Lectures on Analytic Function Spaces and their Applications (Fields Institute Monographs, 39) 3031335716, 9783031335716

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Fields Institute Monographs 39 The Fields Institute for Research in Mathematical Sciences

Javad Mashreghi Editor

Lectures on Analytic Function Spaces and their Applications

Fields Institute Monographs VOLUME 39 Fields Institute Editorial Board: Deirdre Haskell, Deputy Director of the Fields Institute, Toronto, ON, Canada Lisa Jeffrey, University of Toronto, Toronto, ON, Canada Winnie Li, Penn State University, University Park, PA, USA V. Kumar Murty, Director of the Fields Institute, Toronto, ON, Canada Ravi Vakil, Stanford University, Stanford, CA, USA

The Fields Institute is a centre for research in the mathematical sciences, located in Toronto, Canada. The Institute’s mission is to advance global mathematical activity in the areas of research, education and innovation. The Fields Institute is supported by the Ontario Ministry of Training, Colleges and Universities, the Natural Sciences and Engineering Research Council of Canada, and seven Principal Sponsoring Universities in Ontario (Carleton, McMaster, Ottawa, Queen’s, Toronto, Waterloo, Western and York), as well as by a growing list of Affiliate Universities in Canada, the U.S. and Europe, and several commercial and industrial partners.

More information about this series at http://www.springer.com/series/10502

Javad Mashreghi Editor

Lectures on Analytic Function Spaces and their Applications

The Fields Institute for Research in the Mathematical Sciences

Editor Javad Mashreghi Department of Mathematics and Statistics Laval University Québec City, QC, Canada

ISSN 1069-5273 ISSN 2194-3079 (electronic) Fields Institute Monographs ISBN 978-3-031-33571-6 ISBN 978-3-031-33572-3 (eBook) https://doi.org/10.1007/978-3-031-33572-3 © 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 Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The focus program on Analytic Function Spaces and their Applications took place at Fields Institute from July 1 to December 31, 2021. Hilbert spaces of analytic functions form one of the pillars of complex analysis. These spaces have a rich structure and for more than a century have been studied by many prominent mathematicians. They also have several essential applications in other fields of mathematics and engineering, e.g., robust control engineering, signal and image processing, and theory of communication. The most important Hilbert space of analytic functions is the Hardy class .H 2 . However, its close cousins, e.g., the Bergman space .A2 , the Dirichlet space .D, the model subspaces .K , and the de Branges–Rovnyak spaces .H(b), have also been the center of attention in the past two decades. Studying the Hilbert spaces of analytic functions and the operators acting on them, as well as their applications in other parts of mathematics or engineering, was the main subject of this program. During the program, the world leading experts discussed on function spaces and the new achievements and future venues of research on analytic function spaces, their operators, and their applications in other domains. With more than 250 hours of lectures by prominent mathematicians, a wide variety of topics were covered. More explicitly, there were mini-courses and workshops on (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi)

Hardy Spaces Dirichlet Spaces Bergman Spaces Model Spaces Interpolation and Sampling Riesz Bases, Frames and Signal Processing Bounded Mean Oscillation de Branges-Rovnyak Spaces Operators on Function Spaces Truncated Toeplitz Operators Blaschke Products and Inner Functions v

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(xii) (xiii) (xiv) (xv) (xvi)

Preface

Discrete and Continuous Semigroups of Composition Operators The Corona Problem Non-commutative Function Theory Drury-Arveson Space Convergence of Scattering Data and Non-linear Fourier Transform

At the end of each week, there was a high-profile colloquium talk on the current topic. The program also contained two semester-long advanced courses on (i) Schramm Loewner Evolution and Lattice Models and (ii) Reproducing Kernel Hilbert Space of Analytic Functions. The current volume features the mini-courses given on Hardy Spaces, Dirichlet Spaces, Bergman Spaces, Model Spaces, Operators on Function Spaces, Truncated Toeplitz Operators, Semigroups of weighted composition operators on spaces of holomorphic functions, the Corona Problem, Non-commutative Function Theory, and Drury-Arveson Space. However, videos of almost all the mini-courses, advanced courses, plenary talks, colloquium talks, and contributed talks are available at: http://www.fields.utoronto.ca/activities/21-22/function August 2022

Javad Mashreghi

Contents

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Hardy Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Javad Mashreghi 1.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Boundary Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Approximate Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 An Overview of Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Banach Algebra H ∞ (D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The Hardy-Hilbert Space H 2 (D) . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 The Banach Space H p (D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Analysis and Synthesis of Functions in Hardy Spaces . . . . . . . . . . . . . 1.3.1 Synthesis: Various Convergence Types . . . . . . . . . . . . . . . . . . . . 1.3.2 Analysis: Poisson Representations . . . . . . . . . . . . . . . . . . . . . . . . 1.4 An Overview of Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Harmonic Hardy Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Analytic Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Dirichlet Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomas Ransford 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 What is the Dirichlet Space? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 History and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 What to Study? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Where to Find Out More About D? . . . . . . . . . . . . . . . . . . . . . . . 2.2 Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Capacity of Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Capacity of General Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Equilibrium Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Boundary Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 7 9 9 10 14 16 16 32 43 44 50 53 55 55 55 55 56 56 56 56 57 57 58 58 vii

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2.3.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Beurling’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Capacitary Weak-Type and Strong-Type Inequalities. . . . . 2.3.4 Douglas’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Exponential Approach Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Carleson’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.7 Some Further Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The Three Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Boundary Zero Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Arguments of Zero Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Some Further Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Carleson Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Characterization of Carleson Measures. . . . . . . . . . . . . . . . . . . . 2.5.5 Multipliers and Reproducing Kernels . . . . . . . . . . . . . . . . . . . . . 2.5.6 Pick Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.7 Interpolating Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.8 Factorization Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.9 Some Further Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conformal Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Characterization of D via Möbius Invariance . . . . . . . . . . . . . 2.6.3 Composition Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Weighted Composition Operators . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Some Further Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Weighted Dirichlet Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 The Dα Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 The Dμ Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Properties of Dμ (Richter–Sundberg [43]) . . . . . . . . . . . . . . . . 2.7.4 Hadamard Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.5 Special Cases of Hadamard Multipliers . . . . . . . . . . . . . . . . . . . 2.7.6 Some Further Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Shift-Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 The Shift Operator on Dμ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 Invariant Subspaces of (Mz , D) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Cyclic Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 Cyclic Invariant Subspaces and Boundary Zeros . . . . . . . . . 2.8.6 Brown–Shields Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.7 Some Further Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Bergman Space of the Unit Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefan Richter 3.1 Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Aspects of the Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Bergman Versus Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Weighted Bergman Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Disc Automorphisms and Change of Variables . . . . . . . . . . . 3.2.4 Pointwise Bounds and Reproducing Property . . . . . . . . . . . . . 3.2.5 Reproducing Kernel and the Bergman Projection. . . . . . . . . 3.2.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Zero Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Invariant Subspaces of Infinite Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 From Infinite Index to the Invariant Subspace Problem . . 3.4.2 The Index of Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Spectral Synthesis and Dominating Sets . . . . . . . . . . . . . . . . . . 3.4.4 Mz∗ |I ()⊥ Can Be Similar to a Diagonal Normal Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 An Interpolating Sequence That Is Dominating. . . . . . . . . . . 3.4.6 Further Results About Invariant Subspaces with High Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Bergman Inner Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 An Extremal Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Contractive Divisors Via the Biharmonic Function . . . . . . . 3.5.3 The Reproducing Kernel Approach. . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Further Contractive Divisor Results . . . . . . . . . . . . . . . . . . . . . . . 3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 The Schur Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Carleson’s Interpolation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Positive Definite Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephan Ramon Garcia 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Hardy Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Inner Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Canonical Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Bounded Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Model Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Basic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Reproducing Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Finite-Dimensional Model Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Stability Under Co-Analytic Toeplitz Operators . . . . . . . . . . 4.4 The Compressed Shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.4.1 The Livšic–Möller Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Model Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Density Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Bases for Model Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Takenaka–Malmquist–Walsh Basis . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Riesz Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Continuability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Pseudocontinuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Conjugation on Model Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Conjugations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 The Model Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Associated Inner Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4 Generators of Ku . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Aleksandrov–Clark Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Herglotz’ Theorem and Clark Measures. . . . . . . . . . . . . . . . . . . 4.9.2 Clark Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.3 Deeper Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 4.10 Explicit Description of Ku . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Quaternionic Structure of 2 × 2 Inner Functions . . . . . . . . . . . . . . . . . . . 4.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130 131 133 134 135 136 137 137 138 139 139 140 141 142 143 144 145 147 148 150 151 152

Operators on Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . William Ross 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Hilbert Spaces of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 General Operator Theory Concepts . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 The Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 The Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Volterra Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Basic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Numerical Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Commutant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Square Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Invariant Subspaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Complex Symmetric Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Generalized Volterra Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The Cesàro Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Basic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 The Numerical Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Subnormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 The Commutant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 156 156 157 160 162 162 163 164 164 165 166 167 167 168 168 170 170 171 171

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5.6.3 Invariant Subspaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Square Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.5 Generalized Cesaro Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Multiplication Operators on L2 (T) . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 The Bilateral Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Multiplication Operators on H 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.4 The Unilateral Shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.5 Toeplitz Operators on H 2 (D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.6 Toeplitz Operators on H 2 (C+ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.7 Toeplitz Operators on Other Spaces . . . . . . . . . . . . . . . . . . . . . . . 5.8 Hankel Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Hankel Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Hankel Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 The Norm of a Hankel Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.4 Bounded Mean Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 The Hilbert Matrix Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Another Setting for Hankel Operators . . . . . . . . . . . . . . . . . . . . . 5.9.2 Back to Multiplication Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Fourier and Hilbert Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.1 Plancherel’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.2 The Spectrum of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.3 The Hilbert Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.4 Spectrum of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Further Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172 172 173 173 173 174 175 177 179 180 180 181 181 183 184 184 185 186 186 187 187 187 188 189 190 191

Truncated Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emmanuel Fricain 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Function Spaces, Multiplication Operators and Their Cognates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Model Spaces and One Component Inner Functions . . . . . 6.2.3 Carleson Measures for the Hardy Spaces and for the Model Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Truncated Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Definition of Truncated Toeplitz Operators . . . . . . . . . . . . . . . 6.3.2 An Equivalent Definition and Some Basic Properties . . . . 6.4 Why Studying Truncated Toeplitz Operators? . . . . . . . . . . . . . . . . . . . . . . 6.4.1 The Sz.-Nagy–Foias Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 The Commutant of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 The Nevanlinna–Pick Interpolation Problem . . . . . . . . . . . . . . 6.4.4 A Link with Truncated Wiener-Hopf Operators . . . . . . . . . . 6.5 The Class of Symbols for a Truncated Toeplitz Operator . . . . . . . . . .

195 195 196 196 198 202 204 204 206 209 209 211 215 217 219

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Algebraic Characterization of Truncated Toeplitz Operators . . . . . . Complex Symmetric Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Truncated Toeplitz Operators Are Complex Symmetric . . 6.7.2 Another Characterization of Truncated Toeplitz Operators and New Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Norm of a Truncated Toeplitz Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Spectral Properties of A ϕ ............................................ 6.10 Finite Rank Truncated Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Compact Truncated Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Problem of the Existence of a Bounded Symbol. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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Semigroups of Weighted Composition Operators on Spaces of Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isabelle Chalendar and Jonathan R. Partington 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Strongly Continuous Semigroups of Operators: Definition and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Analytic Semiflows on a Domain and Models for Semiflows on D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Models for Analytic Flows on D . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 C0 -Semigroups of Composition Operators . . . . . . . . . . . . . . . . 7.2.5 Spaces on Which Semigroups Are Not C0 . . . . . . . . . . . . . . . . 7.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Universal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Change of Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Asymptotic Behaviour of T n or Tt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 The Discrete Unweighted Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 The Continuous Unweighted Case. . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Weighted Composition Operators . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Isometry and Similarity to Isometry . . . . . . . . . . . . . . . . . . . . . . . 7.4.5 Generators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Compactness and Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Immediate and Eventual Compactness . . . . . . . . . . . . . . . . . . . . 7.5.2 Compact Analytic Semigroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 An Outlook on C+ and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 The Right Halfplane C+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 The Complex Plane C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

222 225 225 228 232 234 236 241 247 252 255 255 256 256 258 259 261 261 262 262 263 263 264 264 267 269 269 270 271 271 272 274 274 275 279

The Corona Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Alexander Brudnyi 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 ¯ 8.2 Banach-Valued ∂-Equations on the Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

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8.2.1 Interpolating Sequences for H ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Proof of Theorem 8.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Particular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Carleson’s Corona Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Structure of the Maximal Ideal Space of H ∞ . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Gleason Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Structure of Ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Structure of Ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Banach-Valued Corona Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Operator Completion Problem for H ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285 286 287 287 292 293 295 295 295 297 299 302 306

A Brief Introduction to Noncommutative Function Theory . . . . . . . . . . . Michael T. Jury 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 nc Sets and nc Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 nc Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Locally Bounded nc Functions Are Differentiable . . . . . . . . . . . . . . . . . 9.4 Topologies and Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Square-Summable nc Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Reproducing Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 The Connection with the Drury-Arveson Space . . . . . . . . . . 9.7 nc Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 .................................. 9.8 The d-Shift and Multipliers of Hnc 9.9 Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.1 Nevanlinna-Pick Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.2 nc Measures and Cauchy Integrals. . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

309

An Invitation to the Drury–Arveson Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Hartz 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Several Definitions of Hd2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Hardy Space Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Discovering the Drury–Arveson Space . . . . . . . . . . . . . . . . . . . . 10.2.3 Power Series Description of the Drury–Arveson Space . . 10.2.4 RKHS Description of the Drury–Arveson Space . . . . . . . . . 10.2.5 Function Theory Description of the Drury–Arveson Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.6 The Drury–Arveson Space as a Member of a Scale of Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.7 The Non-commutative Approach . . . . . . . . . . . . . . . . . . . . . . . . . .

347

309 310 312 314 315 320 321 325 330 331 334 338 343 343 344 345

347 349 349 349 352 353 355 357 358

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Contents

10.3

Multipliers and Operator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Function Theory of Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Dilation and von Neumann’s Inequality . . . . . . . . . . . . . . . . . . . 10.3.4 The Non-commutative Approach to Multipliers . . . . . . . . . . 10.3.5 The Toeplitz Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.6 Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Complete Pick Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Pick’s Theorem and Complete Pick Spaces . . . . . . . . . . . . . . . 10.4.2 Characterizing Complete Pick Spaces . . . . . . . . . . . . . . . . . . . . . 10.4.3 Universality of the Drury–Arveson Space . . . . . . . . . . . . . . . . . 10.5 Selected Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Maximal Ideal Space and Corona Theorem . . . . . . . . . . . . . . . 10.5.2 Interpolating Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 Weak Products and Hankel Operators . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

362 362 365 370 375 376 381 382 382 388 392 396 396 401 405 407

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Contributors

Alexander Brudnyi University of Calgary, Calgary, AB, Canada Isabelle Chalendar Université Gustave Eiffel, Laboratoire d’Analyse et de Mathématiques Appliquées, Champs-sur-Marne, France Emmanuel Fricain Université de Lille, Laboratoire Paul Painlevé, UFR de Mathématiques Bâtiment M2, Villeneuve d’Ascq Cédex, France Stephan Ramon Garcia Pomona College, Department of Mathematics and Statistics, Claremont, CA, USA Michael Hartz Fachrichtung Mathematik, Universität des Saarlandes, Saarbrücken, Germany Michael T. Jury Department of Mathematics, University of Florida, Gainesville, FL, USA Javad Mashreghi Department of Mathematics and Statistics, Laval University, Québec City, QC, Canada Jonathan R. Partington School of Mathematics, University of Leeds, Leeds, Yorkshire, UK Thomas Ransford Université Laval, Département de mathématiques et de statistique, Pav. Vachon, Cité Universitaire, Québec, QC, Canada Stefan Richter Department of Mathematics, University of Tennessee, Knoxville, TN, USA William Ross Department of Mathematics and Statistics, University of Richmond, Richmond, VA, USA

xv

Chapter 1

Hardy Spaces Javad Mashreghi

2020 Mathematics Subject Classification 30H10, 30J05, 30J10, 31A20, 42A50

1.1 Preliminaries The open unit disc is denoted by .D = {z ∈ C : |z| < 1} and its boundary by T = ∂D = {z ∈ C : |z| = 1}; .Hol(D) is the family of all holomorphic functions on .D. We also inevitably need to consider the family of harmonic functions .har(D) on .D. .

1.1.1 Boundary Values Fix .ζ ∈ T and let f be an analytic function defined on .D. One of the most basic questions in function theory is to explore .limz→ζ f (z) as z approaches to .ζ from within .D. To answer this question, it is essential to know how z approaches .ζ . Here are some possibilities. (i) No restriction: putting no restriction on the approach region, except of course staying inside .D, is precisely the definition of continuity of f , as function defined on .D, at the boundary point .ζ . This is a very strong type of convergence, and hence rarely appears in general theorems in function space theory. (ii) Radial limit: on the other extreme, we may restrict z to approach .ζ on the radii passing through the origin and .ζ . This is the most restrictive approach method

The author was partially supported by a Discovery NSERC grant. J. Mashreghi () Department of Mathematics and Statistics, Laval University, Québec City, QC, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mashreghi (ed.), Lectures on Analytic Function Spaces and their Applications, Fields Institute Monographs 39, https://doi.org/10.1007/978-3-031-33572-3_1

1

2

J. Mashreghi

The Stoltz domain .c (eiθ0 )

and if the limit .

lim f (rζ )

r→1−

exists, we say that f has a radial limit at the point .ζ . We write .f ∗ (ζ ) to denote this boundary value. Note that if .f ∗ (ζ ) exists, at least almost everywhere on .T, then, on one hand, we have the analytic function f which lives on .D and, on the other hand, we have its boundary function .f ∗ which lives on .T. The interplay between f and .f ∗ is one of the gems of function space theory. In this note, we keep both notations to clarify the distinction between two functions. However, it is more common that both functions be denoted by f . (iii) Non-tangential limit: To relax a bit the previous ultra restrictive approach, we may let z approaches .ζ on a triangular region anchored at .ζ . More explicitly, let c (ζ ) := {z ∈ D : |z − ζ |  c(1 − |z|)}.

.

If, for each fixed .c > 1, the limit .

lim f (z)

z→ζ z∈c (ζ )

exists, we say that f has a non-tangential limit at .ζ and, as in the previous case, we denote this limit by .f ∗ (ζ ). As c runs from 1 to .∞, the opening angel ranges from .2α = 0 (radial approach) to .2α = π (non-restrictive approach). For functions in any Hardy space, the nan-tangential limit exists almost everywhere. However, if we allow a wider region, the limit may fail to exists. (iv) Tangential approach regions: In some function spaces, e.g., the Dirichlet space, functions exhibit a better behavior and, naively speaking, the approach region

1 Hardy Spaces

3

can be wider than the triangles considered in the previous case. To provide a more rigorous definition, we may rewrite the definition of .c as follows: ϕ (ζ ) := {z ∈ D : |z − ζ |  ϕ(1 − |z|)},

.

where .ϕ(t) = ct. In a more general setting, we consider other families of .ϕ’s. For example, .ϕ(t) = ct 1/2 gives a family of discs which are tangent to .ζ from within .D. An exponentially tangent region is given by .ϕ(t) = c/ log(1/t). Both approach regions appear in studying the boundary behavior of functions in the Dirichlet space. The function f on .D gives birth to .f ∗ on .T. Reciprocally, knowing .f ∗ on .T, how can we recover f .D? This is usually done via the Poisson integral formula f (z) =

.

1 2π



2π

0

1 − |z|2 ∗ f (ζ ) dθ. |z − ζ |2

In most cases, this reconstruction process works. However, there are exception which we will address below. There is also another way to interpret a function defined on .D. For each fixed .r < 1, let fr (ζ ) := f (rζ ),

ζ ∈ T.

.

Hence, we interpret f as a family of functions .(fr )0r 1. Let .r and .s be respectively the conjugate exponents of r and s, i.e. .

1 1 + =1 r r

1 1 + = 1. s s

and

Then, according to the definition of p, we have .

1 1 1 + + = 1. r s p

Fix .eiθ ∈ T. Then we write |f (eiτ )g(ei(t−τ ) )| = |g(ei(t−τ ) )|

.

1− ps

× |f (eiτ )|

1− pr

r

s

× |f (eiτ )| p |g(ei(t−τ ) )| p .

The generalized Hölder’s inequality then implies

.

1 2π



π −π

 |f (eiτ ) g(ei(t−τ ) )| dτ 

1 2π





π

−π

|g(ei(t−τ ) )|

r (1− ps )

 1 dτ

r

 1 π s 1 s (1− pr ) |f (eiτ )| dτ 2π −π  1  π p 1 iτ r i(t−τ ) s . × |f (e )| |g(e )| dτ 2π −π 

×

It is easy to see .r (1 − s/p) = s and .s (1 − r/p) = r. Thus, |(f ∗ g)(e )| 

.

it

s/r gs

r/s f r



1 2π



π

−π

Hence, by Fubini’s theorem,  f ∗ gp =

.

1 2π



π −π

1 |(f ∗ g)(e )| dt it

p

p

1 |f (e )| |g(e iτ

r

i(t−τ ) s

)| dτ

p

.

1 Hardy Spaces

7



s/r gs

r/s f r

=

s/r gs

r/s f r





1 (2π )2

π



−π

s/p × gs

π −π

r/p f r

1 |f (e )| |g(e iτ

r

i(t−τ ) s

p

)| dt dτ

= f r gs .  

1.1.3 Approximate Identities We saw that .L1 (T) is a commutative Banach algebra without a unit. To overcome this shortcoming, we consider a family of integrable functions .{ κ }, which in the limit behaves like the identity element of convolution. This is made precise below. Let . κ ∈ L1 (T), where in our studies the index .κ ranges over either the set of integers .{1, 2, 3, . . . } or over the interval .[0, 1). Therefore, .limκ will mean either .limn→∞ or .limr→1− , depending on the case. The family .{ κ } is an approximate identity if: (I) (normalization) for all .κ, 1 . 2π



2π

κ (eit ) dt = 1;

0

(II) (uniformly .L1 -bounded) the quantity  C = C( ) := sup

.

κ

1 2π



2π

 | κ (eit )| dt

0

is finite; (III) (concentration around .t = 0) for each fixed .δ, .0 < δ < π ,  . lim | κ (eit )| dt = 0. κ

δ|t|π

If . κ (eit )  0, for all .κ and for all .eit ∈ T, then .{ κ } is a positive approximate identity. In this case, (II) is automatically fulfilled with .C = 1. Two celebrated examples of positive approximate identities are as follows. Our main example of a positive approximate identity, which plays a decisive role in studying Hardy spaces, is the Poisson kernel Pr (eit ) =

.

∞  1 − r2 = r |n| eint , 1 + r 2 − 2r cos t n=−∞

0  r < 1.

8

J. Mashreghi

The Poisson kernel .Pr for .r = 0.2, 0.5, 0.8

The Fejér kernel .Kn for .n = 5, 10, 15

The second is the Fejér kernel Kn (eit ) =

.

1 n+1



sin( (n+1)t 2 ) sin( 2t )

2 =

n   k=−n

1−

|k| n+1

 eikt ,

n  0.

Fejér kernel is a major tool in approximation theory. It provides a very satisfactory response for many polynomial approximation problems. The above two specific examples also satisfy the following property, which is stronger than the condition (III):

1 Hardy Spaces

9

(IV) For each fixed .δ, .0 < δ < π ,  .

 sup | κ (eit )| = 0.

lim κ

δ|t|π

Several other celebrated approximate identities also satisfy this stronger property. It is easy to see that, for each fixed .n ∈ Z, .

κ (n) = 1. lim κ

Recall that, by Theorem 1.1.1, the Fourier series of . κ ∗ μ and . κ ∗ f are respectively ∞  .

κ (n) μ(n) ˆ einθ

∞ 

and

n=−∞

κ (n) fˆ(n) einθ .

n=−∞

These are the weighted Fourier series of .μ and f and, in the limit, they approach the Fourier transform of .μ and f . Our goal is to clarify in what sense they approach [2, 9].

1.2 An Overview of Hardy Spaces In this section we provide an overview of Hardy spaces. We give the preliminary definitions as well as some essential results. In the next section we go into further detail and discuss some tools and techniques and sketch some proofs.

1.2.1 The Banach Algebra H ∞ (D) The easiest Hardy space to define, and yet containing the most profound results, is H ∞ (D). It consists of all bounded holomorphic functions in .Hol(D). Hence, it is natural to define

.

f H ∞ (D) := sup |f (z)|.

.

z∈D

The definition maybe rephrased as H ∞ (D) := {f ∈ Hol(D) : f H ∞ (D) < ∞}.

.

10

J. Mashreghi

One of the most essential results of bounded analytic functions in the following result of Fatou [2, 4]. Theorem 1.2.1 Let .f ∈ H ∞ (T ). Then the boundary values f ∗ (eiθ ) = lim f (reiθ )

.

r→1

exist for almost all .eiθ ∈ T.

1.2.2 The Hardy-Hilbert Space H 2 (D) There are various ways to define the Hardy–Hilbert space .H 2 . Each way has its own merits and is suitable for some applications. In particular, the Hilbert space structure of .H 2 is rewarding in easily establishing several results, which are otherwise more difficult to handle (.p = 2), e.g., the M. Riesz result on the boundedness of Hilbert transform. In the following, we discuss four seemingly different approaches to define the norm in .H 2 (D). Then we will distinguish the right route to define .H p (D) spaces for other values of p. • Taylor or Fourier coefficients: The most straightforward way to define .H 2 (D) is to say that it is a ‘copy’ of the sequence space .2 inside .Hol(D). More precisely, each function .f ∈ Hol(D) has the unique Taylor series expansion f (z) =

∞ 

.

an zn ,

z ∈ D.

(1.4)

n=0

We use this representation to define

f H 2 (D) :=

∞ 

.

1 2

|an |2

.

(1.5)

n=0

Then the Hardy–Hilbert space .H 2 (D) is the collection of all analytic functions f on .D for which .f H 2 (D) < ∞. There is another interpretation for the formula (1.5). According to variation of Fatou’s theorem (Theorem 1.2.1), the boundary values f ∗ (eiθ ) = lim f (reiθ )

.

r→1

exist for almost all .eiθ ∈ T. Moreover, the connection between the harmonic analysis on .T and functions space theory on .D is provided via the relation

1 Hardy Spaces

11

f ∗ (n) =

.

⎧ ⎨ an if n  0, ⎩

0 if n < 0.

Then the Parseval identity reveals that  ∗

f L2 (T) =

.

1

∞ 

|f ∗ (n)|2

2

=

∞ 

n=−∞

1 2

|an |

= f H 2 (D) .

2

(1.6)

n=0

In other words, there is an isometry between the Hardy space .H 2 (D) and a certain subspace of .L2 (T). Conversely, let .ϕ ∈ L2 (T) be such that .ϕ(n) ˆ = 0, for all .n  −1. Then the analytic function f (z) =

∞ 

.

n ϕ(n)z ˆ ,

z ∈ D,

n=0

is well-defined and, according to the definition (1.5), .f ∈ H 2 (D). Once more, Parseval’s identity says that .f H 2 (D) = ϕL2 (T) . Moreover, the uniqueness theorem for Fourier coefficients also ensures that .ϕ = f ∗ almost everywhere on .T. The image of .H 2 (D) in .L2 (T) under the above mapping is denoted by .H 2 (T). In other words, we may also consider .H 2 (T) as the subspace of .L2 (T) consisting of elements whose negative spectrum is vanishing. It is a common practice, in particular if the distinction between .D and .T is not essential, to write .H 2 for the Hardy–Hilbert space. In the definition (1.5), it makes sense to replace 2 by p and study the corresponding space. However, by doing so we do not obtain the classical Hardy space p p .H (D). The outcome is a new family of analytic functions which is denoted by . A [1]. In other words, .H p (D) is not isomorphic isometric to the sequence space .p if .p = 2. To overcome this obstacle, we need to consider the integral means of analytic functions. • Integral means: Let  m2 (f, r) =

.

1 2π



2π

|f (re )| dθ iθ

2

 21 ,

0  r < 1.

0

If we use the Taylor expansion (1.4) and plug it into the above formula, again by Parseval’s identity, we obtain

m2 (f, r) =

∞ 

.

n=0

1 2

|an | r

2 2n

,

0  r < 1.

12

J. Mashreghi

Hence, .m(f, r, 2) is an increasing function of r and the monotone convergence theorem (discrete version) says that  sup m2 (f, r) = lim m2 (f, r) =

.

r→1

0r κ(ε) and for all t. This means that . κ ∗ f converges uniformly to f on T.  

.

20

J. Mashreghi

Corollary 1.3.3 Let .u ∈ C(T) and ⎧  2π 1 1 − r2 ⎪ ⎪ u(eit ) dt if 0  r < 1, ⎨ 2π 0 1 + r 2 − 2r cos(θ − t) iθ .U (re ) = ⎪ ⎪ ⎩ u(eiθ ) if r = 1. Then (i) (ii) (iii) (iv)

U is continuous on .D, U is harmonic on .D, for each .0  r < 1, .Ur L∞ (T)  uL∞ (T) , − .Ur − uL∞ (T) → 0 as .r → 1 .

Since U (reiθ ) =

∞ 

.

|n| inθ u(n)r ˆ e ,

0  r < 1.

(1.17)

n=−∞

the above result says that the Abel–Poisson means of the Fourier series of a continuous function converge uniformly on .T to the same function. Corollary 1.3.4 (Weierstrass–Fejér) Let .f ∈ C(T) and, for .n  0, let  n   |k| 1− fˆ(k)eikt , n+1

pn (eit ) =

.

eit ∈ T.

k=−n

Then pn L∞ (T)  f L∞ (T) ,

.

n  0,

and .

lim pn L∞ (T) = f L∞ (T) ,

n→∞

and, moreover, .

lim pn − f L∞ (T) = 0.

n→∞

The Weierstrass classical results says that the polynomials are dense in the space of continuous functions. The importance of Corollary 1.3.4 is that it provides a simple constructive method to obtain the approximating polynomials.

1 Hardy Spaces

1.3.1.3

21

Weak*-Convergence of Dilates: Construction in h1 (D)

Given a Borel measure .μ, how does the family . κ ∗ μ approach .μ? This is clarified below. Theorem 1.3.5 Let .{ κ } be an approximate identity, and let .μ ∈ M(T). Then κ ∗ μ ∈ L1 (T) and, for all .κ,

.

 κ ∗ μL1 (T)  C μ,

.

and μ  sup  κ ∗ μL1 (T) .

.

κ

Moreover, .

lim κ

1 2π



2π

 ϕ(eit )( κ ∗ μ)(eit ) dt =

0

T

ϕ(eit ) dμ(eit )

for all .ϕ ∈ C(T), and thus .

lim  κ ∗ μL1 (T) = μ. κ

If, furthermore, .{ κ } is a positive approximate identity then, for all .κ,  κ ∗ μL1 (T)  μ.

.

In technical language, Theorem 1.3.5 says that the measures .dμκ (eit ) = ( κ ∗ μ)(eit ) dt/2π converge to .dμ(eit ) in the weak* topology of .M(T), as the dual space of .C(T). Proof By Theorem 1.1.2, . κ ∗ μ ∈ L1 (T) and  κ ∗ μL1 (T)   κ L1 (T) μ  Cμ.

.

Let .ϕ ∈ C(T), and define .ψ(eit ) = ϕ(e−it ). Then, .ψ ∈ C(T) and by Fubini’s theorem, .

1 2π

 0

2π

   2π 1 ϕ(eit ) κ (ei(t−τ ) ) dμ(eiτ ) dt 2π 0 T     2π 1 κ (ei(t−τ ) )ϕ(eit ) dt dμ(eiτ ) = T 2π 0

ϕ(eit )( κ ∗ μ)(eit ) dt =

22

J. Mashreghi

  =  =

T

T

1 2π



 κ (ei(−τ −s) )ψ(eis ) ds dμ(eiτ )

2π 0

( κ ∗ ψ)(e−iτ ) dμ(eiτ ).

Theorem 1.3.2 assures that .( κ ∗ ψ)(e−iτ ) converges uniformly to .ψ(e−iτ ) on .T. We thus have .

lim κ

1 2π



2π

 ϕ(eit )( κ ∗ μ)(eit ) dt = lim κ

0

 =

 =

T

T

T

( κ ∗ ψ)(e−iτ ) dμ(eiτ )

ψ(e−iτ ) dμ(eiτ ) ϕ(eiτ ) dμ(eiτ ).

Note that since .|μ| is a finite positive Borel measure, we could change the order of limit and integral above. Since



1 .

2π

2π 0

ϕ(e )( κ ∗ μ)(e ) dt

 (sup  κ ∗ μ1 )ϕ∞ , κ it

it

our calculation above implies



iτ iτ

.  κ ∗ μ1 )ϕ∞

ϕ(e ) dμ(e )  (sup κ T

for all .ϕ ∈ C(T). Hence, by the Riesz representation theorem for bounded functionals on .C(T), we conclude that μ  sup  κ ∗ μL1 (T) .

.

κ

The last conclusion also follows, since .C = 1 for positive approximate identities.

 

Corollary 1.3.6 Let .μ ∈ M(T), and let  U (reiθ ) =

.

T

1 + r2

1 − r2 dμ(eit ), − 2r cos(θ − t)

reiθ ∈ D.

Then .U ∈ h1 (D), Ur L1 (T)  μ,

.

0  r < 1,

(1.18)

1 Hardy Spaces

23

and lim Ur L1 (T) = μ,

.

r→1

and, moreover,  .

2π

lim

r→1− 0

ϕ(eit )U (reit )

dt = 2π

 T

ϕ(eit ) dμ(eit )

for all .ϕ ∈ C(T). Corollary 1.3.15 shows how to construct an element of .h1 (D). It also says that the measures .dμr (eit ) = U (reit )dt/2π converge to .dμ(eit ) in the weak*-topology of .M(T), as the dual space of .C(T). It also has some interesting and profound consequences. First, it shows that the family of absolutely continuous measures is weak* dense in .M(T). Second, the identity sup Ur L1 (T) = μ

.

0r 0. Given p .f ∈ L (T), pick .ϕ ∈ C(T) such that .f − ϕp < ε. Hence  κ ∗ f − f Lp (T) =  κ ∗ (f − ϕ) − (f − ϕ) + ( κ ∗ ϕ − ϕ)Lp (T)

.

  κ ∗ (f − ϕ)Lp (T) +f − ϕLp (T) + κ ∗ ϕ−ϕLp (T)  (1 + C)f − ϕLp (T) +  κ ∗ ϕ − ϕLp (T)  (1 + C)ε +  κ ∗ ϕ − ϕL∞ (T) . However, by Theorem 1.3.2, there is an .κ(ε) such that  κ ∗ ϕ − ϕL∞ (T) < ε,

κ > κ(ε).

.

Therefore,  κ ∗ f − f Lp (T) < (2 + C )ε,

.

κ > κ(ε).

The last conclusion also follows since .C = 1 for positive approximate identities. Corollary 1.3.11 Let .u ∈ Lp (T), .1  p < ∞, and 1 .U (re ) = 2π



iθ

0

2π

1 − r2 u(eit ) dt, 1 + r 2 − 2r cos(θ − t)

Then .U ∈ hp (D), and Ur Lp (T)  uLp (T) ,

.

and

0  r < 1,

reiθ ∈ D.

 

26

J. Mashreghi .

lim Ur Lp (T) = uLp (T) ,

r→1

and, moreover, .

lim Ur − uLp (T) = 0.

r→1

Corollary 1.3.11 shows how to construct an element of .hp (D). It also says that the dilates .Ur converge to u in the norm topology of .Lp (T), .1  p < ∞. Corollary 1.3.12 Let .f ∈ Lp (T), .1  p < ∞, and, for .n  0, pn (eit ) =

.

 n   |k| 1− fˆ(k)eikt , n+1

eit ∈ T.

k=−n

Then pn Lp (T)  f Lp (T) ,

.

n  0,

and .

lim pn Lp (T) = f Lp (T) ,

n→∞

and, moreover, .

lim pn − f Lp (T) = 0.

n→∞

Corollary 1.3.12 provides a highly valuable constructive approximation scheme by trigonometric polynomials in .Lp (T) spaces. This fact helps us to prove an important result in Fourier analysis. Corollary 1.3.13 (Riemann–Lebesgue Lemma) Let .f ∈ L1 (T). Then .

lim fˆ(n) = 0.

|n|→∞

Proof Fix .ε > 0. By Corollary 1.3.12, we may take .n0 large enough such that pn0 − f L1 (T) < ε,

.

where .pn0 is the Fejér means of Taylor series of f , i.e., pn0 (eit ) =

.

n0   1− k=−n0

 |k| fˆ(k)eikt , n0 + 1

eit ∈ T.

1 Hardy Spaces

27

Since |pˆ n0 (n) − fˆ(n)|  pn0 − f L1 (T) < ε,

.

n ∈ Z,

and .pˆ n0 (n) = 0 for .|n| > n0 , we conclude |fˆ(n)| < ε,

.

|n| > n0 .  

1.3.1.5

Weak*-Convergence of Dilates: Construction in h∞ (D)

In this last part, we treat .L∞ -functions. We should not expect uniform convergence here. In fact, the uniform limit of continuous function is continuous. However, a general function in .L∞ (T) is not necessarily continuous. Despite this observation, a weak type of convergence holds. Theorem 1.3.14 Let .{ κ } be an approximate identity, and let .f ∈ L∞ (T). Then . κ ∗ f ∈ C(T) and, for all .κ,  κ ∗ f L∞ (T)  C f L∞ (T) ,

.

and f L∞ (T)  sup  κ ∗ f L∞ (T) .

.

κ

Moreover,  .

lim κ

2π



2π

ϕ(eit ) ( κ ∗ f )(eit ) dt =

0

ϕ(eit ) f (eit ) dt

0

for all .ϕ ∈ L1 (T), and thus .

lim  κ ∗ f L∞ (T) = f L∞ (T) . κ

If, furthermore, .{ κ } is a positive approximate identity on .T then, for all .κ,  κ ∗ f L∞ (T)  f L∞ (T) .

.

In technical language, Theorem 1.3.14 says that . κ ∗f converges to f in the weak*topology of .L∞ (T), as the dual space of .L1 (T). Proof By Theorem 1.1.2, . κ ∗ f ∈ C(T) and

28

J. Mashreghi

 κ ∗ f L∞ (T)   κ L1 (T) f L∞ (T)  Cf L∞ (T) .

.

Let .ϕ ∈ L1 (T), and let .ψ(eit ) = ϕ(e−it ). Then, by Fubini’s theorem, 

2π

.

 it

0



2π

ϕ(e )( κ ∗ f )(e ) dt = it

ϕ(e ) 

0 2π

=

κ (e 0



2π

κ (e 

0



0

2π

=



0

2π

i(t−τ )

i(t−τ )

iτ

)f (e ) dτ dt

 )ϕ(e ) dt f (eiτ ) dτ it

 κ (ei(−τ −s) )ψ(eis ) ds f (eiτ ) dτ

0 2π

=



2π

it

( κ ∗ ψ)(e−iτ )f (eiτ ) dτ.

0

According to Theorem 1.3.10, .( κ ∗ ψ)(e−iτ ) converges to .ψ(e−iτ ) in .L1 (T). Hence,  .



2π

κ

2π

ϕ(eit )( κ ∗ f )(eit ) dt = lim

lim

κ

0

 =

( κ ∗ ψ)(e−iτ )f (eiτ ) dτ

0 2π

ψ(e−iτ )f (eiτ ) dτ

0

 =

2π

ϕ(eiτ )f (eiτ ) dτ.

0

Note that since f is bounded, we could change the order of limit and integral. Since



1 .

2π

2π

0

ϕ(e )( κ ∗ f )(e ) dt

 (sup  κ ∗ f L∞ (T) )ϕL1 (T) , κ it

it

we can say





.

0

2π

ϕ(eiτ )f (eiτ ) dτ

 (sup  κ ∗ f L∞ (T) )ϕL1 (T) κ

for all .ϕ ∈ L1 (T). Hence, again by another Riesz representation theorem introducing .L∞ (T) as the dual of .L1 (T), f L∞ (T)  sup  κ ∗ f L∞ (T) .

.

κ

The last conclusion also follows since .C = 1 for positive approximate identities.

 

1 Hardy Spaces

29

Corollary 1.3.15 Let .u ∈ L∞ (T), and let 1 .U (re ) = 2π



2π

iθ

0

1 − r2 u(eit ) dt, 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D.

Then .U ∈ h∞ (D), and Ur L∞ (T)  uL∞ (T) ,

0  r < 1,

.

and .

lim Ur L∞ (T) = uL∞ (T) ,

r→1

and, moreover,  .

2π

lim

r→1−



2π

ϕ(eit )U (reit ) dt =

0

ϕ(eit )u(eit ) dt

0

for all .ϕ ∈ L1 (T). Corollary 1.3.15 shows how to construct an element of .h∞ D). It also says that .Ur converges to u in the weak*-topology of .L∞ (T), as the dual space of .L1 (T). Corollary 1.3.16 Let .f ∈ L∞ (T) and, for .n  0, let  n   |k| 1− .pn (e ) = fˆ(k)eikt , n+1 it

eit ∈ T.

k=−n

Then pn L∞ (T)  f L∞ (T) ,

n  0,

.

and .

lim pn L∞ (T) = f L∞ (T) ,

n→∞

and, moreover,  .

lim

n→∞ 0

2π



2π

ϕ(eit )pn (eit ) dt =

ϕ(eit )f (eit ) dt

0

for all .ϕ ∈ L1 (T). In technical language, Corollary 1.3.16 says that .pn converges to f in the weak*topology of .L∞ (T), as the dual space of .L1 (T).

30

1.3.1.6

J. Mashreghi

L2 -Norm Convergence of Dilates: Construction in h2 (D)

The 2-norm is a special case with many fascination properties. This stems from the fundamental fact that a Hilbert space structure in available this case. To see the effects of having such a structure note that, for the subclass .L2 (T) ⊂ L1 (T), the uniqueness result of Corollary 1.3.8 can be stated as follows. A family .{ ϕκ } in 2 .L (T) is complete if whenever 

2π

.

f (eit )ϕκ (eit ) dt = 0,

for all κ,

0

we can conclude that .f = 0. With this interpretation, the uniqueness theorem says that the sequence .{eint }n∈Z is complete in .L2 (T). Therefore, .L2 (T) equipped with the inner product 1 .f, g = 2π



2π

f (eit )g(eit ) dt

0

is a Hilbert space. In this setting, two functions .f, g ∈ L2 (T) are orthogonal if 2 .f, g = 0. At the same token, a subset .S ⊂ L (T) is an orthonormal set if every element of S is normalized (has norm one) and every two distinct elements are orthogonal to each other. Theorem 1.3.17 (Bessel’s Inequality) Let .f ∈ L2 (T), and let .{ϕκ : κ ∈ I } be an orthonormal family in .L2 (T). Then 

|f, ϕκ |2  f 2L2 (T) .

.

κ∈I

Proof Let g=

.

 f, ϕκ  ϕκ ,

where the sum is over a finite subset of indices .{κ}. Then f, g =

.



f, ϕκ  f, ϕκ  =



|f, ϕκ |2 ,

and similarly, g, g =

.

  |f, ϕκ |2 . f, ϕκ  f, ϕκ  =

Hence f − g22 = f 22 −

.



|f, ϕκ |2 ,

1 Hardy Spaces

31

which gives  .

|f, ϕκ |2  f 2L2 (T) .

Now take the supremum with respect to all such finite sums.

 

If we consider the orthonormal family .{eint }n∈Z , then Bessel’s inequality is written as ∞ 

1 . |fˆ(n)|2  2π n=−∞



2π

|f (eit )|2 dt.

0

However, more is true! Theorem 1.3.18 (Riesz–Fischer, Parseval) Let .(an )n∈Z ∈ 2 (Z). Then there is an 2 .f ∈ L (T) such that fˆ(n) = an

.

for all .n ∈ Z. Moreover, for each .f ∈ L2 (T), ∞  .

|fˆ(n)|2 =

n=−∞

1 2π



2π

|f (eit )|2 dt.

0

Proof Let fn =

n 

.

ak eikt .

k=−n

Hence fˆn (k) = fn , eikt  =

.

⎧ ⎨ ak if n  |k|, ⎩

0 if n < |k|.

Then 2 .fm − fn 2

=

m 

|ak |2 ,

m > n,

|k|=n+1

which shows that .(fn )n1 is a Cauchy sequence in .L2 (T). The heart of proof is here: .L2 (T) is complete. Hence the sequence .(fn )n1 is convergent to and element f in .L2 (T). Note that

32

J. Mashreghi

fˆ(k) = f, eikt  = lim fn , eikt  = ak ,

.

n→∞

k ∈ Z.

Another consequence of .limn→∞ fn − f 2 = 0 is that f 22 = lim fn 22 =

.

n→∞

∞ 

|ak |2 =

k=−∞

∞ 

|fˆ(k)|2 .

k=−∞

  Parseval’s identity can be rewritten as f L2 (T) = fˆ2 (Z) .

.

Hence the Fourier transform .

L2 (T) −→ 2 (Z) f −→ fˆ

is bijective and shows that the Hilbert spaces .L2 (T) and .2 (Z) are isomorphically isometric.

1.3.2 Analysis: Poisson Representations At the heart of representation theorems rests the Poisson representations. In this part, we exclusively study these representation in several harmonic and analytic Hardy spaces. The basic goal is to show that all constructions in the preceding part for different Hardy spaces are unique. More explicitly, every element of the space is obtained uniquely by those constructions.

1.3.2.1

Poisson Representation in h(D)

Let h(D) = {U : ∇ 2 U = 0 on |z| < R, for some R > 1}.

.

The constant R depends on U , but this is not important. To establish the uniqueness of representations, we start with .h(D), which is the smallest subclass in our discussion. Then we develop further and obtain more uniqueness results.

1 Hardy Spaces

33

Theorem 1.3.19 Let .U ∈ h(D). Then U (reiθ ) =

.

1 2π



2π

0

1 − r2 U (eit ) dt, 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D.

(1.19)

Proof By definition, there exists an .R > 1 such that U is harmonic on .{|z| < R}. Moreover, ∞ 

U (reiθ ) =

an r |n| einθ ,

.

reiθ ∈ DR ,

n=−∞

where 1 .an = 2π



2π

U (eit )e−int dt,

n ∈ Z.

0

Plugging back the formula for .an in the summation formula for U gives   2π ∞   1 it −int U (e )e dt r |n| einθ .U (re ) = 2π 0 n=−∞ iθ

=

1 2π



2π 0

  ∞

 r |n| ein(θ−t) U (eit ) dt.

n=−∞

Recall that ∞ 

r |n| ein(θ−t) =

.

n=−∞

1 − r2 . 1 + r 2 − 2r cos(θ − t)  

Hence, the result immediately follows. Poisson Representation in h∞ (D)

1.3.2.2

Since .h(D) ⊂ h∞ (D), the following result is the first generalization of Theorem 1.3.19. For the following result, we provide a detailed proof to show the main ingredients in such reasonings. For subsequent results, a sketch of proof is provided. Theorem 1.3.20 (Fatou) Let .U ∈ h∞ (D). Then there exists a unique .u ∈ L∞ (T) such that U (reiθ ) =

.

1 2π

 0

2π

1 − r2 u(eit ) dt, 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D,

34

J. Mashreghi

and U h∞ (D) = uL∞ (T) .

.

Proof The uniqueness of u follows from Corollary 1.3.7. The representation is a consequence of Theorem 1.3.19 and the following two facts from measure theory: (i) .L∞ (T) is the dual of .L1 (T); (ii) .L1 (T) is separable. Let Un (z) = U

 

.

 1 1− z , n

(n  2).

Then .Un is defined on the disc .{|z| < Rn }, where .Rn = n/(n − 1) > 1. In other words, .Un ∈ h(D). Hence, by Theorem 1.3.23, for each .z ∈ D,    2π  1 1 1 − r2 1− z = Un (eit ) dt. .U n 2π 0 1 + r 2 − 2r cos(θ − t)

(1.20)

We let .n → ∞. The left hand side tends to .U (reit ). Our goal is to show that the limit of the right hand side has an integral representation. Consider the linear functional .n : L1 (T) −→ C defined by n (f ) =

.

1 2π



2π

f (eit )Un (eit ) dt,

n  2.

0

The estimation |n (f )|  Un L∞ (T) f L1 (T)  U L∞ (T) f L1 (T)

.

(1.21)

shows that each .n is a bounded linear functional on .L1 (T) with the uniform norm restriction n   U L∞ (T) .

.

Now we need the separability of .L1 (T). Let .(fn )n1 be a countable dense subset of .L1 (T). By (1.21), there is a subsequence of .{n1j }j 1 such that .

lim n1j (f1 )

j →∞

exists. Again by (1.21), there is a second subsequence of .{n2j }j 1 , of .{n1j }j 1 , such that

1 Hardy Spaces

35

lim n2j (f2 )

.

j →∞

exists. We continue this process. The outcome is, for any .i  1, a subsequence {nij }j 1 of .{n(i−1)j }j 1 such that .limj →∞ nij (fi ) exists. At this point the celebrated idea of Cantor applies and we consider the diagonal subsequence .{nkk }k1 . Since .{nkk }k1 is eventually a subsequence of .{nij }j 1 , the limit .

.

lim nkk (fi )

k→∞

exists for all .i  1. Since .(fi )n1 is dense in .L1 (T), we can go further and show that the last limit actually exists for all .f ∈ L1 (T). Fix .f ∈ L1 (T) and .ε > 0. Hence, there is an element .fi such that .f − fi 1 < ε. By (1.21), we see that |nkk (f ) − nll (f )|  |nkk (f ) − nkk (fi )|

.

+ |nkk (fi ) − nll (fi )| + |nll (fi ) − nll (f )|  2U L∞ (T) f − fi L1 (T) + |nkk (fi ) − nll (fi )|  2U L∞ (T) ε + |nkk (fi ) − nll (fi )|. Pick .k, l large enough so that |nkk (f ) − nll (f )|  (2U L∞ (T) + 1)ε.

.

Therefore, (f ) = lim nkk (f )

.

k→∞

exists for all .f ∈ L1 (T) and |(f )|  U L∞ (T) f L1 (T) .

.

In other words, . is a bounded linear functional on .L1 (T) with   U L∞ (T)

.

Finally, for a fixed .z = reiθ , fz (eit ) = Pr (ei(θ−t) ) =

.

1 + r2

1 − r2 , − 2r cos(θ − t)

(1.22)

36

J. Mashreghi

as a function of t, is in .L1 (T). Hence, by (1.20), we have (fz ) = lim nkk (fz )

.

k→∞

 2π 1 Pr (ei(θ−t) )Unkk (eit ) dt k→∞ 2π 0    1  iθ = U (reiθ ). = lim U 1 − re k→∞ nkk = lim

On the other hand, by Riesz’s theorem for bounded linear functionals on .L1 (T), there is a .u ∈ L∞ (T) such that   U L∞ (T)

(1.23)

.

and 1 .(f ) = 2π



2π

f (eit )u(eit ) dt,

f ∈ L1 (T).

0

Put .f = fz to deduce U (reiθ ) =

.

1 2π



2π

Pr (ei(θ−t) )u(eit ) dt.

0

Moreover, by Corollary 1.3.15, and (1.22) and (1.23), we conclude that .U h∞ (D) = uL∞ (T) .  

1.3.2.3

Poisson Representation in hp (D), (1 < p < ∞)

It is well known that (i) .Lp (T) is the dual of .Lq (T), where .1/p + 1/q = 1; (ii) .Lp (T) is a separable. Hence, there is no wonder that the proof of Theorem 1.3.20 can be slightly modified to obtain the following result. Note that .h(D) ⊂ h∞ (D) ⊂ hp (D). Theorem 1.3.21 Let .U ∈ hp (D), .1 < p < ∞. Then there exists a unique .u ∈ Lp (T) such that 1 .U (re ) = 2π



iθ

and

0

2π

1 − r2 u(eit ) dt, 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D,

1 Hardy Spaces

37

U hp (D) = uLp (T) .

.

To emphasize on the importance of case .p = 2, if .U ∈ h2 (D) then there is a unique .u ∈ L2 (T) such that 1 .U (re ) = 2π



∞  1 − r2 it |n| inθ u(e ) dt = u(n)r ˆ e 1 + r 2 − 2r cos(θ − t) n=−∞

2π

iθ

0

for all .reiθ ∈ D. Hence, by Parseval’s identity, 1 . 2π



2π

|U (reiθ )|2 dθ =

0

∞ 

2 2|n| |u(n)| ˆ r .

n=−∞

Let .r → 1 to conclude the following. Corollary 1.3.22 Let .U ∈ h2 (D). Then there exists a unique .u ∈ L2 (T) such that U (reiθ ) =

.

1 2π



2π

0

1 + r2

1 − r2 u(eit ) dt, − 2r cos(θ − t)

reiθ ∈ D,

and U h2 (D) = uL2 (T) = u ˆ 2 (Z) =

  ∞

.

1 |uˆ n |2

2

.

n=−∞

1.3.2.4

Poisson Representation in h1 (D)

The case .p = 1 is slightly different. In the previous representation theorems, we used the fact that .Lp (T) is the dual of .Lq (T), whenever .1 < p  ∞. However, this result is not valid for .p = 1. To overcome this shortcoming, we replace .L1 (T) by .M(T). By Riesz’s theorem, .M(T) is the dual of .C(T). Hence, the old technique works. However, the penalty is paid in the limiting case: we obtain a measure, not a function. In analogy with previous results, note that h(D) ⊂ h∞ (D) ⊂ hp (D) ⊂ h1 (D).

.

This is our final step in obtaining Poisson representations. Theorem 1.3.23 Let .U ∈ h1 (D). Then there exists a unique .μ ∈ M(T) such that

38

J. Mashreghi

 U (reiθ ) =

.

T

1 − r2 dμ(eit ), 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D,

and U h1 (D) = μM(T) .

.

If U is a positive harmonic function on .D, then 1 . 2π



2π

0

1 |U (re )| dθ = 2π



iθ

2π

U (reiθ ) dθ = U (0),

0

which means .U ∈ h1 (D). Hence, by Theorem 1.3.23, U has a Poisson integral representation by a measure. However, we can say more about the limiting measure. Here, the measures dμn (eit ) = U ((1 − 1/n)eit ) dt/2π

.

are positive and thus their weak* limit .dμ has to be positive too. Therefore, we deduce the following special case of Theorem 1.3.23. Theorem 1.3.24 (Herglotz) Let U be a positive harmonic function on .D. Then there exists a unique finite positive Borel measure .μ on .T such that  U (re ) =

.

iθ

T

1 − r2 dμ(eit ), 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D.

Herglotz’s theorem provides a straightforward proof of Harnack’s inequality. The main feature of Harnack’s inequality is that the constants appearing as the lower and upper bounds do not depend on the function U . Corollary 1.3.25 (Harnack’s Inequality) Let U be a positive harmonic function on .D. Then, for each .reiθ ∈ D, .

1−r 1+r U (0)  U (reiθ )  U (0). 1+r 1−r

Proof By Theorem 1.3.23, U is the Poisson integral of a positive Borel measure .μ. Since .

1+r 1−r 1 − r2   , 1+r 1−r 1 + r 2 − 2r cos(θ − t)

and since .dμ  0, we have

1 Hardy Spaces

.

1−r 1+r

39



 T

dμ(eit ) 

T

1 − r2 1+r dμ(eit )  2 1−r 1 + r − 2r cos(θ − t)

 T

dμ(eit ).

Once more, by Theorem 1.3.23,  .

T

dμ(eit ) = U (0).  

1.3.2.5

Poisson Representation in H p (D), 1 < p  ∞

Since .H p (D) ⊂ hp (D), the previous Poisson representations are valid for analytic spaces too. As a matter of fact, thanks to the analyticity of functions, we can say more in this case. If .f ∈ H p (D), .1 < p  ∞, then there is an .f ∗ ∈ Lp (T) such that ∞ 

f (z) =

.

f ∗ (n)r |n| einθ

n=−∞

=

∞ 

−1 

f ∗ (n)zn +

f ∗ (n)¯z−n ,

z = reiθ ∈ D.

n=−∞

n=0

But, thanks to the interplay of harmonic analysis and complex analysis, the Fourier coefficients of .f ∗ can also be computed by 

r −|n| f ∗ (n) = 2π

2π

.

f (reit )e−int dt,

n ∈ Z, 0 < r < 1.

0

Hence, r −|n| f ∗ (n) = 2π i



.

|z|=r

f (z)z|n+1| dz,

0 < r < 1.

which implies f ∗ (−1) = f ∗ (−2) = · · · = 0,

.

i.e., .f ∈ H p (T). On the other hand, if .f ∈ H p (T) is given, and we define f (z) =

.

1 2π

 0

2π

1 + r2

1 − r2 f (eit ) dt, − 2r cos(θ − t)

z = reiθ ∈ D,

40

J. Mashreghi

then certainly .f ∈ hp (D). However, ∞ 

f (z) =

f ∗ (n)r |n| einθ =

.

n=−∞

∞ 

f ∗ (n)zn

n=0

and thus it represents an analytic function in .H p (D). These observations are gathered in the following Theorem. Theorem 1.3.26 Let f be analytic on the open unit disc. Then .f ∈ H p (D), .1 < p < ∞, if and only if there exists .f ∗ ∈ H p (T) such that f (reiθ ) =

.

1 2π



2π

0

1 − r2 f ∗ (eit ) dt, 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D.

The function .f ∗ is unique and f (z) =

∞ 

.

f ∗ (n)zn ,

z ∈ D.

n=0

Moreover, lim fr − f ∗ Lp (T) = 0

.

r→1

and f H p (D) = f ∗ H p (T) .

.

We have .f ∈ H 2 (D) if and only if .f ∗ ∈ H 2 (T), and in this case f H 2 (D) = f ∗ H 2 (T) = f ∗ 2 (N) =

 ∞

.

|f ∗ (n)|2

1 2

.

n=0

As we witnessed before, the case .p = ∞ needs special attention. Theorem 1.3.27 Let f be analytic on the open unit disc. Then .f ∈ H ∞ (D) if and only if there exists .f ∗ ∈ H ∞ (T) such that 1 .f (re ) = 2π



iθ

0

2π

1 − r2 f ∗ (eit ) dt, 1 + r 2 − 2r cos(θ − t)

The function .f ∗ is unique and

reiθ ∈ D.

1 Hardy Spaces

41

f (z) =

∞ 

.

f ∗ (n)zn ,

z ∈ D.

n=0

Moreover, .fr converges to .f ∗ in the weak*-topology of .L∞ (T), as .r → 1, i.e.,  .

lim

r→1 0

2π

 ϕ(eit )f (reit ) dt =

2π

ϕ(eit )f (eit ) dt

0

for all .ϕ ∈ L1 (T), and f H ∞ (D) = f ∗ H ∞ (T) .

.

1.3.2.6

Poisson Representations in H 1 (D)

The preceding two theorems show that .H p (D) and .H p (T), .1 < p  ∞, are isometrically isomorphic. If we follow the same line of reasoning for .H 1 (D), we conclude that it is isometrically isomorphic to the subclass of measures {μ ∈ M(T) : μ(−1) ˆ = μ(−2) ˆ = · · · = 0},

.

and, via the Poisson integral representation, every such measure creates a unique element of .H 1 (D), i.e.,  f (z) =

∞

.

T

 1 − r2 it n dμ(e ) = μ(n)z ˆ , 1 + r 2 − 2r cos(θ − t)

z = reiθ ∈ D.

n=0

However, this is not the whole truth. According to a celebrated result of F. and M. Riesz, such measures are necessarily absolutely continuous with respect to the Lebesgue measure. Theorem 1.3.28 Let f be analytic on the unit disc .D. Then .f ∈ H 1 (D) if and only if there exists a unique .f ∗ ∈ H 1 (T) such that f (z) =

.

1 2π

 0

2π

∞

 1 − r2 f ∗ (eit ) dt = f ∗ (n)zn 2 1 + r − 2r cos(θ − t)

(1.24)

n=0

for all .z = reiθ ∈ D. The series is uniformly convergent on compact subsets of .D. Moreover, .

lim fr − f ∗ L1 (T) = 0

r→1

42

J. Mashreghi

and f H 1 (D) = f ∗ H 1 (T) .

.

Using the Cauchy integral formula and the norm convergence of .fr to .f ∗ , we obtain another representation theorem for .H 1 (D) functions. Corollary 1.3.29 Let .f ∈ H 1 (D), with boundary values .f ∗ ∈ H 1 (T). Then 

f ∗ (eit ) dt 1 − e−it z

(1.25)

z¯ eit f ∗ (eit ) dt = 0 z¯ eit − 1

(1.26)

f (z) =

.

1 2π

2π 0

and 

2π

.

0

for all .z ∈ D. Proof Fix .z ∈ D. By the Cauchy integral formula, 1 .f (z) = 2π i

 {|ζ |=r}

1 f (ζ ) dζ = ζ −z 2π



2π

0

reit f (reit ) dt. reit − z

for .r > (1 + |z|)/2. Since

reit 1 2

.

reit − z  r − |z|  1 − |z| and, by Theorem 1.3.28, .fr converges to .f ∗ in the .L1 (T) norm, by letting .r → 1 we have 1 r→1 2π



f (z) = lim

.

2π

reit 1 f (reit ) dt = it re − z 2π

0

 0

2π

eit f ∗ (eit ) dt. eit − z

To prove the second formula, note that .

Then take the limit.

1 2π i

 {|ζ |=r}

z¯ f (ζ ) dζ = 0. z¯ ζ − 1  

1 Hardy Spaces

43

1.4 An Overview of Representation Theorems We gather here a complete list of representation theorems for harmonic or analytic functions on the open unit disc .D. As a general phenomenon, the best and most neat results are for .1 < p < ∞. There are some exceptions for .p = 1 and .p = ∞. There are more differences when one studies .0 < p < 1. There are certain topics that we did not cover, but they appear in the following. The definitions are provided below, but for further details we refer to [11]. (i) Frechét spaces: Depending on p, we saw that the Hardy spaces appear as a Banach algebra (.p = ∞), a Hilbert space (.p = 2), or as a Banach space (.p  1). Whenever .p < 1, the quantity . · p fails to be a norm (the triangle inequality is not satisfied). However, p

dp (f, g) := f − gH p (D)

.

is a complete translation invariant metric on .H p (D). Therefore, we obtain a metrizable topological vector space which is not locally convex. Some authors refer them as a Frechét space, some as an F-space (and keep Frechét space for the locally convex spaces). (ii) Harmonic conjugate: Let u be harmonic on the open unit disc .D. Then there is a unique harmonic function v, with .v(0) = 0, such that .u + iv is analytic on .D. We refer to v as the harmonic conjugate by u. We also write v ∗ (eiθ ) = lim v(reiθ )

.

r→1

whenever the radial limit exists at .eiθ ∈ T. (iii) Hilbert transform: Let .f ∈ L1 (T). The Hilbert transform of f , at the point iθ ∈ T, is .e 1 iθ .f˜(e ) := lim ε→0 π



f (eit ) ε|θ−t|π

2 tan( θ−t 2 )

dt

whenever it exists. As a matter of fact, by a profound result of complex analysis which has been extended in many directions, we know that .f˜(eiθ ) exists for almost all .eiθ ∈ T. the interplay between f and .f˜ in different Lebesgue spaces as well as the connection of Hilbert transform to the conjugate functions are important gems of the theory of Hardy spaces.

44

J. Mashreghi

1.4.1 Harmonic Hardy Spaces Case .p = 1: .u ∈ h1 (D) if and only if there exists a Borel measure .μ ∈ M(T) such that  u(reiθ ) =

.

1 + r2

T

1 − r2 dμ(eit ), − 2r cos(θ − t)

reiθ ∈ D.

The measure .μ is unique and u has the series representation ∞ 

u(re ) = iθ

.

μ(n) ˆ r |n| einθ ,

reiθ ∈ D.

n=−∞

The series is absolutely and uniformly convergent on compact subsets of .D. The Fourier transform .μˆ belongs to .∞ (Z) and μ ˆ ∞ (Z)  μM(T) .

.

The correspondence between .h1 (D) and .M(T) is isometric, i.e., uh1 (D) = μM(T) = |μ|(T),

.

where .|μ| is the total variation of .μ. As a subclass, u is a positive harmonic function on .D if and only if .μ is a positive Borel measure on .T. As .r → 1, the family of measures .dμr (eit ) = u(reit ) dt/2π converges to it .dμ(e ) in the weak*-topology of .M(T), i.e., 

 .

ϕ(e ) dμr (e ) = it

lim

r→1 T

it

T

ϕ(eit ) dμ(eit )

for all .ϕ ∈ C(T). The measure .μ decomposes as dμ(eit ) = ϕ(eit ) dt/2π + dσ (eit ),

.

where .ϕ ∈ L1 (T) and .σ is singular with respect to the Lebesgue measure. Then we have u∗ (eiθ ) = lim u(reiθ ) = ϕ(eiθ )

.

r→1

1 Hardy Spaces

45

for almost all .eiθ ∈ T. Therefore, in this case, we cannot recover u from .u∗ . More explicitly, the Poisson integral of the boundary function gives the harmonic extension 

1 2π

reiθ −→

.

2π

1 + r2

0

1 − r2 u∗ (eit ) dt − 2r cos(θ − t)

which is not entirely equal to .u(reiθ ). The missing part is  re

.

iθ

−→

T

1 − r2 dσ (eit ), 1 + r 2 − 2r cos(θ − t)

which is created by the singular measure .σ is missing. As an extreme, but interesting example, consider the positive harmonic function u(reiθ ) :=

.

1 − r2 , 1 + r 2 − 2r cos(θ )

reiθ ∈ D.

Then u∗ (eiθ ) =

.

⎧ ⎨ 0 if 0 < θ < 2π, ⎩

∞ if

θ = 0.

Therefore, the harmonic extension of .u∗ is identically zero and it sheds no light at all on the initial function u that we started with. The harmonic conjugate of u is given by  v(reiθ ) =

.

T

2r sin(θ − t) dμ(eit ) 1 + r 2 − 2r cos(θ − t)

∞ 

=

−i sgn(n) μ(n) ˆ r |n| einθ ,

reiθ ∈ D.

n=−∞

The series is absolutely and uniformly convergent on compact subsets of .D. For almost all .eiθ ∈ T, the radial limits .v ∗ (eiθ ) exists. It is important to note that .u ∈ h1 (D) is not enough to conclude that .v ∈ h1 (D). As another special subclass of .h1 , if u is generated by an absolutely continuous measure, i.e., u(reiθ ) =

.

1 2π



2π 0

1 + r2

1 − r2 ϕ(eit ) dt, − 2r cos(θ − t)

reiθ ∈ D,

46

J. Mashreghi

with .ϕ ∈ L1 (T), then u has more interesting properties. First, the boundary values exist and u∗ (eiθ ) = ϕ(eiθ ),

.

a.e. on T,

Hence, from now on, we write 1 .u(re ) = 2π



2π

iθ

0

1 − r2 u∗ (eit ) dt, 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D,

with .u∗ ∈ L1 (T). Then the convergence of dilates are in the norm lim ur − u∗ L1 (T) = 0,

.

r→1

and, moreover, uh1 (D) = u∗ L1 (T) .

.

Note that by the Riemann–Lebesgue lemma .u∗ ∈ c0 (Z), and a simple estimation shows u∗ ∞ (Z)  uh1 (D) .

.

In this case, the harmonic conjugate of u is given by v(reiθ ) =

.

=

1 2π



2π 0

∞ 

2r sin(θ − t) u∗ (eit ) dt 1 + r 2 − 2r cos(θ − t)

−i sgn(n) u∗ (n) r |n| einθ ,

reiθ ∈ D,

n=−∞

and, for almost all .eiθ ∈ T, v ∗ (eiθ ) = lim v(reiθ ) = u(e ˜ iθ ).

.

r→1

However, we cannot conclude that .u˜ ∈ L1 (T). By Kolmogorov’s theorem, |{|u| ˜ > λ}| = O(1/λ).

.

In technical terms, .u˜ is in weak-.L1 (T), a space which is larger that .L1 (T). By this result, we can easily show that .v ∈ hp (D), .u˜ ∈ Lp (T), .0 < p < 1, and .

lim vr − u ˜ Lp (T) = 0,

r→1

1 Hardy Spaces

47

and u ˜ Lp (T) = vhp (D)  cp u∗ L1 (T) .

.

However, if we know that u and .u˜ both are in .L1 (T), then .v ∈ h1 (D), lim vr − u ˜ L1 (T) = 0

.

r→1

and the harmonic conjugate is also given by 1 .v(re ) = 2π



2π

iθ

0

1 − r2 u(e ˜ it ) dt, 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D.

Moreover, their Fourier coefficients are related as  .u(n) ˜ = −i sgn(n) u(n), ˆ

n ∈ Z.

Case .1 < p < ∞: .u ∈ hp (D), .1 < p < ∞, if and only if there exists .ϕ ∈ Lp (T) such that u(reiθ ) =

.

1 2π



2π

1 + r2

0

1 − r2 ϕ(eit ) dt, − 2r cos(θ − t)

reiθ ∈ D.

For almost all .eiθ ∈ T, u∗ (eiθ ) := lim u(reiθ ) = ϕ(eiθ ).

.

r→1

The boundary function .u∗ is unique and ∞ 

u(reiθ ) =

.

u∗ (n) r |n| einθ ,

reiθ ∈ D.

n=−∞

The series is absolutely and uniformly convergent on compact subsets of .D. We have the norm convergence .

lim ur − u∗ Lp (T) = 0

r→1

and thus uhp (D) = u∗ Lp (T) .

.

48

J. Mashreghi

If .1 < p  2, then .u∗ ∈ q (Z), where .1/p + 1/q = 1 and u∗ q (Z)  uhp (D) .

.

No wonder the special case .p = 2 leads to more interesting results; .u ∈ h2 (D) if and only if .u∗ ∈ L2 (T), and uh2 (D) = u∗ L2 (T) = u∗ 2 (Z) .

.

However, if .u(0) = u∗ (0) = 0, then u∗ L2 (T) = u∗ L2 (T) .

.

This is because of ∗ (n) = −i sgn(n) u∗ (n), u

.

n ∈ Z,

which holds for all .1 < p < ∞. The harmonic conjugate of u is given by v(reiθ ) =

.

=

1 2π 1 2π

 

2r sin(θ − t) u∗ (eit ) dt 1 + r 2 − 2r cos(θ − t)

2π

1 − r2 u∗ (eit ) dt 1 + r 2 − 2r cos(θ − t)

0

0

∞ 

=

2π

−i sgn(n) u∗ (n) r |n| einθ ,

reiθ ∈ D.

n=−∞

The series is absolutely and uniformly convergent on compact subsets of .D and, for almost all .eiθ ∈ T, v ∗ (eiθ ) = lim v(reiθ ) = u∗ (eiθ ).

.

r→1

Moreover, .v ∈ hp (D), .v ∗ = u∗ ∈ Lp (T), .

lim vr − u∗ Lp (T) = 0

r→1

and vhp (D) = v ∗ Lp (T) = u∗ Lp (T)  cp uhp (D) = cp u∗ Lp (T) .

.

1 Hardy Spaces

49

Case .p = ∞: .u ∈ h∞ (D) if and only if there exists .ϕ ∈ L∞ (T) such that u(reiθ ) =

.



1 2π

2π

0

1 − r2 ϕ(eit ) dt, 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D.

For almost all .eiθ ∈ T, u∗ (eiθ ) = lim u(reiθ ) = ϕ(eiθ ).

.

r→1

The boundary function .u∗ is unique and ∞ 

u(reiθ ) =

u∗ (n) r |n| einθ ,

.

reiθ ∈ D.

n=−∞

The series is absolutely and uniformly convergent on compact subsets of .D. As .r → 1, .ur converges to .u∗ in the weak* topology of .L∞ (T), i.e.,  .



2π

g(e ) u(re ) dt = it

lim

r→1 0

it

2π

g(eit ) u∗ (eit ) dt

0

for all .g ∈ L1 (T). Hence, uh∞ (D) = u∗ L∞ (T) .

.

But, as a proper subclass of .h∞ (D), we have .

lim ur − u∗ L∞ (T) = 0

r→1

if and only if .u∗ ∈ C(T). The harmonic conjugate of u is given by v(reiθ ) =

.

1 2π

1 = 2π =

 

∞ 

2π 0

2r sin(θ − t) u∗ (eit ) dt 1 + r 2 − 2r cos(θ − t)

2π 0

1 + r2

1 − r2 u∗ (eit ) dt − 2r cos(θ − t)

−i sgn(n) u∗ (n) r |n| einθ ,

reiθ ∈ D.

n=−∞

The series is absolutely and uniformly convergent on compact subsets of .D. For almost all .eiθ ∈ T,

50

J. Mashreghi

v ∗ (eiθ ) = lim v(reiθ ) = u∗ (eiθ ),

.

r→1

and ∗ (n) = −i sgn(n) u∗ (n), v∗ (n) = u

.

(n ∈ Z).

It is important to note that .u ∈ h∞ (D) does not imply that .v ∈ h∞ (D). At the same token, .u∗ ∈ L∞ (T) does not imply that .u∗ ∈ L∞ (T). However, for real-valued .u∗ , .



1 2π

2π

∗ (eit )|

eλ |u

dt  2 sec( λ u∗ L∞ (T) ),

0

for .0  λ < π/2 u∗ L∞ (T) . The complete picture is provided by Spanne’s theorem: u∗ ∈ L∞ (T)

.

⇒

v ∗ = u∗ ∈ BMO.

1.4.2 Analytic Hardy Spaces In the following, f be analytic on the open unit disc .D.

Case .1  p < ∞: p ∗ p .f ∈ H (D), .1  p < ∞, if and only if there exists .f ∈ H (T) such that f (reiθ ) =

.

1 2π



2π

0

1 − r2 f ∗ (eit ) dt, 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D.

for almost all .eiθ ∈ T, .

lim f (reiθ ) = f ∗ (eiθ ).

r→1

Thus .f ∗ is unique and we also have the representations f (z) =

.

= =

1 2π i 2π 1 2π

  

2π

f ∗ (eit ) dt 1 − e−it z

2π

2r sin(θ − t) f ∗ (eit ) dt + f (0) 1 + r 2 − 2r cos(θ − t)

2π

eit + z Re f ∗ (eit ) dt + iImf (0) eit − z

0

0

0

1 Hardy Spaces

51

= =

i 2π ∞ 



eit + z ∗ it Re f (e ) dt + f (0) eit − z

2π 0

f ∗ (n) zn ,

z ∈ D.

n=0

The series is absolutely and uniformly convergent on compact subsets of .D. We have the norm convergence .

lim fr − f ∗ Lp (T) = 0.

r→1

Hence, f H p (D) = f ∗ Lp (T) .

.

If .f (0) = f ∗ (0) = 0, then f∗ = −if ∗

.

or equivalently,  Re f ∗ = Imf ∗

.

and

∗ = − Re f ∗ . Imf

If we write .f = u+iv (on .D) or .f ∗ = u∗ +iv ∗ (on .T), the above equations become u∗ = v ∗

.

and

v∗ = −u∗ .

If .1  p  2, then .f ∗ ∈ q (Z+ ), where .1/p + 1/q = 1 and f ∗ q (Z+ )  f ∗ Lp (T) = f H p (D) .

.

Once more we get better results when .p = 2: .f ∈ H 2 (D) if and only if .f ∗ ∈ If so,

H 2 (T).

f H 2 (D) = f ∗ L2 (T) = f ∗ 2 (Z+ )

.

(Parseval Identity)

and, moreover, f ∗ L2 (T) = f∗ L2 (T)

.

provided that .f (0) = f∗ (0) = 0.

(Hilbert Identity)

52

J. Mashreghi

Case .p = ∞: ∞ (D) if and only if there exists .f ∗ ∈ H ∞ (T) such that .f ∈ H 1 .f (re ) = 2π



2π

iθ

0

1 − r2 f ∗ (eit ) dt, 1 + r 2 − 2r cos(θ − t)

reiθ ∈ D.

For almost all .eiθ ∈ T, .

lim f (reiθ ) = f ∗ (eiθ ).

r→1

Thus, .f ∗ is unique and f (z) =

.

= = = =

1 2π i 2π 1 2π i 2π ∞ 

   

2π

f ∗ (eit ) dt 1 − e−it z

2π

2r sin(θ − t) f ∗ (eit ) dt + f (0) 1 + r 2 − 2r cos(θ − t)

2π

eit + z Re f ∗ (eit ) dt + iImf (0) eit − z

2π

eit + z ∗ it Re f (e ) dt + f (0) eit − z

0

0

0

0

f ∗ (n) zn ,

z ∈ D.

n=0

The series is absolutely and uniformly convergent on compact subsets of .D. If .f ∗ (0) = 0, then .f∗ = −if ∗ or equivalently,  Re f ∗ = Imf ∗

and

.

∗ = − Re f ∗ . Imf

If we write .f = u+iv (on .D) or .f ∗ = u∗ +iv ∗ (on .T), the above equations become .

u∗ = v ∗

v∗ = −u∗ .

and

As .r → 1, .fr converges to .f ∗ in the weak* topology of .L∞ (T), i.e.,  .

lim

r→1 0

2π

 g(e ) f (re ) dt = it

it

2π

g(eit ) f ∗ (eit ) dt

0

for all .g ∈ L1 (T). Hence, f H ∞ (D) = f ∗ L∞ (T) .

.

1 Hardy Spaces

53

However, we have the stronger convergence .

lim fr − f ∗ L∞ (T) = 0

r→1

if and only if .f ∗ ∈ A(T) = C(T) ∩ H 1 (T). This subclass of .H ∞ is called the disc algebra.

References 1. CHENG, R., MASHREGHI, J., AND ROSS, W. T. Function theory and p spaces, vol. 75 of University Lecture Series. American Mathematical Society, Providence, RI, [2020] ©2020. 2. DUREN, P. L. Theory of H p spaces. Pure and Applied Mathematics, Vol. 38. Academic Press, New York-London, 1970. 3. EL-FALLAH, O., KELLAY, K., MASHREGHI, J., AND RANSFORD, T. A primer on the Dirichlet space, vol. 203 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2014. 4. FATOU, P. Séries trigonométriques et séries de Taylor. Acta Math. 30, 1 (1906), 335–400. 5. GARNETT, J. B. Bounded analytic functions, first ed., vol. 236 of Graduate Texts in Mathematics. Springer, New York, 2007. 6. HARDY, G. H. The mean value of the modulus of an analytic function. Proc. London Math. Soc. 14 (1915), 269–277. 7. KOOSIS, P. Introduction to Hp spaces, second ed., vol. 115 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. 8. LITTLEWOOD, J. E., AND PALEY, R. E. A. C. Theorems on Fourier Series and Power Series. J. London Math. Soc. 6, 3 (1931), 230–233. 9. MARTÍNEZ-AVENDAÑO, R. A., AND ROSENTHAL, P. An introduction to operators on the Hardy-Hilbert space, vol. 237 of Graduate Texts in Mathematics. Springer, New York, 2007. 10. MASHREGHI, J. The rate of increase of mean values of functions in Hardy spaces. J. Aust. Math. Soc. 86, 2 (2009), 199–204. 11. MASHREGHI, J. Representation theorems in Hardy spaces, vol. 74 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2009. 12. MASHREGHI, J., AND RANSFORD, T. Outer functions and uniform integrability. Ann. Acad. Sci. Fenn. Math. 43, 2 (2018), 1023–1026.

Chapter 2

The Dirichlet Space Thomas Ransford

2020 Mathematics Subject Classification 30C15, 30C85, 32A40

2.1 Introduction 2.1.1 What is the Dirichlet Space? The Dirichlet space .D is the set of functions f holomorphic in the unit disk .D whose Dirichlet integral is finite: D(f ) :=

.

1 π

 D

|f  (z)|2 dA(z) < ∞.

  Note that, if .f (z) = k≥0 ak zk , then .D(f ) = k≥0 k|ak |2 . Consequently .D ⊂ H 2 . It follows that .D is a Hilbert space with respect to the norm . · D given by f 2D := f 2H 2 + D(f ) =



.

(k + 1)|ak |2 .

k≥0

2.1.2 History and Motivation Very very brief history of .D:

Supported by grants from NSERC and the Canada Research Chairs Program. T. Ransford () Université Laval, Département de mathématiques et de statistique, Pav. Vachon, Cité Universitaire, Québec, QC, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mashreghi (ed.), Lectures on Analytic Function Spaces and their Applications, Fields Institute Monographs 39, https://doi.org/10.1007/978-3-031-33572-3_2

55

56

T. Ransford

• Beurling (1930’s–1940’s) • Carleson (1950’s–1960’s) • ... Some reasons for studying .D: • • • •

Potential theory, energy, capacity Geometric interpretation, Möbius invariance Weighted shifts, invariant subspaces Borderline case, still many open problems

2.1.3 What to Study? Some topics of interest: • • • • • • •

Boundary behavior Zeros Multipliers Reproducing kernel Interpolation Conformal invariance Shift-invariant subspaces

2.1.4 Where to Find Out More About D? I recommend the survey articles of Ross [45] and of Arcozzi–Rochberg–Sawyer– Wick [6], as well as the monographs of Ef-Fallah–Kellay–Mashreghi–Ransford [24] and of Arcozzi–Rochberg–Sawyer–Wick [7]. These lectures will mostly follow [24], where nearly all the detailed proofs of pre-2014 results may be found.

2.2 Capacity 2.2.1 Energy Let .μ be a finite positive Borel measure on the unit circle .T. The energy of .μ is defined by   I (μ) :=

.

T T

log

2 dμ(λ) dμ(ζ ). |λ − ζ |

2 The Dirichlet Space

57

Note that we may have .I (μ) = +∞. There is a formula for .I (μ) in terms of Fourier coefficients of .μ: I (μ) =

.

 | μ(k)|2 k≥1

k

+ μ(T)2 log 2.

2.2.2 Capacity of Compact Sets The capacity of a compact set .F ⊂ T is defined by c(F ) := 1/ inf{I (μ) : μ is a probability measure on F }.

.

(2.1)

Elementary properties: • .F1 ⊂ F2 ⇒ c(F1 ) ≤ c(F2 ); • .Fn ↓ F ⇒ c(Fn ) ↓ c(F ); • .c(F1 ∪ F2 ) ≤ c(F1 ) + c(F2 ). Examples: • • • •

c(F ) ≤ 1/ log(2/ diam(F )); c(F ) = 0 if F is finite or countable; .c(F ) ≥ 1/ log(2π e/|F |). In particular .c(F ) = 0 ⇒ |F | = 0; .c(F ) > 0 if F is the (circular) middle-third Cantor set. . .

2.2.3 Capacity of General Sets The inner capacity of .E ⊂ T is defined by c(E) := sup{c(F ) : compact F ⊂ E}.

.

The outer capacity of .E ⊂ T is defined by c∗ (E) := inf{c(U ) : open U ⊃ E}.

.

Properties of .c∗ :  • .c∗ (∪n En ) ≤ n c∗ (En ) (not true for .c(·)). • .c∗ (E) = c(E) if E is Borel (Choquet’s capacitability theorem). A property is said to hold quasi-everywhere (q.e.) if it holds outside an E with c∗ (E) = 0.

.

58

T. Ransford

2.2.4 Equilibrium Measures Let F be a compact subset of .T. A measure .μ attaining the infimum in (2.1) is called an equilibrium measure for F . Proposition 2.2.1 If .c(F ) > 0, then F admits a unique equilibrium measure. The following result is sometimes called the fundamental theorem of potential theory. Theorem 2.2.2 (Frostman) Let .μ be the equilibrium measure for F , and let .Vμ be its potential, namely  Vμ (z) :=

.

T

log

2 dμ(ζ ). |z − ζ |

Then .Vμ ≤ 1/c(F ) on .T, and .Vμ = 1/c(F ) q.e. on F .

2.3 Boundary Behavior 2.3.1 Preliminary Remarks Every .f ∈ D has non-tangential limits a.e. on .T (as .f ∈ H 2 ). However, there exists f ∈ D such that .limr→1− |f (r)| = ∞. For example, consider

.

f (z) :=



.

k≥2

zk . k log k

Then D(f ) =



.

k

k≥2

 1 1 < ∞, = 2 k(log k)2 (k log k) k≥2

but .

lim inf f (r) ≥ r→1−

 k≥2

1 = ∞. k log k

2.3.2 Beurling’s Theorem Theorem 2.3.1 (Beurling [9]) If f ∈ D then f has non-tangential limits q.e. on T.

2 The Dirichlet Space

59

Beurling actually proved his result just for radial limits. Beurling’s theorem is sharp in the following sense: Theorem 2.3.2 (Carleson [16]) Given compact E ⊂ T of capacity zero, there exists f ∈ D such that limr→1− |f (rζ )| = ∞ for all ζ ∈ E.

2.3.3 Capacitary Weak-Type and Strong-Type Inequalities For .f ∈ Hol(D) and .ζ ∈ T, we write .f ∗ (ζ ) := limr→1− f (rζ ), whenever this limit exists. Theorem 2.3.3 (Weak-type inequality (Beurling [9])) There exists an absolute constant A such that, for all .f ∈ D, c(|f ∗ | > t) ≤ Af 2D /t 2

.

(t > 0).

Corollary 2.3.4 There exist absolute constants .A, B > 0 such that, for all .f ∈ D, |{|f ∗ | > t}| ≤ Ae

.

−Bt 2 /f 2

D

(t > 0).

