

,Wct

≈

= MWs −,...,t −

(5.5)

where Wcs ,...,Wct = {γ ∈ K W : γ Wc j = Wc j ∀ s  j  t} K W and MWs −,...,t − denotes the Peirce flag manifold relative to W . Now consider the truncated partition λ∗ of length r − . Then the conical function relative to W is ∗

λ NW =

t 

cnjj −n j+1 N

j=s

and there is a holomorphic character ∗

λ χW (γ ) :=

t 

c j (γ c j )n j −n j+1 N

j=s

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Wcs ,...,Wct such that on K W ∗

∗

∗

λ λ (γ ) N λ ◦ γ ∗ = χW NW W

Wcs ,...,Wct . Let Lλ∗ denote the associated holomorphic line bundle on for all γ ∈ K W W ∗ MWs −,...,t − defined as in (5.3). A holomorphic section σ : MWs −,...,t − → LλW is W → C satisfying characterized by its homogeneous lift  σ :K ∗

λ  σ (κγ ) = χW (γ )  σ (κ)

W and γ ∈ K Wcs ,...,Wct . Via the isomorphism (5.5) we obtain a for all κ ∈ K W ∗ ∗ holomorphic pull-back line bundle Lλζ = ∗ LλW such that holomorphic sections ∗ ζ∗ ζ ∗ → C σ : K σ : Ms −,...,t − → Lλζ are characterized by the homogeneous lift  satisfying  σ (κγ ) =

λ∗ χW (γ |W )  σ (κ)

=

t 

c j (γ c j )n j −n j+1  σ (κ) N

j=s

ζ ∗ )Wcs ,...,Wct . All of this holds in particular when ζ = c ζ ∗ and γ ∈ ( K for all κ ∈ K is a tripotent of rank . Theorem 5.4 Write λ ∈ Nr+ in the form (4.4). Let c be a tripotent of rank  with Peirce 0-space W , such that s−1   < s . Consider the truncated partition (

s+1 λ∗ = (λ+1 , . . . , λr ) = (n s(s −) , n s+1

−s )

(t −t−1 )

, . . . , nt

).

Then there is an isomorphism ∗

c

∗

λ J λ ≈ PW −→ (Mc∗s ,...,t , Lλc ) ≈ c ∗

λ defines a holomorphic section σ φ : which is defined as follows: Any φ ∈ PW ∗ c∗ → C is given by the Fischer-Fock σφ : K Mc∗s ,...,t → Lλc , whose homogeneous lift  inner product ∗

λ  σ φ (κ) = (N W |φ ◦ κ|W )

c∗ . for all κ ∈ K Proof Applying Theorem 5.2 to W we have ∗

W

∗

λ PW −−→ (MWs −,...,t − , LλW ) ≈

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H. Upmeier φ

via the map φ → σW with homogeneous lift ∗

φ

λ |φ ◦ κ)  σW (κ) = (N W λ∗ W . Passing to (M c∗ for all κ ∈ K s ,...,t , Lc ) via the isomorphism (5.5), the assertion follows.   ˇ We finally describe a similar isomorphism for non-tripotent points ζ ∈ E  . Write ζ ∗ and . Then h −∗ K c∗ h ∗ = K ζ = hc for some h ∈ K

c∗ )Wcs ,...,Wct h ∗ = ( K ζ ∗ )h −∗ Wcs ,...,h −∗ Wct h −∗ ( K for the Peirce 2-flag h −∗ Wcs ⊂ . . . ⊂ h −∗ Wct ⊂ h −∗ W ⊂ E. Thus we obtain a commuting diagram c∗ /( K c∗ )Wcs ,... Wct K

Mc∗s −,...,t −

h −∗ ·h ∗

h −∗ ζ∗

ζ ∗ )h −∗ Wcs ,h −∗ Wct ζ ∗ /( K K

Ms −,...,t −.

Now the isomorphism ζ

ζ ∗ /( K ζ ∗ )h J λ −→ ( K ζ ≈

−∗ W

cs ,...

h −∗ Wct

∗

, Lλζ )

is defined via the commuting diagram Jλ c

c ≈

∗

c∗ /( K c∗ )Wcs ,...,Wct , Lλc ) ( K ≈ h ∗ ·h −∗

◦h ≈

Jλ ζ

ζ ≈

ζ ∗ /( K ζ ∗ )h ( K

−∗ W

cs ,...,h

−∗ W

ct

∗

, Lλζ ).

This demonstrates that the bundle J λζ depends in an anti-holomorphic way on ζ .  Funding Open Access funding enabled and organized by Projekt DEAL. Data Availability Statement The author confirms that the data supporting the findings of this study are available within the article. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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References 1. Akhiezer, D.: Lie Group Actions in Complex Analysis. Vieweg (1995) 2. Arazy, J.: A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains. Contemp. Math. 185, 7–65 (1995) 3. Arazy, J., Upmeier, H.: Jordan Grassmann manifolds and intertwining operators for weighted Bergman spaces. In: Doria, A. (ed.) Proceedings Cluj-Napoca, pp. 25–53. Cluj University Press (2008) 4. Biswas, S., Misra, G., Putinar, M.: Unitary invariants for Hilbert modules of finite rank. J. Reine Angew. Math. 662, 165–204 (2012) 5. Chu, C.-H.: Bounded Symmetric Domains in Banach Spaces. World Scientific (2021) 6. Cowen, C., Douglas, R.: Complex geometry and operator theory. Acta Math. 141, 187–261 (1978) 7. deConcini, C., Eisenbud, D., Procesi, C.: Young diagrams and determinantal varieties. Invent. Math. 56, 129–165 (1980) 8. Duan, Y., Guo, K.: Dimension formula for localization of Hilbert modules. J. Oper. Theory 62, 439–452 (2009) 9. Douglas, R., Misra, G., Varughese, C.: On quotient modules - the case of arbitrary multiplicity. J. Funct. Anal. 174, 364–398 (2000) 10. Englis, M., Upmeier, H.: Reproducing kernel functions and asymptotic expansions on Jordan-Kepler varieties. Adv. Math. 347, 780–826 (2019) 11. Faraut, J., Korányi, A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88, 64–89 (1990) 12. Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Clarendo Press, Oxford (1994) 13. Fischer, G.: Lineare Faseräume und kohärente Modulgarben über komplexen Räumen. Arch. Math. 18, 609–617 (1967) 14. Guo, K.: Algebraic reduction for Hardy submodules over polydisk algebras. J. Oper. Theory 41, 127– 138 (1999) 15. Korányi, A., Misra, G.: A classification of homogeneous operators in the Cowen–Douglas class. Adv. Math. 226, 5338–5360 (2011) 16. Loos, O.: Jordan Pairs. Springer Lect. Notes Math. 460 (1975) 17. Loos, O.: Bounded Symmetric Domains and Jordan Pairs. Univ. of California, Irvine (1977) 18. Morrow, J., Kodairam, K.: Complex Manifolds. AMS Chelsea Publishing (2006) 19. Neher, E.: An expansion formula for the norm function of a Jordan algebra. Arch. Math. 69, 105–111 (1997) 20. Schmid, W.: Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen. Invent. Math. 9, 61–80 (1969) 21. Upmeier, H.: Jordan algebras and harmonic analysis on symmetric spaces. Am. J. Math. 108, 1–25 (1986) 22. Upmeier, H.: Toeplitz operators on bounded symmetric domains. Trans. Am. Math. Soc. 280, 221–237 (1983) 23. Upmeier, H.: Hilbert modules and singular vector bundles on bounded symmetric domains. J. Reine Angew. Math. (Crelle) 799, 155–187 (2023) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Complex Analysis and Operator Theory (2022) 16:117 https://doi.org/10.1007/s11785-022-01292-x

Complex Analysis and Operator Theory

Functions and Operators in Real, Quaternionic, and Cliffordian Contexts Florian-Horia Vasilescu1 Received: 6 May 2022 / Accepted: 14 October 2022 / Published online: 5 November 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract This is an expository paper mainly based on certain works due to the author himself. After some introductory sections, we discuss the transformation of vector valued stem functions, defined on sets in the complex plane into quaternionic and Cliffordian valued function, using functional calculi, algebraically or derived via a Cauchy type kernel. Then we consider large families of quaternionic and Cliffordian linear operators, regarded as special classes of real linear operators, extended via a complexification procedure, and thus having the spectrum in the complex plane, which permits the construction of functional calculi with adequate analytic functions, in a classical manner, recaptured by restriction. Keywords Real, Hamilton and Clifford algebras · Spectral and Cauchy transformations · Clifford and quaternionic operators · Analytic functional calculus Mathematics Subject Classification Primary 47A10; Secondary 47A20 · 47A60 · 30G35 · 15A66

1 Introduction The aim of the present work is to exhibit an overview of some of the main results from the author’s works [22, 24, 25], adding certain relevant comments and new remarks.

Dedicated to the memory of Jörg Eschmeier. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

B 1

Florian-Horia Vasilescu [email protected] Department of Mathematics, University of Lille, 59655 Villeneuve d’Ascq, France

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From historical point of view, the algebra of quaternions, denoted in the following by H and often called Hamilton algebra, was introduced in mathematics by W. R. Hamilton as early as 1843. It is a unital non commutative division algebra, with numerous applications not only in mathematics but also in physics, and in other domains as well. The celebrated Frobenius theorem, proved in 1877, placed the algebra of quaternions among the only three finite dimensional division algebras over the real numbers, which is a remarkable feature shared with the real and complex fields. In physics, an early suggestion towards a quaternionic quantum mechanics can be found in a paper by G. Birkhoff and J. von Neumann [2], and later, a quaternionic quantum mechanics was firstly developed in [9]. A Clifford algebra is a unital associative algebra, whose elements generalize the real numbers, complex numbers, quaternions and other hypercomplex systems of numbers. Its theory is connected with that of quadratic forms and orthogonal transformations, with applications in many domains, as analysis, geometry and theoretical physics. This concept is named after the English mathematician William Kingdon Clifford, who had several contributions in this area (see the paper [3] as a first reference). In the present work, we use a special type of a Clifford algebra, which is unital and has n generators, denoted by Cn (see Subsect. 3.1). Going back to quaternions, an important investigation in their context has been to find a convenient manner to express the ”analyticity“ of functions depending on them. Among the pioneer contributions in this direction one should mention the works [18] and [10]. More recently, a concept of slice regularity (or hyperholomorphy) for functions of one quaternionic variable has been introduced in [12] (see also [11]), leading to a vast development sythesized in [6], which contains a large list of references. Moreover, functions defined on other domains, in particular on sets in a Clifford agebra Cn , were considered from this point of view (see [7, 13], etc.). As a matter of fact, we shall mainly deal with functions in the Cliffordian context, deriving the corresponding results for the quaternionic context as (almost) particular cases. Unlike in [12] and in the articles following, in an early preprint cited and partially presented in the work [23], the regularity of a quaternionic-valued function was investigated in the context of matrix quaternions. In fact, each matrix quaternion was regarded as a normal operator, having a spectrum, which was used to define various compatible functional calculi, including the analytic one. This discussion was continued and refined to the context of the abstract Hamilton algebra in [22], by embedding the real C ∗ -algebra of quaternions into its complexification, organized as a complex C ∗ -algebra. The new arguments were not only much simpler but the framework was intrisic, that is, it did not depend of any representation of Hamilton’s algebra as a matrix algebra. Our main tools in [22] were the complexification and conjugation, also intensively used in the subsequent works. Similar ideas also appear in some previous works (see for instance [13]) but the systematic use of the concept of spectrum of a quaternion, seemingly not used so far, is essential for our development, leading to a new approach to regularity, equivalent but different from that based on slice regularity, in both quaternionic and Cliffordian contexts. In fact, the regularity of quaternionicvalued functions, or functions with values in a Clifford algebra, becomes a consequence

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of their representation via a Cauchy type integral (see for instance Proposition 1 from [22]). The scientific necessities of the quantum mechanics, as well as those of other domains like partial differential equations, imposed the study of vector spaces, which are modules over the Hamilton and Clifford algebras, and the associated operators, which are linear with respect to these algebras. Unlike in works by other authors, many of them quoted in the monographs [6, 7], in order to construct an analytic functional calculus for quaternionic linear operators, the class of the quaternionic slice regular functions, defined on subsets in the quaternionic algebra, is replaced by a class of vector-valued holomorphic functions, called stem functions, defined on subsets in the complex plane. In fact, these two classes are isomorphic via a Cauchy type transform (see Theorem 6 from [22]), and we use the latter to construct an analytic functional calculus for what are called quaternionic linear operators. Similar properties are valid in the Cliffordian context, which are exhibited in the work [24], and some of them will be recalled in what follows. The spectral theory for quaternionic or Cliffordian linear operators has been already discussed in numerous work, in particular in the monographs [6] and [7], where the construction of an analytic functional calculus (called S-analytic functional calculus) amounts to associate to a fixed quaternionic or Cliffordian linear operator and to a function from the class of slice hyperholomorphic (or slice regular functions), an “extension” using a specific noncommutative kernel. Unlike in these works, our idea is to first consider the case of real operators on real Banach spaces, whose complex spectrum is in the complex plane, and to perform the construction of an analytic functional calculus for them, using some classical ideas. Then, regarding the quaternionic or Clifford operators as particular cases of real ones, this framework is extended to them, showing that the approach from the real case can be adapted to that more intricate situation. Unlike in [6] or [7], the analytic functional calculus is obtained via a Riesz-Dunford-Gelfand formula, defined in a partially commutatative context, rather than the non-commutative Cauchy type formula used by previous authors. This is possible because the S-spectrum, introduced by F. Colombo and I. Sabadini (see [5]), can be replaced by a spectrum in the complex plane. Moreover, we show the analytic functional calculus obtained with some vectorvalued stem functions, defined in the complex plane, is equivalent to the analytic functional calculus obtained with slice holomorphic functions in [6] or [7], in the sense that the images of these functional calculi coincide (see Remark 8 from [25] and Remark 17 from [24]). Let us briefly describe the contents of this work. The next section is dedicated to some (more or less known) results valid in abstract real algrebras. We present with its proof Theorem 1, which reveals the importance for our approach of the functions preserving the conjugacy, and recall the statement of the general analytic functional calculus in real algebras (Theorem 2), which can be used, with no major changes, also in quaternionic and Cliffordian contexts. A Cauchy transformation, a general concept of complex spectrum, and an abstract slice regularity are also mentioned in this section. The third section presents Hamilton’s and Clifford’s algebras, as real algebras, insisting on the concept of the complex spectrum in these frameworks.

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Inspired by some elementary properties of certain particular spectral operators (see [8], Part III), the fourth section starts with a discrete functional calculus, called spectral transformation, applied to some vector valued functions, defined in the complex plane, leading to Clifford or quaternionic valued functions (see Theorem 3), whose properties are stated by Theorem 3. The slice regularity of such functions is also recalled, the vector valued commutative Cauchy kernel of type Riesz-Dunford is considered, and its the regularity is proved. Example 3 shows that what the non-commutative kernel used in most of the preceding works of other authors is given by the Cauchy transform the Riesz-Dunford kernel. The fifth section introduces quaternionic and Clifford operators, discussing their complex spectrum, as compared with the S-spectrum. The final section is dedicated to the analytic functional calculus for Clifford and Hamilton operators, presenting a general statement, and proving the equivalence between the present functional calculus with functions defined in the complex plane and that developed by the previous authors, with functions defined on domains in the Clifford or Hamilton algeberas. Thanks are due to the referee for a careful reading of the manuscript and for several suggestions.

2 Some Preliminary Results in Real Algebras In this section we mainly present some elements of spectral theory in real algebras, that is, algebras over the real field R (see [1, 15, 17, 19, 21] etc.). We also present a general form of what is called in the literature the slice regularity (see [12] for the original concept in the quaternionic context), introduced in real alternative algebras in [14]. Some concrete examples will be later discussed. 2.1 Spectrum and Analytic Functional Calculus Let A be a unital real Banach algebra, and let AC be its complexification, that is, AC = C ⊗R A, identified with the direct sum A + iA, where i is the imaginary unit. Endowed with the norm c = a + b, where c = a + ib, b, c ∈ A, the algebra AC becomes a unital complex Banach algebra. Moreover, the complex field C can be identified with a subalgebra of AC , consisting of multiples of the unit. Using an idea going back to Kaplansky (see [17] or [15]), the (complex) spectrum of an element a ∈ A is given by σC (a) = {u + iv; u, v ∈ R, (u − a)2 + v 2 not invertible in A}.

(1)

Note that the set σC (a) is conjugate symmetric, that is, u + iv ∈ σC (a) if and only if u − iv ∈ σC (a). The map AC  a + ib → a − ib ∈ AC is a conjugation of AC , meanig that it is a real unital automorphism of AC , whose square is the identity. If s = a + ib we usually put s¯ = a − ib for all s ∈ AC . In particular, an element s ∈ AC is invertible if and

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only if the element s¯ ∈ AC is invertible. Of course, the conjugation of AC extends the usual conjugation of the complex plane C. For an element a ∈ AC , we denote by σ (a) its usual spectrum. Note that λ ∈ σ (a) if and only if λ¯ ∈ σ (¯a). When a ∈ A is regarded as an element of AC , formula (1) can be rewritten as σC (a) = {λ ∈ C, λ − a not invertible in AC },

(2)

because |λ|2 − 2a (λ) + a 2 = (λ − a)(λ¯ − a) is not invertible in AC is equivalent to the fact that neither λ − a nor λ¯ − a is invertible in AC . Consequently, the complex spectrum σC (a) coincides with the spectrum σ (a), computed in AC . Having a concept of spectrum, we may also discuss a concept of analytic functional calculus. We start with some necessary notation. If U ⊂ C is an open set, we denote by O(U , AC ) the algebra of all analytic AC valued functions. If U is conjugate symmetric, and AC  a → a¯ ∈ AC is its natural conjugation, we denote by Os (U , AC ) the real subalgebra of O(U , AC ) consisting of those functions f with the property f (ζ¯ ) = f (ζ ) for all ζ ∈ U . Following [14], such functions will be called (AC -valued ) stem functions. In fact, this definition has an old origin, seemingly going back to [10]. When A = R, so AC = C, the space Os (U , C) will be denoted by Os (U ), which is a real algebra. Note that Os (U , AC ) is also a two-sided Os (U )-module. Let U ⊂ C, and let S(U ) be the set {a ∈ AC ; σ (a) ⊂ U }. If U is open, then S(U ) is also open, via the upper semicontinuity of the spectrum (see [8]). We put AC (U ) = {s ∈ AC ; σ (s) ⊂ U }, and consider the usual analytic functional calculus for an element a ∈ AC (U ):  1 O(U , AC )  f → f (a) = f (λ)(λ − a)−1 dλ ∈ AC , 2πi 

where  is the boundary of a Cauchy domain (that is,  is a finite union of Jordan piecewise smooth closed curves) containing σ (a) in its interior. A natural question is to find a condition insuring the inclusion C[ f ](a) ∈ A for f in a sufficiently large subspace of O(U , AC ), whenever a ∈ A such that σC (a) ⊂ U . The next result is a more general version of Theorem 1 from [25] or of Theorem 6 from [24]. Theorem 1 Let U ⊂ C be open and conjugate symmetric. If f ∈ Os (U , AC ), we have f (a) = f (¯a) for all a with σ (a) ⊂ U . Proof The proof follows the lines of the proof of Theorem 1 from [25]. We put ± :=  ∩ C± , where C+ (resp. C− ) equals to {λ ∈ C; λ ≥ 0} (resp. {λ ∈ C; λ ≤ 0}). We write + = ∪mj=1  j+ , where  j+ are the connected components of + . Similarly, we write − = ∪mj=1  j− , where  j− are the connected components of − , and  j− is the reflexion of  j+ with respect of the real axis.   We fix a function f ∈ Os (U , AC )). As  is the boundary of a Cauchy domain, for each index j we have a parametrization φ j : [0, 1] → C, positively oriented, such that

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φ j ([0, 1]) =  j+ . Taking into account that the function t → φ j (t) is a parametrization of  j− negatively oriented, and setting  j =  j+ ∪  j− , we define 1 2πi

f j (a) :=

=

1 2πi

1



f (ζ )(ζ − a)−1 dζ

j

f (φ j (t))(φ j (t) − a)−1 φ j (t)dt

0

1 − 2πi

1

f (φ j (t))(φ j (t) − a)−1 φ j (t)dt.

