Multilinear operator integrals: theory and applications 9783030324056, 9783030324063

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Lecture Notes in Mathematics  2250

Anna Skripka Anna Tomskova

Multilinear Operator Integrals Theory and Applications

Lecture Notes in Mathematics Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Editors: Karin Baur, Leeds Michel Brion, Grenoble Camillo De Lellis, Princeton Alessio Figalli, Zurich Annette Huber, Freiburg Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Cambridge Angela Kunoth, Cologne Ariane Mézard, Paris Mark Podolskij, Aarhus Sylvia Serfaty, New York Gabriele Vezzosi, Florence Anna Wienhard, Heidelberg

2250

More information about this series at http://www.springer.com/series/304

Anna Skripka • Anna Tomskova

Multilinear Operator Integrals Theory and Applications

123

Anna Skripka Department of Mathematics and Statistics University of New Mexico Albuquerque, NM, USA

Anna Tomskova School of Computer Science and Engineering Inha University in Tashkent Tashkent, Uzbekistan

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-030-32405-6 ISBN 978-3-030-32406-3 (eBook) https://doi.org/10.1007/978-3-030-32406-3 Mathematics Subject Classification (2010): Primary: 46L51, 47B49, 47A60, 47A63, 47B10, 47C15, 47A55, 15A60; Secondary: 46N50, 58J30, 46L87, 46G12, 46H10, 47L25, 26A16 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved 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

Preface

A multilinear operator integral is a powerful tool in noncommutative analysis and its applications. Theory underlying multilinear operator integration has been developing since the 1950s, with a number of amazing advancements made in recent years. The field has accumulated many deep theoretical results and important applications, but no book on this beautiful and important subject appeared in the literature. This book provides a brief yet comprehensive treatment of multilinear operator integral techniques and their applications, partially filling the gap in the literature. The exposition is structured to be suitable for both a topics course and a research aid on methods, results, and applications of multilinear operator integrals. We survey on earlier ideas and contributions to the field and then present in greater detail the best up-to-date results and modern methods. The content includes most practical, refined constructions of multiple operator integrals and fundamental technical results along with major applications of this tool to smoothness properties of operator functions (Lipschitz continuity, Hölder continuity, differentiability), approximation of operator functions, spectral shift functions, spectral flow in the setting of noncommutative geometry, quantum differentiability, and differentiability of noncommutative Lp norms. We demonstrate ideas and include proofs in simpler cases, while highly technical proofs are outlined and supplemented with a list of references. We also state selected open problems in the field. Albuquerque, NM, USA Tashkent, Uzbekistan September 2019

Anna Skripka Anna Tomskova

v

Acknowledgements

The authors thank Fedor Sukochev for inspiration to write an overview of multilinear operator integration, which has ultimately grown into this book. The authors are also grateful to the three referees for their valuable comments and suggestions. In particular, Theorems 5.1.12 and 5.1.13, Sects. 5.1.6 and 5.3.7, and the example after Theorem 3.3.11 were suggested by the referees. Research of the first author was supported in part by NSF grant DMS-1554456.

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1

2 Notations and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Spaces of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Divided Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Linear Operators .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Schatten-von Neumann Classes . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Product of Spectral Measures . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Classical Noncommutative Lp -Spaces and Weak Lp -Spaces .. . . . . . 2.7 The Haagerup Lp -Space . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8 Symmetrically Normed Ideals . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9 Traces on L1,∞ (M, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.10 Banach Spaces and Spectral Operators . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.11 Differentiability of Maps on Banach Spaces . . . . .. . . . . . . . . . . . . . . . . . . .

7 7 10 11 12 15 18 22 24 26 29 32

3 Double Operator Integrals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Double Operator Integrals on Finite Matrices . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.2 Relation to Finite-Dimensional Schur Multipliers .. . . . . . . . . . 3.1.3 Properties of Finite Dimensional Double Operator Integrals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Double Operator Integrals on S2 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 Relation to Schur Multipliers on S2 . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.3 Basic Properties of Double Operator Integrals on S2 . . . . . . . 3.3 Double Operator Integrals on Schatten Classes and B(H) . . . . . . . . . . 3.3.1 Daletskii-Krein’s Approach .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.2 Extension from the Double Operator Integral on S2 . . . . . . . . 3.3.3 Approach via Separation of Variables .. . .. . . . . . . . . . . . . . . . . . . . 3.3.4 Approach Without Separation of Variables . . . . . . . . . . . . . . . . . . 3.3.5 Properties of Double Operator Integrals on Sp and B(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

35 35 36 36 37 41 41 43 44 45 45 46 47 50 51 ix

x

Contents

3.4 3.5

3.6

3.3.6 Symbols of Bounded Double Operator Integrals . . . . . . . . . . . . 3.3.7 Transference Principle . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Nonself-adjoint Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Double Operator Integrals on Noncommutative Lp -Spaces .. . . . . . . . 3.5.1 Extension from the Double Operator Integral on L2 (M, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.2 Approach via Separation of Variables .. . .. . . . . . . . . . . . . . . . . . . . 3.5.3 Approach Without Separation of Variables . . . . . . . . . . . . . . . . . . 3.5.4 Properties of Double Operator Integrals on Lp (M, τ ) . . . . . . Double Operator Integrals on Banach Spaces . . . .. . . . . . . . . . . . . . . . . . . .