Theorem 2.3.5 (Strong-type inequality (Hansson [30])) There exists an absolute constant A such that, for all .f ∈ D, 

∞

.

0

c(|f ∗ | > t) t dt ≤ Af 2D .

2.3.4 Douglas’ Formula Theorem 2.3.6 (Douglas [22]) If f ∈ H 2 , then D(f ) =

.

1 4π 2

   ∗  ∗  f (λ) − f (ζ ) 2   |dλ| |dζ |. λ−ζ T T

Corollary 2.3.7 If f ∈ D, then f has oricyclic limits a.e. in T (Fig. 2.1).

60

T. Ransford

Fig. 2.1 Non-tangential approach region (left) and oricyclic approach region (right)

2.3.5 Exponential Approach Region Theorem 2.3.8 (Nagel–Rudin–Shapiro [40]) If f ∈ D then, for a.e. ζ ∈ D, it holds that f (z) → f ∗ (ζ ) as z → ζ in the exponential approach region  |z − ζ | < κ log

.

1 −1 . 1 − |z|

• Approach region is ‘widest possible’. • This is an a.e. result (not q.e.).

2.3.6 Carleson’s Formula Let .f ∈ H 2 with canonical factorization .f = BSO. Let .(an ) be the zeros of B, and .σ be the singular measure of S. Theorem 2.3.9 (Carleson [17])   D(f ) =

.

(|f ∗ (λ)|2 − |f ∗ (ζ )|2 )(log |f ∗ (λ)| − log |f ∗ (ζ )|) |dλ| |dζ | 2π 2π |λ − ζ |2 T T     2 2 |dζ | 1 − |an | . + + dσ (λ) |f ∗ (ζ )|2 2 2 2π T |λ − ζ | T n |ζ − an |

Corollary 2.3.10 If f belongs to .D, then so does its outer factor. Corollary 2.3.11 The only inner functions in .D are finite Blaschke products.

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2.3.7 Some Further Developments • Chang–Marshall theorem [21]: .



sup exp(|f ∗ (eiθ )|2 ) dθ : f (0) = 0, D(f ) ≤ 1 < ∞. T

• Trade-off between approach regions and exceptional sets: Borichev [13], Twomey [53].

2.4 Zeros 2.4.1 Preliminary Remarks A sequence .(zn ) in .D (possibly with repetitions) is: • a zero set for .D if there exists .f ∈ D vanishing on .(zn ) but .f ≡ 0; • a uniqueness set for .D if it is not a zero set. Proposition 2.4.1 If .(zn ) is a zero set for .D, then there is an .f ∈ D vanishing precisely on .(zn ). It is well known that .(zn ) is a zero set for the Hardy space .H 2 iff .

 (1 − |zn |) < ∞. n

What about the Dirichlet space?

2.4.2 The Three Cases Proposition 2.4.2 If (because D ⊂ H 2 ).



n (1

− |zn |) = ∞, then (zn ) is a uniqueness set for D

Theorem 2.4.3 (Shapiro–Shields [47]) If a zero set for D.



n 1/| log(1 − |zn |)|

< ∞, then (zn ) is

Theorem 2.4.4 (Nagel–Rudin–Shapiro [40]) If (zn ) satisfies neither condition, then there exist a zero set (zn ) and a uniqueness set (zn ) with |zn | = |zn | = |zn | for all n. Thus, in this last case, the arguments of (zn ) matter. Back to this later.

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2.4.3 Boundary Zero Sets Let E be a closed subset of .T. It is called a Carleson set if    2 |dζ | < ∞. . log dist(ζ, E) T Theorem 2.4.5 (Carleson [16]) If E is a Carleson set, then there is .f ∈ A1 (D) with .f −1 (0) = E. Theorem 2.4.6 (Carleson [16], Brown–Cohn [14]) If .c(E) = 0, then there exists f ∈ D ∩ A(D) with .f −1 (0) = E.

.

• Neither result implies the other. • Clearly, if .|E| > 0, then E is a boundary uniqueness set for .D. But there also exist closed uniqueness sets E with .|E| = 0.

2.4.4 Arguments of Zero Sets We return to zero sets within .D, now considering their arguments. Theorem 2.4.7 (Caughran [19]) Let .(eiθn ) be a sequence in .T. The following are equivalent:  • .(rn eiθn ) is a zero set for .D whenever . n (1 − rn ) < ∞. • .E := {eiθn : n ≥ 1} is a Carleson set. Example of a Blaschke sequence that is a uniqueness set for .D:  zn := 1 −

.

 1 ei/ log n . n(log n)2

There is still no satisfactory complete characterization of zero sets.

2.4.5 Some Further Developments • Carleson sets as zero sets for A∞ (D): Taylor–Williams [50].

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2.5 Multipliers 2.5.1 Preliminary Remarks Proposition 2.5.1 D is not an algebra. Proof Suppose D is an algebra. By closed graph theorem, D isomorphic to a Banach algebra. For each z ∈ D, the map f → f (z) is a character, so |f (z)| ≤ spectral radius of f . Therefore every f ∈ D is bounded. Contradiction.  

2.5.2 Multipliers A multiplier for .D is a function .ϕ such that .ϕf ∈ D for all .f ∈ D. The set of multipliers is an algebra, denoted by .M(D). It is easy to see that, in the case of the Hardy space, .M(H 2 ) = H ∞ . When is .ϕ a multiplier of .D? • Necessary condition: .ϕ ∈ D ∩ H ∞ . • Sufficient condition: .ϕ  ∈ H ∞ . To completely characterize multipliers, we introduce a new notion.

2.5.3 Carleson Measures A measure .μ on .D is a Carleson measure for .D if .∃C such that  . |f |2 dμ ≤ Cf 2D (f ∈ D). D

With this notion in hand, it is quite easy to characterize multipliers: Proposition 2.5.2 .ϕ ∈ M(D) iff both .ϕ ∈ H ∞ and .|ϕ  |2 dA is a Carleson measure for .D. This begs a new question: how to characterize Carleson measures?

2.5.4 Characterization of Carleson Measures Let .μ be a finite positive measure on .D.

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Fig. 2.2 Carleson box

For an arc .I ⊂ T, define a ‘Carleson box’ (Fig. 2.2) S(I ) := {reiθ : 1 − |I | < r < 1, eiθ ∈ I }.

.

Carleson [18]: .μ is a Carleson measure for .H 2 iff .μ(S(I )) = O(|I |). When is .μ a Carleson measure for .D? Theorem 2.5.3 (Wynn [54]) The condition .μ(S(I )) = O(ψ(|I |)) is: • necessary if .ψ(x) := 1/ log(1/x); • sufficient if .ψ(x) := 1/ log(1/x)(log log(1/x))α with .α > 1. Theorem 2.5.4 (Stegenga [49]) .μ is a Carleson measure for .D iff there is a constant A such that, for every finite set of disjoint closed subarcs .I1 , . . . , In of .T,     μ ∪nj=1 S(Ij ) ≤ Ac ∪nj=1 Ij .

.

2.5.5 Multipliers and Reproducing Kernels If .f ∈ D and .w ∈ D, then .f (w) = f, kw D , where kw (z) :=

.

 1  1 log wz 1 − wz

(w, z ∈ D).

(2.2)

The function .kw is called the reproducing kernel for w. Proposition 2.5.5 Let .ϕ ∈ M(D) and define .Mϕ : D → D by .Mϕ (f ) := ϕf . Then Mϕ∗ (kw ) = ϕ(w)kw

.

(w ∈ D).

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Proof For all .f ∈ D, we have f, Mϕ∗ (kw )D = ϕf, kw D = ϕ(w)f (w)

.

= ϕ(w)f, kw D = f, ϕ(w)kw D .  

2.5.6 Pick Interpolation Given .z1 , . . . , zn ∈ D and .w1 , . . . , wn ∈ D, is there a .ϕ ∈ M(D) with .Mϕ  ≤ 1 such that .ϕ(zj ) = wj for all j ? Theorem 2.5.6 (Agler [1]) A solution .ϕ exists iff the matrix .(1 − w i wj )kzi , kzj D is positive semi-definite. • Necessity is a simple consequence of the preceding proposition. The same argument works for any reproducing kernel Hilbert space. • Sufficiency is a property of the Dirichlet kernel (2.2) (‘Pick property’).

2.5.7 Interpolating Sequences A sequence .(zn )n≥1 in .D is interpolating for .M(D) if .

(ϕ(z1 ), ϕ(z2 ), ϕ(z3 ), . . . ) : ϕ ∈ M(D) = ∞ .

Theorem 2.5.7 (Marshall–Sundberg [36], Bishop [11], Bøe [12]) Let .(zn )n≥1 be a sequence in .D. The following are equivalent: • .(z n )n≥1 is an interpolating sequence for .M(D); • . n δzn /kzn 2 is a Carleson measure for .D and .

sup

n,m n=m

|kzn , kzm D | < 1. kzn D kzm D

2.5.8 Factorization Theorems We say f is cyclic for .D if .M(D)f = D.

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• Clearly f cyclic .⇒ f (z) = 0 for all .z ∈ D. The converse is false. • f is cyclic for .H 2 iff f is an outer function (Beurling). Theorem 2.5.8 (‘Inner-outer’ factorization, Jury–Martin [32]) If .f ∈ D, then f = ϕg, where .ϕ ∈ M(D) and g is cyclic in .D.

.

Theorem 2.5.9 (Smirnov factorization, Aleman–Hartz–McCarthy–Richter [3]) If .f ∈ D, then .f = ϕ1 /ϕ2 , where .ϕ1 , ϕ2 ∈ M(D) and .ϕ2 is cyclic in .D. Corollary 2.5.10 Given .f ∈ D, there exists .ϕ ∈ M(D) with the same zero set. Consequently, the union of two zero sets is again one.

2.5.9 Some Further Developments • Further characterizations of Carleson measures for D: Arcozzi–Rochberg– Sawyer [5]. • Reverse Carleson measures: Fricain–Hartmann-Ross [27]. • Corona problem for M(D): Tolokonnikov [51], Xiao [55], Trent [52].

2.6 Conformal Invariance 2.6.1 Preliminary Remarks Let .ϕ : D → C and .f : ϕ(D) → C be holomorphic functions. Write .nϕ (w) for the number of solutions z of .ϕ(z) = w. Proposition 2.6.1 (Change-of-variable formula) D(f ◦ ϕ) =

.

1 π



|f  (w)|2 nϕ (w) dA(w). ϕ(D)

Corollary 2.6.2 If .ϕ is injective, then .D(ϕ) = (area of ϕ(D))/π . Corollary 2.6.3 If .f ∈ D and .ϕ ∈ Aut(D), then .f ◦ ϕ ∈ D and .D(f ◦ ϕ) = D(f ). This last property more-or-less characterizes .D:

2.6.2 Characterization of D via Möbius Invariance Fix the following notation: • .H := a vector space of holomorphic functions on .D; • .E(f ) := f, f , where .·, · is a semi-inner product on .H.

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Theorem 2.6.4 (Arazy–Fisher [4], slightly modified) Assume that: • • • •

if .f ∈ H and .ϕ ∈ Aut(D), then .f ◦ ϕ ∈ H and .E(f ◦ ϕ) = E(f ); f 2 := |f (0)|2 + E(f ) defines a Hilbert-space norm on .H; convergence in this norm implies pointwise convergence on .D; .H contains a non-constant function. .

Then .H = D and .E(·) ≡ aD(·) some constant .a > 0.

2.6.3 Composition Operators For holomorphic .ϕ : D → D, define .Cϕ : Hol(D) → Hol(D) by Cϕ (f ) := f ◦ ϕ.

.

If .ϕ ∈ Aut(D), then .Cϕ : D → D. For which other .ϕ is this true?  −k 4k Note: If .ϕ(z) := k≥1 2 z , then .ϕ : D → D, but .Cϕ (D) ⊂ D because .ϕ ∈ / D. Theorem 2.6.5 (MacCluer–Shapiro [35])  .Cϕ : D → D ⇐⇒

nϕ dA = O(|I |2 ).

S(I )

Corollary 2.6.6 (El-Fallah–Kellay–Shabankhah–Youssfi [26]) Conditions for Cϕ : D → D:

.

• necessary: .D(ϕ k ) = O(k) as .k → ∞. • sufficient: .D(ϕ k ) = O(1) as .k → ∞.

2.6.4 Weighted Composition Operators Theorem 2.6.7 (Mashreghi–J. Ransford–T. Ransford [38]) Let T Hol(D) be a linear map. The following are equivalent:

: D →

• T maps nowhere-vanishing functions to nowhere-vanishing functions. • There exist holomorphic functions ϕ : D → D and ψ : D → C \ {0} such that Tf = ψ.(f ◦ ϕ)

.

(f ∈ D).

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2.6.5 Some Further Developments • Compact composition operators on D: MacCluer–Shapiro [35]. • Composition operators in S p -classes: Lefèvre, Li, Queffélec, Rodríguez-Piazza [34]. • Geometry of ϕ(D) when Cϕ is Hilbert–Schmidt: Gallardo-Gutiérrez, Gonzalez [28].

2.7 Weighted Dirichlet Spaces 2.7.1 The Dα Spaces For .−1 < α ≤ 1, let .Dα be the set of holomorphic f on .D with 1 .Dα (f ) := π

• • • •

 D

|f  (z)|2 (1 − |z|2 )α dA(z) < ∞.

Properties:   1−α k .Dα ( |ak |2 k ak z )  kk 2 .D0 = D and .D1 ∼ H = If .0 < α < 1, then .Dα is ‘akin’ to .D (using Riesz capacity .cα ). If .−1 < α < 0, then .Dα is a subalgebra of the disk algebra.

2.7.2 The Dμ Spaces Given a finite positive measure .μ on .T, write .P μ for its Poisson integral:  P μ(z) :=

.

T

1 − |z|2 dμ(ζ ) |ζ − z|2

(z ∈ D).

Following Richter [41], we consider .Dμ , the set of holomorphic f on .D such that 1 .Dμ (f ) := π

 D

|f  (z)|2 P μ(z) dA(z) < ∞.

• If .μ = dθ/2π , then .Dμ = D, the classical Dirichlet space. • If .μ = δζ , then .Dμ is called the local Dirichlet space at .ζ , denoted .Dζ .

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Can recover .Dμ (f ) from .Dζ (f ) using Fubini’s theorem:  Dμ (f ) =

.

T

Dζ (f ) dμ(ζ ).

2.7.3 Properties of Dμ (Richter–Sundberg [43]) • Dμ ⊂ H 2 and is Hilbert space w.r.t. f 2D := f 2H 2 + Dμ (f ). μ

• Douglas formula: if f ∈ Dμ , then f ∗ exists μ-a.e. and   Dμ (f ) =

.

T T

|f ∗ (λ) − f ∗ (ζ )|2 |dλ| dμ(ζ ). 2π |λ − ζ |2

Special case: f ∈ Dζ iff f (z) = a + (z − ζ )g(z) where g ∈ H 2 , and then Dζ (f ) = g2H 2 . • Carleson formula for Dμ (f ). • Polynomials are dense in Dμ . • Dμ (fr ) ≤ 4Dμ (f ) (where fr (z) := f (rz)). Can replace 4 by 1 (Sarason [46], using de Branges–Rovnyak spaces).

2.7.4 Hadamard Multipliers The Hadamard product of power series .h(z) := is defined by (h ∗ f )(z) :=



.



k≥0 ck z

k

and .f (z) :=



k≥0 ak z

k

ck ak zk .

k≥0

We say h is a Hadamard multiplier of .Dμ if .h ∗ f ∈ Dμ for all .f ∈ Dμ .  Theorem 2.7.1 (Mashreghi–Ransford [39]) .h(z) := k≥0 ck zk is a Hadamard multiplier of all the .Dμ spaces iff the matrix ⎞ ⎛ c1 c2 − c1 c3 − c2 c4 − c3 . . . ⎜0 c2 c3 − c2 c4 − c3 . . .⎟ ⎟ ⎜ ⎜0 0 c3 c4 − c3 . . .⎟ ⎟ ⎜ .Th := ⎜ ⎟ 0 0 c4 . . .⎟ ⎜0 ⎠ ⎝. . . . .. .. .. .. maps .2 into .2 . In this case .f → h∗f : Dμ → Dμ  ≤ Th : 2 → 2 , with equality whenever = δζ , ζ ∈ T.

.μ

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2.7.5 Special Cases of Hadamard Multipliers Let .f (z) :=



k≥0 ak z

k.

• If .fr (z) := f (rz), then Dμ (fr ) ≤ r 2 (2 − r)Dμ (f ).

.

• If .σn (f )(z) :=

n

k=0 (1 − k/(n + 1))ak z

k,

then

Dμ (σn (f )) ≤ (n/(n + 1))Dμ (f ).

.

Theorem 2.7.2 (Mashreghi–Parisé–Ransford [37]) Let .μ be a finite positive measure on .T. • The Taylor series of .f ∈ Dμ is .(C, α)-summable in .Dμ for all .α > 1/2. • If .μ = δζ , then there is a .f ∈ Dμ whose Taylor series is not .(C, 12 )-summable in .Dμ .

2.7.6 Some Further Developments • Capacities for Dμ : Chacón [20], Guillot [29]. • Estimates for reproducing kernel, capacities in Dμ : El-Fallah, Elmadani, Kellay [23]. • Superharmonic weights: Aleman [2]. • Dμ has the complete Pick property: Shimorin [48].

2.8 Shift-Invariant Subspaces 2.8.1 Preliminary Remarks Fix the following notation. • T is a bounded operator on a Hilbert space .H; • .Lat(T , H) := the lattice of closed T -invariant subspaces of .H; • .Mz := the shift operator (multiplication by z). Theorem 2.8.1 (Beurling [10]) If .M ∈ Lat(Mz , H 2 ) \ {0}, then .M = θ H 2 where .θ is inner. Analogue for .Lat(Mz , D)?

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2.8.2 The Shift Operator on Dμ Write .(T , H) := (Mz , D). Clearly: (1) .T 2 f 2 − 2Tf 2 + f 2 = 0 for all .f ∈ H. (2) .∩n≥0 T n (H) = {0}. (3) .dim(H  T (H)) = 1. It turns out that the same properties hold if .(T , H) := (Mz , Dμ ). Conversely: Theorem 2.8.2 (Richter [41]) Let T be an operator on a Hilbert space .H satisfying (1),(2),(3). Then there exists a unique finite measure .μ on .T such that .(T , H) is unitarily equivalent to .(Mz , Dμ ).

2.8.3 Invariant Subspaces of (Mz , D) Let .M ∈ Lat(Mz , D). • Clearly .(Mz , M) satisfies properties (1),(2). • If .M =  {0}, then property (3) also holds (Richter–Shields [42]). Leads to: Theorem 2.8.3 (Richter [41], Richter–Sundberg [44]) Let .M ∈ Lat(Mz , D) and ϕ ∈ M  Mz (M) with .ϕ ≡ 0. Then:

.

• .ϕ is a multiplier for .D. • .M = ϕDμ where .dμ := |ϕ ∗ |2 dθ . Corollary 2.8.4 .M is cyclic (i.e. singly generated as an invariant subspace).

2.8.4 Cyclic Invariant Subspaces Problem: Given f ∈ D, identify [f ]D , the closed invariant subspace of D generated by f . Theorem 2.8.5 (Richter–Sundberg [44]) Let f ∈ D have inner-outer factorization f = fi fo . Then [f ]D = fi [fo ]D ∩ D = [fo ]D ∩ fi H 2 .

.

It remains to identify [fo ]D . By analogy with what happens for H 2 , we might expect that [fo ]D = D. However, another phenomenon intervenes, that of boundary zeros.

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2.8.5 Cyclic Invariant Subspaces and Boundary Zeros Given .E ⊂ T, write DE := {h ∈ D : h∗ = 0 q.e. on E}.

.

Theorem 2.8.6 (Carleson [16]) .DE is closed in .D. Hence .DE ∈ Lat(Mz , D). Corollary 2.8.7 Let .f ∈ D and let .E := {f ∗ = 0}. Then .[f ]D ⊂ DE . Open Problem Let .f ∈ D be outer and let .E := {f ∗ = 0}. Then do we have .[f ]D = DE ? In particular, if .c(E) = 0, then do we have .[f ]D = D? Special case where .c(E) = 0 is a celebrated conjecture of Brown–Shields.

2.8.6 Brown–Shields Conjecture We say that .f ∈ D is cyclic for .D if .[f ]D = D. Necessary conditions for cyclicity: • f is outer; • .E := {f ∗ = 0} is of capacity zero. Conjecture 2.8.8 (Brown–Shields [15]) These conditions are also sufficient. Partial results (.A(D) denotes the disk algebra): Theorem 2.8.9 (Hedenmalm–Shields [31]) If .f ∈ D ∩ A(D) is outer and if .E := {f = 0} is countable, then f is cyclic. Theorem 2.8.10 (El-Fallah–Kellay–Ransford [25]) If .f ∈ D∩A(D) is outer and if .E := {f = 0} satisfies, for some . > 0,  1

+ .|Et | = O(t ) (t → 0 ) and dt/|Et | = ∞, then f is cyclic.

0

2.8.7 Some Further Developments • Shift-invariant subspaces and cyclicity in Dμ : Richter–Sundberg [44], Guillot [29], El-Fallah–Elmadani–Kellay [23]. • Optimal polynomial approximants: Bénéteau and co-authors ([8] onwards). • Cyclicity in Dirichlet spaces on the bi-disk: Knese–Kosi´nski–Ransford–Sola [33].

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26. EL-FALLAH, O., KELLAY, K., SHABANKHAH, M., AND YOUSSFI, H. Level sets and composition operators on the Dirichlet space. J. Funct. Anal. 260, 6 (2011), 1721–1733. 27. FRICAIN, E., HARTMANN, A., AND ROSS, W. T. A survey on reverse Carleson measures. In Harmonic analysis, function theory, operator theory, and their applications, vol. 19 of Theta Ser. Adv. Math. Theta, Bucharest, 2017, pp. 91–123. 28. GALLARDO-GUTIÉRREZ, E. A., AND GONZÁLEZ, M. J. Exceptional sets and Hilbert-Schmidt composition operators. J. Funct. Anal. 199, 2 (2003), 287–300. 29. GUILLOT, D. Fine boundary behavior and invariant subspaces of harmonically weighted Dirichlet spaces. Complex Anal. Oper. Theory 6, 6 (2012), 1211–1230. 30. HANSSON, K. Imbedding theorems of Sobolev type in potential theory. Math. Scand. 45, 1 (1979), 77–102. 31. HEDENMALM, H., AND SHIELDS, A. Invariant subspaces in Banach spaces of analytic functions. Michigan Math. J. 37, 1 (1990), 91–104. 32. JURY, M. T., AND MARTIN, R. T. W. Factorization in weak products of complete pick spaces. Bull. Lond. Math. Soc. 51, 2 (2019), 223–229. ´ 33. KNESE, G., KOSI NSKI , L., RANSFORD, T. J., AND SOLA, A. A. Cyclic polynomials in anisotropic Dirichlet spaces. J. Anal. Math. 138, 1 (2019), 23–47. 34. LEFÈVRE, P., LI, D., QUEFFÉLEC, H., AND RODRÍGUEZ-PIAZZA, L. Compact composition operators on the Dirichlet space and capacity of sets of contact points. J. Funct. Anal. 264, 4 (2013), 895–919. 35. MACCLUER, B. D., AND SHAPIRO, J. H. Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Canad. J. Math. 38, 4 (1986), 878–906. 36. MARSHALL, D. E., AND SUNDBERG, C. Interpolating sequences for the multipliers of the Dirichlet space. Unpublished manuscript, 1994. 37. MASHREGHI, J., PARISÉ, P.-O., AND RANSFORD, T. Cesàro summability of Taylor series in weighted Dirichlet spaces. Complex Anal. Oper. Theory 15, 1 (2021), Paper No. 7, 8. ˙ 38. MASHREGHI, J., RANSFORD, J., AND RANSFORD, T. A Gleason-Kahane-Zelazko theorem for the Dirichlet space. J. Funct. Anal. 274, 11 (2018), 3254–3262. 39. MASHREGHI, J., AND RANSFORD, T. Hadamard multipliers on weighted Dirichlet spaces. Integral Equations Operator Theory 91, 6 (2019), Paper No. 52, 13. 40. NAGEL, A., RUDIN, W., AND SHAPIRO, J. H. Tangential boundary behavior of functions in Dirichlet-type spaces. Ann. of Math. (2) 116, 2 (1982), 331–360. 41. RICHTER, S. A representation theorem for cyclic analytic two-isometries. Trans. Amer. Math. Soc. 328, 1 (1991), 325–349. 42. RICHTER, S., AND SHIELDS, A. Bounded analytic functions in the Dirichlet space. Math. Z. 198, 2 (1988), 151–159. 43. RICHTER, S., AND SUNDBERG, C. A formula for the local Dirichlet integral. Michigan Math. J. 38, 3 (1991), 355–379. 44. RICHTER, S., AND SUNDBERG, C. Multipliers and invariant subspaces in the Dirichlet space. J. Operator Theory 28, 1 (1992), 167–186. 45. ROSS, W. T. The classical Dirichlet space. In Recent advances in operator-related function theory, vol. 393 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2006, pp. 171–197. 46. SARASON, D. Local Dirichlet spaces as de Branges-Rovnyak spaces. Proc. Amer. Math. Soc. 125, 7 (1997), 2133–2139. 47. SHAPIRO, H. S., AND SHIELDS, A. L. On the zeros of functions with finite Dirichlet integral and some related function spaces. Math. Z. 80 (1962), 217–229. 48. SHIMORIN, S. Complete Nevanlinna-Pick property of Dirichlet-type spaces. J. Funct. Anal. 191, 2 (2002), 276–296. 49. STEGENGA, D. A. Multipliers of the Dirichlet space. Illinois J. Math. 24, 1 (1980), 113–139. 50. TAYLOR, B. A., AND WILLIAMS, D. L. Ideals in rings of analytic functions with smooth boundary values. Canadian J. Math. 22 (1970), 1266–1283. 51. TOLOKONNIKOV, V. A. Carleson’s Blaschke products and Douglas algebras. Algebra i Analiz 3, 4 (1991), 186–197.

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52. TRENT, T. T. A corona theorem for multipliers on Dirichlet space. Integral Equations Operator Theory 49, 1 (2004), 123–139. 53. TWOMEY, J. B. Tangential boundary behaviour of harmonic and holomorphic functions. J. London Math. Soc. (2) 65, 1 (2002), 68–84. 54. WYNN, A. Sufficient conditions for weighted admissibility of operators with applications to Carleson measures and multipliers. Q. J. Math. 62, 3 (2011), 747–770. 55. XIAO, J. The ∂-problem for multipliers of the Sobolev space. Manuscripta Math. 97, 2 (1998), 217–232.

Chapter 3

Bergman Space of the Unit Disc Stefan Richter

2020 Mathematics Subject Classification 30H20, 47B37, 47B38

3.1 Origins In the 1930s Stefan Bergman started investigating properties of conformal mappings based on the following idea [15, 16]. Let . ⊆ C be a region and let .L2a () denote the collection of all analytic functions on . that are square integrable with respect to Lebesgue measure .dA(z) = dxdy, z = x + iy. This is a Hilbert space with inner product  f, g =

f (w)g(w)

.



dA(w) . π

Thus, if .{un } is an orthonormal basis for .L2a (), then   dA(w) f (w)un (w)un (z) . .f (z) = f, un un (z) = π D n n One shows that one can interchange the sum and the integral, thus  f (z) =

f (w)kz (w)

.



dA(w) = f, kz , π

(3.1)

 where .kz (w) = n un (z)un (w). The function .kz (w) is uniquely determined by the reproducing property (3.1), and it is called the Bergman kernel for .. If . is the 1 unit disc .D, then it turns out that .kz (w) = (1−zw) 2.

S. Richter () Department of Mathematics, University of Tennessee, Knoxville, TN, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mashreghi (ed.), Lectures on Analytic Function Spaces and their Applications, Fields Institute Monographs 39, https://doi.org/10.1007/978-3-031-33572-3_3

77

78

S. Richter

Now assume that . = C is simply connected, let .z0 ∈ , and let .ϕ :  → D be the Riemann mapping with .ϕ(z0 ) = 0 and .ϕ  (z0 ) > 0. Exercise Use the change of variables formula 

 f (u)dA(u) =

.



D

f (ϕ(w))|ϕ  (w)|2 dA(w), f ∈ L1 ()

to relate the Bergman kernels for . and for .D. Then use the particular form of the Bergman kernel for .D to show that ϕ  (z) =

.

kz0 (z) . kz0 (z0 )

Thus, estimates on the Bergman kernel can be used to get estimates on .ϕ  , and approximations to .ϕ  can be computed with any orthonormal basis of .L2a (). This approach to conformal mapping also has applications in several complex variables, and to date results about Bergman kernels and Bergman spaces are important in many active research areas.

3.2 Aspects of the Basic Theory 3.2.1 Bergman Versus Hardy Spaces These notes assume familiarity with the theory of Hardy spaces, see for example [26] and [28]. We will write .T = {z ∈ C : |z| = 1}. Let .0 < p < ∞, and let .dA(z) = rdrdt be p 2-dimensional Lebesgue measure. The Bergman space .La is defined to be the set of all analytic functions f on .D such that p



f p =

.

D

|f |p

dA < ∞. π

(3.2)

For .0 < p < ∞ we define the integral means of the analytic function f by 

2π

Mp (f, r) =

.

0

dt |f (re )| 2π it

p

1/p , 0 ≤ r < 1.

A theorem of Hardy’s says that .Mp (f, r) is a nondecreasing function of r and, p p of course, . f H p = sup0≤r p we have . fr q ≈ (1 − r)2−q/p . Thus, for .q > 2p we cannot have q a continuous inclusion of .H p into .La . Estimates by Hardy and Littlewood can be 2p p used to show .H ⊆ La . The sharp norm bound of the inclusion was observed by Vucoti´c [48] 2p

Theorem 3.2.1 If .0 < p < ∞, then .H p ⊆ La and . f 2p ≤ f H p for every p .f ∈ H . Proof Vucoti´c observed that by use of the classical inner-outer factorization one can reduce the general inclusion to the case .H 2 ⊆ L4a , and then he established the result by a calculation with power series and the use of the Cauchy-Schwarz inequality. We show how to derive the latter inequality with reproducing kernel methods. We must show that whenever .f ∈ H 2 has . f H 2 ≤ 1, then . f 44 =

f 2 22 ≤ 1. The inequality will follow, if we show that multiplication by f is a contractive operator from .H 2 into .L2a . We will apply Theorem 3.6.6 to establish this. 1 1 Let .sz (w) = 1−zw and .kz (w) = (1−zw) 2 be the reproducing kernels for the Hardy space .H 2 and the Bergman space .L2a , see (3.1) or Theorem 3.2.13. The inequality . f H 2 ≤ 1 is equivalent to .sz (w)−f (z)f (w) being positive definite (see the remark after Theorem 3.6.6). We multiply this by .sz (w) and observe .(sz (w))2 = kz (w). Thus, the Schur product Theorem 3.6.7 implies that .kz (w) − f (z)f (w)sz (w) is positive definite, hence by Theorem 3.6.6 .Mf is a contractive multiplier from .H 2 into .L2a . 

Recall that every function in any .H p -space has nontangential limits almost everywhere on the unit circle, and this fact is extensively used to develop the Hardy space theory. For the investigations about Bergman space functions new methods will have to be developed. Theorem 3.2.2 There is f ∈



.

p

La

p>0

such that f has a radial limit almost nowhere. Proof A theorem of Zygmund about lacunary series implies that if an increasing sequence ∞ n of integers satisfies .nk+1 /nk > λ > 1 for all k, then the function .f (z) = k radial limits almost nowhere [53, p. 203]. k=1 z is analytic in .D, but has  2n If we take the function .f (z) = ∞ n=1 z , then for .p ≥ 1 we have

f p ≤

∞ 

.

n=1

n

z2 p =

∞   n=1

1 2n p + 2

1/p < ∞. 

80

S. Richter

3.2.2 Weighted Bergman Spaces We will focus on the Bergman spaces of the unit disc as defined above. However, sometimes it is instructive to consider a larger collection of closely related spaces, the weighted Bergman spaces with standard weights. Let .0 < p < ∞ and .α > −1. If .f : D → C is measurable, then define 

f p,α = (α + 1)

.

D

|f (z)|p (1 − |z|2 )α

dA(z) π

1/p .

Note that we required .α > −1 so that dAα (z) = (α + 1)(1 − |z|2 )α

.

dA(z) π

is a finite measure and the measure .Aα is normalized so that . 1 p,α = Aα (D) = 1 for all parameters. p This is a norm for .p ≥ 1 and .dp,α (f, g) = f − g p,α defines a metric for .0 < p < 1. This can be derived from the following lemma. Lemma 3.2.3 For all .u, v ∈ C, .0 < p < ∞ we have |u + v|p ≤ cp (|u|p + |v|p ), cp = max{1, 2p−1 }.

.

p

Proof The function .r(t) = (1+t) 1+t p is continuous on .[0, ∞) with .r(0) = limt→∞ = 1. If .p ≥ 1, then it has maximum .2p−1 which is attained at .t = 1. If .0 < p < 1, then the maximum is 1. The inequality of the lemma is trivial if .u = 0. If .u = 0, then v |u + v|p ≤ |u|p (1 + | |)p u

.

v ≤ max{1, 2p−1 }|u|p (1 + | |p ) u = max{1, 2p−1 }(|u|p + |v|p ). 

We will write .Lp,α (D) for the space of all functions on .D that are p-integrable with respect to .dAα . .Lp,α (D) is complete for all .0 < p < ∞. Definition 3.2.4 Let .α > −1, .0 < p < ∞. The weighted Bergman space is p,α

La

.

= Lp,α (D) ∩ Hol(D).

3 Bergman Space of the Unit Disc

81 p,α

Theorem 3.2.5 Let .0 < p < ∞, .α > −1. Then the polynomials are dense in .La . p,α

Proof For .f ∈ La and .0 < r < 1 define .fr (z) = f (rz). Then .fr is a uniform limit on .D of polynomials (its Taylor polynomials), hence it will be enough to prove that .fr → f in .Lp,α (D) as .r → 1. We first show that . fr p,α → f p,α as .r → 1. If .α ≥ 0, then an easy argument with the change of variables .w = rz and the monotone convergence theorem establishes the claim. In order to prove the claim for all .α > −1 we use again Hardy’s theorem that the integral mean .Mp (f, r) is a nondecreasing function p p of r. Then for each .0 < s < 1 we have .Mp (f, rs) → Mp (f, s) as .r → 1 (by the uniform convergence of .|fr (z)|p → |f (z)|p on .|z| = s) and 

p

1

fr p,α = 2(α + 1)

.

p

Mp (f, rs)(1 − s 2 )α ds.

0

Thus, the monotone convergence theorem now implies . fr p,α → f p,α as .r → 1. Then .gr (z) = cp (|f |p + |fr |p ) − |f − fr |p ≥ 0, and .gr → 2cp |f |p pointwise as .r → 1. Here .cp is the constant from Lemma 3.2.3. Hence one can use Fatou’s Lemma as in the standard proof of the dominated convergence theorem to finish the argument. 

2,α The spaces .La are Hilbert spaces with inner product .f, gα = D f gdAα . If ∞ n ˆ .f (z) = n=0 f (n)z , then one shows

f 22,α =

∞ 

.

an,α

n=0

|fˆ(n)|2 n+1

for positive coefficients .an,α ≈ (n + 1)−(α+1) . In particular, .an,0 =  n ˆ Hence for the partial sums .sN (f )(z) = N n=0 f (n)z we have

f − sN (f ) 22,α =

∞ 

.

1 n+1 .

an,α |fˆ(n)|2 → 0

n=N +1

as .N → ∞. p,α

Theorem 3.2.6 If .1 < p < ∞, .α > −1, and if .f ∈ La , then . f − sN (f ) p,α → 0. This becomes false for .0 < p ≤ 1, see [51]. The proof of the theorem uses the fact that it works in .H p , because for .1 < p < ∞ the Szegö projection is bounded on .Lp (T), and one can express .sN (f ) by use of the Szegö projection, and then use polar coordinates for the integral.

82

S. Richter

3.2.3 Disc Automorphisms and Change of Variables If .|λ| < 1, then let .ϕλ (z) = inverse function.

λ−z . 1−λz

Note that .ϕλ takes .D onto .D and it is its own

Theorem 3.2.7 (Magic Formula) 1 − |ϕλ (z)|2 =

.

(1 − |λ|2 )(1 − |z|2 ) |1 − λz|2

.

Theorem 3.2.8 (Change of Variables) If .w = ϕλ (z), then .

(1 − |λ|2 )2 dA(z) dA(w) dA(z) = |ϕλ (z)|2 = π π π |1 − λz|4 dA(z) (1 − |λ|2 )2 dA(w) = π π |1 − λw|4

Corollary 3.2.9 If .|λ| < 1, then f ∈ Lp,α (D) ⇐⇒ f ◦ ϕλ ∈ Lp,α (D).

.

3.2.4 Pointwise Bounds and Reproducing Property p,α

Theorem 3.2.10 0 < p < ∞, α > −1. If f ∈ La |f (z)| ≤

1

.

(1 − |z|2 )

2+α p

and z ∈ D, then

f p,α .

p,α

(3.3)

p,α

Hence the unit ball of La is a normal family and La is a closed subspace of p,α Lp,α (D). Thus La is a Banach space for p ≥ 1 and a Frechet space if 0 < p < 1. Proof We will first prove the case z = 0. The general case will follow by a change of variables. We again use the theorem of Hardy that the integral mean Mp (f, r) p is nondecreasing in 0 ≤ r < 1, hence |f (0)|p ≤ Mp (f, r) holds for all 0 ≤ r < 1. Note that if p ≥ 1, then this also follows from the mean value property of f and Hoelder’s inequality. We integrate this with respect to the probability measure (α+1)(1−r 2 )α 2rdr to obtain the required inequality for z = 0. If z ∈ D is arbitrary, then we substitute the above for the function g(w) =

.

z−w f ( 1−zw )

(1 − zw)

4+2α p

3 Bergman Space of the Unit Disc

83

and follow it with a change of variables and application of the magic formula. It is a good exercise to verify the details of this calculation. 

Exercise p,α

(a) Show that for each z ∈ D there is f ∈ La previous inequality. p,α

(b) Show that for each f ∈ La

such that equality holds in the

we have lim sup|z|→1 (1 − |z|2 )

2+α p

|f (z)| = 0.

Theorem 3.2.11 If f ∈ L1,α a and if |z| < 1, then  f (z) = (α + 1)

.

D

f (w)

(1 − |w|2 )α dA(w) . (1 − zw)2+α π

Proof The proof proceeds similar to the previous one. Consider z = 0. By it is enough to prove the statement for the density of the polynomials in L1,α a polynomials f . We have already noted that 1 p,α = 1, so the statement holds for constant functions and it will suffice to show that D f (z)(1 − |z|2 )α dA(z) = 0, whenever f is a polynomial with f (0) = 0. In fact, by linearity it is enough to do the case of a monomial of degree ≥ 1, and that case is easily verified by use of polar coordinates. The general case follows by a change of variables argument as in the previous proof. It is left as an exercise. 

The following lemma turns out to be fundamental for a number of estimates in the theory. Lemma 3.2.12 If t > −1 and s > 0, then  .

(1 − |w|2 )t 1 dA(w) ≈ . 2+t+s (1 − |z|2 )s D |1 − zw|

More precisely, the ratio of the two expressions is bounded above and below independently of z ∈ D. Proof That the integral on the left is larger than a constant multiple of the expression on the right follows from Theorem 3.2.10 with p = 2, α = t and f (w) = (1−|z|2 )1+t/2 . There are a number of different proofs of the opposite inequality, all of 1+ t+s (1−zw)

2

which involve some technical estimates. Forelli and Rudin [27] proved a version of the lemma for the unit ball of Cd and they used power series and Stirling’s formula. For the unit disc the approach of Shields and Williams [45, Lemmas 5 and 6] appears to be fairly straightforward. They use polar coordinates and first show that 

2π

.

0

1 |1 − λreiθ |2+t+s

dθ ≈

1 . (1 − r|λ|)1+t+s

84

S. Richter

Then they use integration by parts to obtain the following string of estimates: 

1

.

0

(1 − r 2 )t rdr ≤ C (1 − r|λ|)1+t+s

 

(1 − r)t+1 dr (1 − r|λ|)2+t+s

1

1 dr (1 − r|λ|)1+s

0

≤C 0

≤C

1

1 (1 − |λ|)s

In the integration by parts step, there would be a summand that can be seen to be bounded, because t + 1 > 0. Similarly, the later estimates work, because t + 1 > 0. We note that the case −1 < t < 0 will be used in the proof of the boundedness of the Bergman projection. 

3.2.5 Reproducing Kernel and the Bergman Projection Theorem 3.2.13 Let α > −1. The function k α (w, z) = kzα (w) =

.

1 , z, w ∈ D (1 − zw)2+α

2,α α is the reproducing kernel for L2,α a , i.e. for all z ∈ D the function kz ∈ La satisfies

f (z) = f, kzα α , for all f ∈ L2,α a .

.

Proof This follows immediately from Theorem 3.2.11.



Theorem 3.2.14 Let α > −1. For f ∈ L1,α (D) define  f (w) (1 − |w|2 )α dA(w) f (w) = .Pα f (z) = (α + 1) dAα (w). 2+α π wz)2+α (1 − (1 − zw) D D 

Then Pα f is an analytic function on D. Pα is called the Bergman projection of order α. If f ∈ L2,α (D), then Pα f equals the value of the orthogonal projection of 2,α L (D) onto L2,α a . Proof It is elementary to check that Pα f is analytic in D. For example, one can use Morera’s theorem. If we temporarily write Q for the orthogonal projection of 2,α ∗ α α α L2,α (D) onto L2,α a , then Q = Q and Qkz = kz since kz ∈ La . Hence if f ∈ L2,α (D), then since Qf ∈ L2,α a Theorem 3.2.11 implies

3 Bergman Space of the Unit Disc

85

Qf (z) = Qf, kzα α = f, Q∗ kzα α = f, kzα α = Pα f (z).

.



At this point it is a natural question to wonder whether Pα takes Lp,α (D) into p,α La for other values of p. That turns out to be true, if 1 < p < ∞ and is false for p = 1 and p = ∞. I will now restrict attention to the case α = 0 and write P = P0 for the Bergman projection. Example 3.2.15 If f (z) = Pf (z) = −

z−log z2

1 1−z

1−|z|2 1−z ,

then f ∈ L∞ (D), but one calculates that

, and that function is not bounded in D.

Now, if P were a bounded operator L1 (D) → L1 (D), then its adjoint would be bounded L∞ (D) → L∞ (D). It turns out one can relate this adjoint to the Bergman projection and show that P would have to be bounded on L∞ (D). Thus, P is not bounded on L1 (D). Theorem 3.2.16 (Zaharjuta, Judoviç [50]) Let 1 < p < ∞, then the formula  Pf (z) =

.

f (w) dA(w) D (1 − zw)2 π

defines a bounded linear operator Lp (D) → Lp (D). The following proof has been adapted from Forelli-Rudin [27], also see [11]. It uses the Schur test (Lemma 3.6.1) and the fundamental inequalities from Lemma 3.2.12. Proof If −1 < t < 0, then by Lemma 3.2.12 with s = −t there is C > 0 such that for all |w| < 1  .

(1 − |z|2 )t dA(z) ≤ C(1 − |w|2 )t . D |1 − wz|2 π

(3.4)

1 and set h(z) = (1 − |z|2 )−σ . Let q be the dual index to p, choose 0 < σ < max(p,q) Then both hp and hq are of the form (1 − |z|2 )t for −1 < t < 0. Hence the Schur test and (3.4) implies that the operator P is bounded. 

3.2.6 Duality Recall that it follows from general principles that whenever .1 < p, q < ∞ are dual p p indices, then .(La )∗ = Lq (D)/(La )⊥ . In applications it is an advantage, if one can identify the dual as a specific function space, rather than a quotient space. By now

86

S. Richter

it is a well-known fact that with the boundedness of the projection operator one can establish a duality result. Corollary 3.2.17 For .1 < p < ∞ and . p1 + q1 = 1 we have .(La )∗ = La with equivalence of norms.

More precisely, the map . : g → Lg , Lg (f ) = D f g dA π is a conjugate linear q p ∗ isomorphism from .La onto .(La ) . p

q

Proof It is clear that . is conjugate linear and it follows from Hoelder’s inequality ˆ that . is bounded. If .0 = g ∈ La , then one computes .Lg (zn ) = g(n) n+1 is not 0 p for some n, hence . is 1–1. In order to see that . is onto let .L ∈ (La )∗ . Then by the Hahn-Banach theorem L can be extended to a bounded linear functional on .Lp (D) of the same norm. Thus, there is .h ∈ Lq (D) such that . h q = L ∗

q and .L(f ) = D f h dA π . Set .g = P h, then by the previous theorem .g ∈ La with . Lg ∗ = g q ≤ c h q = c L . We claim that .L = Lg . p Let .f ∈ La and .0 < r < 1. Then by the change of variables .z = ru and the reproducing property we conclude that q

 .

|z| n, then

.

Z m U x, Z n Uy = Z m−n U x, Uy = U x, Z ∗ m−n Uy = 0.

.

Set V =

∞ 

.

Z n U D∗ A∗ n .

n=0

The series converges since . Z = 1 and . A∗ < 1. In fact,

V x 2 =

∞ 

.

Z n U D∗ A∗ n x 2

n=0

=

∞ 

D∗ A∗ n x 2

n=0

= lim

N 

N →∞

A∗ n x 2 − A∗ n+1 x 2

n=0

= lim x 2 − A∗ N +1 x 2 N →∞

= x 2 . We have R∗V =

∞ 

.

Z n−1 U DA∗ n = V A∗ .

n=1

Thus, .ran V is invariant for .R ∗ and .A∗ is unitarily equivalent to .R ∗ | ran V .



Corollary 3.4.4 If T is as above and if T has an invariant subspace of infinite index, then the Sandwich Theorem holds for T . Proof Let .M ∈ Lat T have infinite index, then take .R = T |M and apply Theorem 3.4.3 to any Hilbert space operator A with . A < c and obtain .N ⊆ M

3 Bergman Space of the Unit Disc

93

so that ⎛

∗ ∗ ⎝ .T = 0A 0 0

⎞ ∗ ∗⎠ with respect to H = N ⊕ (M  N) ⊕ M⊥ . ∗

Thus, nontrivial invariant subspaces of A correspond to invariant subspaces of T

 that lie properly between .N and .M.

3.4.2 The Index of Invariant Subspaces Let .B be a Banach space of analytic functions on .D, let .T = (Mz , B) be bounded and such that .T − λ is bounded below for every .λ ∈ D, and assume that the index p of .B is 1. For the purposes of this note think .H = H p or .H = La , .1 ≤ p < ∞. If .M ∈ Lat T , then .(T − λ)|M is bounded below for each .λ ∈ D, hence it is a semi-Fredholm operator and from the properties of the Fredholm index we conclude that indM = −ind(T |M) = −ind((T − λ)|M) = dim M  (z − λ)M

.

for each .λ ∈ D. Here is an elementary argument showing this in the Hilbert space case: If .W ∈ B(H) is bounded below, then .W ∗ W is invertible and .L = (W ∗ W )−1 W ∗ is a left inverse of W , .LW = I . Hence .W ∗ L∗ = I . If .|z| < 1/ L∗ , then the operator ∗ ∗ ∗ ∗ .I − zL is invertible. Hence the identity .W − z = W (I − zL ) implies that ∗ ∗ ∗ ∗ .I −zL takes .ker(W −z) to .ker W in a 1–1 and onto manner. Thus, .dim ker(W − ∗ z) = dim ker W in a neighborhood of 0. We apply this with .W = (Mz − λ)|M to conclude that .dim ker((Mz − λ)|M)∗ = dim M  (z − λ)M is locally constant in .D, hence it must be constant on .D. Write .Z(M) = {λ ∈ D : f (λ) = 0 for every f ∈ M}, the zero set of .M. Then in the Hilbert space case .λ ∈ Z(M), iff .PM kλ = 0. One easily checks that for every invariant subspace .M and every .λ ∈ D one has .PM kλ ∈ M  (z − λ)M. Thus, .indM = 1, if and only if .PM kλ spans .M  (z − λ)M for some .λ ∈ D \ Z(M), and this happens if and only if .PM kλ spans .M  (z − λ)M for all .λ ∈ D \ Z(M). If .f ∈ B, then we write .[f ] for the smallest T -invariant subspace that contains f . We also say that .[f ] is the cyclic invariant subspace generated by f . If . = {λ1 , λ2 , . . . } is a sequence of points in .D, then .I () denotes the zero-based invariant subspace consisting of all functions that are 0 at each .λi counting multiplicities. The following are easy facts. Proposition 3.4.5 Let .B be a Banach space of analytic functions as considered above. Then

94

1. 2. 3. 4.

S. Richter

M = {0} is the only invariant subspace of index 0. If .f = 0, then .[f ] has index 1. If .I () = 0, then it has index 1. If .M, N are invariant subspaces, then

.

indM ∨ N ≤ indM + indN.

.

Thus, Beurling’s characterization of the non-zero invariant subspaces of (Mz , H p ) as all being of the type .M = [ϕ] = ϕH p for some inner function .ϕ implies that all non-zero invariant subspaces of .H p have index 1. Similarly, for the Dirichlet shift .(Mz , D) it was shown in [40] that every nonzero invariant subspace must have index 1. We will see this is no longer true for the Bergman space. If .M is a non-zero invariant subspace, then

.

M=

.



[f ].

f ∈M,f =0

Thus, every nonzero invariant subspace is a span of index one invariant subspaces, and it becomes important to know when the span of index 1 invariant subspaces has index 2. p

Theorem 3.4.6 Let .1 < p < ∞ and .f, g ∈ La , .g = 0. If .f/g has a finite nontangential limit on .E ⊆ T, .|E| > 0, then ind[f ] ∨ [g] = 1.

.

This is due to Aleman, Richter, Ross, see [2]. It was an attempt to understand from a function theory viewpoint the following operator theoretic result of Apostol, Bercovici, Foias, and Pearcy [9]. Theorem 3.4.7 If .B = H as above is a Hilbert space, and if . zn f → 0 for each .f ∈ H, then invariant subspaces of any index must exist. p

For the Bergman spaces .La Hedenmalm [30] showed the existence of subspaces of index 2 by observing that .I (1 ) ∨ I (2 ) has index 2, whenever .1 and .2 are interpolating sequences whose union is a sampling set. Then he used results of Seip to show that such sequences indeed exist in the Bergman spaces. The idea is that the unit spheres of the subspaces .I (1 ) and .I (2 ) have a positive distance from one another, i.e. .

inf{ f − g : f = g = 1, f ∈ I (1 ), g ∈ I (2 )} > 0.

Then .M = I (1 ) + I (2 ) is actually closed and then it is fairly easy to see that .M has index 2. Hedenmalm himself indicated how to modify his construction to prove the existence of invariant subspaces of any finite index, but constructing invariant

3 Bergman Space of the Unit Disc

95

subspaces of infinite index is more subtle. It was done by Hedenmalm, Richter, Seip, p who also establish the .La -version of these results [34]. We will outline how to prove Theorem 3.4.7 for the Bergman shift and related operators. The construction will rely on the concept of dominating sets.

3.4.3 Spectral Synthesis and Dominating Sets We are looking for unusual invariant subspaces. The following definition is relevant in this context. After all, eigenspaces are the simplest kind of invariant subspaces. Definition 3.4.8 Let .A ∈ B(H). We say that A has spectral synthesis, if every invariant subspace of A contains a nonzero eigenvector. Thus, every operator on a finite dimensional space has spectral synthesis. The unilateral shift has no nonzero eigenvectors, so it does not have spectral synthesis for trivial reasons. If .T = (Mz , H), then the reproducing kernels .kλ are eigenvectors for .T ∗ , so one might wonder whether .T ∗ can have spectral synthesis. If .M⊥ is .T ∗ invariant, then .kλ ∈ M⊥ , if and only if .λ ∈ Z(M). Thus, when Beurling proved his theorem characterizing the invariant subspaces of .S = (Mz , H 2 ), he noted that it implies that the backward shift .S ∗ does not have spectral synthesis, since whenever 2 .ϕ is a singular inner function and .M = ϕH , then .Z(M) = ∅. Shortly after Beurling proved his theorem, John Wermer investigated which diagonal normal operators have spectral synthesis [49]. In the special case, where the eigenvalues form a discrete subset of .D, he discovered a condition that also came up in later work of Brown, Shields, and Zeller [18]. Theorem 3.4.9 Let .{λn } ⊆ D be a sequence of distinct points with no limit point inside .D, let .{en } be an orthonormal basis for a Hilbert space .H and let T = diag{λn } =

∞ 

.

λn en ⊗ en

n=1

be the diagonal operator with eigenvalues .{λn }. The following are equivalent: 1. for a.e. .eit ∈ T there is a subsequence .{λnk } of .{λn } that converges to .eit nontangentially, 2. for every .f ∈ H ∞ (D) we have . f ∞ = sup{|f (λn )| :  n ∈ N}, 3. for every .λ ∈ D there is .{an } ∈ 1 such that .p(λ) = ∞ n=1 p(λn )an for every polynomial p, 4. for every .λ ∈ D there are .x, y ∈ H such that .p(λ) = p(T )x, y for every polynomial p, 5. The operator T does not have spectral synthesis.

96

S. Richter

Definition 3.4.10 A sequence .{λn } ⊆ D is called dominating, if it satisfies conditions (i)-(v). Proof We omit the proof of (i).⇔(ii), see [18]. (ii).⇒(iii): We start by noting that .(L1 /H01 )∗ = H ∞ with duality given by

|dz| 1 ∞ .[f ], h = |z|=1 f (z)h(z) 2π for .f ∈ L , h ∈ H . For .λ ∈ D let .Pλ denote the Poisson kernel, .Pλ (z) =

1−|λ|2 . |1−λz|2

Consider the contractive linear map

W : 1 → L1 /H01 , {an } → [



.

an Pλn ].

n

Then .W ∗ : H ∞ → ∞ is given by .W ∗ h = {h(λn )}. The hypothesis (ii) says that ∗ .W is isometric. It then follows by general principles that W is onto. Thus, if .λ ∈ D, then there is .a = {an } ∈ 1 such that .W a = [Pλ ]. (iii) follows. 1 (iii).⇒(ii):  Let .λ ∈ D, then by hypothesis (iii) there∞is .{an } ∈  such that .p(λ) = and .0 < r < 1, then n p(λn )an for every polynomial p. If .f ∈ H .fr (z) = f (rz) can be uniformly approximated by polynomials, hence .f (rλ) =  − n )an . Now let .r → 1 , then the dominated convergence theorem implies n f (rλ  ∞ .f (λ) = n f (λn )an for every .f ∈ H . ∞ k For .f ∈ H and .k ∈ N we have .f ∈ H ∞ . Thus, |f (λ)|k = |



.

n

f k (λn )an | ≤ sup |f (λn )|k n



|an |

n

 1/k and hence .|f (λ)| ≤ supn |f (λn )| and letting .k → ∞ we conclude that n |an | ∞ and .λ ∈ D. Hence (ii) holds. .|f (λ)| ≤ supn |f (λn )| for every .f ∈ H  n xn en , y =  (iii).⇒(iv) follows by factoring .an = xn y n and setting .x = y e . n n n (iv).⇒(v): Choose .λ = λn for all n. Note that we may assume that for each n either .xn = yn = 0 or .xn = 0 and .yn = 0. Take .M = [x]T . Suppose that for some j .ej ∈ M. Then there are polynomials .pk such that .pk (T )x → ej . Then .pk (λ) = pk (T )x, y → yj . Furthermore, .(T − λj )pk (T )x → 0, hence from the hypothesis again we conclude that (λ − λj )pk (λ) = (T − λj )pk (T )x, y → 0.

.

This implies that .yj = 0. But then .xj = 0 and hence .p(T )x ⊥ ej for each polynomial p. That is a contradiction. (v).⇒(iii): Let .λ ∈ D \ {λn }, let .(0) = M ∈ Lat T contain no eigenvector of T , and let .x = {xn } ∈ M, x = 0. Fix j such that .xj = 0. Since .ej ∈ / M we have .y = PM⊥ ej = 0 and .0 =  y 2 = y, ej  = yj . Set .b = {bn } = {xn y n }, then .b ∈ 1 and .bj = 0.  bn , which is analytic in the compleNow consider the function .F (z) = n z−λ n ment of the closure of .{λn }, and has a simple pole at .λj . Thus, F is not identically 0

3 Bergman Space of the Unit Disc

97

 in .D\{λn }. Thus, there is a .p ∈ N0 such that .F (p) (λ) = (−1)p p! n (z−λbn)p+1 = 0, n  but .F (λ) = . . . F (p−1) (λ) = 0. Set .c = n (z−λbn)p+1 and .an = c(λ−λbn )p+1 . Then n

one checks that (iii) holds for this sequence .a = {an } ∈ 1 .

n



An analysis of the proof shows that the existence of the unusual invariant subspace (the one that does not contain eigenvectors) follows from the existence of the .1 -sequence .{an } in condition .(iii). For the existence of invariant subspaces with infinite index we need the following immediate corollary. Corollary 3.4.11 If . ⊆ D is a disjoint union of infinitely many discrete sets .k = ∞ {λnk }∞ n=1 , each of which is dominating, and if for each k we let .Nk = diag{λnk }n=1 , and set .N = ⊕Nk , then for all .λ ∈ D there are .{xk }, {yk } ⊆ H such that p(N)xk , yj  = p(λ)δkj .

.

Note that the switch from .λnk to their complex conjugates does not affect whether or not the sequence is dominating. The following lemma is from [9], and it shows that the conclusion of the previous lemma is precisely what is needed to construct invariant subspaces of infinite index. Lemma 3.4.12 If there are .{fj }, {gj } ⊆ L2a such that p(Mz∗ )gj , fk  = p(0)δj k

.

for every polynomial p, then N=



.

[fn ]

n

has infinite index. Proof Let .PN denote the projection onto .N. We will show that the set .{PN gj } is linearly independent  in .N  zN.  First suppose . nj=1 aj PN gj = 0, then . nj=1 aj gj ∈ N⊥ and n n   0= aj gj , fk  = aj gj , fk  = ak .

.

j =1

j =1

Thus, .{PN gj } is linearly independent. Next we show that .PN gj ∈ N  zN for each j . Clearly it is in .N. If .pi is a polynomial, then PN gj , zpi fi  = gj , zpi fi  = Mz∗ pi (Mz )∗ gj , fi  = 0.

.

Hence .PN gj ⊥

n

i=1 zpi fi

and that implies that .PN gj ⊥ zN.



98

S. Richter

3.4.4 Mz∗ |I ()⊥ Can Be Similar to a Diagonal Normal Operator Let .1 < p < ∞ with dual index q. A sequence of distinct points . = {λn } ⊆ D is p interpolating for .La , if the linear map p

Tp : La → p , f → {(1 − |λn |2 )2/p f (λn )}

.

is bounded and onto. This happens if and only if the adjoint is bounded and bounded q below. One calculates .Tp∗ : q → La , .Tp∗ en = (1−|λn |2 )2/p kλn . Thus, the sequence p is interpolating for .La , if and only if there are constants .c, C > 0 such that c



.

|an |q ≤



n

q

an (1 − |λn |2 )2/p kλn q ≤ C



n

|an |q

(3.6)

n

for all sequences .{an } ∈ q . Notice that Theorem 3.2.10 and the exercise that p followed it show that the functional .Cz : La → C, .Cz (f ) = f (z) has norm 2 −2/p . Cz = (1 − |z| ) . Hence Theorem 3.2.17 implies that there are .c1 , c2 > 0 such that c1 ≤ (1 − |λn |2 )2/p kλn q ≤ c2

.

for all .λ ∈ D. If .p = 2, then Hilbert space vectors .un ∈ H that satisfy the inequalities c



.

|an |2 ≤

n

 n

an un 2H ≤ C



|an |2

n

are called Riesz sequences. Lemma 3.4.13 If . = {λn } ⊆ D is interpolating for .L2a , and if .I () denotes the corresponding zero set based invariant subspace, then .Mz∗ |I ()⊥ is similar to 2 → I ()⊥ such that .N = diag{λn }, i.e. there is an invertible operator .S :  −1 (M ∗ |I ()⊥ )S. .N = S z Proof Since . is an interpolating sequence for .L2a the operator .T2∗ : 2 → ran T2∗ = I ()⊥ is invertible, and one checks that .Mz∗ T2∗ en = T2∗ Nen for each n, hence −1 (M ∗ |I ()⊥ )S with .S = T ∗ . .N = S 

z 2 Thus we can take the .xk and .yj of Corollary 3.4.11 and use the similarity to send them to the vectors .gk and .fj of Lemma 3.4.12. We obtain the following: Corollary 3.4.14 If there is an interpolating sequence for .L2a that is a countable infinite union of mutually disjoint dominating sequences, then invariant subspaces with infinite index exist in .L2a .

3 Bergman Space of the Unit Disc

99

3.4.5 An Interpolating Sequence That Is Dominating p

The interpolating sequences for .La have been characterized by Seip [42]. Here we only need the following fact. Theorem 3.4.15 There is an interpolating sequence .{λn } ⊆ D for .L2a that is a union of infinitely many mutually disjoint dominating sequences. It is easy to see that interpolating sequences are zero sequences. Hence this theorem shows again that there are zero sequences for .L2a that are not Blaschke sequences, since Blaschke sequences cannot be dominating (or else the corresponding Blaschke product would have norm 0). It also shows again that there are functions in .L2a that have nontangential limits almost nowhere. Let f be a nonzero function that is zero at each point of a dominating sequence. If f had nontangential limits a.e. on a set .E ⊆ T, then they would have to be 0 a.e. But if an analytic function f has nontangential limit 0 on a set of positive measure, then .f = 0. That follows from Privalov’s uniqueness theorem. The construction for the proof of Theorem 3.4.15 will proceed as follows: for 2π ij/N for .j = 0, . . . , N − 1. Then .N ∈ N we set .rN = 1 − 1/N, .λj = λN,j = rN e we will choose a integers .Nk → ∞ such that =

∞ N k −1 

.

{λNk ,j }

k=1 j =0

does the job.

It turns out that the sequence is dominating for all choices of .Nk → ∞. In fact, every .eit ∈ T will be a nontangential limit point of the sequence. One way to see

100

S. Richter

that is as follows. For .c > 0 and .λ ∈ D, λ = 0 let .Iλ (c) be the interval in .T that is λ centered at . |λ| and has length .|Iλ (c)| = c(1 − |λ|). Then a sequence .zn converges  nontangentially to .eit , if and only if there is .c > 0 such that .eit ∈ n Izn (c). Let .0 ≤ t < 2π , and let N be fixed. Then there is .0 ≤ jN < N such that jN jN +1 .2π and one verifies that .eit ∈ IN,jN (2π ). This implies that N ≤ t < 2π N it .λN,jN → e nontangentially, and the same is true for any subsequence. It is now also clear that . can be decomposed as a union of infinitely many mutually disjoint sequences each of which is dominating. In order to show that the .Nk can be chosen so that the sequence is interpolating for .L2a we need a few lemmas. The proof of the following lemma is based on what is often referred to as the Koethe-Toeplitz Theorem. In fact, the argument shows that any sequence that is interpolating for the multiplier algebra is interpolating for the Hilbert space (with control on the constants). Lemma 3.4.16 Assume that .Mϕ , multiplication by .ϕ, is bounded on the Hilbert space of analytic functions .H ⊆ Hol(D) for each .ϕ ∈ H ∞ and . Mϕ = ϕ ∞ . If .{λn } is .H ∞ -interpolating with constant K, i.e. if whenever .{wn } is in the unit ball of .∞ , then there is .ϕ ∈ H ∞ with . ϕ ∞ ≤ K and .ϕ(λn ) = wn for each n, then 1 . K

 

1/2 |an |

2

n

≤

 n

 1/2  kλn 2

≤K an |an |

kλn

n

for all sequences of complex coefficients .{an }. Proof It is enough to prove the statement for sequences .{an } such that .an = 0 for only finitely many n. Fix such .an s. Inductively construct a sequence .tn ∈ R such that N  .

n=1

|an |2 =

N  n=1

eitn an

kλn 2

kλn

for each N. Then choose .ϕ, ψ ∈ H ∞ , . ϕ ∞ , ψ ∞ ≤ K and .ϕ(λn ) =   k k eitn , ψ(λn ) = e−itn for each n. Let .f = n an kλλn , g = n eitn an kλλn so that n

n

3 Bergman Space of the Unit Disc

101

   k 2 1/2 . Furthermore we observe that

f = n an kλλn and . g = n |an | n ∗ ∗ .Mϕ g = f and .M f = g. The result follows from ψ

.

.

1 1

g = Mψ∗ f ≤ f = Mϕ ∗ g ≤ K g . K K 

Now we apply the quantitative version of Carleson’s .H ∞ -interpolation theorem, see Theorem 3.6.2 and the remark following it. One checks that each sequence N −1 .{λN,j } j =0 is separated and satisfies the Carleson measure condition with constants .δ and C independent N . Hence Carleson’s theorem and the previous lemma imply that there is .K > 1 such that for all N and all .a0 , . . . , aN −1 ∈ C we have ⎛ .

1 ⎝ K

N −1 

⎞1/2 |aj |2 ⎠

≤

j =0

N −1 

⎛ aj

j =0

kλN,j

kλN,j

≤K⎝

N −1 

⎞1/2 |aj |2 ⎠

.

j =0

For .N ∈ N let MN = {u =

N −1 

.

k=0

ak

kλN,k : a0 , . . . , aN −1 ∈ C},

kλN,k

where the .|λN,k | = rN are chosen as above. Lemma 3.4.17 Let .H = L2a , .kλ (z) = .u ∈ MN , v ∈ MM we have

1 . (1−λz)2

Let .N, M ∈ N, .N < M, then for all  N3 . M

|u, v| ≤ δ u

v , where δ = 4K 2

.

Proof Write .λi = λN,i and .zj = λM,j . Let .u = M−1 kzj j =0 bj kz , where .|λj | = 1 − 1/N, |zj | = 1 − 1/M.

N −1

j

Then .

(1 − |λi |2 )(1 − |zj |2 ) |kλi (zj )| =

kλi

kzj

|1 − λi zj |2 ≤4

(1 − |λi |)(1 − |zj |) (1 − |λi |)2

=4

N M

i=0

kλ

ai kλi and .v = i

102

S. Richter

Hence using Lemma 3.4.16 we obtain |u, v| ≤



.

|ai ||bj |

i,j

≤



|ai ||bj |4

i,j

≤

√

|kλi (zj )|

kλi

kzj

 N





N M 1/2

|ai |2

i

⎛ ⎞1/2  √ N M⎝ |bj |2 ⎠ 4 M j

N3

u

v

M

≤ 4K 2

∞



Choose a sequence .{εi } of positive numbers such that . ≤ 1/2. We will now inductively choose an increasing sequence of natural numbers .{Nj } such that 2 i=1 εi

|ui , uj | ≤ εi εj ui

uj , whenever ui ∈ MNi , uj ∈ MNj , i = j.

.

Set .N1 = 2. Let .m ≥ 1 and assume that we have chosen .N1 < · · · < Nm such that |ui , uj | ≤ εi εj ui

uj , whenever ui ∈ MNi , uj ∈ MNj , i = j and 1 ≤ i, j ≤ m.

.

Choose .Nm+1 > Nm so that  4K √

.

2

Nm3 ≤ εm+1 min εj . 1≤j ≤m Nm+1

Then for all .1 ≤ i ≤ m and .ui ∈ MNi , um+1 ∈ MNm+1 Lemma 3.4.17 implies  Ni3 2

ui

um+1

.|ui , um+1 | ≤ 4K √ Nm+1  N3 2 ≤ 4K √ m ui

um+1

Nm+1 ≤ εi εm+1 ui

um+1

This concludes the construction of the sequence .{Nj }. For .i ≥ 1 let .ui ∈ MNi . Then for each .n ∈ N

3 Bergman Space of the Unit Disc

103

 n  n       2 2 . 

ui −

ui  ≤ |ui , uj |   i=1

i=j

i=1

≤



εi εj ui

uj

i=j

≤

 n 

2 εi ui

i=1

≤

n 

εi2

n 

i=1

ui 2

i=1

1

ui 2 . 2 n

≤

i=1

This implies

.

 1 3

ui 2 .

ui 2 ≤

ui 2 ≤ 2 2 n

n

n

i=1

i=1

i=1

Finally we combine this with Lemma 3.4.16 and obtain .

 1  kzi 2 3  |ai |2 , |ai |2 ≤

ai

≤

kzi

2K 2K i

i

i

whenever .zi are finitely many mutually distinct points in . and .ai ∈ C. That shows that . is interpolating for .L2a . With a similar construction interpolating sequences that are dominating can be p shown to exist in .La and in certain Hilbert spaces of analytic functions that are more 2 general than .La , see [6] and [5].

3.4.6 Further Results About Invariant Subspaces with High Index The previous construction was not explicit in the sense that it did not show us examples of functions in invariant subspaces with infinite index. However, we have some indication where to look for such functions, namely they were contained in ⊥ for some interpolating sequence .. One can get away from interpolating .I () sequences by use of the majorization function that we will now define. If .H is a Hilbert space of analytic functions on .D such that .T = (Mz , H) is bounded, and if .M ∈ Lat T , then for the majorization function is defined on .D by

104

S. Richter

k (z) =

. M

sup{|f (z)| : f ∈ M, f ≤ 1} . sup{|f (z)| : f ∈ H, f ≤ 1}

It is clear that .0 ≤ kM (z) ≤ 1 for all .z ∈ D. It is a general fact in reproducing kernel Hilbert spaces that the normalized reproducing kernel .f = kz / kz maximizes the absolute value at z among the functions in the unit ball. The reproducing kernel

PM kz

for .M is .PM kz (w), hence .kM (z) = k . Furthermore, one verifies that for any z

.f ∈ M we have |f (z)| ≤ kM (z) f

kz .

.

Thus, .kM measures how large functions in .M can be compared with general functions in .H. Also, we see that .kM (z) = 0, if and only if z is a common zero of functions in .M. If .H = H 2 and if .M = ϕH 2 for some inner function .ϕ, then since multiplication by .ϕ is an isometric operator .H 2 → M one checks that .kM (z) = |ϕ(z)| for each .z ∈ D. In particular, we note that .kM (z) has nontangential limit equal to 1 at almost every point of .T. Theorem 3.4.18 Let .B = (Mz , L2a ), .M ∈ Lat B with .indM = 1. If .0 < ε < 1 let ε (M) = {z ∈ D : kM (z) < ε}.

.

Then the following are equivalent: (i) (ii) (iii) (iv) (v)

there is .N ∈ Lat B such that .M ⊆ N and .indN > 1, there is .N ∈ Lat B such that .M ⊆ N and .indN = ∞, for every .f ∈ H ∞ we have . f ∞ = Mf∗ |M⊥ , there is .0 < ε < 1 such that .ε (M) is dominating, for all .0 < ε < 1 the set .ε (M) is dominating.

It turns out that if .N = {w ∈ T : nt − limz→w kM (z) = 1} and .P = {w ∈ T : nt − liminfz→w kM (z) = 0}, then .N ∪ P has full Lebesgue measure in .T, see [4], Theorems A and B. Thus, the equivalence of conditions (iv) and (v). There also is a generalization of this theorem to more general contraction operators .(Mz , H), see [5]. In order to illustrate this theorem consider the following example. Example 3.4.19 If .0 = f ∈ L2a extends to be continuous on .D ∪ I for some arc .I ⊆ T, then every invariant subspace that contains f must have index 1. See [4], Theorem 5.11 for a condition that is much weaker than continuity. But note that it is not enough for f to have nontangential limits on an arbitrary set of positive measure .E ⊆ T, see Theorem 6.6 of [4]. (w) Proof For .z ∈ D we have .gz (w) = kz (w)f ∈ [f ] and using the hypothesis on f

kz

one shows that . gz → |f (w)| as .z → w for any w in the .T-interior of I . Then

3 Bergman Space of the Unit Disc

|f (z)| =

.

105

|gz (z)| ≤ kM (z) gz ≤ gz ,

kz

hence as z approaches points in I the limit of .kM (z) equals 1 at each w in the interior of I where .f (w) = 0. Hence the set .ε ([f ]) cannot be dominating for any .0 < ε < 1. 

We conclude this section with a few more words about Theorem 3.4.18 and its proof. If N is the normal operator with .{λn } on the diagonal, then for every .f ∈ H ∞ we have . f (N) = sup{|f (λn )| : n ∈ N}. Hence condition (ii) of Theorem 3.4.9 can be restated as . f ∞ = f (N) for every .f ∈ H ∞ . The implication (iii).⇒(ii) of Theorem 3.4.18 follows from a Theorem of Bercovici and Chevreau which can be considered a generalization of the implication .(ii) ⇒ (iv) of Theorem 3.4.9, see [13, 20]. The implication (iv).⇒(iii) was inspired by a connection between .kM (z) and the value of . (I − zS)(S − z)−1 −1 , where .S = PM⊥ B|M⊥ , and some known conditions on resolvent growth and condition (iii) [8, 21]. Parts of the theorem are p known to hold for .La , but it is an open question, whether (iii).⇒(ii) holds for .p = 2. The paper [7] comes close to providing a Banach space analogue of [13] and [20]. Thus, it may be relevant to resolve this question.

3.5 Bergman Inner Functions A useful theorem in the theory of Hardy spaces is the existence of an inner-outer factorization. If .f ∈ H p , then .f = ϕg, where .ϕ is inner and g is cyclic in p p .H , i.e. there is a sequence of polynomials .pn such that .pn f → 1 in .H . We p will now discuss the existence of an inner-outer factorization for functions in .La . Unfortunately, the results are weaker and less is known than in the Hardy space situation.

3.5.1 An Extremal Problem p

p

Definition 3.5.1 0 < p < ∞. A function ϕ ∈ La is called La -inner

(or Bergmaninner, if the value of p is clear from the context), if ϕ p = 1 and D zn |ϕ|p dA = 0 for all n ≥ 1.

p Another way of stating that ϕ is La -inner is to say that D q|ϕ|p dA π = q(0) for is a representing measure for the origin. every polynomial q, i.e. the measure |ϕ|p dA π If p = 2, then any function of norm one in M  zM is L2a -inner. We will see p momentarily that La -inner functions are plentiful for all values of p.

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Lemma 3.5.2 If M ∈ Lat(Mz , La ), and if ϕ ∈ M is a function of unit norm that satisfies 0 < |ϕ(0)| = sup{|f (0)| : f ∈ M, f p ≤ 1},

(3.7)

.

then ϕ is Bergman-inner. If p = 2, then ϕ ∈ M  zM. If all functions in M are 0 at the origin, then a similar lemma holds. Indeed, if M = (0), then there is a smallest integer k such that there is a function f ∈ M such that f (k) (0) = 0. In that case we consider the extremal problem sup{|f (k) (0)| : f ∈ M, f p ≤ 1} and its solutions. In the notes we will always assume that k = 0, and leave the cases k > 0 for the reader to work out. Proof First assume p = 2. Then for any f ∈ M we have f = g + zh for some g ∈ M  zM and h ∈ M. Then by orthogonality f 2 ≥ g and since f (0) = g(0) we see that g is a better candidate as solution to the extremal problem unless h = 0. This proves ϕ ∈ M  zM. Now let p be arbitrary. Fix n > 0. For |a| < 1 consider the function f (z) = |ϕ(0)| (1+azn )ϕ(z)

(1+azn )ϕ p . Then f ∈ M and hence |ϕ(0)| ≥ |f (0)| = (1+azn )ϕ p . This implies p

p

that (1 + azn )ϕ p ≥ 1 = ϕ p for all |a| < 1. We have (1 + azn )p/2 = 1 + p n n 2 2 az + O(|az | ). Hence  0≤

.

= =

D  D  D

(|(1 + azn )p/2 |2 − 1)|ϕ|p dA (|1 +

p n az + O(|azn |2 )|2 − 1)|ϕ|p dA 2

(p Re azn + O(|azn |2 )|ϕ|p dA

We write a = reit divide by r, and let r → 0. We obtain  Re e

.

it

D

zn |ϕ|p dA ≥ 0



for all t. That is possible only if D zn |ϕ|p dA = 0.



Of course, if ϕ is a solution to the extremal problem, then so is eit ϕ for each t. We make a normalization to obtain uniqueness in many cases. p

Definition 3.5.3 ϕ is called an extremal function for (0) = M ∈ Lat(Mz , La ), if ϕ (k) (0) = sup{Ref (k) (0) : f ∈ M, f p = 1}.

.

Here k is the smallest integer such that there is f ∈ M with f (k) (0) = 0.

3 Bergman Space of the Unit Disc

107

At this point it would be a good exercise to verify that if ϕ is a classical inner function with ϕ(0) > 0, and if M = ϕH p , then ϕ is the unique solution to sup{Ref (0) : f ∈ M, f H p = 1}. (k) (0) Note that sup{Ref (k) (0) : f ∈ M, f p = 1} = sup{ Ref : f ∈ M}. Thus,

f p by considering reciprocals, it is elementary to check that the extremal problem is equivalent to .

inf{ h p : h ∈ M, h(k) (0) = 1}

and one of the two problems has a solution, if and only if the other one does, and solutions are scalar multiples of one another. The inf-problem is asking to minimize the norm over elements in a convex weakly closed set. Since for 1 < p < ∞ p La is a strictly convex reflexive Banach space, it follows that extremal functions exist and are unique for each invariant subspace. L1a is a strictly convex Banach space, so if an extremal function for M exists, then it must be unique. However, it appears to be an open problem to determine, whether every M ∈ Lat(Mz , L1a ) has an extremal function. For 0 < p < 1 both existence and uniqueness are unclear. However, it turns out that if M is either singly generated or zero based, then it always has a unique extremal function. For example, we show that if M = I () is a zero based invariant subspace, then extremal functions always exist. We consider p the minimization problem and assume that 0 ∈ / . Let fn ∈ La be a minimizing sequence, i.e. suppose that fn ∈ I () and fn (0) = 1 for each n and

fn p → inf{ h p : h ∈ I (), h(0) = 1}.

.

p

The unit ball in La is a normal family, hence some subsequence of fn will converge p locally uniformly to some analytic function ϕ. By Fatou’s lemma ϕ ∈ La and by the local uniform convergence, it will have the correct zeros, i.e. ϕ ∈ M. Then ϕ is a solution to the extremal problem. Example 3.5.4 Let 0 < p < ∞, let λ ∈ D, and consider the invariant subspace p Mλ = {f ∈ La : f (λ) = 0}. Then the extremal function is ϕ(z) = C

.

C=

λ−z 1 − λz

 1+

 2/p p λ−z 1−λ 2 1 − λz

−1/p p λ  1 + (1 − |λ|2 ) 2 |λ|

This formula was found by Osipenko-Stessin, [38] and independently by DurenKhavinson-Shapiro-Sundberg [24]. Further exact formulas are known only for invariant subspaces with a higher order zero at a single point and for invariant subspaces that are generated by a singular atomic inner function with a single point

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S. Richter

mass, see [23]. In [24] it is also proved that extremal functions for zero based invariant subspaces associated with finite zero sets have analytic continuations to a neighborhood of the unit disc. Furthermore, if I () is any zero-based invariant subspace, then the extremal function has an analytic continuation across any arc in the unit circle on which the zeros do not accumulate. This was proved by Sundberg [47].

3.5.2 Contractive Divisors Via the Biharmonic Function Next we recall Green’s formula. We write . = 1 ∂ 2 ( ∂x

∂ − i ∂y )

and .∂ =

∂ ∂z

=

1 ∂ 2 ( ∂x

∂ + i ∂y ).

+

∂2 ∂y 2

Then, if .u, v ∈

= 4∂∂, where .∂ =

∂ ∂z

=

C 2 (D)



 .

∂2 ∂x 2

D

uv − vudA =

∂v ∂u − v ds. ∂n ∂n T u

(3.8)

∂ is the outwardnormal Here ds denotes arclength and . ∂n  derivative.  1−zw  The Green’s function for .D is .G(z, w) = log  z−w . It has the property that

 1 G(z, w)u(w)dA(w), .u(z) = − 2π D whenever .u ∈ C 2 (D) with .u = 0 on .T. Conversely, if . ∈ C(D) ∩ C 2 (D), and if  1 G(z, w) (z)dA(w), .u(z) = G[ ](z) = − 2π D then .u ∈ C(D) ∩ C 2 (D), .u = and .u = 0 on .T. The biharmonic function for .D is   1 (1 − |z|2 )(1 − |w|2 ) 2 2 2 ) . . (z, w) = (1 − |z| )(1 − |w| ) − |z − w| log(1 + 16 |z − w|2 (1−|z| )(1−|w| ) One verifies that . (z, w) = |z−w| ) for .f (x) = x−log(1+x) > 0. 16 f ( |z−w|2 Hence . (z, w) > 0 for .z, w ∈ D. . has the property that 2

2

2

 1

(z, w)2 u(w)dA(w), .u(z) = π D whenever .u ∈ C 4 (D) with .u =

∂u ∂n

= 0 on .T.

Lemma 3.5.5 Let . ∈ C(D) ∩ C 2 (D), define .u = G[ ] and assume .u ∈ C 4 (D).

3 Bergman Space of the Unit Disc

109

If . D h(z) (z)dA(z) = 0 for every harmonic function .h ∈ C 2 (D), then .u = ∂u ∂n = 0 on .T and .u = [ ]. Proof Let .h ∈ C 2 (D) be harmonic on .D. Then since .u = 0 on .T Green’s Theorem implies 0=

.





 D

h dA =

D

(hu − uh)dA =

T

h

∂u ds. ∂n

We can take .h(z) = zn and .h(z) = zn for all .n ≥ 0, hence we conclude . ∂u ∂n = 0 on .T. Hence .u = [2 u] = [ ].

 p

Theorem 3.5.6 (Contractive Divisor Property) If .M ∈ Lat(Mz , La ) is either a zero based invariant subspace with extremal function .ϕ, or if .M = [ϕ] for some p Bergman-inner function .ϕ, then .f ∈ M we have .f/ϕ ∈ La and . fϕ p ≤ f p for all .f ∈ [ϕ]. Hedenmalm was the first to discover contractive divisors in .L2a [29]. His theorem p was extended to .La by Duren, Khavinson, Shapiro, and Sundberg by use of the biharmonic function [23, 24]. We give an outline of the approach. Proof We do the proof under the additional assumption that .|ϕ|p is sufficiently smooth so that we can apply Lemma 3.5.5. As mentioned above that will be the case when .ϕ is the extremal function for a zero based invariant subspace associated with finitely many zeros. For the general case a more careful analysis is required. We take . (z) = |ϕ(z)|p − 1. Then our assumptions imply that we can use Lemma 3.5.5 to conclude that .u = G[ ] = [|ϕ|p ]. Note that .|ϕ|p is subharmonic in .D, hence p .|ϕ| ≥ 0. The positivity of . implies that .u = G[ ] ≥ 0. Let q be an arbitrary polynomial, then by Green’s Theorem p



p

qϕ p − q p =

.

 =  =

D D D

|q|p

dA π

|q|p u

dA π

u|q|p

dA π

since u =

∂u = 0 on T ∂n

≥ 0 since u ≥ 0 and |q|p is subharmonic. This shows that the theorem holds for functions of the form .qϕ for a polynomial q. p If .f ∈ [ϕ], then there is a sequence of polynomials .qn such that .qn ϕ → f in .La . Then .qn (z) → f (z)/ϕ(z) at least for all .z ∈ D where .ϕ(z) = 0. Then by Fatou’s lemma we obtain

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S. Richter p

p

p

p

f/ϕ p ≤ lim inf qn p ≤ lim inf qn ϕ p = f p .

.

n→∞

n→∞

We omit the extra argument needed for the zero based invariant subspaces. Corollary 3.5.7 Let .ϕ ∈ equivalent: (i) (ii) (iii) (iv)

p La



with . ϕ p = 1 and .ϕ(0) > 0. Then the following are

p

ϕ is .La -inner, .ϕ(0) = sup{Ref (0) : f ∈ [ϕ], f = 1}, f . p ≤ f p for each .f ∈ [ϕ], ϕ p . f ϕ p ≤ f H p for all .f ∈ H . .

Proof The implications (ii) .⇒ (i) .⇒ (iii) have been done in previous theorems. p (iii) .⇒ (ii): If .f ∈ [ϕ] with . f = 1, then by hypothesis .f/ϕ ∈ La and by Theorem 3.3    f (0)    .  ϕ(0)  ≤ f/ϕ p ≤ f p ≤ 1. This shows .Ref (0) ≤ |f (0)| ≤ |ϕ(0)| = ϕ(0), i.e. (ii) holds. (i) .⇒ (iv): It is enough to prove the inequality (iv) for polynomials f . Note that the hypothesis (i) says that the measure . |dz| 2π is the sweep into the circle of the . That means that if a continuous functions f on .T has harmonic measure .|ϕ|p dA π



|dz| dA p extension u in .D, then . T f (z) 2π = D u|ϕ| π . Indeed, by definition it holds for all functions of the form .f (eit ) = eint , and the general case follows by an approximation argument. If f is a polynomial, then .|f |p is subharmonic in .D so that .|f (z)|p ≤ P [|f |p ](z), where .P [|f |p ](z) denotes the Poisson integral of .|f |p . Then 

p

f ϕ p =

.