0

Therefore, 1 f j (a) = − 2πi

1

f (φ j (t))(φ j (t) − a¯ )−1 φ j (t)dt

0

+

1 2πi

1

f (φ j (t))(φ j (t) − a¯ )−1 φ j (t)dt.

0

via our assumption on the function f . Consequently, f j (a) = f j (¯a) for all j, and so f (a) =

m  j=1

f j (a) =

m 

f j (¯a) = f (¯a).

j=1

Remark 1 According to Theorem 1, given a conjugate symmetric open set U ∈ C, and setting A(U ) = {a ∈ A; σC (a) ⊂ U }, we have a map Os (U , AC ) × A(U )  ( f , a) → f (a) ∈ A given by 1 O(U , AC )  f → f (a) = 2πi

 

f (λ)(λ − a)−1 dλ

where  is the boundary of a Cauchy domain containing σC (a) in its interior. The properties of the map f → f (a), which can be called the (left) analytic functional calculus of a, are given by the following. Theorem 2 Let A be a unital real Banach algebra and let AC be its complexification. Let also U ⊂ C be a conjugate symmetric open set, and let a ∈ A, with σC (a) ⊂ U . Then the assignment

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Os (U , AC )  f → f (a) ∈ A is an R-linear map, and its restriction Os (U )  f → f (a) ∈ A is a unital real algebra morphism. Moreover, the following properties are true: (U ), we have ( f h)(a) = f (a)h(a). (1) For all f ∈ Os (U , AC ), h ∈ Os m cn ζ n , ζ ∈ C, with cn ∈ A for all n = (2) For every polynomial P(ζ ) = m n=0 n 0, 1, . . . , m, we have P(a) = n=0 cn a . Proof The arguments are more or less standard (see [8]). The R-linearity of the maps Os (U , AC )  f → f (a) ∈ A, Os (U )  f → f (a) ∈ A, is clear. The second one is actually multiplicative, which follows from the multiplicativity of the usual analytic functional calculus of a. The more general property, specifically ( f h)(a) = f (a)h(a) for all f ∈ Os (U , AC ), h ∈ Os (U ) follows from the equalities, 1 2πi



f (ζ )h(ζ )(ζ − a)−1 dζ

0

⎛

⎜ 1 =⎝ 2πi

 0

⎞⎛ 1 ⎟ f (ζ )(ζ − a)−1 dζ ⎠ ⎝ 2πi



⎞ f (η)(η − a)−1 dη⎠ ,



obtained as in the classical case (see [8], Section VII.3), which holds because h is C-valued and commutes with the elements of A. Here , 0 are the boundaries of two Cauchy domains , 0 respectively, such that contains the closure of 0 , and 0 contains σ (a).  n In particular, for every polynomial P(ζ ) = m n=0 cn ζ with cn ∈ A for all n = m n   0, 1, . . . , m, we have P(a) = n=0 cn a ∈ B(V) for all a ∈ A. Definition 1 Let A be a unital real Banach algebra, and let U ⊂ C be a conjugate symmetric open set. Let also AC be the complexification of A, let A(U ) = {a ∈ A; σC (a) ⊂ U }, and let R(A(U ), A) = {C[ f ] : A(U ) → A; C[ f ](a) = f (a), f ∈ Os (U , AC ), a ∈ A(U )}. The R-linear map Os (U , AC )  f → C[ f ] ∈ R(A(U ), A) is called the A-Cauchy transformation on Os (U , AC )

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2.2 Algebra of Real Linear Operators An important unital real algebra is that consisiting of bounded R-linear operators, which will be discussed in the following. Let V be a real vector space. We denote by VC the complexification of V, identified with the direct sum V + iV, where i is the imaginary unit of the complex plane. As in the case of algebras, the map VC  x + i y → x − i y ∈ VC is said to be the (natural) conjugation of VC , which is R-linear, and whose square is the identity. It will be usually denoted by C. If V is a real Banach space, then VC is a complex Banach space, for which we fix the norm x + i y = x + y for all x + i y ∈ VC with x, y ∈ V, where  ∗  is the norm of V. In this way, the conjugation C is an isometry. Let B(V) (resp. B(VC )) be the Banach algebra of all R-linear (resp. C-linear) operators acting on V (resp. of VC ). We have an unital injective algebra morphism B(V)  T → TC ∈ B(VC ) given by TC (x + i y) = T x + i T y for all T ∈ B(V), which R-linear. The operator TC will be called the complex extension of T . The natural conjugation of VC induces a conjugation on B(VC ) via the equality S = C SC for all S ∈ B(VC ). We set Bc (VC ) = {S = S ; S ∈ B(VC )}, which is a unital real algebra. Then we have the direct sum B(VC ) = Bc (VC ) + iBc (VC ), via the remark that S + S , i(S − S ) ∈ Bc (VC ). Clearly, the real algebras B(V) and Bc (VC ) are isomorphic, also as B(V)C and B(VC ). As a particular case of formula (2), the complex spectrum of an operator T ∈ B(V), regarded as an element of B(V)C , is given by σC (T ) = {λ ∈ C, λ − T not invertible in B(V)C }. 2.3 Abstract Slice Regularity Let SA be the set {e ∈ A; e2 = −1, e = 1}, which will be called the imaginary sphere of A, and which is supposed to be nonempty. To simplify the notation, we put S = SA . Each element e ∈ S is said to be an imaginary unit. For such an element e ∈ S, we define the set Ce := {x + ye; x, y ∈ R}, which is isomorphic to the complex plane. Let  ⊂ A, and let F :  → AC . For a fixed imaginary unit e ∈ S, we assume that the set Ce ∩  is nonempty and open, and that the restriction of F to Ce ∩  is differentiable as a functions of x, y, with x + ye ∈ Ce ∩ . For such a function F, we define the operator 1 ∂¯e = 2



∂ ∂ + Re ∂x ∂y

F(x + ye), x + ye ∈ Ce ∩ ,

where Re is the right multiplication of the elements of AC by e. Such a definition, ¯ introducing the classical ∂-operator in a noncommuting framework, firstly appears in [12], in the quaternionic context (see also [4, 6, 7], etc.). For an abstract similar concept, valid in real alternative algebras, see the paper [14]. In this spirit, we say that

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F is slice regular along the imaginary unit e if ∂¯e F(x + ye) = 0 on the set Ce ∩ . When F is slight regular along all e ∈ S, we say that F is slice regular. Unlike in [6] and in other works, we have some reasons, related to the action of certain operators, to use the right slice regularity rather than the left one. Nevertheless, a left slice regularity can also be defined via the left multiplication of the elements of AC by elements from S. In what follows, the right slice regularity will be simply called slice regularity. Example 1 Fixing a ∈ AC , we denote by  the the set of those s ∈ A such that a − s is invertble in AC . Choosing a imaginary unit e ∈ S such that ae = ea, and assuming that Ce ∩  is nonempty, we can write the equalities ∂ (a − x − ye)−1 = (a − x − ye)−2 , ∂x ∂ Rs (a − x − ye)−1 = −(a − x − ye)−2 , ∂y because e2 = −1, and a, e and (a − x − ye)−1 commute in AC . Therefore, ∂¯e ((a − x − ye)−1 ) = 0, implying that the function   s → (a − s)−1 ∈ AC is slice regular.

3 Hamilton and Clifford Real Algebras The results from the previous section will be applied especially to Hamilton and Clifford algebras, regarded as real algebras. In this section we mainly discuss the specific framework of these algebras, dealing especially with elements of spectral theory. The next subsection is inspired by the Subsect.2.1 from [24]. 3.1 Preliminaries for Clifford Algebras In the following, a Clifford algebra, denoted by Cn for a fixed integer n ≥ 0, is a unital associative real algebra having n + 1 generators e0 = 1, e1 , . . . , en , satisfying the relations e2j = −1, e j ek = −ek e j for all j, k = 1, . . . , n, j = k (see also [6, 16, 20, 24] etc.). In particular, C0 = R, C1 = C, and C2 = H, that is, the the real, complex and quaternionic algebras are special cases of Clifford algebras. If Nn = {1, 2, . . . , n}, for every subset J = { j1 , j2 , . . . , j p } ⊂ Nn , with j1 < j2 < · · · < j p and 1 ≤ p ≤ n, we put e J = e j1 e j2 · · · e j p . We use the symbol J ≺ Nn to indicate that J is an oredered set as above. Assuming also that ∅ ≺ Nn , e∅ = 1, e{ j} = e j , j = 1, . . . , n, and that the family {e J } J ≺Nn is a basis of the vector space Cn , an arbitrary element a ∈ Cn can be written as a=



aJ eJ ,

J ≺Nn

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where a J ∈ R are uniquely determined for all J ≺ Nn . To simplify the notation, we also put e0 = e∅ = 1, and a0 = a∅ . The elements of the Clifford algebra Cn will be called Clifford vectors, or briefly, Cl-vectors. The real linear subspace of Cn spanned by {ek }nk=0 will be denoted by Pn , playing an important role in the following. The Cl-vectors from the subspace Pn , which have the form a = a0 + nk=0 ak ek with ak ∈ R for all k = 0, . . . , n, will be called paravectors (as in [6]). The linear subspace Pn will be often identified with the Euclidean space Rn+1 , via the linear isomorphism Pn 

n 

ak ek → (a0 , a1 , . . . , an ) ∈ Rn+1 .

k=0

 For every a = J ∈ Cn we have a decomposition a = (a) + (a), J ≺Nn a J e where (a) = a0 and (a) = ∅= J ≺Nn a J e J , that is, the real part and the imaginary part of the Cl-vector a ∈ Cn , respectively. The algebra Cn has a norm defined by; |a|2 =



a 2J ,

(4)

J ≺Nn

where a is given by (3). The algebra Cn also has an involution Cn  a → a∗ ∈ Cn , which is defined via the conditions e∗j = −e j ( j = 1, . . . , n), r ∗ = r ∈ R, (ab)∗ = b∗ a∗ for all a, b ∈ Cn (see [6], Definition 2.1.11). According to Proposition 2.1.12 from  [6], we therefore have (a∗ )∗ = a, (a + b)∗ = a∗ + b∗ . Particularly, if a = a0 + nj=1 a j e j , then  a∗ = a0 − nj=1 a j e j , for all a ∈ Pn . Considering the complexification Kn = C ⊗R Cn of Cn , identified with the direct sum Cn + iCn , we have a unital algebra with the involution Kn  c = a + ib → c∗ = a∗ − ib∗ ∈ Kn , a, b ∈ Cn , which extends the involution of Cn . As in the abstract case (see Subsect. 2.1), we have a conjugation on the complexification Kn . Specifically, the R-linear map Kn  c = a + ib → c¯ := a − ib ∈ Kn is a conjugation on Kn , so it is a real automorphism of unital algebras, whose square is the identity. Note that the elements of the real subalgebra Cn commute with the complex numbers in the algebra Kn . As we have, a¯ = a if and only if a ∈ Cn , we get a useful criterion to identify the elements of Cn among those from Kn . The next subsection is partially inspired by the corresponding part from Sect. 3 in [24].

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3.2 Spectrum of a Paravector Because Cn is a real algebra, we have natural concept of (complex) spectrum, via the coreresponding concept in the complex algebra Kn , as noticed in the previous section. For an element κ ∈ Cn , we denote by σC (κ) its spectrum, when κ is identified with an element of Kn . Let also ρC (κ) := C \ σC (κ).We are particularly interested in the spectrum of a paravector, which can be completely described. Lemma 1 Let κ ∈ Pn and let λ ∈ C. The following conditions are equivalent: (i) λ ∈ ρC (κ); (ii) λ2 − λ(κ + κ ∗ ) + |κ|2 = 0; (iii) |λ|2 − 2κ (λ) + κ 2 invertible in Cn . Proof Because κ commutes with κ ∗ and with λ, the equivalence (i) ⇔ (ii) follows from the equality (λ − κ ∗ )(λ − κ) = (λ − κ)(λ − κ ∗ ) = λ2 − λ(κ + κ ∗ ) + |κ|2 ∈ C.

(5)

The equivalence (i) ⇔ (iii) follows from the equality (λ − κ)(λ¯ − κ) = |λ|2 − 2κ (λ) + κ 2 ,

(6)

using the fact that λ − κ is invertible if and only if λ¯ − κ is invertible. Remark 2 We follow the lines of Remark 1 from [22]. (1) It follows from Lemma 1 that λ−κ is invertible if and only if the complex number λ2 − 2λ (κ) + |κ|2 is nonnull. Therefore (λ − κ)−1 =

λ2

1 (λ − κ ∗ ). − 2λ (κ) + |κ|2

Hence, the element λ−κ ∈ Kn is not invertible if and only if λ = (κ)±i| (κ)|. In this way, the spectrum of a paravector κ is given by the equality σC (κ) = {s± (κ)}, where s± (κ) = (κ) ± i| (κ)| are the eigenvalues of κ. (2) As usually, the function ρC (κ)  λ → (λ − κ)−1 ∈ Kn is called the resolvent (function) of κ, and it is a Kn -valued analytic function on ρC (κ). (3) Two paravectors κ, τ ∈ Pn have the same spectrum if and only if (κ) = (τ ) and | (κ)| = | (τ )|. In fact, the equality σC (κ) = σC (τ ) is an equivalence relation in Pn , and the equivalence class of an element κ0 = x0 + y0 s0 ∈ Pn is given by {x0 + y0 s; s ∈ Sn }, where Sn the unit sphere of imaginary paravectors.

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(4) Every paravector κ ∈ Pn \ R can be written as κ = x + ys, where x, y are real numbers, with x = (κ), y ∈ {±| (κ)|}, and s ∈ {± (κ)/| (κ)|} ⊂ Sn . Clearly, we have σC (κ) = {x ± i y}, because (κ) = ys, and the spectrum of κ does not depend on s. Thus, for every λ = u + iv ∈ C with u, v ∈ R, we have ¯ for all s ∈ Sn . σ (u + vs) = {λ, λ} (5) Fixing an element s ∈ Sn , we have an isometric R-linear map from the complex plane C into the space Pn , say θs , defined by θs (u + iv) = u + vs, u, v ∈ R. For every subset A ⊂ C, we put As = {x + ys; x, y ∈ R, x + i y ∈ A} = θs (A).

(7)

Note that, if A is open in C, then As is open in the R-vector space Cs ⊂ Pn . The next subsection reproduces parts from Subsect. 2.1 and Section 3 from [22]. 3.3 Hamilton Algebra The algebra of quaternions H, or Hamilton algebra, is the Clifford algebra C2 , generated as an algebra by the family {e0 , e1 , e2 }. We put e1 = j, e2 = k, and setting l = jk, we obtain the equalities jk = −kj = l, kl = −lk = j, lj = −jl = k, jj = kk = ll = −1. The norm of H defined by (4) and given by |x| =

x02 + x12 + x22 + x32 , x = x0 + x1 j + x2 k + x3 l, x0 , x1 , x2 , x3 ∈ R,

is multiplicative, and its involution is H  x = x0 + x1 j + x2 k + x3 l → x∗ = x0 − x1 j − x2 k − x3 l ∈ H, satisfying x∗ x = xx∗ = |x|2 for all x ∈ H. In particular, every element x ∈ H\{0} is invertible, and x−1 = |x|−2 x∗ . As in the case of Cl-vectors, for an arbitrary quaternion x = x0 + x1 j + x2 k + x3 l, x0 , x1 , x2 , x3 ∈ R, we set x = x0 = (x + x∗ )/2, and x = x1 j + x2 k + x3 l = (x − x∗ )/2, that is, the real and the imaginary part of x, respectively. Let us note that the space of paravectors P2 of the Clifford algebra H = C2 is the vector space spanned by {1, j, k} = H, but when working with quaternions, we replace P2 by H. Nevertheless, most of the properties of the paravectors hold true for quaternions. The complexification H + iH of H will be denoted by M. In the complex algebra M we use the natural concept of spectrum, which can be described in the case of quaternions, as in the case of paravectors.

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Remark 3 (1) Because of the identities (λ − x∗ )(λ − x) = (λ − x)(λ − x∗ ) = λ2 − λ(x + x∗ ) + |x|2 ∈ C, for all λ ∈ C and x ∈ H, implying that the element λ − x ∈ M is invertible if and only if the complex number λ2 − 2λ x + x2 is nonnull, we obtain that (λ − x)−1 =

λ2

1 (λ − x∗ ). − 2λ x + |x|2

Therefore, the spectrum of a quaternion x ∈ H is given by the equality σ (x) = {s± (x)}, where s± (x) = x ± i| x| are the eigenvalues of x. (2) As usually, the resolvent set ρ(x) of a quaternion x ∈ H is the set C \ σ (x), while the function ρ(x)  λ → (λ − x)−1 ∈ M is the resolvent (function) of x, which is an M-valued analytic function on ρ(x). (3) Note that two quaternions x, y ∈ H have the same spectrum if and only if x = y and | x| = | y|. Moreover, the equality σ (x) = σ (y) is an equivalence relation. (4) Let S be the unit sphere of imaginary quaternions. Every quaternion q ∈ H \ R can be written as q = x + ys, where x, y are real numbers, with x = q, y = ±| q|, and s = ± q/| q| ∈ S, and we have σ (q) = {x ± i y}. In fact, for every λ = u + iv ∈ C with u, v ∈ R, we have σ (u + vs) = {λ, λ¯ } for all s ∈ S. Remark 4 The properties of interest of the quaternionic algebra H are not always direct consequences of those of the Clifford algebra C2 . Indeed, in this case of H, we are mainly interested in H-valued functions, defined on subsets of H. The algebra C2 is generated by {1, e1 , e2 }, and the vector space P2 generated by this set is strictly included in H. As the algebra H is also generated by the set {1, e1 , e2 , e3 }, where e3 = e1 e2 , it is isomorphic to the quotient of the algebra C3 by the two-sided ideal generated by e3 − e1 e2 . Consequently, a separate approach concerning the quaternion algebra H (as in [6]), rather than that in the framework of Clifford algebras, is often more appropriate. Note also that for C = C1 we have P1 = C.

4 Spectral Transformation of Some Vector-Valued Stem Functions The space of stem Kn -valued functions, defined on subsets of the complex plane, may be associated with spaces of functions, defined on subsets of Pn , taking values in the Clifford algebra Cn , using spectral methods and functional calculi. This operation may be regarded either as an ”extension“ or as a general functional calculus, with arbitrary functions. Similar results are also valid for some M-valued functions. Nevertheless, we shall mainly deal with Kn -valued functions, as developed in [24], Sect. 4, providing references also for the case of M-valued functions whenever necessary, which can be found in [22].