4 Multiple Operator Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Multiple Operator Integrals on Finite Matrices . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.2 Relation to Multilinear Schur Multipliers .. . . . . . . . . . . . . . . . . . . 4.1.3 Properties of Finite Dimensional Multiple Operator Integrals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.4 Estimates of Multiple Operator Integrals via Double Operator Integrals . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Multiple Operator Integrals on S2 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.1 Pavlov’s Approach . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 Coine-Le Merdy-Sukochev’s Approach ... . . . . . . . . . . . . . . . . . . . 4.3 Multiple Operator Integrals on Schatten Classes and B(H). . . . . . . . . 4.3.1 Approach via Separation of Variables .. . .. . . . . . . . . . . . . . . . . . . . 4.3.2 Approach Without Separation of Variables . . . . . . . . . . . . . . . . . . 4.3.3 Properties of Multiple Operator Integrals on Sp and B(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.4 Nonself-adjoint Case . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.5 Change of Variables . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Multiple Operator Integrals on Noncommutative and Weak Lp -Spaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 Approach via Separation of Variables .. . .. . . . . . . . . . . . . . . . . . . . 4.4.2 Approach Without Separation of Variables . . . . . . . . . . . . . . . . . . 4.4.3 Properties of Multiple Operator Integrals on Lp,∞ (M, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Operator Lipschitz Functions . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.1 Commutator and Lipschitz Estimates in S2 .. . . . . . . . . . . . . . . . . 5.1.2 Commutator and Lipschitz Estimates in Sp and B(H) . . . . . 5.1.3 Commutator and Lipschitz Estimates: Nonself-adjoint Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.4 Lipschitz Type Estimates in Noncommutative Lp -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.5 Lipschitz Type Estimates in Banach Spaces . . . . . . . . . . . . . . . . . 5.1.6 Operator I-Lipschitz Functions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

54 58 60 60 61 61 61 62 63 65 65 65 66 67 74 74 74 75 77 77 79 81 93 99 100 100 100 101 113 113 114 115 119 121 122 123

Contents

5.2 5.3

5.4

5.5

5.6 5.7 5.8

xi

Operator Hölder Functions .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Differentiation of Operator Functions . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.1 Differentiation of Matrix Functions . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.2 Differentiation in B(H) Along Multiplicative Paths of Unitaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.3 Differentiation in B(H) and S1 Along Linear Paths of Self-adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.4 Differentiation in Sp Along Linear Paths of Self-adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.5 Differentiation of Functions of Contractive and Dissipative Operators . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.6 Differentiation in Noncommutative Lp -Spaces .. . . . . . . . . . . . . 5.3.7 Gâteaux and Fréchet I-Differentiable Functions . . . . . . . . . . . . Taylor Approximation of Operator Functions .. . .. . . . . . . . . . . . . . . . . . . . 5.4.1 Taylor Remainders of Matrix Functions .. . . . . . . . . . . . . . . . . . . . 5.4.2 Taylor Remainders for Perturbations in Sp and B(H) . . . . . . 5.4.3 Taylor Remainders for Unsummable Perturbations . . . . . . . . . Spectral Shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5.1 Spectral Shift Function for Self-adjoint Operators . . . . . . . . . . 5.5.2 Spectral Shift Function for Nonself-adjoint Operators . . . . . . 5.5.3 Spectral Shift Measure in the Setting of von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Spectral Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Quantum Differentiability .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Differentiation of Noncommutative Lp -Norms . .. . . . . . . . . . . . . . . . . . . .