 ≤ =

D

D  T

|f |p |ϕ|p

dA π

P [|f |p ](z)|ϕ|p |f |p

dA π

|dz| 2π

p

= f H p (iv).⇒ (i): This follows by a variational argument along the lines of the proof of p Lemma 3.5.2. Let .n > 0 and .|a| < 1. Then the hypothesis implies . (1 + azn )ϕ p ≤ p n n p n 2

1 + az H p . As above we use .|1 + az | = 1 + pReaz + O(|a| ), subtract 1 on

each side, divide by .|a|, and let .|a| → 0. We obtain .Reeit D zn |ϕ|p dA π ≤ 0 for all t. This implies (i). 

3 Bergman Space of the Unit Disc

111

In Theorem 3.5.6 it was shown that the contractive divisor property holds for zero based invariant subspaces, but it was left open whether it held for every singly generated invariant subspace. The following is from [3]. p

Theorem 3.5.8 Let .0 < p < ∞. If .M ∈ Lat(Mz , La ) is either zero-based or singly generated, then .M = [ϕ] for the extremal function .ϕ of .M. If .p = 2, then it was shown in [3] that every .M ∈ Lat(Mz , L2a ) is of the form .M = [M  zM], i.e. .M is generated as an invariant subspace by the Bergmaninner functions in .M  zM. p

Corollary 3.5.9 (Bergman Inner-Outer Factorization) If .f ∈ La , .f = 0, then .f = ϕh, where (i) (ii) (iii) (iv)

p

ϕ, h ∈ La , p h is cyclic in .La , .ϕ is Bergman-inner the inequality . q fϕ p ≤ qf p holds for every polynomial q. .

Properties (i)–(iv) determine .ϕ and h uniquely, but it was shown that conditions (i)–(iii) do not, see [17]. Proof We use Theorem 3.5.8 to choose a Bergman-inner function .ϕ such that .[f ] = p [ϕ], then by Theorem 3.5.6 condition (iv) is satisfied and .h = f/ϕ ∈ La . We have to show that h is cyclic. Since .ϕ ∈ [f ], there will be a sequence of polynomials such that .pn f → ϕ. Then by property (iv) we have

pn h − 1 p =

.

pn f − ϕ

p ≤ pn f − ϕ p → 0. ϕ 

p Lat(Mz , La )

Corollary 3.5.10 Let .0 < p < ∞. If .(0) = M ∈ is either zero-based p or singly generated, and if .ϕ is the extremal function for .M, then .H p ⊆ M ϕ ⊆ La with contractive inclusions. This follows easily from Theorem 3.5.6, Corollary 3.5.7, and Theorem 3.5.8. It is interesting to contrast this with the situation for the invariant subspaces of the 2 Dirichlet shift, where we have .D ⊆ M ϕ ⊆ H , whenever .M is any nonzero invariant subspace of .(Mz , D) and .ϕ is its extremal function, see [41].

3.5.3 The Reproducing Kernel Approach For .p = 2 Shimorin has an approach to Theorem 3.5.8 that leads to a simple proof. These methods can also be used to prove Theorem 3.5.6 in the Hilbert space case. The proofs are based on the theory of reproducing kernels. We have collected a few

112

S. Richter

basic facts in the appendix. For more general facts about reproducing kernel Hilbert spaces see e.g. [10] or [39]. We start with a version of Shimorin’s Theorem, see [46]. Theorem 3.5.11 Let .K be a Hilbert space of analytic functions on .D with reproducing kernel .kz (w). Suppose • .(Mz , K) is bounded, • .zK is closed in .K, and • .dim K  zK = 1. If .ϕ ∈ K  zK with . ϕ = 1, then there is a positive definite function .uz (w) = u(z, w) such that kz (w) =

.

ϕ(z)ϕ(w) − zwuz (w) , (1 − zw)2

if and only if

g + zf 2 ≤ 2( f 2 + zg 2 ) for all f, g ∈ K.

.

We have deliberately stated the Theorem in such a way that it can immediately be applied to index 1 invariant subspaces of .(Mz , L2a ). The parts of the hypothesis that .(Mz , K) be bounded and .zK be closed in .K can also be derived from either one of the two conditions of the conclusion of the Theorem. We should also mention that Shimorin’s paper contains another version, which applies that invariant subspaces of .L2a with arbitrary index. takes Proof The hypothesis implies that .f − f, ϕϕ ∈ zK, hence .Lf = f −f,ϕϕ z .K into .K and the closed graph theorem can be used to show that L is bounded. Furthermore, we see that .f = f, ϕϕ + zLf is an orthogonal decomposition and . zLf ≤ f . Suppose that . g + zf 2 ≤ 2( f 2 + zg 2 ) for all .f, g ∈ K. Then for .h, f ∈ K we have

Lh + zf 2 ≤ 2( f 2 + zLh 2 )

.

= 2( f 2 + h 2 ). √ 2. Hence so does its Thus, the operator .(L Mz ) : K ⊕ K → K has norm .≤  ∗ L and we conclude that .A = 2I − LL∗ − Mz Mz∗ ≥ 0. This implies adjoint . Mz∗ that .uz (w) = Akz , kw  is positive definite. Well-known properties of reproducing kernels imply .Mz Mz∗ kz , kw  = zwkz (w). Furthermore, we calculate

3 Bergman Space of the Unit Disc

113

(L∗ kz )(w) = kz , Lkw 

.

= kz , =

kw − ϕ(w)ϕ  z

kz (w) − ϕ(z)ϕ(w) . z

The fact that .Lϕ = 0 now implies that (LL∗ kz )(w) =

.

kz (w) − ϕ(z)ϕ(w) . zw

This leads to the identity uz (w) = 2kz (w) −

.

kz (w) − ϕ(z)ϕ(w) − zwkz (w), zw

which is equivalent to kz (w) =

.

ϕ(z)ϕ(w) − zwuz (w) (1 − zw)2

Conversely, assume that the reproducing kernel is of the form as required in the theorem with .uz (w) ≥ 0. We can reverse the above calculations and conclude that ∗ ∗ .(2I − LL − Mz Mz )kz , kw  = uz (w) is positive definite. Hence if .z1 , . . . , zn ∈ D and .a1 , . . . , an ∈ C, then

L∗

n 

.

i=1

ai kzi 2 + Mz∗

n  i=1

ai kzi 2 ≤ 2

n 

ai kzi 2 .

i=1

Finite linear of reproducing kernels are dense in .K, hence the column  combinations  √ L∗ : K → K ⊕ K has norm less than or equal to . 2. Taking the operator . Mz∗ adjoint we see that for all .g, f ∈ K the identity .Lzg = g implies . g + zf 2 ≤ 2( f 2 + zg 2 ). 

Corollary 3.5.12 Let .M ∈ Lat(Mz , L2a ) with .indM = 1. If .ϕ ∈ M  zM with . ϕ = 1, then .M = [ϕ] and . f/ϕ ≤ f for all .f ∈ M. Proof (See [37]) We have .[ϕ] ⊆ M. Finite linear combinations of reproducing kernels are dense in any Hilbert function space, hence by Lemma 3.6.5 it will suffice to show that .PM kz ∈ [ϕ] for each .z ∈ D. Note that .K = L2a satisfies the hypothesis of Shimorin’s theorem and has reproducing kernel as in the theorem with .ϕ(z) = 1 and .uz (w) = 0. Hence

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S. Richter

g + zf 2 ≤ 2( f 2 + zg 2 ) for all f, g ∈ L2a .

.

Of course, this implies that the inequality holds for all .f, g ∈ M. Then by another application of Shimorin’s Theorem we conclude that PM kz (w) =

.

ϕ(z)ϕ(w) − zwuz (w) (1 − zw)2

for some positive definite function .uz (w). Taking .z = w we see that |ϕ(z)|2 − |z|2 uz (z) = (1 − |z|2 )2 PM kz , kz  = (1 − |z|2 )2 PM kz 2 ≥ 0.

.

(3.9)

Then by the Cauchy-Schwarz inequality for positive definite functions (Lemma 3.6.4) we have |zwuz (w)|2 ≤ |z|2 |w|2 uz (z)uw (w) ≤ |ϕ(z)|2 |ϕ(w)|2 for all z, w ∈ D.

.

Now fix .z ∈ D. If . PM kz = 0, then it is clear that .PM kz ∈ [ϕ]. Thus, we assume that . PM kz = 0. Then (3.9) implies that .ϕ(z) = 0 and the function vz (w) =

.

zwuz (w) ϕ(z)ϕ(w)

= 1 − (1 − zw)2

(PM kz )(w) ϕ(z)ϕ(w)

is a meromorphic function of w and .|vz (w)| ≤ 1 for all .w ∈ D except perhaps at points where .ϕ(w) = 0. But those points must be removable singularities and we may assume that .vz ∈ H ∞ (D). This implies that PM kz (w) =

.

1 − vz (w) ϕ(z)ϕ(w) (1 − zw)2

and hence .PM kz is an .H ∞ -multiple of .ϕ, thus .PM kz ∈ [ϕ]. Finally we show the contractive divisor property. By Theorem 3.6.6 we have to check that kz (w) −

.

PM kz (w) ϕ(z)ϕ(w)

=

zwuz (w) ϕ(z)ϕ(w)(1 − zw)2

is positive definite, and that follows from the Schur product theorem, which says that the pointwise product of positive definite functions is positive definite (see Theorem 3.6.7). 

3 Bergman Space of the Unit Disc

115

3.5.4 Further Contractive Divisor Results In [31, 32] Hedenmalm, Jacobson, and Shimorin were able to use an approach using certain weighted biharmonic functions to show the following stronger version of the expansive multiplier property: Theorem 3.5.13 Let .0 < p < ∞ and let .1 ⊆ 2 ⊆ D be two zero-sequences for p La . For .i = 1, 2 let .ϕi be the Bergman inner function for the zero-based invariant subspace .I (i ). Then

.

ϕ1 f p ≤ ϕ2 f p for all polynomials f.

.

For .p = 2 the paper [37] by McCullough and Richter contains a proof of this result that is based on the observation that a version of Shimorin’s theorem 3.5.11 holds for all spaces of analytic functions whose reproducing kernel is of the form bz (w) =

.

1 1 − ϕ(z)ϕ(w)(1 − uz (w))

for some analytic function .ϕ and some positive definite function .uz (w) such that bz (z) → ∞ as .|z| → 1, and that for every index 1 invariant subspace .M of .L2a with Bergman inner function .ϕ there is such a kernel .bz (w) such that .PM kz (w) = bz (w)ϕ(z)ϕ(w). Let . ⊆ D be a finite zero set, let .ϕ be the extremal function for the zero-based p p invariant subspace .I () ⊆ La . If .z0 ∈ D, then . ∪ {z0 } is an .La -zero sequence, and we let .ϕ0 be the extremal function for .I ( ∪ {z0 }). Then .G = ϕ0 /ϕ may be p z0 −z . considered to be the analogue for .La of a single Blaschke factor .ϕz0 (z) = |zz00 | 1−z 0z Of course, G will depend on p, on .z0 , and on .. We mentioned earlier that .ϕ and .ϕ0 extend to be analytic in a neighborhood of the closed unit disc. From the original contractive divisor property and an approximate identity argument one shows that .|ϕ(z)|, |ϕ0 (z)| ≥ 1 for all .|z| = 1. Hence G extends to be analytic in a neighborhood of the closed unit disc, and, in fact, .G = ϕz0 (z)f for some classical outer function f . What else can one say about f ? By use of another approximate identity argument and Theorem 3.5.13 one can show that .|f (z)| ≥ 1 for all z in the closure of D, see [33, p. 270, Exercise 6] . For .p = 2 Aleman-Richter, [1] use the reproducing kernel approach of [37] to show that .

1 ≤ Ref (z) ≤ |f (z)| ≤ 1 +

.

1 − |λ|2 |1 − λz|

≤ 3.

It is a question of Hedenmalm’s whether the function G is always starlike, i.e. is univalent and maps onto a starlike region.

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3.6 Appendix 3.6.1 The Schur Test Lemma 3.6.1 (Schur Test) Let (X, μ) be a σ -finite measure space, let 1 < p, q < ∞ be dual indices, and let K(x, y) ≥ 0 be μ × μ measurable. If there are C1 , C2 > 0 and a μ -measurable h > 0 such that  K(x, y)hq (y)dμ(y) ≤ C1 hq (x) a.e. x and

.



X

K(x, y)hp (x)dμ(x) ≤ C2 hp (y) a.e. y, X

then the integral operator  TK f (x) =

K(x, y)f (y)dμ(y)

.

X 1/q

1/p

is bounded Lp (μ) → Lp (μ) with TK ≤ C1 C2 . See e.g. [52], Theorem 3.6.

3.6.2 Carleson’s Interpolation Theorem Theorem 3.6.2 (Carleson’s H ∞ -interpolation Theorem) Let  = {λn } ⊆ D be a sequence of distinct points. Then the following are equivalent: (i) There is K > 0 such that  is H ∞ -interpolating with constant K, i.e. if whenever {wn } is in the unit ball of ∞ , then there is ϕ ∈ H ∞ with ϕ ∞ ≤ K and ϕ(λn ) = wn for each n, (ii) There are C, δ > 0 such that    λn −λm  (a)  1−λ  ≥ δ for all n = m, i.e. the sequence is separated, and  n λm (b) λn ∈Sh (1 − |λj |) ≤ Ch for every Carleson box Sh (Carleson measure condition). In fact, the constants of the theorem depend on each other, but not on the sequences. In particular, the fact that we will need is that for every pair δ, C > 0, there is a K > 0 such that whenever  = {λn } ⊆ D is a sequence of distinct points that satisfies (ii) with the given δ and C, then it satisfies condition (i) with the constant K, see [28], Chapter 7.

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3.6.3 Positive Definite Functions Let X be a set, and let .F(X) = {f : X → C} be the collection of all complex-valued functions on X. We say that .H is a Hilbert function space on X, if .H ⊆ F(X) is a Hilbert space such that the inclusion from .H to .F(X) is 1–1, and such that for each .z ∈ X the evaluation functional .f → f (z) is continuous. Thus, if .H is a Hilbert function space on X, then by the Riesz representation theorem for each .z ∈ X the evaluation functional can be represented by a unique element .kz ∈ H so that .f (z) = f, kz  for each .f ∈ H. .kz (w) = kz , kw  is called the reproducing kernel for .H. Note that . kz 2 = kz , kz  = kz (z) for every .z ∈ X, and that . kz = 0, if and only if every function in .H is 0 at z. Definition 3.6.3 Let X be a set. A function .u : X × X → C is called positive definite, if for all .n ∈ N, all .z1 , . . . , zn ∈ X, and all .a1 , . . . , an ∈ C we have  .

ai a j u(zi , zj ) ≥ 0.

i,j

We will write .u ≥ 0. Thus, .u ≥ 0, if and only if for all .n ∈ N and all .z1 , . . . , zn ∈ X the .n × n matrix with entries .uij = u(zi , zj ) is positive semi definite. If .kz (w) is the reproducing kernel for a Hilbert function space .H, then .u(z, w) = kz (w) is positive definite. Indeed, we have  .

i,j

ai a j kzi (zj ) =

 i,j

ai a j kzi , kzj  =



ai kzi 2 ≥ 0.

i

The Moore-Aronszajn Theorem states that the converse to the above statement is true as well: If .u(z, w) is positive definite, then there is a unique Hilbert function space on X such that .kz (w) = u(z, w) is its reproducing kernel. This last fact will imply the following lemma, but one can also give a more elementary proof. Lemma 3.6.4 If .u : X × X → C is positive definite, then .u(z, z) ≥ 0, .|u(z, w)|2 ≤ u(z, z)u(w, w), and .u(z, w) = u(w, z) for all .z, w ∈ X. Proof .u(z, z) ≥ 0 follows by considering .1 × 1 matrices with .z1 = z, that the other two conditions are true follows by considering .2 × 2 matrices with .z1 = z, .z2 = w. 

Lemma 3.6.5 If .H is a Hilbert function space with reproducing kernel .kw (z), if M ⊆ H is a subspace, then .M is a Hilbert function space, and the function .(PM kw )(z) is the reproducing kernel for .M. .

Proof It is clear the .M is a Hilbert function space, and if .z ∈ X, then .PM kz ∈ M and it satisfies that for all .f ∈ M we have .f (z) = f, kz  = PM f, kz  = f, PM kz . 

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Theorem 3.6.6 If .H(k) and .H(s) are reproducing kernel Hilbert spaces on X, if F : X → C, then

.

MF H(s)→H(k) ≤ 1

.

if and only if kz (w) − F (z)F (w)sz (w)

.

is positive definite. A special case that deserves to be mentioned separately, is when .sz (w) = 1. Then .H(s) = C and the theorem states that . F H(k) ≤ 1, if and only if kz (w) − F (z)F (w) is positive definite.

.

Proof It is clear that . MF H(s)→H(k) ≤ 1, if and only if . MF∗ H(k)→H(s) ≤ 1, and the latter condition is equivalent to

g 2H(k) − MF∗ g 2H(s) ≥ 0

.

for all g that are finite linear combinations of reproducing kernels. That condition is seen to be precisely that .kz (w) − F (z)F (w)sz (w) ≥ 0. 

Theorem 3.6.7 (Schur Product Theorem) If .u, v : X × X → C are positive definite, then so is their product .k(z, w) = u(z, w)v(z, w). Proof An easy proof can be based on the observation that the positive definite functions .u, v can be written as “sums of squares”.  More precisely, by the MooreAronszajn theorem we can write .u(z, w) = α uα (z)uα (w) and .v(z, w) =  v (z)v (w), where . {u } is an orthonormal basis for .H(u) and .{vβ } is an β α β β orthonormal basis for .H(v). Then .kz (w) is also a sum of this type and that is easily seen to be positive definite. If one is bothered by the possibly infinite sums, then remember that positive definiteness is defined in terms of certain square matrices being positive semidefinite, so we may always restrict attention to finite subsets of X, and then the corresponding Hilbert spaces will be finite dimensional. 

References 1. ALEMAN, A., AND RICHTER, S. Single point extremal functions in Bergman-type spaces. Indiana Univ. Math. J. 51, 3 (2002), 581–605. 2. ALEMAN, A., RICHTER, S., AND ROSS, W. T. Pseudocontinuations and the backward shift. Indiana Univ. Math. J. 47, 1 (1998), 223–276. 3. ALEMAN, A., RICHTER, S., AND SUNDBERG, C. Beurling’s theorem for the Bergman space. Acta Math. 177, 2 (1996), 275–310.

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4. ALEMAN, A., RICHTER, S., AND SUNDBERG, C. The majorization function and the index of invariant subspaces in the Bergman spaces. J. Anal. Math. 86 (2002), 139–182. 5. ALEMAN, A., RICHTER, S., AND SUNDBERG, C. Analytic contractions, nontangential limits, and the index of invariant subspaces. Trans. Amer. Math. Soc. 359, 7 (2007), 3369–3407. 6. ALEMAN, A., RICHTER, S., AND SUNDBERG, C. Nontangential limits in P t (μ)-spaces and the index of invariant subspaces. Ann. of Math. (2) 169, 2 (2009), 449–490. 7. AMBROZIE, C., AND MÜLLER, V. Invariant subspaces for polynomially bounded operators. J. Funct. Anal. 213, 2 (2004), 321–345. 8. APOSTOL, C. Ultraweakly closed operator algebras. J. Operator Theory 2, 1 (1979), 49–61. 9. APOSTOL, C., BERCOVICI, H., FOIAS, C., AND PEARCY, C. Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra. I. J. Funct. Anal. 63, 3 (1985), 369–404. 10. ARONSZAJN, N. Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 (1950), 337–404. 11. AXLER, S. Bergman spaces and their operators. In Surveys of some recent results in operator theory, Vol. I, vol. 171 of Pitman Res. Notes Math. Ser. Longman Science and Technology, Harlow, 1988, pp. 1–50. 12. BEKOLLÉ, D., AND BONAMI, A. Inégalités à poids pour le noyau de Bergman. C. R. Acad. Sci. Paris Sér. A-B 286, 18 (1978), A775–A778. 13. BERCOVICI, H. Factorization theorems and the structure of operators on Hilbert space. Ann. of Math. (2) 128, 2 (1988), 399–413. 14. BERCOVICI, H., FOIAS, C., AND PEARCY, C. Dilation theory and systems of simultaneous equations in the predual of an operator algebra. I. Michigan Math. J. 30, 3 (1983), 335–354. 15. BERGMAN, S. The kernel function and conformal mapping, revised ed. Mathematical Surveys, No. V. American Mathematical Society, Providence, R.I., 1970. 16. BERGMANN, S. Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande, I. J. Reine Angew. Math. 1933, 169 (1933), 1–42. 17. BORICHEV, A., AND HEDENMALM, H. Harmonic functions of maximal growth: invertibility and cyclicity in Bergman spaces. J. Amer. Math. Soc. 10, 4 (1997), 761–796. 18. BROWN, L., SHIELDS, A., AND ZELLER, K. On absolutely convergent exponential sums. Trans. Amer. Math. Soc. 96 (1960), 162–183. 19. CARADUS, S. R. Universal operators and invariant subspaces. Proc. Amer. Math. Soc. 23 (1969), 526–527. 20. CHEVREAU, B. Sur les contractions à calcul fonctionnel isométrique. II. J. Operator Theory 20, 2 (1988), 269–293. 21. CHEVREAU, B., AND PEARCY, C. Growth conditions on the resolvent and membership in the classes A and Aℵ0 . J. Operator Theory 16, 2 (1986), 375–385. 22. COIFMAN, R. R., ROCHBERG, R., AND WEISS, G. Factorization theorems for Hardy spaces in several variables. Ann. of Math. (2) 103, 3 (1976), 611–635. 23. DUREN, P., KHAVINSON, D., SHAPIRO, H. S., AND SUNDBERG, C. Contractive zero-divisors in Bergman spaces. Pacific J. Math. 157, 1 (1993), 37–56. 24. DUREN, P., KHAVINSON, D., SHAPIRO, H. S., AND SUNDBERG, C. Invariant subspaces in Bergman spaces and the biharmonic equation. Michigan Math. J. 41, 2 (1994), 247–259. 25. DUREN, P., AND SCHUSTER, A. Bergman spaces, vol. 100 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2004. 26. DUREN, P. L. Theory of H p spaces. Pure and Applied Mathematics, Vol. 38. Academic Press, New York-London, 1970. 27. FORELLI, F., AND RUDIN, W. Projections on spaces of holomorphic functions in balls. Indiana Univ. Math. J. 24 (1974/75), 593–602. 28. GARNETT, J. B. Bounded analytic functions, first ed., vol. 236 of Graduate Texts in Mathematics. Springer, New York, 2007. 29. HEDENMALM, H. A factorization theorem for square area-integrable analytic functions. J. Reine Angew. Math. 422 (1991), 45–68. 30. HEDENMALM, H. An invariant subspace of the Bergman space having the codimension two property. J. Reine Angew. Math. 443 (1993), 1–9.

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31. HEDENMALM, H., JAKOBSSON, S., AND SHIMORIN, S. A maximum principle à la Hadamard for biharmonic operators with applications to the Bergman spaces. C. R. Acad. Sci. Paris Sér. I Math. 328, 11 (1999), 973–978. 32. HEDENMALM, H., JAKOBSSON, S., AND SHIMORIN, S. A biharmonic maximum principle for hyperbolic surfaces. J. Reine Angew. Math. 550 (2002), 25–75. 33. HEDENMALM, H., KORENBLUM, B., AND ZHU, K. Theory of Bergman spaces, vol. 199 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. 34. HEDENMALM, H., RICHTER, S., AND SEIP, K. Interpolating sequences and invariant subspaces of given index in the Bergman spaces. J. Reine Angew. Math. 477 (1996), 13–30. 35. HOROWITZ, C. Zeros of functions in the Bergman spaces. Duke Math. J. 41 (1974), 693–710. 36. KORENBLUM, B. An extension of the Nevanlinna theory. Acta Math. 135, 3–4 (1975), 187– 219. 37. MCCULLOUGH, S., AND RICHTER, S. Bergman-type reproducing kernels, contractive divisors, and dilations. J. Funct. Anal. 190, 2 (2002), 447–480. 38. OSIPENKO, K. Y., AND STESIN, M. I. Recovery problems in Hardy and Bergman spaces. Mat. Zametki 49, 4 (1991), 95–104, 159. 39. PAULSEN, V. I., AND RAGHUPATHI, M. An introduction to the theory of reproducing kernel Hilbert spaces, vol. 152 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. 40. RICHTER, S., AND SHIELDS, A. Bounded analytic functions in the Dirichlet space. Math. Z. 198, 2 (1988), 151–159. 41. RICHTER, S., AND SUNDBERG, C. Invariant subspaces of the Dirichlet shift and pseudocontinuations. Trans. Amer. Math. Soc. 341, 2 (1994), 863–879. 42. SEIP, K. Beurling type density theorems in the unit disk. Invent. Math. 113, 1 (1993), 21–39. 43. SEIP, K. On Korenblum’s density condition for the zero sequences of A−α . J. Anal. Math. 67 (1995), 307–322. 44. SHAPIRO, H. S., AND SHIELDS, A. L. On the zeros of functions with finite Dirichlet integral and some related function spaces. Math. Z. 80 (1962), 217–229. 45. SHIELDS, A. L., AND WILLIAMS, D. L. Bonded projections, duality, and multipliers in spaces of analytic functions. Trans. Amer. Math. Soc. 162 (1971), 287–302. 46. SHIMORIN, S. Wold-type decompositions and wandering subspaces for operators close to isometries. J. Reine Angew. Math. 531 (2001), 147–189. 47. SUNDBERG, C. Analytic continuability of Bergman inner functions. Michigan Math. J. 44, 2 (1997), 399–407. 48. VUKOTI C´ , D. The isoperimetric inequality and a theorem of Hardy and Littlewood. Amer. Math. Monthly 110, 6 (2003), 532–536. 49. WERMER, J. On invariant subspaces of normal operators. Proc. Amer. Math. Soc. 3 (1952), 270–277. 50. ZAHARJUTA, V. P., AND JUDOVIÇ, V. I. The general form of a linear functional in Hp . Uspehi Mat. Nauk 19, 2 (116) (1964), 139–142. 51. ZHU, K. Duality of Bloch spaces and norm convergence of Taylor series. Michigan Math. J. 38, 1 (1991), 89–101. 52. ZHU, K. Operator theory in function spaces, second ed., vol. 138 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2007. 53. ZYGMUND, A. Trigonometric series: Vols. I, II. Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions.

Chapter 4

Model Spaces Stephan Ramon Garcia

2020 Mathematics Subject Classification 47B35, 46E22, 47B32, 30J05, 30J10, 30J15, 47B91

4.1 Introduction Model spaces arise as backward-shift invariant subspaces of the Hardy space .H 2 . A closer inspection reveals that they are of independent interest for their functiontheoretic behavior and operator-theoretic connections. For example, functions in model spaces are characterized by a form of generalized analytic continuation and compressions of the unilateral shift to model spaces provide concrete functional models for a certain class of Hilbert-space contractions. This chapter is a brief and selective overview of the theory of model spaces, an invitation to the novice rather than a refresher for the experienced. These notes parallel the series of three introductory lectures on model spaces given by the author as part of the “Focus Program on Analytic Function Spaces and Their Applications” which was held virtually at the Fields Institute in 2021. The author gave a similar series of lectures in 2013 at the Centre de Recherches Mathématiques. Those presentations later evolved into a survey article [24] (coauthored with W. T. Ross) and a book devoted to model spaces [21] (coauthored with J. Mashreghi and W. T. Ross). Both sources are good starting points for readers looking for more information than this brief sketch provides. This chapter, which owes much to [21, 24], is part of a volume that covers many related topics, both introductory and advanced. Consequently, we do not reinvent the wheel or step on the toes of the other lecturers. Overall, we present the material in an economical fashion and in roughly the same order and context as they appeared in the lectures. We keep things short and suppress many of the proofs since they can

S. R. Garcia () Pomona College, Department of Mathematics and Statistics, Claremont, CA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mashreghi (ed.), Lectures on Analytic Function Spaces and their Applications, Fields Institute Monographs 39, https://doi.org/10.1007/978-3-031-33572-3_4

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be found elsewhere. In many cases, we present finite-dimensional examples, for we find them the friendliest point of entry for most topics. Although experts may be unimpressed by this approach, we hope that the novice will find it instructive and informative. In a brief survey such as this, many topics get overlooked. We focus here on model spaces themselves, rather than operators that interact with them, although we consider shift operators and a few other examples in so far as they illuminate the structure of model spaces and their elements. On the other hand, commutant lifting, interpolation problems, Hankel operators, and truncated Toeplitz operators do not appear. Fortunately, these topics and much more are covered in other chapters in this volume. More thorough treatments of model spaces can be found in [1, 8, 10, 21, 41– 43, 47, 48] and the references therein.

4.2 Preliminaries Model spaces involve Hardy spaces on the open unit disk .D. This material is classical and the details are in [14, 27, 32, 33, 38, 39, 44]. We assume that the reader is familiar with the Lebesgue spaces .L2 = L2 (T) and .L∞ = L∞ (T) on the unit circle .T, and with the Hilbert space .2 (Z) of square-summable complex sequences indexed by the set of integers .Z.

4.2.1 The Hardy Space We identify each .f ∈ L2 with its sequence .(f(n))n∈Z of Fourier coefficients in 2 . (Z). Let m denote normalized Lebesgue measure on .T. If something holds malmost everywhere on .T, we often say that it holds “a.e. on .T.” Parseval’s theorem says that  f 2L2 =

.

T

|f (ζ )|2 dm(ζ ) =



 2 |f(n)|2 = (f(n))n∈Z 2 (Z) .

n∈Z

The Hardy space .H 2 is the set of all analytic functions on .D such that  f H 2 = lim

.

r→1−

T

1 2 |f (rζ )|2 dm(ζ ) < ∞;

the integral means above increase with .r ∈ [0, 1). For each .f ∈ H 2 , the nontangential limiting value .f (ζ ) exists (and is finite) for m-almost every .ζ ∈ T. The resulting boundary function, also denoted f , belongs to .L2 and satisfies

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f L2 = f H 2 . Moreover,

.

f (ζ ) =

∞ 

.

f(n)ζ n

n=0

m-almost everywhere on .T, in which the nth Fourier coefficient .f(n) equals the nth Taylor coefficient for f . Consequently, we drop the subscripts and regard .H 2 as a subspace of .L2 (the word “subspace” here means “norm-closed linear manifold”). As such, .H 2 is a Hilbert space with inner product  f, gH 2 =

.

T

f (ζ )g(ζ ) dm(ζ ) =

∞ 

g (n) = f, gL2 . f(n)

n=0

When the Hilbert space in question (either .H 2 or .L2 ) is clear from context, we may drop the subscript from the inner product. For .1  p < ∞, analogous definitions exist for the Hardy spaces .H p , which are identified with subspaces of .Lp . For .p = ∞, we let .H ∞ denote the algebra of bounded analytic functions on .D endowed with the supremum norm. The boundary function of an .H ∞ function belongs to .L∞ and satisfies .f H ∞ = f L∞ . Consequently, we regard .H ∞ as a Banach subalgebra of .L∞ . For .λ ∈ D, observe that .f (λ) = f, cλ H 2 , in which cλ (z) =

.

1 1 − λz

is the Cauchy kernel (also known as the Szeg˝o kernel); it is the reproducing kernel for .H 2 . In particular, note that f ⊥ cλ

⇐⇒

.

f (λ) = 0.

The derivatives of the Cauchy kernels also play an important role since  f (n) (λ) = f, cλ(n) ,

.

in which cλ(n) (z) =

.

n!λ

n

(1 − λz)n+1

is the nth derivative of .cλ with respect to z.

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If .λ1 , λ2 , . . . , λn ∈ D are distinct, then the corresponding set of Cauchy kernels is linearly independent. Indeed, suppose that . ni=1 ai cλi = 0. Then n  .

n   ai f (λi ) = f, ai cλi = 0

i=1

i=1

for all .f ∈ H 2 . Apply this to a Lagrange interpolation polynomial f such that .f (λi ) = ai for .1  i  n and confirm that .a1 = a2 = · · · = an = 0.

4.2.2 Inner Functions An inner function is a bounded analytic function u on .D such that .|u(ζ )| = 1 for ma.e. .ζ in .T. For example, a unimodular constant function is an inner function, as is a monomial function .zn with .n  1. We describe other examples of inner functions below. If . = (λn )n1 is a (possibly finite) sequence in .D\{0} and N is a nonnegative integer, then the Blaschke product B (z) = zN

.

|λn | λn − z λn 1 − λn z

n1

converges locally uniformly on .D if and only if the Blaschke condition  .

(1 − |λn |) < ∞

n1

holds. Each such Blaschke product is an inner function; . is a Blaschke sequence. If the sequence . is finite, then .B is a finite Blaschke product [23]. Its degree is N plus the number of terms in ., according to multiplicity. If .μ is a positive finite singular Borel measure on .T, then .

   ζ +z dμ(ζ ) Sμ (z) = exp − ζ −z

is a singular inner function. It is, as its name suggests, inner. As an abuse of language, a function of the form .eiγ B or .eiγ Sμ with .γ ∈ R is often called a “Blaschke product” or “singular inner function,” respectively. Theorem 4.2.1 Each inner function factors uniquely as .eiγ B Sμ , in which .γ ∈ [0, 2π ), . is a Blaschke sequence, and .μ is a positive finite singular Borel measure on .T. Any such product is inner.

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4.2.3 Canonical Factorization An outer function is an analytic function F on .D of the form  F (z) = e

.

iγ

exp

T

 ζ +z ϕ(ζ ) dm(ζ ) , ζ −z

in which .γ ∈ R and .ϕ ∈ L1 (T) is real valued. Up to a unimodular constant factor, an outer function is determined by its modulus on .T since  .

log |F (z)| =

T

Pz (ζ )ϕ(ζ ) dm(ζ ),

in which Pz (ζ ) =

.

1 − |z|2 |ζ − z|2

is the Poisson kernel of the unit disk. Therefore, .log |F (z)| is the harmonic extension to .D of .ϕ : T → R, from which it follows that .ϕ = log |F | a.e. on .T. Each f in .H 2 that does not vanish identically satisfies  .

T

log |f (ζ )| dm(ζ ) > −∞.

Thus, there is an outer function corresponding to the boundary data .ϕ = log |f |. The preceding material leads to the canonical factorization of .H 2 functions. Theorem 4.2.2 Each f in .H 2 \{0} has a unique factorization .f = BSF , in which B is a Blaschke product, S is a singular inner function, and F is an outer function in .H 2 . Moreover, any product of this form belongs to .H 2 . An analogous factorization theorem holds for .H p with .1  p < ∞, with the outer factor of .f ∈ H p \{0} belonging to .H p .

4.2.4 Bounded Type Let f be meromorphic on .D. Then f is of bounded type if it is the quotient of two functions in .H ∞ . In particular, a function of bounded type has finite nontangential limits a.e. on .T. The set of all functions of bounded type, the Nevanlinna class, is denoted .N. We say that f is in the Smirnov class .N+ if it is the quotient of two bounded analytic functions, in which the denominator is an outer function. Observe that a function in the Smirnov class is analytic, as opposed to merely meromorphic, on .D. One can show that .N+ contains each of the .H p spaces. Unlike the .H p spaces

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with .1  p < ∞, both .N and .N+ are algebras; in particular, they are closed under sums and products.

4.3 Model Spaces The model space corresponding to a nonconstant inner function u is the orthogonal complement Ku = (uH 2 )⊥

.

in .H 2 of the subspace uH 2 = {uh : h ∈ H 2 }.

.

Note that .Ku contains the constant functions if and only if .u(0) = 0. The definition of .Ku is somewhat indirect since it is characterized as the orthogonal complement of a more concrete subspace of .H 2 . Fortunately, finitedimensional model spaces are explicitly realizable as certain spaces of rational functions. Infinite-dimensional model spaces take more work to understand; these are explored in later sections as new ideas and techniques are introduced.

4.3.1 Basic Properties The nonzero invariant subspaces for the unilateral shift (or forward shift) [Sf ](z) = zf (z)

.

on .H 2 are the subspaces .uH 2 for u inner; this is due to Beurling. Thus, the model spaces are the proper invariant subspaces of .H 2 for the backward shift [S ∗ f ](z) =

.

f (z) − f (0) , z

which is the adjoint of S; that is, .Sf, g = f, S ∗ g for all .f, g ∈ H 2 . The reason for the “shift” nomenclature is apparent if one thinks in terms of Taylor coefficients since S(a0 , a1 , a2 , . . .) = (0, a0 , a1 , a2 , . . .)

.

and S ∗ (a0 , a1 , a2 , . . .) = (a1 , a2 , a3 , a4 , . . .).

.

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127

To understand how model spaces fit inside each other, we must examine the divisibility of inner functions. Let .u, v be nonconstant inner functions. Then u divides v, denoted .u|v, if .v/u ∈ H 2 ; in this case, the quotient is an inner function. The containment of model spaces reflects the factorization of inner functions: ⇐⇒

u|v

.

Ku ⊆ Kv .

(4.1)

We regard .H 2 as a subspace of .L2 in the next proposition, which describes .Ku in terms of boundary functions on .T. This interplay between analytic functions on 2 .D and their boundary values on .T, in the context of the larger ambient space .L , is common in the study of model spaces. Proposition 4.3.1 Let u be a nonconstant inner function. In terms of boundary functions, Ku = H 2 ∩ uzH 2 .

.

Proof Let .f ∈ H 2 . Since .uu = 1 a.e. on .T, we see that .f ⊥ uH 2 if and only if 2 .uf ⊥ H ; that is, .uf ∈ zH 2 . Therefore, .f ∈ Ku if and only if .f ∈ uzH 2 .   Observe that f ∈ Ku

.

⇐⇒

f = uzgfor some g ∈ H 2 .

In fact, one can show that .g ∈ Ku as well. This observation provides a conjugatelinear map from .Ku to itself and serves as the starting point for the material in Sects. 4.8 and 4.10.

4.3.2 Reproducing Kernels Let .g = uh, in which u is inner and .h ∈ H 2 . A computation confirms that the reproducing kernel for .uH 2 is .u(λ)u(z)cλ (z). For .f ∈ Ku , it follows that f (λ) = f, cλ  = f, cλ  − u(λ)f, ucλ  = f, (1 − u(λ)u)cλ  = f, kλ ,

.

in which the reproducing kernel kλ (z) =

.

1 − u(λ)u(z) 1 − λz

belongs to .Ku . Indeed, for .h ∈ H 2 ,  uh, kλ = u(λ)h(λ) − u(λ)h, cλ  = u(λ)h(λ) − u(λ)h(λ) = 0.

.

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Moreover, each .kλ is an outer function that is invertible in .H ∞ . The kernel furnishes the orthogonal projection .Pu from .L2 onto .Ku . Proposition 4.3.2 .(Pu f )(λ) = f, kλ  for each .f ∈ L2 and .λ ∈ D. Proof Since .Pu = Pu∗ , we have .f, kλ  = f, Pu kλ  = Pu f, kλ  = (Pu f )(λ).

 

4.3.3 Finite-Dimensional Model Spaces The concrete description of finite-dimensional model spaces comes down to the following question: when does a Cauchy kernel belong to a model space? Since 2 .uh, cλ  = u(λ)h(λ) for all .h ∈ H , it follows that cλ ∈ Ku

.

⇐⇒

u(λ) = 0.

More can be said if u is a finite Blaschke product [23, Ch. 12]. Proposition 4.3.3 If u is a finite Blaschke product with zeros .λ1 , λ2 , . . . , λn , then  Ku =

.

a0 + a1 z + · · · + an−1 zn−1 (1 − λ1 z)(1 − λ2 z) · · · (1 − λn z)

: a0 , a1 , . . . , an−1

 ∈C .

(4.2)

In particular, if .u(z) = zn , then .Ku is the space of polynomials of degree at most .n − 1. Proof Suppose that u has only simple zeros. The remarks above ensure that .

span{cλ1 , cλ2 , . . . , cλn } ⊆ Ku .

If .f (λi ) = f, cλi  = 0 for .i = 1, 2, . . . , n, then .u|f and .f ∈ uH 2 . Thus, .

span{cλ1 , cλ2 , . . . , cλn }⊥ ⊆ uH 2 = K⊥ u.

Thus, .Ku = span{cλ1 , cλ2 , . . . , cλn } and the desired representation (4.2) follows via a partial-fractions argument. If .λ is a zero of u of multiplicity m, one must replace (m−1)   .cλ with its derivatives .cλ , c , c , . . . , c .   λ λ λ The proof of the previous theorem provides a useful description of finitedimensional model spaces in terms of Cauchy kernels and their derivatives. Proposition 4.3.4 If u is finite Blaschke product with distinct zeros .z1 , z2 , . . . , zd of multiplicities .m1 , m2 , . . . , md , respectively, then   Ku = span cz(i i −1) : 1  i  d, 1  i  mi .

.

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A little more can be said: the finite-dimensional model spaces are precisely those that arise from finite Blaschke products. Proposition 4.3.5 .dim Ku < ∞ if and only if u is a finite Blaschke product. Proof We need only show that .Ku is infinite dimensional whenever u is not a finite Blaschke product. If u has an infinite Blaschke factor, then .Ku contains an infinite linearly independent set of Cauchy kernels. If u has a singular inner factor S, then (4.1) shows that .Ku contains the infinite chain of distinct subspaces .KS 1/n for .n  1. In either case, .dim Ku = ∞.  

4.3.4 Stability Under Co-Analytic Toeplitz Operators Recall that we use boundary functions to regard .H 2 as a subspace of .L2 . Let P denote the Riesz projection, the orthogonal projection from .L2 onto .H 2 . As an orthogonal projection, P is selfadjoint (.P = P ∗ ) and idempotent .(P 2 = P ). In terms of Fourier coefficients, we have P (. . . , a−1 , a0 , a1 , . . .) = (a0 , a1 , . . .),

.

in which the box indicates the position of the zeroth Fourier coefficient. With .z = eiθ , for example, one deduces that P (1 + 2 cos θ ) = P (e−iθ + 1 + eiθ ) = 1 + eiθ = 1 + z.

.

Since .P = P ∗ and .cλ ∈ H 2 , we have [Pf ](λ) = Pf, cλ  = f, P cλ  = f, cλ 

.

for .λ ∈ D and .f ∈ L2 . We can write the inner product as an integral and obtain a Cauchy-integral representation of the Riesz projection. The Riesz projection permits us to make the following important definition. Definition 4.3.6 The Toeplitz operator .Tϕ : H 2 → H 2 with symbol .ϕ ∈ L∞ is Tϕ (f ) = P (ϕf ).

.

If .ϕ ∈ H ∞ , then .Tϕ is an analytic Toeplitz operator; if .ϕ ∈ H ∞ , then .Tϕ is a co-analytic Toeplitz operator. Since Toeplitz operators appear in more detail in other chapters, we focus here only on the basics that are needed at the moment. The adjoint of .Tϕ is Tϕ∗ = Tϕ

.

(4.3)

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since P fixes .H 2 and Tϕ∗ f, g = f, Tϕ g = f, P (ϕg) = Pf, ϕg = f, ϕg

.

= ϕf, g = ϕf, P g = P (ϕf ), g = Tϕ f, g. The unilateral shift (multiplication by z) is .Tz . It enjoys an analytic functional calculus in the following sense. If .ϕ ∈ H ∞ , then one defines .ϕ(Tz ) = Tϕ and proves that the operator norm of .Tϕ equals .ϕ∞ . This respects, and extends, the standard polynomial functional calculus. In particular, one regards analytic Toeplitz operators as analytic functions of .Tz . The backward shift is .Tz¯ . Being .Tz¯ -invariant already, the spaces .Ku are also invariant under co-analytic Toeplitz operators. Proposition 4.3.7 If .ϕ ∈ H ∞ and u is inner, then .Tϕ Ku ⊆ Ku . Proof For .f ∈ Ku and .h ∈ H 2 , it follows from (4.3) that Tϕ f, uh = f, Tϕ (uh) = f, P (ϕuh) = f, uϕh = 0.

.

 

As a consequence of the previous proposition, model spaces are stable under the removal of inner factors. In the context of Banach spaces of analytic functions, this property is sometimes called the F -property. Corollary 4.3.8 Let u be a nonconstant inner function. If .f ∈ Ku , .θ is inner, and f/θ ∈ H 2 , then .f/θ belongs to .Ku . In particular, the outer factor of a function in .Ku belongs to .Ku . .

Proof Observe that .Tθ f = P (θ f ) = P (f/θ ) = f/θ since .f/θ ∈ H 2 .

 

4.4 The Compressed Shift The compression of the unilateral shift to a model space is an operator of fundamental importance. This is highlighted by the Sz.-Nagy-Foia¸s theorem (Theorem 4.4.2), which provides a concrete functional model for a certain class of Hilbert-space contractions. But first we consider the Livšic–Möller theorem, which offers a detailed spectral description of the compressed shift.

4.4.1 The Livšic–Möller Theorem Let u be a nonconstant inner function and let .Pu denote the orthogonal projection from .L2 onto .Ku ; see Proposition 4.3.2 for a concrete representation. The com-

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pressed shift is the operator .Auz : Ku → Ku defined by Auz f = Pu (zf ).

.

It is the compression of the unilateral shift to .Ku , that is, (Auz )n = Auzn

.

for .n  0. We write .Az in place of .Auz when the inner function u is clear from context. One can show that .Auz is unitarily equivalent to .S ∗ |Kv = Azv¯ , in which v(z) = u(z)

.

is also inner. Consequently, one may work with the restricted backward shift or the compressed shift with no loss of generality. The spectrum .σ (Az ) of the compressed shift equals the spectrum   σ (u) = λ ∈ D− : lim inf |u(z)| = 0

.

z→λ z∈D

of the inner function u. The zeros of u and their accumulation points on .T belong to σ (u), as does the support of the singular measure in the singular inner factor of u. The following theorem characterizes the spectrum of an inner function in terms of its canonical factorization.

.

Theorem 4.4.1 (Livšic–Möller [35, 40]) If .u = B Sμ , where .B is a Blaschke product with zero sequence . and .Sμ is a singular inner function with singular measure .μ, then (i) .σ (Az ) = σ (u) = − ∪ supp μ, and (ii) .σp (Az ) = . Here .supp μ denotes the support of the measure .μ. The spectrum of the compressed shift closely mirrors the function-theoretic properties of the corresponding inner function. For example, the eigenvalues of a compressed shift comprise a (possibly empty) Blaschke sequence.

4.4.2 Model Operators Let .H denote a separable complex Hilbert space and let .B(H) denote the set of bounded linear operators on .H. Suppose .T ∈ B(H) is a contraction, that is, .T  

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1. Then .T = K ⊕ U , in which U is unitary and K is a completely nonunitary (CNU) contraction [48, p. 8]. That is, there is no reducing subspace for K upon which K is unitary. Since the spectral theorem addresses most questions about unitary operators, we can focus on CNU contractions. Let .∼ = denote unitary equivalence (of operators or Hilbert spaces). Theorem 4.4.2 (Sz.-Nagy-Foia¸s) Let .T ∈ B(H) be a contraction. If (i) .T n x → 0 for all .x ∈ H, and (ii) .rank(I − T ∗ T ) = rank(I − T T ∗ ) = 1, then there exists an inner function .u such that .T ∼ = S ∗ |Ku . √ Proof The defect operator .D = I − T ∗ T has rank 1, so .ran D ∼ = C and hence 2 . For each .n  0, a computation yields  = ∞ ran D ∼ we identify .H H = n=0 n  .

DT j x2 = x2 − T n+1 x2 .

j =0

j 2 2 Then (i) implies that . ∞ j =0 DT x = x for all .x ∈ H. Therefore, x = (Dx, DT x, DT 2 x, DT 3 x, . . .)

.

is an isometric embedding of .H into .H 2 . Since . is an isometry, its image 2 ∗ .H is closed in .H . It is .S -invariant by construction and hence there are three possibilities: .H = Ku for some nonconstant inner function u, .H = H 2 , or .H = {0}. Since . is an isometry, the third case is impossible. For all .x ∈ H, we have T x = (DT x, DT 2 x, DT 3 x, . . .) = S ∗ x.

.

Let .U : H → H denote the unitary operator obtained from the isometry . by reducing its codomain from .H 2 to .H. Then U T = (S ∗ |H )U ;

.

that is, .T ∼ = S ∗ |H . Then (ii) ensures that .H = H 2 since otherwise .T ∼ = S ∗ and 1 = rank(I − T T ∗ ) = rank(I − S ∗ S) = 0.

.

Therefore, .T ∼ = S ∗ |Ku for some nonconstant inner function u.

 

The previous theorem is existential in the sense that the model space .Ku is produced simply by appealing to the .S ∗ -invariance of .H. However, in some cases the unitary equivalence guaranteed by Theorem 4.4.2 can be seen more or less explicitly. We first require a few words about partial isometries.

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A partial isometry is a .T ∈ B(H) that is isometric on the orthogonal complement of its kernel. For example, unitary operators, the unilateral shift, and the backward shift are partial isometries. It is known that T is a partial isometry if and only if ∗ ∗ .T T is an orthogonal projection; in this case, .T T is the orthogonal projection onto ⊥ ∗ .(ker T ) . Moreover, T is a partial isometry if and only if .T is. A compressed shift u .Az is a partial isometry if and only if .u(0) = 0. A special case of Theorem 4.4.2 can be established directly with a result of Halmos and McLaughlin: two partial isometries on finite-dimensional spaces with one-dimensional kernels are unitarily equivalent if and only if they have the same eigenvalues and same multiplicities [30] (for similarity, see [26]). Theorem 4.4.3 Suppose that .T ∈ B(Cn ) is a partial isometry, .dim ker T = 1, and .σ (T ) ⊆ D. If .0, λ2 , λ3 , . . . , λn are the eigenvalues of T , repeated according to multiplicity, then T is unitarily equivalent to the compressed shift on .Ku , in which u(z) = z

.

n

λi − z i=2

1 − λi z

.

Proof Let u be as above and consider .Auz . It is a partial isometry on .Ku that maps the orthogonal complement of .ker Az = span{u/z} isometrically onto .{1}⊥ ; that span and orthogonal complement take place in .Ku . The unitary equivalence follows from the Halmos–McLaughlin theorem and the fact that the eigenvalues of .Auz are .0, λ2 , λ3 , . . . , λn , repeated according to multiplicity (see Sect. 4.6.1).  

4.5 Density Results Do model spaces contain convenient dense sets? A few results in this direction lie close to the surface, while others are significantly deeper. The linear span of the reproducing kernels  for .Ku is dense in .Ku . In fact, one need not use every kernel. In what follows, . denotes the closed linear span of the set that follows it. Proposition 4.5.1 Let

u be a nonconstant inner function and  . be a sequence of distinct points in .D. If . λ∈ (1 − |λ|) diverges, then .Ku = {kλ : λ ∈ }.  Proof Note that .f ∈ Ku is orthogonal to . {kλ : λ ∈ } if and only if f vanishes on .. Now recall that . is the zero sequence of a nonconstant .H 2 function if and only if . satisfies the Blaschke condition.   The sufficient condition above is not necessary; see the proof of Proposition 4.5.4. Since each .kλ belongs to .Ku ∩ H ∞ , we obtain the next result. Proposition 4.5.2 The linear manifold .Ku ∩ H ∞ is dense in .Ku . Let .A = H ∞ ∩ C(D− ) denote the disk algebra, the Banach algebra (with the supremum norm) of bounded analytic functions on .D that admit a continuous

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extension to the closed disk .D− . It is not obvious, say if u is a singular inner function, that .Ku ∩ A = ∅. However, much more is true. Theorem 4.5.3 (Aleksandrov [5]) If u is an inner function, .Ku ∩A is dense in .Ku . If u is a singular inner function, then .Ku need not contain a function whose derivative is continuous on .D− . Dyakonov and Khavinson showed that this occurs whenever the corresponding singular measure vanishes on all Beurling–Carleson sets [15]. Recall that a closed set .E ⊂ T of Lebesgue measure zero is a Beurling– Carleson set if its complement, when written as a countable union of disjoint maximal open arcs .In , satisfies  .

 |In | log

n1

1 |In |

 < ∞.

Let .An denote the set of analytic functions on .D whose derivatives of order n extend continuously to .T and define A∞ =

∞ 

.

An .

n=1

A recent result of Limani and Malman shows that .A∞ ∩ Ku is dense in .Ku if and only if the singular measure from the singular inner factor of u is concentrated (in a manner that can be made precise) on Beurling–Carleson sets [34]. The following related result has an elementary proof. Proposition 4.5.4 If u is a Blaschke product, .Ku has a dense subset whose elements are continuous on .D− and whose boundary functions are infinitely differentiable on .T. Proof We assume that u has simple zeros .λn for .n  1. Suppose that .f ∈ Ku satisfies .f, kλn  = 0 for all n.  Then .f (λn ) = 0 for .n  1, and hence .u|f ; that is, .f ∈ uH 2 = K⊥ u . Thus, . {kλ1 , kλ2 , . . .} = Ku . Since .kλn (z) = cλn (z), the second statement follows. For a general Blaschke product, include the appropriate   derivatives of the kernel functions in the spanning set.

4.6 Bases for Model Spaces The finite-dimensional model spaces can be concretely described (Proposition 4.3.3). These model spaces correspond to the finite Blaschke products and their elements are certain explicit rational functions. If u has a singular-inner or infinite-Blaschke factor, then its elements are more opaque. At least for infiniteBlaschke products, we would like to have convenient bases to work with. There are several options available.

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Before we begin, we should mention that orthonormal bases for general model spaces are sometimes provided by Theorem 4.9.4 below: if a Clark unitary operator has pure point spectrum, then its normalized eigenvectors form an orthonormal basis for .Ku . However, this topic requires a bit more setup and is postponed until we have the relevant background material.

4.6.1 Takenaka–Malmquist–Walsh Basis Let u be a Blaschke product with zeros .(λn )n1 . For .w ∈ D, define bw (z) =

.

z−w . 1 − wz

This is a Möbius transformation that maps .D onto .D and .T onto .T. Define the Takenaka–Malmquist–Walsh basis .(fn )n1 by  f1 (z) =

.

1 − |λ1 |2

1 − λ1 z

and

fk (z) =

 k−1

i=1

bλi

  1 − |λ |2 k 1 − λk z

for .k  2. Then .(fn )n1 is an orthonormal basis for .Ku ; if u has only simple zeros, then it is essentially the Gram–Schmidt orthonormalization of the sequence of reproducing kernels .(kλn )n1 . With respect to the Takenaka–Malmquist–Walsh basis, the matrix representation of .Auz is lower triangular with .λ1 , λ2 , . . . along the main diagonal. For .i  j , the .(i, j ) matrix entry is Auz fj , fi  = Pu (zfj ), fi  = zfj , fi    −1 i−1  1 − |λj |2   1 − |λ |2   j i , bλ bλ = z λ z 1 − λ z 1 − j i =1 =1        i−1 j −1 = 1 − |λi |2 1 − |λj |2 z =1 bλ cλi =1 bλ cλj ,      j −1 = 1 − |λi |2 1 − |λj |2 z =i bλ cλj , cλi   −1  1 − |λi |2 1 − |λj |2  j λi bλ (λi ) = 1 − λj λi =i  0 if j > i, = λi if j = i.

.

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In particular, the eigenvalues of .Auz are the zeros of u, repeated according to their multiplicities. The matrix entries with .i > j can be computed in a similar fashion, although the computation is not as enlightening.

4.6.2 Riesz Bases Another approach is to relax the notion of “basis.”  A linearly independent sequence (xn )n1 in a Hilbert space .H is a Riesz basis if . {x1 , x2 , . . .} = H and there exist .M1 , M2 > 0 such that for all finite sequences .a1 , a2 , . . . , an ∈ C, .

M1

n 

.

n n 2     |ai |2   ai xi   M2 |ai |2 .

i=1

i=1

i=1

A Riesz basis is the image of an orthonormal basis under a bounded invertible linear operator. One can show that .(xn )n1 is a Riesz basis if and only if the Gram matrix ∞ 2 .[xj , xi ] i,j =1 is bounded and invertible as an operator on . (N). ∞ A sequence . = (λn )n=1 of distinct points in .D is uniformly separated if   ∞  

 λi − λj  . inf   > 0.  1 − λj λi  i1 j =1 j =i

This is the Carleson condition. Loosely put, it says that . is remarkably sparse with respect to the hyperbolic metric on .D. The relevance of the Carleson condition to the Riesz-basis problem for model spaces stems from the next result. Theorem 4.6.1 If u is a Blaschke product with simple zeros .λ1 , λ2 , . . ., then kλn . = kλn 



1 − |λn |2

1 − λn z

comprise a Riesz basis for .Ku if and only if . = (λn )∞ n=1 is uniformly separated. Example 4.6.2 One way to create a uniformly separated sequence is to insist that |λn | → 1 exponentially fast. If there is a .c ∈ (0, 1) such that

.

1 − |λn+1 |  c(1 − |λn |)

.

for .n  1, then . = (λn )∞ n=1 is uniformly separated. If .0  λ1 < λ2 < · · · , then this exponential-separation condition is equivalent to uniform separation. Observe that .Ku contains .S ∗n u for .n  1. Even though these backward shifts of u need not be linearly independent (they cannot be if u is a finite Blaschke product),

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they can still act like an orthornormal basis for .Ku in certain ways. The following proposition says that they form a tight frame for .Ku . Proposition 4.6.3 If u is inner, then .S ∗n u ∈ Ku for .n  1. For each .f ∈ Ku , f =

.



f, S ∗n uS ∗n u

and

f 2 =

n1



|f, S ∗n u|2 .

n1

4.7 Continuability In general, a function in .H 2 has nontangential limits m-a.e. on .T, but it need not be analytically continuable across any arc of .T. In contrast, functions in model spaces are typically more well behaved. They often possess analytic continuations to significantly larger domains and they always enjoy a sort of generalized analytic continuation to the exterior of the closed disk .D− .

4.7.1 Analytic Continuation Let .u = B Sμ , in which .B is a Blaschke product with zero sequence . and .Sμ is a singular inner function with singular measure .μ. Recall that the spectrum of u is σ (u) = − ∪ supp μ.

.

Let . C = C ∪ {∞} denote the extended complex plane and let De = {|z| > 1} ∪ {∞}

.

C. One can show that u is analytically continuable to be the complement of .D− in .  .C\{1/z : z ∈ σ (u)}. In fact, more is true. C\{1/z : z ∈ σ (u)}. Proposition 4.7.1 Each .f ∈ Ku is analytically continuable to . For a given .Ku , there may be specific points on .T at which every .f ∈ Ku has a nontangential limit. Characterizing these points requires a definition. If u is inner and .ζ ∈ T, then u has an angular derivative in the sense of Carathéodory (ADC) at  .ζ if the nontangential limits of u and .u exist at .ζ and .|u(ζ )| = 1. Theorem 4.7.2 (Ahern–Clark [2, 3]) Let .u = B Sμ , where .B is a Blaschke product with zero sequence ., .Sμ is a singular inner function with singular measure .μ, and .ζ ∈ T. The following are equivalent. (i) Every .f ∈ Ku has a nontangential limit at .ζ . (ii) For every .f ∈ Ku , .f (λ) is bounded as .λ → ζ nontangentially.

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(iii) u has an ADC at .ζ . (iv) The nontangential limit of u at .ζ exists and .kζ (z) =  1 − |λ|2  dμ(ξ ) < ∞. + (v) . 2 |ζ − λ|2 T |ξ − ζ |

1 − u(ζ )u(z) 1 − ζz

∈ H 2.

λ∈

The function .kζ in (iv) is a boundary kernel. Under the equivalent conditions of the previous theorem, .kζ belongs to .Ku (the dominated convergence theorem ensures that .krζ → kζ as .r → 1− ) and .f (ζ ) = f, kζ  for .f ∈ Ku . It is of interest to contrast condition (v) with a theorem of Frostman [16]: u has a nontangential limit of modulus one at .ζ in .T if .

 1 − |λn |  dμ(ξ ) + < ∞. |ζ − λn | T |ξ − ζ |

n1

4.7.2 Pseudocontinuation Fundamental to the study of model spaces is a generalized version of analytic continuation. Later, in Sects. 4.8 and 4.10, we will find that it is more convenient to think of this in terms of a certain involution on the corresponding model space. This ultimately leads to an explicit function-theoretic description of model spaces and an explanation for the origin of pseudocontinuability. Definition 4.7.3 Let f and .f be meromorphic on .D and .De , respectively. If the nontangential limit of f (from .D) agrees with the nontangential limit of .f (from  are pseudocontinuations of each other. .De ) a.e. on .T, then f and .f A rational function on .D whose poles are in .De is pseudocontinuable. Inner functions are pseudocontinuable to .De via Schwarz reflection through .T:  u(z) =

1

.

u(1/z)

.

On the other hand, .exp(z) is not pseudocontinuable since its analytic continuation to .De is not meromorphic—it has an essential singularity at .∞. Pseudocontinuations are compatible with analytic continuations in the following sense: if a function has both an analytic continuation and pseudocontinuation to .De , they must be the same. This follows from Privalov’s uniqueness theorem, which asserts that if .f : D → C is analytic and has nontangential limits equal to zero on a subset of .T of positive measure, then f is identically zero [46]. For example, .log(1− z) is not pseudocontinuable to .De because the winding singularity at .z = 1 prohibits a potential pseudocontinuation to .De from equaling its analytic continuation there.

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Let H 2 (De ) = {f (1/z) : f ∈ H 2 }

.

denote the Hardy space of the extended exterior disk. The following result demonstrates the connection between model spaces and pseudocontinuability. Theorem 4.7.4 (Douglas–Shapiro–Shields [13]) Let .f ∈ H 2 . Then .f ∈ Ku if and only if .f/u has a pseudocontinuation .F ∈ H 2 (De ) such that .F (∞) = 0. Proof .(⇒) If .f ∈ Ku , then there is a .g ∈ H 2 such that .f = gzu a.e. on .T. Define F (z) =

.

1 1 ∈ H 2 (De ) g z z

and note that .F (∞) = 0. Moreover, .F = f/u a.e. on .T; that is, F is a pseudocontinuation of .f/u. (⇐) Let .F ∈ H 2 (De ) be a pseudocontinuation of .f/u and .F (∞) = 0. Then

.

F (z) =

.

1 1 h z z

for some .h ∈ H 2 and .f/u = F a.e. on .T. Moreover, .g(z) = h(z) ∈ H 2 and   .f = gzu a.e. on .T. Proposition 4.3.1 ensures that .f ∈ Ku . The noncyclic vectors for the backward shift on .H 2 can also be characterized in terms of pseudocontinuability: .f ∈ H 2 is noncyclic for .S ∗ if and only if f is pseudocontinuable of bounded type (PCBT). By this we mean that f has a pseudocontinuation to .De that is a quotient of bounded analytic functions on .De . Thus, the sum of noncyclic functions is noncyclic, as is their product, so long as it belongs to .H 2 .

4.8 Conjugation on Model Spaces Model spaces host a natural conjugate-linear involution that clarifies the origin of pseudocontinuations (see Sect. 4.10) and interacts in a natural way with the compressed shift and their relatives.

4.8.1 Conjugations A function .C : H → H on a complex Hilbert space .H is a conjugation if C is conjugate linear, involutive (.C 2 = I ), and isometric. The polarization identity and

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conjugate-linearity ensure that the third condition is equivalent to Cx, Cy = y, x

.

for all .x, y ∈ H. If C is a conjugation on .H, a Zorn’s-lemma argument provides an orthonormal basis of .H whose elements are each fixed by C; such a basis is a C-real orthonormal basis. Representing C with respect to such a basis shows that C is unitarily equivalent to the canonical conjugation (z1 , z2 , . . .) → (z1 , z2 , . . .)

.

on an .2 -space of the appropriate dimension. Thus, any two conjugations on the same Hilbert space are unitarily equivalent. Curiously, every unitary operator is the product of two conjugations [28].

4.8.2 The Model Conjugation The natural conjugation carried by .Ku is not easily describable without first realizing .Ku as a subset of .L2 via Proposition 4.3.1. Proposition 4.8.1 The function .C : Ku → Ku defined in terms of boundary functions by Cf = f zu

.

is a conjugation. In particular, f and Cf share the same outer factor. Proof Recall that .Ku = H 2 ∩ uzH 2 . Since .|u| = 1 a.e. on .T, it follows that C preserves outer factors and is conjugate-linear, isometric, and involutive. At first, we only have .Cf = f zu ∈ L2 . If f is orthogonal to .uH 2 , then Cf, zh = f zu, zh = uh, f  = 0

.

for h in .H 2 , and hence .Cf ∈ H 2 . Similarly, Cf, uh = f zu, uh = f z, h = 0,

.

from which it follows that .Cf ∈ Ku .

 

Example 4.8.2 If u is a finite Blaschke product with zeros .λ1 , λ2 , . . . , λn , repeated according to multiplicity, then a computation confirms that  C

.

a0 + a1 z + · · · + an−1 zn−1 n i=1 (1 − λi z)

=

an−1 + an−2 z + · · · + a0 zn−1 . n i=1 (1 − λi z)

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Thus, C reverses and conjugates the numerator in the representation (4.2). Example 4.8.3 The conjugate of a reproducing kernel is a difference quotient:  [Ckλ ](z) =

.

1 − u(λ)u(z) 1 − λz

zu(z) =

u(z) − u(λ) 1 − u(λ)u(z) u(z) · = . 1 − λz z z−λ

Note that .Ckλ is an inner multiple of the outer function .kλ : bu(λ) (u(z)) u(z) − u(λ) = kλ (z). z−λ bλ (z)

[Ckλ ](z) =

.

(4.4)

The difference quotient is the reproducing kernel for conjugate functions, in the sense that .[Cf ](λ) = Cf, kλ  = Ckλ , f  because C is isometric and .C 2 = I . Of course, keep in mind that the map .f →  [Cf ](λ) is conjugate linear. Let u have an ADC at .ζ ∈ T. Then the boundary kernel kζ (z) =

.

1 − u(ζ )u(z) 1 − ζz

belongs to .H 2 (Theorem 4.7.2) and, in fact, to .Ku . It is almost self-conjugate since [Ckζ ](z) =

.

u(z) − u(ζ ) = ζ u(ζ )kζ (z). z−ζ

In fact, there is a scalar multiple of .kζ that is actually self-conjugate. Example 4.8.4 Let u be a finite Blaschke product with n zeros (let one of them be 0), repeated according to multiplicity, and fix .α ∈ T. Then .u(ζ ) = α has precisely n distinct solutions .ζ1 , ζ2 , . . . , ζn on .T [23, Lem. 3.4.3]. The functions i

e 2 (arg α−arg ζj ) 1 − αu(z)  · .eζj (z) = |u (ζj )| 1 − ζj z

(4.5)

for .j = 1, 2, . . . , n form a C-real orthonormal basis of .Ku . See Example 4.9.6 for an example involving a singular inner function.

4.8.3 Associated Inner Functions Let .f = If F be the inner-outer factorization of .f ∈ Ku . Then .g = Cf has the same outer factor as f , so .g = Ig F for some inner function .Ig . Since .g = f zu

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a.e. on .T, it follows that .Ig F = If F zu and hence If Ig F = F zu.

.

The inner function IF = If Ig =

.

F zu F

is the associated inner function of F with respect to u. It is uniquely determined by u and F . Conversely, if .Iv and .Iw are inner functions and .Iv Iw = IF , then Iv F = Iw F zu,

.

so .v = Iv F and .w = Iw F belong to .Ku = H 2 ∩ uzH 2 and satisfy .Cv = w. Recall from Corollary 4.3.8 that the outer factor of a function in .Ku belongs to .Ku . Also observe that .Ku contains an inner function if and only if .u(0) = 0. Proposition 4.8.5 Let .F ∈ Ku be the outer factor of some function in .Ku . The set of all functions in .Ku with outer factor F is Ou (F ) = {θ F : θ inner and θ |IF }.

.

Define a partial ordering on .Ou (F ) by .θ1 F  θ2 F if and only if .θ1 |θ2 . With respect to this ordering, F and .IF F are minimal and maximal, respectively. Moreover, C restricts to an order-reversing bijection from .Ou (F ) to itself. Example 4.8.6 From (4.4) we see that any function in .Ku with outer factor .kλ is of the form .vkλ , in which v is an inner function that divides .bu(λ) (u(z))/bλ (z).

4.8.4 Generators of Ku We say that f generates .Ku if .

! {S ∗n f : n  0} = Ku ;

that is, if .Ku is the smallest backward-shift invariant subspace of .H 2 that contains f . In particular, a generator of .Ku must belong to .Ku . One can show that f generates .Ku if u and the inner factor of Cf are relatively prime; that is, if they have no nonconstant common inner divisors. For example, if .λ ∈ D [Ckλ ](z) =

.

u(z) − u(λ) z−λ

generates .Ku since its conjugate .kλ is outer.

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Example 4.8.7 Each outer function .F ∈ Ku is the sum of two outer generators of Ku . Write

.

1 (1 + IF )F + i 2

F =

.



1 (1 − IF )F 2i



and observe that the “real” and “imaginary” parts in the above are self-conjugate outer functions.

4.9 Aleksandrov–Clark Theory The Aleksandrov–Clark theory of rank-one unitary perturbations of the compressed shift is one of the deepest and most fruitful portions of model-space theory. We begin with a finite-dimensional example. Example 4.9.1 Let u be a finite Blaschke product of degree n such that .u(0) = 0. Consider the compressed shift .Az f = Pu (zf ), in which .Pu denotes the orthogonal projection from .L2 onto .Ku ; see Sect. 4.4. Since .

rank(I − A∗z Az ) = rank(I − Az A∗z ) = 1,

we see that .Az is “almost” unitary. For .g, h ∈ Ku , define the rank-one operator (g ⊗ h)f = f, hg. Then for each .α ∈ T, the Clark operator

.

Uα = Auz + α(k0 ⊗ Ck0 )

.

is unitary, C-symmetric (that is, .Uα = CUα∗ C), and satisfies .Uα eζi = ζi eζi for .i = 1, 2, . . . , n, where u−1 ({α}) = {ζ1 , ζ2 , . . . , ζn }

.

and the C-real orthonormal basis .(eζi )ni=1 is defined by (4.5). In particular, .Uα is unitarily equivalent to the multiplication operator .Mz on .L2 (μα ), where μα =

.

1 (δζ + δζ2 + · · · + δζn ) n 1

is a probability measure on .T. This is a concrete spectral representation of the corresponding Clark operator; see Sect. 4.9.2.

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4.9.1 Herglotz’ Theorem and Clark Measures Generalizing the previous example requires some work. If .μ is a probability measure on .T, then  ζ +z dμ(ζ ) .Fμ (z) = T ζ −z is analytic on .D and .Fμ (0) = 1. Moreover,  .

Re Fμ (z) =

T

1 − |z|2 dμ(ζ ) > 0 |ζ − z|2

for .z ∈ D. A result of Herglotz says that the converse holds. Theorem 4.9.2 (Herglotz) If F is analytic on .D, .F (0) = 1, and .Re F > 0 on .D, then there is a unique Borel probability measure .μ on .T such that .F = Fμ . Let u be inner and .u(0) = 0. For each .α ∈ T, 1 + αu(z) 1 − αu(z)

F (z) =

.

is analytic on .D. Moreover, .F (0) = 1 and  .

Re

1 + αu(z) 1 − αu(z)

 =

1 − |u(z)|2 >0 |α − u(z)|2

on .D. Theorem 4.9.2 provides a unique probability measure .σα on .T such that .

1 + αu(z) = 1 − αu(z)

 T

ζ +z dσα (ζ ). ζ −z

The measures .σα for .α ∈ T are the Clark measures corresponding to u. The next theorem requires a definition. A carrier for .σα is a Borel set .C ⊆ T such that .σα (A ∩ C) = σα (A) for all Borel .A ⊆ T. Carriers need not be unique, nor must they equal the support of the measure. Proposition 4.9.3 Let u be an inner function and .u(0) = 0. Then (i) .σα ⊥ m for all .α ∈ T. (ii) .σα ⊥ σβ for .α = β.  (iii) A carrier for .σα is . ζ ∈ T : lim u(rζ ) = α . r→1−

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(iv) .σα has a point mass at .ζ if and only if .u(ζ ) = α and u has an ADC at .ζ . Moreover, σα ({ζ }) =

.

1 |u (ζ )|

.

More generally, one can let .u ∈ H ∞ with .u(0) = 0 and .u∞  1. The resulting Aleksandrov–Clark measures need not be singular with respect to Lebesgue measure. However, they are still of great interest. For example,  (Af )(α) =

.

T

f (ζ ) dμα (ζ )

extends boundedly from .C(T) to .H 2 , and it is the adjoint of the composition operator with symbol u [4]. Since coverage of these more general measures would take us away from the theory of model spaces, we must refrain from discussing them further here.

4.9.2 Clark Operators We can now generalize Example 4.9.1. For inner u with .u(0) = 0, the compressed shift .Az f = Pu (zf ) is a contraction. For .α ∈ T, the Clark operator Uα = Az + α(k0 ⊗ Ck0 )

.

(4.6)

on .Ku is C-symmetric (as is .Az ) and unitary. Its spectral representation is intimately related to the function-theoretic properties of u. Note that .k0 = 1 (the constant function) and .Ck0 = u/z. Theorem 4.9.4 (Clark [11]) Let u be an inner function with .u(0) = 0. Let .σα be the unique singular Borel probability measure on .T satisfying .

1 + αu(z) = 1 − αu(z)



ζ +z dσα (ζ ). ζ −z

(i) .Uα is a cyclic unitary operator (with cyclic vector 1) whose spectral measure .σα is carried by the Borel set .{ζ ∈ T : limr→1− u(rζ ) = α}. (ii) .σp (Uα ) = {ζ ∈ T : u(ζ ) = α and |u (ζ )| < ∞}. The corresponding eigenvectors are the boundary kernels .kζ .  f (ζ ) dσα (ζ ) is a unitary operator from .L2 (σα ) (iii) .(Vα f )(z) = (1 − αu(z)) 1 − ζz onto .Ku .

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(iv) .Uα = Vα Zα Vα∗ , in which .Zα : L2 (σα ) → L2 (σα ) is the multiplication operator .(Zα f )(ζ ) = ζf (ζ ). For the sake of illustration, let us return to the finite-dimensional setting, in which the details of Theorem 4.9.4 are more transparent than in the infinite-dimensional situation. Although we may miss some analytic subtleties, the concreteness and explicitness of a simple example outweigh the drawbacks.  Example 4.9.5 Let .u = z3 , so that .Ku = {1, z, z2 }. Note that .C1 = z2 , Az 1 = z,

Az z = z2 ,

.

and

Az z2 = 0.

For .α ∈ T, the operator .Uα = Az + α(1 ⊗ z2 ) is unitary and satisfies Uα 1 = z,

Uα z = z 2 ,

.

and

Uα z2 = α.

Thus, the matrix representations of .Az and .Uα with respect to the basis .β = {1, z, z2 } are ⎡

⎤ 000 .[Az ]β = ⎣1 0 0⎦ 010

⎡

⎤ 00α [Uα ]β = ⎣1 0 0 ⎦ . 010

and

The eigenvalues of .Uα are the solutions to .z3 = α; denote them .ω1 , ω2 , ω3 . Corresponding eigenvectors are the boundary kernels kωi (z) =

.

1 − ωi 3 z3 = 1 + ωi z + ωi 2 z2 ; 1 − ωi z

they can be normalized to obtain a C-real orthonormal basis for .Kz3 . The probability measure (which is singular with respect to Lebesgue measure on .T) σα = 13 δω1 + 13 δω2 + 13 δω3

.

satisfies .

1 + αz3 = 1 − αz3



ζ +z dσα (ζ ). ζ −z

Consider the unitary operator .Vα : L2 (σα ) → Kz3 defined by  (Vα f )(z) = (1 − αz3 )

.

f (ζ ) 1 − ζz

dσα (ζ ).

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For .f = c1 χω1 + c2 χω2 + c3 χω3 ∈ L2 (σα ), observe that 

1/3 1/3 1/3 + c2 + c3 .(Vf )(z) = (1 − αz ) c1 1 − ω1 z 1 − ω2 z 1 − ω3 z 3

 ∈ Ku .

Indeed, it is a polynomial of degree at most 2 because each .ωi3 = α; a partialfractions argument and Proposition 4.3.3 confirm the result. Example 4.9.6 Consider the singular inner function  u(z) = exp

.

z+1 z−1



corresponding to a point mass at 1. Proposition 4.7.1 ensures that u is analytically continuable across .T\{1}. Fix .α ∈ T and define .Eα = {ζ : u(ζ ) = α}; it is countable and accumulates only at 1. The Clark measure .σα is discrete and σα ({ζ }) =

.

|ζ − 1|2 2

for .ζ ∈ Eα . Therefore, σα =

.

 |ζ − 1|2 δζ , 2

ζ ∈Eα

in which .δζ denotes the point mass at .ζ ∈ T. The corresponding boundary kernels form an orthonormal basis of .Ku ; they are eigenvectors of a Clark unitary operator with pure point spectrum. One can multiply these vectors by suitable unimodular constants in order to obtain a C-real orthonormal basis; the relevant computations can be found in [20].

4.9.3 Deeper Results Since .Vα f ∈ Ku for every .f ∈ L2 (σα ), it has nontangential limits m-a.e. on .T. To put the next result in context, recall that for .E ⊆ T closed with .m(E) = 0, there is an .f ∈ H 2 (which can be taken to be inner) that fails to have nontangential limits anywhere on E [36, 37]. Theorem 4.9.7 (Poltoratski [45]) Let .f ∈ L2 (σα ). Then .limr→1− (Vα f )(rζ ) = f (ζ ) for .σα -a.e. .ζ ∈ T. The Clark measures arising from an inner function provide a disintegration of Lebesgue measure [21, Sect. 11.4].

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Theorem 4.9.8 (Aleksandrov [4]) Let u be an inner function. For .f ∈ L1 ,  α →

f (ζ )dσα (ζ )

.

is defined for m-a.e. .α ∈ T and is Lebesgue integrable. Moreover,  



.

T

T



f (ζ ) dσα (ζ ) dm(α) =

T

f (ξ ) dm(ξ ).

Proof We sketch the proof in the simpler case .f ∈ C(T). Let .Pz (ζ ) denote the Poisson kernel for .D. Take the real part in the definition of .σα and obtain   .

T

 

 T

Pz (ζ ) dσα (ζ ) dm(α) =  =  =

T

T

T

1 − |u(z)|2 |α − u(z)|2

 dm(α)

Pu(z) (α) dm(α) = 1 Pz (ζ ) dm(ζ ).

Since the set of finite linear combinations of Poisson kernels is dense in .C(T) [21, Prop. 1.2.20], the desired result follows.   Permitting the more general Aleksandrov–Clark measures mentioned above, one can describe when .Ku embeds isometrically into .L2 (μ). Theorem 4.9.9 (Aleksandrov [6]) Let u be an inner function and .μ a positive Borel measure. Then .Ku embeds isometrically into .L2 (μ) if and only if there exists a .ϕ ∈ H ∞ with .ϕ∞  1 such that .μ is the Aleksandrov–Clark measure for .ϕu at 1.

p

4.10 Explicit Description of Ku

The conjugation carried by a model space permits one to describe its elements in a concrete, function-theoretic manner. Each .f ∈ Ku is of the form .f = a + ib, in which the functions a=

.

1 (f + Cf ) 2

and

b=

1 (f − Cf ) 2i

are fixed by C. Thus, we need only characterize those .f ∈ Ku such that .Cf = f , or equivalently, f = f zu

.

(4.7)

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a.e. on .T. For the sake of simplicity, suppose that .u(ζ ) = ζ for some .ζ in .T. This can be arranged by multiplying u by a suitable unimodular constant; both inner functions yield the same model space. If u has an ADC at .ζ , then .kζ ∈ Ku ; if not, then at the very least it belongs to the Smirnov class .N+ , the space of quotients of bounded analytic functions with outer denominator. A computation reveals that kζ = kζ zu

.

a.e. on .T (this is where the hypothesis .u(ζ ) = ζ is used). Substitute this into (4.7) and obtain f/kζ = f/kζ .

.

Therefore, .f ∈ Ku satisfies .f = Cf if and only if .f/kζ ∈ N+ is real a.e. on .T. Definition 4.10.1 .f ∈ N+ is a real Smirnov function if it is real valued almost everywhere on .T. The set of all real Smirnov functions is denoted .R+ . Fortunately, .R+ is well studied and its members can be described explicitly; see the exposition in [22]. It costs us little to consider the spaces p

Ku = H p ∩ zH p

.

for .p ∈ [1, ∞), since the proofs are essentially the same. We summarize the previous material as the following theorem [17, 18, 20, 22]. Theorem 4.10.2 For .1  p < ∞,   p Ku = (a + ib)kζ : a, b ∈ R+ ∩ H p

.

C[(a + ib)kζ ] = (a − ib)kζ .

and

The previous theorem suggests having a closer look at real Smirnov functions. A Möbius-transformation argument provides a simple representation. Theorem 4.10.3 (Helson [31]) If .f ∈ R+ , then there exist inner functions .ψ1 , ψ2 such that .ψ1 − ψ2 is outer and f =i

.

ψ1 + ψ2 . ψ1 − ψ2

Proof The function .τ (z) = i(1 + z)(1 − z)−1 maps .D onto the upper half-plane. Since .τ −1 ◦ f is a quotient of .H ∞ functions and unimodular a.e. on .T, .τ −1 ◦ f = ψ1 /ψ2 is a quotient of inner functions. Thus, f has the desired form.   p

The previous theorems say that each .f ∈ Ku can be expressed algebraically using u and at most four other inner functions. In particular, the phenomenon of pseudocontinuation arises from Schwarz reflections of inner functions.

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Another description of .R+ originates in [17, 25]; see [22] for a thorough exposition, including many worked examples and remarks about various subtleties that arise. First note that if .f = If F is the inner-outer factorization of a function in + .R , then   (1 − If )2 F −4If .If F = , −4 (1 − If )2 where the first term is in .R+ and has the same inner factor as f , and the second factor is an outer function in .R+ . This can be unpacked as follows. Theorem 4.10.4 Each .f ∈ R+ factors as a locally uniformly convergent product f = ρ · K(ϕ0 ) ·

.

∞

T (ϕn ) , T (ϕ−n )

n=1

in which .ϕ±n are inner functions, .ρ ∈ R, K(z) =

.

−4z , (1 − z)2

and

T (z) = i

1 − iz . 1 + iz

4.11 Quaternionic Structure of 2 × 2 Inner Functions Conjugations on model spaces can yield insights about matrix inner functions. The following result originates in [19]; another exposition is [21, Thm. 8.6.2]. Theorem 4.11.1 Let u be an inner function, .a, b, c, d ∈ H ∞ , and =

.

& ' a −b . c d

(4.8)

Then . is unitary almost everywhere on .T and .det  = u if and only if (i) .a, b, c, d belong to .Kzu . (ii) .|a|2 + |b|2 = 1 a.e. on .T. (iii) .Ca = d and .Cb = c, where .C : Kzu → Kzu is the conjugation .Cf = f u. This is analogous to the representation of quaternions of unit modulus via .2 × 2 complex matrices. The representation above has been used to study the characteristic function of a complex symmetric contraction [9] and the reducibility of .C0 (2) operators [49]. A local version has also been considered [29]. A special case of the Darlington synthesis problem from systems theory asks whether, given .a ∈ H ∞ , there are .b, c, d ∈ H ∞ such that the analytic matrix-

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valued function (4.8) is inner; that is, unitary a.e. on .T. Arov and Douglas–Helton connected this to pseudocontinuability long ago [21, Thm. 8.6.5]. A modern explanation relies upon the next lemma [21, Lem. 8.6.4]. Lemma 4.11.2 If .a ∈ Kzu and .a∞  1, then there exists a function .b ∈ Kzu such that .|a|2 + |b|2 = 1 a.e. on .T. Proof If a is inner, let .b = 0. Now suppose that a is not inner. Since .Ca = au, we have u − aCa = u(1 − |a|2 )

.

a.e. on .T. Consequently, .u − aCa ∈ H ∞ \ {0} and hence 

 .

T

log(1 − |a|2 ) dm =

T

log |u − aCa| dm > −∞

by Garcia et al. [21, Cor. 3.3.18]. Thus, there is an outer function .h ∈ H ∞ such that 2 2 .|h| = 1 − |a| a.e. on .T. Since u − aCa = u|h|2

.

belongs to .H ∞ and has the same modulus as .h2 , there is an inner function v such that .vh2 = u|h|2 . Since .log |h| is integrable, h does not vanish on a set of positive measure. Thus, .vh = hu a.e. on .T. We conclude that .b = vh ∈ Kzu and .|a|2 +|b|2 = 1 a.e. on .T.   Theorem 4.11.3 (Arov [7], Douglas–Helton [12]) Let .a ∈ H ∞ . The Darlington synthesis problem with data .a ∈ H ∞ is solvable if and only if .a∞  1 and a is PCBT. Proof If the Darlington problem with data .a ∈ H ∞ is solvable, Theorem 4.11.1 ensures that .a∞  1 and a is PCBT. Now suppose that a is PCBT and .a∞  1. Then .a ∈ Ku ⊂ Kzu for some inner function u. The previous lemma provides 2 2 .b ∈ Kzu such that .|a| + |b| = 1 a.e. on .T. Now fill out . via Theorem 4.11.1.   Theorem 4.11.1 presents an intriguing question: Can one develop an concrete and explicit factorization theory for .2 × 2 inner functions based upon the “quaternionic” representation (4.8)?