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4.1 Stem Functions and Spectral Extensions The main idea of the following approach comes from the theory of spectral operators (see [8], Part III), and it can be applied to define an appropriate functional calculus, also useful in the Cliffordian (or quaternionic) context. Specifically, regarding the algebra Kn as a (complex) Banach space, and considering the Banach space B(Kn ) of all linear operators acting on Kn , the operator L κ , κ ∈ Pn , which is the left multiplication operator on Kn by the paravector κ, is a particular case of a scalar type operator, as defined in [8], Part III, XV.4.1. Its resolution of the identity consists of four projections {0, P± (κ),I}, including the null operator 0 and the identity I, where P± (κ) are the spectral projections of L κ , whose spectrum coincides with that of κ, and its integral representation is given by L κ = s+ (κ)P+ (κ) + s− (κ)P− (κ) ∈ B(Kn ), provided by Corollary 1. For every function f : σ (κ) → C we may define the operator f (L κ ) = f (s+ (κ))P+ (κ) + f (s− (κ))P− (κ) ∈ B(Kn ),

(8)

which provides a functional calculus with arbitrary functions on the spectrum. One can even replace the scalar function f by a function F : σ (κ) → B(Kn ), getting what may be called a ”left functional calculus“, not multiplicative, in general. It is this idea which leads us to try to define some Cn -valued functions on subsets of Pn via certain Kn -valued functions, defined on subsets of C. The next concept is is quoted from [24], Definition 2(2). Definition 2 A subset A ⊂ Pn is said to be spectrally saturated (as in [22], Definition 2) if whenever σ (θ ) = σ (κ) for some θ ∈ Pn and κ ∈ A, we also have θ ∈ A. For an arbitrary A ⊂ Pn , we put S(A) = ∪κ∈A σ (κ) ⊂ C. Conversely, for an arbitrary subset S ⊂ C, we put Sσ = {κ ∈ Pn ; σ (κ) ⊂ S}. From [22], Remark 4, we quote the following properties. Remark 5 (1) If A ⊂ Pn is spectrally saturated, then S = S(A) is conjugate symmetric, and conversely, if S ⊂ C is conjugate symmetric, then Sσ is spectrally saturated, which can be easily seen. Moreover, the assignment S → Sσ is injective. Similarly, the assignment A → S(A) is injective and A = Sσ if and only if S = S(A). (2) If  ⊂ Pn is an open spectraly saturated set, then S() ⊂ C is open, using a direct argument. Conversely, if U ⊂ C is open and conjugate symmetric, the set Uσ is also open via the upper semi-continuity of the spectrum (see [8], Part I, Lemma VII.6.3.). An important particular case is when U = Dr := {ζ ∈ C; |ζ | < r }, for some r > 0. Then Uσ = {κ ∈ Pn ; |κ| < r }. Indeed, if |κ| < r and θ has the property σ (κ) = σ (θ ), from the equality { (κ) ± i| (κ)|} = { (θ ) ± i| (θ )|} it follows that |θ | < r .

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(3) A subset  ⊂ Cn is said to be axially symmetric if for every κ0 = u 0 + v0 s0 ∈  with u 0 , v0 ∈ R and s0 ∈ Sn , we also have κ = u 0 + v0 s ∈  for all s ∈ Sn . This concept is introduced in [6], Definition 2.2.17. In fact, we have the following. Lemma 2 A subset  ⊂ Cn is axially symmetric if and only if it is spectrally saturated. The assertion follows easily from the fact that the equality σ (κ) = σ (τ ) is an equivalence relation in Pn (see Remark 2(3)). We prefer to use the expression ”spectrally saturated set“ to designate an ”axially symmetric set“, because the former name is more compatible with our spectral approach. As in the general case (see Sect. 2), the algebra Kn is endowed with a conjugation given by a¯ = b − ic, when a = b + ic, with b, c ∈ Cn . Note also that, because C is a subalgebra of Kn , the conjugation of Kn restricted to C is precisely the usual complex conjugation. Let U ⊂ C be conjugate symmetric, and let F : U → Kn be a Kn -valued stem function (that is, F(λ¯ ) = F(λ) for all λ ∈ U ). For an arbitrary conjugate symmetric subset U ⊂ C, we put S(U , Kn ) = {F : U → Kn ; F(ζ¯ ) = F(ζ ), ζ ∈ U },

(9)

that is, the R-vector space of all Kn -valued stem functions on U . Replacing Kn by C, we denote by S(U ) the real algebra of all C-valued stem functions, which is an Rsubalgebra in S(U , Kn ). In addition, the space S(U , Kn ) is a two-sided S(U )-module. Remark 6 As in [25], Remark 4, every paravector s ∈ Sn may be associated with two elements ι± (s) = (1 ∓ is)/2 in Kn , which are commuting idempotents such that ι+ (s) + ι− (s) = 1 and ι+ (s)ι− (s) = 0. For this reason, setting Ks± = ι± (s)Cn , we have a direct sum decomposition Kn = Ks+ + Ks− . Explicitly, if a = u + iv, with u, v ∈ Cn , the equation ι+ (s)x + ι− (s)y = a has the unique solution x = u + sv, y = u − sv ∈ Cn , because s−1 = −s. ˜ κ| ˜ −1 , where κ˜ = (κ), the In particular, if κ ∈ Pn and (κ) = 0, setting sκ˜ = κ| elements ι± (sκ˜ ) are idempotents, as above. The next result provides explicit formulas of the spectral projections (see [8], Part I, Section VII.1) associated to the element κ ∈ Pn , regarded as a left multiplication operator on Kn . They are not trivial only if κ ∈ Pn \ R because if κ ∈ R, its spectrum is this real singleton, and the only spectral projection is the identity. The statement of the result corresponding to Lemma 1 from [25] looks like that: Lemma 3 Let κ ∈ Pn \ R be fixed. The spectral projections associated to the eigenvalues s± (κ) are given by P± (κ)a = ι± (sκ˜ )a, a ∈ Kn .

(10)

Moreover, P+ (κ)P− (κ) = P− (κ)P+ (κ) = 0, and P+ (κ) + P− (κ) is the identity on Kn . When κ ∈ R, the corresponding spectral projection is the identity on Kn .

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Corollary 1 For every κ ∈ Pn and a ∈ Kn we have L κ a = s+ (κ)P+ (κ)a + s− (κ)P− (κ)a. We now define the concept of spectral transformation for some K-valued functions (see Definition 4 from [24]). Definition 3 Let U ⊂ C be conjugate symmetric. For every F : U → Kn and all κ ∈ Uσ we define a function Fσ : Uσ → Kn , via the assignment Uσ \ R  κ → Fσ (κ) = F(s+ (κ))ι+ (sκ˜ ) + F(s− (κ))ι− (sκ˜ ) ∈ Kn ,

(11)

where κ˜ = (κ), sκ˜ = |κ| ˜ −1 κ, ˜ and ι± (sκ˜ ) = 2−1 (1 ∓ isκ˜ ), and Fσ (r ) = F(r ), if r ∈ Uσ ∩ R. Formula (11) is strongly related to formula (8), via Remark 6. Theorem 3 Let U ⊂ C be a conjugate symmetric subset, and let F : U → Kn . The element Fσ (κ) belongs to Cn for all κ ∈ Uσ if and only if F ∈ S(U , Kn ). For the proof we refer to Theorem 1 from [24]. Corollary 2 Let U ⊂ C be a conjugate symmetric subset, and let f : U → C. The following conditions are equivalent; (1) f ∈ S(U ); (2) f σ (κ) belongs to Cs , and f σ (κ ∗ ) = f σ (κ)∗ for all κ ∈ Uσ ∩ Cs , where Cs = {u + vs; u, v ∈ R}, and s ∈ Sn . This result is Corollary 2 from [24]. Remark 7 This is a description of the zeros of the functions obtained via Theorem 3, corresponding to Remark 7 from [24]. We shall sketch the proof of one inclusion. Let U ⊂ C be a conjugate symmetric set and let F ∈ S(U , Kn ) be arbitrary. ˜ + We describe the zeros of Fσ in the following way. If Fσ (κ) = F(s+ (κ))ι+ (κ) ˜ = 0, we deduce that F(s+ (κ)) = 0 and F(s− (κ)) = 0, via a manipuF(s− (κ))ι− (κ) ˜ Consequently, setting Z(F) := {λ ∈ U ; F(λ) = 0}, lation with the idempotents ι± (κ). and Z(Fσ ) := {κ ∈ Uσ ; Fσ (κ) = 0}, we must have Z(Fσ ) = {κ ∈ Uσ ; σ (κ) ⊂ Z(F)}. For every subset  ⊂ Pn , we denote by F(, Cn ) the set of all Cn -valued functions on . Let also IF(, Cn ) = {g : F(, Cn ); g(κ ∗ ) = g(κ)∗ ∈ Cs , κ ∈  ∩ Cs , s ∈ Sn }, (12) which is a unital commutative subalgebra of the algebra F(, Cn ). The functions from the space IF(, Pn ) are similar to those called intrinsic functions, appering in [6], Definition 3.5.1, or in [7], Definition 2.1.2.

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The next result provides a Cn -valued general functional calculus for arbitrary stem functions, stated and proved in [24] as Theorem 2 (for the quaternionic case see [22], Theorem 3). Theorem 4 Let  ⊂ Pn be a spectrally saturated set, and let U = S(). The map S(U , Kn )  F → Fσ ∈ F(, Cn ) is R-linear, injective, and has the property (F f )σ = Fσ f σ for all F ∈ S(U , Kn ) and f ∈ S(U ). Moreover, the restricted map S(U )  f → f σ ∈ IF(, Cn ) is unital and multiplicative. Proof We give the proof of this result because of its simplicity. The map F → Fσ is clearly R-linear. The injectivity of this map follows from Remark 7. Note also that Fσ (κ) f σ (κ) = (F(s+ (κ))ι+ (sκ˜ ) + F(s− (κ))ι− (sκ˜ )) ×( f (s+ (κ))ι+ (sκ˜ ) + f (s− (κ))ι− (sκ˜ ) = (F f )(s+ (κ))ι+ (sκ˜ ) + (F f )(s− (κ))ι− (sκ˜ ) = (F f )σ (κ), because f is complex valued, and using the properties of the idempotents ι± (sκ˜ ) In particular, this computation shows that if f , g ∈ S(U ), and so f σ , gσ ∈ IF(, Cn ) by Corollary 2, we have ( f g)σ = f σ gσ = gσ f σ , thus the map f → f σ is multiplicative. It is also clearly unital.   4.2 Slice Regular Kn - and M- Valued Functions We have dealt so far with arbitrary stem functions. We continue our discussion with stem functions having regularity properties. We first discuss the adaptation of the abstract concept of slice regularity, as introduced in Subsect. 2.3, to the case of Kn and M-valued functions. Fixing a Clifford algebra Cn , the subspace Pn ⊂ Cn of paravectors plays an important role in this context. We are particularly interested in Cn -valued functions defined on open subsets of Pn (which is identified with Rn+1 ). For Kn -valued functions defined on subsets of Pn , the concept of slice regularity is defined as follows (see also [6, 24]). As before, let Sn be the unit sphere of imaginary elements of Pn . It is clear that s∗ = −s, s2 = −1, s−1 = −s, and |s| = 1 for all s ∈ Sn . Moreover, every nonnull paravector a can be written as a = (a) + |a|sa , with sa = |a|−1 (a) ∈ Sn . Now, let  ⊂ Pn be an open set, and let F :  → Kn be a differentiable function. In the spirit of [6], we say that F is right slice regular on  if for all s ∈ Sn ,  ∂ 1 ∂ + Rs F(x + ys) = 0, ∂¯s F(x + ys) := 2 ∂x ∂y Reprinted from the journal

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on the set  ∩ (R + Rs), where Rs is the right multiplication of the elements of Cn by s. Although we prefer the right slice regularity, a left slice regularity can also be defined via the left multiplication of the elements of Kn by elements from Sn . In what follows, the right slice regularity will be simply called slice regularity. We are particularly interested in the slice regularity of Cn -valued functions, but the concept is valid for Kn -valued functions and plays an important role in our discussion. For M valued functions, similar results are valid (see Subsect. 2.3 from [22]).  m Example 2 (1) The convergent series of the form m≥0 am κ , on balls {κ ∈ Pn ; |κ| < r }, with r > 0 and am ∈ Kn for all m ≥ 0, are Kn -valued slice regular on their domain of definition. Of course, when am ∈ Cn , such functions are Cn -valued slice regular on their domain of definition. (2) We give an example slightly different from Example 1. As in Example 2 from [24], or Example 2 from [22], we use the following (see Definition 1 from [24] or Definition 1 from [22]). Definition 4 The Kn -valued Cauchy kernel on the open set  ⊂ Pn is given by ρ(κ) ×   (ζ, κ) → (ζ − κ)−1 ∈ Kn .

(13)

The Kn -valued Cauchy kernel on the open set  ⊂ Pn is slice regular. Specifically, choosing an arbitrary relatively open set V ⊂ ∩(R+Rs), and fixing ζ ∈ ∩κ∈V ρ(κ), we can write for κ ∈ V the equalities ∂ (ζ − x − ys)−1 = (ζ − x − ys)−2 , ∂x ∂ Rs (ζ − x − ys)−1 = −(ζ − x − ys)−2 , ∂y because s2 = −1, and ζ , s and (ζ − x − ys)−1 commute in Kn , implying the assertion. 4.3 A Cauchy Transformation in the Clifford or Hamilton Algebra Context Having the Kn -valued Cauchy kernel (see Definition 4), we may introduce a concept of a Cauchy transform (as in [24], Definition 5; see also [22] in the quaternionic context), whose some useful properties will be recalled in this subsection. For a given open set U ⊂ C, we recall that O(U , Kn ) is the complex algebra of all Kn -valued analytic functions on U , and if U is also conjugate symmetric, Os (U , Kn ) is the real subalgebra of O(U , Kn ) consisting of all stem functions from O(U , Kn ). Because C ⊂ Kn , we have O(U ) ⊂ O(U , Kn ), where O(U ) is the complex algebra of all complex-valued analytic functions on the open set U . Similarly, when U ⊂ C is also conjugate symmetric, Os (U ) ⊂ Os (U , Kn ), where Os (U ) is the real subalgebra consisting of all functions f from O(U ) which are stem functions.

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Definition 5 Let U ⊂ C be a conjugate symmetric open set, and let F ∈ O(U , Kn ). For every κ ∈ Uσ we set C[F](κ) =

1 2πi



F(ζ )(ζ − κ)−1 dζ,

(14)



where  is the boundary of a Cauchy domain in U containing the spectrum σ (κ). The function C[F] : Uσ → Kn is called the (Kn -valued) Cauchy transform of the function F ∈ O(U , Kn ). Clearly, the function C[F] does not depend on the choice of . We shall put R(Uσ , Kn ) = {C[F]; F ∈ O(U , Kn )}.

(15)

Proposition 1 Let U ⊂ C be open and conjugate symmetric, and let F ∈ O(U , Kn ). Then function C[F] ∈ R(Uσ , Kn ) is slice regular on Uσ . For the proof see Proposition 1 from [24]. Let  ⊂ Pn be a spectrally saturated open set, and let U = S() ⊂ C, which is conjugate symmetric and also open. We introduce the notation Rs,n () = {C[ f ]; f ∈ Os (U )}, Rs (, Cn ) = {C[F]; F ∈ Os (U , Kn )}, which are R-vector spaces. In fact, these R-vector spaces have some important properties, as already noticed in a quaternionic version of the next theorem (see Theorem 5 in [22]). Theorem 5 Let  ⊂ Pn be a spectrally saturated open set, and let U ⊂ C be given by Uσ = . The space Rs,n () is a unital commutative R-algebra, the space Rs (, Cn ) is a right Rs,n ()-module, the linear map Os (U , Kn )  F → C[F] ∈ Rs (, Cn ) is a right module isomorphism, and its restriction Os (U )  f → C[ f ] ∈ Rs,n () is an R-algebra isomorphism. m n Moreover, for every polynomial P(ζ m) = nn=0 an ζ , ζ ∈ C, with an ∈ Cn for all n = 0, 1, . . . , m, we have Pσ (κ) = n=0 an κ ∈ Cn for all κ ∈ Pn . Remark 8 If F ∈ Os (U , Kn ), then C[F] takes values actually in Cn . As a version of Definition 1, the map F → C[F] is a Cauchy transformation, defined on Os (U , Kn ). Let us also remark that the element C[F](κ) is equal to Fσ (κ) for all κ ∈ , where tha latter is defined by formula (11). This follows from the proof of Theorem 3 from

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[24], as noticed in Remark 9 from the same paper, showing the compatibiliry of those constructions. It is also useful to mention that the space Rs (, Cn ) coincides with the space of all slice regular functions on , as given by Theorem 5 from [24]. Similar assertions are also valid in the quaternionic context. See for instance Theorem 6 from [22]. Similar results, valid in a quaternionic context, can be found in Sect. 5, from [22]. Example 3 This result shows that the non-commutative Cauchy kernel from [6] is given by the Cauchy transform of the complex Cauchy kernel associated to a paravector. Let s, a ∈ Pn with σ (s) ∩ σ (a) = ∅, and so s = a. In particular, the paravector s2 − 2 (a)s + |a|2 is invertible by Lemma 1, because if ζ = a + i| a| ∈ σ (a) then ζ ∈ / σ (s). Let us consider the equality  2 ∗ 2 −1 , S −1 R (a, s) = −(s − a ) s − 2 (a)s + |a| which is the right noncommutative Cauchy kernel (see [6], Definition 2.7.5 for the left version of this kernel). Note also that the function ρ(a)  ζ → (ζ − a)−1 ∈ Kn is in the space Os (ρ(a), Kn ), because (ζ − a)−1 = (ζ¯ − a)−1 . We can show the equality −S −1 R (a, s) =

1 2πi



(ζ − a)−1 (ζ − s)−1 dζ,

s

where s surrounds a Cauchy domain containing σ (s), whose closure is disjoint of σ (a). Indeed, 1 2πi



(ζ − a)−1 (ζ − s)−1 dζ

s

1 = 2πi =

1 2πi



[(ζ − a)(ζ − a∗ )]−1 (ζ − a∗ )(ζ − s)−1 dζ

s



(ζ 2 − 2ζ (a) + |a|2 )−1 (ζ − a∗ )(ζ − s)−1 dζ

s ∗

= (s − a )(s2 − 2 (a)s + |a|2 )−1 , showing that the kernel S −1 R (s, a) is the Cauchy transform of the function ρ(a)  ζ  → −(ζ − a)−1 ∈ Kn .

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5 Clifford and Quaternionic Operators, and Their Spectrum 5.1 Clifford and Quaternionic Spaces, and Their Operators As in [24] (see also [6]), by a Clifford space (or a Cl-space) we mean a two-sided module over a given Clifford algebra Cn . A Clifford space is, in particular, a real vector space, more precisely described in the following. Let Cn be a fixed Clifford algebra, and let V be a real vector space. The space V is said to be a right Cl-space if it is a right Cn - module, that is, there exists in V a right multiplication with the elements of Cn , such that x1 = x, (x + y)a = xa + ya, x(a + b) = xa + xb, x(ab) = (xa)b for all x, y ∈ V and a, b ∈ Cn . Similarly, we may define a concept of left Cl-space, replacing the right multiplication of the vectors of V with the elements of Cn by the left multiplication. A vector space which is simultaneously left and right Cl-space will be simply called a Cl-space. As before, if V is a real or complex Banach space, we denote by B(V) the algebra of all real or complex bounded linear operators, respectively. Let us recall that if V be a real Banach space, then VC = V + iV is its complexification, endowed with the norm x + i y = x + y, for all x, y ∈ V, where  ∗  is the norm of V. We denote by C the conjugation on VC , that is, the map C(x + i y) = x − i y for all x, y ∈ V, which is an R-linear map whose square is the identity. If V is a right Cl-space which is also a Banach space with the norm  ∗ , such that xa ≤ K x|a| for all x ∈ V and a ∈ Cn , where K is a positive constant, then V is said to be a right Banch Cl-space. In a similar way, one defines the concept of a left Banach Cl-space. A real Banach space V will be said to be a Banach Cl-space if it is simultaneously a right and a left Banach Cl-space. Let V be a fixed Banach Cl-space. An operator T ∈ B⇐V⇒ is said to be right Cllinear if T (xa) = T (x)a for all x ∈ V and a ∈ Cn . The set of right Cl-linear operators will be denoted by Br (V), which is, in particular, a unital real Banach algebra. We shall denote by Ra (resp. L a ) the right (resp. left) multiplication operator of the elements of V with the Cl-vector a ∈ Cn . It is clear that Ra , L a ∈ B(V) for all a ∈ Cn . Note also that Br (V ) = {T ∈ B(V); T Ra = Ra T , a ∈ Cn }. The elements of the algebra Br (V) will be sometimes called right Clifford (or Cl-) operators. As we work especially with such operators, the word ”right“ will be usually omitted. Note that all operators L a , a ∈ Cn , are Cl-operators. Consider again the complexification VC of V. Because V is a Cn -bimodule, the space VC is actually a two-sided Kn -module, via the multiplications (a + ib)(x + i y) = ax − by + i(ay + bx), (x + i y)(a + ib) = xa − yb + i(ya + xb), for all a, b ∈ Cn , x, y ∈ V.