4.12 Conclusion In a three-hour minicourse on a broad subject such as model spaces, a great many topics must be omitted, largely upon the inclination and preferences of the speaker. Even the recent text [21] is not comprehensive. Fortunately, minicourses on some

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of the “missing topics” were presented as part of the focus program. For example, de Branges–Rovnyak spaces (close cousins of model spaces), interpolation theory, Riesz bases, deeper results about inner functions, operators on function spaces (e.g., Toeplitz and Hankel operators), and applications to mathematical physics were all covered. Consequently, the reader need not despair and they may look to the other chapters in this volume, or to [8, 10, 21, 41–43, 47, 48], for further connections and explorations. Acknowledgments Author partially supported by NSF Grants DMS-1800123 and DMS2054002.

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43. NIKOLSKI, N. Operators, functions, and systems: an easy reading. Vol. 2, vol. 93 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2002. Model operators and systems, Translated from the French by Andreas Hartmann and revised by the author. 44. NIKOLSKI, N. Hardy spaces, french ed., vol. 179 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2019. 45. POLTORATSKI ˘I, A. G. Boundary behavior of pseudocontinuable functions. Algebra i Analiz 5, 2 (1993), 189–210. 46. PRIWALOW, I. I. Randeigenschaften analytischer Funktionen. Hochschulbücher für Mathematik, Band 25. VEB Deutscher Verlag der Wissenschaften, Berlin, 1956. Zweite, unter Redaktion von A. I. Markuschewitsch überarbeitete und ergänzte Auflage. 47. ROSS, W. T., AND SHAPIRO, H. S. Generalized analytic continuation, vol. 25 of University Lecture Series. American Mathematical Society, Providence, RI, 2002. 48. SZ.-NAGY, B., FOIAS, C., BERCOVICI, H., AND KÉRCHY, L. Harmonic analysis of operators on Hilbert space, second ed. Universitext. Springer, New York, 2010. 49. ZHU, S., YANG, Y., LI, R., AND LU, Y. The reducibility of C0 (2) operators. J. Math. Anal. Appl. 482, 2 (2020), 123570, 17.

Chapter 5

Operators on Function Spaces William Ross

5.1 Introduction This is a set of lecture notes to accompany a series of talks given as part of the Fields Institute session on Operators on Function Spaces from July–December 2021. These notes are also part of a book project by myself, Stephan R. Garcia, and Javad Mashreghi titled Operator Theory by Example (Oxford University Press). Many of the technical details as well as other examples of operators on function spaces can be found there. Operator theory on function spaces is important since it can be used to represent abstract operators on Hilbert spaces in a concrete setting. This change of viewpoint enables one to obtain more information about an operator. For example, a version of the spectral theorem says that any bounded normal operator is unitarily equivalent to a multiplication operator .Mϕ f = ϕf on an .L2 (X, μ) space. From here one can glean information about normal operators by looking at this very specific tangible class of multiplication operators. As another example, if one considers the shift operator .Sen = en+1 on the sequence space .2 = 2 (N0 ) (where .en is the standard basis vector for .2 ), one seems at a loss to come up with examples of invariant subspaces for S, besides the obvious ones defined by the closed linear span of .{ek : k  N }. However, when one views S as the multiplication operator .(Sf )(z) = zf (z) on a certain Hilbert space of analytic functions on the open unit disk .D (the Hardy space .H 2 ), more invariant subspaces appear and one is off to characterize them all—as Beurling did in 1949. This set of notes explores a selection of well-studied operators on Hilbert spaces to give the novice a taste of the subject and encourage them to explore these ideas

W. Ross () Department of Mathematics and Statistics, University of Richmond, Richmond, VA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mashreghi (ed.), Lectures on Analytic Function Spaces and their Applications, Fields Institute Monographs 39, https://doi.org/10.1007/978-3-031-33572-3_5

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further. As this is a sample of operators on function spaces, I do not intend to be thorough on covering both the breadth as well as the depth of the subject. As such, these are a collection of operators that I find interesting and have encountered at the many conferences I have attended over the years. To get the most out of these notes, the reader should be familiar with the basics of real, complex, and functional analysis. Perhaps have the books [48, 49] at their side while reading these notes. In addition, in order to make things more manageable, I restrict myself to the operators on two basic Hilbert spaces, the Lebesgue space 2 2 .L (T) and the Hardy space .H . I also only sample results concerning the Volterra, Cesàro, Toeplitz, Hankel, Fourier transform, and Hilbert transform operators on these spaces. These spaces and operators are easy to describe which allows me to quickly get to some meaningful results. As a matter of apology, I’m leaving out a multi-volume treatise one can write on the subject. Indeed, there are many other Hilbert function spaces one can cover as well as many more interesting operators on these function spaces. Of course, there is the whole arena of Banach spaces of functions and their corresponding operators which would take up another several volumes. Indeed, even the operators I do cover, are classical and there is a large literature on each of them. I will do my best to point out good books the reader can consider in order to learn more.

5.2 Hilbert Spaces of Functions 5.2.1 General Operator Theory Concepts A Hilbert space .H is a complex √ vector space with an inner product .x, y and corresponding norm .x = x, x such that .H is Cauchy complete. If .H is separable (contains a countable dense set), then there is a sequence .(un )∞ n=1 ⊆ H, such that .um , un  = δmn and x=

.

∞  x, un un

for all x ∈ H.

n=1

The sum above converges in the norm of .H. Such a sequence .(un )∞ n=1 is an orthonormal basis for .H. Moreover, x =

.

2

∞ 

|x, un |2 .

n=1

This is a general version of Parseval’s theorem from Fourier analysis,

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A linear transformation .T : H → H is bounded if T  := sup T x

.

x=1

if finite. The quantity .T  is called the norm of T and the set of all bounded operators on .H is denoted by .B(H). One can check that .B(H) is a vector space and in fact an algebra that is Cauchy complete with respect to the operator norm. Some simple facts about the norm are: .S + T   S + T ; .ST   ST ; .aT  = |a|T . There are two other topologies one sees on .B(H). There is the strong operator topology where a sequence .(Tn )∞ n=1 ⊆ B(H) converges to .T ∈ B(H) strongly if .Tn x − T x → 0 for every .x ∈ H. There is also the weak operator topology where .Tn → T weakly if .Tn x, y → T x, y for every .x, y ∈ H. For .T ∈ B(H), the adjoint .T ∗ is the unique element of .B(H) satisfying ∗ .T x, y = x, T y for every .x, y ∈ H. Some basic facts about the adjoint are: ∗ ∗∗ .T  = T ; .T = T ; .(aT + bS)∗ = aT ∗ + bS ∗ ; .T ∗ T  = T 2 . An operator .T ∈ B(H) is compact whenever .(T xn )∞ n=1 has a convergent subsequence for any bounded sequence .(xn )∞ n=1 in .H. Any finite-rank operator is compact and every compact operator is the norm limit of a sequence of finite rank operators. The compact operators form a norm-closed ideal in .B(H). Furthermore, T is compact if and only if .T ∗ is compact. For .T ∈ B(H), the point spectrum .σp (T ) is the set of .λ ∈ C such that .ker(T − λI ) =  {0}. These are the eigenvalues of T . The spectrum .σ (T ) of T is the set of .λ ∈ C for which .T − λI is not invertible. Some basic facts about the spectrum are: ∗ .σ (T ) = ∅; .σ (T ) is compact; .σp (T ) ⊆ σ (T ); .σ (T ) = {λ : λ ∈ σ (T )}.

5.2.2 The Lebesgue Spaces In these next few sections, we discuss some Hilbert spaces consisting of functions. The technical details of this section are found in standard real analysis texts such as [47, 48]. One of the most basic Hilbert spaces is .L2 [0, 1] whose norm is denoted by .f L2 [0,1] and inner product by .f, gL2 [0,1] . Note that .f L2 [0,1] = 0 if and only if f is zero almost everywhere and so, without much fanfare, we identify two functions when they are equal almost everywhere. The quantity above is a norm and, by the Riesz–Fisher theorem, .L2 [0, 1] is (Cauchy) complete. Important in this survey are the following orthonormal bases for .L2 [0, 1]. Proposition 5.2.1 The following sets are orthonormal bases for .L2 [0, 1]. (a) .en (x) = e2π inx , where .n ∈ Z.  2n + 1  √ (b) .fn = 2 cos π x , where .n ∈ N0 . 2  2n + √ 1  π x , where .n ∈ N0 . (c) .gn = 2 sin 2

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Another important Lebesgue space is .L2 (T) = L2 (T, m), where .T is the unit circle and .dm = dt/2π is normalized Lebesgue measure on .T. A calculation with integrals shows that the functions .(ξ n )n∈Z , form an orthonormal basis for .L2 (T) and thus every .f ∈ L2 (T) can be written as ∞ 

f =

f(n)ξ n ,

.

n=−∞

where .f(n) = f, ξ n L2 (T) are the Fourier coefficients of f . The convergence of the series above is in .L2 (T), meaning N      f(n)ξ n  f −

.

L2 (T)

n=−N

→0

as N → ∞.

Parseval’s theorem establishes a unitary operator (norm preserving and surjective linear transformation) .f → (f(n))n∈Z from .L2 (T) to the sequence space .2 (Z). Our last important Lebesgue space is .L2 (R). Two orthonormal bases for .L2 (R) are formed as follows. For .x, t ∈ R, write e−

.

x2 2 2 +2xt−t

=

∞ 

hn (x)

n=0

tn . n!

x2

The function .hn is .e− 2 times the nth Hermite polynomial and is called the nth Hermite function. The first few are x2

h0 (x) = e− 2 ,

.

x2

h1 (x) = e− 2 2x, h2 (x) = e−

x2  2 4x 2

 −2 ,

h3 (x) = e−

x2  2 8x 3

 − 12x ,

h4 (x) = e−

x2  2 16x 4

 − 48x 2 + 12x .

hn (x) , hn L2 (R)

where n ∈ N0 ,

Proposition 5.2.2 The functions Hn (x) =

.

form an orthonormal basis for .L2 (R).

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The orthonormal basis above will be useful when discussing the Fourier transform. Another useful orthonormal basis for .L2 (R), which comes into play when studying the Hilbert transform, stems from the orthonormal basis .(ξ n )n∈Z for .L2 (T) discussed earlier. The linear fractional transformation c(x) =

.

x−i x+i

maps .R onto .T\{1} and a change of variables shows that 1 1 f (c(x)) (Uf )(x) = √ π x+i

.

is a unitary operator from .L2 (T) onto .L2 (R). Furthermore, a computation shows that 1 (x − i)n (U ξ n )(x) = √ , π (x + i)n+1

n  0,

1 (x + i)n−1 n , (U ξ )(x) = √ π (x − i)n

n  1.

.

and .

Since .(ξ n )n∈Z is an orthonormal basis for .L2 (T) and U is unitary, we have the following. Proposition 5.2.3 The vectors .(U ξ n )(x), where .n ∈ Z, form an orthonormal basis for .L2 (R). We mentioned Fourier series for .L2 (T). For .L2 (R) this gets replaced by the Fourier transform 1 .(Ff )(x) = √ 2π



∞ −∞

f (t)e−itx dt.

For .f ∈ L1 (R) ∩ L2 (R), this integral converges absolutely. For general .f ∈ L2 (R) this integral converges in the mean in that  A   1   f (t)e−itx dt  2 → 0 Ff − √ L (R) 2π −A

.

as A → ∞.

Plancherel’s theorem says that .F is a unitary operator on .L2 (R) and 1 (F ∗ f )(x) = √ 2π



∞

.

−∞

f (t)eitx dt.

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5.2.3 The Hardy Spaces The Hardy space is one of the first, and most fruitful, Hilbert spaces of analytic functions to be studied. Several wonderful texts for this are [12, 35, 38, 44]. There are two ways one can view the Hardy space. The first is to view .H 2 as a subspace of .L2 (T) by defining H 2 = {f ∈ L2 (T) : f(n) = 0 for all n < 0}.

.

Equivalently, .H 2 is the subspace of .L2 (T) of functions f with Fourier series f =

∞ 

.

f(n)ξ n .

n=0

Since f 2L2 (T) =

∞ 

.

|f(n)|2

n=0

 n  is finite, this implies that the power series . ∞ n=0 f (n)z converges for all .z ∈ D = {z ∈ C : |z| < 1} and thus defines an analytic function .f (z) on .D by .f (z) =  ∞  n n=0 f (n)z . Furthermore, f (rξ ) =

∞ 

.

f(n)r r ξ n ,

0  r < 1,

ξ ∈ T,

n=0

and the above is the Fourier series of the function .ξ → f (rξ ). This function has Fourier coefficients .(r n f(n))∞ n=0 and so Parseval’s theorem says that  .

T

|f (rξ )|2 dm(ξ ) =

∞ 

r 2n |f(n)|2 .

n=0

  2 The quantity above increases as .r → 1− and bounded above by . ∞ n=0 |f (n)| . 2 Thus, every .f ∈ H can be viewed as an analytic function on .D satisfying  .

sup

0r 0 for all r > 0.

.

The essential range is always closed. When .ϕ is continuous on .T, .Rϕ = ϕ(T), the range of .ϕ. Proposition 5.7.1 The following hold for .ϕ ∈ L∞ (T). (a) .Mϕ∗ = Mϕ . (b) .σ (Mϕ ) = Rϕ . Multiplication operators are normal operators in that .Mϕ Mϕ∗ = Mϕ∗ Mϕ . In fact, the spectral theorem says that any normal operator can be written as a multiplication operator on some .L2 (X, μ) space.

5.7.2 The Bilateral Shift An important multiplication operator is the bilateral shift .(Mf )(ξ ) = ξf (ξ ) on .L2 (T). With respect to the orthonormal basis .(ξ n )n∈Z for .L2 (T), the matrix representation of M is

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⎡

. . .. .. .. ⎢ . . . . ⎢··· 0 0 0 ⎢ ⎢··· 1 0 0 ⎢ ⎢ ⎢··· 0 1 0 .⎢ ⎢··· 0 0 1 ⎢ ⎢··· 0 0 0 ⎢ ⎢··· 0 0 0 ⎣ .. .. .. . . .

.. . 0 0 0 0 1 0 .. .

.. . 0 0 0 0 0 1 .. .

⎤ .. . ⎥ 0 ···⎥ ⎥ 0 ···⎥ ⎥ ⎥ 0 ···⎥ ⎥. 0 ···⎥ ⎥ 0 ···⎥ ⎥ 0 ···⎥ ⎦ .. . . . .

Note that M is unitary. Since .Rξ = T, then .σ (M) = T. Many important things are known about M. For example, one knows the .∗-cyclic vectors .f ∈ L2 (T). Proposition 5.7.2 The following are equivalent for .f ∈ L2 (T). (a) .span{M k f : k ∈ Z} = L2 (T). (b) .|f | > 0 almost everywhere. The proof of this is essentially Stone–Weierstrass theorem concerning the density of the trigonometric polynomials in .L2 (T). The harder problem, determining the cyclic vectors, that is the .f ∈ L2 (T) for which .span{M k f : k  0} = L2 (T), involves a theorem of Szego [29]. Proposition 5.7.3 The following are equivalent for .f ∈ L2 (T). (a) .span{M k f : k  0} = L2 (T). (b) .|f | > 0 almost everywhere and .log |f | ∈ L1 (T). The invariant subspaces are also known due to work of Wiener and Helson [28]. Theorem 5.7.4 Let .S ⊆ L2 (T) be an invariant subspace for M. (a) If .MS = S, then there is a measurable set .E ⊆ T such that .S = χE L2 (T). (b) If .MS  S, then there is a .ϕ ∈ L∞ (T) with .|ϕ| = 1 almost everywhere such that .S = ϕH 2 (T). If .AM = MA, then A commutes with multiplication by any trigonometric polynomial (which are dense in .L2 (T)). This can be used to prove the following. Corollary 5.7.5 .{M} = {Mϕ : ϕ ∈ L∞ (T)}.

5.7.3 Multiplication Operators on H 2 What are the multipliers of .H 2 (.ϕH 2 ⊆ H 2 )? Since the constant functions belong to .H 2 , any multiplier of .H 2 must belong to .H 2 . So the question becomes, what 2 2 2 .ϕ ∈ H satisfy .ϕH ⊆ H ?

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Proposition 5.7.6 For .ϕ ∈ H 2 , the following are equivalent. (a) .ϕH 2 ⊆ H 2 . (b) .ϕ ∈ H ∞ . Furthermore, when .ϕ ∈ H ∞ the operator .Tϕ f = ϕf is bounded on .H 2 with .Tϕ  = ϕ∞ . The operator .Tϕ will be a special example of a Toeplitz operator (to be discussed 2 in a moment). With respect to the orthonormal basis .(zn )∞ n=0 for .H , .Tϕ has the (lower triangular) matrix representation ⎡

a0 ⎢a ⎢ 1 ⎢ ⎢a2 .⎢ ⎢a3 ⎢ ⎢a4 ⎣ .. .

0 a0 a1 a2 a3 .. .

0 0 a0 a1 a2 .. .

0 0 0 a0 a1 .. .

0 0 0 0 a0 .. .

⎤ ··· · · ·⎥ ⎥ ⎥ · · ·⎥ ⎥ · · ·⎥ , ⎥ · · ·⎥ ⎦ .. .

 n where .ϕ(z) = ∞ n=0 an z . The adjoint of .Mϕ on .L2 (T) was .Mϕ . The adjoint of .Tϕ is a bit more tricky since multiplication by the function .ϕ (for .ϕ ∈ H ∞ ) does not take .H 2 to itself (unless .ϕ is a constant function). An important operator needed here is the orthogonal projection .P+ from .L2 (T) onto .H 2 , called the Riesz projection. It is defined via Fourier series by P+

∞  

.

n=−∞

∞   f(n)ξ n = f(n)ξ n . n=0

One can show that .P+2 = P+ and that .P+∗ = P+ . Proposition 5.7.7 For .ϕ ∈ H ∞ , .Tϕ∗ f = P+ (ϕf ). Proof For any .f, g ∈ H 2 , Tϕ f, g = P+ (ϕf ), g

.

= ϕf, P+ g = ϕf, g = f, ϕg = P+ f, ϕg = f, P+ (ϕg). This shows the desired formula.

 

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5.7.4 The Unilateral Shift A particularly important multiplication operator on .H 2 is the unilateral shift .(Sf )(z) = zf (z). Notice that S is an isometry and, with respect to the orthonormal 2 basis .(zn )∞ n=0 for .H , S has the matrix representation ⎡ 0 ⎢1 ⎢ ⎢ . ⎢0 ⎢ ⎣0 .. .

0 0 1 0 .. .

0 0 0 1 .. .

⎤ 0 ··· 0 · · ·⎥ ⎥ ⎥ 0 · · ·⎥ . ⎥ 0 · · ·⎦ .. . . . .

One can work out its adjoint (S ∗ f )(z) =

.

f (z) − f (0) z

which has the matrix representation ⎡ 0 ⎢0 ⎢ ⎢ . ⎢0 ⎢ ⎣0 .. .

1 0 0 0 .. .

0 1 0 0 .. .

⎤ 0 ··· 0 · · ·⎥ ⎥ ⎥ 1 · · ·⎥ . ⎥ 0 · · ·⎦ .. . . . .

Note that .S = Tz and .S ∗ = Tz . Some spectral results about S and .S ∗ are the following. Proposition 5.7.8 The following hold for the unilateral shift S and its adjoint .S ∗ . (a) .σ (S) = σ (S ∗ ) = D. (b) .σp (S) = ∅. (c) .σp (S ∗ ) = D. One of the most important theorems in operator theory is this result of Beurling [12, 29] that describes the invariant subspaces for S. Theorem 5.7.9 Let .M ⊆ H 2 , .M = {0}. Then the following are equivalent. (a) .SM ⊆ M. (b) .M = H 2 for some inner function .. In the above, an inner function is a . ∈ H 2 for which .|| = 1 almost everywhere on .T. Examples of inner functions include the unimodular constants; the monomials n .{z : n ∈ N0 }; the Blaschke products

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B(z) =

.

∞  aj aj − z , |aj | 1 − aj z

j =1

 where . ∞ j =1 (1 − |aj |) (which makes the infinite product above converge); and the singular inner functions 



Sμ (z) = exp −

.

T

 ξ +z dμ(ξ ) , ξ −z

where .μ is a finite positive Borel measure on .T that is singular with respect to Lebesgue measure m; as well as products of all four of these types of functions. An important factorization theorem says that any inner function can be written as n .γ z BSμ , where .γ ∈ T. The commutant of the bilateral shift M is .{Mϕ : ϕ ∈ L∞ (T)}. For the commutant of the unilateral shift S, we have the following. Proposition 5.7.10 .{S} = {Tϕ : ϕ ∈ H ∞ }. The elements of the commutant also form the multipliers of .H 2 (Proposition 5.7.6). The invariant subspaces of .S ∗ are .(H 2 )⊥ , where . is inner. However, one desires a more tangible description of them. In a way, they can be somewhat difficult to describe. The following result is a step in the right direction. Theorem 5.7.11 For an inner function ., the following are equivalent for .f ∈ H 2. (a) .f ∈ (H 2 )⊥ . (b) There is a .G ∈ H 2 ( C\D) which vanishes at .∞ such that .

lim

r→1−

f (rξ ) = lim G(sξ )  s→1+

for almost every .ξ ∈ T. (c) There is a .g ∈ H 2 such that .f (ξ ) = (ξ )ξg(ξ ) for almost every .ξ ∈ T. This brings up the whole topic of pseudocontinuations which can be further investigated in [46]. Theorem 5.7.11 is a bit abstract and not readily applicable. Perhaps by way of a specific example, let us consider the case where . is a Blaschke product with simple zeros .(λn )∞ n=1 . Example 5.7.12 If B is a Blaschke product with simple zeros, then (BH 2 )⊥ = span

.



1 1 − λk z

 :k1 .

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5.7.5 Toeplitz Operators on H 2 (D) A classical investigation dating back to Toeplitz explores the boundedness of the matrix operators ⎡

a0 ⎢a ⎢ 1 ⎢ ⎢a2 .T (a) = ⎢ ⎢a3 ⎢ ⎢a4 ⎣ .. .

a−1 a0 a1 a2 a3 .. .

a−2 a−1 a0 a1 a2 .. .

a−3 a−2 a−1 a0 a1 .. .

a−4 a−3 a−2 a−1 a0 .. .

⎤ ··· · · ·⎥ ⎥ ⎥ · · ·⎥ ⎥ · · ·⎥ , ⎥ · · ·⎥ ⎦ .. .

where .a = (aj )j ∈Z . Notice how these matrices are constant on each of the diagonals. The boundedness of such matrices (viewed as operators on .2 by left multiplication), called Toeplitz matrices, comes from the following result of Toeplitz (for the selfadjoint case) and Hartman and Winter (general case). Theorem 5.7.13 The matrix .T (a) defines a bounded operator on .2 if and only if there is a .ϕ ∈ L∞ (T) such that .aj =  ϕ (j ) for all .j ∈ Z. Observe that .T (a) is the lower right-hand corner of the matrix of the multiplication operator .Mϕ from (5.3). Inspired by this, Brown and Halmos studied operators on .H 2 that yield the same matrix. They do this as follows. For .ϕ ∈ L∞ (T) define the Toeplitz operator .Tϕ on .H 2 by Tϕ f = P+ (ϕf ),

.

f ∈ H 2.

Notice what is happening here. The product .ϕf belongs to .L2 (T), but not necessarily in .H 2 , and thus the need for the Riesz projection onto .H 2 . Theorem 5.7.14 (Brown–Halmos [6]) The Toeplitz operator .Tϕ is bounded on .H 2 if and only if .ϕ ∈ L∞ . Furthermore, .Tϕ  = ϕ∞ and matrix representation of n ∞ is the Toeplitz matrix .T (a), where .Tϕ with respect to the orthonormal basis .(z ) n=0 .a = ( ϕ (j ))j ∈Z . The proof of Proposition 5.7.7 yields the following. Proposition 5.7.15 .Tϕ∗ = Tϕ . Here are a few facts concerning the spectrum of a Toeplitz operator [10]. Theorem 5.7.16 Let .ϕ ∈ L∞ (T). (a) (b) (c) (d)

σ (Tϕ ) is connected. Rϕ ⊆ σ (Tϕ ) ⊆ co(Rϕ ), where .co(Rϕ ) is the closed convex hull of .Rϕ . If .ϕ is real valued, then .σ (Tϕ ) = [essinf ϕ, esssup ϕ]. If .ϕ ∈ H ∞ , then .σ (Tϕ ) = ϕ(D).

. .

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A beautiful theorem of Brown and Halmos answers the question when an operator on .H 2 is a Toeplitz operator. Recall that S is the unilateral shift in .H 2 . Theorem 5.7.17 (Brown–Halmos [6]) For .A ∈ B(H 2 ) the following are equivalent. (a) A is a Toeplitz operator. (b) .S ∗ AS = A. Here are some results as to when a Toeplitz operator is zero or when a product of Toeplitz operators is zero [4]. Theorem 5.7.18 For .ϕ ∈ L∞ (T) the following are equivalent (i) .ϕ = 0 almost everywhere. (ii) .Tϕ = 0. (iii) .Tϕ is compact. For .ϕ1 , ϕ2 , . . . , ϕn ∈ L∞ (T), the following are equivalent (i) .ϕj = 0 almost everywhere for some j (ii) .Tϕ1 Tϕ2 · · · Tϕn is finite rank. (iii) .Tϕ1 Tϕ2 · · · Tϕn = 0

5.7.6 Toeplitz Operators on H 2 (C+ ) There is an analogous theory for Toeplitz operators .T on .H 2 (C+ ) where . ∈ L∞ (R) and T f = PH 2 (C+ ) f.

.

In the above, .PH 2 (C+ ) is the orthogonal projection of .L2 (R) onto .H 2 (C+ ). Most of the results and the proofs for Toeplitz operators on .H 2 carry over to the Toeplitz operators on .H 2 (C+ ). There are a few results that look a little different. For example, the Brown–Halmos theorem becomes .A ∈ B(H 2 (C+ )) is a Toeplitz operator if and only if .A = Te−iλt ATeiλt for all .λ > 0.

5.7.7 Toeplitz Operators on Other Spaces We would be remiss if we failed to mention that Toeplitz operators have been studied on other Hilbert spaces of analytic functions such as the Bergman and Dirichlet spaces. See the final section for a definition of these spaces. The results are often different that what happens on the Hardy space. In particular, one can have nonzero compact operators on the Bergman space.

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5.8 Hankel Operators In this section we explore a class of patterned matrices of the form ⎡

α1 ⎢α2 ⎢ .⎢ ⎣α3 .. .

α2 α3 α4 .. .

α3 α4 α5 .. .

⎤ ··· · · ·⎥ ⎥ · · ·⎥ ⎦ .. .

which are called Hankel matrices. Probably the most famous of these is the classical Hilbert matrix ⎡

1

1 2 1 3 1 4 1 5

1 3 1 4 1 5 1 6

1 4 1 5 1 6 1 7

⎢1 ⎢2 ⎢ ⎢1 .H = ⎢ ⎢3 ⎢1 ⎣4 .. .. .. .. . . . .

···

⎤

⎥ ··· ⎥ ⎥ ⎥ ··· ⎥. ⎥ ··· ⎥ ⎦ .. .

As with Toeplitz matrices, one can recast these matrices as operators on function spaces. Two good sources for this are [40, 41]. An inequality of Hilbert says that H is a bounded operator on .2 with .H  = π . Further work of Magnus shows that .σ (H ) = [0, π ] and .σp (H ) = ∅. A result of Rosenblum [45] shows that H is unitarily equivalent to the multiplication operator .Mπ/ cosh(π ) on .L2 (R+ ).

5.8.1 Hankel Matrices For a vector .a = (an )n∈Z , start with the doubly infinite Hankel matrix ⎡

.. ⎢ . . ⎢ ··· a 4 ⎢ ⎢ ··· a ⎢ 3 ⎢ .H (a) = ⎢ · · · a2 ⎢ ⎢ · · · a1 ⎢ ⎢ · · · a0 ⎣ .. . ..

.. . a3 a2 a1 a0 a−1 .. .

.. . a2 a1 a0 a−1 a−2 .. .

.. . a1 a0 a−1 a−2 a−3 .. .

.. . a0 a−1 a−2 a−3 a−4 .. .

⎤ ⎥ ··· ⎥ ⎥ ··· ⎥ ⎥ ⎥ ··· ⎥ ⎥ ··· ⎥ ⎥ ··· ⎥ ⎦ .. .

and observe that the entries are constant on the reverse diagonals. If

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⎡

. . .. .. .. ⎢ . . . . ⎢ ··· 0 0 0 ⎢ ⎢ ··· 0 0 0 ⎢ ⎢ .F = ⎢ · · · 0 0 1 ⎢ ⎢ ··· 0 1 0 ⎢ ⎢ ··· 1 0 0 ⎣ .. .. .. . . .

.. . 0 1 0 0 0 .. .

.. . 1 ··· 0 ··· 0 ··· 0 ··· 0 ··· .. . . . .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

one shows that F is a unitary operator on .2 (Z) with .H (a) = F M(a), where ⎡

.. ⎢ . . ⎢ · · · a0 ⎢ ⎢··· a 1 ⎢ ⎢ ⎢ · · · a2 .M(a) = ⎢ ⎢ · · · a3 ⎢ ⎢ · · · a4 ⎢ ⎢ · · · a5 ⎣ .. .

.. .

.. .

.. .

.. .

.. .

a−1 a0 a1 a2 a3 a4 .. .

a−2 a−1 a0 a1 a2 a3 .. .

a−3 a−2 a−1 a0 a1 a2 .. .

a−4 a−3 a−2 a−1 a0 a1 .. .

a−5 a−4 a−3 a−2 a−1 a0 .. .

..

⎤ ⎥ ···⎥ ⎥ ···⎥ ⎥ ⎥ ···⎥ ⎥. ···⎥ ⎥ ···⎥ ⎥ ···⎥ ⎦ .. .

Notice how .M(a) is the matrix representation of the multiplication operator .Mϕ on .L2 (T) where .a = ( ϕ (n))n∈Z . From our earlier discussion, this says that .M(a) defines a bounded operator on .2 (Z) if and only if the vector .a is the sequence of Fourier coefficients of a bounded function .ϕ. The following theorem of Nehari [37] determines when a singly infinite Hankel matrix is bounded on .2 . Theorem 5.8.1 (Nehari) The matrix ⎡

a−1 ⎢a ⎢ −2 ⎢ ⎢a−3 .⎢ ⎢a−4 ⎢ ⎢a−5 ⎣ .. .

a−2 a−3 a−4 a−5 a−6 .. .

a−3 a−4 a−5 a−6 a−7 .. .

a−4 a−5 a−6 a−7 a−8 .. .

a−5 a−6 a−7 a−8 a−9 .. .

⎤ ··· · · ·⎥ ⎥ ⎥ · · ·⎥ ⎥ · · ·⎥ ⎥ · · ·⎥ ⎦ .. .

defines a bounded operator on .2 if and only if there is a .ϕ ∈ L∞ (T) such that .an =  ϕ (n) for all .n < 0.

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5.8.2 Hankel Operators To give us an idea of the context of the results from the previous section, let us put Hankel matrices in the context of operator theory on function spaces. As with Toeplitz operators, we will view .H 2 as a subspace of functions on .L2 (T) (those with vanishing negative Fourier coefficients). Define .H02 = ξ H 2 . The space .H02 will be the .L2 (T) functions with vanishing positive Fourier coefficients. Since the Riesz projection .P+ is the orthogonal projection of .L2 (T) onto .H 2 and .L2 (T) = H 2 ⊕H02 , I − P+ will be the orthogonal projection of .L2 (T) onto .H02 .

.

For .ϕ ∈ L∞ (T), define the Hankel operator .Hϕ : H 2 → H02 by Hϕ f = (I − P+ )(ϕf ).

.

m

2 ∞ 2 Notice how .(ξ n )∞ n=0 and .(ξ )m=1 are orthonormal bases for .H and .H0 respectively. Then

Hϕ ξ n , ξ −m  = P− (ϕξ n ), ξ −m 

.

= ϕξ n , P− (ξ −m ) = ϕξ n , ξ −m  = ϕ, ξ −m−n  = ϕ (−m − n). Thus, the entry in position .(m, n) of the matrix representation of .Hϕ with respect to the two bases above is . ϕ (−m − n), where .m  1 and .n  0. In other words, .Hϕ is represented by the infinite Hankel matrix from Theorem 5.8.1. Notice that .Hϕ = 0 if and only of .ϕ ∈ H ∞ . Furthermore, a result of Kronecker [34] says that the Hankel matrix ⎡

a−1 ⎢a ⎢ −2 ⎢ ⎢a−3 .⎢ ⎢a−4 ⎢ ⎢a−5 ⎣ .. .

a−2 a−3 a−4 a−5 a−6 .. .

a−3 a−4 a−5 a−6 a−7 .. .

a−4 a−5 a−6 a−7 a−8 .. .

a−5 a−6 a−7 a−8 a−9 .. .

has finite rank if and only of the function .f (z) = One can use this to prove the following.

⎤ ··· · · ·⎥ ⎥ ⎥ · · ·⎥ ⎥ · · ·⎥ ⎥ · · ·⎥ ⎦ .. .

∞

k=1 a−k z

k

Theorem 5.8.2 For .ϕ ∈ L∞ (T) the following are equivalent.

is a rational function.

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(i) .Hϕ has finite rank. (ii) .(I − P+ )ϕ is a rational function. (iii) There is a finite Blaschke product B such that .Bϕ ∈ H ∞ . Deeper results classify the compact Hankel operators. Theorem 5.8.3 For .ϕ ∈ L∞ (T) the following are equivalent. (i) .Hϕ is compact. (ii) .ϕ ∈ H ∞ + C(T). There is also a version of the Brown–Halmos theorem for Hankel operators. Theorem 5.8.4 Suppose that .A ∈ B(H 2 , H02 ), S is the unilateral shift on .H 2 , and M is the bilateral shift on .L2 (T). Then A is a Hankel operator if and only if .(I − P+ )MA = AS.

5.8.3 The Norm of a Hankel Operator Notice how the definition of the Hankel operator .Hϕ only depends on the negative Fourier coefficients of .ϕ ∈ L∞ (T). Furthermore, Hϕ f  = (I − P+ )(ϕf )  ϕf   ϕ∞ f .

.

Thus, .Hϕ   ϕ∞ . However, for .ψ ∈ H ∞ , .Hϕ = Hϕ−ψ and thus .Hϕ  = Hϕ−ψ   ϕ − ψ∞ . We conclude that Hϕ   inf∞ ϕ − ψ∞ = dist(ϕ, H ∞ ).

.

ψ∈H

A theorem of Nehari rounds out the picture. Theorem 5.8.5 (Nehari) For .ϕ ∈ L∞ (T), .Hϕ  = dist(ϕ, H ∞ ). Furthermore, there is a .ψ ∈ H ∞ such that .Hϕ  = ϕ − ψ∞ .

5.8.4 Bounded Mean Oscillation One can state Nehari’s theorem in a more tangible way. Theorem 5.8.6 A Hankel matrix ⎡

α1 ⎢α2 ⎢ .⎢ ⎣α3 .. .

α2 α3 α4 .. .

α3 α4 α5 .. .

⎤ ··· · · ·⎥ ⎥ · · ·⎥ ⎦ .. .

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 k defines a bounded operator on .2 if and only if .ϕ(z) = ∞ k=0 αk ξ is a function of bounded mean oscillation on .T. This operator is compact if and only of .ϕ is of vanishing mean oscillation on .T. A function .ϕ ∈ L1 (T) is of bounded mean oscillation if 1 . sup I ⊆T m(I )

 |f − fI |dm < ∞. I

In the above, I is an arc of .T and 1 .fI = m(I )

 f dm I

is the mean of f on I . The space of functions with bounded mean oscillation is denoted by .BMO. One can show that .L∞ (T) ⊆ BMO. However, the reverse containment is not true since .log |p| ∈ BMO, whenever p is a trigonometric polynomial with .p ≡ 0. A function .ϕ ∈ L1 (T) is of vanishing mean oscillation (.VMO) if .

lim sup

a→0+ |I |a

1 m(I )

 |f − fI |dm = 0. I

Note that .C(T) ⊆ VMO. A classical theorem of Riesz says that when .1 < p < ∞, the Riesz projection satisfies .P+ Lp (T) ⊆ Lp (T) (in fact .P+ Lp (T) = H p , the .Lp version of the Hardy space). This no longer holds when .p = ∞. Here one has .P+ L∞ (T) ⊆ BMO and .P+ C(T) ⊆ VMO.

5.9 The Hilbert Matrix Again Recall the Hilbert matrix ⎡

1

1 2 1 3 1 4 1 5

1 3 1 4 1 5 1 6

1 4 1 5 1 6 1 7

⎢1 ⎢2 ⎢ ⎢1 .H = ⎢ ⎢3 ⎢1 ⎣4 .. .. .. .. . . . .

···

⎤

⎥ ··· ⎥ ⎥ ⎥ ··· ⎥ ⎥ ··· ⎥ ⎦ .. .

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Note that f (z) =

∞ 

.

n=0

 1  1 n 1 z = log n+1 z 1−z

belongs to .BMOA and P ϕ = f,

.

ϕ(eit ) = ie−it (π − t).

5.9.1 Another Setting for Hankel Operators All of the operators, except the Hankel operators, studied in theses notes go from a Hilbert space to itself. There is a version of Hankel operators from .H 2 to itself. Here one needs to make a few adjustments. Define the operator J on .L2 (T) by 2 .(Jf )(ξ ) = ξ f (ξ ) and note that J is unitary. Furthermore, .J = I ; .J P+ J = I −P+ ; 2 2 2 2 .J H = H ; .J H = H . 0 0 For .ϕ ∈ L∞ define .ϕ : H 2 → H 2 by .ϕ = J Hϕ . Here one can show that 2 ∞ .ϕ  = Hϕ ; .A ∈ B(H ) is equal to .ϕ for some .ϕ ∈ L (T) if and only if ∗ .S A = AS; .ϕψ = ϕ Tψ + Tξ J ϕ ψ .

5.9.2 Back to Multiplication Operators For .ϕ ∈ L∞ (T), we explored the multiplication operator .Mϕ f = ϕf on .L2 (T). If we decompose .L2 (T) as .L2 (T) = H 2 ⊕ H02 , then we can write .Mϕ in matrix form 

 Tϕ Hϕ∗ .Mϕ = . Hϕ Sϕ In the above, .Tϕ is a Toeplitz operator, .Hϕ is the Hankel operator, and .Sϕ : H02 → H02 is defined by .Sϕ f = (I − P+ )(ϕf ), .f ∈ H02 , is the dual of a Toeplitz operator. For .ϕ, ψ ∈ L∞ (T) one can work out the matrix product of .Mϕ and .Mψ to prove ∗ = T H ∗ + H ∗S . the useful formulas .Tϕψ = Tϕ Tψ + Hϕ∗ Hψ and .Hϕψ ϕ ψ ϕ ψ

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5.10 Fourier and Hilbert Transforms 5.10.1 Plancherel’s Theorem Although we already discussed the Fourier transform .F, let us cover some of the operator-theoretic properties of .F. Recall that 

1 (Ff )(x) = lim √ A→∞ 2π

A

.

−A

f (t)e−itx dx,

where we understand the above limit to be in the mean; .

 A   1   lim Ff − √ f (t)e−itx dx  2 = 0. A→∞ L (R) 2π −A

If .f ∈ L1 (R) ∩ L2 (R), then 1 (Ff )(x) = √ 2π



∞

.

−∞

f (t)e−itx dx

for every .x ∈ R. In fact, .(Ff )(±∞) = 0 (which is the Riemann–Lebesgue lemma). The important theorem here is due to Plancherel. Theorem 5.10.1 (Plancherel) The Fourier transform is a unitary operator on L2 (R) with

.

1 (F ∗ f )(x) = √ 2π



∞

.

−∞

f (t)eixt dx.

5.10.2 The Spectrum of F To begin to understand the spectrum of .F, observe that the operator .(Uf )(x) = f (−x) is unitary on .L2 (R) and .U 2 = I . Moreover, .F ∗ = U F and thus .F4 = I . One can verify the identity 1 (F3 + zF2 + z2 F + z3 I ) 1 − z4

(F − zI )−1 =

.

for all .z ∈  {±1, ±i} and hence .σ (F) ⊆ {±1, ±i}. Another calculation shows that FHn = (−i)n Hn ,

.

where .Hn are the normalized Hermite functions from Proposition 5.2.2. The summary result here is the following.

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Proposition 5.10.2 .σ (F) = σp (F) = {1, −1, i, −i}. With respect to the orthonormal basis of (normalized) Hermite functions, the matrix representation of .F is ⎡ 1 ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ . ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎣ .. .

0 −i 0 0 0 0 0 0 .. .

0 00 0 0 00 0 −1 0 0 0 0 i 0 0 0 01 0 0 0 0 −i 0 00 0 0 00 0 .. .. .. .. . . . .

0 0 0 0 0 0 −1 0 .. .

⎤ 0 ··· 0 · · ·⎥ ⎥ 0 · · ·⎥ ⎥ ⎥ 0 · · ·⎥ ⎥ 0 · · ·⎥ . ⎥ 0 · · ·⎥ ⎥ 0 · · ·⎥ ⎥ i · · ·⎥ ⎦ .. . . . .

5.10.3 The Hilbert Transform For .g ∈ L2 (R), consider 1 .(Hg)(x) = PV π



∞ −∞

g(t) 1 dt = lim x−t π →0+

 |t|>

g(t) dt, x−t

the Hilbert transform of g. The principal value is needed here since the integrand is not necessarily convergent [21]. This integral initially appeared in the study of harmonic conjugates. Indeed, the function,  (Pg)(z) :=

∞

.

−∞

Re

 1 1  g(t)dt, iπ t − z

called the Poisson integral of g, is harmonic on .C+ and has a radial limit as .z → x equal to .g(x) or almost every .x ∈ R. Informally speaking, the above function solves the Dirichlet problem with data g. The function  (Qg)(z) :=

∞

.

−∞

Im

 1 1  g(t)dt iπ t − z

is harmonic on .C+ and is the (unique up to an additive constant) harmonic conjugate of .Pg. In fact, (Pg)(z) + i(Qg)(z) =

.

1 iπ



∞ −∞

g(t) dt. t −z

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This last integral is known as the Cauchy transform of g. Some rather delicate limiting arguments will prove the following. Proposition 5.10.3 For .g ∈ L2 (R), .

lim (Qg)(x + iy) = (Hg)(x)

y→0+

for almost every .x ∈ R. The important theorem here is the following. We include a proof since it relates to the previous section on the Fourier transform .F on .L2 (R). Theorem 5.10.4 The Hilbert transform .H is a unitary operator on .L2 (R) such that 2 .H = −I . Proof For .f ∈ L2 (R), let .(Wf )(x) = −i sgn(x)f (x). Note that W is isometric on .L2 (R) and an integral calculation with the Fourier transform .F shows that .H = F∗ W F. This shows that .H is unitary and .H2 = −I .  

5.10.4 Spectrum of H We end this section with a description of the spectrum of the Hilbert transform. Theorem 5.10.5 .σ (H) = σp (H) = {−i, i}. Proof By the proof of Theorem 5.10.4, .H is unitarily equivalent (via the Fourier transform) to the multiplication operator .(Wf )(x) = −i sgn(x)f (x) on .L2 (R). An earlier discussion shows that .σ (W ) = σp (W ) is the essential range of the symbol .−i sgn(x) which is equal to .{−i, i}.   As with the Fourier transform, we can give the complete spectral decomposition of .H. Consider the following functions: .

1 (x − i)n vn (x) = √ , π (x + i)n+1

n  0,

1 (x + i)n−1 wn (x) = √ , π (x − i)n

n < 0.

and .

As stated in Proposition 5.2.3, .{vn : n  0} ∪ {wn : n < 0} is an orthonormal basis for .L2 (R). Using some computational facts about the Hilbert transform, one can see that   . ker(H + iI ) = {vn : n  0} and ker(H − iI ) = {wn : n  1}.

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The matrix representation of .H with respect to this basis is the doubly infinite diagonal matrix ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ .⎢ ⎢ ⎢ ⎢ ⎢ ⎣

..

. 0 0 0 0 0 .. .

00 i 0 0i 00 00 00 .. .. . .

0 0 0 −i 0 0 .. .

0 0 0 0 −i 0 .. .

0 0 0 0 0 −i .. .

⎤ · · ·⎥ · · ·⎥ ⎥ · · ·⎥ ⎥ ⎥ · · ·⎥ . ⎥ · · ·⎥ ⎥ · · ·⎥ ⎦ .. .

Although we did not get too much into this here, the mapping properties of the Hilbert transform were a rich source of study. For .1 < p < ∞, .HLp (R) ⊆ Lp (R) (but this does not hold when .p = 1 or .p = ∞). Furthermore, the Hilbert transform preserves certain Lipschitz classes of functions on .R.

5.11 Further Explorations One is free to explore the operators covered in these notes beyond the Hardy space H 2 . Indeed, there are two other common spaces to discuss. Notice how an analytic n 2 function .f (z) = ∞ n=0 an z on .D belongs to .H when

.

f 2H 2 =

∞ 

.

|an |2

n=0

is finite. Two related spaces where people have studied these operators are the Bergman space .A2 of analytic functions f (as above) with f 2A2 =

.

∞  |an |2 n+1 n=0

is finite and the Dirichlet space of analytic functions with f 2D =

∞ 

.

|an |2 (n + 1)

n=0

finite. Notice that .D  H 2  A2 . The shift operators, Toeplitz, and Hankel operators have been explored on these spaces and due the differences in the function theory of the ambient spaces .H 2 , D, and .A2 , the operator theory for these spaces is different. Good sources to learn more about operators on other function spaces besides .H 2 are [11, 13, 27, 57].

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Conspicuously missing from our discussion are the composition operators Cϕ f = f ◦ ϕ, where .ϕ is an analytic self map of .D, on various spaces of analytic functions on .D. For .H 2 , these operators have a long history and much is understood about them [8, 52]. More recent work covers many of these operators on Hilbert spaces of analytic functions on the polydisk .Dn and the unit ball in .Cn . The operators have the same definitions but the results are sometimes different due to the more complicated geometry stemming from the several variable setting. Of course there is the whole topic of model theory for contractions which realizes certain contractions on Hilbert spaces as the compression of the shift S to subHardy Hilbert spaces. A particular example of this is the compression of the shift operator S on .H 2 to .(H 2 )⊥ , where . is an inner function. These are used to model contractions T on Hilbert spaces for which .rank(I − T ∗ T ) = rank(I − T T ∗ ) = 1 and .T n → 0 strongly. Other examples involve the de Branges–Rovnyak spaces [15, 16].

.

References 1. AGMON, S. Sur un problème de translations. C. R. Acad. Sci. Paris 229 (1949), 540–542. 2. ALEMAN, A., AND CIMA, J. A. An integral operator on H p and Hardy’s inequality. J. Anal. Math. 85 (2001), 157–176. 3. ALEMAN, A., AND KORENBLUM, B. Volterra invariant subspaces of H p . Bull. Sci. Math. 132, 6 (2008), 510–528. 4. ALEMAN, A., AND VUKOTI C´ , D. Zero products of Toeplitz operators. Duke Math. J. 148, 3 (2009), 373–403. 5. BÖTTCHER, A., AND SILBERMANN, B. Analysis of Toeplitz operators, second ed. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2006. Prepared jointly with Alexei Karlovich. 6. BROWN, A., AND HALMOS, P. R. Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213 (1963/1964), 89–102. 7. BROWN, A., HALMOS, P. R., AND SHIELDS, A. L. Cesàro operators. Acta Sci. Math. (Szeged) 26 (1965), 125–137. 8. COWEN, C. C., AND MACCLUER, B. D. Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. 9. DONOGHUE, JR., W. F. The lattice of invariant subspaces of a completely continuous quasinilpotent transformation. Pacific J. Math. 7 (1957), 1031–1035. 10. DOUGLAS, R. G. Banach algebra techniques in operator theory. Pure and Applied Mathematics, Vol. 49. Academic Press, New York-London, 1972. 11. DUREN, P., AND SCHUSTER, A. Bergman Spaces, vol. 100 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2004. 12. DUREN, P. L. Theory of H p spaces. Academic Press, New York, 1970. 13. EL-FALLAH, O., KELLAY, K., MASHREGHI, J., AND RANSFORD, T. A Primer on the Dirichlet Space, vol. 203 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2014. 14. ERDOS, J. A. The commutant of the Volterra operator. Integral Equations Operator Theory 5, 1 (1982), 127–130. 15. FRICAIN, E., AND MASHREGHI, J. The Theory of H (b) spaces, Volume 1, vol. 20 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2014.

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16. FRICAIN, E., AND MASHREGHI, J. The Theory of H (b) spaces, Volume 2, vol. 21 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2015. 17. GARCIA, S. R. Conjugation and Clark operators. In Recent advances in operator-related function theory, vol. 393 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2006, pp. 67– 111. 18. GARCIA, S. R., PRODAN, E., AND PUTINAR, M. Mathematical and physical aspects of complex symmetric operators. J. Phys. A 47, 35 (2014), 353001, 54. 19. GARCIA, S. R., AND PUTINAR, M. Complex symmetric operators and applications. Trans. Amer. Math. Soc. 358, 3 (2006), 1285–1315. 20. GARCIA, S. R., AND PUTINAR, M. Complex symmetric operators and applications. II. Trans. Amer. Math. Soc. 359, 8 (2007), 3913–3931. 21. GARNETT, J. Bounded analytic functions, first ed., vol. 236 of Graduate Texts in Mathematics. Springer, New York, 2007. 22. HALMOS, P. R. A Hilbert space problem book, second ed., vol. 19 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1982. Encyclopedia of Mathematics and its Applications, 17. 23. HARDY, G. H. Divergent Series. Oxford, at the Clarendon Press, 1949. 24. HARDY, G. H., LITTLEWOOD, J. E., AND PÓLYA, G. Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988. Reprint of the 1952 edition. 25. HARTMAN, P., AND WINTNER, A. On the spectra of Toeplitz’s matrices. Amer. J. Math. 72 (1950), 359–366. 26. HARTMAN, P., AND WINTNER, A. The spectra of Toeplitz’s matrices. Amer. J. Math. 76 (1954), 867–882. 27. HEDENMALM, H., KORENBLUM, B., AND ZHU, K. Theory of Bergman spaces, vol. 199 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. 28. HELSON, H. Lectures on invariant subspaces. Academic Press, New York-London, 1964. 29. HOFFMAN, K. Banach spaces of analytic functions. Dover Publications Inc., New York, 1988. Reprint of the 1962 original. 30. HUPERT, L., AND LEGGETT, A. On the square roots of infinite matrices. Amer. Math. Monthly 96, 1 (1989), 34–38. 31. KHADKHUU, L., AND TSEDENBAYAR, D. On the numerical range and numerical radius of the Volterra operator. Izv. Irkutsk. Gos. Univ. Ser. Mat. 24 (2018), 102–108. 32. KRIETE, T. L., AND TRUTT, D. On the Cesàro operator. Indiana Univ. Math. J. 24 (1974/75), 197–214. 33. KRIETE, III, T. L., AND TRUTT, D. The Cesàro operator in l 2 is subnormal. Amer. J. Math. 93 (1971), 215–225. 34. KRONECKER, L. Zur Theorie der Elimination einer Variablen aus zwei algebraischen Gleichungen. Monatsber. Königl. Preussischen Akad. Wiss. (Berlin) (1881), 535–600. 35. MASHREGHI, J. Representation theorems in Hardy spaces, vol. 74 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2009. 36. MASHREGHI, J., PTAK, M., AND ROSS, W. T. Square roots of some classical operators. ArXiv e-prints (2021). 37. NEHARI, Z. On bounded bilinear forms. Ann. of Math. (2) 65 (1957), 153–162. 38. NIKOLSKI, N. Hardy spaces, French ed., vol. 179 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2019. 39. NIKOLSKI, N. Toeplitz matrices and operators, French ed., vol. 182 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2020. 40. PARTINGTON, J. R. An introduction to Hankel operators, vol. 13 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1988. 41. PELLER, V. V. Hankel operators and their applications. Springer Monographs in Mathematics. Springer-Verlag, New York, 2003. 42. PERSSON, A.-M. On the spectrum of the Cesàro operator on spaces of analytic functions. J. Math. Anal. Appl. 340, 2 (2008), 1180–1203.

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43. POMMERENKE, C. Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation. Comment. Math. Helv. 52, 4 (1977), 591–602. 44. RADJAVI, H., AND ROSENTHAL, P. Invariant subspaces, second ed. Dover Publications, Inc., Mineola, NY, 2003. 45. ROSENBLUM, M. On the Hilbert matrix. II. Proc. Amer. Math. Soc. 9 (1958), 581–585. 46. ROSS, W. T., AND SHAPIRO, H. S. Generalized analytic continuation, vol. 25 of University Lecture Series. American Mathematical Society, Providence, RI, 2002. 47. ROYDEN, H. L. Real analysis, third ed. Macmillan Publishing Company, New York, 1988. 48. RUDIN, W. Real and Complex Analysis, third ed. McGraw-Hill Book Co., New York, 1987. 49. RUDIN, W. Functional analysis, second ed. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. 50. SARASON, D. A remark on the Volterra operator. J. Math. Anal. Appl. 12 (1965), 244–246. 51. SARASON, D. Generalized interpolation in H ∞ . Trans. Amer. Math. Soc. 127 (1967), 179–203. 52. SHAPIRO, J. H. Composition operators and classical function theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. 53. SHAPIRO, J. H. Volterra adventures, vol. 85 of Student Mathematical Library. American Mathematical Society, Providence, RI, 2018. 54. SHIELDS, A. L., AND WALLEN, L. J. The commutants of certain Hilbert space operators. Indiana Univ. Math. J. 20 (1970/71), 777–788. 55. TOEPLITZ, O. Zur theorie der quadratischen Formen von unendlichvielen Veränderlichen. Nachr. Kön. Ges. Wiss. Göttingen (1910), 489–506. 56. TOEPLITZ, O. Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen Veränderlichen. Math. Ann. 70, 3 (1911), 351–376. 57. ZHU, K. Operator theory in function spaces, second ed., vol. 138 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2007. 58. ZYGMUND, A. Trigonometric series. Vol. I, II, third ed. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002. With a foreword by Robert A. Fefferman.

Chapter 6

Truncated Toeplitz Operators Emmanuel Fricain

6.1 Introduction This is a set of lecture notes to accompany a series of talks given as part of the Fields Institute session on Truncated Toeplitz Operators from July–December 2021. To prepare these notes, we intensely use two recent surveys on truncated Toeplitz operators [16, 38], as well as [32, Chapter 13] and the founding paper [52]. Truncated Toeplitz operators on model spaces have been formally introduced by Sarason in [52], although some special cases have long ago appeared in literature, most notably as model operators for completely nonunitary contractions with defect numbers one and for their commutant. This new area of study has been recently very active and many open questions posed by Sarason in [52] have now been solved. See [6–9, 17, 20, 33, 38, 39, 53, 54]. Nevertheless, there are still basic and interesting questions which remain mysterious. The truncated Toeplitz operators live on the model spaces .K , which are the closed invariant subspaces for the backward shift operator .S ∗ acting on the Hardy space .H 2 (see Sect. 6.2 for precise definitions). Given a model space .K and a function .ϕ ∈ L2 = L2 (T), the truncated Toeplitz operator .A ϕ (or simply .Aϕ if there is no ambiguity regarding the model space) is defined on a dense subspace of .K as the compression to .K of multiplication by .ϕ. The function .ϕ is then called a symbol of the operator. Note that the symbol is never uniquely defined by the operator. From this and other points of view the truncated Toeplitz operators have much more in common with Hankel Operators than with Toeplitz operators.

E. Fricain () Université de Lille, Laboratoire Paul Painlevé, UFR de Mathématiques Bâtiment M2, Villeneuve d’Ascq Cédex, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mashreghi (ed.), Lectures on Analytic Function Spaces and their Applications, Fields Institute Monographs 39, https://doi.org/10.1007/978-3-031-33572-3_6

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We intend to give a short introduction to this fascinating area of research. Our objective is not to be exhaustive but rather to make discover the different techniques and the beauty of this theory through some key results. The structure of the paper is the following. After a preliminary section with generalities about Hardy spaces, model spaces, one-component inner functions (which will play a special role here), Toeplitz and Hankel operators, and Carleson measures, first for the whole .H 2 and then for model spaces, truncated Toeplitz operators are introduced in Sect. 6.3. Then, in Sect. 6.4, we discuss why it is worth studying truncated Toeplitz operators. We explain how they appear in several natural problems in operator theory and complex function theory. In Sect. 6.5, we describe the class of symbols, because one difficulty with truncated Toeplitz operators, compared to the classical Toeplitz operator, is that the symbol is never unique. In Sects. 6.6 and 6.7, we give several useful characterization of truncated Toeplitz operators and discuss some interesting connections with another important class of operators, the so-called complex symmetric operators. In Sect. 6.8, we give some estimates on the norm of a truncated Toeplitz operator, while in Sect. 6.9, we obtain a complete description of the spectrum of a truncated Toeplitz operator associated to a symbol in .H ∞ . In Sects. 6.9 and 6.10, we discuss the class of finite rank and compact truncated Toeplitz operators. Finally, in the last section, we discuss the important problem of the existence of a bounded symbol.

6.2 Preliminaries For the content of this section, [26, 43] are classical references for general facts about Hardy spaces, while [46] can be used for Toeplitz and Hankel operators and [32] for model spaces. We recall the main definitions and properties but we assume that the reader is a little familiar with the theory of Hardy spaces and Toeplitz operators

6.2.1 Function Spaces, Multiplication Operators and Their Cognates Recall that for .1  p < +∞, the Hardy space .H p of the open unit disk .D = {z ∈ C : |z| < 1} is the space of analytic functions f on .D satisfying .f p < +∞, where 

2π

f p = sup

.

0r 0. λ∈D∪E()

(c) For .λ ∈ D, we have 2

Ckλ p  kλ p  2kλ p ,

.

(6.6)

where .C = P −1 Lp →Lp is a constant which depends only on . and p. Also, if 2  p p .ζ ∈ E(), then .k ζ ∈ L if and only if .kζ ∈ L and (6.6) holds for .λ = ζ . Proof The proof of .(a) is immediate using definition. For the proof of .(b), note that, for .λ ∈ D ∪ E(), we have 1−|(0)|  |1−(0)(λ)| = |k0 (λ)|  k0 2 kλ 2 = (1−|(0)|2 )1/2 kλ 2 ,

.

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which implies   .kλ 2



1 − |(0)| 1 + |(0)|

1/2 . 2

It remains to prove .(c). Using (6.4), we have .kλ = (1 + (λ))kλ , whence 2  p .P k λ = kλ . Thus the result follows from the fact that .P is bounded on .L and from the trivial estimate .|1 + (λ)(z)|  2, .z ∈ T. It is easy to check that the above arguments also hold when .λ = ζ ∈ E().   Lemma 6.2.3 Let . be an inner function. Then (i) .K2 = K ⊕ K . 1 . In particular, if .f, g ∈ K and f or g is also bounded, then (ii) .K · K ⊂ K  2 .fg ∈ K2 . Proof (1) Observe that H 2 = K ⊕ H 2 = K ⊕ (K ⊕ H 2 ) = K ⊕ K ⊕ 2 H 2 ,

.

which implies that .K2 = H 2 2 H 2 = K ⊕ K . (2) Let .f, g ∈ K , and write .f = f1 , .g = g1 , with .f1 , g1 ∈ H02 . Then 1 1 2 .f g ∈ H and since .f1 g1 ∈ H , we also have .fg =  f1 g1 . That means that 0 1 . Finally, observe that if f or g is bounded, then .f f g ∈ H 1 ∩ 2 H01 = K 1 2 2 2 2 or .g1 is also bounded and then .f g ∈ H ∩  H = K2 .  

.

 )2 ∈ K . It follows from Lemma 6.2.3 that for every .λ ∈ D, .(k 2 A useful tool in the theory of model spaces is the notion of conjugation. Recall that a map C from a Hilbert space .H into itself is a conjugation on .H if C is anti-linear, isometric and involutive, meaning that the following properties are satisfied:

(i) for every .x, y ∈ H and every .λ ∈ C, .C(λx + y) = λCx + Cy; (ii) for every .x, y ∈ H, .Cx, Cy = y, x; (iii) .C 2 = I . It is easy to check that the map .C defined on .L2 by C f = ¯zf¯;

(6.7)

.

is a conjugation on .L2 which has the convenient supplementary property of mapping .K precisely onto .K . In other words, its restriction to .K is a conjugation on .K . When convenient we shall write .f˜ for .C f . It can be easily verified that (z) − (λ)  , k λ (z) = z−λ

.

z, λ ∈ D.

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 ∗ In particular, .k 0 = S . Moreover, when . has an angular derivative in the sense of Carathéodory at .ζ ∈ T, we have

(z) − (ζ )  = ζ¯ (ζ )kζ (z), k ζ (z) = z−ζ

.

z ∈ D.

In view of their main role in the study of operators on model spaces, we end this subsection by a discussion on a particular class of inner functions. Fix a number .0 < ε < 1, and define (, ε) = {z ∈ D : |(z)| < ε}.

(6.8)

.

The function . is called one-component if there exists a value of .ε for which (, ε) is connected. (If this happens, then .(, δ) is connected for every .ε < δ < 1.) One-component functions have been introduced by Cohn [23]. An extensive study of these functions appears in [2, 5]; all results quoted below appear in [5]. See also [18, 19] for more recent results on this interesting class. The above definition is not very transparent. In fact, one-component functions are rather special: a first immediate reason is that they must satisfy .m(σ ()) = 0. This condition, of course, is not sufficient, but it suggests examining some simple cases. The set .σ () is of course empty for finite Blaschke products, which are onecomponent. The next simplest case is when .σ () consists of just one point. One

.

z+ζ

can prove easily that the elementary singular inner functions .(z) = e z−ζ (for .ζ ∈ T) are indeed one-component. Suppose then that . is a Blaschke product whose zeros .an tend nontangentially to a single point .ζ ∈ T. If .

|ζ − an+1 | > 0, n≥1 |ζ − an | inf

(6.9)

then . is one-component. So, in particular, if .0 < r < 1 and . is the Blaschke product with zeros .1 − r n , .n  1, then . is one-component. If condition (6.9) is not satisfied, then usually . is not one-component. A detailed discussion of such Blaschke products is given in [5], including the determination of Carleson measures for such model spaces (see Sect. 6.2.3).

6.2.3 Carleson Measures for the Hardy Spaces and for the Model Spaces Let us discuss first some objects related to the Hardy space; we will afterwards see what analogous facts are true for the case of model spaces.

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A finite measure .μ on .D is called a Carleson measure if .H 2 ⊂ L2 (|μ|) (such an inclusion, if it exists, is automatically continuous). It is known that this is equivalent to .H p ⊂ Lp (|μ|) for all .1 ≤ p < ∞. Carleson measures can also be characterized by a geometrical condition, as follows. For an arc .I ⊂ T such that .|I | := m(I ) < 1 we define its associated Carleson window S(I ) = {z ∈ D : 1 − |I | < |z| < 1 and z/|z| ∈ I }.

.

Then .μ is a Carleson measure if and only if .

sup I

|μ|(S(I )) < ∞. |I |

(6.10)

Condition (6.10) is called the Carleson condition. Note that Carleson measures cannot have mass on the unit circle (intervals containing a Lebesgue point of the corresponding density would contradict the condition (6.10)). These measures and condition (6.10) appeared in the famous work of Carleson on .H ∞ interpolating sequences [13, 14]. Analogous results may be proved concerning compactness. In this case the relevant notion is that of vanishing Carleson measure, which is defined by the property .

|μ|(S(I )) = 0. |I |→0 |I | lim

(6.11)

Then the embedding .H p ⊂ Lp (|μ|) is compact if and only if .μ is a vanishing Carleson measure. Similar questions for model spaces have been developed starting with the papers [23, 24] and [55]; however, the results in this case are less complete. Let us introduce first some notations. For .1 ≤ p < ∞, define .

p

Cp () = {μ finite measure on T : K → Lp (|μ|) is bounded},

It is clear that .Cp () is a complex vectorial subspace of the complex measures 1 , it on the unit circle. Using the relations .K2 = K ⊕ K and .K · K ⊂ K 2 2 2 is easy to see that .C2 ( ) = C2 () and .C1 ( ) ⊂ C2 (). Example 6.2.4 Let .ζ ∈ E(). Then the Dirac measure .δζ at point .ζ belongs to C2 (). Note that this is in contrast with Carleson measures for the Hardy spaces.

.

It is natural to look for geometric conditions to characterize these classes. Things are, however, more complicated, and the results are only partial. We start by fixing a number .0 < ε < 1; then the .(, ε)-Carleson condition asserts that .

sup I

|μ|(S(I )) < ∞, |I |

(6.12)

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where the supremum is taken only over the intervals .|I | such that .S(I ) ∩ (, ε) = ∅. (Remember that .(, ε) is given by (6.8).) It is then proved in [55] that if .μ satisfies the .(, ε)-Carleson condition, then p the embedding .K ⊂ Lp (|μ|) is continuous. The converse is true if . is onecomponent; in which case the embedding condition does not depend on p, while fulfilling of the .(, ε)-Carleson condition does not depend on .0 < ε < 1 (see Theorem 6.2.5 below). As concerns the general case, it is shown by Aleksandrov [5] that if the converse is true for some .1 ≤ p < ∞, then . is one-component. Also, . is one-component if and only if the embedding condition does not depend on p. More precisely, the next theorem is proved in [5] (note that a version of this result for .p ∈ (1, ∞) already appears in [55]). Theorem 6.2.5 The following are equivalent for an inner function .: (i) . is one-component. (ii) For some .0 < p < ∞ and .0 < ε < 1, .Cp () concides with the class of measures that satisfy the .(, ε)-Carleson condition. (iii) For all .0 < p < ∞ and .0 < ε < 1, .Cp () concides with the class of measures that satisfy the .(, ε)-Carleson condition. (iv) The class .Cp () does not depend on .p ∈ (0, ∞). In particular, if . is one component, then so is .2 , whence .C1 (2 ) = C2 (2 ) = C2 (). Observe that the .(, ε)-Carleson condition is less rigid than the Carleson condition (6.10) because we only need to test condition (6.10) on a subclass of Carleson windows. This can be explained by the fact that functions in .K are in general more regular than an arbitrary .H 2 function; in particular, they can be analytically continued in a neighborhood of .T \ σ (), and .(, ε) ∩ T ⊂ σ (). Note that a general characterization of .C2 () has recently been obtained in [41]; however, the geometric content of this result is not easy to see.

6.3 Truncated Toeplitz Operators 6.3.1 Definition of Truncated Toeplitz Operators Let . be a (non constant) inner function and .ϕ ∈ L2 . The truncated Toeplitz operator .Aϕ = A ϕ , introduced by Sarason in [52], is a densely defined, possibly unbounded operator on .K . Its domain contains .K ∩ H ∞ , on which it acts by the formula Aϕ f = P (ϕf ),

.

f ∈ K ∩ H ∞ .

(6.13)

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205

Note that .K ∩ H ∞ is dense in .K because it contains the reproducing kernels .kλ , since we trivially have the following estimate: |kλ (z)| 

.

2 , 1 − |λ|

z, λ ∈ D.

If .Aϕ thus defined extends to a bounded operator, that operator is called a TTO. The class of all TTOs on .K is denoted by .T(). In other words, T() = {Aϕ : ϕ ∈ L2 and Aϕ ∈ L(K )}.

.

Example 6.3.1 When .ϕ ∈ L∞ , then .ϕf ∈ L2 for all .f ∈ K and so P (ϕf )2  ϕf 2  ϕ∞ f 2 .

.

In particular, .Aϕ is a TTO and .Aϕ   ϕ∞ . Example 6.3.2 The case when .ϕ(z) = z will play a special role. It will be denoted by .S and is called the model operator on .K (see Sect. 6.4.1). In other words, S (f ) = P (zf ),

.

f ∈ K .

It is well-known that, in general, Toeplitz operators do not commute. This is also the case for TTO but in a special case, TTO commute. Proposition 6.3.3 Let .ϕ, ψ ∈ H ∞ . Then .Aϕ Aψ = Aψ Aϕ = Aϕψ . Proof Let .ϕ, ψ ∈ H ∞ . Since .ϕH 2 ⊂ H 2 , for every .f ∈ K , we have Aϕ Aψ f = P (ϕP ψf ) = P (ϕψf ) = Aϕψ f.

.

Therefore .Aϕ Aψ = Aϕψ . The second relation .Aψ Aϕ = Aϕψ immediately follows by changing the role of .ϕ and .ψ.   In particular, according to Proposition 6.3.3, if .ϕ ∈ H ∞ , then  A ϕ S = S Aϕ .

.

∞ In other words, .A ϕ is in the commutant of .S when .ϕ ∈ H . An important result says that the converse is true (see Sect. 6.4.2).

Example 6.3.4 When .(z) = zn , for some .n  1, the space .K consists of polynomials of degree less or equal to .n − 1 and the set .{1, z, . . . , zn−1 } forms an orthonormal basis for .K . Moreover, any truncated Toeplitz operator .Aϕ , when represented with respect to this basis, yields a Toeplitz matrix. Indeed, for .0 

206

E. Fricain

j, k  n − 1, we have Aϕ zj , zk 2 = P (ϕzj ), zk 2 = ϕzj , zk 2 .

= ϕ, zk−j 2 = ϕ (k − j ), and so the matrix representation of .Aϕ with respect to the basis .{1, z, . . . , zn−1 } is the Toeplitz matrix ⎛

⎞ ϕ(−1) ˆ ϕ(−2) ˆ . . . . . . ϕ(−n ˆ + 1) .. ⎜ ⎟ .. ⎜ ϕ(1) ⎟ . . ϕ(0) ˆ ϕ(−1) ˆ ⎜ ˆ ⎟ ⎜ ⎟ . . . . . . . . ⎜ ϕ(2) ⎟ . . . ˆ ϕ(1) ˆ . ⎟ .⎜ ⎜ ⎟ .. .. .. .. ⎜ ⎟ . . . . ϕ(−1) ˆ ϕ(−2) ˆ ⎜ ⎟ ⎜ ⎟ . . .. . . ϕ(1) ⎝ ⎠ ˆ ϕ(0) ˆ ϕ(−1) ˆ . . . ϕ(2) ˆ ϕ(1) ˆ ϕ(0) ˆ ϕ(−n ˆ + 1) . . . ϕ(0) ˆ

It is worth pointing out in this particular case that since .K is a finite dimensional space, then .Aϕ ∈ T() for all .ϕ ∈ L2 . Conversely, any .n × n Toeplitz matrix gives rise to a truncated Toeplitz operator on .K .

6.3.2 An Equivalent Definition and Some Basic Properties The operator .Aϕ can alternatively be understood as follows. Note that the orthogonal projection .P can be seen as an integral operator on .L2 by writing that, for .g ∈ L2 , we have    λ ∈ D. .(P g)(λ) = P g, kλ 2 = g, kλ 2 = gkλ dm, T

Now, since for each compact subset K of .D, there exists a constant .CK such that .

sup |kλ (ζ )|  CK ,

λ∈K

and since for fixed .ζ ∈ T, the function .λ −→ kλ (ζ ) is analytic on .D, the preceding integral formula still makes sense, and defines an analytic function on .D, even when g belongs to the larger space .L1 . Moreover, for any .ϕ ∈ L2 , we have .ϕf ∈ L1 for

6 Truncated Toeplitz Operators

207

all .f ∈ K . Thus we can define the linear transformation Aϕ : K −→ O(D)

.

(here .O(D) denotes the analytic function on .D) by the integral formula  (Aϕ f )(λ) =

.

T

ϕf kλ dm,

f ∈ K , λ ∈ D.

(6.14)

Moreover, the Cauchy–Schwarz inequality yields |(Aϕ f )(λ)|  f 2 ϕ2 kλ ∞ ,

.

λ ∈ D.

(6.15)

Proposition 6.3.5 Let . be inner, let .ϕ ∈ L2 and let .Aϕ defined by (6.14). If .Aϕ f ∈ K for every .f ∈ K , then .Aϕ is bounded on .K . Furthermore, in that case, the definitions in (6.13) and (6.14) for .Aϕ coincide. Proof To show that the operator defined by (6.14) is bounded, we will use the closed graph theorem. Suppose that .(fn )n1 is a sequence in .K satisfying .fn → f in .K and .Aϕ fn → g in .K . First, it follows from (6.15) that, for every .λ ∈ D, we have (Aϕ fn )(λ) → (Aϕ f )(λ).

.

Second, since .Aϕ fn → g in .K , we also have that .(Aϕ fn )(λ) → g(λ) for every λ ∈ D. Thus .Aϕ f = g and by the closed graph theorem, we conclude that .Aϕ is bounded on .K . For the proof of the second part of the proposition, observe that the bounded operators defined by the formulas in (6.13) and (6.14) agree on the dense   set .K ∩ H ∞ and therefore must be equal.

.

A basic computation shows that the adjoint of a TTO is itself a TTO. Proposition 6.3.6 If .ϕ ∈ L2 such that .Aϕ ∈ T(), then .A∗ϕ ∈ T() and A∗ϕ = Aϕ .

.

Proof Let .f, g ∈ K ∩ H ∞ . We have f, A∗ϕ g2 = P (ϕf ), g2 = ϕf, g2 .

= f, ϕg2 = f, P (ϕg)2 = f, Aϕ g2 .

That proves that .A∗ϕ = Aϕ , and in particular .A∗ϕ ∈ T().

 

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E. Fricain

Example 6.3.7 When .ϕ is in .H ∞ , it turns out that .H 2 is obviously invariant with respect to .Tϕ = Mϕ , and so .K is invariant with respect to .Tϕ∗ = Tϕ . It then follows that A∗ϕ = Tϕ |K .

.

In particular, ∗ S = S ∗ |K .

.

Remark 6.3.8 The space .T() forms a linear space since we easily have αAϕ + βAψ = Aαϕ+βψ ,

.

α, β ∈ C, ϕ, ψ ∈ L2 .

However, the product of two truncated Toeplitz operators is not always a truncated Toeplitz operator. See Example 6.10.4. For the class of symbols in .H ∞ , the TTO enable to define a functional calculus for .S . Proposition 6.3.9 (The .H ∞ Functional Calculus of Sz.-Nagy–Foias) Let .T : H ∞ −→ L(K ) defined by .T (ϕ) = Aϕ , .ϕ ∈ H ∞ . Then T is a contractive morphism of Banach algebra which extends the polynomial calculus for .S . In other words, for every .ϕ, ψ ∈ H ∞ , for every .α, β ∈ C and for every .p ∈ C[X], we have (i) (ii) (iii) (iv)

T (αϕ + βψ) = αT (ϕ) + βT (ψ); T (ϕψ) = T (ϕ)T (ψ); .T (p) = p(S ); .T (ϕ)  ϕ∞ . . .

Proof (i) follows from Remark 6.3.8, (ii) follows from Proposition 6.3.3 and (iv) follows from Example 6.3.1. So the only point to check is (iii). Let p be a  k polynomial, .p(z) = N a k=0 k z . Using (1), we have T (p) =

N 

.

ak T (zk ),

k=0

and by (2), .T (zk ) = T (z)k . But .T (z) = Az = S . Hence T (p) =

N 

.

k=0

k ak S = p(S ).

 

In Sect. 6.9, we will see that this functional calculus satisfies a spectral mapping theorem for the subclass of symbols in .H ∞ ∩ C(T).

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209

As we have just seen, the class of bounded analytic symbols already generates a rich class of operators. Thus, the reader might wonder why we bother defining .Aϕ for .ϕ ∈ L2 (and possibly unbounded) when one can define .Aϕ everywhere on .K if .ϕ ∈ L∞ . We did not go through all this trouble when defining Toeplitz operators on .H 2 . Indeed for Toeplitz operators, the symbol is unique in that .Tϕ1 = Tϕ2 if and only if .ϕ1 = ϕ2 almost everywhere. Furthermore, one can show that for a symbol 2 ∞ , has a bounded .ϕ in .L , the densely defined operator .Tϕ f = P+ (ϕf ), .f ∈ H 2 ∞ extension to .H if and only if .ϕ ∈ L . For truncated Toeplitz operators, the symbol is never unique (see Theorem 6.5.1). Moreover, as will see in Sect. 6.12, there are 2 .Aϕ with .ϕ ∈ L that extend to bounded operators on .K but for which there is no bounded symbol that represents .Aϕ . Moreover, it is well known that the set of bounded Toeplitz operators on .H 2 forms a weakly closed linear space in .L(H 2 ). To have a similar result for truncated Toeplitz operators (see Corollary 6.6.4), we need to include such unusual operators with symbols in .L2 .

6.4 Why Studying Truncated Toeplitz Operators? Interest in truncated Toeplitz operators has been very strong over the past 15 years. Besides the fascinating structure of these operators, they can be seen as a natural generalization of Toeplitz matrices (see Example 6.3.4). On the other hand, they naturally appear in several questions of operator theory and functions theory.

6.4.1 The Sz.-Nagy–Foias Model One of the main reasons that model spaces and truncated Toeplitz operators are worthy of study in their own rights stems from the so-called model theory developed by Sz.-Nagy–Foias, which shows that a wide range of Hilbert space contractions can be realized concretely as .S = A z on some model space .K . Theorem 6.4.1 (Sz.-Nagy–Foias) Let .T : H −→ H be a Hilbert space contraction. Assume that .

lim T n xH = 0,

n→∞

(x ∈ H)

and that the operators .I − T ∗ T and .I − T T ∗ are of rank one. Then there exists an ∗ = S ∗ |K . inner function . such that T is unitarily equivalent to .S  Proof Let .DT be the unique positive square root of the positive operator .I − T ∗ T , i.e. .DT = (I − T ∗ T )1/2 . The key point of the proof is the following simple

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E. Fricain

observation. Let x be any vector in .H. Then we have DT T n x2H = DT2 T n x, T n xH

.

= (I − T ∗ T )T n x, T n xH = T n x2H − T n+1 x2H , which, using the fact that .T n x −→ 0, implies that the sequence .(DT T n x)n0 is in .2 and its .2 -norm is equal to .x. Note that .DT T n x belongs to the range of .DT , which is of dimension one and thus can be identified by .C. Now, define a linear map U from .H into .H 2 by (U x)(z) =

.

∞  (DT T n x)zn ,

(z ∈ D).

n=0

Since the sequence .(DT T n x)n0 is in .2 and of norm .x, the map U is welldefined and isometric from .H onto its range E, which is a closed subspace of .H 2 . Then we have ∞  .(U T x)(z) = (DT T n+1 x)zn n=0



=S

∗

∞  (DT T n+1 x)zn+1 DT T x +



n=0

∞   = S∗ (DT T n x)zn n=0 ∗

= (S U x)(z),

(z ∈ D),

which reveals that .U T = S ∗ U . This relations implies two things. Firstly, E is a closed .S ∗ -invariant subspace of .H 2 and, secondly, T is unitarily equivalent to .S ∗ |E. Note that E cannot be equal to .H 2 . Indeed, if .E = H 2 , then we would obtain that the rank of .I − T T ∗ would be the rank of .I − S ∗ S which is zero (because S is a contraction on .H 2 ), which contradicts the assumption that .I − T T ∗ is of rank one. Therefore, by Beurling’s theorem, there exists an inner function . such that ∗ ∗ .E = K and this proves that T is unitarily equivalent to .S |K = S .    We can restate the Sz.-Nagy–Foias theorem as follows. Corollary 6.4.2 (Sz.-Nagy–Foias) Let .T : H −→ H be a Hilbert space contraction. Assume that .

lim T ∗ n xH = 0,

n→∞

(x ∈ H)

6 Truncated Toeplitz Operators

211

and that the operators .I − T ∗ T and .I − T T ∗ are of rank one. Then there exists an inner function . such that T is unitarily equivalent to .S .

6.4.2 The Commutant of S We discuss now the commutant of .S , that is the set {S } = {A ∈ L(K ) : AS = S A}.

.

The description of this commutant was obtained by Sarason [51]. His work was motivated by the study of interpolation problems for bounded analytic functions on the open unit disc. The description of .{S } is a deep result known as the commutant lifting theorem for .S . The term “commutant lifting” stems from the following phenomenon. To find solutions to the operator equation AS = S A

.

(6.16)

for an .A ∈ L(K ), we “lift it” to the operator equation BS = SB,

.

(6.17)

where S is the shift operator on the larger space .H 2 and .B ∈ L(H 2 ). Then, we apply a theorem of Brown–Halmos saying that B must be a Toeplitz operator .Tϕ with .ϕ ∈ H ∞ . We then return to (6.16) to prove that .A = A ϕ . The main difficulty is to lift equation (6.16) into equation (6.17). This can be proved using dilation theory, further developed by Sz.-Nagy–Foias in an abstract context. In this note, following [45, Section 3.1.9], we will present another approach, based on Nehari’s theorem, which uses a link between the commutant of .M and Hankel operators. This link is precisely stated in the following lemma. Lemma 6.4.3 Let . be an inner function, and let .A ∈ L(K ). Define .

A∗ : H 2 −→ H−2 f −→ AP f.

Then the following assertions are equivalent. (i) .AS = S A. (ii) .A∗ is a Hankel operator. Proof First, let us verify that .A∗ is a well-defined operator from .H 2 into .H−2 . Indeed, since .K = H 2 ∩  H−2 , we have .AP H 2 ⊂  H−2 , which implies that 2 2 .AP H ⊂ H− .

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E. Fricain

Moreover, AS = S A ⇐⇒ AP S = P SA,

(on K ),

.

⇐⇒ AP SP = P SAP ,

(on H 2 ).

Since .zH 2 ⊂ H 2 , we have .P S(I − P ) = 0, whence .P SP = P S and AP SP = AP S,

.

(on H 2 ).

Recall that .P = P−  (on .H 2 ), where .P− = I − P+ . Thus AS = S A ⇐⇒ AP S = P SAP

.

(on H 2 ),

⇐⇒ AP S = P− SAP ⇐⇒ AP S = P− ZAP ⇐⇒ A∗ S = P− ZA∗ . Here Z is the shift operator (i.e. the multiplication by the independent variable z) on .L2 (T). The latter condition exactly means that .A∗ is a Hankel operator (see   Sect. 6.2.1). As we will see in Sect. 6.5, the symbol of a truncated Toeplitz operator is never unique. In the proof of the commutant lifting theorem, we will need a special case of this. ∞ Lemma 6.4.4 Let .ϕ ∈ H ∞ . Then .A ϕ = 0 if and only if .ϕ ∈ H .

Proof Assume that .ϕ = g for some .g ∈ H ∞ . Then, for each .f ∈ K , we have A ϕ f = P (ϕf ) = P (f g) = 0,

.

because .fg ∈ H 2 = (K )⊥ . Conversely, assume that .A ϕ = 0. Since for each 2 2 .f ∈ H , .ϕPH 2 f ∈ H , we get P (ϕf ) = P (ϕP f ) + P (ϕPH 2 f ) = A ϕ P f = 0,

.

which implies that .ϕH 2 ⊂ H 2 . In particular, .ϕ = g, for some .g ∈ H 2 . But, taking the absolute values of both sides shows that .|ϕ| = |g| a.e. on .T. Hence ∞ , and finally .ϕ ∈ H ∞ . .g ∈ H   We are now ready for the commutant lifting theorem for the operator .S . Theorem 6.4.5 (Sarason, [51]) Let . be an inner function, and let .A ∈ {S } , i.e. ∞ such that .A = A . .A ∈ L(K ) and .AS = S A. Then there exists .ϕ ∈ H ϕ Moreover, the following assertions hold.

6 Truncated Toeplitz Operators

213

∞ (i) For any representation .A = A ϕ , with .ϕ ∈ H , we have

A = dist(ϕ, H ∞ ).

.

(ii) ∞ A = inf{ϕ∞ : A = A ϕ with ϕ ∈ H }.

.

(iii) There exists a particular choice .ϕ ∈ H ∞ such that A = A ϕ

.

and

A = ϕ∞ .

Proof According to Lemma 6.4.3, the operator .A∗ = AP is a Hankel operator. Since A is bounded, we have A∗ f 2 = AP f 2 = AP f 2  A f 2 ,

.

(f ∈ H 2 ),

which implies that .A∗ is bounded and .A∗   A. Furthermore, Af 2 = AP f 2 = AP f 2 = A∗ f 2  A∗  f 2 ,

.

(f ∈ K ),

whence .A  A∗ . Therefore, AL(K ) = A∗ L(H 2 ,H 2 ) . −

.

Nehari’s theorem implies that there exists a function .η ∈ L∞ (T) such that .A∗ = Hη and .A∗  = η∞ = dist(η, H ∞ ). Hence, A = A∗  = η∞ = dist(η, H ∞ ).

.