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As in Subsect. 2.2, for every T ∈ B(V), we consider its complex extension to VC given by TC (x + i y) = T x + i T y, for all x, y ∈ V, which is actually C-linear, so TC ∈ B(VC ). Moreover, the map B(V)  T → TC ∈ B(VC ) is a unital injective morphism of real algebras. Moreover, if T ∈ Br (V), the operator TC is right Kn -linear, that is TC ((x + i y)(a + ib)) = TC (x + i y)(a + ib) for all a + ib ∈ Kn , x + i y ∈ VC , via a direct computation. The left and right multiplications with a ∈ Cn on VC will be still denoted by L a , Ra , respectively, as elements of B(VC ). We set Br (VC ) = {S ∈ B(VC ); S Ra = Ra S, a ∈ Cn }, which is a unital complex algebra, consisting of all right Kn -linear operators on VC , containing all operators L a , a ∈ Cn . It is easily seen that if T ∈ Br (V), then TC ∈ Br (VC ). Assuming that V is a Banach Cl-space implies that Br (V) is a unital real Banach Cl-algebra (that is, a Banach algebra which also a Banach Cl-space), via the algebraic operations (aT )(x) = aT (x), and (T a)(x) = T (ax) for all a ∈ Cn and x ∈ V. The complexification Br (V)C of Br (V) is, in particular, a unital complex Banach algebra, with the product (T1 +i T2 )(S1 +i S2 ) = T1 S1 −T2 S2 +i(T1 S2 +T2 S1 ), T1 , T2 , S1 , S2 ∈ Br (V), and a fixed norm, say (T1 + i T2 ) = T1  + T2 , T1 , T2 ∈ Br (V). Also note that the complex numbers, regarded as elements of Br (V)C , commute with the elements of Br (V). Remark 9 As in Subsect. 2.2, for every S ∈ B(VC ) we put S = C SC ∈ B(VC ), and S → S is a conjugate linear automorphism of the algebra B(VC ), whose square is the identity operator. In fact, the map S → S is a conjugation of B(V), induced by C. Moreover, S = S if and only if S(V) ⊂ V. In particular, we have S = S1 + i S2 with S j (V) ⊂ V, j = 1, 2, uniquely determined. Its action on the space VC is given by S(x + i y) = S1 x − S2 y + i(S1 y + S2 x) for all x, y ∈ V. Because C Ra = Ra C for all a ∈ Pn , it follows that if S ∈ Br (VC ), then S ∈ r B (VC ). Moreover, we have (S + S )(V) ⊂ V, i(S − S )(V) ⊂ V, and (TC ) = TC for all T ∈ Br (V). Note also that the map Br (V)  T → TC ∈ {S = S ; S ∈ Br (VC )} is actually a real unital algebra isomorphism, since its surjectivity follows from the equality (S|V)C = S whenever S = S ∈ Br (VC ). This implies that the algebras Br (VC ) and Br (V)C are isomorphic. This isomorphism is given by the assignment Br (V)C  T1 + i T2 → T1C + i T2C ∈ Br (VC ) which is is actually an algebra isomorphism, via a direct calculation. The continuity of this assignment is also clear, and therefore it is a Banach algebra isomorphism. For this reason, may identify the algebras Br (VC ) and Br (V)C . As already noticed above, the real algebras Br (V) and {S ∈ Br (VC ); S = S } may and will be also identified.

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The operators from the algebra Br (V) will be sometimes called Clifford operators, or simply Cl-operators. The (complex) spectrum of an operator T ∈ B r (V) is defined by σC (T ) = σ (TC ). Looking at Definition 3.1.4 from [6] (see also Definition 6 from [24], we give the folowing. Definition 6 For a given operator T ∈ B r (V), we have σCl (T ) := {κ ∈ Pn ; T 2 − 2 (κ)T + |κ|2 ) not invertible} The set σCl (T ) is call it the Clifford (or Cl-)spectrum of T . The complement ρCl (T ) = Pn \ σCl (T ) is called the Clifford (or Cl-)resolvent of T . Note that, if a ∈ σCl (T ), then {b ∈ Pn ; σ (b) = σ (a)} ⊂ σCl (T ). In other words; the subset σCl (T ) spectrally saturated (see Definition 2(2)). This concept is related to that of S-spectrum, defined in [6]. Definition 6 is given only for historical reasons. In fact, we mainly use the classical concept of complex spectrum. Indeed, since every operator T ∈ Br (V) is, in particular, R-linear, we also have a complex resolvent, defined by ρC (T ) = {λ ∈ C; (T 2 − 2 (λ)T + |λ|2 )−1 ∈ Br (V)

= {λ ∈ C; (λ − TC )−1 ∈ Br (VC )} = ρ(TC ),

and the associated complex spectrum σC (T ) = σ (TC ) as well, which will be mainly used in the following. Note that both sets σC (T ) and ρC (T ) are conjugate symmetric. There exists a strong connexion between σCl (T ) and σC (T ). Specifically, we can prove the following (see Lamma 4 from [24]). Lemma 4 For every T ∈ Br (V) we have the equalities σCl (T ) = {a ∈ Pn ; σC (T ) ∩ σ (a) = ∅}.

(16)

σC (T ) = {λ ∈ σ (a); a ∈ σCl (T )}.

(17)

and

Remark 10 (1) As expected, the set σCl (T ) is nonempty and bounded, which follows from Lemma 4. In fact, we have the equality σCl (T ) = { (λ) + | (λ)|s; λ ∈ σC (T ), s ∈ Sn }. It is also closed, as a consequence of Definition 6, because the set of invertible elements in Br (V) is open. (2) For quaternionic operators we have similar definitions and assertions. A complete description of this case can be found in [25].

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6 Analytic Functional Calculus for Clifford and Quaternionic Operators Remark 11 If V is a Banach Cl-space, in particular a real Banach space, each operator T ∈ Br (V) has a complex spectrum σC (T ), and so one can use the classical RieszDunford functional calculus, actually replacing the scalar-valued analytic functions by operator-valued analytic ones, which is a well known idea. Specifically, if T ∈ Br (V), then TC ∈ Br (VC ), and for later use, if U ⊃ σ (TC ) is an open set in C and F : U → B(VC ) is analytic, the (left) Riesz-Dunford analytic functional calculus is given by the formula  1 F(ζ )(ζ − TC )−1 dζ, F(TC ) = 2πi 

where  is the boundary of a Cauchy domain containing σ (TC ) in U . Moreover, since σ (TC ) is conjugate symmetric, we may and shall assume that both U and  are conjugate symmetric. A natural question is to find an appropriate condition to have F(TC ) = F(TC ), which implies the invariance of V under F(TC ). This is given by the following (see also Theorem 6 from [24]). Theorem 6 Let U ⊂ C be open and conjugate symmetric. If F ∈ Os (U , B(VC )), we have F(TC ) = F(TC ) for all T ∈ B(V) with σC (T ) ⊂ U . Moreover, if F ∈ Os (U , Br (VC )), and T ∈ Br (V), then F(TC ) ∈ Br (VC ). This assertion is also a particular case of Theorem 1 from Subsect. 2.1, obtained for A = Br (V) (see also Theorem 6 from [25]). The following result expresses the (left) analytic functional calculus of a given operator from Br (V) with Br (V)C -valued stem functions, obtained as a particular case of Theorem 2 from Subsect. 2.1, when A = Br (V). It can be found as Theorem 7 from [24], and it is a version of Theorem 4 from [25], proved in a quaternionic context. When F ∈ Os (U , Br (VC )), and T ∈ Br (V), we set F(T ) := F(TC )|V, where V is regarded as a real subspace of VC , which is invariant under F(TC ), as stated in Theorem 5. Theorem 7 Let V be a Banach Cl-space, let U ⊂ C be a conjugate symmetric open set, and let T ∈ Br (V), with σC (T ) ⊂ U . Then the assignment Os (U , Br (V)C )  F → F(T ) ∈ Br (V) is an R-linear map, and the map Os (U )  f → f (T ) ∈ Br (V) is a unital real algebra morphism. Moreover, the following properties hold true:

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(1) for all F ∈ Os (U , Br (V)C ), f ∈Os (U ), we have (F f )(T ) = F(T ) f (T ). m n with An ∈ Br (V) for all (2) for every polynomial P(ζ ) = n=0 mAn ζ , ζn ∈ C, r n = 0, 1, . . . , m, we have P(T ) = n=0 An T ∈ B (V). Corollary 3 Let V be a Banach Cl-space, let U ⊂ C be a conjugate symmetric open set, and let T ∈ Br (V), with σC (T ) ⊂ U . There exists an assignment Os (U , Kn )  F → F(T ) ∈ Br (V), which is an R-linear map, such that (1) for all F ∈ Os (U , Kn ), f ∈ Os (U ), we have (F f )(T ) = F(T ) f (T ). n (2) for every polynomial P(ζ ) = m n=0 an ζ , ζ ∈ C, with an ∈ Cn for all n = m n 0, 1, . . . , m, we have P(T ) = n=0 an T ∈ Br (V). Because the algebra Os (U , Kn ) can be regarded as a subalgebra of the algebra Os (U , Br (V)C ), whose elements are identified with left multiplication operators, this corollary is a direct consequence of Theorem 7. See also Corollary 3 from [24], and Theorem 5 from [25], stated and proved in the quaternionic context. Remark 12 The space Rs (, Cn ), introduced in Subsect. 4.3, can be independently defined, and it consists of the set of all Cn -valued functions, which are slice monogenic in the sense of [6], Definition 2.2.2 (or slice regular, as called in this work). They are used in [6] to define a functional calculus for tuples of not necessarily commuting real linear operators. Specifically, with a slightly modified notation, given an arbitrary family(T0 , T1 , . . . , Tn ), acting on the real space V, it is associated with the operator T = nj=0 T j ⊗ e j , acting on the two-sided Cn -module Vn = V ⊗R Cn . In fact, the symbol ”⊗” may (and will) be omitted. Moreover, as alluded in [6], page 83, we may work on a Banach Cl- space V, and using operators from Br (V). Roughly speaking, after fixing a Clifford operator, each regular Cn -valued function defined in a neighborhood  of its Cl-spectrum is associated with another Clifford operator, replacing formally the paravector variable with that operator. This constraction is explained in Chapter 3 of [6]. For an operator T ∈ Br (V), the right S-resolvent is defined via the formula ∗ 2 −1 S −1 R (s, T ) = −(T − s )(T − 2 (s)T + |s|) , s ∈ ρCl (T )

(18)

(which is the right version of formula (3.5) from [6]; see also formula (4.47) from [6]). Fixing an element κ ∈ Sn , and a spectrally saturated open set  ⊂ Pn , for  ∈ Rs (, Cn ) one sets  1 (s)dsκ S −1 (19) (T ) = R (s, T ), 2π κ

where κ consists of a finite family of closed curves, piecewise smooth, positively oriented, being the boundary of the set κ = {s = u + vκ ∈ ; u, v ∈ R}, where  ⊂  is a spectrally saturated open set containing σCl (T ), and dsκ = −κdu ∧ dv.

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Formula (19) is a slight extension of the (right) functional calculus, as defined in [6], Theorem 3.3.2 (see also formula (4.54) from [6]). Our Corollary 3 constructs, in particular, an analytic functional calculus with functions from Os (U , Kn ), where U is a neighborhood of the complex spectrum of a given Cliffordian operator, leading to another Clifford operator, replacing formally the complex variable with that operator. We can show that those functional calculi are equivalent. This is a consequence of the isomorphism of the spaces Os (U , Kn ) and Rs (Uσ , Cn ), given by Theorem 5 (see also Remark 8). Let us give a direct argument concerning the equivalence of those analytic functional calculi. Because the space VC is also a Cl-space, we may apply these formulas to the extended operator TC ∈ Br (VC ), replacing T by TC in formulas (18) and (19). In fact, using the properties of the morphism T → TC (see beginning of Section 7), we −1 deduce that S −1 R (s, T )C = S R (s, TC ). For the function  ∈ Rs (, Cn ) there exists a function F ∈ Os (, Kn ) such that Fσ = , by Theorem 5. Denoting by κ the boundary of a Cauchy domain in C containing the compact set ∪{σ (s); s ∈ κ }, we can write

(TC ) =

=

1 2π

 κ

1 2πi

 κ

⎛ ⎜ 1 ⎝ 2πi



⎞ ⎟ F(ζ )(ζ − s)−1 dζ ⎠ dsκ S −1 R (s, TC )

κ

⎛

⎜ 1 F(ζ ) ⎝ 2π



⎞ ⎟ (ζ − s)−1 dsκ S −1 R (s, TC )⎠ dζ.

κ

It follows from the complex linearity of S −1 R (s, TC ), and via an argument similar to that for getting formula (4.49) in [6], that −1 (ζ − s)S −1 R (s, TC ) = S R (s, TC )(ζ − TC ) − 1,

whence −1 −1 + (ζ − s)−1 (ζ − TC )−1 , (ζ − s)−1 S −1 R (s, TC ) = S R (s, TC )(ζ − TC )

and therefore, 1 2π +



(ζ − s)−1 dsκ S −1 R (s, TC ) =

κ

1 2π



1 2π



−1 dsκ S −1 R (s, TC )(ζ − TC )

κ

(ζ − s)−1 dsκ (ζ − TC )−1 = (ζ − TC )−1 ,

κ

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because 1 2π



dsκ S −1 R (s, TC )

κ

1 = 1 and 2π



(ζ − s)−1 dsκ = 0,

κ

as in Theorem 4.8.11 from [6], since the Kn -valued function s → (ζ − s)−1 is analytic in a neighborhood of the set κ ⊂ Cκ for each ζ ∈ κ , respectively. Therefore (TC ) = (T )C = F(TC ) = F(T )C , implying (T ) = F(T ). Conversely, choosing a function F ∈ Os (, Kn ), and denoting by  ∈ Rs (, Cn ) its Cauchy transform, the previous computation in reverse order shows that (T ) = F(T ). Consequently, for a fixed T ∈ Br (V), the maps  : Rs (, Cn ) → Br (V), with () = (T ), and  : Os (, Kn ) → Br (V), with (F) = F(T ), we must have the equality  =  ◦ C[∗], where C[∗] is the Cauchy transform. Remark 13 Unlike in [6, 7], our approach permits to obtain a version of the spectral mapping theorem in a classical stile, via direct arguments. Recalling that Rs,n () is the subalgebra of Rs (, Cn ) whose elements are also in IF(, Cn ) (see Theorem 5), for every operator T ∈ Br (V) and every function  ∈ Rs,n () one has σCl ((T )) = (σCl (T )), via Theorem 3.5.9 from [6]. Using our approach, for every function f ∈ Os (U ), one has f (σC (T )) = σC ( f (T )), directlly from the corresponding (classical) spectral mapping theorem in [8]. This result is parallel to that from [6] mentioned above, also giving an explanation for the former, via the isomorphism of the spaces Os (U ) and Rs,n () Data Availibility All data generated and analysed during this study are included in this published article.

References 1. Baskakov, A.G., Zagorskii, A.S.: Spectral theory of linear relations on real Banach spaces. Math. Notes (Russian: Matematicheskie Zametki) 81(1), 15–27 (2007) 2. Birkhoff, G., von Neumann, J.: The Logic of quantum machanics. Ann. Math. 37, 823 (1936) 3. Clifford, W.K.: Preliminary sketch of bi-quaternions. Proc. Lond. Math. Soc. 1(381–395), s1-4 (1871) 4. Colombo, F., Gentili, G., Sabadini, I.D.: Struppa, extension results for slice regular functions of a quaternionic variable. Adv. Math. 222, 1793–1808 (2009) 5. Colombo, F., Sabadini, I., Struppa, D.C.: Slice monogenic functions. Israel J. Math. 171, 385–403 (2009). https://doi.org/10.1007/s11856-009-0055-4 6. Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative functional calculus, theory and applications of slice hyperholomorphic functions, progress in mathematics, vol. 28. Birkhäuser/Springer Basel AG, Basel (2011) 7. Colombo, F., Gantner, J., Kimsey, D. P.: Spectral Theory on the S-spectrum for Quaternionic Operators, Birkhäuser, (2018) 8. Dunford, N., N., Schwartz, J. T.: Linear Operators, Part I: General Theory, Interscience Publishers, New York, London,: Part III: Spectral Operators, p. 1971. Wiley Interscience, New York, London, Sydney, Toronto (1958) 9. Finkelstein, D., Jauch, J.M., Schiminovich, S., Speiser, D.: Foundations of quaternion quantum mechanics. J. Math. Phys. 3(207), 207–220 (1962) 10. Fueter, R.: Über eine Hartogs’schen Satz, Comment. Math. Helv., 12: 75-80, (1939/40) 11. Gentili, G., Stoppato, C., Struppa, D.C.: Regular functions of a quaternionic variable. Springer Monographs in Mathematics, Springer/Heidelberg (2013)

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F. -H. Vasilescu 12. Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216, 279–301 (2007) 13. Ghiloni, R., Moretti, V., Perotti, A.: Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys. 25, 4 (2013) 14. Ghiloni, R., Perotti, A.: Slice regular functions on real alternative algebras. Adv. Math. 226, 1662–1691 (2011) 15. Ingelstam, L.: Real Banach algebras. Ark. Mat. 5, 239–270 (1964) 16. Jefferies, B.: Spectral properties of noncommuting operators. Lecture Notes in Mathematics, vol. 1843. Springer-Verlag, Berlin (2004) 17. Kaplansky, I.: Normed algebras. Duke. Math. J. 16, 399–418 (1949) 18. Moisil, G., Theodorescu, N.: Fonctions holomorphes dans l’espace. Mathematica (Cluj) 5, 142–159 (1931) 19. Palmer, T.W.: Real C ∗ -algebras. Pacific J. Math. 35(1), 195–204 (1970) 20. Porteous, I.R.: Topological geometry. Van Nostrand Reinhold Company, New York-TorontoMelbourne (1969) 21. Rosenberg, J.: Structure and applications of real C ∗ -algebras. Contemp. Math. 671, 235 (2016) 22. Vasilescu, F.-H.: Quaternionic regularity via analytic functional calculus. Integral Equ. Oper. Theory 92, 18 (2020). https://doi.org/10.1007/s00020-020-2574-7 23. Vasilescu, F.-H.: Analytic functional calculus in quaternionic framework. Math. Rep. 23(33), 1–2 (2021) 24. Vasilescu, F.-H.: Spectrum and analytic functional calculus for clifford operators via stem functions. Concr. Oper. 8, 90–113 (2021) 25. Vasilescu, F.-H.: Spectrum and analytic functional calculus in real and quaternionic frameworks. Pure Appl. Funct. Anal. 7(1), 389–407 (2022) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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Complex Analysis and Operator Theory (2022) 16: 74 https://doi.org/10.1007/s11785-022-01252-5

Complex Analysis and Operator Theory

Yet Another Approach to Poly-Bergman Spaces Nikolai Vasilevski1 Received: 17 May 2022 / Accepted: 1 June 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract We present an alternative characterization of the poly-Bergman and true-polyBergman function spaces on the unit disk D, based on the pure isometry acting on the standard L 2 -space over D and the representation of this isometry in terms of differential operators. As an application, we give the representation of the Landau magnetic Hamiltonian in its action in L 2 -space over D. The true-poly-Bergman spaces appear there as the corresponding eigenspaces. Keywords Poly-Bergman · True-poly-Bergman · Pure isomerty · Landau magnetic Hamiltonian Mathematics Subject Classification Primary 30D60; Secondary 30G30 · 30H20

1 Introduction The poly-Bergman function spaces on the unit disk D are an important particular case of the polyanalytic function spaces, those that consist of n-differentiable functions ϕ satisfying the equation ∂nϕ = 0, ∂z n

n ∈ N.

We mention [1–3, 6, 7, 9, 10, 13, 17, 18], where various properties and numerous applications of the polyanalytic functions were studied.

Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.