Since .η ∈ L2 (T) = H 2 ⊕ H−2 , we can write η = ϕ + ψ,

.

where .ϕ ∈ H 2 and .ψ ∈ H−2 . But, P− η = Hη  = A∗  = AP  = 0,

.

which means .ψ = P− η = 0. Hence, η = ϕ ∈ H 2 ∩ L∞ = H ∞ .

.

(6.18)

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E. Fricain

Rewrite the last identity as .η = ϕ, with .ϕ ∈ H ∞ . This is a rewarding representation. In fact, for .f ∈ K , we have Af = (AP f )

.

= A∗ f = Hϕ f = P− ϕf = P ϕf = A ϕ f, whence .A = A ϕ . Moreover, according to (6.18) and using the fact that .|| = 1 a.e. on .T, we have A = ϕ∞ = dist(ϕ, H ∞ ).

.

Since .A ϕ = A, we have

(6.19)  

∞ A = ϕ∞  inf{h∞ : A = A h , with h ∈ H }.

.

 If .h ∈ H ∞ is such that .A h = A = Aϕ , then according to Lemma 6.4.4, .h − ϕ ∈ ∞ H . In other words, there exists .g ∈ H ∞ such that .h = ϕ + g. Thus, .

dist(h, H ∞ ) = dist(ϕ, H ∞ ) = A,

and h∞ = ϕ + g  dist(ϕ, H ∞ ) = A.

.

Hence, .

∞ inf{h∞ : A = A h , with h ∈ H }  A,

which proves that ∞ ∞ A = inf{h∞ : A = A h , with h ∈ H } = dist(ϕ, H ).

.

Finally, (6.19) shows that the infimum is attained, which ends the proof.

6 Truncated Toeplitz Operators

215

6.4.3 The Nevanlinna–Pick Interpolation Problem We will now explain how we can use the commutant lifting theorem to solve the Nevanlinna-Pick interpolation problem. Given n points .λ1 , λ2 , . . . , λn in the open unit disc .D, and n points .ω1 , ω2 , . . . , ωn in the complex plane, we would like to know if there exists a function f in the closed unit ball of .H ∞ interpolating the points .λi to the points .ωi , that is f (λi ) = ωi ,

(1  i  n).

.

Before giving the answer to this question, we need an additional property of truncated Toeplitz operators with symbols in .H ∞ . Lemma 6.4.6 Let .ϕ ∈ H ∞ and .λ ∈ D such that .(λ) = 0. Then Aϕ kλ = ϕ(λ)kλ .

.

Proof Observe that .kλ = kλ since .(λ) = 0. In particular, .kλ ∈ K . Using Proposition 6.3.6, we have, for every .f ∈ K , Aϕ kλ , f 2 =kλ , Aϕ f 2 =kλ , ϕf 2 .

=ϕ(λ)f (λ) =ϕ(λ)kλ , f 2 .

Since this is true for every .f ∈ K , we conclude that .Aϕ kλ = ϕ(λ)kλ .

 

The following result answers the question of Nevanlinna–Pick interpolation. Theorem 6.4.7 (Nevanlinna–Pick) Let .(λi )1in be n distinct points in .D, and let .(ωi )1in be complex numbers. Then the following are equivalent. (i) There exists a function f in .H ∞ such that f (λi ) = ωi ,

.

(1  i  n),

and, moreover, .f ∞  1. (ii) The matrix .Q = (Qj,k )1j,kn , where Qj,k =

.

is nonnegative.

1 − ωj ωk 1 − λj λk

,

(j, k = 1, . . . , n),

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E. Fricain

Proof We do not follow the original proof but the one based on the commutant lifting theorem of Sarason. We start by introducing the operator T which is exploited in the proof of equivalence. Let B be the (finite) Blaschke product associated to the sequence .(λi )1in . It is not difficult to check that the sequence of reproducing kernels .(kλi )1in forms a basis of .KB . Note that .KB is finite dimensional. Hence we can consider the linear bounded operator T from .KB into itself defined by T kλi = ωi kλi ,

(1  i  n).

.

(6.20)

Using Lemma 6.4.6 for .SB∗ = AB z¯ , we have T SB∗ kλi = λi ωi kλi = SB∗ T kλi ,

(1  i  n).

.

Since .(kλi )1in is a basis of .KB , we deduce that .T SB∗ = SB∗ T . Hence, Theorem 6.4.5 ensures that there exists .f ∈ H ∞ such that .T = AB and f¯ T  = min{g∞ : T = AgB¯ , g ∈ H ∞ }.

.

(6.21)

For any function g which fulfills .T = AgB¯ , we have T kλi = AB g¯ kλi = g(λi )kλi ,

.

(1  i  n).

Therefore, g(λi ) = ωi

.

(1  i  n).

(6.22)

To establish the connection between Q and T , let .ai ∈ C, .1  i  n. Then, using the operator T , we can write 

ai aj

.

1i,j n

1 − ωi ωj 1 − λi λj

=



  ai aj kλi , kλj 2 − T kλi , T kλj 2

1i,j n

=



ai aj (I − T ∗ T )kλi , kλj 2

1i,j n

= (I − T ∗ T )h, h2 , n where .h = i=1 ai kλi . Hence, the matrix Q is nonnegative if and only if the operator .I − T ∗ T is positive, which is equivalent to say that T is a contraction. ∞ , .f  .(i) ⇒ (ii): Assume that there exists .f ∈ H ∞  1, such that .f (λi ) = ωi , .1  i  n. Hence, using (6.20) and Lemma 6.4.6, we have k , T kλi = ωi kλi = f (λi )kλi = AB f¯ λi

.

(1  i  n).

6 Truncated Toeplitz Operators

217

Thus we get .T = AB . Now, (6.21) implies that f¯ T   f ∞  1,

.

i.e. T is a contraction, and thus Q is nonnegative. .(ii) ⇒ (i): Assume that Q is nonnegative. Thus, T is a contraction. By (6.21), we know that there exists a function .f ∈ H ∞ , .f ∞  1 such that .T = AB . Thus, f¯ by (6.22), .f (λi ) = ωi , .1  i  n, which gives .(i).  

6.4.4 A Link with Truncated Wiener-Hopf Operators It turns out that truncated Toeplitz operators on the model space .K are closed connected with another class of operators, the truncated Wiener–Hopf operators, which arise naturally in projection methods to solve certain convolution equations. See [40]. Let us explain the link between these two important classes of operators. Let .ϕ ∈ L1 (R), .a > 0, and let  (Wϕ f )(x) =

2a

.

ϕ(x − t)f (t), dt,

x ∈ (0, 2a),

0

for .f ∈ L∞ (0, 2a). It is easy to see that .Wϕ f ∈ L2 (0, 2a), and then .Wϕ is a densely defined operator on .L2 (0, 2a). If .Wϕ extends to a bounded operator on .L2 (0, 2a), then it is called a truncated Wiener–Hopf operator. To explain the link between a truncated Wiener–Hopf operator and a truncated Toeplitz operator, it will be easier to go into the upper half-plane. We denote by 2 .H (C+ ) the Hardy space of the upper-half plane .C+ = {z ∈ C : Im(z) > 0} and if . is an inner function in .C+ , the associated model space .K is defined as 2 2 .K = H (C+ ) H (C+ ). Finally, we recall that if .F is the Fourier transform, the Paley–Wiener Theorem says that .F−1 L2 (R+ ) = H 2 (C+ ). Proposition 6.4.8 Let .a (z) = eiaz , .a > 0, and let .ϕ in the Schwartz class .S(R). Then, −1 2a FA = W ϕ. ϕ F

.

Proof Denote by .Ua the (unitary) operator of translation by a on .L2 (R), i.e. (Ua f )(x) = f (x + a), .f ∈ L2 (R). Write

.

Ua L2 (R+ ) =L2 (−a, +∞) = L2 (−a, a) ⊕ L2 (a, +∞) .

=L2 (−a, a) ⊕ U−a L2 (R+ ).

218

E. Fricain

Apply the inverse of the Fourier transform and use the Paley–Wiener Theorem and the well-known facts that .F −1 Ua = a F −1 and .F;−1 U−a = a F −1 to get a H 2 (C+ ) = F −1 L2 (−a, a) ⊕ a H 2 (C+ ).

.

Multiply this equation by .a gives K2a = a F −1 L2 (−a, a) = a F −1 χ(−a,a) L2 (R).

.

It is easy to check that the operator .T = a F −1 χ(−a,a) is a partial isometry on 2 ∗ is the orthogonal projection onto .Im(T ) = K .L (R). Hence .T T 2a (see [27, Theorem 7.22] for this result on partial isometries). We deduce a nice formula for the orthogonal projection onto .K2a : P2a = a F −1 χ(−a,a) Fa .

.

Let .f ∈ L2 (0, 2a). Then, we have −1 2a FA f =FP2a ϕF −1 f ϕ F

=Fa F −1 χ(−a,a) F(a ϕF −1 f ) .

=U−a χ(−a,a) F(ϕF −1 Ua f ) =U−a χ(−a,a) ϕ ∗ Ua f.

Let .x ∈ R. We get

.

−1 2a (FA f )(x) =χ(−a,a) (x − a)( ϕ ∗ Ua f )(x − a) ϕ F  ∞ ϕ (x − a − y)f (y + a) dy =χ(0,2a) (x)



−∞ 2a

=χ(0,2a) (x)

ϕ (x − t)f (t) dt.

0

In other words, .FAϕ 2a F −1 = W ϕ. 

 

Proposition 6.4.8 says that the class of truncated Wiener–Hopf operators (at least for a class of symbols in the Schwartz class) is unitary equivalent to a particular class of truncated Toeplitz operators (associated to the model space with inner function .2a ). Therefore, truncated Toeplitz operators can be viewed as a generalization of truncated Wiener–Hopf operators.

6 Truncated Toeplitz Operators

219

6.5 The Class of Symbols for a Truncated Toeplitz Operator We know that the symbol .ϕ of a Toeplitz operator .Tϕ is unique. As we already noticed (see Lemma 6.4.4), the story for truncated Toeplitz operators is much different. Theorem 6.5.1 (Sarason, [52]) Let . be an inner function. A truncated Toeplitz operator .Aϕ is identically zero if and only if .ϕ ∈ H 2 + H 2 . Consequently, Aϕ1 = Aϕ2 ⇐⇒ ϕ1 − ϕ2 ∈ H 2 + H 2 .

.

Proof Suppose that .ϕ = h1 +h2 , where .h1 , h2 ∈ H 2 . Then, for .f ∈ K ∩H ∞ , we have ϕf = f h1 + h2 f.

.

The first term belongs to .H 2 and so .P (f h1 ) = 0. Moreover, since .K = H 2 ∩ zH 2 ⊂ zH 2 , the second term belongs to .zH 2 , and thus, we also have .P (f h2 ) = 0. Hence Aϕ (f ) = P (ϕf ) = P (f h1 ) + P (f h2 ) = 0.

.

Conversely, suppose that .Aϕ (f ) = 0 for every .f ∈ K ∩ H ∞ . Write the symbol 2 2 .ϕ ∈ L as .ϕ = ψ + χ , where .ψ, χ ∈ H . In particular, we have .Aψ = −Aχ . 2 Now, using the fact that .ψ and .χ are in .H , it is not difficult to check the following identities on .K ∩ H ∞ Aψ S = S Aψ

.

and

∗ ∗ Aχ S = S Aψ .

∗ . We now In particular, we deduce that .Aψ and .Aχ commute with both .S and .S need the following well-known identity ∗ I − S S = k0 ⊗ k0 .

.

To check (6.23), note that for .f ∈ K , we have ∗ (I − S S )f =f − P (SP (S ∗ f ))

=f − P (SS ∗ f ) .

=f − P (f − f (0)) =f (0)P (1) = f (0)k0 =(k0 ⊗ k0 )f.

(6.23)

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E. Fricain

Now use (6.23) to deduce Aψ (k0 ⊗ k0 ) = (k0 ⊗ k0 )Aψ .

.

If we evaluate this operator identity at .k0 , we get k0 22 Aψ k0 = Aψ k0 , k0 2 k0 .

.

In particular, we have .Aψ k0 = ck0 for some constant .c ∈ C. From here we can write 0 =(Aψ − cI )k0 =P (ψ − c)(1 − (0)) .

=P (ψ − c) − (0)P ((ψ − c)) =P (ψ − c), the last equality following from the fact that .(ψ − c) ∈ H 2 . Thus .ψ − c ∈ H 2 . In particular, if we apply the first part of the proof, we deduce .Aψ = cI . But since 2 .Aχ = −Aψ = −cI , repeating the same arguments yields to .χ + c ∈ H , and so .χ + c ∈ H 2 . Therefore ϕ = ψ + χ = (ψ − c) + (χ + c) ∈ H 2 + H 2 ,

.

 

which completes the proof.

In particular, the preceding result tells us that there are always infinitely many symbols (many of them unbounded) which represent the same truncated Toeplitz operator. On the other hand, if we denote by .S = L2 (H 2 + H 2 ), it follows from Theorem 6.5.1 that every truncated Toeplitz operator has a unique symbol .ϕ ∈ S . This space is sometimes called the space of standard symbols. In the following result, we gather from [7, Lemma 3.1] some interesting properties of this space. Some of them will be used in Sect. 6.12 for the construction of a bounded truncated Toeplitz operator that has no bounded symbols. Lemma 6.5.2 ([7], Lemma 3.1) Let . be an inner function and let .Q (respectively .PS ) the orthogonal projection onto .K ⊕ zK (respectively onto .S ). Then: 2

(i) .Q () =  − (0) ; (ii) we have K ⊕ zK = S ⊕ Cq ,

.

where .q = Q ()−1 2 Q (); (iii) .Q and .PS are bounded on .Lp for .1 < p < ∞. In particular, .q ∈ Lp .

6 Truncated Toeplitz Operators

221

Proof (a) Note that zK = zH 2 ∩ zH 2 = zH 2 ∩ H 2 = (H 2 ∩ zH 2 ) = K .

.

Therefore, we have K ⊕ zK = K ⊕ K ,

.

and then, .Q = P + M P M . In particular, we have Q () =P () + M P M () = P ((0)) + M P 1 .

2

=((0) + )(1 − (0)) =  − (0) , and (a) is proved. (b) Since .L2 = H 2 ⊕ H02 ⊕ K ⊕ zK , if follows that .S ⊂ K ⊕ zK , and thus K ⊕ zK = Q (S + H 2 + H02 + C) = S ⊕ CQ (),

.

(6.24)

which proves (b). Note that according to (a), one easily see that .Q () ≡ 0. (c) If follows from the identity .Q = P + M P M , that .Q is bounded on .Lp for all .1 < p < ∞. Further, according to (b), we have PS = Q − q ⊗ q ,

.

(6.25)

and since .q belongs to .L∞ , we deduce that .PS is also bounded on .Lp for .1 < p < ∞.   Another interesting result observed by Sarason [52] is the following. Corollary 6.5.3 Let .ϕ ∈ L2 . Then there exists a pair of functions .ϕ1 , ϕ2 ∈ K such that .Aϕ = Aϕ1 +ϕ2 . Moreover, if .ϕ1 , ϕ2 is one such pair, the most general such pair equals .ϕ1 + ck0 , .ϕ2 − ck ¯ 0 , where c is a complex number. In particular, .ϕ1 and .ϕ2 are uniquely determined if we fix the value of one of them at the origin. Proof Decompose the symbol .ϕ as .ϕ = ψ1 + ψ2 , with .ψ1 , ψ2 ∈ H 2 . Define 2 .ϕ = P ψ , . = 1, 2. Observe that .ψ1 − ϕ1 and .ψ2 − ϕ2 both belong to .H . Hence ϕ − (ϕ1 + ϕ2 ) = (ψ1 − ϕ1 ) + (ψ2 − ϕ2 ) ∈ H 2 + H 2 .

.

Theorem 6.5.1 now ensures that .Aϕ = Aϕ1 +ϕ2 .

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E. Fricain

Moreover, suppose that .ϕ1 , ϕ2 are in .K and .Aϕ = Aϕ1 +ϕ2 . Then, according to Theorem 6.5.1, we get that .(ϕ1 −ϕ1 )+(ϕ2 −ϕ2 ) is in .H 2 +H 2 . In particular, .ϕ1 − ϕ1 ∈ H 2 + H 2 . Observe now that the orthogonal projection .P maps .H 2 + H 2 onto .Ck0 , which gives that .ϕ1 = ϕ1 + ck0 for some complex number c. Similarly, there exists a constant .d ∈ C such that .ϕ2 = ϕ2 + dk0 . We deduce that .ck0 + dk0 is in .H 2 + H 2 . But observe that ¯ ck0 + dk0 = c + d¯ − c(0) − d(0),

.

which yields .c + d¯ ∈ H 2 + H 2 . Take now any .f ∈ K , .f = 0. According to Theorem 6.5.1, we get ¯ ) = (c + d)f, ¯ 0 = Ac+d¯ (f ) = P ((c + d)f

.

which implies that .c + d¯ = 0, or .d = −c. ¯ Therefore, .ϕ1 = ϕ1 + ck0 and .ϕ2 = ϕ2 − ck ¯ 0 .    Example 6.5.4 Let .(z) = z3 and note that .K = {1, z, z2 }. As already mentioned (see Example 6.3.4), in that case, the truncated Toeplitz operator on .K can be viewed as operators whose matrix with respect to the orthonormal basis 2 .{1, z, z } is a Toeplitz matrix. Suppose that A is a generic .3 × 3 Toeplitz matrix ⎛ ⎞ ad e . ⎝b a d ⎠ . cba Then consider .ϕ(z) = e¯z2 + d z¯ + a + bz + cz2 . An easy computation shows that .A = Aϕ . The function .ϕ is the standard symbol for A and we have .ϕ = ϕ + ψ, ¯ + ez ¯ 2. where .ϕ, ψ ∈ K are defined as .ϕ(z) = a2 + bz + cz2 and .ψ(z) = a2¯ + dz

6.6 Algebraic Characterization of Truncated Toeplitz Operators A well known result of Brown–Halmos says that .T ∈ L(H 2 ) is a Toeplitz operator if and only if .T = ST S ∗ , where S is the shift operator on .H 2 . Sarason obtained an analogue of this result for truncated Toeplitz operators, where the compressed shift (or the model operator) .S plays the role of S. Theorem 6.6.1 (Sarason, [52]) A bounded operator A on .K belongs to .T() if and only if there are functions .ϕ, ψ ∈ K such that ∗ A = S AS + ϕ ⊗ k0 + k0 ⊗ ψ.

.

6 Truncated Toeplitz Operators

223

In that case, we have .A = Aϕ+ψ . The proof of this result is based on two lemmas. Lemma 6.6.2 For .ϕ, ψ ∈ K , we have ∗ Aϕ+ψ − S Aϕ+ψ S = ϕ ⊗ k0 + k0 ⊗ ψ.

.

∗ commutes with .A (see ProposiProof Since .S commutes with .Aϕ and .S ψ tion 6.3.3), we have ∗ ∗ ∗ = Aϕ (I − S S ) + (I − S S )Aψ . Aϕ+ψ − S Aϕ+ψ S

.

∗ = k  ⊗ k  , which yields By (6.23), we have .I − S S 0 0 ∗ Aϕ+ψ − S Aϕ+ψ S = Aϕ k0 ⊗ k0 + k0 ⊗ A∗ψ k0

.

Observe now that, since .ϕ ∈ K , we have .Aϕ k0 = P (ϕ − (0)ϕ) = ϕ. ∗ = A , and so .A∗ k  = A k  = ψ, According to Proposition 6.3.6, we have .Aψ ψ 0 ψ ψ 0 which gives the result.   Lemma 6.6.3 Let .ϕ, ψ ∈ K . For .f, g ∈ K ∩ H ∞ , we have Aϕ+ψ f, g2 =

.

∞    n  n n n  f, S k0 2 S ϕ, g, 2 + f, S ψ2 S k0 , g2 . n=0

Proof By Lemma 6.6.2, ∗ = ϕ ⊗ k0 + k0 ⊗ ψ. Aϕ+ψ − S Aϕ+ψ S

.

Thus, for any integer .n  0, n+1 ∗ n+1 n n  n  n ∗n n S Aϕ+ψ S = S ϕ ⊗ S k0 + S k0 ⊗ S ψ. − S Aϕ+ψ S

.

Now sum both sides of the equation above from .n = 0 to .n = N to get Aϕ+ψ =

.

N    n N +1 n  n  n ∗ N +1 k0 + S k0 ⊗ S ψ + S Aϕ+ψ S . S ϕ ⊗ S n=0

Thus, for .f, g ∈ K ∩ H ∞ , Aϕ+ψ f, g2 = .

N    n  n n n  f, S k0 2 S ϕ, g, 2 + f, S ψ2 S k0 , g2 n=0

∗ N +1 ∗ N +1 +Aϕ+ψ S f, S g2 .

224

E. Fricain

It remains to show that the last summand on the right tends to 0 as .N → ∞. Using ∗ , observe that the fact that .Aψ and .Aϕ commute with .S ∗ N +1 ∗ N +1 ∗ N +1 ∗ N +1 ∗ N +1 ∗ N +1 Aϕ+ψ S f, S g2 = S f, S Aϕ g2 + S Aψ f, S g2 .

.

∗ N → 0, as .N → ∞, in the strong The desired conclusion now follows because .S n 2 operator topology (remind that for any .h(z) = ∞ n=0 an z in .H , we have

S

.

∗N

h22

2  ∞ ∞     n−1  = an z  = |an |2 → 0,   n=N

2

as N → ∞.)

n=N

Proof of Theorem 6.6.1 It follows from Lemma 6.6.2 and Corollary 6.5.3 that every operator in .T() satisfies the condition of the theorem. Suppose, conversely, that A is a bounded operator on .K that satisfies the condition ∗ A − S AS = ϕ ⊗ k0 + k0 ⊗ ψ,

.

with .ϕ, ψ ∈ K . Arguing as in the proof of Lemma 6.6.3, we deduce that, for every integer .N  0, A=

.

N   n  N +1 n  n  n ∗ N +1 S ϕ ⊗ S k0 + S k0 ⊗ S ψ + S AS . n=0

∗ N → 0, .N → ∞, in the strong operator topology to see Again use the fact that .S that

A=

.

∞   n  n  n  n S ϕ ⊗ S k0 + S k0 ⊗ S ψ , n=0

where the series converges in the strong operator topology. Finally, by Lemma 6.6.3, we can conclude that .A = Aϕ+ψ , and in particular, .A ∈ T().   Theorem 6.6.1 admits an interesting corollary. Corollary 6.6.4 (Sarason, [52]) The set .T() is closed in the weak operator topology. Proof Suppose the net .(A(α) ) in .T() converges weakly to the bounded operator A. By Theorem 6.6.1, for each index .α, there are functions .ϕα , ψα in .K such that ∗ A(α) − S A(α) S = ϕα ⊗ k0 + k0 ⊗ ψα .

.

(6.26)

6 Truncated Toeplitz Operators

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Moreover, by Corollary 6.5.3, the function .ψα can be taken to satisfy .ψα (0) = 0. Then we have ∗  A(α) k0 − S A(α) S k0 = k0 22 ϕα − k0 , ψα 2 k0 = k0 22 ϕα .

.

It then follows that the net .(ϕα ) converges weakly, say to a function .ϕ in .K . The net .(ϕα ⊗ k0 ) thus converges in the weak operator topology, and so by (6.26), the net .(k0 ⊗ ψα ) also converges in the weak operator topology, implying that the net .(ψα ) converges weakly, say to a function .ψ in .K . Passing to the limit in (6.26), we obtain ∗ A − S AS = ϕ ⊗ k0 + k0 ⊗ ψ,

.

and it follows by Theorem 6.6.1 that .A = Aϕ+ψ is in .T().

 

Remark 6.6.5 When .K is finite dimensional (which means that . is a finite Blaschke product), one can get more specific results using matrix representations. It has already been noticed (see Examples 6.3.4 and 6.5.4) that if .(z) = zN , then the corresponding truncated Toeplitz operators are the Toeplitz matrices (with respect to the standard orthonormal basis .{1, z, . . . , zN −1 } of .K ). For more general finite Blaschke products, we have the following result from [22]. Theorem 6.6.6 (Cima–Ross–Wogen) Let . be a finite Blaschke product of degree n with distincts zeros .λ1 , λ2 , . . . , λn and let A be any operator on .K . If .MA = (ri,j )1i,j n is the matrix representation of A with respect to the basis .{kλ1 , kλ2 , . . . , kλn }, then .A ∈ T() if and only if ri,j

.

 (λ1 ) =   (λi )



r1,i (λ1 − λi ) + ri,j (λj − λ1 ) λj − λi

 ,

for .1  i, j  n and .i = j .

6.7 Complex Symmetric Operators 6.7.1 Truncated Toeplitz Operators Are Complex Symmetric A conjugation on a complex Hilbert space .H is a map .C : H → H that is conjugatelinear, involutive (.C 2 = I ) and isometric (.Cx, Cy = y, x for every .x, y ∈ H). We now say that .T ∈ L(H) is C-symmetric if .T = CT ∗ C and complex symmetric if there exists a conjugation C on .H with respect to which T is C-symmetric. This class of operators received a lot of attention during the past few years, in particular due to the works of Garcia–Putinar [34, 35]. Among important examples of complex symmetric operators, we have the Volterra operator, normal operators, Toeplitz matrices,....

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However, note that Toeplitz operators cannot be complex symmetric in general. Indeed, if an operator T is complex symmetric, then, in particular, it should satisfy .

dim ker T = dim ker T ∗ .

But, if for instance .T = Tϕ , with .ϕ ∈ H ∞ , .ϕ ≡ 0, then .ker Tϕ = {0} and .ker Tϕ∗ = {0} (because .Tϕ∗ kλ = Tϕ kλ = ϕ(λ)kλ ). Thus, for every .ϕ ∈ H ∞ , .ϕ ≡ 0, .Tϕ is not complex symmetric. However, the situation is dramatically different for TTO. It turns out that truncated Toeplitz operators on .K are .C -symmetric, where .C is the natural conjugation on .K introduced in (6.7). Theorem 6.7.1 (Sarason, [52]) Let . be an inner function and .Aϕ ∈ T(). Then we have .Aϕ = C A∗ϕ C . Proof According to Proposition 6.3.6, for every .f, g ∈ K ∩ H ∞ , we have C A∗ϕ C f, g2 =C g, A∗ϕ C f 2 = C g, P (ϕC f )2 =C g, ϕC f 2 = zg, ϕzf 2 .

=g, ϕf 2 = ϕf, g2 = P (ϕf ), g2 =Aϕ (f ), g2 .

Hence .Aϕ = C A∗ϕ C .

 

Using the conjugation .C , we can give an analogue of the characterization of T() given in Theorem 6.6.1.

.

Corollary 6.7.2 Let . be an inner function and let .A ∈ L(K ). Then .A ∈ T() if and only if there are functions .ϕ, ψ ∈ K such that ∗ A − S AS = ϕ ⊗ S ∗  + S ∗  ⊗ ψ.

.

(6.27)

Moreover, in that case, we have .A = AC ψ+C ϕ . Proof Assume that A satisfies (6.27). If we apply on the left and on the right .C to both sides of (6.27), we get ∗ C AC − C S AS C = C ϕ ⊗ C S ∗  + C S ∗  ⊗ C ψ.

.

∗ , we have .C S ∗ = S C , which gives According to Theorem 6.7.1 applied to .S     ∗ C AC − S C AC S = C ϕ ⊗ C S ∗  + C S ∗  ⊗ C ψ.

.

But on .T, we have C S ∗  = ¯z(¯z( − (0))) = ( − (0)) = 1 − (0) = k0 .

.

6 Truncated Toeplitz Operators

227

Thus, if .B = C AC , we obtain ∗ B − S BS = C ϕ ⊗ k0 + k0 ⊗ C ψ.

.

(6.28)

Then it follows from Theorem 6.6.1 that .B ∈ T(). In particular, there exists .χ ∈ L2 such that .B = Aχ . Now Theorem 6.7.1 implies that A = C BC = C Aχ C = A∗χ = Aχ .

.

In other words, .A ∈ T(). Conversely, assume that .A ∈ T(). Then .B = C AC ∈ T() and the arguments above can be reversed showing that A must satisfy (6.27). Moreover, according to (6.28) and Theorem 6.6.1, we have .C AC = B = AC ϕ+C ψ . It follows that .A = C AC ϕ+C ψ C , and we get from Theorem 6.7.2 that A = A∗C

.

 ϕ+C ψ

= AC ψ+C ϕ .

 

It should be noted that the matrix representation of a truncated Toeplitz operators Aϕ with respect to a modified Clark basis is complex symmetric (i.e. self-transpose). This fact was first observed in [34] and developed further in [31]. It is suspected that the truncated Toeplitz operators might serve as some sort of model operator for various classes of complex symmetric operators. An increasing long list of complex symmetric operators have been proven to be unitarily equivalent to truncated Toeplitz operators. For example, Sarason [50] proved that the Volterra operator on .L2 (0, 1), a standard example of complex symmetric operator, is unitarily equivalent to a truncated Toeplitz operator acting on the space .K corresponding to the atomic inner function .(z) = exp − 1+z 1−z . Among other first examples of complex symmetric operators being unitarily equivalent to truncated Toeplitz operators are: rank-one operators, .2 × 2 matrices, normal operators, for .k ∈ N ∪ {∞} the k-fold inflation of a finite Toeplitz matrix. See [38] for details and further examples. All these results yield to the following still open questions: .

Is every complex symmetric operator unitarily equivalent to a truncated Toeplitz operator? If not, which ones are?

One can also explore when a bounded operator is similar to a truncated Toeplitz operator. In finite dimensions, it is proved in [20] that every operator on a finite dimensional space is similar to a co-analytic truncated Toeplitz operator.

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Which operators are similar to truncated Toeplitz operators in infinite dimensional spaces?

6.7.2 Another Characterization of Truncated Toeplitz Operators and New Examples We now discuss another interesting characterization of truncated Toeplitz operators, also given by Sarason [52], which yields to new examples of truncated Toeplitz operators. We first start with three simple lemmas. In the following, we should understand ∗ ⊥ as the orthogonal complement of .CS ∗  in .K , that is .(S ∗ )⊥ = K

.(S )   ∗ CS . Lemma 6.7.3 Let .f ∈ H 2 . Then (f ∈ K and Sf ∈ K ) ⇐⇒ f ∈ (S ∗ )⊥ .

.

Proof First let us assume that f and Sf belong to .K . We thus have f, S ∗ 2 = Sf, 2 = 0.

.

Hence .f ⊥ CS ∗ . Conversely, assume that .f ∈ K CS ∗ . Remember that .C (k0 ) = S ∗ , and then .k0 = C (S ∗ ). We thus have 0 = f, S ∗ 2 = C (S ∗ ), C f 2 = k0 , C f 2 = (C f )(0).

.

In other words, .C f = zg, for some .g ∈ H 2 . But since .g = S ∗ (C f ) and .S ∗ K ⊂ K , the function g belongs indeed to .K . Then we get zf = zC (zg) = zz2 g = zg = C g.

.

In particular, we deduce that .zf ∈ K .

 

The second result is a simple functional analysis result. Lemma 6.7.4 Let X be a Banach space, V a dense subspace of X, and let .ϕ : X −→ C be a linear continuous functional on X, .ϕ ≡ 0. Then (i) there exists .v0 ∈ V such that .ϕ(v0 ) = 1. (ii) .ker ϕ ∩ V is dense in .ker ϕ.

6 Truncated Toeplitz Operators

229

Proof (1) Since V is dense in X, .ϕ|V is not identically zero, because otherwise by continuity .ϕ would be identically zero, which contradicts the hypothesis. Then, in particular, there is a vector .v ∈ V such that .λ = ϕ(v) = 0. Now take .v0 = v/λ. (2) Let .w ∈ ker ϕ. Then we can find a sequence .(wn )n in V such that .wn −wX → 0, as .n → ∞. Observe now that |ϕ(wn )| = |ϕ(wn − w)|  ϕwn − wX ,

.

which implies that .ϕ(wn ) → 0, as .n → ∞. Applying (1), let .v0 ∈ V such that ϕ(v0 ) = 1, and define .un = wn − ϕ(wn )v0 . We easily see that .un ∈ V ∩ ker ϕ, and

.

un − vX  wn − vX + |ϕ(wn )|v0 X → 0, as n → ∞.

.

Lemma 6.7.5 Let . be an inner function. Then 1. .K ∩ H ∞ ∩ (S ∗ )⊥ is dense in .(S ∗ )⊥ . 2. .K ∩ A(D) ∩ (S ∗ )⊥ is dense in .(S ∗ )⊥ . Proof Of course (2) implies (1) and so we only prove (2). Recall that by Aleksandrov’s theorem (see [21, Section 8.5]), the space .A(D) ∩ K is dense in .K . It remains to apply Lemma 6.7.4 to .V = A(D) ∩ K and to the linear continuous functional ϕ : K −→ C h −→ h, S ∗ 2 ,

.

for which .ker ϕ = (S ∗ )⊥ . We can now give a new characterization of operators in .T(). Theorem 6.7.6 (Sarason, [52]) Let .A ∈ L(K ). Then .A ∈ T() if and only if it satisfies the implication f, Sf ∈ K ⇒ ASf, Sf 2 = Af, f 2 .

.

(6.29)

Proof First, let us assume that .A ∈ T(). According to Corollary 6.5.3, there are ϕ, ψ ∈ K such that .A = Aϕ+ψ¯ . To prove that A satisfies (6.29), it is sufficient, according to Lemma 6.7.3 and Lemma 6.7.5, to check that

.

ASf, Sf 2 = Af, f 2 ,

.

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for every .f ∈ K ∩ H ∞ ∩ (S ∗ )⊥ . We have ASf, Sf 2 = Aϕ+ψ Sf, Sf 2 = Aϕ Sf, Sf 2 + Sf, Aψ Sf 2 .

.

Since .f ∈ (S ∗ )⊥ , according to Lemma 6.7.3, .Sf ∈ K , and thus Aϕ Sf, Sf 2 = P (ϕzf ), zf 2 = ϕzf, zf 2 = ϕf, f 2 = Aϕ f, f 2 .

.

Similarly, we have Sf, Aψ Sf, 2 = zf, P (ψzf )2 = zf, ψzf 2 = ψf, f 2 = Aψ f, f 2 .

.

Therefore, we deduce ASf, Sf 2 = Aϕ (f ), f 2 + Aψ f, f 2 = Af, f 2 .

.

For the other direction, suppose that A satisfies (6.29). We shall prove that .A ∈ ∗ AS . Note that for every T() by showing that A satisfies (6.27). Let .B = A − S  ∗ ⊥ function .f ∈ (S ) , we have .S f = Sf (by Lemma 6.7.3), and then ∗ Bf, f 2 =Af, f 2 − S AS f, f 2

=Af, f 2 − AS f, S f 2 .

=Af, f 2 − ASf, Sf 2 =0.

By the polarization identity, we get Bf, g2 = 0

.

whenever f, g ∈ (S ∗ )⊥ .

Note that the orthogonal projection in .K with range .(S ∗ )⊥ equals .I − cS ∗  ⊗ S ∗ , where .c = S ∗ −2 2 . We have then (I − cS ∗  ⊗ S ∗ )B(I − cS ∗  ⊗ S ∗ ) = 0,

.

implying ∗ A−S AS = B = cBS ∗ ⊗S ∗ +cS ∗ ⊗B ∗ S ∗ −c2 BS ∗ , S ∗ 2 S ∗ ⊗S ∗ .

.

Defining .ϕ = cBS ∗  − c2 BS ∗ , S ∗ 2 S ∗  and .ψ = cB ¯ ∗ S ∗ , we obtain ∗ A − S AS = ϕ ⊗ S ∗  + S ∗  ⊗ ψ.

.

Corollary 6.7.2 finally implies that .A ∈ T().

 

6 Truncated Toeplitz Operators

231

Using the above characterization, we can give another way of building bounded truncated Toeplitz operators. For .μ ∈ C2 () (see Sect. 6.2.3), denote by .μ the sesquilinear functional on .K × K defined by  μ (f, g) =

.

T

f g¯ dμ.

Observe that    .  (f, g)  μ



1/2  |f | d|μ|

1/2 |g| d|μ|

2

T

2

T

 cf 2 g2 ,

because .K → L2 (|μ|). In particular, there exists a bounded operator .Aμ = A μ on .K such that   .Aμ f, g2 = f g¯ dμ. T

The operator .Aμ is another type of truncated Toeplitz operator. Theorem 6.7.7 (Sarason, [52]) Let .μ ∈ C2 (). Then .Aμ ∈ T(). Proof According to Theorem 6.7.6 and Lemma 6.7.5, it is sufficient to show that for every .f ∈ A(D) ∩ K ∩ (S ∗ )⊥ , we have Aμ (Sf ), Sf 2 = Aμ (f ), f 2 .

.

But since .|Sf | = |f | on .T, we get  Aμ (Sf ), Sf 2 =

 |Sf | dμ = 2

.

T

T

|f |2 dμ = Aμ (f ), f 2 .

That proves that .Aμ ∈ T(). Example 6.7.8 Assume that .ζ ∈ T is an angular point of Carathéodory for .. Then, we know (see Sect. 6.2.2) that .f −  → f (ζ ) is continuous on .K . In other words, the Dirac measure .δζ , at point .ζ , is in .C2 (). By Theorem 6.7.7, the operator .Aδζ is in .T(). Observe that if .f, g ∈ K , then  Aδζ f, g2 =

.

T

f gdδζ = f (ζ )g(ζ ) = f, kζ 2 kζ , g2 ,

where .kζ is the reproducing kernel of .K at point .ζ . In other words, .Aδζ f = f, kζ 2 kζ , .f ∈ K , meaning that Aδζ = kζ ⊗ kζ .

.

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Thus, in particular, .Aδζ is a rank-one TTO. We will come back in Sect. 6.10 to this operator. A natural question, posed by Sarason [52, page 513], is whether the converse of Theorem 6.7.7 holds. In other words, does every bounded truncated Toeplitz operators arise from a measure .μ ∈ C2 ()? This question was settled in the affirmative in a beautiful paper by Baranov et al. [6]. Theorem 6.7.9 (Baranov–Bessonov–Kapustin) Let .A ∈ L(K ). Then .A ∈ T() if and only if .A = Aμ for some .μ ∈ C2 (). In the proof of this result (and in many results of [6]), a key role is played by the following Banach space X defined by X=

 

.

xk yk : xk , yk ∈ K ,



 xk 2 yk 2 < ∞ ,

(6.30)

k

k

where the norm in X is defined as the  infimum of representations of the elements in the form . k xk yk .



.

k

xk 2 yk 2 over all

6.8 Norm of a Truncated Toeplitz Operator For Toeplitz operators, a classical result of Brown–Halmos says that if .ϕ ∈ L∞ , then .Tϕ is bounded and .Tϕ  = ϕ∞ . In contrast to this, we can say little more than Aϕ   ϕ∞ ,

.

(6.31)

for general truncated Toeplitz operators with bounded symbols .ϕ. For instance, observe that .A = 0 and so A  < ∞ = 1.

.

Nevertheless using Theorem 6.5.1, we can easily improve the estimate (6.31) and obtain Aϕ   inf{ψ∞ : ϕ − ψ ∈ H 2 + H 2 }.

.

The fact that there are many symbols that represent the same truncated Toeplitz operator makes the problem of computing the norm difficult. However, it is possible to give lower estimates of .Aϕ  for general .ϕ in .L2 . Although a variety of lower bounds on .Aϕ  are provided in [37], we focus here on perhaps the most useful of these. Remind the definition of the Poisson integral in (6.1).

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233

Theorem 6.8.1 (Garcia–Ross, [37]) Let . be an inner function and .ϕ ∈ L2 . Then .

sup{|(Pϕ)(λ)| : λ ∈ D, (λ) = 0}  Aϕ ,

where the supremum above is regarded as 0 if . never vanishes on .D. Proof Let .λ ∈ D such that .(λ) = 0. Then .kλ = kλ and we have       kλ kλ , 2  Aϕ   Aϕ kλ 2 kλ 2 =(1 − |λ|2 )|P (ϕkλ ), kλ 2 | .

=(1 − |λ|2 )|ϕkλ , kλ 2 |     ϕ(ξ ) 2  dm(ξ ) =(1 − |λ| )  2 T |ξ − λ| =|(Pϕ)(λ)|.

Taking the supremum over all .λ in .D such that .(λ) = 0 gives the result.

 

Corollary 6.8.2 Let . be an inner function whose zeros accumulate almost everywhere on .T and let .ϕ ∈ C(T). Then .Aϕ  = ϕ∞ . Proof Let .ζ be an accumulation point of .−1 ({0}). Then there exists a sequence .(λn )n1 ⊂ D, .(λn ) = 0 such that .λn → ζ , .n → ∞. According to Theorem 6.8.1, for every .n  1, we have |(Pϕ)(λn )|  Aϕ .

.

Now, since .ϕ ∈ C(T), letting n go to .∞, we get from (6.2) |ϕ(ζ )|  Aϕ .

.

But, this is true for almost every .ζ ∈ T, which yields to .ϕ∞  Aϕ . Now combining with (6.31) gives the result.   Example 6.8.3 Let .(tn )n1 be a dense sequence in .[0, 2π ] and let .λn =  1 1 − n2 eitn , .n  1. We easily see that .(λn )n1 is a Blaschke sequence and if we denote by B its associated Blaschke product, then its boundary spectrum .σ (B) is equal to .T. Indeed, if .eit ∈ T, .t ∈ [0, 2π ], we have by classical estimates |eit − λn |  |eit − eitn | +

.

1 1  |t − tn | + 2 . 2 n n

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Since .(tn )n1 is dense sequence in .[0, 2π ], there exists a subsequence .(tn )1 such that .tn → t, as . → ∞. Hence .λn → eit , as . → ∞. Now according to Corollary 6.8.2, for every .ϕ ∈ C(T), we have .AB ϕ  = ϕ∞ . The norm of some truncated Toeplitz operators can be related to the norm of some Hankel operators and certain classical extremal problems from function theory. We refer the reader to the papers [15, 36] for more on this. Moreover, if . is a finite Blaschke product (so that the corresponding model space .K is finite dimensional) and .ϕ belongs to .H ∞ , then straightforward residue computations allow us to represent .Aϕ with respect to any of the classical orthonormal bases of .K (the Takenaka basis, the Clark basis,...). In any case, one can readily compute the norm of .Aϕ by computing the norm of one of its matrix representations. This approach was undertaken in the paper [36].

6.9 Spectral Properties of A ϕ The Livsic–Möeller Theorem identified the spectrum of the model operator .S with the spectrum of the inner function . (see (6.3)). The following result of P. Fuhrmann [28, 29] generalizes this for the analytic truncated Toeplitz operators .Aϕ , where .ϕ ∈ H ∞ . As we will see, the proof depends crucially on the famous Corona Theorem of L. Carleson. Theorem 6.9.1 (Fuhrmann) Let . be an inner function and .ϕ ∈ H ∞ . Then σ (Aϕ ) = {λ ∈ C : inf (|(z)| + |ϕ(z) − λ|) = 0}.

.

z∈D

Proof Let .λ ∈ C satisfying .infz∈D (|(z)| + |ϕ(z) − λ|) = 0. Then there exists a sequence .(zn )n1 ⊂ D such that .|(zn )| → 0 and .|ϕ(zn ) − λ| → 0 as .n → ∞. Since .ϕ ∈ H ∞ , we have .A∗ϕ = Tϕ |K and so (A∗ϕ − λI )kzn 2 = Tϕ−λ kzn 2 = P+ ((ϕ − λ)(1 − (zn ))kzn )2

.

 P+ ((ϕ − λ)kzn )2 + |(zn )|P+ ((ϕ − λ)kzn )2 Now recall that for every .ψ ∈ H ∞ , we have .P+ (ψkλ ) = ψ(λ)kλ . Then it follows (A∗ϕ − λI )kzn 2  ϕ(zn ) − λ)kzn 2 + (ϕ∞ + |λ|)(zn )|kzn 2 .

 C(|ϕ(zn ) − λ| + |(zn )|)kzn 2 ,

6 Truncated Toeplitz Operators

235

where .C = ϕ∞ + |λ| + 1. Now dividing both terms by .kzn 2 gives    kzn  kz 2  ∗  . (Aϕ − λI )   C(|ϕ(zn ) − λ| + |(zn )|) n .    kzn 2 kzn 2 2

k 

Observe now that . kzn 2 = (1 − |(zn )|2 )−1/2 → 1 as .n → +∞, which finally zn 2

gives that

   kzn   ∗  . (Aϕ − λI )  → 0, n → ∞.   kzn 2  2

In particular, .

inf{(A∗ϕ − λI )f 2 : f ∈ K , f 2 = 1} = 0.

Hence .A∗ϕ − λI is not bounded below and hence not invertible. Thus λ ∈ σ (A∗ϕ ) = σ (Aϕ ),

.

and so {λ ∈ C : inf (|(z)| + |ϕ(z) − λ|) = 0} ⊂ σ (Aϕ ).

.

z∈D

To prove the reverse inclusion, we will now show that .

inf (|(z)| + |ϕ(z) − λ|) > 0 ⇒ λ ∈ / σ (Aϕ ).

z∈D

According to Carleson Corona Theorem, when .infz∈D (|(z)| + |ϕ(z) − λ|) > 0, there exists .f, g ∈ H ∞ such that f (z)(z) + g(z)(ϕ(z) − λ) = 1,

.

z ∈ D.

Hence, by Proposition 6.3.3, we get Af A + Ag (Aϕ − λI ) = I.

.

But .A = 0 (see Lemma 6.4.4). Hence Ag (Aϕ − λ) = I.

.

Since .Ag and .Aϕ commute because g and .ϕ are in .H ∞ , we also have .(Aϕ −λI )Ag = I , and so .Aϕ − λI is invertible, showing that .λ ∈ / σ (Aϕ ).  

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Remind from (6.3) the definition of .s(), the spectrum of an inner function .. Corollary 6.9.2 Let . be an inner function and .ϕ ∈ H ∞ ∩ C(T). Then σ (Aϕ ) = ϕ(s()).

.

Proof Let .λ ∈ C. According to Theorem 6.9.1, .λ ∈ σ (Aϕ ) if and only if there exists a sequence .(zn )n1 ⊂ D satisfying (zn ) → 0

.

and

ϕ(zn ) → λ,

as n → ∞.

Passing to a subsequence if necessary, we may assume that .(zn )n1 converges to some point .z0 ∈ D− (the closed unit disc). By definition, .z0 ∈ s() and since .ϕ is continuous on .D− , we have .ϕ(zn ) → ϕ(z0 ), .n → ∞. In particular, .λ = ϕ(z0 ) ∈ ϕ(s()). Hence .σ (Aϕ ) ⊂ ϕ(s()). Conversely, if .λ ∈ ϕ(s()), then there is .z0 ∈ s(), such that .λ = ϕ(z0 ). In particular, there is a sequence .(zn )n1 ⊂ D such that .zn → z0 and .(zn ) → 0, as .n → ∞. By continuity of .ϕ, we have once more that .ϕ(zn ) → ϕ(z0 ) = λ and so .λ ∈ σ (Aϕ ).   The natural polynomial functional calculus .p(S ) = Ap can be extended to H ∞ in such a way that the symbol map .ϕ −→ ϕ(S ) := Aϕ is linear, contractive and multiplicative (see Proposition 6.3.9). The above corollary says that we have a spectral mapping theorem for symbols in .H ∞ ∩ C(T). Note that in the case when .ϕ ∈ H ∞ , it is possible also to give an alternate description of .σ (Aϕ ) in terms of cluster sets for .ϕ. See [32, page 294–296] for details. We can also describe the point spectrum of .Aϕ and is adjoint. See [32, Corollaries 13.15, 13.17 and 13.18] or [27, Theorem 14.34 and Corollary 14.35]. Let us mention also another interesting new approach by Camara and Partington [12]. They give invertibility and Fredholmness criterion for truncated Toeplitz p operators in model spaces .K of the upper half-plane, .1 < p < ∞ with essentially bounded symbols in a class including the algebra .C(R) + H ∞ , as well as sums of analytic and anti-analytic functions satisfying a .-separation condition. Their approach is based on the equivalence after extension of truncated Toeplitz operators to Toeplitz operators with .2 × 2 matrix symbols. Let us conclude this section by saying that in general determine the spectrum of a truncated Toeplitz operator is a problem highly non trivial. We only know the answer for particular class of symbols and a lot remain to do.

.

6.10 Finite Rank Truncated Toeplitz Operators Brown and Halmos [11] have showed that there are no nonzero compact Toeplitz operators on .H 2 . In contrast, there are many example of finite rank (hence compact) truncated Toeplitz operators. In fact, the rank-one truncated Toeplitz operators were

6 Truncated Toeplitz Operators

237

first identified by Sarason [52, Theorem 5.1]. For the proof of this result, we need the following two lemmas. Remind from Sect. 6.2.2 the notation .f for .f ∈ K . Lemma 6.10.1 Let . be an inner function. (a) For .λ ∈ D, we have ∗   S kλ = λkλ − (λ)k 0

.

and     S k λ = λkλ − (λ)k0 .

.

(b) For .λ ∈ D \ {0}, we have S kλ =

.

1 λ

kλ −

1 λ

k0

and ∗ S kλ =

.

1  1 . kλ − k λ 0 λ

Proof (a) For the first equality, note that for every .f, g ∈ H ∞ , we have .S ∗ (f g) = f S ∗ (g) + g(0)S ∗ f , which yields to ∗  S kλ = S ∗ ((1 − (λ))kλ ) = (1 − (λ)S ∗ kλ + kλ (0)S ∗ (1 − (λ)).

.

But .S ∗ kλ = λkλ and .S ∗  = C k0 , and so ∗  S kλ = λ(1 − (λ))kλ − (λ)C k0 = λkλ − (λ)C k0 .

.

∗ (see Theorem 6.7.1), we obtain the second equality by Since .S C = C S applying .C to the first one:

S C kλ = C (λkλ − (λ)C k0 ) .

= λC kλ − (λ)k0 .

(b) For .λ ∈ D \ {0}, we have S kλ = P (zkλ ) = P (z(1 − (λ))kλ ) = P (zkλ ).

.

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E. Fricain

Since .(1 − λz)kλ = 1, we have zkλ =

.

1 λ

(kλ − 1),

and we obtain S kλ =

.

1 λ

P (kλ − 1) =

1 λ

(kλ − k0 ),

which is the first equality. As in (a), the second equality is obtained from the first through an application of .C .   Lemma 6.10.2 Let . be an inner function. Assume that . has an angular derivative in the sense of Carathéodory at the point .ζ ∈ T. Then the equalities of the preceding lemma hold with .ζ in place of .λ. Proof Since . has an angular derivative in the sense of Carathéodory at the point ζ ∈ T, it is known that .kλ → kζ , as .λ tends nontangentially to .ζ , in the .H 2 norm. Now, we obtain the conclusion by letting .λ tend nontangentially to .ζ in the equalities of Lemma 6.10.1.

.

Theorem 6.10.3 (Sarason, [52]) Let . be an inner function. The operators  (i) .kλ ⊗ k λ =A   (ii) .k λ ⊗ kλ = A

(iii)

 .k ζ

⊗

kζ

 z¯ −λ¯  z−λ

for .λ ∈ D, for .λ ∈ D,

= Ak  +k  −1 , where . has an angular derivative in the sense of ζ

ζ

Carathéodory at .ζ ∈ T

are truncated Toeplitz operators having rank one. Proof (i) We consider first a point .λ ∈ D \ {0} and apply the criterion of Theorem 6.6.1  to the operator .kλ ⊗ k λ . By Lemma 6.10.1,    ∗  S (kλ ⊗ k λ )S = S kλ ⊗ S kλ     1  1    − (λ)k  ⊗ λ k k k = − λ 0 0 λ . λ λ (λ)  (λ)      = kλ ⊗ k kλ ⊗ k0 + k0 ⊗ k0 . λ − k0 ⊗ kλ − λ λ Thus (λ)        ∗  ((kλ − k0 ) ⊗ k0 ). kλ ⊗ k λ − S (kλ ⊗ kλ )S = k0 ⊗ kλ + λ

.

6 Truncated Toeplitz Operators

239

 By Theorem 6.6.1, .kλ ⊗ k λ = Aϕ with symbol

(λ)

ϕ=

.

λ

 (kλ − k0 ) + k λ.

According to Theorem 6.5.1, we have Ak  = A1−(0) = A1

Ak  = A(1−(λ))kλ = Akλ ,

and

.

0

λ

 whence .kλ ⊗ k λ = Aψ , with .ψ, written as a function of the variable .z ∈ T, equals to

ψ(z) =

(λ)

=

(λ)

λ

 (kλ (z) − 1) + k λ (z)



λ

.

=(λ) =

 (z) − (λ) −1 + z¯ − λ¯ 1 − λz 1

z 1 − λz

z(z) 1 − λz

=

+

z((z) − (λ)) 1 − λz

(z) z¯ − λ¯

 Taking the limit as .λ → 0, we find that .k0 ⊗ k 0 is the truncated Toeplitz

operator with symbol . (z) z¯ . (ii) Conjugating the identity proved in (a) conclusion, we find that for every .λ ∈ D    ∗   ∗ k λ ⊗ kλ = (kλ ⊗ kλ ) = Aψ = Aψ ,

.

where .ψ(z) = (z) z−λ . (iii) Let .ζ be a point of .T at which . has an angular derivative in the sense of Carathéodory. By Lemma 6.10.2, the first part of the proof of (i) can be  repeated with .ζ in place of .λ to show that .kζ ⊗ k ζ is the truncated Toeplitz operator with symbols

.

(ζ )   (kζ − k0 ) + k ζ . ζ¯

    ¯ Using that .k ζ = ζ (ζ )kζ , we see that .kζ ⊗ kζ is the truncated Toeplitz

operator with symbol .kζ + kζ − k0 . As above in the proof of (i), we can replace .k0 by 1, obtaining the symbol .kζ + kζ − 1.

 

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E. Fricain

It turns out that Sarason proved [52, Theorem 5.1] that Theorem 6.10.3 gives all the possible rank-one operators in .T(). In other words, any truncated Toeplitz operators of rank one is a nonzero scalar multiple of one of the above. We should also mention the somewhat more involved results of Sarason [52, Theorems 6.1 and 6.2] which identify a variety of natural finite-rank truncated Toeplitz operators. Finally, the full classification of the finite-rank truncated Toeplitz operators was given by Bessonov in [8]. We would like to finish this section by giving an example of a rank-two operator 2 / T(). .A ∈ T() such that .A ∈ Example 6.10.4 Let . be an inner function such that .dim(K )  3. Consider .A =     2 / T(). k0 ⊗ k 0 + k0 ⊗ k0 . Then .A ∈ T() but .A ∈ Indeed, the fact that .A ∈ T() follows directly from Theorem 6.10.3. Let us now check that .A2 is not a truncated Toeplitz operator. Argue by absurd, and assume that 2 .A ∈ T(). We have           A2 = k0 , k 0 2 (k0 ⊗ k0 ) + k0 , k0 2 (k0 ⊗ k0 ) .

   2   2    + k 0 2 (k0 ⊗ k0 ) + k0 2 (k0 ⊗ k0 ).

       As .k0 ⊗ k 0 and .k0 ⊗ k0 are in .T() and .k0 2 = k 2 (because .C is an  ⊗k  is in .T(). It now isometry), it follows that the operator .B := k  ⊗ k  + k 0

0

0

0

∗ has rank one or two. By follows from Theorem 6.6.1 that the operator .B − S BS Lemma 6.10.1

.

     ∗   S (k 0 ⊗ k0 )S =S k0 ⊗ S k0 =(−(0)k0 ) ⊗ (−(0)k0 ) = |(0)|2 k0 ⊗ k0 ,

and so  ∗ 2       B − S BS = k 0 ⊗ k0 + (1 − |(0)| )k0 ⊗ k0 − S k0 ⊗ S k0 .

.

Observe that .k0 is a cyclic vector for the operator .S . Indeed, let .f ∈ K and n k  for all non negative integers n. Then, we have assume that .f ⊥ S 0 0 = f, P (zn k0 )2 = f, zn (1 − (0))2 = f, zn 2

.

which says that all the Taylor coefficients of f at the origin vanish, hence that .f = 0.  In particular, we get that .k0 and .S k0 are linearly independent. If .k 0 is not a   ∗ linear combination of .k0 and .S k0 , then the operator .B − S BS would have  = rank three which is not possible. Hence there are scalar a and b such that .k 0

ak0 + bS k0 . Applying .S to the last equality and using Lemma 6.10.1 again,

6 Truncated Toeplitz Operators

241

we get .

2  − (0)k0 = aS k0 + bS k0 .

2 k  is a linear Since .k0 and .S k0 are linearly independent, .b = 0. Hence .S 0    combination of .k0 and .S k0 . By the cyclicity of .k0 for .S , we obtain that 2 .dim(K ) = 2, a contradiction. Therefore, .A is not in .T().

6.11 Compact Truncated Toeplitz Operators Surprisingly enough, the first result about compactness of truncated Toeplitz operators dates from 1970. In [1, Section 5] Ahern–Clark introduced what are, in our terminology, truncated Toeplitz operators with continuous symbol, and they proved the following result. Theorem 6.11.1 (Ahern–Clark) Let . be an inner function and .ϕ ∈ C(T). Then Aϕ is compact if and only if .ϕ|σ () ≡ 0.

.

This result has been rediscovered more recently in [39]; see also [38]. We follow the proof given in [39]. Proof Suppose that .ϕ|σ () ≡ 0. Let .ε > 0 and pick .ψ ∈ C(T) such that .ψ vanishes on an open set containing .σ () and .ϕ − ψ∞  ε. Since .Aϕ − Aψ   ϕ−ψ∞  ε, and the set of compact operators is norm closed in .L(K ), it suffices to show that .Aψ is compact. Take a sequence .(fn )n1 in .K that tends weakly to zero, and let us check that .Aψ fn → 0 in norm. To this end, let K denote the closure of .ψ −1 (C \ {0}), and observe that .K ⊂ T \ σ (). Then, we know that for every .ζ ∈ K, the function kζ (z) =

.

1 − (ζ )(z) , 1 − ζ¯ z

belongs to .K and for every .f ∈ K , we have f (ζ ) = f, kζ 2

.

and kζ 22 =

.

1 − |(ζ )|2 = | (ζ )|. 1 − |ζ |2

In particular, since .(fn )n converges weakly to zero in .K , we have .fn (ζ ) = fn , kζ 2 → 0 as .n → ∞ and there exists a constant .C > 0 such that for every

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E. Fricain

n  1, .fn 2  C. Since . is analytic on a neighborhood of K, we obtain

.

 |fn (ζ )| = |fn , kζ 2 |  fn 2 kζ 2  C sup | (ζ )| < ∞,

.

ζ ∈K

(6.32)

for every .ζ ∈ K. By the Dominated Convergence Theorem, it follows that  Aψ fn 22 = P (ψfn )22  ψfn 22 =

|ψ|2 |fn |2 dm → 0,

.

K

as .n → ∞, whence .Aψ fn tends to zero in norm, as desired. Conversely, suppose that .ϕ ∈ C(T) and .Aϕ is compact. Let κλ =

.

kλ kλ 2

be the normalized reproducing kernel for .K at point .λ, and define 1 − |λ|2 .Fλ (z) = 1 − |(λ)|2

   1 − (λ)(z) 2     ,   1 − λ¯ z

Observe that .Fλ (z)  0 and .

1 2π



2π

Fλ (eit ) dt = 1.

0

Suppose that .ξ = eiα ∈ σ (). By definition, there is a sequence .(λn )n1 ⊂ D such that .λn → ξ and .|(λn )| → 0. If .|t − α|  δ > 0, then Fλn (eit )  Cδ

.

1 − |λn |2 , 1 − |(λn )|2

for some positive absolute constant .Cδ . In particular, since .|(λn )| → 0, as .n → ∞, we get .

sup Fλn (eit ) → 0,

|t−α|δ

as n → ∞.

(6.33)

First we show that    2π   1 it it  ϕ(e )Fλn (e ) dt  = 0. . lim ϕ(ξ ) − n→∞  2π 0

(6.34)

6 Truncated Toeplitz Operators

243

To do this, note that    2π   it it ϕ(ξ ) − 1 = 1 ϕ(e )F (e ) dt λn   2π 2π 0

   

0

1  2π 1 + 2π

.



2π

1 2π

  (ϕ(ξ ) − ϕ(eit ))Fλn (eit ) dt 





|t−α|δ

|t−α|δ

|ϕ(ξ ) − ϕ(eit )|Fλn (eit ) dt+ |ϕ(ξ ) − ϕ(eit )|Fλn (eit ) dt



|t−α|δ

|ϕ(ξ ) − ϕ(eit )|Fλn (eit ) dt+

+ 2ϕ∞ sup Fλn (eit ). |t−α|δ

The first integral can be made small by the absolute continuity of .ϕ (choosing an appropriate .δ) and the fact that .Fλn always integrate to one. Once .δ > 0 is fixed, the second term goes to zero by (6.33). This verifies (6.34). Next, we show that  . lim ϕFλn dm = 0. (6.35) n→∞ T

We need the fact that .κλn → 0 weakly in .K . To prove this, note that, for every z ∈ D, we have

.

κλn (z) =

.

(1 − |λn |2 )1/2 1 − (λn )(z) , (1 − |(λn )|2 )1/2 1 − λn z

and since .λn → ζ ∈ T and .(λn ) → 0, .n → ∞, we get that κλn (z) → 0,

.

as n → ∞.

On the other hand, since .(κλn )n1 is norm bounded (it is of norm 1 for every n), we deduce that .κλn → 0 weakly in .H 2 , whence in .K . Now to verify (6.35), observe that .

     ϕFλ dm = |ϕκλ , κλ 2 | = |P (ϕκλ ), κλ 2 | = |Aϕ κλ , κλ 2 |. n n n n n n n   T

Then, an application of Cauchy-Schwartz inequality yields       .  ϕFλn dm  Aϕ κλn 2 κλn 2 = Aϕ κλn 2 . T

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E. Fricain

Now use the fact that .Aϕ is compact and .κλn → 0 weakly as .n → ∞ to conclude that .Aϕ κλn 2 → 0. This proves (6.35). Combining (6.34) with (6.35) shows that .ϕ(ξ ) = 0 for every .ξ ∈ σ (), and completes the proof the theorem.   Remark 6.11.2 Note that Theorem 6.11.1 admits a more precise version: if .ϕ ∈ C(T), then the essential spectrum of .Aϕ is σe (Aϕ ) = ϕ(σ ()),

.

(6.36)

and Aϕ e = sup |ϕ(ζ )|.

.

ζ ∈σ ()

This can be proved using the explicit triangularization theory developed by Ahern and Clark in [1] (see also the exposition in [46, Lecture V]). Recently, Garcia– Ross–Wogen gave an algebraic proof of these formulae using the unital .C ∗ -algebra generated by .S . Thinking of Hartmann’s theorem, it seems plausible to believe that continuous symbols play for compact TTOs the role played by bounded symbols for general TTOs. However, as we will see in the next section, there exist inner functions . for which even rank-one operators might not have bounded symbols (not to speak about continuous). So we have to consider only certain classes of inner functions, suggested by the boundedness results in the previous section. In this sense one has the following result proved by Bessonov [9]. Theorem 6.11.3 (Bessonov) Let . be an inner function and let .ϕ ∈ C(T) + H ∞ . Then σe (Aϕ ) = ϕ(σ ()).

.

Here we should understand .ϕ(σ ()) as   ϕ(σ ()) = ζ ∈ C : lim inf (|(Pϕ)(z) − ζ | + |(z)| = 0 ,

.

z∈D,|z|→1

where .Pϕ is the Poisson extension of .ϕ, see (6.1). The main step in the proof of Theorem 6.11.3 is an application of a corona Theorem for the algebra .H ∞ + C(T) obtained by Mortini and Wick [44]. If we impose some conditions on the inner function, one can get some characterization of compact truncated Toeplitz operators. Remind here, from Sect. 6.2.3, the classes of Carleson measures .Cp (), .p  1.

6 Truncated Toeplitz Operators

245

Theorem 6.11.4 (Bessonov) Let . be an inner function such that .C1 (2 ) = C2 (), and let A be a truncated Toeplitz operator on .K . Then the following are equivalent: (i) A is compact. (ii) .A = Aϕ for some .ϕ ∈ C(T). In particular, this characterization of compact TTO applies when . is one component. One can see that instead of .C(T) the main role is played by .C(T). We give below some ideas about the connection between these two classes. Theorem 6.11.5 (Chalendar-Fricain-Timotin, [16]) Let . be an inner function such that .C2 () = C1 (2 ) and .m(σ ()) = 0, and let .A ∈ T(). The following are equivalent: (i) A is compact. (ii) .A = Aϕ for some .ϕ ∈ C(T) with .ϕ|σ () = 0. Proof (ii). ⇒ (i) is proved in Theorem 6.11.1. Suppose now (i) is true. By Theorem 6.11.4, there is a .ψ ∈ C(T) such that .A = Aψ . By the Rudin–Carleson interpolation theorem (see, for instance, [30, Theorem II.12.6]), there exists a function .ψ1 ∈ C(T) ∩ H ∞ (that is, in the disk algebra) such that .ψ|σ () = ψ1 |σ (). Then one checks easily that .ϕ = (ψ −ψ1 ) is continuous on .T, .ϕ|σ () = 0, and .Aϕ = Aψ (since .Aψ1 = 0).   In particular, Theorem 6.11.5 applies to the case when . one-component, since for such functions we have .C2 () = C1 (2 ) and .m(ρ()) = 0 [3, Theorem 6.4]. We also have the following result which is contained in [9, Proposition 2.1]; it was revisited in [16]. Proposition 6.11.6 (Bessonov) Let . be an inner function. (i) If .ϕ ∈ C(T) + H ∞ , then .Aϕ is compact. (ii) If .ϕ ∈ C(T) + H ∞ , then the converse is also true. Proof (i) Assume that .ϕ ∈ C(T) + H ∞ . Hence .ϕ ∈ C(T) + H ∞ , which implies by Hartmann’s theorem that .Hϕ is compact. On the other hand, since .C(T)+H ∞ is an algebra and . ∈ H ∞ ⊂ C(T) + H ∞ , we deduce that the function .ϕ = ϕ also belongs to .C(T) + H ∞ . Another application of Hartmann’s theorem gives that the operator .Hϕ is also compact. The fact that .Aϕ is compact follows now from the relation Aϕ = (Hϕ − Hϕ )|K .

.

(6.37)

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E. Fricain

To prove (6.37), observe that for every .f ∈ K , we have (Hϕ − Hϕ )f =P− (ϕf ) − P− (ϕf ) =(ϕf − P+ (ϕf )) − (ϕf − P+ (ϕf ) =P+ (ϕf ) − P+ (ϕf )

.

=P (ϕf ) =Aϕ (f ). (ii) Assume now that .ϕ ∈ C(T)+H ∞ and also that .Aϕ is compact. Then, by (6.37), the operator .Hϕ |K = Aϕ + Hϕ |K is also compact, whence .Hϕ |K is compact. Let us now remark that Hϕ = Hϕ PH 2 + Hϕ P |H 2 ,

.

(6.38)

where .PH 2 denotes the orthogonal projection from .H 2 onto .H 2 . Indeed, let 2 2 .f ∈ H , and decompose f as .f = g1 + g2 , where .g1 ∈ H and .g2 ∈ K . Then Hϕ f =P− (ϕg1 ) + P− (ϕg2 ) =P− (ϕg1 ) + P− (ϕg2 )

.

=Hϕ g1 + Hϕ g2 , and it remains to observe that .g1 = PH 2 f and .g2 = P f . Now it follows from (6.38) that .Hϕ is compact, and Hartmann’s theorem implies that .ϕ ∈ C(T) + H ∞ .   Finally, let us mention that the problem of deciding when certain truncated Toeplitz operators are in Schatten–von-Neumann classes .Sp has no clear solution, yet, even in the usually simple case of the Hilbert–Schmidt ideal. In [42], Lopatto– Rochberg give criteria for particular cases. For the case of one-component inner functions ., a conjecture is proposed in [9] for the characterization of Schatten– von Neumann TTOs in .K . It states essentially that a truncated Toeplitz operator 1/p is in .Sp if and only if it has at least one symbol .ϕ in the Besov space .Bpp (note that this would not necessarily be the standard symbol). This last space admits several equivalent characterizations; for instance, if we define, for .τ ∈ T, .τ f (z) = f (τ z) − f (z), then  1/p .Bp,p



= f ∈L : p

T

 p τ f p dm(τ ) < ∞ . |1 − τ |2

6 Truncated Toeplitz Operators

247

The conjecture is suggested by the similar result in the case of Hankel operators [47, z+1 Chapter 6]. It is true if .(z) = e z−1 , as shown by Bessonov in [10]. Bessonov also proposes some alternate characterizations in terms of Clark measures.

6.12 Problem of the Existence of a Bounded Symbol Recall that a famous result of Brown–Halmos says that a Toeplitz operator .Tϕ on H 2 is bounded if and only if its symbol .ϕ is in .L∞ . Moreover, we have .Tϕ  = ϕ∞ . In other words, the map .ϕ → Tϕ is isometric from .L∞ onto the space of bounded Toeplitz operator on .H 2 . In the case of truncated Toeplitz operators, the ∞ into .T(). It is then natural to ask map .ϕ → A ϕ is again contractive from .L whether it is onto, that is whether any bounded truncated Toeplitz operator on .K possesses an .L∞ symbol. This question was addressed by Sarason in [52]. As one may expect, the answer will depend on the inner function .. In [7], Baranov et al. give an answer to this question by constructing an example of a rank-one truncated Toeplitz operator that has no bounded symbols. The construction is based on the following crucial lemma.

.

Lemma 6.12.1 ([7], Lemma 5.2) Let . be an inner function and .1 < p < ∞. There exists a constant C depending only on . and p such that if .ϕ, ψ ∈ L2 are two symbols for the same truncated Toeplitz operator, with .ϕ ∈ K ⊕ z¯ K , then ϕp  C(ψp + ϕ2 ).

.

In particular, if .ψ ∈ Lp , then .ϕ ∈ Lp . Proof Remind that .S = L2 (H 2 + H 2 ). By hypothesis and Theorem 6.5.1, we have .PS ϕ = PS ψ; therefore, using (6.25), ϕ = Q ϕ = PS ϕ + ϕ, q 2 q = PS ψ + ϕ, q 2 q .

.

By Lemma 6.5.2, we have PS ψp  C1 ψp ,

.

while ϕ, q 2 q p  ϕ2 q p ,

.

whence the lemma follows.

 

In Theorem 6.10.3, it is proved that if .ζ ∈ E() (remind that it means that . has an angular derivative in the sense of Carathéodory at .ζ ), then .kζ ⊗ kζ is a rank-one

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E. Fricain

truncated Toeplitz operator with symbol .kζ + kζ − 1. Lemma 6.12.1 suggests to look for the symbol in .K ⊕ zK . 2

Lemma 6.12.2 Let .ζ ∈ E() and define .ϕζ = ζ (ζ )zkζ . Then .ϕζ belongs to   .K ⊕ zK and .Aϕζ = k ζ ⊗ kζ . 2

Proof First note that if .ζ ∈ E(), then by Lemma 6.2.2, .ζ ∈ E(2 ) and .kζ ∈ 2

K2 . Now, using that .K2 = K ⊕ K , we can write .kζ = f + g, with .f, g ∈ K . Then 2

zkζ = (zf + zg) = zf + zg = C (f ) + zg.

.

Since .C (f ) ∈ K , we deduce that .ϕζ ∈ K ⊕ zK . It remains to prove that .ϕζ is a symbol for .kζ ⊗ kζ . According to Theorem 6.5.1, this is equivalent to .ϕζ − (kζ + kζ − 1) ∈ H 2 + H 2 . First note that for almost all points .z, ζ ∈ T, we have 1 .

1 − ζz

+

ζz 1 1 = + = 1. 1 − zζ ζ z − 1 1 − zζ

Hence ϕζ − (kζ + kζ − 1) = ζ (ζ )(z)z  −

.

1 − 2 (ζ )2 (z) 1 − ζz

1 − (ζ )(z) 1 − ζz

= ζ (ζ )(z)z

1 − 2 (ζ )2 (z) (ζ )(z) (ζ )(z) + + 1 − ζz 1 − ζz 1 − ζz (ζ )(z) (ζ )(z) (ζ )(z) + + 1 − ζz 1 − ζz 1 − ζz

=

(ζ )(z)

=

(ζ )(z) (1 − ζ z) 1 − ζz

ζz − 1

 1 − (ζ )(z) + −1 1 − ζz

− ζz

= (ζ )(z). Therefore .ϕζ − (kζ + kζ − 1) ∈ H 2 and then .Aϕζ = Ak  +k  −1 = kζ ⊗ kζ .   ζ

ζ

We are now ready to answer negatively to the question of Sarason.

6 Truncated Toeplitz Operators

249

Theorem 6.12.3 (Baranov-Chalendar-Fricain-Mashreghi-Timotin, [7]) Suppose that . is an inner function which has an angular derivative in the sense of Carathéodory at .ζ ∈ T. Let .p ∈ (2, ∞). Then, the following are equivalent: (i) the bounded truncated Toeplitz operator .kζ ⊗ kζ has a symbol .ψ ∈ Lp ; (ii) .kζ ∈ Lp . In particular, if .kζ ∈ / Lp , for some .p ∈ (2, ∞), then .kζ ⊗kζ is a bounded truncated Toeplitz operator with no bounded symbol. Proof According to Lemma 6.12.2, a symbol for the operator .kζ ⊗ kζ is .ϕζ = 2

(ζ )ζ zkζ . Since by Lemma 6.2.2, .ϕ ∈ Lp if and only if .kζ ∈ Lp , we obtain that (ii) implies (i). Conversely, assume that the bounded truncated Toeplitz operator   p .k ζ ⊗ kζ has a symbol .ψ ∈ L . Since .ϕζ is a symbol in .K ⊕ zK , we may then apply Lemma 6.12.1 and obtain that .ϕ ∈ Lp . Once again according to Lemma 6.2.2, we get that .kζ ∈ Lp , which proves that (i) implies (ii).   To obtain a bounded truncated Toeplitz operator with no bounded symbol, it is now sufficient to have a point .ζ ∈ T such that (6.5) is true for .p = 2 but not for some strictly larger value of p. It is easy to give concrete examples, as, for instance: (i) a Blaschke product with zeros .ak accumulating to the point 1, and such that  1 − |ak |2

.

k

|1 − ak |2

< ∞,

(ii) a singular function .σ =  .

k

 1 − |ak |2 = ∞ for some p > 2; |1 − ak |p k



k ck δζk

ck < ∞, |1 − ζk |2

 with . k ck < ∞, .ζk → 1, and

 k

ck = ∞ for some p > 2. |1 − ζk |p

As we have just seen, one can construct some inner functions . for which there exists some bounded truncated Toeplitz operators with no bounded symbols. On the contrary,  it is also proved in [7] that for the singular inner function .1 (z) =  exp z+1 z−1 , every bounded truncated Toeplitz operator has a bounded symbol. This follows from results obtained by Rochberg [48] on the Paley–Wiener space. That of course yields the following question: can we characterize inner functions . for which every bounded truncated Toeplitz operator on .K has a bounded symbol? In [6], Baranov–Bessonov–Kapustin gave interesting characterizations of such inner functions in terms of set .Cp (), and also some factorization properties. We will now detail a little bit some of their results. Theorem 6.12.4 (Baranov–Bessonov–Kapustin, [6]) Let .A ∈ T(). Then A admits a bounded symbol if and only if .A = Aμ for some .μ ∈ C1 (2 ). This result should be compared with Theorem 6.7.9.

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Proof Let us assume that .A = Aϕ , where .ϕ ∈ L∞ . Then .A = Aμ with .dμ := 1 , we have ϕ dm. Notice now that for every .f ∈ K 2 

 .

T

|f | d|μ| =

|f ||ϕ| dm  ϕ∞ f 1 ,

T

1 → L1 (|μ|), proving that .μ ∈ C (2 ). whence .K 1 2 Conversely, assume that .A = Aμ , where .μ ∈ C1 (2 ). Consider .E the subspace of .L1 formed by functions which are finite sums of functions of the forms .xk yk , with .xk , yk ∈ K , and define the functional . on .E by

 :f −  →

.

T

f dμ,

Let us check that . is well defined and continuous (if .E is equipped with the .L1 norm). Note that for every .xk , yk ∈ K , we have .xk yk = xk zyk , where .yk = C (yk ) ∈ K ⊂ H 2 . In particular, .¯zxk yk = xk yk ∈ H 1 . On the other hand, writing .xk = zxk , we also have ¯zxk yk = 2 z2 xk yk ∈ 2 zH 1 .

.

1 . Moreover, since .μ ∈ C (2 ) ⊂ C (), it Thus .¯zxk yk ∈ H 1 ∩ 2 zH 1 = K 1 2 2 can be proved (see [6, Proposition 3.2]) that .|| = 1, .|μ|-almost everywhere. Thus, 1 , and we obtain for any .f ∈ E, we have .¯zf ∈ K 2



 .

T

|f | d|μ| =

T

|¯zf | d|μ|  Cf 1 .

In particular, . is well defined and continuous, and so, by Hahn-Banach Theorem, it can be continuously extended to .L1 . Hence, there exists a function .ϕ ∈ L∞ such that   . f dμ = (f ) = f ϕ dm, ∀f ∈ E. T

T

It follows that, for every .x, y ∈ K ,  Aμ x, y2 =

.

T

 x y¯ dμ = (x y) ¯ =

and thus .Aμ = Aϕ , with .ϕ ∈ L∞ .

T

xyϕ dm = Aϕ x, y2 ,  

It follows immediately from Theorems 6.12.4 and 6.7.9 that if .C1 (2 ) = C2 (), then every bounded truncated Toeplitz operator admits a bounded symbol. Baranov– Bessonov–Kapustin have showed that the converse is also true and they also make

6 Truncated Toeplitz Operators

251

an interesting connection with a factorization problem involving the space X (see (6.30)). Theorem 6.12.5 (Baranov–Bessonov–Kapustin, [6]) Let . be an inner function. The following assertions are equivalent: (i) every bounded truncated Toeplitz operator on .K admits a bounded symbol; (ii) .C1 (2 ) = C2 (2 ); (iii) for any .f ∈ H 1 ∩ 2 z2 H 1 , there exists .xk , yk ∈ K with f =



.

xk yk

and

k



xk 2 yk 2 < ∞.

k

If . is a one-component inner function, then all classes .Cp () coincide (see [5, Theorem 1.4]). Moreover, if . is a one-component inner function, the .2 is, too, hence .C1 (2 ) = C2 (2 ). An immediate consequence of Theorem 6.12.5 is then the following. Corollary 6.12.6 Let . be a one-component inner function. Then every bounded truncated Toeplitz operator on .K admits a bounded symbol.

Does the converse is true? That means: assume that every bounded truncated Toeplitz operator on .K admits a bounded symbol. Does it follows that . is one component?

Let us notice that it is shown in [4, Theorem 8] that the condition .C1 (2 ) = C2 (2 ) implies that all the Clark measures .σα , .α ∈ T, (associated to .) are discrete. It implies immediately another class of counterexamples to the existence of a bounded symbol. Corollary 6.12.7 Let . be an inner function and assume that for some .α ∈ T, the Clark measure .σα is not discrete. Then there exist a bounded truncated Toeplitz operator on .K that do not admit a bounded symbol. Let .μ be a continuous singular measure. It is well known (see [32, Chapter 11]) that there exists an inner function . such that 1 − |(z)2 | . = |1 − (z)|2

 T

1 − |z|2 dμ(ζ ). |z − ζ |2

In particular, we get that .μ is the Clark measure .σ1 associated to . at point 1. By Corollary 6.12.7, we know that there exist a bounded truncated Toeplitz operator on .K that does not admit a bounded symbol.

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References 1. AHERN, P. R., AND CLARK, D. N. On functions orthogonal to invariant subspaces. Acta Math. 124 (1970), 191–204. 2. ALEKSANDROV, A. On embedding theorems for coinvariant subspaces of the shift operator I. In Complex analysis, operators and related topics, vol. 113 of Oper. Theory Adv. Appl. Birkhäuser, Basel, 2000, pp. 45–64. 3. ALEKSANDROV, A. B. Inner functions and related spaces of pseudocontinuable functions. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 170, Issled. Linein. Oper. Teorii Funktsii. 17 (1989), 7–33, 321. English translation in J. Soviet Math. 63 (1993), no. 2, 115–129. 4. ALEKSANDROV, A. B. On the existence of angular boundary values of pseudocontinuable functions. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222, Issled. po Linein. Oper. i Teor. Funktsii. 23 (1995), 5–17, 307. 5. ALEKSANDROV, A. B. Embedding theorems for coinvariant subspaces of the shift operator. II. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 262, Issled. po Linein. Oper. i Teor. Funkts. 27 (1999), 5–48, 231. English translation in J. Math. Sci. (New York) 110 (2002), no. 5, 2907–2929. 6. BARANOV, A., BESSONOV, R., AND KAPUSTIN, V. Symbols of truncated Toeplitz operators. J. Funct. Anal. 261, 12 (2011), 3437–3456. 7. BARANOV, A., CHALENDAR, I., FRICAIN, E., MASHREGHI, J., AND TIMOTIN, D. Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators. J. Funct. Anal. 259, 10 (2010), 2673–2701. 8. BESSONOV, R. V. Truncated Toeplitz operators of finite rank. Proc. Amer. Math. Soc. 142, 4 (2014), 1301–1313. 9. BESSONOV, R. V. Fredholmness and compactness of truncated Toeplitz and Hankel operators. Integral Equations Operator Theory 82, 4 (2015), 451–467. 10. BESSONOV, R. V. Schatten properties of Toeplitz operators on the Paley-Wiener space. Ann. Inst. Fourier (Grenoble) 68, 1 (2018), 195–215. 11. BROWN, A., AND HALMOS, P. R. Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213 (1963/64), 89–102. 12. CÂMARA, M. C., AND PARTINGTON, J. R. Spectral properties of truncated Toeplitz operators by equivalence after extension. J. Math. Anal. Appl. 433, 2 (2016), 762–784. 13. CARLESON, L. An interpolation problem for bounded analytic functions. Amer. J. Math. 80 (1958), 921–930. 14. CARLESON, L. Interpolations by bounded analytic functions and the corona problem. Ann. of Math. (2) 76 (1962), 547–559. 15. CHALENDAR, I., FRICAIN, E., AND TIMOTIN, D. On an extremal problem of Garcia and Ross. Oper. Matrices 3, 4 (2009), 541–546. 16. CHALENDAR, I., FRICAIN, E., AND TIMOTIN, D. A survey of some recent results on truncated Toeplitz operators. In Recent progress on operator theory and approximation in spaces of analytic functions, vol. 679 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2016, pp. 59–77. 17. CHALENDAR, I., AND TIMOTIN, D. Commutation relations for truncated Toeplitz operators. Oper. Matrices 8, 3 (2014), 877–888. 18. CIMA, J., AND MORTINI, R. One-component inner functions. Complex Anal. Synerg. 3, 1 (2017), Paper No. 2, 15. 19. CIMA, J., AND MORTINI, R. One-component inner functions II. In Advancements in complex analysis—from theory to practice. Springer, Cham, [2020] ©2020, pp. 39–49. 20. CIMA, J. A., GARCIA, S. R., ROSS, W. T., AND WOGEN, W. R. Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity. Indiana Univ. Math. J. 59, 2 (2010), 595–620.

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Chapter 7

Semigroups of Weighted Composition Operators on Spaces of Holomorphic Functions Isabelle Chalendar and Jonathan R. Partington

2020 Mathematics Subject Classification 30H10, 30H20, 30D05, 47B33, 47D06

7.1 Introduction This paper is based on three hours of lectures given by the first author in the “Focus Program on Analytic Function Spaces and their Applications” July 1–December 31, 2021, organized by the Fields Institute for Research in Mathematical Sciences. The goal of this paper is to give an introduction to the properties of discrete and continuous .C0 -semigroups of (weighted) composition operators on various spaces of analytic functions. To that aim we detail the structure of semiflows of analytic functions on the open unit disc .D and their generators, which provides information on the generators of continuous semigroups of composition operators on Banach spaces X embedding in .Hol(D), the Fréchet space of holomorphic functions on .D. An initial motivation for studying such semigroups is a better understanding of an specific universal operator. Moreover, adding a weight to a composition operator is motivated by the fact that such operators describe isometries on non-Hilbertian Hardy spaces and they appear automatically when the Banach spaces X are replaced by Banach spaces of holomorphic functions on another domain such as the right half-plane. Thanks to the analysis of spectral properties, we deduce the asymptotic behaviour of discrete semigroups of composition operators on various Banach spaces such as the Hardy spaces, the weighted Bergman spaces, Bloch type spaces or standard

I. Chalendar () Université Gustave Eiffel, Laboratoire d’Analyse et de Mathématiques Appliquées, Champs-sur-Marne, France e-mail: [email protected] J. R. Partington School of Mathematics, University of Leeds, Leeds, Yorkshire, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mashreghi (ed.), Lectures on Analytic Function Spaces and their Applications, Fields Institute Monographs 39, https://doi.org/10.1007/978-3-031-33572-3_7

255

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weighted Bergman space of infinite order. As a byproduct we obtain characterization of the properties of isometry and similarity to isometry, still for composition operators. We then describe the limit at infinity for continuous semigroups of composition operators. Compactness (immediate and eventual) and analyticity of semigroups of composition operators are then considered on the Hardy space .H 2 (D), even though other classes of Banach spaces may also be considered. References are given for more complete information. Finally, we provide some perspectives for semigroups of composition operators on .H 2 (C+ ), where .C+ is the right-half-plane, as well as an analysis of semigroups of composition operators on the Fock space. The latter case can be treated in a complete way since the non-trivial semiflows involved are necessarily expressed using polynomials of degree one.