B 1

Nikolai Vasilevski [email protected] Department of Mathematics, CINVESTAV, Mexico City, Mexico

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The paper is devoted to an alternative to the previously used characterisations of the poly-Bergman and true-poly-Bergman function spaces on the unit disk, and is organised as follows. In Sect. 2, we introduce some notation and recall the facts being used further on. In Sect. 3, we summarize the known (see, e.g. [4, 12, 19]) characterizations and properties of the n-poly-Bergman spaces A2n , the closed subspaces of the Hilbert space L 2 (D, d A) with probability Lebesgue measure, that consist of n-polyanalytic functions, as well as the true-n-poly-Bergman function spaces A2(n) = A2n  A2n−1 . The main content of the paper starts in Sect. 4, where we propose a new approach based on the pure isometry acting on L 2 (D, d A), which we define initially by its action on the basis elements of L 2 (D, d A). We recover the characterization and the properties of the spaces of our interest, A2n and A2(n) in term of our isometry. Then we find its invariant, basis-independent representation, using the differential operators introduced by A. Wünsche [22, Sections 4 and 6]. In the last Sect. 5, we look on discussed in the paper approach through the so-called extended Fock space formalism. By that me mean the representation of the Heisenberg algebra (the three-dimensional algebra H3 = {a, a† , 1} with commutators [a, a† ] = 1 and [a, 1] = [a† , 1] = 0) in our Hilbert space L 2 (D, d A), where, unlike the classical situation, [5, Chapter 5, Section 5.2], dim ker a > 1. We start with the result [19, Theorem 2.18] stating that the existence of a pure isometry in a separable Hilbert space is equivalent to the existence of the formally adjoint lowering and raising operators a and a† , that satisfy the desired properties. Based on the formulas of Sect. 4, we describe the operators a and a† in their invariant, basis independent form. We show then that the operator a† a, acting in L 2 (D, d A), is unitarily equivalent to the Landau magnetic Hamiltonian (see e.g. [13, 16]) 2 =− ∂ +z ∂ ,  ∂z ∂z ∂z

acting in L 2 (C, dμ), with the Gaussian measure dμ. It is interesting to observe that in  the eigenspaces, that correspond to the eigenvalues λn = n − 1, both cases, a† a and , are nothing but the true-n-polyanalytic spaces, true-n-poly-Bergman in the case of L 2 (D, d A), and true-n-poly-Fock in the case of L 2 (C, dμ).

2 Preliminaries Introduce the Hilbert space L 2 (D, d A), with the probability measure d A(z) = π1 d xd y, z = x + i y. Recall, see e.g. [4, Section 3], [22, Section 3], that the system of functions

e p,q (z, z) =



p+q +1

min{ p,q}  k=0

(−1)k

( p + q − k)! z p−k z q−k . k!( p − k)!(q − k)!

p, q ∈ Z+ ,

forms the orthonormal basis in L 2 (D, d A).

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Note that the leading term of e p,q is a multiple of the monomial m p,q = z p z q , while the monomial m p,q itself is a linear combination of the basis elements e p−,q− , with  = 0, 1, . . . , min{ p, q}, see e.g. [19, Formula (2.7)]. The following mutually adjoint two-dimensional singular integral operators on L 2 (D, d A)  (SD f )(z) = −

D

f (w) d A(w) and (SD∗ f )(z) = − (w − z)2

 D

f (w) d A(w) (w − z)2

(2.1)

turn out to be useful in the study of the poly-Bergman spaces, which was firstly observed by A. Dzhuraev [6, Chapter 2, Section 5]. Finally we recall the standard notation. The linear span of finite number H1 , H2 ,…,Hn of linearly independent subspaces of a Hilbert space H is called the direct sum and is denoted by H1  H2  · · ·  Hn , in case when they are additionally pairwise orthogonal, we write H1 ⊕ H2 ⊕ · · · ⊕ Hn , and call it the orthogonal sum. Note that, given even just two closed subspaces H1 and H2 with trivial intersection, H1 ∩ H2 = {0}, (equivalently being linearly independent), their direct sum H1  H2 may not be closed. It depends on the so-called minimal angle between them. Recall (see e.g. [8]) that the minimal angle ϕ (m) (H1 , H2 ) between two closed subspaces H1 and H2 of a Hilbert space H is defined as cos ϕ (m) (H1 , H2 ) = sup {|x, y| : x ∈ H1 , y ∈ H2 and x = y = 1} , and that ([8, Lemma 1]) the direct sum of two closed subspaces, that intersect only by zero, is closed if and only if the minimal angle between them is grater then zero.

3 Poly-Bergman Spaces We deal with the functions defined on the unit disk D in the complex plane. Recall that an n-differentiable function ϕ is called n-polyanalytic if it satisfies the equation ∂nϕ = 0, ∂z n Reprinted from the journal

n ∈ N.

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As known, the above condition is equivalent ([3, Section 1.1]) to the following its representations ϕ=

n−1 

zq fq ,

(3.1)

q=0

where all f q are analytic functions. We denote by On = On (D) the linear space of all n-polyanalytic functions in D. The main objects of our study are the spaces A2n = A2n (D) := On ∩ L 2 (D, d A), which consist of all n-polyanalytic functions from L 2 (D, d A). These spaces are known to be closed in L 2 (D, d A), and are called n-poly-Bergman spaces. Note that the space A2 := A21 is the classical Bergman subspace of L 2 (D, d A), consisting of analytic functions. Obviously, for each n ∈ N, we have A2n  A2n+1 . We introduce thus the true-npoly-Bergman spaces A2(n) = A2(n) (D) by A2(n) = A2n  A2n−1 = A2n ∩ (A2n−1 )⊥ , with the convention A2(1) = A21 = A2 . Once the main objects have been introduced, let us recall their different characterizations. First of all (see e.g. [12, Formula (3.3)] and [19, Corollary 3.6]) we have the following description of the n-poly-Bergman space   A2n = ker(SD∗ )n = span e p,q : p ∈ Z+ , q = 0, 1, . . . , n − 1   = span m p,q : p ∈ Z+ , q = 0, 1, . . . , n − 1 ,

(3.2)

and (see e.g. [4, Corollary 6], [19, Section 4]) the following description of the true-npoly-Bergman space   A2(n) = span e p,q : p ∈ Z+ , q = n − 1 . At this point it would be appropriate to make the following observation. We introduce first the following (obviously closed in L 2 (D, d A)) subspaces A[n] = z n−1 A2 = {z n−1 f (z) : f ∈ A2 },

n ∈ N.

Any finite number of such spaces is linearly independent. This can be easily checked ∂ to a linear combination of their by consecutive applying the appropriate powers of ∂z elements. Then obviously A[1]  A[2]  · · ·  A[n] ⊂ A2n ,

864

n ∈ N.

(3.3)

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Lemma 3.1 The space A[1]  A[2]  · · ·  A[n] is not closed and dense on A2n , so that clos (A[1]  A[2]  · · ·  A[n] ) = A2n . Proof The reason of not being closed is in the zero minimal angle between each two of the subspaces A[k] , k = 1, . . . , n. Indeed, √ let 1k−1≤  k < q ≤ n, and = k +  z z ∈ A[k] and y = consider the following norm 1 elements x  √  + 2q − k z q−1 z +q−k ∈ A[q] . Then,  cos ϕ (m) (A[k] , A[q] ) = lim x , y  = (k + )( + 2q − k)z k−1 z  , z q−1 z +q−k  →∞ √ (k + )( + 2q − k) = 1, = lim →∞ +q or ϕ (m) (A[k] , A[q] ) = 0. Further, the space A[1]  A[2]  · · ·  A[n] is dense in the direct sum of the two closed subspaces, L n−1  A[n] , where L n−1 = clos (A[1]  A[2]  . . .  A[n−1] ). Take now k = 1 and q = n, then x ∈ L n−1 and y ∈ A[n] . The above calculation ensures that ϕ (m) (L n−1 , A[n] ) = 0, and thus the not closedness of A[1]  A[2]  . . .  A[n] follows from [8, Lemma 1]. Finally, (3.2) implies (3.3).   Of course each function ϕ ∈ A2n admits the representation (3.1), but the point is that the terms z q f q do not necessarily belong to L 2 (D, d A). As an instructive example ([20, Example 3.2]), we consider A22

ϕ=

 p∈N

 1 e p,1 = z p+1



p∈N

 p+2 e p,0 − p+1

p∈N

√

p( p + 2) e p−1,0 , p+1

where each term in the right-hand side does not belong to L 2 (D, d A). Further ([11, Theorem 4.3]), L 2 (D, d A) =



A2(n) ,

(3.4)

n∈N

which in our context directly follows from (3.2) and the description of the orthonormal basis in L 2 (D, d A). We mention as well that the orthogonal projection Pn : L 2 (D, d A) → A2n is given ([11, Theorem 2.3]) by Pn = I − (SD )n (SD∗ )n , and that the following isometric isomorphisms between true-poly-Bergman spaces ([11, Theorem 3.5]), are given by the operators SD and SD∗ , SD : A2(n)  L n −→ A2(n+1) ,

SD∗ : A2(n+1) −→ A2(n)  L n ,

for each n ∈ N, and where L n = {αz n−1 : α ∈ C}.

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4 Pure Isometries Approach Following [19, Section 3], we introduce two pure isometries (unilateral shifts) on L 2 (D, d A), defining them on basis elements e p,q , p, q ∈ Z+ , by V : e p,q −→ e p,q+1

and

 : e p,q −→ e p+1,q . V

(4.1)

All the main properties of the poly-Bergman spaces and formulas for the related operators of the previous section can be easily reformulated now in terms of the above isometries. We summarize them in the following lemma. Lemma 4.1 The following statements hold A2n = ker(V ∗ )n−1 =

n−1

V k (A2 ),

k=)

A2(n)

=V

n−1

(A ), 2

Pn = I − V n−1 (V ∗ )n−1 , ∗ SD = V V V ∗ . SD∗ = V , the last Proof First three equalities follow directly from the definitions of V and V two follow from [19, Formula (3.8) and Section 4].   Corollary 4.2 For each n ∈ N, we have the following isometric isomorphisms V : A2(n) −→ A2(n+1) , V ∗ : A2(n+1) −→ A2(n) , V n−1 : A2 −→ A2(n) , (V ∗ )n−1 : A2(n) −→ A2 . Their action is given by V : A2(n)  f =

 p∈Z+

∗

V :

A2(n+1)



 f =

V

:A  f = 2



c p e p,n−1 ∈ A2(n) ,

p∈Z+

p∈Z+

(V ∗ )n−1 : A2(n)  f =



c p e p,n −→

c p e p,0 −→



c p e p,n ∈ A2(n+1) ,

p∈Z+

p∈Z+

n−1



c p e p,n−1 −→



c p e p,n−1 ∈ A2(n) ,

p∈Z+

c p e p,n−1 −→

p∈Z+



c p e p,0 ∈ A2 .

p∈Z+

866

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A slight inconvenience of the above formulas is that the isometries are given in their action on the basis elements of L 2 (D, d A). And our next task is to find their invariant, basis independent form. To do this we will use the operators introduced by A. Wünsche [22, Sections 4 and 6] for the unweighted case, α = 0. Also we will keep using our notation e p,q = e p,q (z, z) for the elements of the orthonormal basis in L 2 (D, d A). On the dense in L 2 (D, d A) domain D, which consists of all finite linear combinations of the basis elements e p,q , or, which is the same, all finite linear combinations of the monomials m p,q , we define the operators (see [22, Formulas (4.9) and (4.13)]) ∂ ∂ ∂ ∂ ∂ ∂ −z = z− z=z − z + I, ∂z ∂z ∂z ∂z ∂z ∂z



2 2 ∂ ∂ ∂ ∂ ∂2 ∂2 + z + z = −4 + z+ z−I H = −4 ∂z∂z ∂z ∂z ∂z∂z ∂z ∂z L=z

By [22, Formulas (4.8) and (4.10)]), the operators L and H act on the basis elements as follows Le p,q = ( p − q)e p,q

H e p,q = ( p + q + 1)2 e p,q .

and

Note that these and subsequent formulas characterizing the action of the involved operators on the basis elements can be alternatively verified by the direct calculations. Note that all eigenvalues of the diagonal operator H are positive, and thus all powers H s , with s ∈ R, are well defined, in particular 1

H 2 e p,q = ( p + q + 1)e p,q . Then it is convenient to introduce two other diagonal operators, densely defined on D, L (1) L (2)



1 1 ∂ 1 1 ∂ 2 2 H +L −1 = H +z − z , = 2 2 ∂z ∂z



1 1 ∂ 1 1 ∂ 2 2 H −L −1 = H +z − z , = 2 2 ∂z ∂z

which act of the basis elements by L (1) e p,q = pe p,q

and

L (2) e p,q = qe p,q .

(4.2)

Note that the kernel of the operator L (2) coincides with A2 , while on the orthogonal complement (A2 )⊥ = L 2 (D, d A)  A2 the operator L (2) is diagonal, whose all −1

2 eigenvalues are positive. In what follows we will use the operators L −1 (2) and L (2) ,

being well defined on (A2 )⊥ . By the same reasoning, the operator L −1 (1) is well defined Reprinted from the journal

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2 )⊥ = L 2 (D, d A)  A 2 , where A 2 = span {e0,q : q ∈ Z+ } is the closed on (A subspace of L 2 (D, d A) of all anti-analytic functions. We also need the following operators ([22, Formulas (6.12)-(6.13)]), densely defined on D,   1 1 ∂ (1) H − 4 and K − = H 4 z L (1) + (1 − zz) ∂z   1 1 ∂ (1) K + = H 4 z L (1) − (1 − zz) H − 4 , ∂z   1 1 ∂ (2) H − 4 and K − = H 4 z L (2) + (1 − zz) ∂z   1 1 ∂ (2) K + = H 4 z L (2) − (1 − zz) H − 4 . ∂z

By formulas [22, Formulas (6.3) and (6.4)]), the above operators act on the basis elements as follows (1) (1) K− e p,q = pe p−1,q and K + e p,q = ( p + 1)e p+1,q , (2)

(2)

K − e p,q = qe p,q−1 and K + e p,q = (q + 1)e p,q+1 .  in the invariant, basis The above permits us to represent the isometries V and V independent form. Proposition 4.3 The following representations hold   1 ∂ (2) −1 14 (1 − zz) K = L H z L − H−4 , V = L −1 (2) (2) + (2) ∂z    (2) 1 − 14 −1 4 z L (2) + (1 − zz) ∂ K − L −1 L (2) , ∗ (2) = H ∂z H V = 0,   1 ∂ (1) −1 41  = L −1 K + (1 − zz) z L = L H − H−4 , V (1) (1) (1) ∂z    (1) 1 −1 − 14 −1 4 z L (1) + (1 − zz) ∂ K L = H L (1) , − (1) ∗ ∂z H  V = 0,

on (A2 )⊥ on A2

 2 )⊥ on (A 2 on A

,

.

Proof Follows directly from the action of the operators involved on the basis elements  with further extension by continuity from the dense set D to the whole L 2 (D, d A).  The above proposition permits us to give the basis independent form of all formulas from Lemma 4.1 and Corollary 4.2.

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For example, for each n ∈ N, we have the following isometric isomorphisms   1 1 ∂ 4 (1 − zz) L −1 H z L − H − 4 : A2(n) −→ A2(n+1) , (2) (2) ∂z

 n−1  ∂ −1 41 − 14 L (2) H z L (2) − (1 − zz) H : A2 −→ A2(n) . ∂z Another example represents the two-dimensional singular integral operator SD , defined in (2.1), in an alternative form involving differential operators ⎧  ⎨ L −1 K (2) K (1) L −1 = L −1 L z − (2) 2 − (2) + (1) (2) SD = ⎩0,

∂ ∂z (1 −

  ∂ 2 ⊥ L −1 zz) 2z L (1) + (1 − zz) ∂z (1) , on (A ) 2 on A

5 Extended Fock Space Formalism Point of View In this section we apply to our case of interest, the space L 2 (D, d A) and poly-Bergman subspaces, the following general result. Theorem 5.1 ([19, Theorem 2.18]) Let H be a separable infinite dimensional Hilbert space. Then the following statements are equivalent: (1) there is a pure isometry V in H; (2) the Hilbert space H admits the orthogonal sum decomposition H=

∞

H(n) ,

(5.1)

n=1

where all H(n) have the same (finite or infinite) dimension; (3) there are two formally adjoint lowering and raising operators a and a† , that act invariantly on a common domain dense in H, such that the following commutation relation holds [a, a† ] = I ,

(5.2)

the set L (1) = ker a is a closed subspace of H, and the set of finite linear combinations of elements from all spaces L (n) := (a† )n−1 L (1) is dense in H. Moreover, the subspaces H(n) in (5.1) are related to the operators V , a and a† as follows H(1) = ker V ∗ = ker a and H(n) = V n−1 (ker V ∗ ) = (a† )n−1 (ker a), for n > 1. The item (3) of the theorem in fact describe the representation of the Heisenberg algebra (the three-dimensional algebra H3 = {a, a† , 1} with commutators [a, a† ] = 1

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and [a, 1] = [a† , 1] = 0) in the Hilbert space H, where, unlike the classical situation, [5, Chapter 5, Section 5.2], dim ker a > 1. We refer to this representation as to the extended Fock space formalism, for details see [19, Section 2]. In our case, by (3.4) we have the following orthogonal sum decomposition L 2 (D, d A) =



A2(n) ,

n∈N

onto subspaces with the same (infinite) dimension. The pure isometry V , defined in (4.1), obeys the properties ker V ∗ = A2 = A2(1) and A2(n) = V n−1 (ker V ∗ ). The formally adjoint lowering and raising operators a and a† are defined then on the common domain D dense in L 2 (D, d A) as follows a : e p,q

√ √ q V ∗ e p,q = qe p,q−1 , p ∈ Z+ , q > 0 −→ 0, p ∈ Z+ , q = 0

(5.3)

and a† : e p,q −→

  q + 1V e p,q = q + 1e p,q+1 , p ∈ Z+ , q ∈ Z+ .

(5.4)

Note that the operators a and a† satisfy on D the commutation relation [a, a† ] = I . Further, it is easy to figure out that thus defined operators can be extended to the common domain D0 dense in L 2 (D, d A),  D0 =

f =

∞ 

f k e pk ,qk ∈ L 2 (D, d A) :

k=1

∞ 

 qk | f k | < ∞ , 2

k=1

by af =

∞  √ qk f k e pk ,qk −1 , with qk − 1 = 0 if qk = 0, k=1

∞   † a f = qk + 1 f k e pk ,qk +1 , k=1

where they are mutually adjoint, and the set of finite linear combinations of elements from all spaces (a† )n−1 ker a is dense in L 2 (D, d A). Proposition 4.3 permits us to give invariant, basis independent, form of a and a† : a=

⎧ ⎨

  1 1 1 −1 (2) − 2 ∂ K− H − 4 L (2)2 , on A⊥ L (2) = H 4 z L (2) + (1 − zz) ∂z

⎩0,

  1 1 ∂ − 1 (2) −1 a† = L (2)2 K + = L (2)2 H 4 L (2) z − (1 − zz) H − 4 . ∂z 870

on A

,

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Yet Another Approach to poly-Bergman Spaces

That is we realize the extended Fock space formalism in the space L 2 (D, d A), where the operators a and a† are given by the above formulas. Recall (see e.g. [14, Chapter I, Section 1]) that a pure isometry is determined up to a unitary equivalence by its multiplicity, i.e., two pure isometries V1 and V2 , acting on separable Hilbert spaces H1 and H2 respectively, are unitary equivalent, that is there exists a unitary operator U : H1 → H2 such that V2 = U V1 U ∗ , if and only if dim ker V1∗ = dim ker V2∗ . This is equivalent (Theorem 5.1) to the fact that all, up to a unitary equivalence, realizations of the extended Fock space formalism are determined by a singe parameter d = dim ker a = dim ker V ∗ ∈ N ∪ ∞. In our case d = dim ker a = dim ker V ∗ = ∞. Another, and in a sense the most simple, realization of the extended Fock space formalism with d = ∞ is as follows (for proofs and details see [20, Section 4], see also [18]). Let L 2 (C, dμ) be the Hilbert space of square-integrable on C functions with the Gaussian measure dμ(z) = e−z·¯z d A(z). As known, the elements e p,q (z, z) =



p!q!

min{ p,q}  k=0

(−1)k z p−k z q−k , k!( p − k)!(q − k)!

p, q ∈ Z+

form an orthonormal basis in L 2 (C, dμ). The corresponding formally adjoint lowering and raising operators [18, Formula (2.4)] have the form a=

∂ , ∂z

∂ + z. ∂z

a† = −

They act invariantly on the common dense in L 2 (C, dμ) domain D0 , being the span of all basis elements e p,q , with p, q ∈ Z+ . Furthermore they satisfy the commutation relation [a, a† ] = I . Recall that n-polyanalytic space in L 2 (C, dμ), called n-poly-Fock space, is given by

Fn2

∂ = ker ∂z

n = span {e p,q : p ∈ Z+ , q = 0, 1, . . . , n − 1},

2 space is given by F 2 = F 2  F 2 . Moreover, and [18] the true-n-poly-Fock F(n) n n−1 (n)

L 2 (C, dμ) =



2 F(n) .

n∈N

Having the same value d = ∞, these two different realizations of the extended Fock space formalism have to be unitarily equivalent. Lemma 5.2 The unitary operator U : L 2 (C, dμ) → L 2 (D, d A), defined on basis elements by U : e p,q −→ e p,q ,

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p, q ∈ Z+ ,

N. Vasilevski

and then extended by continuity to the whole L 2 (C, dμ), provides the unitary equivalence of the above two different realizations of the extended Fock space formalism. Proof By [18, Theorem 2.9], we have √ qe p,q−1 , p ∈ Z+ , q > 0 a : e p,q −→ and 0, p ∈ Z+ , q = 0  a† : e p,q −→ q + 1e p,q+1 , p ∈ Z+ , q ∈ Z+ . We check now that a = U aU ∗ and a† = U a† U ∗ . On the basis elements we have √ √ ae p,q = U aU ∗ e p,q = U ae p,q = U qe p,q−1 = qe p,q−1 , q > 0,   a† e p,q = U a† U ∗ e p,q = U a† e p,q = U q + 1e p,q+1 = q + 1e p,q+1 ,  

which coincides with (5.3) and (5.4).

Our interest to these two realizations of the extended Fock space formalism is caused, in particular, by the so-called Landau magnetic Hamiltonian (see [13] and [16] for details). It is the operator 2  = − ∂ + z ∂ ( = a† a ),  ∂z ∂z ∂z 2 is densely defined in L 2 (C, dμ). It easy to see that each true-poly-Fock space F(n)  , and invariant for the action of 

 2 = (n − 1)I , | F (n)

n ∈ N.

 consists of infinitely many equidistant eigenThat is, the spectrum of the operator  values, each of infinite multiplicity (Landau levels), they are of the form λn = n − 1, n ∈ N, and the corresponding eigenspaces are the true-polyanalytic spaces in 2 . L 2 (C, dμ), the true-n-poly-Fock spaces F(n) Consider now its unitarily equivalent counterpart in the space L 2 (D, d A):  ∗.  = a† a = U a† aU ∗ = U U  The second formula in (4.2) and (4.3) imply that ⎡

 = L (2) = 

∂2



1⎣ ∂ ∂ −4 + z + z 2 ∂z∂z ∂z ∂z

872

⎤

2  21 +z

∂ ∂ − z⎦ . ∂z ∂z

(5.5)

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This operator can be extended from D to a self-adjoint operator on the domain  D = #

f =

∞ 

f n ∈ L 2 (D, d A) : f n ∈

A2(n)

n=1

and

∞ 

 (n − 1) f n < ∞ , 2

n=1

where it acts by the rule  = a† a : 



f n −→

n∈N



(n − 1) f n , with

f n ∈ A2(n) .

n∈N

That is, the Landau magnetic Hamiltonian realized in the space L 2 (D, d A) has the form (5.5), it preserves true-poly-Bergman spaces, being on each A2(n) just a scalar operator  | 2 = (n − 1)I ,  A (n)

n ∈ N.

 , as it should be, consists of the eigenvalues λn = n − 1, n ∈ N, The spectrum of  and the corresponding eigenspaces are again the true-polyanalytic spaces, being in this case the true-n-poly-Bergman spaces A2(n) . Acknowledgements This work was partially supported by CONACYT grants 280732 and FORDECYTPRONACES/61517/2020238630, Mexico. Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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N. Vasilevski 14. Nagy, B.Sz., Foias, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space. Revised and Enlarged Edition Springer (2010) 15. Turbiner, A.V.: Lie algebras in Fock space. In: ‘Complex Analysis and Related Topics’, “Operator theory: Advances and Applications”, v. 114, p.265-284 (1999) 16. Turbiner, A.V., Vasilevski, N.L.: Poly-analytic functions and representation theory. Complex Analysis and Operator Theory 15, 110 (2021). (24 p) 17. Vasilevski, N.L.: On the structure of Bergman and poly-Bergman spaces. Integral Equ. Oper. Theory 33, 471–488 (1999) 18. Vasilevski, N.L.: Poly-Fock spaces. Operator Theory. Advances and Applications 117, 371–386 (2000) 19. Vasilevski, N.: On the poly-analytic and anti-poly-analytic function spaces, Preprint (2021) 20. Vasilevski, N.: Extended Fock space formalism and polyanalytic functions, Preprint (2022) 21. Weidmann, J.: Linear Operators in Hilbert Spaces. Springer Verlag (1980) 22. Wünsche, A.: Generalized Zernike or disc polynomials. J. Comput. Appl. Math. 174(1), 135–163 (2005) 23. Zhu, K.: Analysis on Fock spaces. Springer (2012) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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Complex Analysis and Operator Theory (2023) 17:37 https://doi.org/10.1007/s11785-023-01335-x

Complex Analysis and Operator Theory

Quantum Permutation Matrices Moritz Weber1 Received: 2 June 2022 / Accepted: 20 January 2023 / Published online: 15 March 2023 © The Author(s) 2023

Abstract Quantum permutations arise in many aspects of modern “quantum mathematics”. However, the aim of this article is to detach these objects from their context and to give a friendly introduction purely within operator theory. We define quantum permutation matrices as matrices whose entries are operators on Hilbert spaces; they obey certain assumptions generalizing classical permutation matrices. We give a number of examples and we list many open problems. We then put them back in their original context and give an overview of their use in several branches of mathematics, such as quantum groups, quantum information theory, graph theory and free probability theory.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quantum Permutation Matrices in Operator Theory . . . . . . . . . . . . . 2.1 Definition of Quantum Permutation Matrices . . . . . . . . . . . . . . 2.2 Link to Classical Permutation Matrices . . . . . . . . . . . . . . . . . 2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Quantum Permutation Matrices and Quantum Isomorphisms of Graphs 3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Faithful Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dedicated to Jörg Eschmeier who sadly passed away in 2021. Communicated by Mihai Putinar. This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht. This work has been supported by the SFB-TRR 195, by the Heisenberg program of the DFG (German Research Foundation) and by OPUS LAP: quantum groups, graphs and symmetries via representation theory jointly funded by DFG and NCN (Poland).

A previous version of this chapter was published Open Access under a Creative Commons Attribution 4.0 International License at http://link.springer.com/10.1007/s11785-023-01335-x.

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Moritz Weber [email protected] Fachbereich Mathematik, Saarland University, Postfach 151150, 66041 Saarbrücken, Germany

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M. Weber 3.2 Hilbert Sudoku/SudoQ . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Quantum Symmetries of Graphs . . . . . . . . . . . . . . . . . . . . . 3.4 Quantum Sinkhorn Algorithm . . . . . . . . . . . . . . . . . . . . . . 3.5 Intermediate Quantum Permutations . . . . . . . . . . . . . . . . . . . 3.6 More Examples, Constructions and Quantum Transposition Matrices . . 4 Quantum Permutation Matrices in Their Broader Context and Use . . . . . . 4.1 Quantum Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Quantum Groups and Quantum Permutation Groups . . . . . . . . . . . 4.3 Quantum Symmetries of Graphs and Quantum Isomorphisms of Graphs 4.4 Quantum Information Theory . . . . . . . . . . . . . . . . . . . . . . . 4.5 Free Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction One of the most basic symmetry operations in mathematics, is given by permutations: Take N points x1 , . . . , x N listed in some order, and permute them. We may capture permutations in various ways in mathematics, and one way to do so is by using permutation matrices. Recall that a permutation matrix is an N × N -matrix σ ∈ M N ({0, 1}), (a) whose entries σi j are either 0 or 1, (b) such that each column and each row contains exactly one 1, all other entries being 0. Here is an example of a 4 × 4 permutation matrix: ⎛ 0 ⎜0 σ =⎜ ⎝1 0

1 0 0 0

0 0 0 1

⎞ 0 1⎟ ⎟ 0⎠ 0

Such an N × N permutation matrix acts on C N by permuting the canonical basis vectors e1 , . . . , e N ∈ C N . So, in a way, we identify our N points x1 , . . . , x N from above with the basis vectors e1 , . . . , e N and permute them by letting the matrix σ act on them – we identify the N -elementary set X = {x1 , . . . , x N } with the N -dimensional space C N . Now, this is a very common theme in “quantum mathematics”: We identify a classical space with another object having some more “functional analytic” properties—and this allows us to define and study “quantum versions” of this classical space. What exactly do we mean by this? Let us postpone this discussion to Sect. 4.1 and let us define quantum permutation matrices right away. A quantum permutation matrix is an N × N -matrix u ∈ M N (B(H )), where H is some Hilbert space, such that (a) all entries u i j ∈ B(H ) are orthogonal projections (i.e. u i j = u i∗j = u i2j ), N N u ik = k=1 u k j = 1 for all i, j = 1, . . . , N . (b) and k=1 Actually, an easy calculation (see Lemma 2.2) shows that the latter implies (b’) u ik u jk = u ki u k j = 0 for all i, j, k = 1, . . . , N with i = j.

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Here is an example of a 4 × 4-quantum permutation matrix with H = C2 : ⎛ 0 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 u=⎜ ⎜ 1 ⎜ ⎜ 0 ⎜ ⎝ 0 0

0 0

0 0

0 0

0 1

11 0 1 1

0 1 −1 0 1 2 −1 1 0 00 0 0 0 0 00 1 00 0 1 2

0 0

0 0

0 1

0 0

1 2

1 −1 −1 1

1 11 2 11

00 0 0

00 00

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Let us make a few observations: Firstly, the entries of any (classical) permutation matrix σ ∈ M N ({0, 1}) satisfy the axioms of a quantum permutation matrix—a permutation matrix is a quantum permutation matrix, with H = C (and thus B(C) = C). Secondly, the entries u i j ∈ B(H ) of a quantum permutation matrix do not need to commute, as the above example shows. Thirdly, the above matrix u is “quantum” indeed: While the above classical permutation matrix σ sends the first particle to the third one (meaning, it maps e1 to e3 ), the above quantum permutation matrix u sends the first particle a little bit to the third, and a little bit to the fourth – in the sense that 2 2 2 the matrix u ∈ M4 (M2 (C)) acting on C2 ⊕ C2 ⊕

C ⊕ C sends the first copy of C 10 to some part of the third copy (namely to C2 ⊆ C2 ) and to some part of the 00

00 fourth copy (namely to C2 ⊆ C2 ). Quantum, isn’t it? 01 However, the “quantum” aspect of the matrix will be neglected in the main part of the article: • In Sect. 2, we will study quantum permutation matrices as such, in the realm of operator theory, and list a number of open problems. • In Sect. 4, we will then explain how quantum permutation matrices fit into the broader context of quantum groups, quantum information, graph theory and free probability, and again list a number of open problems. And we give references and sketch the history of the field. With the present article, we hope to provide a friendly access to quantum permutation matrices (also known as magic unitaries) for people interested in operator theory. In our functional analysis research seminar at Saarland University, we sometimes had a “friendly clash of cultures” between the groups of Jörg Eschmeier (together with Ernst Albrecht, Gerd Wittstock and Heinz König, and later also Michael Hartz’s group) and the more operator algebraic ones by Roland Speicher and myself. However, a common ground was to consider operators on Hilbert spaces and to try to understand their properties. While Jörg was more interested in (tuples of) commuting operators, we were more interested in noncommuting operators—but in the end, we all dealt with operators on Hilbert spaces, to some extent. And maybe, Hardy spaces and their generalizations or complex analysis can be helpful when studying quantum permutation matrices at some point, in Jörg’s spirit?

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The quantum permutation matrices we are presenting here are interestingly diverse objects: They can be seen as an array of operators, containing certain tuples of commuting operators (within one row or one column, for instance), but also involving noncommutativity between other entries. It is good to think back to those happy times, when we had our interestingly diverse research seminar sessions at Saarland University, containing tuples from Jörg’s group and tuples from Roland’s and my groups, each group asking in the sense of: – “But how about the Hardy space?” – “And how about noncommutativity?”

2 Quantum Permutation Matrices in Operator Theory In this section, we define and study quantum permutation matrices in the realm of operator theory and we list a number of open problems. See Sect. 4 for the historical origins. 2.1 Definition of Quantum Permutation Matrices Let us define quantum permutation matrices first. Definition 2.1 Let N ∈ N and let H be a (complex) Hilbert space. A quantum permutation matrix (also called magic unitary) is a matrix u ∈ M N (B(H )) consisting of entries u i j ∈ B(H ), i, j = 1, . . . , N , such that (a) u i j = u i∗j = u i2j for all i, j = 1, . . . , N (i.e., the entries are projections) N N u ik = k=1 u k j = 1 for all i, j = 1, . . . , N with i = j. (b) and k=1 ⎞ u 11 u 12 · · · u 1N ⎜ u 21 u 22 · · · u 2N ⎟ ⎟ ⎜ u=⎜ ⎟ ∈ M N (B(H )) .. ⎠ ⎝ . u N 1 u 2N · · · u N N ⎛

N H ). In So, by definition, a quantum permutation matrix is an operator in B( k=1 some sense, it “permutes” the N copies of H , just as a classical permutation matrix permutes N copies of C. If H is finite-dimensional of dimension d, we have u ∈ M N (Md (C)), i.e. u is an N × N -matrix whose entries are matrices themselves (of size d × d). See Sect. 2.3 for examples. Let us derive further (well-known) relations amongst the entries of quantum permutation matrices. Lemma 2.2 For any quantum permutation matrix, we have u ik u jk = u ki u k j = 0 for all i, j, k = 1, . . . , N with i = j.

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Proof Projections summing up to 1 (or to simply to a projection) need to be mutually orthogonal. This follows from positivity. Indeed, if p1 , . . . , p N are projections auch  that k pk = 1, then for any j = 1, . . . , N , N

( pi p j )∗ ( pi p j ) =

i=1,i= j

N ( pi p j )∗ ( pi p j ) − p j i=1

= pj

N

 pi

p j − p j = p j − p j = 0.

i=1

Hence, the positive elements ( pi p j )∗ ( pi p j ) with i = j sum up to zero, which means   that each of these summands needs to be zero; hence pi p j = 0 if i = j. Note that in case one wants to define quantum permutation matrices in general one should better add the relations from Lemma 2.2, as orthogonality is not implied in a general ∗ -algebra [14, Rem. 4.10]. ∗ -algebras,

2.2 Link to Classical Permutation Matrices Let us study the case dim H = 1 in Definition 2.1. In that case, we obtain classical permutation matrices. Lemma 2.3 Let u ∈ M N (B(H )) be a quantum permutation matrix and let dim H = 1. Then u is a (classical) permutation matrix. Proof If dim H = 1, we have H = C and B(H ) = C. Thus, the entries u i j of u are scalars. Now, since u i j = u i∗j , they are actually real, and as u i2j = u i j , we infer u i j ∈ {0, 1}. In each row (and in each column), by Lemma 2.2, there is at most one nontrivial entry, and by (b) of Definition 2.1, there is exactly one. These are the defining properties of a permutation matrix.   Also, if all entries u i j commute, we are in the classical situation.1 Lemma 2.4 Let u ∈ M N (B(H )) be a quantum permutation matrix and let H be finitedimensional of dimension d ∈ N. Assume that all entries u i j ∈ B(H ) commute, for all i, j = 1, . . . , N . Then, there are permutation matrices σ1 , . . . , σd ∈ M N (C), such that u is unitarily equivalent to ⎛ σ1 ⎜ σ2 ⎜ ⎜ .. ⎝ .

⎞ ⎟   ⎟ σt ∈ M N (C) ⊆ M N d (C) ∼ ⎟= = M N (Md (C)). ⎠ σd

t=1,...,d

t=1,...,d

1 As an alternative to Lemma 2.4, one may show that for commuting entries u , there are permutation ij  matrices σ1 , . . . , σd such that u = dt=1 σt ⊗ at ∈ M N (C) ⊗ B(H ) for at = u 1σt (1) ...u N σt (N ) ∈ B(H ).

[68].

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Proof We identify B(H ) with Md (C). Special case. For convenience, let us prove a special case first, assuming that the rank of each u i j is at most one. Fix the first row u 11 , u 12 , . . . , u 1N ∈ Md (C) of u. These are mutually orthogonal projections summing up to 1, by Definition 2.1 (and Lemma 2.2). We thus find a unitary W ∈ Md (C) such that each w1 j := W u 1 j W ∗ ∈ Md (C) is diagonal, for all j = 1, . . . , N . We pick an orthonormal basis e1 , . . . , ed of Cd according to this diagonalization and we denote by pk , for k = 1, . . . , d, the projection onto the one-dimensional space spanned by the basis vector ek . Since the rank of each w1 j is at most one, we have {w1 j | j = 1, . . . , N } = { pk | k = 1, . . . , d}. Now, by assumption, each element wi j := W u i j W ∗ ∈ Md (C), i, j = 1, . . . , N commutes with all w1m , m = 1, . . . , N , i.e. it commutes with all pk , k = 1, . . . , d. Hence, each wi j is a diagonal projection matrix of rank at most one and we have {wi j | i, j = 1, . . . , N } = { pk | k = 1, . . . , d}. Denote by σt ∈ M N ({0, 1}), t = 1, . . . , d the matrix with (σt )i j := (wi j )tt , i.e. σt consists in the t-th diagonal entries of all wi j ’s. Since the matrix w formed by the elements wi j , i, j = 1, . . . , N is a quantum permutation matrix, we infer that σ1 , . . . , σd are permutation matrices. General case. We now drop the assumption that the rank of each u i j is at most one and we prove the general case. Again, we find a unitary W1 ∈ Md (C) such that each (1) w1 j := W1 u 1 j W1∗ ∈ Md (C) is diagonal, for all j = 1, . . . , N . Define the quantum (1)

(1)

permutation matrix w(1) with entries wi j := W1 u i j W1∗ . Since each wi j commutes (1)

with all elements from the first row of w(1) , each wi j is block diagonal (but not necessarily diagonal, in contrast to the above special case) with respect to the blocks of the first row of w(1) . Now, consider the second row of w (1) . Again, these are projections summing up to (2) (1) 1, and we find a unitary W2 ∈ Md (C) such that each w2 j := W2 w2 j W2∗ ∈ Md (C) is diagonal, for all j = 1, . . . , N . The crucial point is, that we may choose W2 to be block diagonal with respect to the blocks of the first row of w(1) , since the second row of w (1) is block diagonal with respect to the first row of w (1) . Thus, defining the (2) (1) quantum permutation matrix w (2) with entries wi j := W2 wi j W2∗ , we note that both the first and the second row of w (2) are diagonal, since W2 being block diagonal with respect to the first row did not change the first row. Iterating, we may eventually diagonalize all elements of the matrix and we end up with a quantum permutation matrix w = (wi j ), which is unitarily equivalent to u and whose entries wi j are all diagonal, i.e. each wi j is a diagonal matrix with entries 0 or