7.2 Background 7.2.1 Strongly Continuous Semigroups of Operators: Definition and Characterization We recall some of the standard facts about one-parameter semigroups of operators, which may be found in many places, such as [29] and [43]. Definition 7.2.1 A semigroup .(Tt )t≥0 of operators on a Banach space X is a family of bounded operators satisfying: (i) .T0 = Id, the identity operator, and (ii) .Tt+s = Tt Ts for all .s, t ≥ 0. If, in addition, it satisfies: (iii) .Tt x → x as .t → 0+ for all .x ∈ X, then it is called a strongly continuous or .C0 -semigroup. A uniformly continuous semigroup .(Tt )t≥0 is one satisfying .Tt − Id → 0 as t → 0+ . Associated with this is the notion of an infinitesimal generator, or simply generator. We define an (in general unbounded) operator A whose domain is

.

D(A) := {x ∈ X : lim

.

t→0+

Tt x − x t

exists},

and then Ax := lim

.

t→0+

Tt x − x t

for

x ∈ D(A).

Moreover, the generator of a .C0 -semigroup characterizes completely a semigroup, that is two .C0 -semigroups are equal if and only if their generators are equal.

7 Semigroups of Weighted Composition Operators on Spaces of Holomorphic. . .

257

By the uniform boundedness principle, each .C0 -semigroup is uniformly bounded on each compact interval. As a corollary, for every .C0 -semigroup .(Tt )t0 , there exists .w ∈ R and .M  1 such that Tt   Mewt for all t  0.

.

Contractive .C0 -semigroups are the ones for which one can take .M = 1 and w = 0, whereas quasicontractive .C0 -semigroups are the ones for which one can take .M = 1 and w is arbitrary. The domain .D(A) of the generator A of a .C0 -semigroup is always dense in X (and moreover .(A, D(A)) is a closed operator). It is then natural to characterize the linear operators .(A, D(A)) that are the generator of .C0 -semigroups. For semigroups of contractions on Hilbert spaces, Lumer and Phillips, in 1961, provided a beautiful criterion [38] (see also [29, Theorem 3.15]).

.

Theorem 7.2.2 (Lumer–Phillips) Let .(A, D(A)) be a linear operator with dense domain on a Hilbert space H . The following assertions are equivalent: (i) A generates a .C0 -semigroup of contractions; (ii) there exists .λ > 0 such that .(λId − A)D(A) = H and for all .x ∈ D(A), .

ReAx, x  0;

(iii) for all .λ > 0 we have .(λId − A)D(A) = H and for all .x ∈ D(A), .

ReAx, x  0.

Since .(Tt )t0 is a .C0 -semigroup of quasicontractions on a Hilbert space H if and only if there exists .w  0 such that .(e−wt Tt )t0 is a .C0 -semigroup of contractions, it is then easy to deduce a version of the Lumer–Phillips Theorem for .C0 -quasicontractions. Theorem 7.2.3 Let .(A, D(A)) be a linear operator with dense domain on a Hilbert space H . The following assertions are equivalent: (i) A generates a .C0 -semigroup of quasicontractions; (ii) there exists .λ > 0 such that .(λId − A)D(A) = H and .

sup

ReAx, x < ∞;

x∈D(A),x1

(iii) for all .λ > 0 we have .(λId − A)D(A) = H and .

sup x∈D(A),x1

ReAx, x < ∞.

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For generators of .C0 -semigroups which are not necessarily quasicontractive, another beautiful criterion involving the growth of the resolvent is due to Hille and Yosida [29, Theorem 3.8]. Theorem 7.2.4 Let A be a linear operator defined on a linear subspace .D(A) of the Banach space X, .w ∈ R and .M > 0. Then A generates a .C0 -semigroup .(Tt )t0 that satisfies Tt   Mewt

.

if and only if (a) A is closed and .D(A) is dense in X; (b) every real .λ > w belongs to the resolvent set of A and for such .λ and for all positive integers n, (λId − A)−n  

.

M . (λ − ω)n

We shall restrict ourselves to Banach spaces X of functions that are holomorphic on a domain . (usually the unit disc .D but sometimes the right half-plane .C+ or the complex plane .C) and satisfying the condition that point evaluations .f → f (z) are continuous for all .z ∈ . Assuming this, we have that norm convergence of a sequence .(fn ) to f implies local uniform convergence (uniform convergence on compact subsets of .). Recall that for suitable .ϕ :  →  holomorphic, the composition operator .Cϕ : X → X is defined by .(Cϕ f )(z) = f (ϕ(z)), for .f ∈ X and .z ∈  (assuming that .Cϕ maps X boundedly into X, an issue which will be discussed later). Likewise, for suitable w holomorphic on ., the weighted composition operator .Ww,ϕ on X is defined by .(Ww,ϕ f )(z) = w(z)f (ϕ(z)). The main theme of this paper is to study .C0 -semigroups of (weighted) composition operators. However, we may also look at composition semigroups from a non-operatorial point of view (for example, as in [16]).

7.2.2 Analytic Semiflows on a Domain and Models for Semiflows on D Definition 7.2.5 A continuous analytic semiflow on a domain  is a family (ϕt )t≥0 of holomorphic mappings from  to itself satisfying: (i) ϕ0 (z) = z for all z ∈ , (ii) ϕt+s = ϕt ◦ ϕs for all s, t ≥ 0, and (iii) for all z ∈ , the mapping t → ϕt (z) is continuous on [0, ∞).

7 Semigroups of Weighted Composition Operators on Spaces of Holomorphic. . .

259

Remark 7.2.6 A family (ϕt )t≥0 of holomorphic mappings from  to itself satisfying only (i) and (ii) is called an algebraic semiflow. Such an algebraic semiflow (ϕt )t≥0 is continuous on D if and only if there exists a ∈ D such that limt→0 ϕt (a) = 1 (see [16, Thm. 8.1.16]). In this situation there exists a unique holomorphic function G :  → C such that .

∂ϕt (z) = G(ϕt (z)), z ∈  and t ∈ [0, ∞). ∂t

This function is called the infinitesimal generator of the analytic semiflow (ϕt )t≥0 on  and we denote by Gen() the set of all infinitesimal generator of analytic semiflows on . There are several complete characterizations of Gen(D) in [16, Chap. 10], which is discussed in more details in Sect. 7.4.2. Analytic semiflows on D can be partitioned into two classes, depending on the localization of their Denjoy–Wolff point α, discussed below in Sect. 7.4 (see [1], [24, Chap. 2] and [16, Chap. 8]). If α ∈ D, by conjugating by the automorphism bα , where bα (z) :=

.

α−z , 1 − αz

we may suppose without loss of generality that α = 0. In this case there is a semiflow model ϕt (z) = h−1 (e−ct h(z)),

.

where c ∈ C with Re c ≥ 0, and h : D →  is a conformal bijection between D and a domain  ⊂ C, with h(0) = 0 and  is spiral-like or star-like (if c is real), in the sense that e−ct w ∈  for all

.

w ∈  and

t ≥ 0.

If α ∈ T, then there exists a conformal map h from D onto a domain  such that  + it ⊂  for all t  0, and there is a semiflow model ϕt (z) = h−1 (h(z) + it).

.

7.2.3 Models for Analytic Flows on D This subsection relies heavily on Subsection 8.2 in [16].

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Definition 7.2.7 A family .(ϕt )t0 of analytic selfmaps of .D is a called a continuous (algebraic) flow if (a) .ϕt is an automorphism of .D for all .t  0; (b) .(ϕt )t0 is a continuous (algebraic) semiflow. If .ϕt is an automorphism for all .t  0, we can introduce the notation .ϕ−t := ϕt−1 for all .t  0 and then observe that (c) .ϕs+t = ϕs ◦ ϕt for all .s, t ∈ R; (d) for all .z ∈ D, the mapping .t → ϕt (z) is continuous on .R if .(ϕt )t0 is a continuous flow. In fact, (d) is equivalent to the continuity of .t → ϕt ∈ Hol(D) on .R, where Hol(D) is endowed with the topology of the uniform convergence on compacta of .D. Here is a characterization of continuous flows in the set of continuous semiflows [16, Thm. 8.2.4].

.

Theorem 7.2.8 Let .(ϕt )t0 be a continuous (algebraic) semiflow on .D. Then it is a continuous (algebraic) flow if and only if there exists .t0 > 0 such that .ϕt0 is an automorphism. The following theorem [16, Thm. 8.2.6] is an explicit description of all the continuous flows on .D. Theorem 7.2.9 Let .(ϕt )t0 be a nontrivial continuous flow on .D. Then .(ϕt )t0 has one of the following three mutually exclusive forms: (1) There exists .α ∈ D and .w ∈ R \ {0} such that ϕt (z) =

.

(e−iwt − |α|2 )z + α(1 − e−iwt ) , t  0, z ∈ D. α(e−iwt − 1)z + 1 − |α|2 e−iwt

Moreover it is the unique continuous flow of elliptic automorphisms for which ϕt (α) = α for all t and .ϕt (α) = e−iwt . (2) There exist .α1 , α2 ∈ T, .α1 = α2 and .c > 0 such that .

ϕt (z) =

.

(α2 − α1 ect )z + α1 α2 (ect − 1) , t  0, z ∈ D. (1 − ect )z + α2 ect − α1

Moreover it is the unique continuous flow of hyperbolic automorphisms for which .ϕt (αi ) = αi for all t (.i = 1, 2) and .ϕt (α1 ) = e−ct . (3) There exist .α ∈ T and .w ∈ R \ {0} such that ϕt (z) =

.

(1 − iwt)z + iwαt . −iwαtz + 1 + iwt

Moreover it is the unique continuous flow of parabolic automorphisms for which ϕt (α) = α for all t and .ϕt (α) = 2itwα.

.

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7.2.4 C0 -Semigroups of Composition Operators Clearly a semiflow .(ϕt )t≥0 induces a semigroup of composition operators (on .D these are bounded, by Littlewood’s subordination theorem [37]), and the following condition gives a way of testing the strong continuity. Proposition 7.2.10 Let E be a dense subspace of a Banach space X and .(Tt )t≥0 a semigroup of bounded operators on X such that there exists a .δ > 0 with .sup0≤t≤δ Tt  < ∞. Then .(Tt ) is a .C0 -semigroup on X if and only if .limt→0 Tt f − f X = 0 for all .f ∈ E. In particular, if the polynomials are dense in X then it is enough to check that limt→0 Tt en − en X = 0 for all .n = 0, 1, 2, . . ., where .en (z) = zn .

.

Proof Clearly, the “only if” condition holds. Conversely, if .limt→0 Tt f −f X = 0 for all .f ∈ E, let .M := sup0≤t≤δ Tt , and let . > 0 be given and .f ∈ X. We may . Then find a .p ∈ E such that .f − p < 2(M + 1) Tt f − f  ≤ Tt f − Tt p + Tt p − p + p − f 

.

≤ Mf − p + /2 + p − f  < for sufficiently small t.

 

7.2.5 Spaces on Which Semigroups Are Not C0 In Proposition 7.2.10 we have seen a sufficient condition for a semiflow to induce a .C0 -semigroup of composition operators, and in the Hardy, Dirichlet and Bergman spaces we do indeed arrive at such a semigroup. Recently, Gallardo-Gutiérrez et al. [32] have shown that weighted composition operators .(Wwt ,ϕt )t≥0 do not form a nontrivial .C0 -semigroup on spaces X satisfying ∞ ⊂ X ⊂ B, where .B is the Bloch space. This includes the case .X = BMOA. .H The proof is based on estimates for derivatives of interpolating Blaschke products. For composition operators, and with .X = H ∞ and .B, this result was shown earlier by Blasco et al. [15] with an argument involving the Dunford–Pettis property. For spaces between .H ∞ and .B, the result for composition operators was given by Anderson et al. [2] using geometric function theory.

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7.3 Motivation 7.3.1 Universal Operators Rota [45, 46] introduced the concept of a universal operator. Definition 7.3.1 An operator .U ∈ L(H ) is universal if for all nonzero .T ∈ L(H ) there is a closed subspace .M = {0} of H , an isomorphism .J : M → H and a −1 (λT )J . .λ ∈ C \ {0} such that .U M ⊂ M and .U |M = J That is, a universal operator is a “model” for all .T ∈ L(H ). Universal operators are of interest in the study of the invariant subspace problem, whether every operator on a separable infinite-dimensional Hilbert space has a nontrivial closed invariant subspace. This has a positive solution if and only every minimal invariant subspace of a given universal operator is finite-dimensional. We refer to [21] for more details and examples. Caradus [17] gives a convenient sufficient condition for a Hilbert space operator to be universal. Theorem 7.3.2 Let .U ∈ L(H ) be such that: (i) .dim ker U = ∞; and (ii) U is surjective. Then U is universal. Consider now the hyperbolic automorphisms ϕr (z) =

.

z+r 1 + rz

with .r ∈ (−1, 1). It is helpful to consider them using the parametrization ψt (z) := ϕr (z)

.

with

r=

1 − e−t 1 + e−t

for .t ∈ R. It was shown by Nordgren et al. [42] that for .r = 0 the operator .Cϕr − Id is universal on .H 2 . Of course .Cϕr and .Cϕr − Id have the same invariant subspaces. A simpler proof using the fact that .Cϕr can be embedded in a .C0 -group was given by Cowen and Gallardo-Gutiérrez [25]. Some work on the invariant subspaces of such operators is due to Matache [39], Mortini [41] and Gallardo-Gutiérrez and Gorkin [31].

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7.3.2 Isometries Another application of weighted composition operators goes back to Banach [11], who showed that every surjective isometry F of the space .C(K) of continuous complex functions on a compact metric space K has the form F (f ) = w(f ◦ ϕ),

.

where .w ∈ C(K) satisfies .|w| = 1 and .ϕ is a homeomorphism of K. The Hardy space .H 2 (D) is a Hilbert space, and thus has many (linear) isometries; however for other p with .1 < p < ∞ there are relatively few, and they are expressible as weighted composition operators. In [27] deLeeuw et al. gave a description of the isometric surjections of .H 1 (D), which arise from conformal mappings of .D onto .D. Moreover, Forelli [30] gave the following theorem, which does not assume surjectivity. Theorem 7.3.3 Suppose that .p = 2 and that .T : H p (D) → H p (D) is a linear isometry. Then there are a non-constant inner function .ϕ and a function .F ∈ H p (D) such that Tf = F (f ◦ ϕ)

.

(7.1)

for .f ∈ H p (D).

7.3.3 Change of Domain It is well known that composition operators on the Hardy space .H 2 (C+ ) of the right half-plane are unitarily equivalent to weighted composition operators on .H 2 (D). For example the following explicit formula is given in [20]. Proposition 7.3.4 Let M denote the self-inverse bijection from .D onto .C+ given 1−z by .M(z) = , and let . : C+ → C+ be holomorphic. Then the composition 1+z operator .C on .H 2 (C+ ) is unitarily equivalent to the operator .L : H 2 (D) → H 2 (D) defined by L f (z) =

.

1 + (z) f ( (z)), 1+z

where . = M ◦  ◦ M. So for example the .C0 -group .(Tt )t∈R on .H 2 (C+ ) given by .Tt g(z) = g(et z) 2 .(z ∈ C+ ) for .g ∈ H (C+ ) is unitarily equivalent to the weighted composition

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group .(St )t∈R given by St f (z) =

.

2 f 1 + z + et (1 − z)



 1 + z − et (1 − z) . 1 + z + et (1 − z)

Formulae for general domains are given, for example, in [35, Prop. 2.1]. If W is a weighted composition operator between two Hardy–Smirnoff spaces .E 2 (1 ) and .E 2 (2 ), with .1 and .2 conformally equivalent to the disc .D, then W is unitarily equivalent to a weighted composition operator on .H 2 (D). Similar formulae are given for Bergman spaces.

7.4 Asymptotic Behaviour of T n or Tt 7.4.1 The Discrete Unweighted Case For a fixed composition operator .Cϕ there are several possible modes of convergence for the sequence .(Cϕn )n≥1 , some of which we now list in progressively weaker order. • Norm convergence. There exists an operator .P ∈ L(X) such that .Cϕn −P  → 0. • Strong convergence. There exists an operator .P ∈ L(X) such that .Cϕn x − P x → 0 for all .x ∈ X. • Weak convergence. There exists an operator .P ∈ L(X) such that .Cϕn x → P x weakly for all .x ∈ X. In each case P is the projection onto .fix(Cϕ ) := {x ∈ X : Cϕ x = x} along the decomposition X = fix(Cϕ ) ⊕ Im(Id − Cϕ ).

.

The following theorem from [7] helps with the analysis. Theorem 7.4.1 Let .T ∈ L(X) with .supn T n  < ∞. Then the following are equivalent. (i) .P := lim T n exists and P is a finite-rank operator. (ii) (a) The essential spectral radius .re (T ) satisfies .re (T ) < 1; (b) .σp (T ) ∩ T ⊂ {1}; and (c) if .1 ∈ σ (T ) then 1 is a pole of the resolvent .(zId − T )−1 of order 1. In this case P is the residue at 1. We sketch the proof. Proof (i) . ⇒ (ii): Let .X1 = P X, .X2 = (Id − P )X and .Ti = TXi for .i = 1, 2.

7 Semigroups of Weighted Composition Operators on Spaces of Holomorphic. . .

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Then .T2n L(X2 ) → 0 as .n → ∞, and hence .r(T2 ) < 1. Since .σ (Y ) = σ (T1 ) ∪ σ (T2 ) = {1} ∪ σ (T2 ) and  −1

(λId − T )

.

=

1 λ−1

(λId − T2

on X1 , )−1

on X2 ,

we see that (ii) follows. (ii) . ⇒ (i): Let P be the residue at 1, and let .X1 = P X, .X2 = (Id − P )X and .Ti = TXi for .i = 1, 2. Then .σ (T1 ) = {1} and .σ (T2 ) = σ (T ) \ {1} by (a) and (b). Thus .r(T2 ) < 1 and so .T2n L(X2 ) → 0. It follows from (c) that .T1 is diagonalisable and thus .T1 = Id.   Recall that .X → Hol(D) means that X embeds continuously in .Hol(D), which means that, for all .λ ∈ D, .δλ : f → f (λ) from X to .C is bounded. Arendt and Batty [3] have given criteria for strong convergence. More recently, from [5] we mention the following theorem. Theorem 7.4.2 Let .ϕ : D → D be holomorphic, .Cϕ ∈ L(X), where X is a Banach space such that .X → Hol(D). Then .(Cϕn )n converges uniformly if and only if .re (Cϕ ) < 1. So it remains to study the essential spectral radius of .Cϕ . To this end we write ϕ (n) = ϕ ◦ . . . ϕ .    n factors

.

7.4.1.1

Denjoy–Wolff Theory

A mapping .ϕ : D → D is an elliptic automorphism if it has the form ϕ = ψa ◦ Rθ ◦ ψa ,

.

a−z = ψa−1 (z), and .Rθ (z) = eiθ z with .θ ∈ R. 1 − az Then the classical Denjoy–Wolff theorem states that provided .ϕ is not an elliptic automorphism, the sequence .(ϕ (n) )n converges uniformly to some .α ∈ D on each compact subset of .D. Such an .α is called the Denjoy–Wolff point of .ϕ and is sometimes denoted by .DW(ϕ). where .a ∈ D and .ψa (z) =

Case 1 .|α| = 1. Theorem 7.4.3 Suppose that .C[z] ⊂ X and suppose that each .L ∈ X with n .L(en ) = L(e1 ) has the form .L(f ) = δz0 (f ) := f (z0 ) for some .z0 ∈ D. Then n n .supn Cϕ  = ∞. Therefore even weak convergence of .(Cϕ ) cannot occur.

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Proof If .supn Cϕn  = M < ∞ then |f (α)| = lim |f (ϕ (n) (0))| ≤ δ0 Mf ,

.

n→∞

so .δα = δz0 for some .z0 ∈ D, which is absurd.

 

Case 2 .|α| < 1. The natural candidate for P is given by .Pf = f (α) = δα (f )1, a rank-one operator. Theorem 7.4.4 For composition operators on .H p (D), with .1 ≤ p < ∞, the sequence .(Cϕn ) converges uniformly and strongly if and only if .|α| < 1 and .ϕ is not inner. It converges weakly if and only if .|α| < 1. We proved that .re (Cϕ ) < 1 on .H 2 if and only if .ϕ is not inner and .|α| < 1. When .ϕ is inner and .|α| < 1 then .Cϕ is similar to an isometry (this is a necessary and sufficient condition [12]). We deduce the result for .H p (D) from a theorem of Shapiro [48], namely .re,H 2 (Cϕ ))2 = re,H p (Cϕ ))p . For .p = 1 there is the inequality “.>”. Hence .re,H 2 (Cϕ )) < 1 implies that .re,H p (Cϕ )) < 1. We may summarise the results obtained on various Banach spaces in a table [5, 6]. Space p (D) p .Aβ (D), .β > −1, .1 ≤ p < ∞ γ .B0 , B ∞ .Hν , .0 < p < ∞ p

Uniform not inner, .|α| < 1 .|α| < 1

Strong not inner, .|α| < 1 .|α| < 1

Weak 0 one has .|ϕt0 (ξ )| = 1 on a set of positive measure; then .Cϕt0 is not compact on .H 2 (D) and so the semigroup .(Cϕt )t≥0 is not immediately compact. Proof For the Hardy space, this follows since the weakly null sequence .(en )n≥0 with .en (z) = zn is mapped into .(ϕtn0 ), which does not converge to 0 in norm.   We shall now give a sufficient condition for immediate compactness of a semigroup of composition operators, in terms of the associated function G. First, we recall a classical necessary and sufficient condition for compactness of a composition operator .Cϕ in the case when .ϕ is univalent [24, pp. 132, 139]. Theorem 7.5.3 For .ϕ : D → D analytic and univalent, the composition operator Cϕ is compact on .H 2 (D) if and only if

.

.

lim

z→ξ

1 − |z|2 =0 1 − |ϕ(z)|2

for all .ξ ∈ T. The following proposition collects together standard results on trace-class composition operators [24, p. 149].

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Proposition 7.5.4 For .ϕ : D → D analytic with .ϕ∞ < 1, the composition operator .Cϕ is trace-class on .H 2 (D). Here is an example showing that immediate and eventual compactness are different even for semigroups of composition operators on .H 2 (D). Example 7.5.5 Let h be the Riemann map from .D onto the starlike region  := D ∪ {z ∈ C : 0 < Re(z) < 2 and 0 < Im(z) < 1},

.

with .h(0) = 0. Since .∂ is a Jordan curve, the Carathéodory theorem [44, Thm 2.6, p. 24] implies that h extends continuously to .T. Let .ϕt (z) = h−1 (e−t h(z)). Note that for .0 < t < log 2, .ϕt (T) intersects .T on a set of positive measure, and thus, .Cϕt is not compact by Proposition 7.5.2. Moreover, for .t > log 2, .ϕt ∞ < 1, and therefore .Cϕt is compact (actually traceclass). Figure 7.1 represents the image of .ϕt for different values of t.

7.5.2 Compact Analytic Semigroups A .C0 -semigroup T will be called analytic (or holomorphic) if there exists a sector θ = {reiα , r ∈ R+ , |α| < θ} with .θ ∈ (0, π2 ] and an analytic mapping .T : θ → L(X) such that .T is an extension of T and

.

.

sup T(ξ ) < ∞.

ξ ∈θ ∩D

t=0

0 < t < ln 2

e−t h(z)

ϕt (z) = h−1 (e−t h(z))

Fig. 7.1 Representations of .ϕt (D) for different values of t

t > ln 2

7 Semigroups of Weighted Composition Operators on Spaces of Holomorphic. . .

273

In both cases, the generator of T (or .T) will be the linear operator A defined by   T (t)x − x exists .D(A) = x ∈ X, lim Rt→0 t and, for all .x ∈ D(A), T (t)x − x . Rt→0 t

Ax = lim

.

In the particular case of analytic semigroups, the compactness is equivalent to the compactness of the resolvent, by Theorem 7.5.1, since the analyticity implies the uniform continuity [29, p. 109]. Remark 7.5.6 For an analytic semigroup .(T (t))t0 , being eventually compact is equivalent to being immediately compact. Indeed, consider Q, the quotient map from the bounded linear operators on a Hilbert space .L(H ) onto the Calkin algebra (the quotient of .L(H ) by the compact operators). Then .(QT (t))t0 is an analytic semigroup which vanishes for .t > 0 large enough, and therefore vanishes identically (this observation is attributed to W. Arendt). We may include that remark in the following result. Theorem 7.5.7 Let .G : D → C be a holomorphic function such that the operator A defined by .Af (z) = G(z)f  (z) with dense domain .D(A) ⊂ H 2 (D) generates an analytic semigroup .(T (t))t0 of composition operators. Then the following assertions are equivalent: (i) .(T (t))t0 is immediately compact; (ii) .(T (t))t0 is eventually   compact;   (iii) .∀ξ ∈ T, .limz∈D,z→ξ  G(z) z−ξ  = ∞. Theorem 7.5.8 Let .G : D → C be a holomorphic function such that the operator A defined by .Af (z) = G(z)f  (z) has dense domain .D(A) ⊂ H 2 (D). The operator A generates an analytic semigroup of composition operators on .H 2 (D) if and only if there exists .θ ∈ (0, π2 ) such that for all .α ∈ {−θ, 0, θ } .

lim sup Re(eiα zG(z)) ≤ 0 for all ξ ∈ T. z∈D,z→ξ

Using the semiflow model, we have the following result. Theorem 7.5.9 Let .(Cϕt )t≥0 be an immediately compact analytic semigroup on H 2 (D). Then the following conditions are equivalent:

.

1. There exists a .t0 > 0 such that .ϕt0 ∞ < 1; 2. For all .t > 0 one has .ϕt ∞ < 1.

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Therefore, if there exists a .t0 > 0 such that .ϕt0 ∞ < 1, then .(Cϕt )t≥0 is immediately trace-class. It is of interest to consider the relation between immediate compactness and analyticity for a .C0 -semigroup of composition operators: this is because compactness of a semigroup .(T (t))t≥0 is implied by compactness of the resolvent together with norm-continuity at all points .t > 0, as in Theorem 7.5.1. Example 7.5.10 Consider G(z) =

.

2z , z−1

Now the image of the unit disc under .zG(z) is contained in the left half-plane so the operator .A : f → Gf  generates a non-analytic .C0 -semigroup of composition operators .(Cϕt )t≥0 on .H 2 (D). On the other hand, it can be shown that .Cϕt is compact—even trace-class—for each .t > 0. For we have the equation ϕt (z)e−ϕt (z) = e−2t ze−z .

.

Now the function .z → ze−z is injective on .D; this follows from the argument principle, for the image of .T is easily seen to be a simple Jordan curve. It follows that .ϕt ∞ < 1 for all .t > 0, and so .Cϕt is trace-class. Example 7.5.11 The semigroup corresponding to .G(z) = (1 − z)2 is analytic but not immediately compact. For ϕt (z) =

.

(1 − t)z + t −tz + 1 + t

the Denjoy–Wolff point is 1, so the semigroup cannot be immediately compact. The analyticity follows on calculating .zG(z) for .z = eiθ . We obtain 2 .−4 sin (θ/2), which gives the result by Theorem 7.5.8.

7.6 An Outlook on C+ and C 7.6.1 The Right Halfplane C+ Unlike in the case of the disc, not all composition operators on .H 2 (C+) are bounded, and there are no compact composition operators. The following theorem was given by Elliott and Jury [28] (see also [40]). Recall that the angular derivative .ϕ  (∞) of a self-map of .C+ is defined by ϕ  (∞) = lim

.

z→∞

z ϕ(z)

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Theorem 7.6.1 Let .ϕ : C+ → C+ be holomorphic. The composition operator .Cϕ is bounded on .H 2 (C+ ) if and only √ if .ϕ has finite angular derivative .0 < λ < ∞ at infinity, in which case .Cϕ  = λ. We also have for the essential norm that 2 .Cϕ e = Cϕ  so that there are no compact composition operators on .H (C+ ). Berkson and Porta [13] gave the following criterion for an analytic function G to generate a one-parameter semiflow of analytic mappings from .C+ into itself, namely, solutions to the initial value problem .

∂ϕt (z) = G(ϕt (z)), ∂t

ϕ0 (z) = z,

namely the condition x

.

∂(Re G) ≤ Re G ∂x

on C+ ,

where as usual .x = Re z. In this case the semigroup with generator .A : f → Gf  consists of composition operators. Arvanitidis [8] showed that a necessary and sufficient condition for these composition operators to be bounded is that the non-tangential limit .δ :=

limz→∞ G(z)/z exists, in which case .Cϕ  = e−δt/2 and the semigroup is t quasicontractive. In [10] it was shown that for G holomorphic on .C+ a necessary condition for the operator .A : f → Gf  to generate a quasicontractive semigroup is .

Re G(z) > −∞, Re z

inf

z∈C+

and for a contractive semigroup this infimum is non-negative.

7.6.2 The Complex Plane C For .C we present material on the Fock space from [23]. The Fock space is arguably one of the most important spaces of entire functions, and it is possible to give a complete answer to several questions involving (weighted) composition operators. For .1 ≤ ν < ∞ the Fock space .F ν is defined to be the space of entire functions f on .C such that the norm  f ν :=

.

ν 2π

C

ν −νz2 /2

|f (z)| e

1/ν dm(z)

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is finite. The space .F2 is a Hilbert space with orthonormal basis .(e˜n )∞ n=0 defined by zn e˜n (z) = √ . n!

.

7.6.2.1

Composition Operators

Boundedness of composition operators on .F2 was characterized in [19] as follows. Theorem 7.6.2 For an entire function .ϕ the composition operator .Cϕ is bounded on .F2 if and only if either .ϕ(z) = az + b with .|a| < 1 and .b ∈ C or .ϕ(z) = az with .|a| = 1. In the case where .|a| < 1 the operator .Cϕ is compact. Thus there are relatively few bounded composition operators on .F2 and most natural questions can be answered easily. When we come to weighted composition operators there will be much more to say. The discussion here is based mostly on [23], although some results may also be found in [47], which concentrates on spectra and mean ergodicity. Considering now iterates .(Cϕn )n we have ⎧ ⎨f a n z + n .Cϕ f (z) = ⎩f (z)

1−a n 1−a b

if a = 1, if a = 1.

We now wish to use Theorem 7.4.1 again, so we need to know whether .Cϕ is powerbounded. The following formula is due to [19] in the case .ν = 2, and [26] in general. 

1 |b|2 .Cϕ  = exp 4 1 − |a|2

 if ϕ(z) = az + b

with

|a| < 1.

Theorem 7.6.3 The asymptotics of iteration of bounded composition operators on F ν are as follows.

.

(i) If .ϕ(z) = az with .|a| = 1 and .a = 1, then .(Cϕn ) consists of unitary operators and does not converge even weakly. (ii) If .ϕ(z) = az + b with .|a| < 1 then .(Cϕn ) converges in norm to the operator   b . .T : f → f 1−a The proof is short, and we include it here. Proof 1. In the first case we have that .Cϕn f (z) = f (a n z) and for .f (z) = z there is clearly no convergence.

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277

2. In the second case ϕ (n) (z) = a n z +

.

1 − an b =: an + bn z, say. 1−a

and |bn |2 |b|2 → |1 − a|2 1 − |an |2

.

so .Cϕ is power bounded. Also .re (Cϕ ) = 0 since .Cϕ is compact. For the point spectrum, suppose that .λ is an eigenvalue and f an eigenvector. Then f (az + b) = λf (z)

.

and so .f (an + bn z) = λn f (z), which implies that λn f (z) → f (b/(1 − a)).

.

(7.4)

This means that f is identically zero if .|λ| = 1 and .λ = 1. Finally if .1 ∈ σ (Cϕ ) then 1 is an eigenvalue since .Cϕ is compact. By (7.4) f is constant, and .dim ker(Cϕ − Id) = 1. Hence 1 is a pole of the resolvent of order at most 1. Indeed, assume that the pole order of 1 is larger than 1. Looking at the Jordan normal form of .T0 := Cϕ |X0 , where .X0 = P F ν and P the residue, we see that there exists .f ∈ F ν such that .(Id − Cϕ )f = 1C . Evaluating at .z = b 1−a we obtain a contradiction. Thus the pole order is 1. It follows that P is the projection onto .ker(Cϕ − Id) = C1C along .{f ∈ F ν : f (b/(1 − a)) = 0}. Thus n .Pf = f (b/(1 − a))1C . This completes the proof that .Cϕ → P is norm thanks to   Theorem 7.4.1. Similarly, we can characterise .C0 -semigroups of bounded composition operators. Theorem 7.6.4 A .C0 -semigroup .(Cϕt )t≥0 of bounded composition of .F ν satisfies one of the following conditions for .t > 0. (i) .ϕt (z) = eλt z + C(eλt − 1) for some .λ ∈ C− = {z ∈ C : Re z < 0} and .C ∈ C. (ii) .ϕt (z) = eλt z for some .λ ∈ iR. The generator is given by Af (z) = λ(z + C)f  (z),

.

where .C = 0 in case (2). Moreover .(Cϕt )t≥0 converges in norm a .t → ∞ if and only if .λ ∈ C− , in which case the limit T satisfies .Tf = f (−C).

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Using a theorem from [4] we get: Theorem 7.6.5 Every .C0 -semigroup on .F2 with generator of the form .Af = Gf  for some .G ∈ Hol(C) is a semigroup of composition operators with generator satisfying .

G(z) = az + b, with Re a < 0, G(z) = az, with a ∈ iR.

or

Moreover, the condition .

lim sup Re zG(z) ≤ 0 |z|→∞

(7.5)

is necessary and sufficient for such G.

7.6.2.2

Weighted Composition Operators

For weighted composition operators .Ww,ϕ we have the result of Le [36], extended by Hai and Khoi [34] as follows: Proposition 7.6.6 The weighted composition operator .Ww,ϕ with w not identically zero is bounded on .F ν if and only if both the following conditions hold: • .w ∈ F ν ; 2 2 • .M(w, ϕ) := supz∈C |w(z)|2 e(|ϕ(z)| −|z| ) < ∞. Moreover, in this case .ϕ(z) = az + b with .|a| ≤ 1. If .|a| = 1 then we also have w(z) = w(0)e−baz

.

(7.6)

for .z ∈ C. See also [18] for a recent discussion of these results. Note that from [34] we have the useful inequality   . M(w, ϕ) ≤ Ww,ϕ  ≤ M(w, ϕ)/|a|. Thus for power-boundedness of .Ww,ϕ , it is possible to give a complete result for |a| = 1.

.

Theorem 7.6.7 Suppose that .ϕ(z) = az + b with .|a| = 1, then .Ww,ϕ is power2 bounded on .F ν if and only if .|w(0)| ≤ e−|b| /2 . The case .|a| < 1 is apparently unsolved, although this result for non-vanishing w (as in the case of semigroups) can be found in [18].

7 Semigroups of Weighted Composition Operators on Spaces of Holomorphic. . .

279

Theorem 7.6.8 For .ϕ(z) = az + b with .|a| < 1 and w nonvanishing the operator Ww,ϕ is bounded on .F ν if and only if

.

w(z) = ep+qz+rz

2

.

and either: (i) .|r| < β/2, in which case .Ww,ϕ is compact on .F ν , or (ii) .|r| = β/2 and, with .t = q + ba, one has either .t = 0 or else .t = 0 and β t2 ν .r = − 2 |t|2 . In case (ii) .Ww,ϕ is not compact on .F . Finally, we can use this to characterise .C0 -semigroups on the Fock space [23]. Theorem 7.6.9 A .C0 -semigroup .Tt f (z) = wt (z)f (ϕt (z)) (.t ≥ 0) of weighted composition operators on the Fock space .F ν has one of the following two expressions: (i) .ϕt (z) = exp(λt)z + C(exp(λt) − 1) for some .λ ∈ C− and .C ∈ C; in which 2 case .wt = ept +qt z+rt z , where explicit formulae for .pt , .qt and .rt can be given. (ii) .ϕt (z) = exp(λt)z + C(exp(λt) − 1) for some .λ ∈ iR and .C ∈ C, in which case μt |C|2 (eλt −1) for .wt (z) = wt (0) exp(C(exp(λt) − 1)z) and moreover .wt (0) = e e some .μ ∈ C.

References 1. M. Abate. Iteration theory of holomorphic maps on taut manifolds. Commenda di Rende (Italy): Mediterranean Press, 1989. 2. A. Anderson, M. Jovovic, and W. Smith. Composition semigroups on BMOA and H ∞ . J. Math. Anal. Appl., 449(1):843–852, 2017. 3. W. Arendt and C. J. K. Batty. Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc., 306(2):837–852, 1988. 4. W. Arendt and I. Chalendar. Generators of semigroups on Banach spaces inducing holomorphic semiflows. Israel J. Math., 229(1):165–179, 2019. 5. W. Arendt, I. Chalendar, M. Kumar, and S. Srivastava. Asymptotic behaviour of the powers of composition operators on Banach spaces of holomorphic functions. Indiana Univ. Math. J., 67(4):1571–1595, 2018. 6. W. Arendt, I. Chalendar, M. Kumar, and S. Srivastava. Powers of composition operators: asymptotic behaviour on Bergman, Dirichlet and Bloch spaces. J. Aust. Math. Soc., 108(3):289–320, 2020. 7. W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck. One-parameter semigroups of positive operators, volume 1184 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. 8. A. G. Arvanitidis. Semigroups of composition operators on Hardy spaces of the half-plane. Acta Sci. Math. (Szeged), 81(1–2):293–308, 2015. 9. C. Avicou, I. Chalendar, and J. R. Partington. A class of quasicontractive semigroups acting on Hardy and Dirichlet space. J. Evol. Equ., 15(3):647–665, 2015. 10. C. Avicou, I. Chalendar, and J. R. Partington. Analyticity and compactness of semigroups of composition operators. J. Math. Anal. Appl., 437(1):545–560, 2016.

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11. S. Banach. Theory of linear operations, volume 38 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1987. Translated from the French by F. Jellett, With comments by A. Pełczy´nski and Cz. Bessaga. 12. F. Bayart. Similarity to an isometry of a composition operator. Proc. Amer. Math. Soc., 131(6):1789–1791 (electronic), 2003. 13. E. Berkson and H. Porta. Semigroups of analytic functions and composition operators. Michigan Math. J., 25(1):101–115, 1978. 14. E. Bernard. Weighted composition semigroups on Banach spaces of holomorphic functions. Arch. Math., 119(2):167–178, 2022. 15. O. Blasco, M.D. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis, and A.G. Siskakis. Semigroups of composition operators and integral operators in spaces of analytic functions. Ann. Acad. Sci. Fenn. Math., 38(1):67–89, 2013. 16. F. Bracci, M. D. Contreras, and S. Díaz-Madrigal. Continuous semigroups of holomorphic self-maps of the unit disc. Springer Monographs in Mathematics. Springer, Cham, 2020. 17. S. R. Caradus. Universal operators and invariant subspaces. Proc. Amer. Math. Soc., 23:526– 527, 1969. 18. T. Carroll and C. Gilmore. Weighted composition operators on Fock spaces and their dynamics. J. Math. Anal. Appl., 502(1):125234, 2021. 19. B. J. Carswell, B. D. MacCluer, and A. Schuster. Composition operators on the Fock space. Acta Sci. Math. (Szeged), 69(3–4):871–887, 2003. 20. I. Chalendar and J. R. Partington. On the structure of invariant subspaces for isometric composition operators on H 2 (D) and H 2 (C+ ). Arch. Math. (Basel), 81(2):193–207, 2003. 21. I. Chalendar and J. R. Partington. Modern approaches to the invariant-subspace problem, volume 188 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2011. 22. I. Chalendar and J. R. Partington. Weighted composition operators: isometries and asymptotic behaviour. J. Operator Theory, 86(1):189–201, 2021. 23. I. Chalendar and J.R. Partington. Weighted composition operators on the Fock space: iteration and semigroups. Acta Sci. Math. (Szeged), 89(1–2):93–108, 2023. 24. C. C. Cowen and B. D. MacCluer. Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. 25. C.C. Cowen and E.A. Gallardo-Gutiérrez. A new proof of a Nordgren, Rosenthal and Wintrobe theorem on universal operators. In Problems and recent methods in operator theory, volume 687 of Contemp. Math., pages 97–102. Amer. Math. Soc., Providence, RI, 2017. 26. J. Dai. The norm of composition operators on the Fock space. Complex Var. Elliptic Equ., 64(9):1608–1616, 2019. 27. K. de Leeuw, W. Rudin, and J. Wermer. The isometries of some function spaces. Proc. Amer. Math. Soc., 11:694–698, 1960. 28. S. Elliott and M. T. Jury. Composition operators on Hardy spaces of a half-plane. Bull. Lond. Math. Soc., 44(3):489–495, 2012. 29. K.-J. Engel and R. Nagel. A short course on operator semigroups. Universitext. Springer, New York, 2006. 30. F. Forelli. The isometries of H p . Canad. J. Math., 16:721–728, 1964. 31. E.A. Gallardo-Gutiérrez and P. Gorkin. Minimal invariant subspaces for composition operators. J. Math. Pures Appl. (9), 95(3):245–259, 2011. 32. E.A. Gallardo-Gutiérrez, A.G. Siskakis, and D.V. Yakubovich. Generators of C0 -semigroups of weighted composition operators. Israel J. Math., 255(1):63–80, 2023. 33. E.A. Gallardo-Gutiérrez and D.V. Yakubovich. On generators of C0 -semigroups of composition operators. Israel J. Math., 229(1):487–500, 2019. 34. P. V. Hai and L. H. Khoi. Boundedness and compactness of weighted composition operators on Fock spaces Fp (C). Acta Math. Vietnam., 41(3):531–537, 2016. 35. R. Kumar and J. R. Partington. Weighted composition operators on Hardy and Bergman spaces. In Recent advances in operator theory, operator algebras, and their applications, volume 153 of Oper. Theory Adv. Appl., pages 157–167. Birkhäuser, Basel, 2005.

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Chapter 8

The Corona Problem Alexander Brudnyi

2020 Mathematics Subject Classification 0D55. 30H05

8.1 Introduction Let A be a unital commutative complex Banach algebra. The maximal ideal space of A denoted by .M(A) is the set of nonzero homomorphisms .A → C. Every element of .M(A) is a continuous linear functional on A of norm .≤ 1, i.e., .M(A) is a subset of the closed unit ball of the dual space .A∗ of A. It is a compact Hausdorff space in the weak.∗ topology of .A∗ . Let .C(M(A)) be the Banach space of complex continuous functions on .M(A) equipped with supremum norm. The Gelfand transform . ˆ : A → C(M(A)), .a(ξ ˆ ) := ξ(a), is a nonincreasing-norm morphism of Banach algebras. The kernel of the Gelfand transform is the Jacobson radical of A, denoted .J (A). An algebra A is semisimple if .J (A) = {0}. Let A be a closed unital subalgebra of the algebra .Cb (X) of bounded complex functions on a Hausdorff space X equipped with supremum norm separating points of X. Then A is semisimple and the Gelfand transform is an isometric embedding. Moreover, there is a continuous embedding .ι : X → M(A) taking .x ∈ X to the evaluation homomorphism .f → f (x), .f ∈ A. The complement of the closure of .ι(X) in .M(A) is called the corona. The corona problem is: Given X and .A ⊂ Cb (X) to determine whether the corresponding corona is empty.

Research is supported in part by NSERC. A. Brudnyi () University of Calgary, Calgary, AB, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mashreghi (ed.), Lectures on Analytic Function Spaces and their Applications, Fields Institute Monographs 39, https://doi.org/10.1007/978-3-031-33572-3_8

283

284

A. Brudnyi

This problem can be equivalently reformulated as follows, see, e.g., [Ga]: A collection .f1 , . . . , fn of functions in A satisfies the corona condition if 1 ≥ max |fj (x)|  δ > 0 for all x ∈ X.

.

1j n

(8.1)

The corona problem being solvable (i.e., the corona is empty) means that for all n ∈ N and .f1 , . . . , fn satisfying the corona condition, the Bezout equation

.

f1 g1 + · · · + fn gn = 1

.

(8.2)

has a solution .g1 , . . . , gn ∈ A. For .A = H ∞ (:= H ∞ (D)), the algebra of bounded holomorphic functions on the open unit disk .D ⊂ C, the corona problem was posed by S. Kakutani in 1941. The complement of the closure of .ι(D) in .M(H ∞ ) was called the corona by D. Newman [34] (as in this case there would have been a set of maximal ideals suggestive of the sun’s corona if the complement failed to be empty). Newman showed that the corona problem in the disk is equivalent to a certain interpolation problem [10, p. 548]. The latter was solved by L. Carleson [10] in 1962. The proof of the Carleson corona theorem was subsequently simplified by L. Hörmander [24] who used the Koszul ¯ complex technique to reduce it to a .∂-problem on .D. Along these lines the simplest proof of the corona theorem was obtained by T. Wolff, see, e.g., [19]. Following the appearance of Carleson’s proof, a number of authors have proved the corona theorem for finite bordered Riemann surfaces, e.g., [1, 2, 15, 16, 22, 37, 39–41]. The corona problem was first solved for a class of infinitely connected domains (the, so-called, “roadrunner” domains) by M. Behrens [4], [5] (his results were further extended in [11, 12, 32]), and for a class of finitely sheeted covering Riemann surfaces by M. Nakai [31]. Also, it was shown that there are non-planar Riemann surfaces for which the corona is non-trivial (see, e.g., [3, 17, 18, 26, 27] and references therein). This is due to a structure that in a sense makes the surface seem higher dimensional. So there is a hope that the restriction to the Riemann sphere might prevent this obstacle. However, the general problem for planar domains is still open as is the problem in several variables for the ball and polydisk. (In fact, there are no known examples of domains in .Cn , .n  2, without corona.) In this direction, Gamelin [18] has shown that the corona problem for planar domains is local in the sense that it depends only on the behavior of the domain locally about each boundary point. At present, one of the strongest corona theorems for planar domains is due to Moore [M]. It states that the corona is empty for any domain with boundary contained in the graph of a .C 1+ function. This result is the extension of the earlier result of Jones and Garnett [GJ] for a Denjoy domain (i.e., a domain with boundary contained in .R). Among other results, it is worth mentioning recent results of Handy [21] establishing the corona theorem for complements of certain square Cantor sets and of NewDelman [33] who proves the corona theorem for the complement of a subset of a Lipschitz graph of homogeneous type.

8 The Corona Problem

285

In this chapter, we present the solution of the corona problem for a class of bounded Banach-valued holomorphic functions on .D and formulate some results related to the operator corona problem.

¯ 8.2 Banach-Valued ∂-Equations on the Disk 8.2.1 Interpolating Sequences for H ∞ Recall that the pseudohyperbolic metric on .D is defined by    z−w  ,  .ρ(z, w) :=  1 − wz ¯ 

z, w ∈ D.

A sequence .{zn } ⊂ D is said to be interpolating for .H ∞ if every interpolation problem f (zn ) = an ,

.

n ≥ 1,

(8.3)

with a bounded sequence .{an } ⊂ C has a solution .f ∈ H ∞ . By the Banach open mapping theorem, there is a constant M such that problem (8.3) has a solution .f ∈ H ∞ satisfying

f H ∞ ≤ M sup |an |.

.

n∈N

The smallest possible M is said to be the constant of interpolation of .{zn }. Clearly, every finite subset of .D is an interpolating sequence for .H ∞ . In general, by the Carleson theorem, see, e.g., [19, Ch. VII, Thm. 1.1], .ζ = {zn } ⊂ D is interpolating for .H ∞ if and only if for some .δ > 0 δ(ζ ) := inf



.

k

ρ(zj , zk ) ≥ δ.

(8.4)

j, j =k

Moreover, the constant of interpolation .Mζ of .ζ satisfies (here .δ(ζ ) = 1 if .ζ consists of one element.) ⎧  2 ⎫ √ ⎨ ⎬ 2 1+ 1−δ e 1 2e ≤ Mζ ≤ min log 2 , . (8.5) . ⎩δ ⎭ δ δ δ (The first term in the braces is an upper bound of the P. Jones linear interpolation operator [25, Thm. 6] obtained in [51] by elementary arguments, and the second one is due to Earl [14, Thm. 2].)

286

A. Brudnyi

If .ζ = {zn } ⊂ D is interpolating for .H ∞ , then the Blaschke product .Bζ having simple zeros at points of .ζ , Bζ (z) := .



μ(zn )

n

z − zn , 1 − z¯n z

z μ(z) := − |z|

z ∈ D, (8.6)

if z = 0

and

μ(0) := 1,

is said to be interpolating. Remark 8.2.1 Note that  .

ρ(zj , zk ) = (1 − |zk |2 )Bζ (zk ).

j, j =k

8.2.2 Main Result A subset S of a metric space .(M, d) is said to be .-separated if .d(x, y) ≥  for all x, y ∈ S, .x = y. A maximal .-separated subset of M is said to be an .-chain. Thus, if .S ⊂ M is an .-chain, then S is .-separated and for every .z ∈ M \ S there is .x ∈ S such that .d(z, x) < . Existence of .-chains follows from the Zorn lemma. A closed subset with a nonempty interior .K ⊂ D such that an .ε-chain of K, . ∈ (0, 1), with respect to the pseudohyperbolic metric .ρ is an interpolating sequence for .H ∞ is said to be a quasi-interpolating set.1 Let X be a complex Banach space. By .Cρ∞ (K, X) we denote the set of X-valued ∞ .C functions f on .D with support in K uniformly continuous with respect to the pseudohyperbolic metric .ρ. We equip .Cρ∞ (K, X) with the norm .

f := sup f (z) X .

.

z∈D

Let us formulate the main result used in the proof of the corona theorem. Theorem 8.2.2 Given a quasi-interpolating set K, there exists a continuous linear ∞ ∞ operator .LX K : Cρ (K, X) → Cρ (D, X) with norm bounded by a constant depending on K only such that for every .f ∈ Cρ∞ (K, X), .

1 In

∂ X 1 (LK f ) = f. ∂ z¯ 1 − |z|2

this case every chain of K with respect to .ρ is an interpolating sequence for .H ∞ .

8 The Corona Problem

287

Moreover, if .f ∈ Cρ∞ (K, X) is such that .f (K)  X,2 then .(LX K f )(D)  X. Remark 8.2.3 In fact, if .{zn } is a . 12 -chain in K such that .

inf k



ρ(zj , zk ) ≥ δ > 0,

j, j =k

then one can show that . LX K is bounded from above by a constant depending on .δ only.

8.3 Proof of Theorem 8.2.2 8.3.1 Particular Case We only outline the main ideas of the proof which goes along the lines of the proof of [7, Thm. 3.5]. We use the following auxiliary result. Lemma 8.3.1 ([19, Ch. X, Lm. 1.4]) Let B be an interpolating Blaschke product with zeros .{zn } such that .

inf (1 − |zn |2 )|B (zn )| ≥ δ > 0. n

There exist .λ = λ(δ), .0 < λ < 1, and .r = r(δ), .0 < r < 1, .

lim λ(δ) = 1 and

δ→1

lim r(δ) = 1,

δ→1

such that the set .B −1 (Dr ) = {z ∈ C : |B(z)| < r}, where .Dr := {w ∈ C : |w| < r}, is the union of pairwise disjoint domains .Vn with .zn ∈ Vn satisfying Vn ⊂ {z : ρ(z, zn ) < λ}.

.

Moreover, B maps every .Vn biholomorpically onto .Dr and for .|w| < r the function Bw (z) =

.

B(z) − w , 1 − wB(z) ¯

z ∈ D,

is the product of an interpolating Blaschke product with one zero in each .Vn and a constant of modulus 1.

2 I.e.,

the closure of .f (K) is a compact subset of X.

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A. Brudnyi

By .bn : Dr → Vn we denote the holomorphic map inverse to .B|Vn . Let N be the set of indices of .{zn }. Proposition 8.3.2 ([7, Prop. 3.7]) There exist a positive .ε ≤ r and functions .fj ∈ H ∞ (D × Dε ) such that fj (bj (w), w) = 1,

.

.

fj (bk (w), w) = 0,

|fj (z, w)| ≤ 2M,

k = j,

(8.7)

(z, w) ∈ D × Dε ,

(8.8)

j

where M :=

sup

.

{aj } ∞ (N) ≤1

inf{ f H ∞ ; f ∈ H ∞ , f (zj ) = aj , j ∈ N }

is the constant of interpolation for .{zj }. Proof Due to [19, Ch. VII, Th. 2.1] there exist functions .gj ∈ H ∞ such that gj (zj ) = 1,

.

gj (zk ) = 0,

k = j,

and



|gj (z)| ≤ M,

z ∈ D.

j

Consider a bounded linear operator .L : ∞ (N ) → H ∞ of norm . L = M defined by the formula L(a)(z) :=



.

aj gj (z),

a = {aj }j ∈N ,

z ∈ D.

j ∈N

Let .R(w) : H ∞ → ∞ (N ) be the restriction operator to .b(w) = {bj (w)}j ∈N , .w ∈ Dr . Then ⎧ ⎫ ⎨ ⎬ .(R(w) ◦ L)(a) := aj gj (bk (w)) . ⎩ ⎭ j ∈N

k∈N

This implies that .P (w) := R(w) ◦ L : ∞ (N ) → ∞ (N ), .w ∈ Dε , is a family of bounded operators of norms .≤ M holomorphically depending on w and such that .P (0) = id. The Cauchy estimates for derivatives of bounded holomorphic functions r M yield . dP dw (w) ≤ r−|w| . In particular, for .|w| ≤ ε := 3M we have      P (w) − 1 :=  P (w) − P (0)  ≤ |w|

.

M 1 ≤ . r − |w| 2

Hence, the operator .P (w) is invertible and . P (w)−1 ≤ 2.

8 The Corona Problem

289

We set ˆ L(w) := L ◦ P (w)−1 ,

.

w ∈ Dε .

ˆ Then the linear operator .L(w) : ∞ (N ) → H ∞ is continuous, holomorphically ˆ ˆ depends on .w ∈ Dε and . L(w) ≤ 2M. Moreover, .R(w) ◦ L(w) = id. We define ˆ fj (·, w) := L(w)(δ j ),

.

j ∈ N;

(8.9)

here .δj = {δij }i∈N ∈ ∞ (N ), .δij = 1 if .i = j and 0 otherwise. Clearly, .{fj }j ∈N satisfies the required properties.   For a complex Banach space X and the set .D×Ds we denote by .Cbh, u (D×Ds , X) the Banach space of bounded X-valued continuous functions on .D × Ds holomorphic with respect to the first coordinate and uniformly continuous with respect to the second one equipped with the norm

f :=

.

f (z, w) X .

sup (z,w)∈D×Ds

In addition, we denote by .Cb (Ds ×N, X) the space of bounded X-valued continuous functions on .Ds × N. Let . B := {(z, w) ∈ D2 : w = B(z)} be the graph of B in .D2 = D × D, and let .R : f → f | B be the restriction operator. Proposition 8.3.3 For .ε as in Proposition 8.3.2, there exists a bounded linear operator .S : Cb (D 2ε × N, X) → Cbh, u (D × D 2ε , X) of norm .≤ 2M such that (R ◦S)(g)(bj (w), w) = g(w, j )

.

for all (w, j ) ∈ D 2ε ×N, g ∈ Cb (D 2ε ×N, X).

Proof We define S(g)(z, w) :=



.

fj (z, w)g(w, j )

(8.10)

j

with .fj as in Proposition 8.3.2. Then the required result follows from (8.7) and (8.8).   We retain notation of Propositions 8.3.2 and 8.3.3. Denote by .Eε (X) the vector space of X-valued .C ∞ (0, 1) forms .ω = f d z¯ on −1 (D ε ) such that functions .f˜(w, j ) := f (b (w)) d b¯j (w) , .D with supports in .B j d w¯ 2 .(w, j ) ∈ D ε × N, are in .Cb (D ε × N, X). 2 2

290

A. Brudnyi

Proposition 8.3.4 There exists a linear operator .G : Eε (X) → Cρ∞ (D, X) such that for .ω = f d z¯ ∈ Eε (X) (1) ¯ ∂G(ω) = ω,

.

(2) .

sup G(ω)(z) X ≤ 2M z∈D

sup

(w,j )∈D ε ×N

f˜(w, j ) X ,

2

(3) {(G(ω) ◦ bj )(w) : (w, j ) ∈ D 2ε × N} ∈ Cb1 (D 2ε × N, X);

.

(4) .G(ω)|D\B −1 (D ε ) is the uniform limit of a sequence of X-valued functions of the 2

form . li=0 hi B −i with .hi ∈ H ∞ (D, X). Hereafter .H ∞ (U, X) denotes the Banach space of bounded X-valued holomorphic functions F on an open subset .U ⊂ D with norm . F := supz∈U F (z) X . Proof We represent .ω as a .(0, 1) form on .D × N with values in X replacing z by bj (w) in each .Vj . Then, by the assumptions of the proposition, we obtain the form ¯ see (8.10). .f˜d w ¯ with .f˜ ∈ Cb (D 2ε × N, X). Let us consider the form .ω˜ := S(f˜)d w, Since .supp ω˜ ⊂ D 2ε , it can be regarded as a continuous .(0, 1) form on .D with values .

in .H ∞ (D, X). We define a linear operator .I : Cbh, u (D2 , X) → Cbh, u (D2 , X) by the formula   g(z, ξ ) 1 dξ ∧ d ξ¯ . .I (g)(z, w) = 2π i D ξ −w

Remark 8.3.5 Passing in the above integral to the polar coordinates .ξ = reiϕ we get 1 .I (g)(z, w) = 2π

 

g(z, reiϕ + w)e−iϕ dr ∧ dϕ. w+D

From this formula one easily deduce that if g has a relatively compact image, then I (g) has a relatively compact image as well.

.

Now, see, e.g., [19, Ch. VIII.1], (in the distributional sense) .

∂I (g) =g ∂ w¯

and

sup (z,w)∈D2

I (g)(z, w) X ≤

sup (z,w)∈D2

g(z, w) X .

8 The Corona Problem

291

Finally, we set G(ω)(z) := I (S(f˜))(z, B(z)),

.

z ∈ D.

Since .I (S(f˜)) depends holomorphically on the first coordinate, ¯ ¯ ∂I (S(f˜))(z, w)  d B(z) ∂G(ω) d B(z) := = S(f˜)(z, B(z)) w=B(z) · ∂ z¯ ∂ w¯ d z¯ d z¯ ¯ . d b¯j (w)  d B(z) = f (z) on each Vj =(f ◦ bj )(w) w=B(z) · d z¯ d w¯ (in the distributional sense). (Here, we have used that .bj is the map inverse to B on .Vj .) In particular, since f is .C ∞ by the hypothesis, the latter implies that .G(ω) is .C ∞ as well. Next, properties (2) and (3) follow from the corresponding properties of the operator I . To prove (4) note that .I (S(f˜))(z, w), .z ∈ D, .w ∈ C \ D 2ε , can be regarded as a continuous up to the boundary bounded holomorphic function in w with values in .H ∞ (D, X). Applying the Cauchy integral formula to this function (integrating over the boundary .{w ∈ C : |w| = 2ε }) we approximate it uniformly on

l −i ε .C\Dε/2 by a sequence of functions of the form . i=0 hi (z)w , .(z, w) ∈ D×C\D 2 , ∞ (D, X), .l ∈ N. Replacing w by .B(z) we get the required result. .hi ∈ H   Note that since .B(zj ) = 0 and . B H ∞ = 1, by the Schwarz lemma the open pseudohyperbolic ball .B(zj , s) centered at .zj of radius .s ≤ r is a subset of .bj (Ds ) ⊂ Vj (⊂ B(zj , λ)). Let Z be the closure of the set .j ∈ N B(zj , 2ε ). Proposition 8.3.6 Let .f ∈ Cρ∞ (Z, X). Then the form ˜ j ) := element of .Eε . In addition, for .g(w, sup

.

(ω,j )∈D ε × N r 3M

≤

rδ 3

f (z) d z¯ 1−|z|2

d b¯ (w) g(bj (w)) djw¯ , .(w, j )

=: gd z¯ is an

∈ D 2ε × N,

g(w, ˜ j ) X ≤ c(δ) sup f (z) X .

2

Recall that .ε :=

.

z∈D

(see (8.5)).

z+zj 1+z¯j z , .z ∈ D, be the Möbius transformation of .D sending 0 gj−1 (Vj ) ⊂ Dλ , .s ≤ r. We set .hj = gj−1 ◦ bj : Dr → Dλ and

Proof Let .gj (z) :=

to .zj . Then .Ds ⊂ fj := f ◦ gj |Dλ . Then .supp fj ⊂ D 2ε .

.

292

A. Brudnyi

We have g(w, ˜ j ) :=

d g¯ j d h¯ j (w) f (bj (w)) d b¯j (w) fj (hj (w)) · = (hj (w)) · 2 2 d w¯ d z¯ d w¯ 1 − |bj (w)| 1 − |gj (hj (w))|

.

=fj (hj (w)) ·

d h¯ j (w) 1 . · 2 d w¯ 1 + zj hj (w) 1 − |hj (w)| 1 + z¯ j hj (w)

·

(8.11) This and the Cauchy estimates for the derivative of .hj imply that sup

.

(ω,j )∈D ε ×N

g(w, ˜ j ) X ≤

2

1 1−

ε2 4

4λ sup f (z) X . ε z∈D

(8.12)

Also, since the functions hj (w)

.

and

d h¯ j (w) 1 , · d w¯ 1 + zj hj (w) 1 − |hj (w)|2 1 + z¯ j hj (w)

·

(w, j ) ∈ D 2ε × N,

belong to .Cb (D 2ε × N), in order to compete the proof of the proposition it suffices to check that the (bounded) function .fj (z), .(z, j ) ∈ D 2ε × N, belongs to .Cb (D 2ε × N, X). This is clearly true as f is uniformly continuous with respect to .ρ by the definition of the class .Cρ∞ (Z, X). This completes the proof of the proposition.  

8.3.2 General Case The proof of Theorem 8.2.2 can be reduced to the case considered in the previous section. Specifically, there are a finite cover of the quasi-interpolating set K by quasi-interpolating sets .K1 , . . . , Kl , interpolating Blaschke products .B1 , . . . , Bl and functions .fi ∈ Cρ∞ (Ki , X) such that (i) f = f1 + · · · + fl ;

.

fi (z) (ii) Every .(0, 1) form . 1−|z| 2 dz, .z ∈ D, belongs to the space .Eεi (X), where .εi is the .ε of the previous section for .B := Bi ; (iii)

fi ≤ f

.

for all

i.

8 The Corona Problem

293

The decomposition in (i) is defined by a suitable .Cρ∞ partition of unity subordinate to the cover .∪i Ki of K, see [7] for details. Let .Gi : Eεi (X) → Cρ∞ (D, X) be the linear operator of Proposition 8.3.4 for .B := Bi , then the required operator .LX K is given by LX Kf =

l

.

Gi fi ,

f ∈ Cρ∞ (K, X).

i=1

We leave the details to the readers.

8.4 Carleson’s Corona Theorem We use the following result. Theorem 8.4.1 Suppose .f ∈ H ∞ with . f H ∞ ≤ 1. Given .δ > 0 there are an 1 .ε = ε(δ) ∈ (0, δ), a quasi-interpolating set .Kδ ⊂ D having a . -chain .{zn } ⊂ ε 2 such that  . inf ρ(zj , zk ) ≥ γ (δ) > 0, k

j, j =k

and a function .ψ ∈ C ∞ (D), .0 ≤ ψ ≤ 1, satisfying .D \ Kδ ⊂ ψ −1 ({0, 1}) and (i) {z ∈ D : |f (z)| ≥ δ} ⊂ ψ −1 (1) ∩ (D \ Kδ );

.

(ii) {z ∈ D : |f (z)| ≤ ε} ⊂ ψ −1 (0) ∩ (D \ Kδ );

.

(iii) .dψ(z) =

g(z) dz 1−|z|2

+

h(z) d z¯ , 1−|z|2

where .g, h ∈ Cρ∞ (Kδ ), .max{ g , h } ≤ c(δ).

(The result is due to Carleson [10], see also Ziskind [52] and [19, Ch. VIII.5] for details.) The proof of the Carleson corona theorem is based on Theorem 8.4.1 and some algebraic construction known as the Koszul complex (see [19, Ch. VIII]). Specifically, suppose .f1 , . . . , fn ∈ H ∞ satisfy the corona condition 1 ≥ max |fj (z)|  δ > 0 for all z ∈ D.

.

1j n

(8.13)

294

A. Brudnyi

For such .δ and each .fj , Theorem 8.4.1 gives a function .ψj ∈ C ∞ (D) and the corresponding quasi-interpolating set .Kδ,j . We set ϕj :=

.

Then the function .|ϕj | ≤

1 ε

fj

ψj

n

i=1 ψi

.

and n .

ϕj fj = 1.

j =1

Let us consider equations .

∂bj k ∂ϕk = ϕj , ∂ z¯ ∂ z¯

1 ≤ j, k ≤ n.

(8.14)

h (z)

jk ∞ Here the right-hand side is of the form . 1−|z| 2 , .z ∈ D, where .hj k ∈ Cρ (Kδ ), .Kδ := n ∪j =1 Kδ,j , and . hj k ≤ C(δ). Since .Kδ is a quasi-interpolating set as well having a 1 . -chain such that 2

.

inf k



ρ(zj , zk ) ≥ γ (δ, n) > 0,

j, j =k

Theorem 8.2.2 (see also Remark 8.2.3) guarantees existence of a solution .bj k of (8.14) such that . bj k := supD |bj k | ≤ C(δ, n). We define gj = ϕj +

.

n (bj k − bkj )fk ,

1 ≤ j ≤ n.

k=1

Then .gj ∈ H ∞ , . gj H ∞ ≤

1 ε

 n), and + 2nC(δ, n) =: C(δ, n .

j =1

as required.

fj gj = 1,

8 The Corona Problem

295

8.5 Structure of the Maximal Ideal Space of H ∞ 8.5.1 Gleason Parts For .x, y ∈ M(H ∞ ) the formula ρ(x, y) := sup{|fˆ(y)| ; f ∈ H ∞ , fˆ(x) = 0, f H ∞ ≤ 1}

.

gives an extension of the pseudohyperbolic metric .ρ on .D to .M(H ∞ ). The Gleason part of .x ∈ M(H ∞ ) is then defined by .π(x) := {y ∈ M(H ∞ ) ; ρ(x, y) < 1}. For ∞ ) we have .π(x) = π(y) or .π(x)∩π(y) = ∅. Hoffman’s classification .x, y ∈ M(H of Gleason parts [20] shows that there are only two cases: either .π(x) = {x} or .π(x) is an analytic disk. The former case means that there is a continuous one-to-one and onto map .Lx : D → π(x) such that .fˆ ◦ Lx ∈ H ∞ for every .f ∈ H ∞ . Moreover, any analytic disk is contained in a Gleason part and any maximal (i.e., not contained in any other) analytic disk is a Gleason part. By .Ma and .Ms we denote the sets of all non-trivial (analytic disks) and trivial (one-pointed) Gleason parts, respectively. It is known that .Ma ⊂ M(H ∞ ) is open. Hoffman proved that .π(x) ⊂ Ma if and only if x belongs to the closure of some interpolating sequence in .D.

8.5.2 Structure of Ma In [6] .Ma is described as a fibre bundle over a fixed compact Riemann surface S of genus .≥ 2. (This description works with any Riemann surface of genus .≥ 2 since an orbit of the action of the fundamental group of the surface on .D by deck transformations is an .ε-chain, .ε ∈ (0, 1), in .D with respect to the pseudohyperbolic metric. In particular, .Ma and .Ms can be defined as Gleason parts of limit points of any such chain.) Specifically, let G be the fundamental group of S. Let . ∞ (G) be the Banach algebra of bounded complex-valued functions on G with pointwise multiplication ˇ and supremum norm. By .βG we denote the Stone-Cech compactification of G, i.e., ∞ the maximal ideal space of . (G) equipped with the Gelfand topology. The universal covering .r : D → S is a principal fibre bundle with fibre G. Namely, there exists a finite open cover .U = (Ui )i∈I of S by sets biholomorphic to .D and a locally constant cocycle .g¯ = {gij } ∈ Z 1 (U; G) such that .D is biholomorphic to the quotient space of the disjoint union .V = i∈I Ui × G by the equivalence relation Ui × G  (x, g) ∼ (x, ggij ) ∈ Uj × G.

.

The identification space is a fibre bundle with projection .r : D → S induced by projections .Ui × G → Ui , see, e.g., [23, Ch. 1].

296

A. Brudnyi

Next, the right action of G on itself by multiplications is extended to the right continuous action of G on .βG. Let .r˜ : E(S, βG) → S be the associated with this action bundle on S with fibre .βG constructed by cocycle .g. ¯ Then .E(S, βG) is a compact Hausdorff space homeomorphic to the quotient space of the disjoint union  = i∈I Ui × βG by the equivalence relation .V Ui × βG  (x, ξ ) ∼ (x, ξgij ) ∈ Uj × βG.

.

The projection .r˜ : E(S, βG) → S is induced by projections .Ui × βG → Ui .  induced by the embedding .G → Note that there is a natural embedding .V → V βG. This embedding commutes with the corresponding equivalence relations and so determines an embedding of .D into .E(S, βG) as an open dense subset.  induced Similarly, for each .ξ ∈ βG there exists a continuous injection .V → V by the injection .G → βG, .g → ξg, commuting with the corresponding equivalence relations. Thus it determines a continuous injective map .iξ : D → E(S, βG). Let .XG := βG/G be the set of co-sets with respect to the right action of G on .βG. Then .iξ1 (D) = iξ2 (D) if and only if .ξ1 and .ξ2 determine the same element of .XG . If .ξ represents an element .x ∈ XG , then we write .ix (D) instead of .iξ (D). In particular, .E(S, βG) = x∈XG ix (D). Let .U ⊂ E(S, βG) be open. We say that a function .f ∈ C(U ) is holomorphic if .f |U ∩D is holomorphic in the usual sense. The set of holomorphic on U functions is denoted by .O(U ). It was shown in [6, Th. 2.1] that each .h ∈ H ∞ (U ∩ D) is extended to a unique holomorphic function .hˆ ∈ O(U ). In particular, the restriction map .O(E(S, βG)) → H ∞ is an isometry of Banach algebras. Thus the quotient space of .E(S, βG) (equipped with the factor topology) by the equivalence relation x ∼ y ⇔ f (x) = f (y)

.

for all f ∈ O(E(S, βG))

is homeomorphic to .M(H ∞ ). By q we denote the quotient map .E(S, βG) → M(H ∞ ). A sequence .{gn } ⊂ G is said to be interpolating if .{gn (0)} ⊂ D is interpolating for .H ∞ (here G acts on .D by Möbius transformations). Let .Gin ⊂ βG be the union of closures of all interpolating sequences in G. It was shown that .Gin is an open dense subset of .βG invariant with respect to the right action of G. The associated with this action bundle .E(S, Gin ) on S with fibre .Gin constructed by the cocycle .g¯ ∈ Z 1 (U; G) is an open dense subbundle of .E(S, βG) containing .D. It was established in [6] that q maps  .E(S, Gin ) homeomorphically onto .Ma so that for each .ξ ∈ Gin the set .q iξ (D) coincides with the Gleason part   .π q(iξ (0)) . Also, for distinct .x, y ∈ E(S, βG) with .x ∈ E(S, Gin ) there exists .f ∈ O(E(S, βG)) such that .f (x) = f (y). Thus .q(x) = x for all .x ∈ E(S, Gin ), i.e., .E(S, Gin ) = Ma . It is worth noting that every bounded uniformly continuous (Lipschitz) with respect to the metric .ρ function f on .D admits a continuous extension .fˆ to

8 The Corona Problem

297

E(S, βG) (and, in particular, to .Ma ) so that for every map .iξ : D → E(S, βG) the function .fˆ ◦ iξ is uniformly continuous (Lipschitz) with respect to .ρ on .D. From the definition of .E(S, βG) that  follows  for a simply connected open subset .U ⊂ S restriction .E(S, βG)|U := r˜ −1 (U ) is a trivial bundle, i.e., there exists an isomorphism of bundles (with fibre .βG) .

ϕ : E(S, βG)|U → U × βG,

.

ϕ(x) := (˜r (x), ϕ(x)), ˜

x ∈ E(S, βG)|U ,

mapping .r˜ −1 (U ) ∩ D biholomorphically onto .U × G. A subset .W ⊂ r˜ −1 (U ) of the form .RU,H := ϕ −1 (U × H ), .H ⊂ βG, is called rectangular. The base of topology on .E(S, Gin ) (:= Ma ) consists of rectangular sets .RU,H with .U ⊂ S biholomorphic to .D and .H ⊂ Gin being the closure of an interpolating sequence in G (so H is a clopen subset of .βG). Another base of topology on .Ma ˆ is given by sets of the form .{x ∈ Ma ; |B(x)| < ε}, where B is an interpolating Blaschke product. This follows from the fact that for a sufficiently small .ε the set −1 (D ) ⊂ D, .D := {z ∈ D : |z| < ε}, is biholomorphic to .D × B −1 (0), see [19, .B ε ε ε ˆ Ch. X, Lm. 1.4]. Hence, .{x ∈ Ma ; |B(x)| < ε} is biholomorphic to .Dε × Bˆ −1 (0).

8.5.3 Structure of Ms It was proved by Suárez [42] that the set .Ms of trivial Gleason parts is totally disconnected, i.e., .dim Ms = 0 (because .Ms is compact). Since this important property is used in the sequel, we present an alternative short proof of the Suárez result given in [8]. A topological space X is totally disconnected if any subset of X containing more than two points is disconnected. If X is a compact Hausdorff space, then it is totally disconnected if and only if .dim X = 0 (see, e.g., [30] for basic results of the dimension theory). For a continuous function .g : X → C we set .SX (g; c) := {x ∈ X : |g(x)| < c}. By .clX we denote closure in X. In the next result, .gˆ ∈ C(M(H ∞ )) stands for the (continuous) extension of .g ∈ H ∞ to .M(H ∞ ) by means of the Gelfand transform. Also, we equip the set of trivial Gleason parts .Ms ⊂ M(H ∞ ) with the induced topology. Let .f ∈ H ∞ \ {0}, . f ∞ = 1, be such that .fˆ(x) = 0 for some .x ∈ Ms . Lemma 8.5.1 Given .c > 0 there is a clopen subset .C ⊂ SMs (fˆ; c) containing x. Proof We apply Theorem 8.4.1 to f and .δ := 2c . Then we obtain .ε = ε(δ) ∈ (0, δ), ∞ .Kδ ⊂ D and .ψ ∈ C (D) satisfying the statement of the theorem.

298

A. Brudnyi

According to the definition of a quasi-interpolating set and the Hoffman theorem [20], clM(H ∞ ) (Kδ ) ⊂ Ma ,

(8.15)

.

see [8, Thm. 2.1] for details. Consider the open set .U := M(H ∞ ) \ clM(H ∞ ) (Kδ ). By definition, U ∩ D = D \ Kδ .

.

Also, the open set .U ∩ D is dense in U by the corona theorem. Let us show that the function .ψ admits a continuous extension to U . Since this problem is of a local nature, it suffices to show that given an open set .V  U , .ψ|V ∩D admits a continuous extension to V . Indeed, property (iii) of Theorem 8.4.1 and property (4) of Proposition 8.3.4 used in the proof of Theorem 8.2.2 show that there ¯ = ∂ψ ¯ is a bounded function .κ ∈ C ∞ (D) such that .∂κ V ∩D is the uniform

r and

.κ| s limit of a sequence of bounded functions of the form . s=1 li=1 his (z)Bs−i (z), .z ∈ V ∩ D, .r ∈ N , where all .Bs are interpolating Blaschke products with zeros outside an open pseudohyperbolic neighbourhood of .clD (V ∩ D) and all .his ∈ H ∞ , see the proof of [7, Thm. 3.5] for details. Since all .his and .Bs admit continuous extensions to .M(H ∞ ) (via the Gelfand transform), .κ|V ∩D admits a continuous extension to V . Since .ψ = κ + f for some .f ∈ H ∞ , .ψ|V ∩D admits a continuous extension to V as well. Finally, since V is an arbitrary relatively compact open subset of U and .U ∩ D is dense in U , local extensions of .ψ|V ∩D to V glue together and give the required extension .ψˆ of .ψU ∩D to U . Further, due to (8.15) and Theorem 8.4.1, .Ms ⊂ U and .ψˆ is a continuous ˆ = 0} function on U with values in .{0, 1}. In particular, .C := {y ∈ Ms : ψ(y) ˆ ˆ is a clopen subset of .Ms containing .SMs (f ; ε) ( x). Moreover, .ψ = 1 on ∞ ) : |fˆ(y)| > δ}. Then, since .δ = c , .C ⊂ S   .{y ∈ M(H Ms (fˆ; c). 2 Corollary 8.5.2 .Ms is totally disconnected. Proof By the definition of the Gelfand topology, any open neighbourhood of .x ∈ Ms in .Ms contains an open neighbourhood of the form n  .

{SMs (fˆi ; ci ) : fˆi (x) = 0, fi H ∞ = 1, ci ∈ (0, 1)},

n ∈ N.

i=1

In turn, each of the latter sets contains a clopen neighbourhood of x by Lemma 8.5.1. Thus, .Ms has the base of topology consisting of clopen sets, i.e., .Ms is totally disconnected.  

8 The Corona Problem

299

8.6 Banach-Valued Corona Problem The fact that .Ms is totally disconnected is used in the proof of the following result. Lemma 8.6.1 Given a finite open cover .U of .M(H ∞ ) there are an open finite refinement .V = (Vi )i∈I of .U and a continuous partition .{ρi }i∈I subordinate to .V such that (a) All nonempty .Vi ∩ Vj , .i = j , are relatively compact subsets of .Ma ; (b) Every .ρi |D ∈ C ∞ (D) and dρi =

.

hi (z) gi (z) dz + d z¯ , 2 1 − |z| 1 − |z|2

z ∈ D,

where .gi , hi ∈ Cρ∞ (D). For the proof see [7, Prop. 3.2, Lm. 4.1]. Let .H ∞ (A) := H ∞ (D, A) be the Banach algebra of bounded holomorphic functions on .D with images in a commutative complex unital Banach algebra A. Using the developed technique we prove the following result. Theorem 8.6.2 Let .f1 , . . . , fm , f ∈ H ∞ (A). Suppose there exists a finite open cover .(Uj )1≤j ≤ of .M(H ∞ ) such that for every .1 ≤ j ≤ the function .f |Uj ∩D belongs to the ideal .Ij ⊂ H ∞ (Uj ∩ D, A) generated by functions 2 belongs to the ideal .I ⊂ H ∞ (A) generated by .f1 |Uj ∩D , . . . , fm |Uj ∩D . Then .f .f1 , . . . , fm . Proof The proof repeats the arguments of the proof of [7, Thm. 1.11]. Let .U = (Uj )1≤j ≤ be a finite open cover of .M(H ∞ ) satisfying assumptions of the theorem. Passing to a refinement of .U we may replace it by a cover .V = (Vj )1≤j ≤k satisfying conditions of Lemma 8.6.1. Let .{ρj }1≤j ≤k be the partition of unity of the lemma subordinate to the cover .V. By the hypothesis of the theorem, there exists a family of functions .gij ∈ H ∞ (Vj ∩ D, A), .1 ≤ i ≤ m, .1 ≤ j ≤ k, such that f |Vj ∩D =

m

.

on each

gij fi

Vj ∩ D.

i=1

We set ci,rs := gir − gis

on

.

Vr ∩ Vs ∩ D = ∅,

and then hir :=



.

s

(ρs |D )ci,rs ,

(8.16)

300

A. Brudnyi

where the sum is taken over all s for which .Vs ∩ Vr = ∅. By the definition, .hir are bounded functions in .C ∞ (Vr , A). Moreover, hir − his = ci,rs

.

on

Vr ∩ Vs ∩ D = ∅.

Further, define hi := gir − hir

.

Vr ∩ D.

on

¯ i is a .C ∞ AClearly, each .hi is a bounded function in .C ∞ (D, A). In particular, .∂h valued .(0, 1) form on .D with support in the closure, say, W , of the union of all nonempty .Vr ∩ Vs ∩ D. Since by Lemma 8.6.1 all nonempty .Vs ∩ Vr  Ma , the ¯ i = Hi (z)2 , .z ∈ D, set W is quasi-interpolating. Also, one can easily check that .∂h 1−|z| ∞ where .Hi ∈ Cρ (W, A). According to (8.16) we have f =

m

.

hi fi .

i=1

Now, we apply the arguments similar to those in [19, Ch. VIII, Th. 2.1]. Specifically, if we set gi := f hi +

m

.

ais fs ,

s=1

where ais = bis − bsi ∈ C ∞ (D, A)

.

are bounded and

¯ is = hi ∂h ¯ s, ∂b

(8.17)

then m .

gi fi = f 2

and

i=1

¯ hi ) + ¯ i = ∂(f .∂g

m

¯ i + hi ∂¯ ¯ s − hs ∂h ¯ i ) = f ∂h fs · (hi ∂h

s=1

¯ i − ∂h

m s=1

fs hs = 0.

 m s=1

 fs hs

8 The Corona Problem

301

Hence, .gi ∈ H ∞ (A) and so .f 2 belongs to the ideal .I ⊂ H ∞ (A) generated by .f1 , . . . , fm . Thus, to complete the proof of the theorem it remains to prove existence of ¯ is = hi ∂h ¯ s. bounded solutions of equations .∂b ¯ s = hi (s)Hs2(z) , .z ∈ D, where .hi Hs ∈ Cρ∞ (W, A). To this end, note that .hi ∂h 1−|z| Then according to Theorem 8.2.2 the required bounded solutions .bis of the above equations exist. The proof of the theorem is complete.   ∞ (A) ⊂ H ∞ (A) be the Banach subalgebra of holomorphic functions Let .Hcomp with relatively compact images.3 Then the analog of Theorem 8.6.2 is valid for ∞ (A). The proof is exactly the same as above because the ideals of algebra .Hcomp operator of Theorem 8.2.2 preserves relative compactness of images of A-valued functions. In what follows, by 1 we denote the unit of A as well as the constant function of value .1 ∈ A. Using Theorem 8.6.2 we prove the following result. ∞ (A). The Bezout equation Theorem 8.6.3 Let .f1 , . . . , fn ∈ Hcomp

f1 g1 + · · · + fn gn = 1

.

(8.18)

∞ (A) if and only if for every .z ∈ D there exist .g , . . . , g is solvable in .Hcomp 1,z n,z ∈ A such that

f1 (z)g1,z + · · · + fn (z)gn,z = 1 and

.

sup max giz A =: M < ∞. z∈D

i

(8.19)

∞ (A) admits a continuous extension Proof It is easy to check that every .f ∈ Hcomp ∞ ), A). .fˆ ∈ C(M(H Suppose that condition (8.19) is satisfied. Let .x ∈ M(H ∞ ) and .Ux ⊂ M(H ∞ ) be an open neighbourhood of x such that

.

max

y∈Ux

3 The

n i=1

1 .

fˆi (x) − fˆi (y) A ≤ 4M

∞ has been introduced in [49]. notation .Hcomp

(8.20)

302

A. Brudnyi

Let .x˜ ∈ Ux ∩ D. Then for all .z ∈ Ux ∩ D we have  n    n  n                fi (z)gi,x˜ − 1 ≤  fi (z) − fi (x) ˜ gi,x˜  +  fi (x)g ˜ i,x˜ − 1        .

i=1

i=1

A

≤M

i=1

A

A

1 1 = . 2M 2

This implies that .hx (z) := ni=1 fi (z)gi,x˜ , .z ∈ Ux ∩ D, is an invertible function in ∞ .Hcomp (Ux ∩ D, A). Its inverse is given by the uniformly convergent series h−1 x :=

∞

.

(1 − hx )j .

j =0

We set gi,x (z) := fi (z)h−1 x (z),

.

z ∈ Ux ∩ D,

1 ≤ i ≤ n.

∞ (U ∩ D, A) and Then each .gi,x ∈ Hcomp x n .

fi (z)gi,x = 1

on Ux ∩ D.

i=1

In this way, we obtain a finite cover of .D by sets of the form .Ux ∩ D on which the corresponding local Bezout equations are solvable. Applying the ‘relatively compact’ version of Theorem 8.6.2 in this situation for the trivial ideal .I = ∞ (A) and local ideals .H ∞ (U ∩ D, A) generated by .f , . . . , f on .U ∩ D Hcomp 1 n x comp x we obtain the required result. The converse statement of the theorem is trivial.   As a corollary of the previous results we obtain (see [7, Thm. 1.12] for details). Theorem 8.6.4 ∞ M(Hcomp (A)) ∼ = M(H ∞ ) × M(A).

.