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1. Now, denote by σt ∈ M N ({0, 1}), t = 1, . . . , d the matrix with (σt )i j := (wi j )tt , which yields permutation matrices σ1 , . . . , σd , since w is a quantum permutation matrix.   We observe, that Lemma 2.3 is a special case of Lemma 2.4 with d = 1. The next lemma shows, that if we want to go beyond the classical case, we need to choose N ≥ 4. Lemma 2.5 Let u ∈ M N (B(H )) be a quantum permutation matrix with N ≤ 3. Then the entries of u all commute. Proof If N = 1 or N = 2, the situation is trivial (in the latter case: u 12 = 1 − u 11 , u 21 = 1 − u 11 , u 22 = u 11 ). For N = 3, we copy the following proof from [53, Sect. 2.2]. Consider u i j and u kl . We want to show that they commute. They do, if i = k or j = l, by Lemma 2.2, or since u i j = u kl , in case both equations hold, i = k and j = l. If now i = k and j = l, there is some m with m = j and m = l such that { j, l, m} = {1, 2, 3}. Since i = k, we have by Lemma 2.2 u i j u k j u im = 0,

u i j u km u im = 0,

u i j u kl u il = 0,

which yields, by Definition 2.1(b), u i j u kl u im = u i j (u k j + u kl + u km )u im = u i j u im = 0 and u i j u kl = u i j u kl (u i j + u il + u im ) = u i j u kl u i j = (u i j u kl u i j )∗ = (u i j u kl )∗ = u kl u i j .   2.3 Examples Let us now come to truly non-classical examples. Example 2.6 Let p, q ∈ B(H ) be projections. The following is a (well-known) 4 × 4 quantum permutation matrix. ⎛

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Note that p and q need not commute. As a concrete example, let us rearrange the example from the introduction: ⎛

0 0

0 1

0 0

0 00

1 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ u = ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0

0 0 1 0 0 0 0 0

0 1

0 0

0 0

0 0



00 00

00 00

1 11 2 11

1 −1 1 2 −1 1

⎞ 00 ⎟ 00 ⎟ ⎟ 00 ⎟ ⎟ 00

⎟ ⎟ 1 −1 ⎟ 1 ⎟ 2 −1 1 ⎟ ⎟ 11 ⎠ 1 2

11

For this matrix, the entries do not commute and the assumptions of Lemma 2.4 are violated. Indeed, this matrix cannot be written as a direct sum of classical permutation matrices (since the entries of a direct sum of permutation matrices all commute). In order to have more examples, let us mention a construction by Woronowicz, see for instance [49, Def. 3.1] Definition 2.7 Given two matrices u, v ∈ M N (B(H )) their Woronowicz tensor product is defined as u k v :=

N

Ei j ⊗

i, j=1

N

 u ik ⊗ vk j .

k=1

It is easy to see that if u and v are quantum permutation matrices, so is u k v. Example 2.8 The following example is taken from [49, Ex. 3.12], where p, p , q, q ∈ B(H ) may be any projections. ⎛

p ⊗ p ⎜ ⎜ (1 − p) ⊗ p ⎜ ⎜ q ⊗ (1 − p ) ⎝

(1 − p) ⊗ q

p ⊗ (1 − p )

(1 − p) ⊗ (1 − q )

p ⊗ q

(1 − p) ⊗ (1 − p )

p ⊗ (1 − q )

(1 − q) ⊗ (1 − q )

q ⊗ p

(1 − q) ⊗ q

q ⊗ (1 − q )

(1 − q) ⊗ p

q ⊗ q

(1 − q) ⊗ (1 − p )

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

In fact, this matrix arises as ⎛

⎞ ⎞ ⎛ p 0 1− p 0 0 1 − p 0 p

⎜1 − p 0 ⎜ p 0 ⎟ 0 p 0 ⎟ ⎜ ⎟ ⎟ k ⎜1 − p

⎝ 0 ⎠ ⎝ q 0 1−q 0 1 − q ⎠ 0 q 0 1 − q 0 q 0 1−q 0 q

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Example 2.9 And another one from [49, Ex. 3.13], again coming from taking the Woronowicz tensor product. Note that, as in the previous example, we may replace all p by p in the second tensor leg, and by p

in the third one, and the same for q. ⎛

⎞ p⊗ p⊗ p p ⊗ (1 − p) ⊗ q p ⊗ p ⊗ (1 − p) p ⊗ (1 − p) ⊗ (1 − q) ⎜ +(1 − p) ⊗ q ⊗ (1 − p) +(1 − p) ⊗ (1 − q) ⊗ (1 − q) +(1 − p) ⊗ q ⊗ p +(1 − p) ⊗ (1 − q) ⊗ q ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (1 − p) ⊗ p ⊗ p (1 − p) ⊗ (1 − p) ⊗ q (1 − p) ⊗ p ⊗ (1 − p) (1 − p) ⊗ (1 − p) ⊗ (1 − q) ⎜ ⎟ ⎜ ⎟ + p ⊗ q ⊗ (1 − p) + p ⊗ (1 − q) ⊗ (1 − q) + p ⊗ q ⊗ p + p ⊗ (1 − q) ⊗ q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ q ⊗ (1 − p) ⊗ p q ⊗ p⊗q q ⊗ (1 − p) ⊗ (1 − p) q ⊗ p ⊗ (1 − q) ⎜ ⎟ ⎜+(1 − q) ⊗ (1 − q) ⊗ (1 − p) ⎟ +(1 − q) ⊗ q ⊗ (1 − q) +(1 − q) ⊗ (1 − q) ⊗ p +(1 − q) ⊗ q ⊗ q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ (1 − q) ⊗ (1 − p) ⊗ (1 − p) (1 − q) ⊗ p ⊗ (1 − q) ⎠ (1 − q) ⊗ (1 − p) ⊗ p (1 − q) ⊗ p ⊗ q +q ⊗ (1 − q) ⊗ (1 − p) +q ⊗ q ⊗ (1 − q) +q ⊗ (1 − q) ⊗ p +q ⊗ q ⊗ q

Example 2.10 The following example is taken from [13, Sect. 2]. Let







10 i 0 0 1 0i g1 := , g2 := , g3 := , g4 := , 01 0 −i −1 0 i 0 be the Pauli matrices. For a unitary matrix x ∈ U2 ⊆ M2 (C), we view the matrix gi xg j ∈ M2 (C) as a vector in C4 . Let wi(x) j ∈ M4 (C) be the rank one projection 4 ∼ C . Then, wx = (w (x) ) is a quantum permutation onto the vector gi xg j ∈ M2 (C) = ij matrix. Also, x → wx , as a function in C(U2 , M4 (C)), is a quantum permutation matrix. Example 2.11 The following example is taken from [11, Def. 4.1] which is linked to quantum Latin squares. Let H be an N -dimensional Hilbert space and let ξi j ∈ H be vectors, for i, j = 1, . . . , N , such that • for every i = 1, . . . , N , the set {ξi j | j = 1, . . . , N } forms an orthonormal basis of H , • and for every j = 1, . . . , N , the set {ξi j | i = 1, . . . , N } forms an orthonormal basis of H . Let pi j ∈ B(H ) be the rank one projection onto the vector ξi j . Then, p = ( pi j ) forms a quantum permutation matrix. In [11], amongst others the case is studied when h ∈ M N (C) is a complex Hadamard matrix: Denote its rows by h 1 , . . . , h N . They may be viewed as invertible elements in the algebra C N and considering the vectors ξi j := h i /h j ∈ C N and their rank one projections pi j , we may construct a quantum permutation matrix. See also [31]. 2.4 Quantum Permutation Matrices and Quantum Isomorphisms of Graphs There are quantum permutation matrices which generalize isomorphisms of graphs, see their use in Sect. 4.3. Given a finite simple graph  = (V , E), we denote by i ∼ j, if two vertices i, j ∈ V are connected by an edge from E, and i  j otherwise. We specify i ∼1 j or i ∼2 j in case there are two graphs 1 and 2 .

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Definition 2.12 Let N ∈ N and let H be a Hilbert space. Let 1 = (V , E 1 ) and 2 = (V , E 2 ) be finite simple graphs, with V = {1, . . . , N }. A quantum isomorphism matrix of 1 and 2 is a matrix u ∈ M N (B(H )) consisting of entries u i j ∈ B(H ), i, j = 1, . . . , N , such that (a) u i j = u i∗j = u i2j for all i, j = 1, . . . , N , N N (b) k=1 u ik = k=1 u k j = 1 for all i, j = 1, . . . , N with i  = j, (c) and u i j u kl = u kl u i j = 0 if i ∼2 k and j 1 l, (d) as well as u i j u kl = u kl u i j = 0 if i 2 k and j ∼1 l. If  = 1 = 2 , we say that u is a quantum automorphism matrix of  in that case. Hence, a quantum isomorphism matrix is a quantum permutation matrix (by (a) and (b)). Another way of expressing relations (c) and (d) is to say, that u A1 = A2 u holds, where Ak ∈ M N ({0, 1}) is the adjacency matrix of k , k = 1, 2, i.e. (A1 )i j = 1, if i ∼1 j and (A1 )i j = 0, if i 1 j. Let us prove it. Lemma 2.13 Let u be a quantum permutation matrix. We have u A1 = A2 u if and only if relations (c) and (d) of Definition 2.12 hold. Proof Since

u is =



u is (A1 )sl = (u A1 )il

s

s: s∼1 l

and

u tl =



(A2 )it u tl = (A2 u)il ,

t

t: i∼2 t

we observe that u A1 = A2 u holds if and only if s: s∼1 l

u is =



u tl .

t: i∼2 t

Thus, assuming u A1 = A2 u, we obtain in case i ∼2 k and j 1 l u i j u kl =



u i j (u tl u kl ) =

t: i∼2 t



(u i j u is )u kl = 0.

s: s∼1 l

Here, we used Lemma 2.2 to show that u tl u kl = δtk u kl and u i j u is = δ js u i j , i.e. t: i∼2 t u tl u kl = u kl (since k appears in the sum, because i ∼2 k), whereas  s: s∼1 l u i j u is = 0, since j does not appear in the sum. Likewise, we obtain in case i 2 k and j ∼1 l u i j u kl =



(u i j u is )u kl =

s: s∼1 l



u i j (u tl u kl ) = 0.

t: i∼2 t

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Conversely, assuming relations (c) and (d) of Definition 2.12, we infer s: s∼1 l

u is =

s: s∼1 l

t

u is u tl =





u is u tl =

s t: i∼2 t

s: s∼1 l t: i∼2 t



u is u tl =

u tl .

t: i∼2 t

  As in Sect. 2.2, we see that Definition 2.12 generalizes the classical situation: Given graphs 1 and 2 , assume that σ ∈ S N is an isomorphism between them. We then have σ A1 σ −1 = A2 , or equivalently σ A1 = A2 σ . Example 2.14 Let  be the undirected graph on V = {1, 2, 3, 4} given by: 1

2

3

4

The matrices from Example 2.6 are quantum automorphism matrices of this graph.

3 Open Problems There are many open problems related with quantum permutation matrices. Let us state these problems first, as problems in operator theory, and let us shift the background information to the next section. 3.1 Faithful Models We don’t have a (finite-dimensional2 ) faithful model of quantum permutation matrices, for N ≥ 5. The task is to find a quantum permutation matrix u with u i j ∈ B(H ) on some finite-dimensional Hilbert space H , such that for any polynomial p in the N 2 entries and for any other quantum permutation matrix v, if p(u) = p(u 11 , u 12 , . . . , u N N ) = 0, then also p(v) = 0. In other words, if a polynomial relation holds for u, then it holds for any other v, too. Problem 3.1 Find a faithful model (for polynomial relations3 ) of quantum permutation matrices of size N . There are models for N = 4: The map x → wx of Example 2.10 produces a faithful model, see [13]. See also [7, 11, 25, 27] for more on (not necessarily faithful) models, in particular [27, Conj. 5.7] for a conjecture on inner faithfulness of some 2 We will see in Sect. 4.2 that we may define a universal C ∗ -algebra A (N ) generated by the entries of a S

quantum permutation matrix. By the noncommutative Gelfand-Naimark Theorem, we may represent this C ∗ -algebra faithfully on some Hilbert space – but it might be an infinite-dimensional one. 3 More generally, we are interested in faithful, finite-dimensional ∗ -representations of A (N ). S

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matrix model for quantum permutation matrices of any size N ≥ 4. See also [6] for more on the case N = 4. As a more concrete problem, consider the matrix of Example 2.6. It is not a faithful model. Indeed, the polynomial p(u) = u 13 vanishes for the matrix in Example 2.6, but not for the matrix in Example 2.8. Also the matrix from the latter example is not a faithful model. Indeed, consider the polynomial p(u) = u 11 u 23 . It vanishes for the matrix in Example 2.8 (as p(u) = ( p ⊗ p )((1 − p) ⊗ (1 − p )) = 0 for that matrix), but not for the matrix in Example 2.9, as    p(u) = p ⊗ p ⊗ p + (1 − p) ⊗ q ⊗ (1 − p) (1 − p) ⊗ p ⊗ (1 − p) + p ⊗ q ⊗ p = p ⊗ pq ⊗ p + (1 − p) ⊗ qp ⊗ (1 − p) = 0 for suitable choices of the projections. Let us rephrase the questions from [49, Question 4.10, 4.11]. Problem 3.2 Is the matrix in Example 2.9 a faithful model of size N = 4? ⎛

⎞ p⊗ p⊗ p p ⊗ (1 − p) ⊗ q p ⊗ p ⊗ (1 − p) p ⊗ (1 − p) ⊗ (1 − q) ⎜ +(1 − p) ⊗ q ⊗ (1 − p) +(1 − p) ⊗ (1 − q) ⊗ (1 − q) +(1 − p) ⊗ q ⊗ p +(1 − p) ⊗ (1 − q) ⊗ q ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (1 − p) ⊗ p ⊗ p (1 − p) ⊗ (1 − p) ⊗ q (1 − p) ⊗ p ⊗ (1 − p) (1 − p) ⊗ (1 − p) ⊗ (1 − q) ⎜ ⎟ ⎜ ⎟ + p ⊗ q ⊗ (1 − p) + p ⊗ (1 − q) ⊗ (1 − q) + p ⊗ q ⊗ p + p ⊗ (1 − q) ⊗ q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ q ⊗ (1 − p) ⊗ p q ⊗ p⊗q q ⊗ (1 − p) ⊗ (1 − p) q ⊗ p ⊗ (1 − q) ⎜ ⎟ ⎜+(1 − q) ⊗ (1 − q) ⊗ (1 − p) ⎟ +(1 − q) ⊗ q ⊗ (1 − q) +(1 − q) ⊗ (1 − q) ⊗ p +(1 − q) ⊗ q ⊗ q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ (1 − q) ⊗ (1 − p) ⊗ (1 − p) (1 − q) ⊗ p ⊗ (1 − q) ⎠ (1 − q) ⊗ (1 − p) ⊗ p (1 − q) ⊗ p ⊗ q +q ⊗ (1 − q) ⊗ (1 − p) +q ⊗ q ⊗ (1 − q) +q ⊗ (1 − q) ⊗ p +q ⊗ q ⊗ q

There are hints [39] that this is indeed the case! However, there is no proof yet. 3.2 Hilbert Sudoku/SudoQ An interesting problem is the following.  NGiven a rectangular array of projections u ik = 1 for all i = 1, . . . , m and such (u i j )i=1,...,m; j=1,...,N , m < N such that k=1 that u i j u k j = 0 for all i = k and all j = 1, . . . , N . Problem 3.3 Can we fill up a given rectangular matrix to a quantum permutation matrix? More precisely, can we find further projections (u i j )i=m+1,...,N ; j=1,...,N such that u = (u i j )i, j=1,...,N is a quantum permutation matrix? Let us give a concrete example. Suppose the following array is given: u 11 u 21

u 12 u 22

u 13 u 23

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u 14 u 24

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All u i j are projections, we have u 11 +u 12 +u 13 +u 14 = 1 and u 21 +u 22 +u 23 +u 24 = 1 as well as u 1 j u 2 j = 0 for j = 1, 2, 3, 4. Problem 3.4 Can you always fill up this array to a 4×4 quantum permutation matrix? Or can you give a counter example of eight projections such that the above array may not be filled up? The answer is unknown. (And it is yes, if all projections commute, see Lemma 2.4.) See [32, Sect. 5] for the source of this problem and see [31] for the links to Hadamard matrices and Example 2.11. As a side remark, such a rectangular matrix needs to be filled up to a quadratic matrix – we may not fill it up to a rectangular matrix4 As a starting point for Problem 3.4, you might want to investigate the following question. Problem 3.5 Given two projections p, q ∈ B(H ), classify how they can be decomposed into projections p = p1 + p2 and q = q1 + q2 such that p1 ⊥ q1 and p2 ⊥ q2 . This might be useful firstly, when investigating how the given rectangular array of Problem 3.4 may look like in general, with p1 = u 11 , p2 = u 21 , q1 = u 12 and q2 = u 22 ; secondly given such a rectangular array, you must find a decomposition of p := 1 − (u 11 + u 21 ) and q := 1 − (u 12 + u 22 ) into p = u 31 + u 41 and q = u 32 + u 42 on the way to solve Problem 3.4; and thirdly, this shall also help when trying to find counterexamples. Recall also Halmos’s investigation of two projections in generic position [46]. In fact, you may even think of more complicated situations: Think of a quantum permutation matrix with “empty spots”. Can you always fill it up to a complete quantum permutation matrix? You might want to call this game “Hilbert Sudoku” or “SudoQ” [62]. Problem 3.6 Is there a Hilbert Sudoku consisting in entries u i j which all commute, such that no “classical” solution exists (i.e. no completion to a quantum permutation matrix such that all entries commute) but a nonclassical one (some of the additional u i j do not commute)? See also [62, Ex. 4.3, 4.4, Conj. 4.2, Conj. on page 3] for more on the Hilbert Sudoku game. The Hilbert sudoku problem (in its rectangular form) is linked to the following classical situation. Suppose, we have two finite, simple graphs 1 = (V1 , E 1 ) and 2 = (V2 , E 2 ) with m = |V1 | ≤ |V2 | = N . Assume we have an injective graph 4 There cannot be a “rectangular quantum permutation matrix”: Given a matrix (u ) i j i=1,...,m; j=1,...,N of N  projections u i j such that k=1 u ik = 1 and m u = 1, we have k j k=1

m·1=

m N i=1 j=1

ui j =

N m

u i j = N · 1,

j=1 i=1

so m = N holds. Acknowledgements to Alexander Mang for this short and sweet argument. See also [1, after Lemma 4.1].

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homomorphism ϕ : 1 → 2 , i.e. if i ∼1 j, then ϕ(i) ∼2 ϕ( j). We can write this homomorphism as an array (σi j )i=1,...,m; j=1,...,N with σi j = 1, if ϕ(i) = j and zero N otherwise. Since ϕ is defined on all of V1 , we have k=1 σik = 1 for all i = 1, . . . , m. Since ϕ is injective, we have u i j u k j = 0 for all i = k and all j. And since 1 and the subgraph ϕ(1 ) of 2 are isomorphic, it is easy to fill up the graph 1 to a larger graph 1 which is then isomorphic to 2 . So, we may rephrase the question of filling up rectangular arrays (u i j )i=1,...,m; j=1,...,N to quantum permutation matrices: Suppose that the u i j in that rectangular array also satisfy the relations (c) and (d) of Definition 2.12 for two graphs 1 and 2 . In that case, we have a graph 1 which we may view as a subgraph of 2 to some extent (it is quantum isomorphic to a subgraph of 2 ). Problem 3.7 Can we complete the graph 1 to a graph 1 which is quantum isomorphic to 2 ? This is unclear! 3.3 Quantum Symmetries of Graphs There is a whole community interested in the following question: Given a finite, simple graph , is there a quantum automorphism matrix of  (in the sense of Definition 2.12) whose entries do not all commute (i.e. there are some i, j, k, l such that u i j u kl = u kl u i j )? In that case, we say that the graph has quantum symmetries. On the other hand, if all quantum automorphism matrices of  have commuting entries, we say that  has no quantum symmetries. As an example, take the complete graph (no self-edges, no multiple edges, undirected) on four vertices. It does have quantum symmetries. Indeed, any 4 × 4 quantum permutation matrix is a quantum automorphism of the complete graph (note that the relations (c) and (d) of Definition 2.12 are redundant for the complete graph), and the matrix of Example 2.6 has noncommuting entries. The same holds true for the graph in Example 2.14, it has quantum symmetries. Adding one edge to that graph however, we obtain a graph without quantum symmetries. See also [36, 73] for more on quantum symmetries and these examples of simple graphs on a small number of vertices. For quite a few simple graphs, the question of the existence of quantum symmetries is settled. See also Sect. 4.3. However, there are also many open questions in this regard. For instance, how about the Johnson graph J (6, 3), the Tutte 12-cage or the graph constructed from the linear constraint system on K 3,3 – do these graphs have quantum symmetries? [71, Ch. 8.2] Problem 3.8 Can you find a quantum permutation matrix with noncommuting entries which is a quantum automorphism matrix for one of these graphs? Another interesting question in that context: Problem 3.9 Can you find an asymmetric graph  (i.e.a graph whose automorphism group is trivial) and a quantum automorphism matrix of  such that one of its entries u i j with i = j is nonzero?

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We have no clue whether such a graph exists, but it would be a very exciting example. See Sect. 4.3. In fact, it seems that certain automorphism groups are “quantum excluding” [26]. Generalizing the above question, we may ask: Problem 3.10 Can you find a graph  whose automorphism group is either the trivial one {e}, a cyclic group Zk or the symmetric group S3 (also for the alternating groups An , in particular for A5 , it is open) and a quantum automorphism matrix of  such that one of its entries u i j with i = j is nonzero? And yet another question: Problem 3.11 Can you find two simple connected asymmetric graphs 1 and 2 which are non-isomorphic, but for which a quantum isomorphism matrix as in Definition 2.12 exists? Actually, solving Problem 3.11 in the affirmative solves Problem 3.9 in the affirmative by taking  as the disjoint union of 1 and 2 . We know that there are such graphs as in Problem 3.11, if we drop the asymmetry assumption: There are non-isomorphic graphs which are quantum isomorphic [53, Sect. 4.4]. However, the smallest such example we know has 24 vertices. Problem 3.12 Can you find two graphs 1 and 2 , each having less than 24 vertices, which are non-isomorphic, but for which a quantum isomorphism matrix as in Definition 2.12 exists? What is the smallest such example? It is known to experts that the smallest example must have at least 16 vertices. [68] 3.4 Quantum Sinkhorn Algorithm Sinkhorn’s algorithm, in a nutshell, is a procedure to construct bistochastic matrices: Take an arbitrary orthogonal matrix A ∈ M N (R) and normalize the rows such that their entries sum up to one, for each row. Then do the same for the columns. Oh no, you just destroyed the row sums – their sum might now differ from one! Nevermind, simply normalize again the rows, then the columns, then the rows etc. Eventually, magically, this will converge to abistochastic matrix, i.e. to an orthogonal matrix  B ∈ M N (R) such that k Bik = k Bk j = 1 for all i and j. In [27], a Sinkhorn type algorithm for quantum permutation matrices has been presented. Pick N 2 rank one projections and put them in an N × N matrix. Then normalize alternatingly the rows and the columns. In [64], we refined this algorithm: We may also insert a graph  in this algorithm and adjust the normalization procedure so that the iterations tend to respect the graph better and better. When restricting our algorithm to a friendly subclass of vertex-transitive graphs (to so called quasi Cayley graphs), we obtain very good results: The algorithm predicts with high accuracy whether or not the given graph has quantum symmetries. However, unlike in the case of the classical Sinkhorn algorithm, we are unable to prove convergence (althought we observe it).

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Problem 3.13 Does this quantum Sinkhorn algorithm converge? This is related to many other such questions in operator theory regarding the stability property: Given a sequence of operators which approximately almost satisfy a certain condition – can we find a limit object, which actually precisely satisfies this condition? We don’t know for the quantum Sinkhorn algorithm. 3.5 Intermediate Quantum Permutations The next open problem we want to present here has been puzzling the quantum group community for quite a while. We rephrase it in terms of operator theory, which looks a bit cumbersome, since we avoid the language of quantum groups and C ∗ -algebras. Problem 3.14 Given N ≥ 6. Find (or disprove the existence of) (a) a polynomial p in the N 2 entries of a N × N matrix, (b) and two N × N quantum permutation matrices u and v each with some noncommuting entries (at least two entries shall not commute), such that (c) (d) (e) (f)

p(σ ) = 0 for all permutation matrices σ ∈ M N ({0, 1}), p(u) = 0 (i.e. the polynomial relation vanishes on the entries of u), p(v) = 0 (it does not vanish on the entries of v), and whenever we take any quantum  N permutation matrix w with p(w) = 0, then wik ⊗ wk j . also p(w ) = 0, where wi j := k=1

In that case, you just proved the existence of a famous intermediate quantum permutation group, see Sect. 4.2. 3.6 More Examples, Constructions and Quantum Transposition Matrices Finally, we need more examples of quantum permutation matrices. Problem 3.15 Is there a good machine for constructing quantum permutation matrices? Is there a nice subclass of quantum permutation matrices, which may be studied separately? Possibly on some nice Hilbert spaces, like functional Hilbert spaces, Hardy spaces, Bergman spaces, Fock spaces? See [7, 13, 27, 49] for some models of quantum permutation matrices. Actually, recall that Examples 2.8 and 2.9 come from the Woronowicz tensor product. In fact, the class of quantum permutation matrices is closed under taking the Woronowicz tensor product, conjugation with a diagonal unitary W ⊕ . . . ⊕ W (as in Lemma 2.4), or other operations, see for instance [11, Def. 3.6]. Now, the theory of classical permutation matrices allows for a nice generating set: Transpositions – every permutation matrix may be written as a product of transpositions. We do not have an analog for quantum permutation matrices.

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Problem 3.16 Are there “quantum transposition matrices” such that every quantum permutation matrix can be constructed from a tuple of quantum transposition matrices? Are there natural building blocks of quantum permutation matrices?

4 Quantum Permutation Matrices in Their Broader Context and Use In this final section, we briefly sketch the context of the above objects and problems and we give references for further reading. We try to be very selective here in order to keep it short and simple. In particular, we cannot reflect the whole variety of the existing research on these topics. 4.1 Quantum Mathematics The notion “quantum mathematics” is not well-established yet, but it is used from time to time to subsume various areas of mathematics losely related to quantum physics. A common theme is noncommutativity, i.e. we consider algebraic structures where we might have x y = yx for some elements x and y. In the 1930s and 1940s, the theories of von Neumann algebras [58, 59] and of C ∗ -algebras [35, 42] have been founded, as a starting point of “noncommutative analysis” or “quantum analysis”. 4.1.1 Gelfand (Naimark) Philosophy The famous Gelfand-Naimark Theorem from 1943 [42] (see also [24, Thm. II.2.2.4]) states that a unital C ∗ -algebra is commutative if and only if it is isomorphic to an algebra of continuous functions on a compact Hausdorff space. This is even a functorial/categorial relation and we may thus identify commutative C ∗ -algebras with compact spaces – and view noncommutative C ∗ -algebras as analogues of “noncommutative” compact spaces, in some sense. This Gelfand duality might look a bit strange when one sees it for the first time, but it is just the beginning of a whole philosophy of “quantum” versions of classical theories. 4.1.2 Extensions of the Gelfand Philosophy While the theory of C ∗ -algebras can be viewed, to some extent, as noncommutative (or “quantum”) topology [41], the theory of von Neumann algebras is viewed as noncommutative measure theory [2, Sect. 3], [24, Ch. III]; there is Connes’s noncommutative (differential) geometry [33], Voiculescu’s free probability [57, 63, 77, 78] (as a counter part to probability theory; you might also consider the community of “quantum probability” here), and there is Woronowicz’s theory of compact quantum groups [65, 75, 81, 83]. They are all building on this Gelfand philosophy: commutative algebras correspond to the classical situation, while noncommutative ones correspond to their quantum counterpart. To some extent, you may also add quantum information theory (in analogy to information theory) [61, 80] and Taylor’s free analysis/noncommutative function theory (in analogy to complex analysis) [52, 74] to this family – although

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the latter ones rely little on Gelfand duality. However, all these theories found their pioneers in the 1980s (besides von Neumann algebras: in the 1930s and C ∗ -algebras: in the 1940s) and more and more interdependences have been revealed in the past few decades. So this is why it might be the time to subsume all these theories under the name “quantum mathematics” and to view it as a deeply intervowen branch of modern mathematics. See also [82] for a very brief introduction and overview, or [38]. 4.2 Quantum Groups and Quantum Permutation Groups In the context of “quantum mathematics”, the role of symmetries is played by quantum groups rather than by groups. 4.2.1 Quantum Groups In the 1980s, Woronowicz [83, 84] defined compact quantum groups and he showed that this class naturally generalizes the class of compact groups. See also the books [65, 75]. Let us present the slightly easier definition of a compact matrix quantum group [83]. See also the introductory notes [81]. Definition 4.1 Let N ∈ N. A compact matrix quantum group is a pair G = (A, u) with u = (u i j )i, j=1,...,N such that (a) A is a unital C ∗ -algebra which is generated by the elements u i j ∈ A, with i, j = 1, . . . , N , (b) u = (u i j ) and u¯ = (u i∗j ) are invertible matrices in M N (A), N ∗ (c) the map  : A → A ⊗min A given by (u i j ) = k=1 u ik ⊗ u k j is a homomorphism. Woronowicz proved a Gelfand-Naimark type theorem for compact matrix quantum groups (A, u): If the C ∗ -algebra A is commutative, then A is isomorphic to the algebra of continuous functions C(G) on some compact matrix group G ⊆ G L N (C), the generators u i j are then the evaluation maps of the matrix entries, and  arises from matrix multiplication (hence, from the group operation of G). See [75, Prop. 5.1.3] for the more general statement on compact quantum groups. Let us mention, that there also other (strongly related) notions of quantum groups, mostly in a purely algebraic setting rather than in Woronowicz’s analytic one, see for instance [50]. See [75, Sect. 5.4] for some links. 4.2.2 Quantum Permutation Groups In the 1990s, Sh. Wang [79] defined S N+ , a quantum version of the symmetric group S N . It is given by the universal C ∗ -algebra A S (N ) := C ∗ (u i j , i, j = 1, . . . , N | u i j = u i∗j = u i2j ,

k

892

u ik =



u k j = 1).

k

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So, it is the universal C ∗ -algebra generated by the entries of a “universal quantum permutation matrix”. If we add commutativity of all generators to this C ∗ -algebra, we obtain a commutative C ∗ -algebra, which is – by Gelfand-Naimark’s Theorem – isomorphic to the algebra of continuous functions on some compact space. Which space? S N , the symmetric group! So, in Woronowicz’s theory (which also takes the group structure on S N into account, in Gelfand duality), S N+ is a reasonable quantum counterpart of S N . 4.2.3 Quantum Permutation Matrices as Representations of AS (N); Further Reading Now, we immediately see that any quantum permutation matrix, as defined in Definition 2.1, gives rise to a ∗ -representation of A S (N ), so Sect. 2 basically boils down to the representation theory of this C ∗ -algebra. This is also the origin of quantum permutation matrices, better known under the name magic unitaries. They can also be viewed as generalized latin squares. It is impossible to list the whole literature on S N+ , but here is a small collection: [6, 10–12, 15, 20, 22, 23, 28, 30, 37, 47, 79]. See in +. particular the survey [9]. See [45, 72] for a discussion on possible definitions of S∞ See also [54] for another point of view on quantum permutations. 4.2.4 Operator Algebras Associated to SN+ Note that the C ∗ -algebras A S (N )—or more generally, the C ∗ -algebras A associated to compact matrix quantum groups G = (A, u) – lead to very interesting and deeply studied examples of operator algebras. In fact, any compact quantum group possesses a distinguished state, the Haar state, see for instance [75, Thm. 5.1.6, Def. 5.1.9]. So, by GNS construction, we obtain a reduced version of any quantum group, and also a von Neumann algebra associated to it. Now, these objects are at the same time very intricate, and yet tractable due to their extra structure, with striking connections to operator algebras associated with classical discrete groups. See for instance [81, Sect. 7.3, Sect. 7.4.6] for a short and incomplete (and also slightly outdated) overview on these operator algebraic aspects. See also the literature mentioned in the previous subsubsection. 4.2.5 “Easy” Quantum Groups The representation theory of S N+ is quite combinatorial using partitions of sets, similar to Brauer diagrams and Schur-Weyl duality. See the work on “easy” quantum groups for this larger class of “combinatorial quantum groups” containing S N+ ; here is some excerpt: [17, 18, 29, 40, 43, 60, 67, 76] Here is an introduction to the field: [81]. 4.2.6 Intermediate Quantum Permutation Groups For the famous question on the existence of intermediate quantum permutation groups mentioned in Sect. 3.5, let us state it here more properly: Problem 4.2 Is there some N ∈ N and a quantum group G such that S N  G  S N+ ? Reprinted from the journal

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The answer is no for N ≤ 5, see [4, 15], and it is unknown for N ≥ 6. Providing a polynomial p and a solution to the question in Sect. 3.5 would produce such an intermediate quantum group G whose associated C ∗ -algebra is the quotient of C(S N+ ) by the relations p = 0. 4.2.7 SN+ as a Symmetry Object By the way, just like S N is the symmetry object of N points within the category of groups (i.e. it is the maximal group acting on N points), S N+ is the symmetry object of N points within the theory of quantum groups: We may define actions of quantum groups on N points (after identifying N points with C N in the sense of Gelfand duality), and we observe that S N+ is the maximal object acting on it [79, 81]. As S N+ contains S N , due to a natural definition of what containment means here, we have more ways of quantum permuting points than just permuting them. The quantum world has a richer notion of symmetry! 4.3 Quantum Symmetries of Graphs and Quantum Isomorphisms of Graphs Let us comment on further use of quantum permutation matrices as symmetry objects. 4.3.1 Quantum Automorphism Groups of Graphs In 2005, Banica defined the quantum automorphism group of a finite graph [3, 21]. It is given by the quotient of the above C ∗ -algebra A S (N ) by the relations (c) and (d) of Definition 2.12. It naturally generalizes the automorphism group of a simple finite graph  = ({1, . . . , N }, E), which in turn is given by Aut() = {σ ∈ S N | σ A = Aσ }, where A ∈ M N ({0, 1}) is the adjacency matrix of . The quantum automorphism group of  contains the automorphism group, and we say that  has quantum symmetries in case this is a strict containment; this definition is consistent with the one given in Sect. 3.3. Again, the quantum world has a richer notion of symmetry—for a graph having quantum symmetries, we have more ways of quantum permuting its vertices than just permuting them. The literature on quantum symmetries is growing rapidly these days, and here is a short collection: [3, 5, 8, 21, 44, 69–71, 73]. Let us also mention some Erdös-Renyi type results in this context: [53, Thm. 3.15] and [34, 48]. 4.3.2 Quantum Isomorphism of Graphs Closely linked is the concept of quantum isomorphism of two graphs. We say that two graphs are quantum isomorphic, if a quantum isomorphism (in the sense of Definition 2.12) between them exists. Surprisingly, there are graphs, which are non-isomorphic

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but quantum isomorphic, see [53, Sect. 4.4]. This is striking—the quantum isomorphism class of a graph is larger than its isomorphism class, in general! This is also very interesting in the context of graph homomorphism counts. In the 1960s, Lovasz showed, that two graphs 1 and 2 are isomorphic, if and only if for all graphs  , the graph homomorphism counts from  to 1 and from  to 2 coincide. Do we really need all graphs  ? May we restrict to graph homomorphism counts for a smaller class, say planar graphs  for instance? In 2019, Mancinska and Roberson proved a Quantum Lovasz Theorem [56] (or rather [55] for the full version): Two graphs 1 and 2 are quantum isomorphic, if and only if for all planar graphs  , the graph homomorphism counts from  to 1 and from  to 2 coincide. As there are graphs, which are quantum isomorphic but not isomorphic, we may not restrict to planar graphs in Lovasz’s Theorem. The result by Mancinska and Roberson is exciting in many ways: Not only is it a strong theorem in graph theory—it has been achieved by means from quantum group theory and quantum information, revealing a nice interplay between these three fields. 4.4 Quantum Information Theory Let us elaborate more on the link to quantum information theory. 4.4.1 Graph Isomorphism Game Consider the nonlocal game, as described in [53, 55]: Given two graphs 1 = (V1 , E 1 ) and 2 = (V2 , E 2 ) with |V1 | = |V2 | = N , a referee passes a vertex x A ∈ X to Alice and a vertex x B ∈ X to Bob; here, X := V1  V2 is the disjoint union of V1 and V2 . Alice replies with a vertex y A ∈ X which is not from the same graph as x A , and likewise Bob replies with y B ∈ X different from the graph where y A is from. The players win the game, if (1) the set {x A , x B , y A , y B } has four elements, two of them being from V1 (let us call them j and l) and two of them from V2 (calling them i and k) and we have j ∼1 l if and only if i ∼2 k, (2) or the set {x A , x B , y A , y B } has two elements, one from V1 and one from V2 . 4.4.2 Classical Winning Strategy There is a perfect strategy (i.e. a strategy with which they may always win regardless of the referee’s input), if and only if 1 and 2 are isomorphic. Indeed, in that case let ϕ : V1 → V2 be an isomorphism of 1 and 2 and instruct Alice to reply with ϕ(x A ) ∈ V2 in case x A ∈ V1 and with ϕ −1 (x A ) ∈ V1 in case x A ∈ V2 ; likewise for Bob. 4.4.3 Quantum Winning Strategy Now, may Alice and Bob increase their chances to win when applying a quantum strategy? Technically, they are now performing quantum measurements, on a shared

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entangled state. And the answer is: Yes, they can! If the given graphs 1 and 2 are non-isomorphic but quantum isomorphic, there is no classical winning strategy, but there is a perfect quantum strategy. The strategy comes from a quantum isomorphism matrix, as in Definition 2.12, of course. So, the existence of perfect quantum strategies for this game in quantum information theory is linked with quantum permutation matrices (and hence also with quantum groups), and also with quantum isomorphisms of graphs and a quantum version of Lovasz’s Theorem from graph theory. Beautiful, isn’t it? See also [14, 66] for more on such links. 4.5 Free Probability Theory Let us mention another use of quantum permutation matrices, or rather of the quantum permutation group S N+ . 4.5.1 Classical de Finetti Theorem In probability theory, De Finetti’s Theorem may be stated as follows: Given a sequence (xn )n∈N of real random variables, this sequence is iid (independent, identically distributed) over the tail algebra if and only if it is exchangeable (i.e. its distribution is invariant under the action of the symmetric groups S N , N ∈ N on finite tuples of the sequence). 4.5.2 Free de Finetti Theorem Now, in free probability theory [57, 63, 77, 78], there is a notion of free independence, a kind of a noncommutative counterpart of classical independence, for noncommuting random variables. And just as S N is the distributional symmetry object for classical (conditional) independence, S N+ is the distributional symmetry object for (conditional) free independence—Köstler and Speicher’s De Finetti Theorem [51] states: Given a sequence (xn )n∈N of selfadjoint noncommutative random variables, this sequence is freely independent and identically distributed over the tail algebra if and only if it is quantum exchangeable (i.e. its distribution is invariant under the action of the quantum permutation groups S N+ , N ∈ N on finite tuples of the sequence). 4.5.3 More on de Finetti Theorems and Other Stochastic Aspects This is another instance of the interplay between various fields of “quantum mathematics”: Just as groups provide the correct symmetries for probability theory, the correct symmetries for free probability are provided by quantum groups. See for instance [19], [16, Sect. 1.1] for more on such de Finetti theorems, or [18] for other stochastic aspects of S N+ . Acknowledgements Acknowledgements go to Nicolas Faroß, Daniel Gromada, Michael Hartz, Luca Junk, Alexander Mang, Julien Schanz, Simon Schmidt, Roland Speicher and Adam Skalski for comments on an earlier version of this article.

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Quantum Permutation Matrices Funding Open Access funding enabled and organized by Projekt DEAL. Data Availability Statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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