8.7 Operator Completion Problem for H ∞ ∞ , see [9]. In this part we present some results on the Oka-Grauert theory for .Hcomp In what follows, for a complex Banach algebra .A with unit 1 we denote by .A−1 −1 the group of invertible elements. It is a complex Banach Lie group. By .A−1 0 ⊂A

8 The Corona Problem

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we denote the connected component containing 1. Also, by .A−1 l = {a ∈ A : ∃ b ∈ A such that ba = 1} we denote the set of left-invertible elements of .A and by 2 .id A = {a ∈ A : a = a} the set of idempotents of .A. −1 −1 Group .A−1 acts on .A−1 l by left multiplications: .A0 × Al  (g, a) → ga ∈ 0 −1 under this action is a complex Lie subgroup A−1 l . The stabilizer of .a ∈ Al −1 −1 .A (a) ⊂ A of elements g satisfying .ga = a. 0 0 −1 In turn, .A0 acts on .id A by similarity transformations: .A−1 0 × id A  (g, a) → gag −1 ∈ id A, and the stabilizer of .a ∈ id A under this action is a complex Lie −1 −1 = a. subgroup .A−1 0 (a) ⊂ A0 of elements g satisfying .gag Let A be a complex Banach algebra with unit 1. The next result describes prop∞ (A). erties of idempotents and left invertible elements of the algebra .A := Hcomp Theorem 8.7.1 ([9, Thms. 4.4 (b) 4.8 (a), 4.9]) ∞ (A)) be such that the stabilizer .A−1 (F (0)) ⊂ A−1 of .F (0) ∈ (i) Let .F ∈ id(Hcomp 0 0 ∞ (A))−1 such that .GF G−1 = id A is connected, then there exists .G ∈ (Hcomp 0 F (0). ∞ (A))−1 be such that the stabilizer .A−1 (F (0)) ⊂ A−1 of (ii) Let .F ∈ (Hcomp l 0 0 −1 −1 ∞ .F (0) ∈ A is connected, then there exists .G ∈ (Hcomp (A)) such that l 0 .GF = F (0). ∞ (A))−1 .4 So the question remains as to how to characterize the elements of .(Hcomp l The following general result holds.

Theorem 8.7.2 ([9, Thm. 4.9]) Let .A be a complex unital Banach algebra. A map ∞ (A) has a left inverse .G ∈ H ∞ (A) if and only if for every .z ∈ D there F ∈ Hcomp comp exists a left inverse .Gz of .F (z) such that .supz∈D Gz A < ∞.

.

In what follows, .L(X1 , X2 ) denotes the Banach space of bounded linear operators between complex Banach spaces .X1 and .X2 . Theorem 8.7.2 is related to the following problem posed by Sz.-Nagy in 1978: Problem 8.7.3 Suppose .F ∈ H ∞ (L(H1 , H2 )), .H1 , H2 are separable Hilbert spaces, satisfies . F (z)x ≥ δ x for all .x ∈ H1 , .z ∈ D, where .δ > 0 is a constant. Does there exist .G ∈ H ∞ (L(H2 , H1 )) such that .G(z)F (z) = IH1 for all .z ∈ D? This problem is of importance in operator theory (angles between invariant subspaces, unconditionally convergent spectral decompositions) and in control theory (the stabilization problem). It is also related to the study of submodules of .H ∞ and to many other subjects of analysis, see [35, 36, 43, 44, 50] and references therein. Obviously, the condition imposed on F is necessary since it implies existence of a uniformly bounded family of left inverses of .F (z), .z ∈ D. The question is whether this condition is sufficient for the existence of a bounded holomorphic left inverse of F . In general, the answer is known to be negative (see

4 The

problem was first considered in [49].

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[45, 46, 48] and references therein); it is positive as soon as .dim H1 < ∞ or F is a “small” perturbation of a left invertible function .F0 ∈ H ∞ (L(H1 , H2 )) (e.g., if .F − F0 belongs to .H ∞ (L(H1 , H2 )) with values in the class of Hilbert Schmidt ∞ (L(H , H )) due to Theorem 8.7.2. operators), see [47], or if .F ∈ Hcomp 1 2 Example 8.7.4 Let .L(X) be the Banach algebra of bounded linear operators on a complex Banach space X equipped with the operator norm. By .IX ∈ L(X) we denote the identity operator and by .GL(X) ⊂ L(X) the set of invertible bounded linear operators on X. Clearly, .L(X)−1 := GL(X). By .GL0 (X) ⊂ GL(X) we denote the connected component of .IX . 0 ⊕ X1 , Each .E ∈ id L(X) determines a direct sum decomposition .X = XE E 0 := ker E and .X 1 := ker (I − E). It is easily seen that the stabilizer where .XE X E k .GL0 (X)(E) ⊂ GL0 (X) consists of operators .B ∈ GL0 (X) such that .B(X ) ⊂ E k k XE , .k = 0, 1. In particular, restrictions of operators in .GL0 (X)(E) to .XE determine 0)⊕ a monomorphism of complex Banach Lie groups .SE : GL0 (X)(E) → GL(XE 1 ). Moreover, .S is an isomorphism if .GL(X k ), .k = 0, 1, are connected. GL(XE E E Now, Theorem 8.7.1(1) leads to the following statement: ∞ (L(X))) be such that for .E := F (0) ∈ id L(X), groups (A) Let .F ∈ id(Hcomp −1 0 1 ∞ .GL(X ) and .GL(X ) are connected. Then there exists .G ∈ (Hcomp (L(X))) E E 0 −1 such that .GF G = E.

In particular, this is valid if X is one of the spaces: a finite-dimensional space, a Hilbert space, .c0 or . p , .1 ≤ p ≤ ∞. Indeed, .GL(X) is connected if .dimC X < ∞ and contractible for other spaces of the list, see, e.g., [29] and references therein. Moreover, each subspace of a Hilbert space is Hilbert, and if X is either .c0 or . p , .1 ≤ p ≤ ∞, then its infinite-dimensional complemented subspaces are isomorphic to X as well, see [28, 38]. This provides the required condition in (A). (Note also that there are complex Banach spaces X with disconnected groups .GL(X), see, e.g., [13].) Further, each .E ∈ L(X)−1 determines a complemented subspace .X1 := l ran E ⊂ X isomorphic to X. Then the stabilizer .GL0 (X)(E) ⊂ GL0 (X) of E consists of operators .B ∈ GL0 (X) such that .B|X1 = IX1 . If .X2 ⊂ X is a complement to .X1 , then each .B ∈ GL0 (X)(E) has a form  B=

.

IX1 C 0 D

 ,

where

D ∈ GL(X2 ) and C ∈ L(X2 , X1 ).

(8.21)

Thus .GL0 (X)(E) is homotopy equivalent to the subgroup of .GL(X2 ) (∼ = GL(X/X1 )) of operators D such that .diag(IX1 , D) ∈ GL0 (X). In particular, this subgroup coincides with .GL(X2 ) if the latter is connected.

8 The Corona Problem

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Now, Theorem 8.7.1(2) leads to the following statement: ∞ (L(X)))−1 be such that for .E := F (0) ∈ L(X)−1 the corre(B) Let .F ∈ (Hcomp l l sponding group .GL(X/X1 ) is connected. Then there is .G ∈ (Hcomp (L(X)))−1 0 such that .F = GE.

Let us identify X with .X1 by E and regard F as an element of ∞ (L(X , X ⊕ X )) and E as an operator in .L(X , X ⊕ X ). Then Hcomp 1 1 2 1 1 2

.

 F =

.

F1 F2



 G=

,

G11 G12 G21 G22



 and

E=

IX1 0

 ,

∞ (L(X , X )) and .G ∈ H ∞ (L(X )), .i = 1, 2, .G where .Fi ∈ Hcomp 1 i ii i ij ∈ comp ∞ Hcomp (L(Xj , Xi )), .i, j ∈ {1, 2}, .i = j . Now, equation .F = GE implies that

F1 = G11 ,

.

F2 = G21 ,

∞ (L(X ⊕ X )))−1 . i.e., F extends to an invertible element .G ∈ (Hcomp 1 2 0

Thus, statement (B) and Theorem 8.7.2 yield: ∞ (L(Y , Y )), where .Y , .i = 1, 2, are complex Banach (B. ) Suppose .F ∈ Hcomp 1 2 i spaces is such that for each .z ∈ D there exists a left inverse .Gz of .F (z) satisfying .

sup Gz < ∞. z∈D

Assume that .GL(Y ), .Y := ker G0 , is connected. Then there exist .H ∈ ∞ (L(Y ⊕ Y, Y )), .G ∈ H ∞ (L(Y , Y ⊕ Y )) such that for all .x ∈ D, Hcomp 1 2 2 1 comp H (x)G(x) = IY2 ,

.

G(x)H (x) = IY1 ⊕Y

and

H (x)|Y1 = F (x).

(Statement (B. ) is obtained from (B) if we set .X1 := Y1 , .X2 := Y and .X := Y2 .) The proof of Theorem 8.7.1 is based on the following result established in [9, Cor. 2.7]. Let G be a connected complex Banach Lie group with unit 1. Consider a finite open cover .{Ui }i∈I of .M(H ∞ ). Suppose that .{Gij }i,j ∈I , is a holomorphic 1-cocycle on the cover .(Ui ∩ D)i∈I of .D, with a relatively compact image in G, i.e., each .Gij ∈ O(Ui ∩ Uj ∩ D, G) is such that the closure of its image in G is compact and Gii = 1,

.

Gij = G−1 ji

and

Gij Gj k = Gik

on

Ui ∩ Uj ∩ Uk ∩ D = ∅.

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Then there are .Gi ∈ O(Ui , G), .i ∈ I , with relatively compact images in G such that Gij = G−1 i Gj

.

on

Ui ∩ Uj ∩ D = ∅.

Acknowledgments Research is supported in part by NSERC.

References 1. N. L. Alling, A proof of the corona conjecture for finite open Riemann surfaces, Bull. Amer. Math. Soc. 70 (1964), 110–112. 2. N. L. Alling, Extensions of meromorphic function rings over non-compact Riemann surfaces, I, Math. Z. 89 (1965), 273–299. 3. D. E. Barrett, and J. Diller, A new construction of Riemann surfaces with corona. J. Geom. Anal. 8 (1998), 341–347. 4. M. Behrens, On the corona problem for a class of infinitely connected domains, Bull. Amer. Math. Soc. 76 (1970), 387–391. 5. M. Behrens, The maximal ideal space of algebras of bounded analytic functions on infinitely connected domains, Trans. Amer. Math. Soc. 161 (1971), 358–380. 6. A. Brudnyi, Topology of the maximal ideal space of H ∞ , J. Funct. Anal. 189 (2002), 21–52. 7. A. Brudnyi, Banach-valued holomorphic functions on the maximal ideal space of H ∞ , Invent. Math. 193 (2013) 187–227. 8. A. Brudnyi, Topology of the maximal ideal space of H ∞ revisited, Adv. Math. 299 (2016), 931–939. 9. A. Brudnyi, Oka principle on the maximal ideal space of H ∞ , St. Petersburg Math. J. , 31 (5) (2020), 769–817. 10. L. Carleson, Interpolation of bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547–559. 11. W. Deeb, A class of infinitely connected domain and the corona, Trans. Amer. Math. Soc. 31 (1977), 101–106. 12. W. Deeb and D.R. Wilken,  domains and the corona, Trans. Amer. Math. Soc. 231 (1977), 107–115. 13. A. Douady, Une espace de Banach dont le groupe linéaire n’est pas connexe, Indag. Math. 68 (1965), 787–789. 14. J. P. Earl, On the interpolation of bounded sequences by bounded functions, J. London Math. Soc. 2 (1970), 544–548. 15. C. J. Earle and A. Marden, Projections to automorphic functions, Proc. Amer. Math. Soc. 19 (1968), 274–278. 16. F. Forelli, Bounded holomorphic functions and projections. Illinois J. Math. 10 (1966), 367– 380. 17. T. W. Gamelin, Uniform algebras and Jensen measures, London Math. Soc. Lecture Notes Series 32. Cambridge Univ. Press, Cambridge-New York, 1978. 18. T. W. Gamelin, Localization of the corona problem, Pacific J. Math. 34 (1970), 73–81. 19. J. B. Garnett, Bounded analytic functions, Academic Press, 1981. 20. K. Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. 86 (1967), 74–111. 21. J. Handy, The corona theorem on the complement of certain square Cantor sets. J. Anal. Math. 108 (1) (2009), 1–18. 22. M. Hara and M. Nakay, Corona theorem with bounds for finitely sheeted disks, Tohoku Math. J. 37 (1985), 225–240. 23. F. Hirzebruch, Topological methods in Algebraic Geometry, Springer-Verlag, New York, 1966.

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24. L. Hörmander, Generators for some rings of analytic functions, Bull. Amer. Math. Soc. 73 (1967), 943–949. ¯ 25. P. W. Jones, L∞ estimates for the ∂-problem in a half-plane, Acta Math. 150 no. 1–2 (1983), 137–152. 26. P. W. Jones and D. Marshall, Critical points of Green’s functions, harmonic measure and the corona theorem, Ark. Mat. 23 (1985), 281–314. 27. F. Lárusson, Holomorphic functions of slow growth on nested covering spaces of compact manifolds, Canad. J. Math. 52 (2000), 982–998. 28. J. Lindenstrauss, On complemented subspaces of m, Israel J. Math. 5 (1967), 153–156. 29. B. S. Mityagin, The homotopy structure of the linear group of a Banach space, Russ. Math. Surv. 25 (5) (1970), 59–103. 30. K. Nagami, Dimension Theory, Academic Press, New York, 1970. 31. M. Nakay, The corona problem on finitely sheeted covering surfaces, Nagoya Math. J. 92 (1983), 163–173. 32. J. Narita, A remark on the corona problem for plane domains, J. Math. Kyoto Univ. 25 (1985), 293–298. 33. B. M. NewDelman, Homogeneous subsets of a Lipschitz graph and the Corona theorem, Publ. Mat. 55 (2011), 93–121. 34. D. J. Newman, Interpolation in H ∞ (D), Trans. Amer. Math. Soc. 92 no. 2 (1959), 502–505. 35. N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002, Translated from the French by Andreas Hartmann. 36. N. K. Nikolski, Treatise on the shift operator, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer–Verlag, Berlin, 1986, Spectral function theory, With an appendix by S. V. Hrušˇcev [S. V. Khrushchëv] and V. V. Peller, Translated from the Russian by Jaak Peetre. 37. B. Oh, A short proof of Hara and Nakai’s theorem, Proc. Amer. Math. Soc. 136 (2008), 4385– 4388. 38. A. Pełczy´nski, Projections in Certain Banach Spaces, Studia Math. 19 (1960), 209–228. 39. E. L. Stout, Two theorems concerning functions holomorphic on multiply connected domains, Bull. Amer. Math. Soc. 69 (1963), 527–530. 40. E. L. Stout, Bounded holomorphic functions on finite Riemann surfaces, Trans. Amer. Math. Soc. 120 (1965), 255–285. 41. E. L. Stout, On some algebras of analytic function on finite open Riemann surfaces, Math. Z. 92 (1966), 366–379. Corrections to: On some algebras of analytic function on finite open Riemann surfaces. Math. Z. 95 (1967), 403–404. 42. D. Suárez, Trivial Gleason parts and the topological stable rank of H ∞ , Amer. J. Math. 118 (1996), 879–904. 43. S. R. Treil, Geometric methods in spectral theory of vector-valued functions: some recent results, Toeplitz operators and spectral function theory, Oper. Theory Adv. Appl., vol. 42, Birkhäuser, Basel, 1989, pp. 209–280. 44. S. R. Treil, Unconditional bases of invariant subspaces of a contraction with finite defects, Indiana Univ. Math. J. 46 (1997), no. 4, 1021–1054. 45. S. Treil, Geometric methods in spectral theory of vector-valued functions: Some recent results, in Toeplitz Operators and Spectral Function Theory, Oper. Theory Adv. Appl., vol. 42, Birkhäuser Verlag, Basel 1989, pp. 209–280. 46. S. Treil, Lower bounds in the matrix corona theorem and the codimension one conjecture, GAFA 14 5 (2004), 1118–1133. 47. S. Treil, An operator Corona theorem, Indiana Univ. Math. J. 53 (2004), no. 6, 1763–1780. 48. S. Treil and B. Wick, Analytic projections, Corona Problem and geometry of holomorphic vector bundles, J. Amer. Math. Soc. 22 (2009), no. 1, 55–76. 49. P. Vitse, A tensor product approach to the operator corona theorem, J. Operator Theory 50 (2003), 179–208.

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50. M. Vidyasagar. Control system synthesis: a factorization approach, MIT Press Series in Signal Processing, Optimization, and Control 7, MIT Press, Cambridge, MA, 1985. 51. S. A. Vinogradov, E. A. Gorin and S. Y. Khrushchev, Interpolation in H ∞ along P. Jones’ lines. (in Russian), Zapiski Nauchnyh Seminarov LOMI 113 (1981), 212–214. 52. S. Ziskind, Interpolating sequences and the Shilov boundary of H ∞ (), J. Funct. Anal. 21 (1976), no. 4, 380–388.

Chapter 9

A Brief Introduction to Noncommutative Function Theory Michael T. Jury

2020 Mathematics Subject Classification 46L52, 46E22, 47A13, 16S38

9.1 Introduction The phrase “noncommutative function theory” refers to a body of mathematical ideas aimed at extending complex analysis to the realm of functions of noncommuting variables. Much as analytic functions in a planar domain might be viewed as generalized polynomials, a “noncommutative (nc) function” is an object which can be thought of as a generalized nc polynomial, that is, a polynomial in noncommuting indeterminates, with coefficients in some field (in these notes, we will work exclusively over the complex numbers .C). The particular notion of nc function we will work with originated in the 1970’s in the work of J.L. Taylor, though it took rather a long time to gain traction. At present it is a quite active area of analysis, with connections to operator theory, operator algebras, free probability, real algebraic geometry, matrix analysis, etc. In these notes we will take a very narrow and selective view: after some brief introductory material we will highlight only those aspects of the theory that will be most recognizable to workers in the classical theory of function spaces (in keeping with the theme of the Focus Program). We have no intention of being exhaustive or encyclopedic; these notes are meant only as a very rapid introduction to some of the key ideas. Many statements are given without proof; we provide proofs only when they are both elementary and serve to illustrate an important idea or technique. We will begin with the definitions and basic properties of nc sets and nc functions, and give examples. For the subsequent development, as we have said, we will not try to touch on all the areas in which nc functions arise (such as free probability, free semialgebraic geometry, etc.), but instead will focus on those aspects of the theory

M. T. Jury () Department of Mathematics, University of Florida, Gainesville, FL, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mashreghi (ed.), Lectures on Analytic Function Spaces and their Applications, Fields Institute Monographs 39, https://doi.org/10.1007/978-3-031-33572-3_9

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that will be most recognizable to function theorists. Thus, we will consider spaces of nc functions (analogous to spaces of holomorphic functions, such as the Hardy space in the disk) and examine familiar objects and problems (multipliers, reproducing kernels, inner-outer factorization, and so on) in this setting. We will try to highlight the central role of the notion of a “realization” in this theory, and indicate some applications of the nc machinery to problems in ordinary function spaces (in particular the Drury-Arveson space and spaces with complete Nevanlinna-Pick kernel).

9.2 nc Sets and nc Functions We have said that polynomials in noncommuting indeterminates should count as “noncommutative analytic functions,” if such things are to exist. Polynomials in a single variable are, of course, naturally viewed as functions on the coefficient field .C. The passage from algebra to analysis consists in topologizing .C and viewing these polynomials as continuous (in fact, differentiable functions). What about nc polynomials? If we have a formal expression in noncommuting indeterminates .x1 , x2 , such as p(x1 , x2 ) = 1 + x1 x2 − x2 x1 − 2x12 x2 + 3x1 x2 x1 ,

.

we would like to view it as a function (and then, hopefully, as a continuous or differentiable function). Since we want to remember that the indeterminates are not meant to commute, it will not be adequate to evaluate p at scalar arguments. Rather, we need a source of noncommutativity, and it is natural to plug in square matrices .X1 , X2 . We could plug in square matrices of arbitrary sizes .n × n, and there seems to be no reason to favor one size over another, so it will probably be best to think of p as a graded function, acting separately on .1 × 1 matrices, .2 × 2, etc. To see how we might go beyond polynomials, we should first fix our understanding of what sorts of sets our functions should be defined on. Since we already have in mind the idea of evaluating functions at square matrices, let us begin there. First, some notation: let .Mn (C) be the set of .n × n matrices, and for .d  1 put Mdn := {(X1 , . . . , Xd ) : Xj ∈ Mn (C)}

.

We now let Md =

∞ 

.

Mdn ,

n=1

we call this the matrix universe. It is a graded set, with a “level” .Mdn at each size n. An nc polynomial defines a graded function .p : Md → M1 , i.e. for each level n

9 A Brief Introduction to Noncommutative Function Theory

311

we have a function p[n] : Mdn → Mn

.

obtained by evaluating the nc polynomial at .n × n matrix arguments in the usual way. Fundamental to everything that follows is the observation that the functions .p[n] at the different levels n interact with each other. In particular, let X = (X1 , . . . , Xd ) ∈ Mdn

.

and Y = (Y1 , . . . , Yd ) ∈ Mdm ,

.

and form their direct sum  X⊕Y =

.

   Xd 0 X1 0 ,..., ∈ Mdn+m . 0 Y1 0 Yd

Then we have, evidently, p(X ⊕ Y ) =

.

  p(X) 0 0 p(Y )

(that is, p respects direct sums). There is another simple (but very important) algebraic property: let X = (X1 , . . . , Xd ) ∈ Mdn

.

and let S be an invertible .n × n matrix. We can conjugate X by the similarity S: S −1 XS = (S −1 X1 S, . . . , S −1 Xd S) ∈ Mdn

.

Then p(S −1 XS) = S −1 p(X)S.

.

(that is, p respects similarities). To define more general nc functions, we will want these same properties to hold but, just as we study holomorphic functions that are not entire, we should not insist that our nc functions be defined on the entire matrix universe .Md , but just on an “nc-set”:

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Definition 9.2.1 An nc-set . ⊂ Md is a graded set . = n ⊂ Mdn ,

∞

n=1 n ,

with

n = 1, 2, . . . ,

.

such that . respects direct sums: that is, if .X ∈ n and .Y ∈ m then X ⊕ Y ∈ n+m .

.

An nc-domain is an nc-set . such that .n is open in .Mdn at each level n (here we give .Mdn the usual topology it inherits as a complex vector space of dimension 2 .dn ).

9.2.1 Examples We present a few simple examples of nc domains, the row ball in particular will play an important role in what follows. • The row ball: at each level n we form the set n = {X = (X1 , . . . , Xd ) ∈ Mdn : X1 X1∗ + · · · + Xd Xd∗  < 1}.

.

(In other words, these are the .X ∈ Mdn for which the .n × dn-matrix .

  X1 X2 · · · Xd

has norm less than 1.) It is readily checked that this is an nc domain. At level 1 we have 1 = {z = (z1 , . . . , zd ) : |z1 |2 + · · · + |zd |2 < 1},

.

the open unit ball of .Cd . • The column ball: similarly, put n = {X = (X1 , . . . , Xd ) ∈ Mdn : X1∗ X1 + · · · + Xd∗ Xd  < 1}.

.

(In other words, the .dn × n-matrix ⎡

⎤ X1 ⎢X2 ⎥ ⎢ ⎥ .⎢ . ⎥ ⎣ .. ⎦ Xd

9 A Brief Introduction to Noncommutative Function Theory

313

has norm less than 1.) At level 1 we have 1 = {z = (z1 , . . . , zd ) : |z1 |2 + · · · + |zd |2 < 1},

.

the open unit ball of .Cd . (So, the row ball and column ball coincide at level 1—but they are different at level 2, as easy examples show. One may also observe that X belongs to the row ball if and only if .X∗ = (X1∗ , . . . , Xd∗ ) belongs to the column ball.) • The nc polydisk: n = {X = (X1 , . . . , Xd ) ∈ Mdn : Xj  < 1, j = 1, . . . d}

.

(In other words, the .dn × dn-matrix ⎡

⎤

X1

.

⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎦

X2 ..

. Xd

has norm less than 1.) At level 1 we have 1 = {z = (z1 , . . . , zd ) : |zj | < 1, j = 1, . . . d},

.

the open unit polydisk .Dd . • The row ball, column ball, and nc polydisk are all examples of polyhedral domains:  = {X ∈ Md : δ(X) < 1}

.

for some matrix polynomial .δ. By a matrix polynomial we mean an .I × J matrix, each entry of which is an nc polynomial. One may quickly verify that any polyhedral domain is an nc set. Each of the examples above (the row ball, column ball, and nc polydisk) is a polyhedral domain, in two variables we would take .δ to be      X1  X1 , .δ(X1 , X2 ) = X1 X2 , X2 X2 respectively. • We will give another important class of examples below, when we consider the domains of nc rational expressions. For example, the set of pairs .(X1 , X2 ) for which .(I − X1 X2 ) is invertible will be an nc domain, called the domain of the nc rational expression .(1 − x1 x2 )−1 .

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9.2.2 nc Functions Now that we understand what kind of sets we are working on, we can consider functions on these sets. Definition 9.2.2 Let . ⊂ Md be an nc-set. An nc-function on . is a graded function f :→M

.

such that: • f respects direct sums: whenever .X ∈ n , Y ∈ m , we have     X 0 f (X) 0 .f = 0 Y 0 f (Y ) • f respects similarities: f (S −1 XS) = S −1 f (X)S

.

whenever X and .S −1 XS belong to .. These definitions go back to the work of Taylor [26–29]. In these notes, our interest will be exclusively in nc functions defined on nc domains. Let us consider some examples of nc functions: • Of course, nc polynomials are nc functions: X1 X22 + 3X2 X1 X2 − X1 X2 + X2 X1 − 5I

.

An nc polynomial is evidently an entire nc function, defined on all of .Md . • nc rational functions: going beyond polynomials, we take inverses, and restrict the domain to those X where the inverse(s) appearing in the expression exist. For example, f (X1 , X2 ) = (X1 X2 − X2 X1 )−1 ,

.

defined on the set  = {(X1 , X2 ) : det(X1 X2 − X2 X1 ) = 0}

.

is an nc rational function. One may check readily that this . is an nc set, and is open at each level n, hence an nc domain. (But . is nonempty only at levels .n > 1. In particular, to allow examples such as this we have not insisted that

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315

nc sets be nonempty at every level.) We will have more to say about nc rational functions below, in Sect. 9.7. • nc power series: a formal power series in noncommuting indeterminates, such as ∞ 

x2m x1 x3n

.

(9.1)

m,n=0

will define an nc function provided it is suitably convergent. For example the series (9.1) will evidently converge whenever both .X2  < 1 and .X3  < 1, and will define an nc function on the resulting nc domain. Elementary estimates will show that the resulting function is locally bounded in this domain. We will examine a particular class of nc power series (those with square-summable coefficients) in much more detail later in these notes, beginning in Sect. 9.5. • Let us look briefly at two sorts of functions of a single matrix variable which are not nc functions. First, the adjoint mapping X → X∗

.

respects direct sums, but not similarities. On the other hand, the trace mapping X → tr(X)I

.

(with the identity chosen the same size as X) will respect similarities, but not direct sums. There is one important method of constructing holomorphic functions in a single variable that does not have an analog in the above list—namely, Cauchy integrals:  f (z) =

.

K

dμ(λ) λ−z

(Here K is a compact set in the complex plane (e.g., a closed contour) and .μ a finite measure on K.) It turns out that one can define nc functions by some (sufficiently abstracted) recipe like this. It will take us too far afield to examine such constructions in detail, but we will make a few comments in Sect. 9.10.

9.3 Locally Bounded nc Functions Are Differentiable So far, our definition of nc function is purely algebraic—respecting direct sums and similarities. But these simple algebraic assumptions turn out to have striking analytic consequences, as expressed in the title of this section. We begin with the following proposition, which already demonstrates the power of these algebraic properties.

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Proposition 9.3.1 Let . be an nc domain and .f :  → M be an nc function. If f[n] : n → Mn is continuous at each level n, then each .f[n] is holomorphic.

.

What we will prove is that each .f[n] is differentiable as a function between the vector spaces .Mdn and .Mn . This implies that .f[n] is holomorphic as a function of .dn2 complex variables (thinking of each entry of each of the matrices as an independent complex variable). Before going into the proof, let’s recall that matrices know about derivatives. In particular, for .2 × 2 matrices with complex entries, we have  2  2   2  xh x 2xh x xh + hx = . = 0 x2 0 x2 0x  3  3 2   3 2  xh x 3x h x x h + xhx + hx 2 = . = 3 0 x 0 x3 0x and so on. In general, for any polynomial p in one variable, we have     xh p(x) p (x)h .p . = 0 p(x) 0x

(9.2)

The next lemma isolates the algebra underlying this observation, and extends in to the general situation. Lemma 9.3.2 (Intertwining Lemma) Let . be an nc set and let f be an nc function on .. For .X ∈ n , Y ∈ m , and .W ∈ Mn×m , if   X XW − W Y . 0 Y belongs to .n+m , then     X XW − W Y f (X) f (X)W − Wf (Y ) .f = . 0 Y 0 f (Y ) Let us note the following special case: for any .X ∈ n , Y ∈ m and any n × m matrix W , if .XW = W Y , then .f (X)W = Wf (Y ). That is, f respects intertwinings. In fact, one can show that if f is a graded function on an nc set, which respects intertwinings, then it must respect similarities and direct sums. So we could use this as an alternative definition of nc function.

.

Proof We begin with a quick matrix calculation. If W is any .n × m matrix, then the matrix   In W . 0 Im

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is invertible, with inverse  −1   In W In −W , . = 0 Im 0 Im and .

 −1      X XW − W Y X 0 I W In W . = n 0 Im 0 Im 0 Y 0 Y

Thus, since f respects direct sums and similarities, we have  −1      X 0 X XW − W Y In W In W .f =f 0 Im 0 Im 0 Y 0 Y −1     X 0 In W In W f 0 Im 0 Im 0 Y −1     f (X) 0 In W In W = 0 Im 0 Im 0 f (Y )   f (X) f (X)W − Wf (Y ) = 0 f (Y )

=

 We can now prove that continuous nc functions are automatically holomorphic. Proof of Proposition 9.3.1 Fix .X ∈ n , .H ∈ Mdn , with .H  small, and fix a small nonzero scalar t. Write     X + tH (X + tH ) 1t − 1t X X + tH H . = 0 X 0 X   X + tH H will belong to .2n for 0 X sufficiently small .H  and t. Then, by the intertwining lemma, with .W = 1t In ,

Since we are working in an nc domain,

f

.

.

    f (X + tH ) 1t [f (X + tH ) − f (X)] X + tH H . = 0 X 0 f (X)

Taking .t → 0, the limit on the left hand side exists, by our assumption that f is continuous at each level. But this means that the limit on the right-hand side also exists, in particular the limit of the upper-right corner entry, which is nothing but the directional derivative .Df (X)[H ]. 

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In particular, from the above  proof  we conclude that whenever .X ∈ n and H is XH sufficiently small (so that . belongs to .2n ) we have 0 X f

.

    XH f (X) Df (X)[H ] = , 0 X 0 f (X)

which generalizes the bit of matrix algebra (9.2) with which we began this section. In fact, the same intertwining idea can be pushed further, to deduce continuity (and hence differentiability) from an even weaker assumption. Let . be an nc domain and f an nc function defined in .. Provisionally, let us say that f is locally bounded if at each level n, for each .X ∈ n there is an open neighborhood .U ⊂ n , containing X, and a real number .M > 0, so that .f (Z)  M for all .Z ∈ U . Proposition 9.3.3 If f is a locally bounded nc function, then each .f[n] : n → Mn is continuous. Proof Let . > 0 and fix .X ∈ n . By  the local  boundedness assumption, there is a X 0 neighborhood .U ⊂ 2n containing . ∈ 2n , and a real number M so that if 0 X .Z ∈ U , then .f (Z)  M. Next, since U is open, there is a .δ > 0 so that if .Y ∈ n and .X − Y  < δ, then1   X (M/)(X − Y ) . ∈ U. 0 Y Using the intertwining lemma again, with .W =

M  I,

we get

   f (X) (M/)(f (X) − f (Y ))    M.  .   0 f (Y ) So, inspecting the upper right corner again, we see that f (X) − f (Y )  

.

if X − Y  < δ. 

Let us work out an example of the nc derivative for polynomials, to see what is going on. In two variables, consider the monomial .x12 x2 . For fixed .H = (H1 , H2 ),

1 By “.X−Y ” we mean any norm on the finite-dimensional vector space of d-tuples .(X , . . . , X ) 1 d

of .n × n matrices (since the dimension is finite, all norms are equivalent); for example we could use the row norm.

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using

.

 2  2  X1 H1 X1 X1 H1 + H1 X1 = 0 X1 0 X12

we find for .p(x1 , x2 ) = x12 x2     2    X2 H2 X1 H1 X2 H2 X1 H1 , = .p 0 X1 0 X2 0 X1 0 X2  2  X1 X2 X12 H2 + X1 H1 X2 + H1 X1 X2 = 0 X12 X2 The derivative of the monomial in the direction H is thus the upper-right corner: Dp(X)[H ] = X12 H2 + X1 H1 X2 + H1 X1 X2

.

What has happened we have taken the monomial .X12 X2 , and successively replaced each occurrence of .Xi with the corresponding .Hi , and summed the results, obtaining an expression linear in each of the .Hi (as it must be, being a directional derivative in the direction H ). It should be evident how the rule extends to general monomials, and thus to all polynomials by linearity. One can also make sense of higher order derivatives—for example, to work out the correct definition of the second derivative, one evaluates f at points of the form ⎛

⎞ XH 0 . ⎝ 0 X K⎠ 0 0 X and reads off the upper right corner entry, which will be a bilinear expression in H and K. Continuing in this fashion, higher order derivatives are defined, and under reasonable hypotheses one can use these higher derivatives to obtain “Taylor– Taylor” series expansions for nc functions f . We refer to the book of Vinnikov and Kaliuzhnyi-Verbovetskyi [17] for a detailed exposition. Let us pause for breath and ask: what would all this look like in one variable? For a domain, let’s use the matrix disk: . = {Z : Z < 1} If f is a locally bounded “nc” function in ., then it’s analytic near 0, has a Taylor–Taylor series, f (Z) =

∞  f (n) (0)

.

n=0

n!

Zn

and after some checking, .f (Z) is given by the classical functional calculus.

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9.4 Topologies and Further Remarks Though we will not have to be too careful about it for what we want to do in the rest of these notes, it is worth making some mention of topologies on nc sets. In the last section, when talking about continuity and local boundedness, we assumed our nc sets . were open at each level, and implicitly giving . the topology generated by open sets (in the usual topology of finite-dimensional vector spaces) at each level separately. We will call this the fine topology. However, one might reasonably ask for (and have cause to use) topologies for which the open sets are themselves nc sets. (The fine topology has open sets which are not nc—indeed, an open set .U ⊂ Mdn is evidently not closed under direct sums.) There are at least two other topologies, which do consist of nc sets, which appear in the literature: Definition 9.4.1 • The free topology on .Md is the topology generated by polyhedral domains .δ(X) < 1. (Here .δ runs over all matrix polynomials of all sizes.) • The fat topology is the topology generated by nc polydiscs: D (X) =

.

  Y ∈ Mdkn : Y − ⊕n X <  n

running over all .X ∈ Md and . > 0. We observe right away that these topologies cannot be Hausdorff—indeed, the requirement  ofbeing closed under direct sums means no nc set can ever separate X 0 . X from . 0 X An nc function which is locally bounded in the free topology is sometimes called a free holomorphic function—note that this condition is stronger than being locally bounded for the fine topology (which was the notion of local boundedness we used in the last section). With the machinery developed thus far, one can begin to develop a rich nc function theory. One place to start would be with nc versions of classical theorems from calculus and complex analysis. It turns out such theorems can indeed be proved, though in many cases there are surprising twists. Since it would be too much of a digression from our main topic (function spaces), we will not give details, but only a brief list and a few references • There is an nc version of the inverse function theorem [12], [19] (valid in the fine topology) and [3] (valid in the fat topology). • Likewise, there are distinct versions of the implicit function theorem (with different hypotheses) in the fine and fat topologies [3]. • In classical complex analysis, we know that holomorphic functions can be locally uniformly approximated by polynomials. Local polynomial approximation is a

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more delicate problem in the nc setting, and the topology matters—for example, it can fail in the fat topology [3]. • There is a strikingly strong nc version of the monodromy theorem from complex analysis [18].

9.5 Square-Summable nc Power Series We turn now to the main goal of these notes, which is to say something about Hilbert spaces of nc functions. Motivated by the theory of Hardy spaces in the disk, thinking of nc functions it is natural to ask ourselves what happens with square-summable nc power series: thus, we consider expressions of the form  .

cα Xα ,

with

α



|cα |2 < +∞.

(9.3)

α

Let us explain the notation. We fix a positive integer .d  1 and consider the free monoid .F+ d consisting of all words in d letters α = i1 i 2 · · · i n

.

where .i1 , . . . , in are drawn from the set .{1, . . . , d}. The number n is called the length of the word, and we write .|α| = n. We also include the empty word .∅, by definition .|∅| = 0. If .α = i1 i2 · · · in is a word and .X = (X1 , . . . , Xd ) is a d-tuple of square matrices, we put Xα := Xi1 Xi2 · · · Xin ,

.

and by convention .X∅ := I (the identity matrix of the same size as X). In the theory of Hardy spaces, the first thing one does is prove that squaresummable power series converge locally uniformly in the disk .{|z| < 1}. We thus ask where the series (9.3) converges. Our first result in this direction (but not the last) is the following: Proposition 9.5.1 The power series (9.3) converges for all d-tuples .X ∈ Md which satisfy X2row := X1 X1∗ + · · · + Xd Xd∗  < 1

.

(that is, for all X in the row ball).

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M. T. Jury

Proof First, let us sort the series by degree, into a sum of homogeneous nc polynomials: ∞  .

k=0

⎛ ⎝



⎞ cα Xα ⎠

(9.4)

|α|=k

Let us now fix a degree k and try to find a useful estimate of the sum  .

cα Xα

|α|=k

We can write this finite sum as a row of X’s times a column of c’s: ⎞ cα I ⎟  ⎜  ⎜cβ I ⎟ cα Xα = Xα Xβ · · · Xγ ⎜ . ⎟ ⎝ .. ⎠ ⎛

.

(9.5)

|α|=k

cγ I where I is the identity matrix of the same size as X. Then ⎛ ⎞  cα I        ⎜cβ I ⎟     ⎜   ⎟ α β γ . cα Xα  ⎜ . ⎟ .     X X ··· X  .   ⎝ ⎠ |α|=k   .   c I  γ ⎛ ⎞1/2    α β =  X X · · · Xγ  ⎝ |cα |2 ⎠

(9.6)

(9.7)

|α|=k

To continue we make the following claim: for any d-tuple of matrices .X = (X1 , . . . , Xd ), and any .k  1, we have    Xα Xβ · · · Xγ   Xkrow .

.

(9.8)

Recall that in the matrix on the right, the indices .α, β, . . . run over all words in d letters of length exactly k. We prove the claim in the case .d = 2 and .k = 2, the reader may easily supply the inductive argument for the general case. Putting  .Y = X1 X1 X1 X2 X2 X1 X2 X2 , we have Y Y ∗ = X1 (X1 X1∗ + X2 X2∗ )X1∗ + X2 (X1 X1∗ + X2 X2∗ )X2∗    X1 X2row I X1∗ + X2 X2row I X2∗

.

 X4row I

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  We conclude that . X1 X1 X1 X2 X2 X1 X1 X1   X2row . With the claim in hand, we can finish the proof. Inserting the estimate we have just obtained into (9.6), we find   ⎛ ⎞1/2        α β γ ⎝ . cα Xα  |cα |2 ⎠     X X ··· X  |α|=k  |α|=k ⎛  Xkrow ⎝



⎞1/2 |cα |2 ⎠

.

|α|=k

Letting the word length k run from m to n, summing, and applying (9.8) and Cauchy-Schwarz, we obtain   ⎛ ⎞1/2   n       . cα Xα  Xkrow ⎝ |cα |2 ⎠   mkn |α|=k  k=m |α|=k  

n 

k=m

1/2 ⎛ X2k row

⎝

n  

⎞1/2 |cα |2 ⎠

k=m |α|=k

We now see immediately that, if .Xrow < 1, the original series will be Cauchy in norm, hence convergent, and we will have the estimate  2       α 2 −1 2 . cα X   (1 − Xrow ) |cα | .  α  α

(9.9) 

Of course, we could have run the proof differently—instead of (9.5), we could have written ⎞ Xα β⎟  ⎜  ⎜X ⎟ cα Xα = cα I cβ I · · · cγ I ⎜ . ⎟ ⎝ .. ⎠ ⎛

.

|α|=k

(9.10)

Xγ

Continuing in this way, we would have deduced convergence of the series again, this time under the assumption that X lies in the column ball: X2col := X1∗ X1 + · · · Xd∗ Xd  < 1,

.

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M. T. Jury

again with the estimate like (9.9), but with .Xcol in place of .Xrow . Thus, we conclude the original . 2 series converges both in the row ball and the column ball, but now we are led naturally to look for a more organized way of performing this estimate. For a matrix point .X = (X1 , . . . , Xd ), let us introduce the following quantity: ⎧ ⎫⎞ 1/N  ⎪ ⎪ ⎨ ⎬     ⎟ ⎜ α 2  .ρ(X) := lim sup ⎝sup c X : |c | = 1 ⎠. α α   ⎪ ⎪ N →∞  ⎩|α|=N ⎭ |α|=N ⎛

(9.11)

Using the estimates above, one quickly sees that .ρ(X)  Xrow and also .ρ(X)  Xcol . More generally, we have: Proposition 9.5.2 The series (9.3) converges whenever .ρ(X) < 1. Proof If .ρ(X) = r < R < 1, then by the definition of .ρ(X) we have that, for all N sufficiently large,   ⎛ ⎞1/2      N⎝ . cα Xα  |cα |2 ⎠ .  R |α|=N  |α|=N Since .R < 1, the same estimates as in the previous proof allow us to control the tail of the series by a geometric series.  While the quantity (9.11) is somewhat daunting in appearance, one may observe that, in one variable, it reduces to the familiar Gelfand formula for the spectral radius of the matrix X: rad(X) = lim XN 1/N .

.

N →∞

The quantity .ρ(X) is called the outer spectral radius of .X = (X1 , . . . , Xd ). In fact, the quantity .ρ(X) can be expressed in terms of the spectral radius of a linear transformation. Namely, if X is an .n × n tuple and .X denotes the linear transformation on .Mn defined by the formula X (T ) :=

d 

.

Xj T Xj∗ ,

j =1

then it can be shown [20, 23] that ρ(X) = (rad(X ))1/2 ,

.

9 A Brief Introduction to Noncommutative Function Theory

325

and also that the linear transformation .X is represented, in a suitable basis, by the n2 × n2 matrix

.

d  .

Xj ⊗ Xj .

j =1

(Here .A denotes the entrywise complex conjugate of the matrix A.) Thus, we have ⎛ ρ(X) = ⎝rad(

d 

.

⎞1/2 Xj ⊗ Xj )⎠

.

j =1

The connection back to row contractions is supplied by the following nc multivariate version of the Rota-Strang theorem, due originally to Popescu [23] (see also Pascoe [20] and Salomon et al. [24]): Theorem 9.5.3 A matrix tuple X is jointly similar to a strict row contraction if and only if .ρ(X) < 1. In other words, the spectral radius ball .{X ∈ Md : ρ(X) < 1} is an nc domain, and it forms the similarity hull of the row ball .{X ∈ Md : Xrow < 1}. (That is, it is the smallest nc domain which contains the row ball, and is invariant under similarities .X → S −1 XS.) It will follow that any nc function on the spectral radius ball is completely determined by its values on the row ball.

9.6 Reproducing Kernels In the last section we saw that an nc power series with square summable coefficients f (X) =



.

α

cα Xα ,



|cα |2 < ∞

(9.12)

α

converges for all X in the row ball, with the estimate f (X) 

.

(1 − X2row )1/2

 

1/2 |cα |

2

.

(9.13)

α

This implies that for such a series, the mapping .X → f (X) defines a (locally bounded) nc function in the row ball. Indeed, at any X where the series converges, .f (X) is a pointwise limit of nc polynomials, so f respects direct sums and similarities. Furthermore, the estimate (9.13) shows that f is uniformly bounded in the nc domain .Xrow  R < 1, for each .0 < R < 1, and hence is locally

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bounded in the row ball. Now, the collection of such functions, represented by square-summable power series, can be made into a Hilbert space in the obvious way, 2 with series expansions using the . 2 -inner product on the coefficients: for .f, g ∈ Hnc f (Z) =



.

g(Z) =

cα Z α ,

α



dα Z α ,

α

we define f, gHnc2 :=



.

cα dα ,

α

so that the norm is f 2 :=

 

.

1/2 |cα |2

.

α

Of course, in the one-variable case this is just the classical Hardy space .H 2 . Let us 2 (we have suppressed the parameter d in the notation). Now, denote our space .Hnc applying the estimate (9.13) again, we see that for each X in row ball, at level n, the map 2 f → f (X) : Hnc → Mn×n (C)

.

2 and the usual topology on .M is continuous (for the norm topology on .Hnc n×n (C)). 2 We are thus tempted to regard .Hnc as a Hilbert space of nc functions, much as we regard .H 2 as a space of analytic functions in the disk. In that case, the (bounded) point evaluations .f → f (z) give rise to a reproducing kernel, which allows us to make use of all the attendant theory of reproducing kernel Hilbert spaces. We may then of course ask for a theory of reproducing kernel nc Hilbert spaces in our present setting. In turns out to be inconvenient to work with the evaluations .f → f (X) at all the different levels n, since we would like to work instead with linear functionals (taking scalar values), in order to have access to the Riesz representation theorem. We cam accomplish this by evaluating matrix entries. Specifically, let us fix a point W in the row ball at some level n, along with arbitrary vectors .u, x ∈ Cn . We then conclude that for each such triple .(W, u, x), 2 by the recipe we obtain a bounded linear functional on .Hnc

f → x ∗ f (W )u.

.

This is evidently a bounded linear functional, being the composition of the bounded linear maps f → f (W )

.

and

f (W ) → x ∗ f (W )u.

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327

Thus, by the Riesz representation theorem, for each such triple there is a vector2 2 2 .KW,u,x ∈ Hnc so that for all .f ∈ Hnc , we have x ∗ f (W )u = f, Kw,u,x .

(9.14)

.

Now, as expected, we can deduce a formula for the coefficients of the nc power series defining .KW,u,x . If f is given by the series (9.12), with coefficients .cα , then using (9.14) we find x ∗ f (W )u =



.

cα x ∗ W α u = f, KW,u,x .

α

It follows that the .αth coefficient in the expansion of .KW,u,x is x ∗ W α u = u∗ (W α )∗ x.

.

Encouraged by our experience with the classical theory of reproducing kernels, we examine the inner product of two such functions K, hoping to obtain a bivariate expression that is “positive” in some sense. Let us fix two triples .(Z, y, v) and .(W, x, u) (at levels n and m of the row ball, respectively) and compute: KW,u,x , KZ,v,y  =



.

α

=y

∗

y ∗ Z α vu∗ (W α )∗ x.

 

(9.15) 

∗

α ∗

Z [vu ](W ) α

x

(9.16)

α

We are thus evaluating the .(x, y) entry of the matrix in parentheses. The inner expression .[vu∗ ] is a rank-one matrix of size .n × m, and the matrices .Z α and .(W α )∗ are of sizes .n × n and .m × m respectively. Thus, for fixed Z and W we can define a linear map on the vector space of .n × m rectangular matrices by the formula 

Z α P (W α )∗ .

(9.17)

KW,u,x , KZ,v,y  = y ∗ K(Z, W )[vu∗ ]x.

(9.18)

P → K(Z, W )[P ] :=

.

α

We can now write .

2A

word about uniqueness—the vector .KW,u,x is of course uniquely determined by the linear functional, but the linear functional is not uniquely determined by the triple .W, u, x (for example, conjugate by a similarity sufficiently close to the identity, or take direct sums with zeroes).

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and consider the bivariate function .K(Z, W ) on the row ball, which takes each pair of points Z (at level n) and W (at level m) to the linear map .K(Z, W ) on .Mn×m just defined. Is there a sense in which this .K(Z, W ) is “positive”? Yes: fix a finite set of triples .(Zi , vi , yi ), and scalars .ci . Then from (9.18) we have 2      .0   ci KZi ,vi ,yi  .  

(9.19)

i



=

ci cj KZi ,vi ,yi , KZj ,vj ,yj .

(9.20)

ci cj yj∗ K(Zj , Zi )[vj vi∗ ]yi

(9.21)

i,j



=

i,j

Suppose now we keep Z and v the same and only allow y to vary, we then see that the matrix yj∗ K(Z, Z)[vv ∗ ]yi

.

must be positive semidefinite. Now allowing the v’s to vary and summing, we see that yj∗ K(Z, Z)[P ]yi

.

is positive semidefinite for any positive matrix P of the same size as Z, since P can be written as a sum of rank one positive matrices .vv ∗ . This means that, if we consider .W = Z at level n, the map .K(Z, Z) is a completely positive map of the matrix algebra .Mn×n (C) to itself. We therefore call .K(Z, W ) a completely positive nc kernel (or cpnc kernel). A general theory of such kernels has been worked out by Ball, Marx, and Vinnikov [8]; we do not attempt to cover the whole theory here, but we can briefly summarize the key points: for any Hilbert space of nc functions on an nc set ., if the evaluations .f → x ∗ f (W )u are bounded, then the function .K(Z, W ) defined by the relation KW,u,x , KZ,v,y  = y ∗ K(Z, W )[uv ∗ ]x

.

is a cpnc kernel on . × . Conversely, if .K(Z, W ) is a cpnc kernel, then there is a Hilbert space of nc functions with K as its kernel, specifically, for triples .(Z, v, y) and .(W, u, x) we will have an nc function .KW,u,x satisfying y ∗ KW,u,x (Z)v = y ∗ K(Z, W )[uv ∗ ]x

.

and Hilbert space of nc functions so that for all f in the space we have x ∗ f (W )u = f, KW,u,x .

.

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329

Moreover if the kernel is locally bounded in a suitable sense, then the nc functions in the Hilbert space will be locally bounded. 2 of nc square-summable power series. We Let us return now to the space .Hnc computed above the cpnc kernel for this space as KW,u,x (Z) =

.

 (u∗ (W α )∗ x)Z α ,

(9.22)

α

but there is a slightly different way to arrange the calculation that will also be helpful. For the coefficients we have u∗ (W α )∗ x = x ∗ W α u = xW u. α

.

(here .W is the matrix whose entries are the complex conjugates of the entries of W ). So, we can slightly rewrite (9.22) as KW,u,x (Z) =



.

= x ∗ W uZ α. α

α

=



∗

= (x ⊗ idn )

α

(9.23)  

 α

W ⊗Z

α

(u ⊗ idn )

(9.24)

α

where .idn denotes the identity map on .n × n matrices. Now, observe that if .A = (A1 , . . . , Ad ), .B = (B1 , . . . , Bd ) are points at level m and n respectively, we have for each integer .k  1 (A1 ⊗ B1 + · · · + Ad ⊗ Bd )k =



.

Aα ⊗ B α .

|α|=k

(Here .⊗ denotes the Kronecker tensor product of matrices.) Thus, our series (9.23) can be expressed in the form 

 ∞  k (W1 ⊗ Z1 + · · · + Wd ⊗ Zd ) (u ⊗ idn ). .KW,u,x (Z) = (x ⊗ idn ) ∗

k=0

Since .Z, W lie in the row ball, using what we have said above about the joint spectral radius, it is not hard to show that the matrix W Z := W1 ⊗ Z1 + · · · + Wd ⊗ Zd

.

has spectral radius strictly less that 1, and therefore the geometric series converges. We obtain finally  −1 (u ⊗ idn ). KW,u,x (Z) = (x ∗ ⊗ idn ) I − W Z

.

(9.25)

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This bears an obvious resemblance to the Szeg˝o kernel .(1 − wz)−1 for the Hardy space .H 2 in the disk.

9.6.1 The Connection with the Drury-Arveson Space As noted earlier, at level 1 the row ball is the Euclidean unit ball in .Cd : Bd := {(z1 , . . . , zd ) : |z1 |2 + · · · + |zd |2 < 1}.

.

2 are locally bounded, they are holomorphic, and so if we Since the functions in .Hnc 2 to the unit ball at level 1, we obtain a vector space of restrict the functions in .Hnc holomorphic functions in the ball. Which functions do we get? One way to approach the question is via the kernels we have just computed—indeed it should be clear that by considering the kernels only at level 1, we obtain a reproducing kernel Hilbert space on the ball .Bd , and some quick checking shows that the restriction map will 2 onto this space. We can compute the resulting kernel be a partial isometry of .Hnc on the ball in closed form. Indeed, working at level 1 we let .w = (w1 , . . . , wd ) and d .z = (z1 , . . . , zd ) be points in the open unit ball .B , and take .x = y = 1. Then the formula (9.25) reduces to

k(z, w) = (1 − z, w)−1

.

(9.26)

where .z, w := z1 w1 + · · · + zd wd is the usual Hermitian inner product in .Cd . The kernel (9.26) is the kernel for the so-called Drury-Arveson space. (Let us remark that one may object slightly, that this doesn’t actually answer the question we posed— restricting to level 1 gives the Drury-Arveson space .Hd2 , though this picture doesn’t actually tell us which holomorphic functions (beyond the finite linear combinations of kernel functions) lie in the space. But there are other descriptions of the DruryArveson space, obtainable by function-theoretic means.) We can quickly see that more is true: since .Hd2 is a reproducing kernel Hilbert space with kernel (9.26), the finite linear combinations of these kernels are dense in .Hd2 . But to the kernel .kw (z) = (1 − z, w)−1 we can associate the “lifted” nc function .Kw (Z) = (1 − wZ)−1 , and by the way everything is constructed, for inner products of kernels at level 1 we have kw , kz H 2 = Kw , Kz Hnc2 ,

.

d

and therefore the “free lift” map .kw → Kw , extends by linearity to a dense subspace 2 . Let us summarize of .Hd2 , and on that dense subspace is an isometry of .Hd2 into .Hnc our discussion in the following proposition:

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2 onto Proposition 9.6.1 Restriction to level 1 induces a partial isometry from .Hnc 2 2 the Drury-Arveson space .Hd . Conversely, every .f ∈ Hd has a unique norm2 . preserving free lift to an nc function .F ∈ Hnc

The reader may verify that the free lift can be computed explicitly at the level of Taylor (or Taylor–Taylor) coefficients: for example, let us consider the monomial 2 .z z2 in the 2-variable Drury-Arveson space. There are evidently three different nc 1 monomials which restrict to this one at level 1, namely Z12 Z2 , Z1 Z2 Z1 , and Z2 Z12 .

.

It turns out that the norm preserving free lift of any monomial is given by averaging (uniformly) over all possible monomial lifts, so for example the free lift of .z12 z2 is .

1 2 Z1 Z2 + Z1 Z2 Z1 + Z2 Z12 . 3

9.7 nc Rational Functions The reproducing kernels (9.25) obtained in the last section are, of course, nc functions in the row ball, and representable as power series (which is how we obtained them). However they also belong to a more special class of nc functions, called the nc rational functions. If we wanted to invent (or discover) a definition of nc rational function, by analogy with the classical, commutative setting, we should count polynomials among the rational functions, as well as expressions obtained from polynomials by allowing multiplicative inversion (subject to restrictions on the domain of the resulting expression, if necessary). We will do basically the same thing in the nc setting: roughly speaking an nc rational expression will be a syntactically correct formal expression involving nc monomials, scalars, the arithmetic signs .+ and .·, parentheses, and inversions .(·)−1 . So, things like (2 − (1 − X1 X2 )−1 )−1 , (X1 X2 − X2 X1 )−1 ,

.

etc.

will be nc rational expressions. The domain of a rational expression will be the set of all points X, at any matrix level, at which the expression is defined, that is, at which all inverses appearing in the expression exist. For example, .(0, 0) is in the domain of the first rational expression above, but not the second. In fact the domain of the second expression is empty at level 1. Our primary interest will be in 2 , by necessity the domain of such rational expressions which define members of .Hnc an expression must contain the row ball, and so in particular must contain 0. Luckily, for rational expressions containing 0 in their domain, there is a good “realization” theory, which turns out to be quite useful.

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A (monic) linear pencil is an expression of the form LA (x1 , . . . , xd ) = In −

d 

.

Aj xj

j =1

for some .n × n matrices .A1 , . . . , Ad and formal indeterminates .x1 , . . . , xd . For a point .X = (X1 , . . . , Xd ) at level m in the matrix universe, we evaluate .LA (x) at X by the recipe LA (X1 , . . . , Xd ) := In ⊗ Im −

d 

.

Aj ⊗ Xj .

(9.27)

j =1

If we fix the .A s, and consider the expression .LA (X)−1 , then we see that the expression in the right hand side of (9.27) will be invertible for all X sufficiently close to 0, at any level. It is then not too hard to show that if we fix vectors .b, c ∈ Cn , then r(X) = (c∗ ⊗ id)(LA (X))−1 (b ⊗ id)

.

resolves to an nc rational expression in the X’s. Importantly, there is a converse for nc rational expressions which contain 0 in their domain: Theorem 9.7.1 ([25, 32]) If .r(x) is a rational expression and .0 ∈ Dom(r(x)), then r(x) has a descriptor realization

.

r(x) = c∗ LA (x)−1 b

(9.28)

.

for some monic linear pencil .LA of some size n, and vectors .b, c ∈ Cn . One may check that the set of all .X ∈ Md for which .LA (X)−1 exists is an nc domain containing 0. Given .r(X), the .A, b, c appearing in the realization are far from unique. We call the realization minimal if the size n of the A’s is as small as possible, among all realizations of .r. It turns out that minimal realizations are unique, up to change of basis. Precisely, if .A, b, c and .A , b , c are two minimal realizations (necessarily of the same size n) of the same .r, then there is an invertible .n × n matrix S so that A = S −1 AS,

.

b = S −1 b,

c = S ∗ c.

Moreover, the rational expression given by a minimal realization has a “maximal” domain among all realizations of the expression in a certain sense, see [32]. If .r(x) is a polynomial, the A’s in the pencil of a minimal realization will be jointly nilpotent (The converse is evident: if the A’s are jointly nilpotent, this means .Aα = 0

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for all sufficiently long words .|α|, so that we can expand (9.28) in a geometric series, where all terms of sufficiently high order vanish, giving a polynomial.). We will not prove the theorem here, but we can sketch out how a realization can be constructed in the case of polynomials (in fact the theorem can be proved by adapting this construction to the general rational case). An nc polynomial is a finite ( sum .p(x) = |α|N = x α . Let V be the vector space spanned by the monomial expressions .x α , .|α|  N (that is, the space of all nc polynomials of degree at most N). For each letter .i = 1, . . . , d there is a “backward shift operator” .Ai which “erases” the letter i when .α = iβ begins with i, and annihilates the word entirely otherwise. (So .A1 x1 x2 = x2 and .A1 x2 x1 = 0, etc.) If .c∗ : V → C is the map that evaluates the scalar term of a polynomial (in other words .c∗ p = p(0)), then one sees quickly that .c∗ Ai p is equal to the coefficient of .x1 , .c∗ A2 A1 p is equal to the coefficient of .x1 x2 , and so forth. By their definition, the .Ai are jointly nilpotent, and then a quick computation verifies that for these .Ai , c, and .b = p, we get a realization of p. 2 , by inspecting the formula we obtained Returning the focus to our space .Hnc for the kernel functions (9.25), we that it is in fact a descriptor realization, in other 2 are in fact nc rational functions. (The words, all of the nc kernel functions in .Hnc realizations provided by the kernel formula will not be minimal in general.) One is led naturally to ask for a characterization of all the nc rational functions which lie 2 . In the classical setting, it is easy to see which rational functions belong to in .Hnc 2 .H (D): certainly they must have no poles in .|z| < 1, and a quick integral estimate shows that a rational function with a pole on the unit circle cannot belong to .L2 of the circle. Thus, a rational function belongs to .H 2 (D) if and only if it has no poles in the closed disk, that is, if and only if it is analytic in some disk of radius .R > 1. 2 functions in terms of In the nc setting, we have no (known) characterization of .Hnc boundary values, so it is perhaps not so clear where to begin. Nonetheless, we have the following theorem: Theorem 9.7.2 ([16]) Let .r be an nc rational function with minimal descriptor realization r(Z) = c∗ (I − AZ)−1 b

.

2 if and only if .ρ(A) < 1. Then .r belongs to the nc Hardy space .Hnc

Here .ρ(A) is the joint spectral radius defined in Sect. 9.5. By Theorem 9.5.3, this means that A is jointly similar to a row contraction, let us call it .W . Thus, when 2 .r(Z) is a rational function in .Hnc we can choose a minimal realization of the form r(Z) = x ∗ (I − W Z)−1 u = KW,x,u

.

with W in the row ball. But this is exactly one of our reproducing kernels! We conclude:

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2 are precisely the Corollary 9.7.3 The nc rational functions belonging to .Hnc reproducing kernels .KW,x,u of (9.25).

This may seem odd when compared to the classical Hardy space—the Szeg˝o kernels .kw (z) = (1 − wz)−1 are rational functions, but of course not every rational function in .H 2 (D) is one of these. What is going on is that we are accustomed to thinking of elements of .H 2 (D) as ordinary analytic functions in the disk, but by the usual functional calculus they can be evaluated on any matrix Z of spectral radius 2 .ρ(Z) < 1. Thus, we can extend .H functions to higher matrix “levels” as well. We would then find that we have continuous evaluations .f → f (Z) at every level, and we could construct reproducing kernels for the evaluations .f → y ∗ f (Z)v just as we have above. It then turns out that order is restored: these “higher-level” kernels will be rational functions at level 1, and every rational function in the Hardy space 2 .H (D) is indeed a reproducing kernel at one of these higher-level points. To see this more explicitly, suppose .r(z) is a rational function in the Hardy space, and suppose for simplicity it has a partial fraction expansion r(z) =

n 

.

k=1

bk 1 − λk z

with the .λk distinct and each .|λk | < 1. If we put .A = diag(λ1 , . . . , λn ), take b to be the column vector .b = (b1 , . . . , bn )t , and .c = (1, 1, . . . , 1)t , then this .A, b, c will be a realization of r, and for any .f ∈ H 2 we will have f, rH 2 = c∗ f (A)b.

.

(If r does not have simple poles, then the matrix A in the realization will not be diagonal, but contain a Jordan block corresponding to each higher-order pole.)

2 9.8 The d-Shift and Multipliers of Hnc

One of the main reasons for the interest in the classical Hardy space .H 2 (D) is that 2 of it provides a functional model for the shift operator. Similarly, our space .Hnc square-summable nc power series, viewed as a space of nc functions, provides a (noncommutative) functional model for the d -shift, which we now define. Recall the free monoid .F+ d from our discussion of nc power series in Sect. 9.5. 2 We form the Hilbert space . (F+ d ) with orthonormal basis .{ξα }α∈F+ . To each of the d

letters .i = 1, . . . , d, we associate an operator .Li acting in . 2 (F+ d ), defining it on the basis vectors .ξw by Li ξα = ξiα

.

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and extending linearly. Since the .ξα form an orthonormal basis, one may quickly check that .Li is an isometry. (Of course, when .d = 1, our space is . 2 (N) and then this operator is the usual unilateral shift.) When .d > 1, we have a system .L1 , . . . , Ld of isometries, and from their definition we see that they have orthogonal ranges. Indeed, the range of .Li is the closed span of those .ξα for which .α = iβ for some word .β, and by orthonormality, when .i = j we have .ξiβ ⊥ξj γ for all words .β and .γ . The fact that the .Li are isometries with orthogonal ranges is expressed algebraically by the operator identities L∗i Lj = δij I.

.

(9.29)

Similarly one may check that the adjoints .L∗i act on the basis vectors as follows: if ∗ .α = iβ, then .L ξα = ξβ , whereas if the word .α does not begin with the letter i, i ∗ then .Li ξα = 0. The relations (9.29) are also equivalent to the statement that the row operator L = (L1 , . . . , Ld ),

.

acting from d copies of . 2 (Fd+ ) into one copy of . 2 (F+ d ), is an isometry. Thus L is sometimes called a row isometry, and we will refer to L as the (left) d-shift. The .Li are sometimes also called the (left) creation operators. One could of course also define right d -shift .Ri , .i = 1, . . . , d by appending letters on the right: Ri ξα = ξαi .

.

The .Ri again from a row isometry. One may also observe that, while the .Li do not commute amongst themselves, we have .Li Rj = Rj Li for all .i, j = 1, . . . , d. The study of the d-shift began with works of Arias and Popescu [4] and independently Davidson and Pitts [10], initially the goal was to develop the operator theory of such systems of isometries on the model of the (deep and successful) operator theory of the unilateral shift operator S. However, this initial work was carried out in the context of operator theory and operator algebras, the connections with noncommutative function theory came later. Of particular interest was the free semigroup algebra .Ld , which is the WOT-closed unital operator algebra generated by .I, L1 , . . . , Ld . The motivation for this, of course, is that for the single shift operator S, this operator algebra is naturally identified with the algebra .H ∞ (D) of bounded analytic functions in the disk. Let us now look at the nc functional model. Classically the shift is modeled by multiplication .f (z) → zf (z) on the Hardy space. We can do something very similar in the nc setting: by its definition, the collection of “monomials” .{Z α } is 2 , and so the operators of multiplication by the nc an orthonormal basis for .Hnc coordinate functions MZi : f (Z) → Zi f (Z),

.

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or in power series form Zi f (Z) = Zi



.

cα Z α =



α

cα Z j α

α

evidently form a system of isometries with orthogonal ranges, unitarily equivalent to the ones just defined. Now, for any nc polynomial p, the multiplication operator .f (Z) → p(Z)f (Z) is 2 . The WOT-closure of this algebra of operators evidently a bounded operator in .Hnc is then naturally identified with the free semigroup algebra .Ld . Do the elements of this abstract closure have a more concrete description? The answer is yes; first we need one definition: 2 if Definition 9.8.1 Say an nc function .ϕ(Z) is a left multiplier of .Hnc 2 Mϕ f := ϕ(Z)f (Z) ∈ Hnc

.

2 whenever f ∈ Hnc

2 if and only if Theorem 9.8.2 ([21, 24]) An nc function .ϕ is a left multiplier of .Hnc .ϕ is bounded in the row ball .Zrow < 1. Moreover in this case we have

1) .Mϕ  = supZrow 1 the space .Hd2 does not support any non-constant isometric multipliers. We also note that by standard arguments, such a factorization becomes available for all spaces with complete Nevanlinnna-Pick (CNP) kernel, such as the Dirichlet space in the unit disk. Of course, the factorizations we have obtained thus far are quite abstract, and indeed there is much still not known about them. Let us push in the functiontheoretic direction: one consequence of Beurling’s theorem, combined with the Riesz inner-outer factorization theory, is that one can characterize the cyclic vectors for the unilateral shift: in the .H 2 picture (the correct picture to work in), these are exactly the outer functions. One can prove easily that a polynomial will be outer (or, equivalently, cyclic for the shift) if and only if it has no zeroes in the unit disk .|z| < 1. However, the purely operator-theoretic formulation of the Arias–Popescu/Davidson–Pitts Beurling theorem gives no clue as to what the cyclic vectors for the d-shift will be. It is not even clear, just from that theorem, which nc polynomials are cyclic. Likewise, one very useful feature of the classical inner-outer factorization theory is that it allows one to characterize the zero sets of .H 2 functions (they are the Blaschke sequences) and to divide out zeroes in a norm-controlled way. At present there is nothing like this available in the nc setting, though we can say a few things. Let us first think about what we might mean by zero sets.

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Definition 9.9.7 Let f be an nc function in some nc domain .. Let .Z ∈ n and let y ∈ Cn be any nonzero vector. We say that the pair .(Z, y) is a (left) detailed zero of f if

.

y ∗ f (Z) = 0.

.

The (left) detailed zero set of f is Z(f ) := {(Z, y) : y ∗ f (Z) = 0}

.

2 , from the definition we see immediately that if .(Z, y) is In the particular case of .Hnc a left detailed zero of f , then it will also be a zero of .f ϕ for any right multiplier .ϕ. Thus, the right invariant subspace generated by f inherits all the left detailed zeroes of f . Likewise, given any collection .Z of pairs .(Z, y), with Z in the row ball, the collection 2 {h ∈ Hnc : y ∗ h(Z) = 0 for all (Z, y) ∈ Z}

.

(9.31)

2 . (To see that .M(Z) is closed, will be a closed, right-invariant subspace of .Hnc recall that the evaluation maps .h → h(Z) are continuous.) Thus we can think of .M(Z) as a “zero-based” invariant subspace. (We must be slightly careful, since these subspaces need not be cyclic, so they are described by a more general form of the nc Beurling theorem than we have given above.) However to get a suitable theorem, we must (it seems) depart slightly from the orthodoxy established thus far and consider also the row ball at the “infinite level.” That is, we consider tuples of operators .Z = (Z1 , . . . , Zd ) on an infinite dimensional Hilbert space .H, which satisfy .Z2row := Z1 Z1∗ +· · · Zd Zd∗  < 1. Treating the infinite level requires some technical subtleties which we will suppress for the sake of this discussion, it will suffice to say that Popescu’s functional calculus allows us to evaluate functions in ∞ .Hrow at points at the infinite level. Modulo these technical subtleties, for a collection of .Z of pairs .(Z, y) where Z is a row contraction in a Hilbert space .H (possibly at the infinite level) and y is a vector in .H, we define 2 M(Z) := {h ∈ Hnc : y ∗ h(Z) = 0 for all (Z, y) ∈ Z}

.

(9.32)

We may similarly redefine (if we are careful) the left detailed zero set to include pairs .(Z, y) at the infinite level. Returning to the classical setting for a moment, when we divide out zeroes of an .H 2 function f , we find a Blaschke product .B(z) with the same zeroes as f , this .B(z) is an inner function which divides f . However it may not be the inner factor of f , since f may also be divisible by a singular inner function, that is, an inner function which is nonvanishing in the disk. The refined form of the inner-outer factorization asserts that every .f ∈ Hd2 factors as f (z) = B(Z)S(Z)F (Z)

.

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where B is Blaschke, S is singular inner, and F is outer, and this factorization is unique up to trivial normalization. Returning to the nc setting, let us make two more definitions. Definition 9.9.8 2 . Let .Z be the left detailed zero set (1) Let . be an isometric left multiplier of .Hnc of . (where we include the infinite level), and let .M(Z) be the right-invariant subspace associated to .Z as in (9.32). We say that . is Blaschke if 2 M(Z) = Hnc .

.

(2) We say that a left multiplier .(Z) is nc singular inner if it is isometric (as a left multiplier), and .(Z) takes invertible values in the row ball (including the infinite level). The condition in item (1) says that . is the Beurling inner function generating the invariant subspace associated to its own zero set. It is then not too hard to show that in the classical setting, an inner function is Blaschke according to our definition if and only if it is a Blaschke product. With these definitions, we have: 2 . Then f can be factored as Theorem 9.9.9 ([15]) Let .f ∈ Hnc

f (z) = B(Z)S(Z)F (Z)

.

where .B(Z) is Blaschke, .S(Z) is nc singular inner, and .F (Z) is right cyclic. At present, the “fine structure” of the factors .B, S, F (if there is any) is still mysterious. In particular, we do not know how to give intrinsic characterizations of any of these three classes of functions. However, we are able to obtain some corollaries, particularly if we start with rational or polynomial f (in which case, it turns out, we do not have to deal with the infinite level). In the classical setting, one can quickly prove, for example, that a rational function in the Hardy space cannot have a singular inner factor, and that the inner and outer factors of a rational function are again rational. As an application of the theorem just stated, one can make similar conclusions in the nc case. 2 . Theorem 9.9.10 ([16]) Let f be an nc rational function in .Hnc

(1) in the nc inner-outer factorization f = F,

.

both . and F are rational, and . is Blaschke. If f is an nc polynomial, then F is also an nc polynomial, with .deg(F )  deg(f ). (2) f is cyclic (left or right) if and only if the matrix .f (Z) is nonsingular for all Z in the row ball.

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Let us sketch a quick proof of item (2). Indeed, first note that if f is a right cyclic 2 , then its left detailed zero set must be empty. (Every function in the vector in .Hnc right invariant subspace inherits the left detailed zeroes of f .) Conversely, suppose .f (Z) is nonsingular in the row ball. That is, the left detailed zero set of f is empty, which means that f cannot have a Blaschke factor. Now if f is rational, by (1) it cannot ever have a singular inner factor, and so by Theorem 9.9.9 it must be outer.

9.10 Further Topics 9.10.1 Nevanlinna-Pick Interpolation The classical Nevanlinna-Pick interpolation problem is the following: given complex numbers .z1 , . . . , zn , with all .|zj | < 1, and another set of complex numbers .w1 , . . . , wn , when does there exist a function f , analytic and bounded by 1 in the disk .|z| < 1, such that .f (zj ) = wj for each .j = 1, . . . , n? The answer is that such an f exists if and only if the Pick matrix  .

1 − wi wj 1 − zi z j

n (9.33) i,j =1

is positive semidefinite. (In the most trivial case, a one-point problem will have a solution as long as .|w| < 1). One source of contemporary interest in such problems is the connection with Hilbert function spaces (see the book [1]). One can evidently pose an analogous problem for nc functions in any reasonable nc domain, for example in the row ball: given points .Z1 , . . . , Zn in the row ball (possibly at different levels), and matrices .W1 , . . . , Wn , when does there exist an nc function f , bounded by 1 in the row ball, such that .f (Zj ) = Wj , .j = 1, . . . , n? Evidently the data must be graded: each .Wj must be the same size as the corresponding .Zj . There are also subtler conditions that do not arise in the classical setting...for example, since nc functions respect direct sums, if the point .Zj consists of diagonal matrices, then the corresponding .Wj must be diagonal. (It turns out more generally that .Wj must belong to the (unital) matrix algebra generated by the coordinate matrices of .Zj .) On the other hand, there is a simplification available in this matricial setting, in the sense that every interpolation problem reduces to a one-point problem: since our solutions must be nc functions, which respect direct sums, we will have .f (Zj ) = Wj for all j if and only if .f (⊕Zj ) = ⊕Wj . Of course, one would hope to find some sort of positivity condition, analogous to the positivity of the Pick matrix, that is necessary and sufficient, and indeed such conditions are available. One approach, due to Ball, Marx, and Vinnikov [9], constructs an nc kernel out of the data and shows that the problem has a solution if and only if this kernel is a CPNC kernel as described above. (This approach is quite general and gives criteria in domains besides the row ball.) A somewhat different approach [5], specific to the

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row ball, provides a single matrix built from the interpolation data, and shows that the problem has a solution if and only if this matrix is positive. On the other hand, Nevanlinna-Pick problems are very closely bound up with the realization theory of contractive functions. Classically, using a factorization of the Pick matrix as a Gram matrix, one can (via the so-called “lurking isometry” technique) read off a realization of a function f solving the problem: f (z) = D + C(1 − Az)−1 Bz

.

for matrices or operators .A, B, C, D acting between suitably chosen spaces. The point is that this can be done in such a way that the block operator  .

AB CD



is a contraction, which implies (after some calculation) that f is bounded by 1 in the disk. (See again [1] for details.) As one should realize by now, the advantage of this approach is that it is readily adaptable to the nc setting. For a very general treatment of Pick interpolation, by means of realizations, in the setting of free holomorphic functions, see [2].

9.10.2 nc Measures and Cauchy Integrals Let us mention one particular way to define Cauchy-integral-like-objects in our context. (There are other related ways, which are important in free probability, [30, 31].) Starting with the d-shift .L = (L1 , . . . , Ld ), we let .Td be the operator system formed by taking the norm closure of the set .{p(L) + q(L)∗ } where .p, q range over all nc polynomials. If we did this for for .d = 1 (the unilateral shift S), the resulting object would be a .C ∗ -algebra, the Toeplitz algebra .T, which is orderisomorphic (as an operator system) to the .C ∗ -algebra .C(T) of continuous functions on the unit circle .T = {|z| = 1}. Therefore, every positive linear functional on .T is identified with a positive measure on the circle. So, in our setting we define a (positive) nc measure to be a positive linear functional on the operator system .Td . Given such a .μ, its nc Cauchy transform is the nc function (Cμ)(Z) := (μ ⊗ idn )(I − L∗ Z)−1 .

.

(As always, we use our abbreviated notation .L∗ Z = L∗1 ⊗Z1 +· · ·+L∗d ⊗Zd .) We are thus thinking of .(I − L∗ Z)−1 as an operator-valued nc Cauchy kernel. It is not hard to check that .Cμ is a locally bounded nc function in the row ball. One can extend this notion in various ways, e.g. to nc Herglotz integrals, so that we have the following theorem of Popescu [22]: an nc function .f (Z) satisfies .f (Z) + f (Z)∗  0 for all

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Z in the row ball if and only if there is a (necessarily unique) positive nc measure .μ on .Td such that f (Z) = (μ ⊗ idn )((I + L∗ Z)(I − L∗ Z)−1 ).

.

From here one can investigate nc Hilbert function spaces associated to .μ, deBranges-Rovnyak type spaces, the theory of Clark measures, etc. See [13]. One significant source of complications, however, is that when .d > 1, the object ∗ .Td is only an operator system, not a .C -algebra.

References 1. AGLER, J., AND MCCARTHY, J. E. Pick interpolation and Hilbert function spaces, vol. 44 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002. 2. AGLER, J., AND MCCARTHY, J. E. Pick interpolation for free holomorphic functions. Amer. J. Math. 137, 6 (2015), 1685–1701. 3. AGLER, J., AND MCCARTHY, J. E. The implicit function theorem and free algebraic sets. Trans. Amer. Math. Soc. 368, 5 (2016), 3157–3175. 4. ARIAS, A., AND POPESCU, G. Factorization and reflexivity on Fock spaces. Integral Equations Operator Theory 23, 3 (1995), 268–286. 5. AUGAT, M., JURY, M. T., AND PASCOE, J. E. Effective noncommutative Nevanlinna-Pick interpolation in the row ball, and applications. J. Math. Anal. Appl. 492, 2 (2020), 124457, 21. 6. BALL, J. A., BOLOTNIKOV, V., AND FANG, Q. Schur-class multipliers on the Fock space: de Branges-Rovnyak reproducing kernel spaces and transfer-function realizations. In Operator theory, structured matrices, and dilations, vol. 7 of Theta Ser. Adv. Math. Theta, Bucharest, 2007, pp. 85–114. 7. BALL, J. A., BOLOTNIKOV, V., AND FANG, Q. Transfer-function realization for multipliers of the Arveson space. J. Math. Anal. Appl. 333, 1 (2007), 68–92. 8. BALL, J. A., MARX, G., AND VINNIKOV, V. Noncommutative reproducing kernel Hilbert spaces. J. Funct. Anal. 271, 7 (2016), 1844–1920. 9. BALL, J. A., MARX, G., AND VINNIKOV, V. Interpolation and transfer-function realization for the noncommutative Schur-Agler class. In Operator theory in different settings and related applications, vol. 262 of Oper. Theory Adv. Appl. Birkhäuser/Springer, Cham, 2018, pp. 23– 116. 10. DAVIDSON, K. R., AND PITTS, D. R. The algebraic structure of non-commutative analytic Toeplitz algebras. Math. Ann. 311, 2 (1998), 275–303. 11. DAVIDSON, K. R., AND PITTS, D. R. Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras. Integral Equations Operator Theory 31, 3 (1998), 321–337. 12. HELTON, J. W., KLEP, I., AND MCCULLOUGH, S. Proper analytic free maps. J. Funct. Anal. 260, 5 (2011), 1476–1490. 13. JURY, M. T., AND MARTIN, R. T. W. Non-commutative Clark measures for the free and abelian Toeplitz algebras. J. Math. Anal. Appl. 456, 2 (2017), 1062–1100. 14. JURY, M. T., AND MARTIN, R. T. W. Factorization in weak products of complete pick spaces. Bull. Lond. Math. Soc. 51, 2 (2019), 223–229. 15. JURY, M. T., MARTIN, R. T. W., AND SHAMOVICH, E. Blaschke-singular-outer factorization of free non-commutative functions. Adv. Math. 384 (2021), Paper No. 107720, 42. 16. JURY, M. T., MARTIN, R. T. W., AND SHAMOVICH, E. Non-commutative rational functions in the full Fock space. Trans. Amer. Math. Soc. 374, 9 (2021), 6727–6749.

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17. KALIUZHNYI-VERBOVETSKYI, D. S., AND VINNIKOV, V. Foundations of free noncommutative function theory, vol. 199 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2014. 18. PASCOE, J. E. Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity. https://arxiv.org/abs/2002.07801. 19. PASCOE, J. E. The inverse function theorem and the Jacobian conjecture for free analysis. Math. Z. 278, 3–4 (2014), 987–994. 20. PASCOE, J. E. The outer spectral radius and dynamics of completely positive maps. Israel J. Math. 244, 2 (2021), 945–969. 21. POPESCU, G. Free holomorphic functions on the unit ball of B(H )n . J. Funct. Anal. 241, 1 (2006), 268–333. 22. POPESCU, G. Free holomorphic functions and interpolation. Math. Ann. 342, 1 (2008), 1–30. 23. POPESCU, G. Similarity problems in noncommutative polydomains. J. Funct. Anal. 267, 11 (2014), 4446–4498. 24. SALOMON, G., SHALIT, O. M., AND SHAMOVICH, E. Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball. Trans. Amer. Math. Soc. 370, 12 (2018), 8639–8690. 25. SCHÜTZENBERGER, M. P. On the definition of a family of automata. Information and Control 4 (1961), 245–270. 26. TAYLOR, J. L. The analytic-functional calculus for several commuting operators. Acta Math. 125 (1970), 1–38. 27. TAYLOR, J. L. A joint spectrum for several commuting operators. J. Functional Analysis 6 (1970), 172–191. 28. TAYLOR, J. L. A general framework for a multi-operator functional calculus. Advances in Math. 9 (1972), 183–252. 29. TAYLOR, J. L. Functions of several noncommuting variables. Bull. Amer. Math. Soc. 79 (1973), 1–34. 30. VOICULESCU, D. Free analysis questions. I. Duality transform for the coalgebra of ∂X : B . Int. Math. Res. Not., 16 (2004), 793–822. 31. VOICULESCU, D.-V. Free analysis questions II: the Grassmannian completion and the series expansions at the origin. J. Reine Angew. Math. 645 (2010), 155–236. ˇ C ˇ , J. Matrix coefficient realization theory of noncommutative rational functions. J. 32. VOL CI Algebra 499 (2018), 397–437.

Chapter 10

An Invitation to the Drury–Arveson Space Michael Hartz

Dedicated to the memory of Jörg Eschmeier

2020 Mathematics Subject Classification 46E22, 47A13, 47A20, 47B32, 47L30

10.1 Introduction Let .Bd = {z ∈ Cd : z2 < 1} denote the Euclidean unit ball in .Cd . The Drury– Arveson space .Hd2 is a space of holomorphic functions on .Bd that generalizes the classical Hardy space .H 2 on the unit disc to several variables. Given the remarkable success of the theory of .H 2 , it is natural to study generalizations to several variables. In fact, there are deeper reasons that lead to the Drury–Arveson space. In these notes, we focus on two particular prominent ones: (1) Operator theory: .Hd2 hosts an analogue of the unilateral shift that is universal for commuting row contractions. (2) Function theory: .Hd2 is a universal space among certain Hilbert spaces of functions (complete Pick spaces). Thus, .Hd2 is a universal object in two separate areas. Just as in the case of .H 2 , the theory of .Hd2 turns out to be very rich because of several different viewpoints on this object. We will see that .Hd2 can be

The author was partially supported by a GIF grant and by the Emmy Noether Program of the German Research Foundation (DFG Grant 466012782). M. Hartz () Fachrichtung Mathematik, Universität des Saarlandes, Saarbrücken, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mashreghi (ed.), Lectures on Analytic Function Spaces and their Applications, Fields Institute Monographs 39, https://doi.org/10.1007/978-3-031-33572-3_10

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understood as (a) (b) (c) (d)

a space of power series, a reproducing kernel Hilbert space with a particularly simple kernel, a particular Besov–Sobolev space, and symmetric Fock space.

Different points of view reveal different aspects of .Hd2 , and often a given problem becomes more tractable by choosing the appropriate point of view. The target audience of these notes includes graduate students and newcomers to the subject. In particular, these notes are intended to be an introduction to the Drury–Arveson space and do not aim to give a comprehensive overview. For further information and somewhat different points of view, the reader is referred to the survey articles by Shalit [145] and by Fang and Xia [77]. Material regarding the operator theoretic and operator algebraic aspects of the Drury–Arveson space can be found in Arveson’s original article [29] and in the book by Chen and Guo [48]. For the appearance of the Drury–Arveson space in the context of complete Pick spaces, the reader is referred to the book by Agler and McCarthy [6]. For a harmonic analysis point of view, see the book by Arcozzi, Rochberg, Sawyer and Wick [22]. In an attempt to make these notes more user friendly, a few deliberate choices were made. Firstly, the presentation is often not the most economical one; rather, the goal is to give an idea of how one might come across certain ideas and how various results are connected. Secondly, instead of discussing results in their most general form, we often restrict attention to instructive special cases. In the same vein, simplifying assumptions are made whenever convenient. We end this introduction by mentioning a few early appearances of the Drury– Arveson space .Hd2 . In comparison to some of the other function spaces covered during the Focus Program, .Hd2 was systematically studied only much later. The name “Drury–Arveson space” comes from a 1978 paper of Drury [67] and a 1998 paper of Arveson [29], both of which were motivated by operator theoretic questions. Arveson’s paper brought .Hd2 to prominence, explicitly realized .Hd2 as a natural reproducing kernel Hilbert space, and connected it to symmetric Fock space. Early appearances of .Hd2 can also be found in papers of Lubin [104, 105] and work of Müller and Vasilescu [113]. Around the turn of the millennium, the Drury– Arveson space and related non-commutative objects were also studied by Davidson and Pitts [60] and by Arias and Popescu [23], and the universal role of .Hd2 in the context of interpolation problems was established by Agler and McCarthy [4, 5]. After all these developments, the subject gained a lot of momentum, and has been very active to this day.

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10.2 Several Definitions of Hd2 10.2.1 Hardy Space Preliminaries As the Drury–Arveson space is a generalization of the classical Hardy space .H 2 , let us recall a few basic facts about .H 2 . At the Focus Program, a mini course on the Hardy space was given by Javad Mashreghi. For more background, see also the books [70, 83, 94, 102, 116]. Let .D = {z ∈ C : |z| < 1} be the open unit disc and let .O(D) be the algebra of all holomorphic functions on .D. A common definition of the Hardy space is   H 2 = f ∈ O(D) : f 2H 2 = sup

2π

.

|f (reit )|2

0≤r −1, there exists .C ≥ 0 such that  . |f |2 |R m ϕ|2 (1 − |z|2 )2m−d dV ≤ Cf 2H 2 for all f ∈ Hd2 . Bd

d

Once again, in the language of harmonic analysis, the second condition above means that |R m ϕ|2 (1 − |z|2 )2m−d dV

.

is a Carleson measure for .Hd2 . Proofs of this result can be found in [119, Theorem 3.7] (see also the later paper [46]) and in [13, Theorem 6.3]. To illustrate how the characterization in Theorem 10.3.10 can be used, we consider the following example. Example 10.3.11 Let .d = 2 and let .ϕ ∈ Mult(H22 ) with |ϕ(z)| ≥ ε > 0

.

for all z ∈ B2 .

Is . ϕ1 ∈ Mult(H22 )? Note that the corresponding question for .d = 1 is trivial, as the multiplier algebra is .H ∞ in this case. For .d = 2, we may use the characterization in Theorem 10.3.10 with .m = 1. Indeed, it is clear that . ϕ1 ∈ H ∞ (B2 ). To check the Carleson measure condition for

10 An Invitation to the Drury–Arveson Space

.

1 ϕ,

369

note that  1   Rϕ  1      =  2  ≤ 2 |Rϕ|. R ϕ ϕ ε

.

Thus, the Carleson measure condition for .ϕ implies the Carleson measure condition for . ϕ1 , so . ϕ1 ∈ Mult(H22 ). The reasoning in the previous example can be extended to higher dimensions. Once again, additional work is required since higher order derivatives complicate matters. Theorem 10.3.12 If .ϕ ∈ Mult(Hd2 ) with .|ϕ| ≥ ε > 0 on .Bd , then . ϕ1 ∈ Mult(Hd2 ). This result can be seen as a very special case of the corona theorem for .Hd2 due to Costea, Sawyer and Wick [53], which we will discuss in Sect. 10.5.1. A direct proof of the result above was found by Fang and Xia [75] and by Richter and Sunkes [132]. To obtain a truly function theoretic characterization of multipliers from Theorem 10.3.10, it remains to find a geometric characterization of Carleson measures. This was achieved by Arcozzi, Rochberg and Sawyer; see [21] for the precise statement. Since known function theoretic characterizations of multipliers can be difficult to work with in practice, it is also desirable to have good necessary and sufficient conditions. Fang and Xia considered the condition .

sup w∈Bd

ϕKw  < ∞, Kw 

1 where .Kw (z) = 1−z,w is the reproducing kernel of .Hd2 , which is clearly necessary for .ϕ ∈ Mult(Hd2 ). In [76], they showed that it is not sufficient. The following sufficient condition was shown by Aleman, McCarthy, Richter and the author.

Theorem 10.3.13 For .f ∈ Hd2 , let Vf (z) = 2f, Kz f  − f 2

.

(z ∈ Bd ).

If .Re Vf is bounded in .Bd , then .f ∈ Mult(Hd2 ). The proof can be found in [11, Corollary 4.6]. The function .Vf in the theorem is called the Sarason function of f . If .d = 1, then .Re Vf is the Poisson integral of .|f |2 . In particular, .f ∈ H ∞ if and only if .Re Vf is bounded if .d = 1. Fang and Xia showed that if .d ≥ 2, then boundedness of .Re Vf is not necessary for 2 .f ∈ Mult(H ), see [78] and also [14, Proposition 8.1]. d

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10.3.3 Dilation and von Neumann’s Inequality As alluded to earlier, the tuple .Mz = (Mz1 , . . . , Mzd ) on the Drury–Arveson space plays a key role in multivariable operator theory. We first recall the relevant one variable theory. More background material on dilation theory can be found for instance in [120, 123, 147]. Throughout, we assume that .H is a complex Hilbert space. The following fundamental result due to von Neumann [155] forms the basis of a rich interplay between operator theory and function theory. Theorem 10.3.14 (von Neumann’s Inequality) Let .T ∈ B(H) with .T  ≤ 1. For every polynomial .p ∈ C[z], p(T ) ≤ sup{|p(z)| : |z| ≤ 1}.

.

There are now many different proofs of von Neumann’s inequality, see for instance [123, Chapter 1], [120, Chapters 1 and 2] and [68]. A particularly short proof uses the following dilation theorem due to Sz.-Nagy [149]. Theorem 10.3.15 (Sz.-Nagy Dilation Theorem) Let .T ∈ B(H) with .T  ≤ 1. Then there exists a Hilbert space .K ⊃ H and a unitary .U ∈ B(K) with  p(T ) = PH p(U )H

.

for all p ∈ C[z].

A dilation U can be written down explicitly, see [142]. Notice that Sz.-Nagy’s dilation theorem reduces the proof of von Neumann’s inequality to the case of unitary operators, which in turn easily follows from basic spectral theory as unitary operators are normal operators whose spectrum is contained in the unit circle. This is the general philosophy behind dilation theory: associate to a given operator on Hilbert space a better behaved operator on a larger Hilbert space. This idea had a profound impact on operator theory, see [120] and [150]. There is a slight variant of the Sz.-Nagy dilation theorem, in which unitary operators are replaced by more general isometries, but the relationship between the original operator and its dilation becomes tighter. Theorem 10.3.16 (Sz.-Nagy Dilation Theorem, Second Version) Let .T ∈ B(H) with .T  ≤ 1. Then there exists a Hilbert space .K ⊃ H and an isometry .V ∈ B(K) such that .V ∗ H ⊂ H and  T ∗ = V ∗ H .

.

Note that in the setting of this result, we in particular find that .p(T ) = PH p(V )H for all .p ∈ C[z]. A proof can be found for instance in [120, Chapter 1]. It is not difficult to deduce the two versions of Sz.-Nagy’s dilation theorem from each

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other. For instance, the first version can be obtained from the second by extending the isometry V to a unitary U on a larger Hilbert space, see again [120, Chapter 1]. The conclusion of the first version of Sz.-Nagy’s dilation theorem is often summarized by saying that “every contraction dilates to a unitary”, whereas the second version says that “every contraction co-extends to an isometry”. In terms of operator matrices, the relationship between T , a unitary dilation U and an isometric co-extension V can be understood as ⎡ ∗ 0 ⎣ .U = ∗T ∗ ∗

⎤ 0 0⎦ ∗

(this is a result of Sarason [140, Lemma 0]) and  V =

.

 T 0 , ∗ ∗

see for instance [6, Chapter 10]. Let us now move into the multivariable realm. Let .T = (T1 , . . . , Td ) be a tuple of commuting operators in .B(H): Ti Tj = Tj Ti

.

(i, j = 1, . . . , d).

Just as there is more than one reasonable extension of the unit disc to higher dimensions, there is more than one reasonable contractivity condition in multivariable operator theory. For instance, one might impose the contractivity condition .Tj  ≤ 1 for all j , i.e. one considers tuples of commuting contractions. The operator theory of these tuples connects to function theory on the polydisc .Dd . There is a large body of literature on commuting contractions. Very briefly, this theory works well for .d = 2, thanks to an extension of Sz.-Nagy’s dilation theorem due to Andô [19]. However, both von Neumann’s inequality and Andô’s theorem fail for .d ≥ 3, which is a significant obstacle in the study of three or more commuting contractions, see [120, Chapter 5]. Here, we will consider a different contractivity condition. Definition 10.3.17 An operator tuple .T = (T1 , . . . , Td ) ∈ B(H)d is said to be a row contraction if the row operator

.

 T1 · · · Td : Hd → H,

(xi )di=1 →

d  i=1

is a contraction.

Ti xi ,

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This is equivalent to demanding that d  .

Tj Tj∗ ≤ I.

j =1

Note that a tuple of scalars is a row contraction if and only if it belongs to the closed (Euclidean) unit ball. Thus, one expects that operator theory of commuting row contractions connects to function theory in the unit ball. This is indeed the case, and we will shortly see that row contractions are intimately related to the Drury– Arveson space. Definition 10.3.18 A spherical unitary is a tuple .U = (U1 , . . . , Ud ) of commuting  normal operators with . di=1 Ui Ui∗ = I . Notice that spherical unitaries in dimension one are simply unitaries. Spherical unitaries are essentially well understood thanks to the multivariable spectral theorem. In particular, every spherical unitary is unitarily equivalent to a direct sum of operator tuples of the form .Mz on .L2 (μ), where .μ is a measure supported on .∂Bd , see [6, Appendix D] and [54, Section II.1]. It is not true that every commuting row contraction dilates to a spherical unitary. Indeed, the tuple .Mz on .Hd2 does not, since the multiplier norm on the Drury–Arveson space is not dominated by the supremum norm on the ball by Proposition 10.3.7, but spherical unitaries U satisfy .p(U ) ≤ p∞ for all polynomials p. Instead, we have the following dilation theorem, which is one of the central results in the theory of the Drury–Arveson space. It explains its special place in multivariable operator theory. Theorem 10.3.19 (Dilation Theorem for .Hd2 ) Let .T = (T1 , . . . , Td ) be a commuting row contraction on .H. Then T co-extends to a tuple of the form .S ⊕ U , where U is spherical unitary and S is a direct sum of copies of .Mz on .Hd2 . Explicitly, the co-extension statement means that there exist Hilbert spaces .K and .E, an isometry .V : H → (Hd2 ⊗ E) ⊕ K and a spherical unitary U on .K such that ((Mzi ⊗ IE ) ⊕ Ui )∗ V = V Ti∗

.

for .i = 1, . . . , d. In this case, identifying .H with a subspace of .(Hd2 ⊗ E) ⊕ K via the isometry V , we see that .H is invariant under .((Mzi ⊗ IE ) ⊕ Ui )∗ and that  Ti∗ = ((Mzi ⊗ IE ) ⊕ Ui )∗ H .

.

In particular,  p(T ) = PH p((Mz ⊗ I ) ⊕ U )H

.

for all p ∈ C[z1 , . . . , zd ].

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Various versions of Theorem 10.3.19 were proved by Drury [67], by Müller and Vasilescu [113] and by Arveson [29]. Notice that the case .d = 1 recovers the second version of Sz.-Nagy’s dilation theorem. Indeed, in dimension one, every operator of the form .S ⊕ U is clearly an isometry, and these are in fact all isometries by the Wold decomposition, see for example [54, Theorem V.2.1]. If .d ≥ 2, then there is no direct analogue of the first version of Sz.-Nagy’s dilation theorem. This is closely related to the fact that the identity representation of the algebra generated by .Mz is a boundary representation in the sense of Arveson; see Lemma 7.13 and the discussion following it in [29]. Just as in one variable, the dilation theorem implies an inequality of von Neumann type. Corollary 10.3.20 (Drury’s von Neumann Inequality) Let .T = (T1 , . . . , Td ) be a commuting row contraction. Then p(T ) ≤ pMult(H 2 )

.

d

for all p ∈ C[z1 , . . . , zd ].

This inequality makes clear the importance of the multiplier norm on the Drury– Arveson space. Note that the inequality is sharp, since we obtain equality by taking 2 .T = Mz on .H . d Drury’s von Neumann inequality can be deduced from Theorem 10.3.19 by using that spherical unitaries U satisfy .p(U ) ≤ p∞ ≤ pMult(H 2 ) for all d polynomials p. We will give a more direct argument. The special case of Theorem 10.3.19 in which the spherical unitary summand is absent is important. The key concept is that of purity. Let .T = (T1 , . . . , Td ) be a row contraction. Let θ : B(H) → B(H),

.

A →

d 

Ti ATi∗ .

i=1

Then .θ (I ) ≤ I , and so .I ≥ θ (I ) ≥ θ 2 (I ) ≥ . . . ≥ 0. We say that T is pure if n .limn→∞ θ (I ) = 0 in the strong operator topology. In the pure case, we obtain the following version of the dilation theorem. Theorem 10.3.21 Every pure commuting row contraction co-extends to a direct sum of copies of .Mz on .Hd2 . This result is sufficient for proving Drury’s von Neumann inequality. Proof of Drury’s von Neumann Inequality If T is a row contraction, then .rT is a pure row contraction for .0 ≤ r < 1. Indeed, θ (I ) ≤ r 2 I

.

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and so θ n (I ) ≤ r 2n I,

.

which even converges to zero in norm as .n → ∞. Now, if .p ∈ C[z], then Theorem 10.3.21 implies that p(rT ) ≤ p(Mz ) = pMult(H 2 ) .

.

d

Now let .r  1.



We now provide a proof of the dilation theorem in the pure case. Proof of Theorem 10.3.21 Since T is a row contraction, we may define d

1/2  . = I − Ti Ti∗ = (I − θ (I ))1/2 . i=1

Let V : H → Hd2 ⊗ H,

.

Vh =

 |α|! zα ⊗ (T ∗ )α h. α! d

α∈N

Using purity, we find that V h2 =

.

N  |α|!  T α 2 (T ∗ )α h, h = lim (θ n (I ) − θ n+1 (I ))h, h N →∞ α! d n=0

α∈N

= h2 − lim θ N +1 (I )h, h = h2 . N →∞

Hence V is an isometry. Finally, the formula for .Mz∗i in Example 10.3.2 shows that |α|! α |α − ei |! α−ei z z = (α − ei )! α!

Mz∗i

.

if .αi = 0, so (Mz∗i ⊗ I )V = V Ti∗ .

.

 Thus, .ran(V ) is invariant under .Mz∗i ⊗ I and .Mz∗i ⊗ I ran(V ) is unitarily equivalent to .Ti∗ . 

We will give a proof of the dilation theorem in the non-pure case in Sect. 10.3.5.

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Remark 10.3.22 Since the multiplier norm of .Hd2 and the supremum norm are not comparable if .d ≥ 2 (Proposition 10.3.7), the right-hand side in Drury’s von Neumann inequality cannot be replaced with a constant times the supremum norm. One might nonetheless ask if this is possible if one only considers row contractive matrix tuples of a fixed matrix size. It was shown by Richter, Shalit and the author that this can be done: For every .d, n ∈ N, there exists a constant .C(d, n) so that for every row contraction T consisting of d commuting .n × n matrices and every polynomial p, the inequality p(T ) ≤ C(d, n) sup |p(z)|

.

z∈Bd

holds. In fact, one can even take .C(d, n) to be independent of d; see [93].

10.3.4 The Non-commutative Approach to Multipliers The approach to the Drury–Arveson space using non-commutative functions outlined in Sect. 10.2.7 is also very useful in the context of multipliers. Define   ∞ 2 Hnc = ∈ Hnc : sup :  (X) < ∞ .

.

X∈Bnc d

∞ defines a left multiplication operator Each . ∈ Hnc 2 2 L : Hnc → Hnc ,

.

F → F,

and .L  = supX∈Bnc  (X). In fact, every left multiplication operator is of d ∞ as a nonthis form; see [139, Section 3] for more details. Thus, we think of .Hnc 2 commutative analogue of .Mult(Hd ). Of particular importance are the left multiplication operators by the variables .Li := Lxi . These are isometries with pairwise orthogonal ranges. The space .{L : ∞ } turns out to be the WOT-closed algebra generated by .L , . . . , L . This

∈ Hnc 1 d algebra is known as the non-commutative analytic Toeplitz algebra .Ld and has been the subject of intense study, in particular by Davidson and Pitts [59–61], by Popescu [126–128] and by Arias and Popescu [23], to only name a few references. It is also one of the Hardy algebras of Muhly and Solel [112]. There is a version of the dilation theorem for .Hd2 in the non-commutative context, due to Bunce [43], Frazho [81] and Popescu [124, 125]. Theorem 10.3.23 Every (not necessarily commuting) row contraction T coextends to a tuple of isometries with pairwise orthogonal ranges. If T is pure, then the co-extension can be chosen to be a direct sum of copies of the tuple 2 .(L1 , . . . , Ld ) on .Hnc .

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∞ and .Mult(H 2 ) is given by the following result of The connection between .Hnc d Davidson and Pitts; see Corollary 2.3 and Section 4 in [60]. It is an analogue of the (much simpler) Hilbert space result Proposition 10.2.15.

Theorem 10.3.24 The map ∞ R : Hnc → Mult(Hd2 ),

.



→ B , d

∞ with is a complete quotient map. If .ϕ ∈ Mult(Hd2 ), then there exists . ∈ Hnc

B = ϕ and . Hnc∞ = ϕMult(H 2 ) .

.

d

d

∞ onto the open unit ball Quotient map means that R maps the open unit ball of .Hnc of .Mult(Hd2 ), and the modifier “completely” means that the same property holds for ∞ ) → M (Mult(H 2 )), defined by applying R entrywise. In all induced maps .Mn (Hnc n d particular, complete quotient maps are completely contractive. For background on these notions, see [120]. The fact that every multiplier has a norm preserving lift can 2 of Frazho [82] and Popescu be deduced from the commutant lifting theorem for .Hnc [124]; see also [58]. There are questions that turn out to be simpler in the non-commutative setting 2 than in the commutative setting of .H 2 . For instance, one of the advantages of .Hnc d in the non-commutative setting is the fact that the operators .Li are isometries with pairwise orthogonal ranges. This leads to a powerful Beurling theorem, due to Arias and Popescu [23, Theorem 2.3] and Davidson and Pitts [61, Theorem 2.1]. Jury and Martin used this non-commutative Beurling theorem to answer open questions in the commutative setting; see [95, 96]. The passage from the non-commutative to the commutative setting is made possible by Proposition 10.2.15 and Theorem 10.3.24.

10.3.5 The Toeplitz Algebra In his proof of the dilation theorem in the non-pure case, Arveson made use of the Toeplitz .C ∗ -algebra .Td , which is defined to be the unital .C ∗ -subalgebra of .B(Hd2 ) generated by .Mz1 , . . . , Mzd . A key result about .Td is the following theorem, again due to Arveson, which is [31, Theorem 5.7]. We let .K denote the ideal of compact operators in .B(Hd2 ). Theorem 10.3.25 There is a short exact sequence of .C ∗ -algebras 0 −→ K −→ Td −→ C(∂Bd ) −→ 0,

.

where the first map is the inclusion and the second map sends .Mzi to .zi .

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Proof As remarked in Example 10.3.2, we have d  .

Mzi Mz∗i = I − P0 ,

(10.15)

i=1

where .P0 denotes the orthogonal projection onto the constant functions. Hence .P0 ∈ Td . A little computation shows that if .p, q are polynomials in d variables, then (Mp P0 Mq∗ )(f ) = f, qp

.

for all f ∈ Hd2 .

Hence .Td contains all rank one operators of the form .·, qp for polynomials p and q, and therefore .K ⊂ Td . It remains to identify the quotient .Td /K. The formula for the action .Mz∗i on monomials in Example 10.3.2 shows that (Mz∗i Mzi − Mzi Mz∗i )zα =

.

α +1 αi α |α| − αi α i − z = z . |α| + 1 |α| |α|(|α| + 1)

for all .α ∈ Nd \ {0}. Hence, with respect to the orthogonal basis of monomials, the operator .Mz∗i Mzi − Mzi Mz∗i is a diagonal operator whose diagonal tends to zero, and so it is compact. Therefore, the quotient .C ∗ -algebra .Td /K is generated by the d commuting normal elements .Ni = Mzi + K and so it is commutative by Fuglede’s theorem; see [135, Theorem 12.16]. (Alternatively, one can compute directly that ∗ ∗ .Mz Mzi − Mzi Mz is compact for all .i, j = 1, . . . , d). j j By the Gelfand–Naimark theorem (see [54, Section I.3] or [31, Section 2.2]), the commutative .C ∗ -algebra .Td /K is isomorphic to .C(K) for some compact Hausdorff space K. More precisely, if K = {(χ (N1 ), . . . , χ (Nd )) : χ character of Td /K} ⊂ Cd ,

.

then the maximal ideal space of .Td /K is identified with K, and modulo this identification, the Gelfand transform takes the form Td /K → C(K),

.

Mzi + K → zi ,

so this map is a .∗-isomorphism. Finally, from (10.15), it follows that K is a non-empty subset of .∂Bd . Unitary invariance of .Hd2 (Proposition 10.2.9) then implies that .K = ∂Bd , which gives the result. 

We now discuss how the dilation theorem in the pure case (Theorem 10.3.21) and Theorem 10.3.25 can be used to prove the dilation theorem in general (Theorem 10.3.19). The starting point is the observation that if T is a row contraction, then .rT is a pure row contraction for all .0 ≤ r < 1, see the proof of Drury’s von Neumann inequality. Thus, the dilation theorem in the pure case yields for each

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M. Hartz

r < 1 a dilation of .rT , and one would like to take some kind of limit as .r  1. Doing this directly is tricky, as the individual dilations are difficult to control. There is a different approach due to Arveson [26], which in general relies on Arveson’s extension theorem [25] and Stinespring’s dilation theorem [148]. A comprehensive exposition of this technique can be found in [120, Chapter 7] and [147, Section 7]. Briefly, Arveson showed that the existence of dilations is equivalent to certain maps being completely contractive or completely positive, and these behave well with respect to taking limits. For instance, we saw that Sz.-Nagy’s dilation theorem implies von Neumann’s inequality. Arveson’s technique allows us to go back and deduce Sz.-Nagy’s dilation theorem from von Neumann’s inequality (for matrix polynomials). In our setting, the argument runs as follows.

.

Proof of Theorem 10.3.19 Let T be a commuting row contraction on .H and let S = span{Mzα (Mzβ )∗ : α, β ∈ Nd } ⊂ Td .

.

For each .0 ≤ r < 1, the tuple .rT is a pure row contraction, so by Theorem 10.3.21, there exists an isometry .Vr : H → Hd2 ⊗ Er such that (Mzi ⊗ I )∗ Vr = Vr rTi∗

.

(10.16)

for .i = 1, . . . , d. Define ϕr : S → B(H),

.

X → Vr∗ XVr .

Then .ϕr is a unital and completely positive, meaning that it maps positive operators to positive operators, and the same is true for all maps .Mn (S) → Mn (B(H)) defined by applying .ϕr entrywise. Moreover (10.16) implies that ϕr (Mzα (Mzβ )∗ ) = (rT )α ((rT )β )∗

.

Hence, for each .X ∈ S, the limit .ϕ(X) := limr→1 ϕr (X) exists, and this defines a unital completely positive map .ϕ : S → B(H) with ϕ(Mzα (Mzβ )∗ ) = T α (T β )∗

.

for all α, β ∈ Nd .

By Arveson’s extension theorem and Stinespring’s dilation theorem (see [120, Theorem 7.5 and Theorem 4.1]), there exist a Hilbert space .L, an isometry .V : H → L and a unital .∗-homomorphism .π : Td → B(L) such that V ∗ π(Mzα (Mzβ )∗ )V = T α (T β )∗

.

for all α, β ∈ Nd .

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We claim that the range of V is invariant under .π(Mzi )∗ for each i. To see this, note that V ∗ π(Mzi Mz∗i )V = Ti Ti∗ = V ∗ π(Mzi )V V ∗ π(Mzi )∗ V

.

and so 0 = V ∗ π(Mzi )(I − V V ∗ )π(Mzi )∗ V

.

= ((I − V V ∗ )π(Mzi )∗ V )∗ ((I − V V ∗ )π(Mzi )∗ V ), hence .(I − V V ∗ )π(Mzi )∗ V = 0. This shows invariance of the range of V under ∗ .π(Mzi ) . Thus, π(Mzi )∗ V = V V ∗ π(Mzi )∗ V = V Ti∗ .

.

It remains to show that the tuple .π(Mz ) is unitarily equivalent to .S ⊕ U , where S is a direct sum of copies of .Mz and U is spherical unitary. To this end, we use the structure of the Toeplitz .C ∗ -algebra observed in Theorem 10.3.25 and basic representation theory of .C ∗ -algebras (see the discussion preceding Theorem I.3.4 in [28]) to conclude that .π : Td → B(L) splits as a direct sum .π = π1 ⊕ π2 . Here, .π1 is induced by a representation of .K and hence is unitarily equivalent to a multiple of the identity representation of .Td on .B(Hd2 ), and .π2 factors through the quotient .Td /K ∼ = C(∂Bd ). Thus, .π1 (Mz ) is unitarily equivalent to a direct sum of copies of .Mz , and .π2 (Mz ) is spherical unitary, as desired. 

Remark 10.3.26 (a) Examination of the proof of Theorem 10.3.25 shows that the operator system .S used in the proof of Theorem 10.3.19 is dense in the .C ∗ -algebra .Td . Thus, the use of Arveson’s extension theorem could be avoided. In turn, this leads to a uniqueness statement about co-extensions of the form .S ⊕ U ; see [29, Theorem 8.5] for details. (b) It was remarked before the proof of Theorem 10.3.19 that Arveson’s technique makes it possible to deduce dilation theorems from von Neumann type inequalities. This can also be done here. Drury’s von Neumann inequality shows that the map Td ⊃ span{Mzα : α ∈ Nd } → B(H),

.

Mzα → T α ,

is contractive, and the same proof shows that it is in fact completely contractive. Applying Arveson’s extension theorem and Stinespring’s dilation theorem to this map and using Theorem 10.3.25 yields a dilation of T of the form .S ⊕ U . The advantage of working with the larger operator system .S in the proof is that we not only obtain a dilation, but a co-extension. This idea already appeared in work of Agler [1]. For the relationship between dilations and co-extensions in

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M. Hartz

the context of reproducing kernel Hilbert spaces, the reader is also referred to [50]. (c) The proof of the dilation theorem in the non-pure case of Müller and Vasilescu, see [113, Theorem 11], does not rely on the representation theory of the Toeplitz ∗ ∗ .C -algebra. Instead, Müller and Vasilescu construct by hand an extension of .T d ∗ ∗ of the form .(Mz ⊗ I ) ⊕ V , where V is a spherical isometry, i.e. . i=1 Vi Vi . They then use an earlier result of Athavale [32] to extend the spherical isometry to a spherical unitary. Another proof of the dilation theorem was given by Richter and Sundberg [130]. Their proof uses Agler’s theory of families and extremal operator tuples [2]. The short exact sequence in Theorem 10.3.25 encodes in particular the basic fact that the tuple .Mz on .Hd2 consists of essentially normal operators, i.e. that .Mz∗i Mzi − Mzi Mz∗i is a compact operator for each .1 ≤ i ≤ d. Let .I ⊂ C[z1 , . . . , zd ] be a homogeneous ideal. To avoid certain trivialities, we assume I to be of infinite co-dimension. By Hilbert’s Nullstellensatz, this is equivalent to demanding that the vanishing locus V (I ) = {z ∈ Cd : p(z) = 0 for all p ∈ I }

.

does not consist only of the origin. On .I ⊥ = Hd2  I , we define the operator tuple I .S = (S1 , . . . , Sd ) by  Si = PI ⊥ Mzi I ⊥ .

.

Since I is invariant under .Mz , the tuple .S I is still a tuple of commuting operators. A famous conjecture of Arveson asserts that .S I should also be essentially normal. In fact, Arveson made a stronger conjecture [30], which was further refined by Douglas [65] in the following form. Conjecture 10.3.27 (Arveson–Douglas Essential Normality Conjecture) The commutators .Sj Sk∗ − Sk∗ Sj belong to the Schatten class .Sp for all .p > dim V (I ) and all .1 ≤ j, k ≤ d. Arveson’s initial motivation came from his work on the what is known as the curvature invariant [30], but there are other reasons for considering this conjecture. For instance, if the conjecture is true, then letting .TI denote the .C ∗ -algebra generated by S, one obtains a short exact sequence 0 −→ K −→ TI −→ C(V (I ) ∩ ∂Bd ) −→ 0,

.

see for instance [86, Section 5]. This would be analogous to the short exact sequence in Theorem 10.3.25. The vision of Arveson and Douglas was, very roughly speaking, to connect operator theory of the tuple .S I to algebraic geometry of the variety .V (I ).

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381

The conjecture remains open in general, but it has been verified in special cases. In particular, Guo and Wang showed the following result; see Proposition 4.2 and Theorem 2.2 in [86], see also [73, Theorem 2.3] and [146, Theorem 4.2]. Theorem 10.3.28 The Arveson–Douglas conjecture holds for .S I in each of the following cases: (a) .d ≤ 3; (b) I is a principal homogeneous ideal. We only mention one more result, which was independently shown by Engliš and Eschmeier [72] and by Douglas, Tang and Yu [66].  Theorem 10.3.29 If I is a radical ideal (i.e. .I = {p : pV (I ) = 0}) and .V (I ) is smooth away from 0, then the Arveson–Douglas conjecture holds for .S I . For a thorough discussion of the Arveson–Douglas essential normality conjecture, see [145, Section 10] and [77, Section 5].

10.3.6 Functional Calculus If T is a tuple of commuting operators, then there is no issue in making sense of p(T ) for a polynomial .p ∈ C[z1 , . . . , zd ]. However, it is often desirable to extend the supply of functions f for which one can make sense of .f (T ); this is called a functional calculus for T . Let .Ad be the multiplier norm closure of the polynomials in .Mult(Hd2 ). Then Drury’s von Neumann inequality immediately implies that every commuting row contraction admits an .Ad -functional calculus. More precisely, we obtain:

.

Theorem 10.3.30 If T is a commuting row contraction on .H, then there exists a unital completely contractive homomorphism Ad → B(H),

.

p → p(T )

(p ∈ C[z1 , . . . , zd ]).

If .d = 1, then the functional calculus above is the disc algebra functional calculus of a contraction. Classically, the next step beyond the disc algebra is .H ∞ . Since functions in .H ∞ only have boundary values almost everywhere on the unit circle, one has to impose an additional condition on the contraction. Indeed, the scalar 1 will not admit a reasonable .H ∞ -functional calculus. Sz.-Nagy and Foias showed that every completely non-unitary contraction (i.e. contraction without unitary direct summand) admits an .H ∞ -functional calculus; see [150, Theorem III.2.1]. This is sufficient for essentially all applications, since every contraction T decomposes as .T = Tcnu ⊕ U for a completely non-unitary contraction .Tcnu and a unitary U , and the unitary part can be analyzed using the spectral theorem. The .H ∞ -functional calculus has been used very successfully for instance in the search for invariant subspaces [41, 42].

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M. Hartz

We have now seen that for higher dimensions and commuting row contractions, the natural replacement for .H ∞ is the multiplier algebra of the Drury–Arveson space. We say that a commuting row contraction T is completely non-unitary if it has no spherical unitary summand. Once again, one can show that every commuting row contraction decomposes as .T = Tcnu ⊕ U for a completely non-unitary tuple .Tcnu and a spherical unitary tuple U ; see [49, Theorem 4.1]. The following theorem, due to Clouâtre and Davidson [49], is then a complete generalization of the Sz.-Nagy–Foias .H ∞ -functional calculus to higher dimensions. Theorem 10.3.31 If T is a completely non-unitary commuting row contraction on .H, then T admits a .Mult(Hd2 )-functional calculus, i.e. there exists a weak-.∗ continuous unital completely contractive homomorphism .

Mult(Hd2 ) → B(H),

p → p(T )

(p ∈ C[z1 , . . . , zd ]).

A different proof, along with a generalization to other reproducing kernel Hilbert spaces, was given by Bickel, McCarthy and the author [37]. Remark 10.3.32 Every pure row contraction is completely non-unitary, but the converse is false, even if .d = 1. For instance, the backwards shift on .2 is completely non-unitary, but not pure. For pure row contractions, the .Mult(Hd2 )-functional calculus can be constructed directly with the help of the dilation theorem (Theorem 10.3.21). Indeed, if T is a pure commuting row contraction on .H, then by the dilation theorem, there exists an isometry .V : H → Hd2 ⊗ E with (Mz∗i ⊗ I )V = V Ti∗

.

(i = 1, . . . , d).

Define

: Mult(Hd2 ) → B(H),

.

ϕ → V ∗ (Mϕ ⊗ I )V .

Using weak-.∗ density of the polynomials in .Mult(Hd2 ), it is straightforward to check that . satisfies the conclusion of Theorem 10.3.31.

10.4 Complete Pick Spaces 10.4.1 Pick’s Theorem and Complete Pick Spaces In addition to its special role in multivariable operator theory, the Drury–Arveson space also plays a central role in the study of a particular class of reproducing kernel Hilbert spaces, called complete Pick spaces. For in-depth information on this topic, see [6]. The definition of complete Pick spaces is motivated by the following classical interpolation theorem from complex analysis due to Pick [122] and Nevanlinna [115].

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383

Theorem 10.4.1 Let .z1 , . . . , zn ∈ D and .λ1 , . . . , λn ∈ C. There exists .ϕ ∈ H ∞ with ϕ(zi ) = λi for 1 ≤ i ≤ n

.

and

ϕ∞ ≤ 1

if and only if the matrix  1 − λ λ n i j 1 − zi zj i,j =1

.

is positive semi-definite. The matrix appearing in Theorem 10.4.1 is nowadays usually called the Pick matrix. Example 10.4.2 (a) If .n = 1, i.e. we have a one point interpolation problem .z1 → λ1 , then the Pick matrix is positive semi-definite if and only if .|λ1 | ≤ 1. This observation also shows that in general, positivity of the Pick matrix implies that .|λi | ≤ 1 for each i, which is obviously a necessary condition for interpolation. (b) Let .n = 2 and consider the two point interpolation problem .zi → λi for .i = 1, 2. Let us assume that .|λi | ≤ 1 for each i. The Pick matrix takes the form ⎡ P =

.

1−|λ1 |2 ⎣ 1−|z1 |2 1−λ2 λ1 1−z2 z1

⎤

1−λ1 λ2 1−z1 z2 ⎦ . 1−|λ2 |2 2 1−|z2 |

Notice that the two diagonal entries are non-negative. We distinguish two cases. If .|λ1 | = 1, then the .(1, 1) entry of P is equal to zero, so P is positive if and only if the off-diagonal entries are equal to zero. Since .|λ1 | = 1, this happens if and only if .λ2 = λ1 . Thus, if .|λ1 | = 1, then the two point interpolation problem has a solution if and only if .λ2 = λ1 . This is nothing else than the maximum modulus principle for holomorphic functions on .D: If .ϕ : D → D is holomorphic and if there exists .z1 ∈ D with .|ϕ(z1 )| = 1, then .ϕ is constant. The case .|λ2 | = 1 is analogous. Next, suppose that .|λ1 |, |λ2 | < 1. By the Hurwitz criterion for positivity, the Pick matrix is positive if and only if .det(P ) ≥ 0. Using the elementary (but very useful) identity 1−

.

(1 − |a|2 )(1 − |b|2 ) |1 − ab|2

 a − b 2   =  1 − ab

we find that .det(P ) ≥ 0 if and only if  λ −λ   z −z  2  2   1  1  ≤ . 1 − z1 z 2 1 − λ1 λ2

.

(a, b ∈ D),

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M. Hartz

These quantities have geometric meaning. Define  z−w    dph (z, w) =   1 − zw

.

(z, w ∈ D).

Then .dph is called the pseudohyperbolic metric on .D; see [83, Section I.1] for background. Therefore, the two point interpolation problem .zi → λi for .i = 1, 2 has a solution if and only if .dph (λ1 , λ2 ) ≤ dph (z1 , z2 ). This statement is usually called the Schwarz–Pick lemma. Pick’s interpolation theorem (Theorem 10.4.1) was proved before Hilbert spaces were abstractly formalized. However, reproducing kernel Hilbert spaces provide significant insight into Pick’s theorem. The basis for this is the following basic multiplier criterion. As usual, if A is a Hermitian matrix, we write .A ≥ 0 to mean that A is positive semi-definite. Proposition 10.4.3 Let .H be an RKHS on X with kernel k. Then .ϕ ∈ Mult(H) with ϕMult(H) ≤ 1 if and only if for all finite .F = {z1 , . . . , zn } ⊂ X,

.

 n k(zi , zj )(1 − ϕ(zi )ϕ(zj )) i,j =1 ≥ 0.

.

(10.17)

Proof The proof rests on the basic identity .Mϕ∗ kz = ϕ(z)kz for .ϕ ∈ Mult(H) and ∗ .z ∈ X; see Remark 10.3.4. Let .ϕ be a multiplier of norm at most one. Then .Mϕ  ≤ 1 and so for all .z1 , . . . , zn ∈ X and .a1 , . . . , an ∈ C, we have n n n 2   2 

2         ϕ(zi )ai kzi  = Mϕ∗ ai kzi  ≤  ai kzi  . 

.

i=1

i=1

i=1

Expanding both sides as a scalar product and rearranging, it follows that n  .

k(zi , zj )(1 − ϕ(zi )ϕ(zj ))aj ai ≥ 0,

i,j =1

which says that (10.17) holds for .F = {z1 , . . . , zn } Conversely, suppose that (10.17) holds for all finite subsets of X. Since linear combinations of kernel functions form a dense subspace of .H, the computation above shows that there exists a contractive linear operator T : H → H,

.

kz → ϕ(z)kz .

As in the proof of Proposition 10.3.3, one shows that .T ∗ = Mϕ , so .ϕ ∈ Mult(H) with .ϕMult(H) ≤ 1. 

Taking .H = H 2 and recalling that .Mult(H 2 ) = H ∞ , it follows that a function ∞ if and only if for every finite set .ϕ : D → C belongs to the unit ball of .H

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385

F = {z1 , . . . , zn } ⊂ D,

.

 1 − ϕ(z )ϕ(z ) n i j ≥ 0. i,j =1 1 − zi z j

.

This proves necessity in Theorem 10.4.1, since the matrix in Theorem 10.4.1 corresponds to the special case when F is the set of interpolation nodes. Conversely, one can use operator theoretic arguments, more specifically commutant lifting, to prove sufficiency in Theorem 10.4.1. This approach to the Nevanlinna–Pick interpolation problem was pioneered by Sarason [141]. We will provide a proof of sufficiency following Theorem 10.4.15 below. Viewing Theorem 10.4.1 as a statement about the reproducing kernel Hilbert space .H 2 raises an obvious question: For which reproducing kernel Hilbert spaces is Pick’s theorem true? Definition 10.4.4 An RKHS .H with kernel k is said to be a Pick space if whenever z1 , . . . , zn ∈ X and .λ1 , . . . , λn ∈ C with

.

[k(zi , zj )(1 − λi λj )]ni,j =1 ≥ 0,

.

then there exists .ϕ ∈ Mult(H) with ϕ(zi ) = λi for 1 ≤ i ≤ n

.

and

ϕMult(H) ≤ 1.

Example 10.4.5 The Hardy space .H 2 is a Pick space by Pick’s interpolation theorem 10.4.1. The Bergman space 2 .La

=

L2a (D)





= f ∈ O(D) :

f 2L2 a

=

D

 |f |2 dA < ∞ ,

where A is the normalized area measure on .D, is not a Pick space. To see this, observe that .Mult(L2a ) = H ∞ , and that the multiplier norm is the supremum norm. Moreover, the reproducing kernel of .L2a is given by k(z, w) =

.

1 (1 − zw)2

(z, w ∈ D);

see, for instance, [69, Section 1.2]. Thus, the reproducing kernel of .L2a is the square of the reproducing kernel of .H 2 , but the multiplier algebras agree. However, it is a simple matter to find points in .D whose Pick matrix with respect to the kernel of .L2a is positive, but whose Pick matrix with respect to the kernel of .H 2 is not positive. For instance, let .z1 = 0, z2 = r ∈ (0, 1) and .λ1 = 0, λ2 = t ∈ (0, 1). Then the Pick matrix with respect to .L2a is

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M. Hartz

 .

1 1

1



1−t 2 (1−r 2 )2

,

which is positive if and only if t 2 ≤ r 2 (2 − r 2 ).

.

Choosing t so that equality holds, we see that .t > r, hence the Pick matrix with respect to .H 2 is not positive. (Alternatively, the two point interpolation problem cannot have a solution by the Schwarz lemma.) Remark 10.4.6 One can show that a Pick space is uniquely determined by its multiplier algebra; see for instance [89, Corollary 3.2] for a precise statement. Thus, there exists at most one Pick space with a given multiplier algebra. This generalizes the argument for the Bergman space in Example 10.4.5. It turns out that one obtains a significantly cleaner theory by not only considering interpolation with scalar targets, but also with matrix targets. This can be regarded as part of a more general principle in functional analysis, namely that demanding that certain properties hold at all matrix levels often has powerful consequences; see for instance [25, 120]. Definition 10.4.7 An RKHS .H with kernel k is said to be a complete Pick space if whenever .r ∈ N, r ≥ 1 and .z1 , . . . , zn ∈ X and .1 , . . . , N ∈ Mr (C) with [k(zi , zj )(I − i ∗j )]ni,j =1 ≥ 0,

.

then there exists . ∈ Mr (Mult(H)) with

(zi ) = λi for 1 ≤ i ≤ n

.

and

 Mr (Mult(H)) ≤ 1.

Instead of saying that .H is a (complete) Pick space, we will also say that the reproducing kernel k is a (complete) Pick kernel. It may not be immediately clear how useful it is to turn Pick’s interpolation theorem into a definition. However, we will see that the complete Pick property of a space .H has very powerful consequences for the function theory and operator theory associated with .H. Moreover, the complete Pick property is satisfied by many familiar RKHS. Before turning to examples of complete Pick spaces and to consequences of the complete Pick property, we consider a reformulation. Let .H be an RKHS on X with kernel k. For .Y ⊂ X, let   HY = {f Y : f ∈ H}

.

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387

 with the quotient norm .gH|Y = inf{f H : f Y = g}. This is an RKHS on Y  with kernel .k Y ×Y ; see for instance [121, Corollary 5.8]. Thus, by definition,  R : H → HY ,

.

 f → f Y ,

is a quotient mapping, meaning  that it maps the open unit ball onto the open unit ⊥ = ball. Note that .ker(R) {ky : y ∈ Y } =: HY , so R induces a unitary operator  between .HY and .HY . On the level of multipliers, we always obtain a complete contraction .

 Mult(H) → Mult(HY ),

 ϕ → ϕ Y .

The Pick property precisely says that one also obtains a quotient mapping on the level of multipliers. Proposition 10.4.8 Let .H be an RKHS on X with kernel k. The following assertions are equivalent: (i) .H is a (complete) Pick space;   (ii) for each finite set .F ⊂ X, the map .Mult(H) → Mult(HF ), ϕ → ϕ F , is a (complete) quotient map;   ) there exists .ϕ ∈ Mult(H) with (iii) for F  each finite .F ⊂ Y and all .ψ ∈ Mult(H  .ϕ  = ψ and .ϕMult(H) = ψMult(H ) (respectively the same property for F F all .r ∈ N, r ≥ 1 and all .ψ ∈ Mr (Mult(HF )). Proof (i) .⇔ (iii) Let .F = {z1 , . . . , zn } be a finite subset of X and let ψ : F → C,

.

zi → λi

(i = 1, . . . , n).

Then Proposition 10.4.3, applied to the space .H|F , shows that .ψMult(H

F

)

≤ 1 if

and only if the Pick matrix for the interpolation problem .zi → λi is positive. Thus, H is a Pick space if and only if (iii) holds, and a similar argument applies in the matrix valued setting. (iii) .⇒ (ii) is trivial.  (ii) .⇒ (iii) By assumption, the restriction map .Mult(H) → Mult(HF ) maps the open unit ball onto the open unit ball. We have to show that the image of the closed unit ball contains the closed unit ball. This follows from a weak-.∗ compactness argument. Recall  from the discussion preceding Proposition 10.3.5 that .Mult(H) and .Mult(H ) carry natural weak-.∗ topologies. With respect to these, the restricF  tion map is weak-.∗–weak-.∗ continuous, since modulo the identification .HF ∼ = before the proposition, the restriction map is given by .Mϕ → HF explained  PHF Mϕ H . By Alaoglu’s theorem, the closed unit ball of .Mult(H) is weak-.∗ F compact, hence so is its image under the restriction map. In particular, the image is norm closed and hence contains the closed unit ball. 

.

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M. Hartz

10.4.2 Characterizing Complete Pick Spaces Complete Pick spaces turn out to have a clean characterization, thanks to a theorem of McCullough [109, 110] and Quiggin [129]. We will use a version of the result due to Agler and McCarthy [4]. Let us say that a function .F : X × X → C is positive if .[F (xi , xj )]ni,j =1 is positive semi-definite for any .x1 , . . . , xn ∈ X. We also write .F ≥ 0. Theorem 10.4.9 Let .H be an RKHS on X with kernel k. Assume that .k(x, y) = 0 for all .x, y ∈ X. Then .H is a complete Pick space if and only if for some .z ∈ X, Fz : X × X → C,

.

(x, y) → 1 −

k(x, z)k(z, y) , k(z, z)k(x, y)

is positive. In this case, .Fz is positive for all .z ∈ X. There are now several proofs of this result, see for instance [6, Chapter 7]. A simple proof of necessity was recently obtained by Knese [101]. Just as in the classical case of .H 2 , it is possible to deduce sufficiency from a suitable commutant lifting theorem; see [36, Section 5] and [18]. In this context, the non-commutative approach mentioned in Sects. 10.2.7 and 10.3.4 is also useful; see [23, 60, Section 4], [58, Section 5] and [55]. We will shortly discuss sufficiency in a bit more detail, but let us first consider some examples. Theorem 10.4.9 becomes even easier to state if one adds a normalization hypothesis. A kernel .k : X × X → C is said to be normalized at .x0 ∈ X if .k(x, x0 ) = 1 for all .x ∈ X. In this case, we say that k (or .H) is normalized. Many kernels of interest are normalized at a point. For instance, the Drury–Arveson kernel is normalized at the origin. In the abstract setting, we think of the normalization point as playing the role of the origin. Any non-vanishing kernel can be normalized by considering k(x, y)k(x0 , x0 )  .k(x, y) = k(x, x0 )k(y, x0 ) for some .x0 ∈ X. This normalization neither changes the multiplier algebra nor positivity of Pick matrices, and so it can usually be assumed without loss of generality. See also [6, Section 2.6] for more discussion. If we set z to be the normalization point in Theorem 10.4.9, then we obtain the following result. Corollary 10.4.10 If k is normalized, then k is a complete Pick kernel if and only if .1 − k1 is positive. The last result can be used to quickly give examples of complete Pick spaces. Corollary 10.4.11 The Drury–Arveson space is a complete Pick space.

10 An Invitation to the Drury–Arveson Space

Proof The function .1 −

1 k(z,w)

389

= z, w is positive.

For .a > 0, let .Ha be the RKHS on .D with kernel encountered these spaces in Sect. 10.2.6.



.

1 (1−zw)a .

We already

Corollary 10.4.12 .Ha is a complete Pick space if and only if .0 < a ≤ 1. Proof We have ! ∞  n−1 a .1 − (1 − zw) = (−1) (zw)n , n a

n=1

" # where . an = a(a−1)...(a−n+1) . A function of this type is positive if and only if all n! coefficients in the sum are non-negative; see for instance the proof of Theorem 7.33 in [6] or [89, Corollary 6.3]. " # If .0 < a ≤ 1, then .(−1)n−1 an ≥ 0 for all n, so .1 − k1 is positive. If .a > 1, then " # 2−1 a < 0, hence .1 − 1 is not positive. 

.(−1) 2 k The spaces .Ha for .a ∈ (0, 1] are weighted Dirichlet spaces. Recall that the classical Dirichlet space is 

D = f ∈ O(D) :



.

D

 |f  |2 dA < ∞ ,

where dA denotes integration with respect to normalized area measure. We equip D with the norm  2 2 .f D = f  2 + |f  |2 dA. H

.

D

Once again, more information on the Dirichlet space can be found in [22, 71]. The following important result of Agler [3] in fact predates Theorem 10.4.9. It initiated the study of complete Pick spaces. Theorem 10.4.13 The classical Dirichlet space is a complete Pick space. Proof The reproducing kernel of .D is given by k(z, w) =

.

∞ 1  1 1 log = (zw)n , zw 1 − zw n+1 n=0

see for instance [71, Theorem 1.2.3]. This kernel is normalized at 0. The coefficients of .1 − 1/k are more difficult to compute than those in Corollary 10.4.12. Instead, the standard approach is to use a Lemma of Kaluza (see [6, Lemma 7.38]), which 1 says that log-convexity of the sequence . n+1 implies that all coefficients of .1 − 1/k are non-negative; whence Corollary 10.4.10 applies. (A sequence .(an ) of positive

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M. Hartz

numbers is log-convex if .an2 ≤ an−1 an+1 for all .n ≥ 1.) For an explicit computation of the coefficients of .1 − 1/k, see for instance [91, Section 5]. 

Remark 10.4.14 The complete Pick property is an isometric property and generally not stable under passing to an equivalent norm. For instance, a frequently used equivalent norm on the Dirichlet space is given by  2 f 2D  = |f (0)| +

.

D

|f  |2 dA.

The reproducing kernel with respect to this norm is  .k(z, w) = 1 + log



1 1

=1+ (zw)n , 1 − zw n ∞

n=1

k is .− 12 < 0, so the Dirichlet and one computes that the coefficient of .(zw)2 in .1−1/ space is not a complete Pick space in the norm . · D . Let us briefly discuss sufficiency in Theorem 10.4.9. For simplicity, we only consider the case of the Drury–Arveson space; in other words, Corollary 10.4.11. The approach to proving the complete Pick property of the Drury–Arveson space we discuss here is to first establish a realization formula for multipliers; see for instance [6, Chapter 8] and [7, Chapter 2] for background on this topic. The following theorem was obtained by Ball, Trent and Vinnikov (see Theorem 2.1 and Theorem 4.1 in [36]), and by Eschmeier and Putinar (see Proposition 1.2 in [74]). Theorem 10.4.15 Let .Y ⊂ Bd and let .ϕ : Y → C. The following statements are equivalent: (i) .ϕMult(H 2 |Y ) ≤ 1. d (ii) There exists a Hilbert space .E and a unitary 

 AB .U = : E ⊕ C → Ed ⊕ C CD with ϕ(z) = D + C(IE − Z(z)A)−1 Z(z)B,

.

 where .Z(z) = z1 . . . zd : Ed → E. Proof (i) .⇒ (ii) The argument is known as a lurking isometry argument. It works as follows. Suppose that .ϕMult(H 2 |Y ) ≤ 1. Proposition 10.4.3 shows that d

(z, w) →

.

1 − ϕ(z)ϕ(w) 1 − z, w

(10.18)

10 An Invitation to the Drury–Arveson Space

391

is positive on .Y × Y . Therefore, this function admits a Kolmogorov decomposition, meaning that there is a Hilbert space .E and a map .h : Y → E so that .

1 − ϕ(z)ϕ(w) = h(w), h(z)E 1 − z, w

for all z, w ∈ Y.

(One can take .E to be the RKHS on Y with reproducing kernel (10.18), and .h(z) to be the kernel at z in .E; see [121, Theorem 2.14] or the first proof of Theorem 2.53 in [6].) Rearranging and collecting positive terms, we see that z, wh(w), h(z) + 1 = h(w), h(z) + ϕ(z)ϕ(w).

.

Now comes the key observation. The last equation says that ⎞ ⎛ ⎞ ⎛ z1 h(z) + $ w1 h(w) ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ . . ⎜ ⎟,⎜ ⎟ ⎝wd h(w)⎠ ⎝zd z(w)⎠ 1

1

, =

! !h(z) h(w) , ϕ(w) ϕ(z) E⊕C

Ed ⊕C

for all .z, w ∈ Y . Therefore, we may define a linear isometry V by ⎞ z1 h(z) ! ⎜ .. ⎟ h(z) ⎜ . ⎟ .V : ⎜ , ⎟ → ϕ(z) ⎝zd h(z)⎠ ⎛

1 initially mapping the closed linear span of vectors appearing on the left onto the closed linear span of vectors appearing on the right. By enlarging .E if necessary, we may extend V to a unitary .Ed ⊕ C → E ⊕ C. Let .U = V ∗ . Decomposing 

 AB .U = : E ⊕ C → Ed ⊕ C, CD we find from the definition of V that A∗ Z(z)∗ h(z) + C ∗ = h(z)

.

and

B ∗ Z(z)∗ h(z) + D ∗ = ϕ(z). Solving the first equation for .h(z), we obtain h(z) = (IE − A∗ Z(z)∗ )−1 C ∗ ;

.

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M. Hartz

observe that .A∗  ≤ 1 since U is unitary and hence .A∗ Z(z)∗  < 1 for all .z ∈ Bd , so .I − A∗ Z(z)∗ is indeed invertible. Substituting the result into the second equation, we conclude that ϕ(z) = D ∗ + B ∗ Z(z)∗ (IE − A∗ Z(z)∗ )−1 C ∗ ,

.

which gives (ii) after taking adjoints. (ii) .⇒ (i) Suppose that (ii) holds and define h(z) = (IE − A∗ Z(z)∗ )−1 C ∗ ∈ E.

.

Reversing the steps in the proof of (i) .⇒ (ii), it follows that .

1 − ϕ(z)ϕ(w) = h(w), h(z)E 1 − z, w

for all z, w ∈ Y,

 hence .ϕ is a multiplier of .Mult(Hd2 Y ) of norm at most one by Proposition 10.4.3. 

Using Theorem 10.4.15, it is a simple matter to show that the Drury–Arveson space satisfies the Pick property. In fact, Theorem 10.4.15 holds in the vector-valued setting, which gives the complete Pick property of .Hd2 . For simplicity, we restrict our attention to the scalar case. Proof of Pick property of .Hd2 Let .Y ⊂ Bd and suppose that .ϕ : Y → C satisfies .ϕ Mult(H 2 |Y ) ≤ 1. Applying (i) .⇒ (ii) of Theorem 10.4.15, we obtain a realization d

ϕ(z) = D + C(IE − Z(z)A)−1 Z(z)B

.

(z ∈ Y ).

The same formula defines an extension . of .ϕ to .Bd ; note that .Z(z)A < 1 for z ∈ Bd , so the inverse exists. By (ii) .⇒ (i) of Theorem 10.4.15, . Mult(H 2 ) ≤ 1.

.

Thus, .Hd2 is a Pick space; see Proposition 10.4.8.

d



Remark 10.4.16 The proof above shows that we may extend multipliers defined on arbitrary subsets of .Bd , not only finite ones. This can also be directly seen from the definition of the (complete) Pick property by means of a weak-.∗ compactness argument.

10.4.3 Universality of the Drury–Arveson Space We saw in the last subsection that the Drury–Arveson space is a complete Pick space. Remarkably, it is much more than an example. If one is willing to include the case .d = ∞, then the Drury–Arveson space is a universal complete Pick space.

10 An Invitation to the Drury–Arveson Space

393

To explain this, recall from Corollary 10.4.10 that a normalized kernel k is a complete Pick kernel if and only if .1 − k1 is positive. (We again only consider normalized kernels for the sake of simplicity.) Applying the Kolmogorov factorization to the function .1 − k1 , we obtain a Hilbert space .E and .b : X → E satisfying .b(x) < 1 for all .x ∈ X and 1−

.

1 = b(x), b(y). k(x, y)

(This is the same procedure as in the proof of Theorem 10.4.15, with the exception that the roles of x and y on the right-hand side are reversed. To achieve this, one may pass from the Hilbert space .E to the conjugate Hilbert space .E; see also the first proof of Theorem 2.53 in [6].) Conversely, if .1 − k1 has the form as above, then it is positive. Thus, we obtain the following theorem, due to Agler and McCarthy [4]. Theorem 10.4.17 An RKHS .H with normalized kernel k is a complete Pick space if and only if there exists a Hilbert space .E and .b : X → E such that .b(x) < 1 for all .x ∈ X and k(x, y) =

.

1 . 1 − b(x), b(y)

Before continuing, we make two more simplifying assumptions: (1) .H separates the points of X, and (2) .H is separable. The first condition implies that .k(·, y1 ) = k(·, y2 ) whenever .y1 , y2 ∈ X with y1 = y2 , hence b is injective. The second condition implies that the space .E can be taken to be separable as well, so we may assume that .E = Cd or .E = 2 . (This follows from the fact that an RKHS is separable if and only if it admits a countable set of uniqueness, and the RKHS with kernel .b(x), b(y) is contained in .H.) Notice that if .E = Cd , then Theorem 10.4.17 expresses the kernel k as a 1 composition of the Drury–Arveson kernel . 1−z,w and the embedding b. If .d = ∞, we need to extend our definition of the Drury–Arveson space.

.

Definition 10.4.18 Let .B∞ = {z ∈ 2 : z2 < ∞}. The Drury–Arveson space 2 is the RKHS on .B with reproducing kernel H∞ ∞

.

K(z, w) =

.

1 . 1 − z, w2

2 is also a complete Pick space. On the level Corollary 10.4.10 shows that .H∞ of function spaces, Theorem 10.4.17 means that there exist .d ∈ N ∪ {∞} and an isometry

394

M. Hartz

V : H → Hd2 ,

.

k(·, y) →

1 . 1 − ·, b(y)

Thus, we obtain an embedding of .H into .Hd2 that sends kernel functions to kernel functions. Taking the adjoint of V , we obtain a composition operator that is coisometric (which is the same as a quotient map in the Hilbert space setting). Thanks to the complete Pick property, this carries over to multiplier algebras, which leads to the following result, again due to Agler and McCarthy [4]. It shows the special role that the Drury–Arveson space plays in the theory of complete Pick spaces. Theorem 10.4.19 (Universality of the Drury–Arveson Space) If .H is a normalized complete Pick space on X, then there exist .1 ≤ d ≤ ∞ and a map .b : X → Bd such that Hd2 → H,

.

f → f ◦ b,

is a co-isometry, and .

Mult(Hd2 ) → Mult(H),

ϕ → ϕ ◦ b,

is a complete quotient map. In fact, if .ϕ ∈ Mult(H), then there exists . ∈ Mult(Hd2 ) with .ϕ = ◦ b and . Mult(H 2 ) = ϕMult(H) . d

Proof The first statement was already explained. The second statement follows from the complete Pick property of .Hd2 . Indeed, let .Y = b(X). If .ϕ ∈ Mult(H), 1 define .ψ : Y → C by .ψ(b(x)) = ϕ(x). The relation .k(x, y) = 1−b(x),b(y)  implies that .ψ ∈ Mult(Hd2 Y ) with .ψMult(H 2 |Y ) = ϕMult(H) , for instance by d

Proposition 10.4.3. By the complete Pick property of .Hd2 , we may extend .ψ to a multiplier of .Hd2 of the same norm, see also Remark 10.4.16. The same proof works for matrix valued multipliers. 

The Agler–McCarthy universality theorem is sometimes summarized as saying that “every complete Pick space is a quotient of the Drury–Arveson space”. Remark 10.4.20 For many spaces of interest, including the classical Dirichlet space and the weighted Dirichlet spaces .Ha for .a ∈ (0, 1) (see Corollary 10.4.12), one has to take .d = ∞ in Theorem 10.4.19. This can be seen by determining the rank of the kernel .1− k1 ; see Proposition 11.8 and Corollary 11.9 in [89] and also [134] for a geometric approach. In the case of the classical Dirichlet space, .d = ∞ is necessary even if one weakens the conclusions in Theorem 10.4.19 in certain ways; see [91] for a precise statement. It is possible to push this line of reasoning even further, which was done by Davidson, Ramsey and Shalit [63]. First, we note that Theorem 10.4.19 can be further reformulated to say that defining .V = b(X), we obtain a unitary

10 An Invitation to the Drury–Arveson Space

 Hd2 V → H,

.

395

f → f ◦ b.

Which subsets V of .Bd are relevant? Definition 10.4.21 If .S ⊂ Hd2 , let V (S) = {z ∈ Bd : f (z) = 0 for all f ∈ S}.

.

If .W ⊂ Bd , let  I (W ) = {f ∈ Hd2 : f W = 0}.

.

We say that .V ⊂ Bd is a variety if .V = V (S) for some .S ⊂ Hd2 . Note that if .d = 1, then the varieties are precisely the Blaschke sequences, along with the entire unit disc. If .1 ≤ d < ∞, then every algebraic variety is a variety in the sense above, as .Hd2 contains all polynomials. Conversely, every variety in the sense above is an analytic variety in .Bd , as .Hd2 consists of holomorphic functions. If .W ⊂ Bd is arbitrary, let .V = V (I (W )) ⊃ W . Clearly, V is the smallest variety containing W . We think of V as an analogue of the Zariski closure in algebraic geometry. It is immediate from the definitions that   Hd2 V → Hd2 W ,

.

f → f |W ,

 is unitary. In particular, every function .Hd2 W has a unique extension to a function  in .Hd2 V . Thus, we obtain the following refined version of Theorem 10.4.19, due to Davidson, Ramsey and Shalit [62]. Theorem 10.4.22 If .H is a normalized complete Pick space on X, then there exists a variety .V ⊂ Bd for some .1 ≤ d ≤ ∞ and a map .b : X → V such that  Cb : Hd2 V → H,

.

f → f ◦ b,

is unitary. In this case, .

 Mult(Hd2 V ) → Mult(H),

ϕ → ϕ ◦ b,

is a completely isometric isomorphism.  Note that if .ϕ ∈ Mult(Hd2 V ), then .Cb Mϕ Cb∗ = Mϕ◦b , so the completely isometric isomorphism if given by conjugation with the unitary .Cb . By the complete Pick property of .Hd2 ,   MV := Mult(Hd2 V ) = Mult(Hd2 )V .

.

396

M. Hartz

This raises the following natural question. Question 10.4.23 What is the relationship between the operator algebra structure of .MV and the geometry of V ? The study of this question was initiated by Davidson, Ramsey and Shalit in [62, 63]. At the Focus Program, Orr Shalit provided a survey of results in this area. For more details, the reader is referred to the survey article [138]. Other references are [57, 88, 89, 92, 99].

10.5 Selected Topics This last section contains a few selected topics regarding the Drury–Arveson space and more generally complete Pick spaces. The selection was made based on which other topics were covered at the Focus Program, but it is admittedly also heavily influenced by the author’s taste. For other selections and perspectives, we once again refer the reader to [145] and [77].

10.5.1 Maximal Ideal Space and Corona Theorem We assume that .d < ∞. The multiplier algebra of the Drury–Arveson space .Hd2 is a unital commutative Banach algebra. Hence Gelfand theory strongly suggests that one should study its maximal ideal space M(Mult(Hd2 )) = {χ : Mult(Hd2 ) → C : χ is linear, multiplicative, = 0},

.

which is a compact Hausdorff space in the weak-.∗ topology. Naturally, we obtain a topological embedding Bd → M(Mult(Hd2 ))

.

by sending .w ∈ Bd to the character of point evaluation at w. In the reverse direction, we obtain a continuous surjection π : M(Mult(Hd2 )) → Bd ,

.

χ → (χ (z1 ), . . . , χ (zd )).

(The fact that .π takes values in .Bd follows from the fact that the coordinate functions form a row contraction and that characters are completely contractive, see [120, Proposition 3.8], or from computation of the spectrum of the tuple of coordinate functions.)

10 An Invitation to the Drury–Arveson Space

397

The following result was shown by Gleason, Richter and Sundberg, see [85, Corollary 4.2]. It shows that Gleason’s problem can be solved in the multiplier algebra of .Hd2 . For background on Gleason’s problem, see [136, Section 6.6]. Theorem 10.5.1 Let .w ∈ Bd and let .ϕ ∈ Mult(Hd2 ). Then there exist .ϕ1 , . . . , ϕd ∈ Mult(Hd2 ) with ϕ(z) − ϕ(w) =

d 

.

(zi − wi )ϕi (z)

for all z ∈ Bd .

i=1

Therefore, .π −1 (w) is the singleton consisting of the character of point evaluation at w. The last sentence of Theorem 10.5.1 is false for .d = ∞; see [56]. Already in the case .d = 1, the fibers of .π over points in the boundary are known to be extremely complicated; see [94, Chapter 10] and [83, Chapter X]. For general .d ≥ 1, the theory of interpolating sequences (specifically Theorem 10.5.15 below) ˇ implies that .π −1 (w) contains a copy of the Stone-Chech remainder .βN \ N for each 2 .w ∈ ∂Bd ; cf. the argument on p. 184 of [83]. In particular, .M(Mult(H )) is not d ℵ 0 metrizable and has cardinality .22 . In summary, .M(Mult(Hd2 )) contains a well understood part, namely the open ball .Bd , and a much more complicated part, namely .π −1 (∂Bd ), i.e. characters in fibers over the boundary. If .d = 1, then Carleson’s corona theorem ([45], see also [83, Chapter VIII]) shows that the unit disc (i.e. the well understood part) is dense in the maximal ideal space of .H ∞ = Mult(H12 ). A deep result of Costea, Sawyer and Wick [53] shows that this remains true for higher d. Theorem 10.5.2 (Corona Theorem for .Hd2 ) .Bd is dense in .M(Mult(Hd2 )). In their proof, Costea, Sawyer and Wick use the complete Pick property of .Hd2 to reduce the original problem involving .Mult(Hd2 ) to a problem about .Hd2 . The point is that problems in .Hd2 are likely more tractable because one can exploit the Hilbert space structure. Costea, Sawyer and Wick then manage to solve the (still extremely difficult) Hilbert space problem using the function theory description of 2 .H , thus proving their corona theorem. This result and its proof showcase again d how different perspectives on the Drury–Arveson space can be helpful. The translation of the corona problem into a Hilbert space problem, as well as the proof of Theorem 10.5.1, rely on a factorization result for multipliers complete Pick spaces, which has proved to be extremely useful for a variety of problems. In the case of .H 2 , it was shown by Leech [103] (see also [97] for a historical discussion). References for the Drury–Arveson space and general complete Pick spaces are [74, Theorem 1.6], [6, Theorem 8.57] and [36]. We write .Mn,m (Mult(H)) for the space of all .n × m matrices with entries in .Mult(H), which we regard as operators from m .H into .Hn . Extending an earlier definition, we say that a function .L : X × X →

398

M. Hartz

Mr (C) is positive if the block matrix .[L(xi , xj )]ni,j =1 is positive semi-definite for any .x1 , . . . , xn ∈ X. Theorem 10.5.3 Let .H be a normalized complete Pick space with kernel k and let

∈ Mr,n (Mult(H)) and . ∈ Mr,m (Mult(H)) be given. The following assertions are equivalent:

.

(i) there exists . ∈ Mn,m (Mult(H)) of norm at most one such that .  = ; (ii) the function .(z, w) → k(z, w)( (z) (w)∗ − (z)(w)∗ ) is positive; ∗ ≥ M M ∗ holds. (iii) the operator inequality .M M   Proof (Sketch) The equivalence of (ii) and (iii) follows by testing the inequality on finite linear combinations of vectors in .Hr whose entries are reproducing kernels. If (i) holds, then ∗ ∗ ∗ ∗ M  M = M M  M M ≤ M M ,

.

so (iii) holds. The main work occurs in the proof of (ii) .⇒ (i); it is here where the complete Pick property is used. This implication can be shown by using a lurking a lurking isometry argument, similar to the proof of Theorem 10.4.15; see [6, Theorem 8.57] for the details. 

Remark 10.5.4 (a) Taking .n = r = m = 1 and . = 1 in Theorem 10.5.3, we essentially recover the basic multiplier criterion (Proposition 10.4.3). (b) In the setting of (iii) of Theorem 10.5.3, the Douglas factorization lemma [64] always yields a contraction .T : Hm → Hn so that .M T = M ; this has nothing to do with multiplication operators or the complete Pick property. The crucial point in Theorem 10.5.3 is that T can be chosen to be a multiplication operator again. Thus, Theorem 10.5.3 can be regarded as a version of the Douglas factorization lemma for the multiplier algebra of a complete Pick space. Under suitable assumptions, the complete Pick property is necessary for the validity of such a Douglas factorization lemma; see [111] for a precise statement. (c) There is a version of Theorem 10.5.3 for infinite matrices; see [6, Theorem 8.57] for the precise statement. Let us see how Leech’s Theorem can be used to show Theorem 10.5.1. Sketch of proof of Theorem 10.5.1 Let .w = 0. We have to show that if .ϕ ∈ Mult(Hd2 ) with .ϕ(0) = 0, then there exist .ϕ1 , . . . , ϕd ∈ Mult(Hd2 ) with .ϕ = d i=1 zi ϕi . We may assume that .ϕMult(H 2 ) ≤ 1. Since .ϕ(0) = 0, the function d

ϕ is a contractive multiplier from .Hd2 into .{f ∈ Hd2 : f (0) = 0}. The reproducing kernel for this last space is .K − 1, where K denotes the reproducing kernel of .Hd2 . A small modification of the basic multiplier criterion (Proposition 10.4.3, see for instance [121, Theorem 5.21] for the general statement) then shows that

.

10 An Invitation to the Drury–Arveson Space

399

(K(z, w) − 1) − K(z, w)ϕ(z)ϕ(w) =

.

1 (z, w − ϕ(z)ϕ(w)) 1 − z, w

is positive as a function of .(z, w). In this setting, we may apply (ii) .⇒ (i) of Theorem 10.5.1 with .n = d, .r = m = 1, . (z) = z1 · · · zd and . = ϕ to find a multiplier . ∈ Md,1 (Mult(Hd2 )) with .ϕ = . Writing . in terms of its components, the result in the case .w = 0 follows. The general case of .w ∈ Bd can be deduced from the case .w = 0 by using biholomorphic automorphisms of the ball; see [85, Corollary 4.2]. Alternatively, the proof above can be extended to arbitrary .w ∈ Bd with the help of some multivariable spectral theory; see [89, Proposition 8.5]. The additional statement follows from the first part. Indeed, let .χ ∈ M(Mult(Hd2 )) with .w = π(χ ) ∈ Bd . Given .ϕ ∈ Mult(Hd2 ), write ϕ(z) − ϕ(w) =

.

d  (zi − wi )ϕi (z) i=1

as in the first part and apply .χ to both sides. Since .χ (zi ) = wi for .1 ≤ i ≤ d and χ (1) = 1, it follows that .χ (ϕ) = ϕ(w). Thus, .χ is the character of evaluation at w. 

.

Remark 10.5.5 Theorem 10.5.1 can also be proved by using the non-commutative approach to multipliers explained in Sect. 10.3.4; see [63, Proposition 3.2] and [59, Corollary 2.6]. Next, we discuss how Leech’s Theorem can be used in the translation of the corona problem into a Hilbert space problem. First, it is a routine exercise in Gelfand theory to translate the corona problem into a more concrete question about multipliers that are jointly bounded below. Proposition 10.5.6 The following are equivalent: (i) .Bd is dense in .M(Mult(Hd2 ));  (ii) whenever .ϕ1 , . . . , ϕn ∈ Mult(Hd2 ) with . ni=1 |ϕi |2 ≥ ε2 > 0, there exist  n 2 .ψ1 , . . . , ψn ∈ Mult(H ) with . i=1 ϕi ψi = 1. d For a proof, see [117, Lemma B.9.2.6] or [83, Theorem V.1.8]. Leech’s theorem makes it possible to translate condition (ii), which involves multipliers, into a condition involving only functions in .Hd2 . This result is often called the Toeplitz corona theorem. In the case of .H 2 , it was proved by Arveson [27]. In the case of the Drury–Arveson space, the result appears in [36, Section 5] and in [74, Theorem 2.2]. Theorem 10.5.7 (Toeplitz Corona Theorem) Let .H be a normalized complete Pick space with kernel k. Let .ϕ1 , . . . , ϕn ∈ Mult(H). The following are equivalent:

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M. Hartz

 (i) there exist .ψ1 , . . . , ψn ∈ Mult(H) with . ni=1 ϕi ψi = 1; (ii) there exists .δ > 0 such that n

 (z, w) → k(z, w) ϕi (z)ϕi (w) − δ 2

.

i=1

is positive.  (iii) the row operator . Mϕ1 · · · Mϕn : Hn → H is surjective.  Proof We may normalize so that the row . := ϕ1 . . . ϕn has multiplier norm at most one. Then Theorem 10.5.3, applied with with .r = m = 1 and . = δ, shows that there exists a contractive column multiplier . ∈ Mn,1 (Mult(H)) so that .  = δ if and only if the mapping in (ii) is positive. This shows the equivalence of (i) and (ii). The equivalence with (iii) then follows from Theorem 10.5.3 and the basic ∗ ≥ δ 2 I for some operator theory fact that .M is surjective if and only if .M M .δ > 0, which is a consequence of the open mapping theorem; see [64] or [135, Theorem 4.13] for a more general Banach space result. (Note that the implication (i) .⇒ (iii) also follows directly from the observation that the column of the .ψi is a right inverse of the row of the .ϕi .) 

Remark 10.5.8 The proof of Theorem 10.5.7 gives quantitative information on the norm of the .ψi . If we normalize so the row of the .ϕi has norm one and .δ is as in (ii), then we may achieve that the norm of the column of the .ψi is at most . 1δ , and this is best possible. In order to prove the corona theorem for .Hd2 , one therefore has to show that if 2 .ϕ1 , . . . , ϕn ∈ Mult(H ) with d n  .

|ϕi |2 ≥ ε2 > 0,

(10.19)

i=1

then there exists .δ > 0 such that n (z, w) →

.

i=1 ϕi (z)ϕi (w) − δ

2

1 − z, w

(10.20)

is positive. This was achieved by Costea, Sawyer and Wick [53]. Note that testing the diagonal .{z = w}, one recovers (10.19) with .ε = δ. Letting  (10.20) on . = ϕ1 · · · ϕn , the passage from (10.19) to (10.20) is equivalent to proving that ∗ M M Kz , Kz  ≥ ε2 Kz 2

.

implies that there exists .δ > 0 such that

for all z ∈ Bd

10 An Invitation to the Drury–Arveson Space ∗ M M f, f  ≥ δ 2 f 2

.

401

for all f ∈ Hd2 .

The philosophy that in order to show certain properties of operators, it should suffice to test the property on kernel functions, is sometimes called the reproducing kernel thesis; see for instance [22, Chapter 4] and [117, Section A.5.8]. Remark 10.5.9 It is natural to ask for the dependence of .δ in (10.20) on .ε in (10.19). By Remark 10.5.8, this leads to bounds on the norm of the corona solutions in terms of the corona data. For a discussion of bounds in the classical case .d = 1, see [118, Appendix 3] and [153]. Given the Toeplitz corona theorem for complete Pick spaces and the Costea– Sawyer–Wick corona theorem for .Hd2 , one may wonder if the corona theorem holds for all complete Pick spaces. Without further assumptions, this is not the case, which was shown by Aleman, McCarthy, Richter and the author; see [10, Theorem 5.5]. Theorem 10.5.10 There exists a normalized complete Pick space .H with the following properties: (a) .H consists of holomorphic functions on .D, (b) .H contains all functions holomorphic in a neighborhood of .D, and (c) .D is not dense in .M(Mult(H)). The proof uses a construction of Salas [137], so the space .H was called the Salas space in [10]. The Salas space is in fact contained in the disc algebra. Observe that condition (b) rules out trivial constructions such as taking the restriction of the Hardy space on .2D to .D. (In [10, Theorem 5.5], a condition that is slightly different from (b) is stated; but (b) follows from the fact that the reproducing kernel of .H is an given by a power series with coefficients .an satisfying . an+1 → 1.)

10.5.2 Interpolating Sequences Next, we will discuss interpolating sequences. The classical definition in the disc is the following. Definition 10.5.11 A sequence .(zn ) in .D is an interpolating sequence (IS) for .H ∞ if the map .H ∞ → ∞ , ϕ → (ϕ(zn )), is surjective. The study of interpolating sequences sheds considerable light on the structure of the maximal ideal space of .H ∞ ; see for instance [83, Chapter X]. They also made an appearance in the description of Douglas algebras, which are subalgebras of ∞ containing .H ∞ . The description is due to Chang [47] and Marshall [108]; see .L also [83, Chapter IX]. For general background on interpolating sequences, see [83, Chapter VII], [143] and [6, Chapter 9].

402

M. Hartz

At first, it is not even obvious that interpolating sequences in .H ∞ exist, but they do, and in fact they were characterized by Carleson [44]. His characterization involves two more conditions: (WS)

(zn ) is weakly separated if there exists .ε > 0 so that

.

 z −z  m   n  ≥ε 1 − zm zn

.

(C)

whenever n = m.

(zn ) satisfies the Carleson measure condition if there exists .C ≥ 0 so that

.

.

 (1 − |zn |2 )|f (zn )|2 ≤ Cf 2H 2

for all f ∈ H 2 .

n

Note that the quantity appearing in condition (WS) is the pseudohyperbolic metric on .D, which we already encountered  in Example 10.4.2. The Carleson measure condition means that the measure . n (1 − |zn |2 )δzn is a Carleson measure for .H 2 . Theorem 10.5.12 (Carleson’s Interpolation Theorem) In .H ∞ , (IS) .⇔ (WS) + (C) The definition of an interpolating sequence, as well as the conditions involved in Carleson’s theorem, naturally carry over to the Drury–Arveson space, and in fact to any complete Pick space. Definition 10.5.13 Let .H be a complete Pick space on X with reproducing kernel k. A sequence .(zn ) in X is said to (IS)

be an interpolating sequence (for .Mult(H)) if the map .

(WS)

is surjective; be weakly separated if there exists .ε > 0 so that 1−

.

(C)

Mult(H) → ∞ , ϕ → (ϕ(zn )),

|k(zn , zm )|2 ≥ε k(zn , zn )k(zm , zm )

whenever n = m.

satisfy the Carleson measure condition if there exists .C ≥ 0 so that .

 |f (zn )|2 ≤ Cf 2H k(z , z ) n n n

for all f ∈ H.

The analogue of Carleson’s theorem in the Dirichlet space was established independently by Bishop [38] and by Marshall and Sundberg [107]. The proof of

10 An Invitation to the Drury–Arveson Space

403

Marshall and Sundberg crucially used the complete Pick property of the Dirichlet space. They established the following theorem, see [107, Corollary 7] and [6, Theorem 9.19]. Recall that a sequence .(xn ) of vectors in a Hilbert space is said to be a Riesz sequence if there exist constants .C1 , C2 > 0 with C1



.

n

2

 

  |αn |2 ≤  αn xn  ≤ C2 |αn |2 n

n

for all sequences .(αn ) of scalars. If the linear span of the .xn is dense, the Riesz sequence is called a Riesz basis. Theorem 10.5.14 Let .H be a normalized complete Pick space on X with kernel k. Then a sequence .(zn ) in X is an interpolating sequence if and only if the normalized reproducing kernels .{kzn /kzn  : n ∈ N} form a Riesz sequence in .H. Proof We prove sufficiency. Suppose that the normalized reproducing kernels form  a Riesz sequence in .H. Let .Z = {zn : n ∈ N} and consider the space .HZ . Then the  restrictions to Z of the normalized reproducing kernels form a Riesz basis of .HZ . Thus, for every .(wn ) ∈ ∞ , we obtain a bounded linear map   Tw : HZ → HZ ,

.

  kzn Z → wn kzn Z .

 The adjoint .Tw∗ is multiplication operator on .HZ with symbol .ϕ given by .ϕ(zn ) = wn . This shows that .

 Mult(HZ ) → ∞ ,

ϕ → (ϕ(zn )),

is surjective. By the complete Pick property of X, the restriction map .Mult(H) →  Mult(HZ ) is surjective, so Z is an interpolating sequence for .Mult(H). Necessity is true independently of the complete Pick property; see for instance [6, Theorem 9.19] for a proof. 

Theorem 10.5.14 is another instance where the complete Pick property of a space H can be used to translate a problem about .Mult(H) into a more tractable Hilbert space problem in .H. Theorem 10.5.14 has the following consequence regarding the existence of interpolating sequences.

.

Theorem 10.5.15 Let .H be a normalized complete Pick space on X with kernel k. If .(zn ) is a sequence in X such that .limn→∞ k(zn , zn ) = ∞, then .(zn ) has subsequence that is interpolating. Proof The assumption implies that .limn→∞ kzn  = ∞. Using the particular 1 form of the kernel .k(z, w) = 1−b(z),b(w) given by Theorem 10.4.17, we see that k (w)

zn limn→∞ k = 0 for all .w ∈ X; hence .(kzn /kzn ) is a sequence of unit vectors zn  in .H that tends to zero weakly. From this, one can recursively extract a subsequence

.

404

M. Hartz

of .(kzn /kzn ) that is a Riesz sequence (see for example [8, Proposition 2.1.3] for a more general statement), so the result follows from Theorem 10.5.14. 

The proof above is taken from [12, Proposition 5.1]. For a direct proof using Pick matrices, see [57, Proposition 9.1] and also [144, Lemma 4.6]. Theorem 10.5.15 in particular yields the existence of interpolating sequences in the Drury–Arveson space and in the Dirichlet space. The existence of interpolating sequences in the Dirichlet space was proved earlier by Axler [33]. After the theorem of Bishop and of Marshall and Sundberg, the scope of the characterization of interpolating sequences was further extended by Bøe [40] (see also [39]), but this still left open the case of the Drury–Arveson space. Using work of Agler and McCarthy [6, Chapter 9], Carleson’s characterization was extended to all complete Pick spaces, including the Drury–Arveson space, by Aleman, Richter, McCarthy and the author [12]. Theorem 10.5.16 In any normalized complete Pick space, (IS) .⇔ (WS) + (C). The first proof of this result used the solution of the Kadison–Singer problem due to Marcus, Spielman and Srivastava [106]. Explicitly, it used an equivalent formulation of the Kadison–Singer problem, known as the Feichtinger conjecture. In the context of interpolating sequences, the Feichtinger conjecture, combined with Theorem 10.5.14, implies that every sequence satisfying the Carleson measure condition is a finite union of interpolating sequences. In the case of .Hd2 for finite d, a proof avoiding the use of the Marcus–Spielman– Srivastava theorem was given in [12, 16]. The key property making this possible is the so-called column-row property of .Hd2 . To explain this property, recall that in .Hd2 , we have    Mz · · · Mz  = 1. d 1

.

On the other hand, one may compute that ⎡ ⎤  Mz1  ⎢ . ⎥ √ . ⎣ . ⎦  = d. .   Mzd In particular, the column norm of a tuple of multiplication operators may exceed the row norm. In fact, one can iterate this example to construct a sequence of multipliers that are bounded as a row, but unbounded as a column; see [16, Subsection 4.2] or [51, Lemma 4.8]. What about the converse? Definition 10.5.17 An RKHS .H satisfies the column-row property with constant c ≥ 1 if

.

10 An Invitation to the Drury–Arveson Space

405

⎡ ⎤  Mϕ1  ⎢   ⎥ . Mϕ Mϕ · · ·  ≤ c  ⎣Mϕ2 ⎦  1 2   .. . for all sequences .(ϕn ) in .Mult(H). It was shown in [12, Remark 3.7] that if .H is a complete Pick space satisfying the column-row property, then one obtains a direct proof of Theorem 10.5.16 that does not rely on the Marcus–Spielman–Srivastava theorem. The column-row property also came up in other contexts, such as in the theory of weak product spaces, see Sect. 10.5.3 below. The first non-trivial example of a complete Pick space satisfying the column-row property was the Dirichlet space, a result due to Trent [154], see also [100]. Generalizing Trent’s argument, it was shown in [16], see also [13], that if .d < ∞, then .Hd2 satisfies the column-row property with some constant .cd . Thus, one obtains a direct proof of Theorem 10.5.16 for .Hd2 . Finally, the column-row property was shown to be automatic for all complete Pick spaces in [90]. Theorem 10.5.18 Every normalized complete Pick space satisfies the column-row property with constant 1.

10.5.3 Weak Products and Hankel Operators The Hardy space .H 2 is typically not studied as an isolated object, but as a member of the scale of .H p spaces. Of particular importance are .H 1 and .H ∞ . The analogue of .H ∞ in the context of the Drury–Arveson space is generally considered to be the multiplier algebra .Mult(Hd2 ). What is an appropriate analogue of .H 1 ? The classical function theory definition of .H 1 is   H 1 = f ∈ O(D) : sup

.

2π

 |f (reit )| dt < ∞ .

0≤r