128 129 129 132 135 137 140 141 142 145 145 149 153 154 155 162 165 166 170 172

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 179 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 189

Chapter 1

Introduction

A multilinear (or multiple) operator integral is a beautiful and powerful tool and one of the main objects of research in noncommutative analysis. In this book we discuss the rich theory of multiple operator integration and its numerous applications, choosing the latter to be our main motivation determining the exposition of the theory. The first double operator integral technique can be detected in K. Löwner’s work on matrix monotone functions in 1934, although the term “operator integral” was coined two decades later. K. Löwner related sign-definiteness of the increment of the matrix function f (A) − f (B) to the sign-definiteness of the matrix (f [1] (λj , μk ))dj,k=1 of the divided difference of f at the spectral points of selfadjoint matrices A and B by interpreting f (A) − f (B) as the Schur product of the matrix (f [1] (λj , μk ))dj,k=1 and the matrix A − B in the eigen bases of A and B. The latter connection can be stated as f (A) − f (B) = TfA,B [1] (A − B),

(1.0.1)

where the linear transformation TfA,B [1] is an example of a double operator integral on

the space of matrices. It is constructed based on the function f [1] of two variables and the spectral data of the matrices A and B. Later the representation (1.0.1) was extended to a much broader setting and found important applications in perturbation theory. In particular, it is utilized in the study of smoothness properties of operator functions and in derivation of various norm bounds for operator functions. The first transformation receiving the name “double operator integral” was introduced by Yu. L. Daletskii and S. G. Krein in their work on differentiation of functions of infinite dimensional operators in 1956. They represented the derivative  d  f (A + tX) of a function f along a path of bounded self-adjoint operators dt t =0 A + tX as  d f (A + tX)t =0 = TfA,A [1] (X), dt © Springer Nature Switzerland AG 2019 A. Skripka, A. Tomskova, Multilinear Operator Integrals, Lecture Notes in Mathematics 2250, https://doi.org/10.1007/978-3-030-32406-3_1

1

2

1 Introduction

where the transformation TfA,A [1] is an iterated Riemann-Stieltjes integral with respect to the spectral family of A and the class of functions f in their approach is assumed to be more restrictive than turned out to be necessary. They  also obtained an d estimate for the operator norm of the derivative dt f (A + tX)t =0 based on the A,A analysis of the transformation Tf [1] . That estimate has been substantially improved and the restriction of f relaxed by means of modern approaches to double operator integration, but the idea of involving operator integrals in the study of differentiation and approximation of operator functions remains vital. In 1964, M. G. Krein noticed a connection between the aforementioned results by K. Löwner and Yu. L. Daletski, S. G. Krein and brought it to the attention of B. S. Birman and M. Z. Solomyak, who later developed several approaches to double operator integration and substantially extended the range of applicability of the method. In particular, they introduced a transformation TϕA,B acting on the Hilbert-Schmidt ideal S2 as an integral of a bounded measurable function ϕ with respect to a product spectral measure arising from the spectral measures of selfadjoint operators A and B. This transformation TϕA,B is a double operator integral in the full sense due to the Hilbert space structure of the space S2 . M. S. Birman and M. Z. Solomyak also introduced a transformation TϕA,B on B(H), the space of bounded linear operators on a Hilbert space H, that is tied to a factorization of the function ϕ(λ, μ) separating the variables λ and μ. The boundedness of TϕA,B on B(H) and on the trace class S1 was characterized via the integral projective tensor product decomposition of ϕ by V. V. Peller in 1985. The latter result is similar to the celebrated Grothendieck’s characterization of Schur multipliers on B(2 ) (see, e.g., [153, Chapter 5]). While the aforementioned constructions of operator integrals along with the perturbation formula (1.0.1) allowed to obtain information about f (A) − f (B), they were insufficient to characterize operator Lipschitz functions on Schattenvon Neumann ideals Sp , 1 < p < ∞. The latter was M. G. Krein’s question stimulating the development of operator integration since 1964 and resolved by D. Potapov and F. Sukochev in 2011. The resolution was achieved by switching from the separation of variables in the symbol ϕ = f [1] of a double operator integral to a different decomposition of f [1] and subsequently applying harmonic analysis of UMD spaces. This approach also gave the best possible bound for the Schatten norm of f (A) − f (B) in terms of A − B, namely, f (A) − f (B)p  cp f Lip A − Bp ,

1 < p < ∞,

(1.0.2)

and applied to all scalar Lipschitz functions f . Various counterexamples show that (1.0.2) does not extend to p = 1. The existing results in the case A − B ∈ S1 either involve a smaller set of admissible scalar functions f and a larger norm of f [1] or they estimate f (A) − f (B) in the larger ideal S1,∞ . As mentioned above, there are several constructions known under the name “double operator integral”, which are denoted by TϕA,B indicating that the function

1 Introduction

3

ϕ, called a symbol, substitutes for an integrand and the spectral measures of the operators A and B substitute for a measure of integration. The properties of the transformation TϕA,B depend on the space where it acts, on type of the symbol ϕ and sometimes on the operators A and B. These parameters and, hence, the specific construction to be used, are frequently stipulated by the class of problems approached with methods of operator integration. Often advantages of different approaches to operator integration should be synthesized for a successful resolution of a problem. In addition to double operator integrals on B(H) and Sp , there exist double operator integrals on B(X), where X is a Banach space, and on the noncommutative Lp -spaces. The latter are useful in the quantized calculus introduced by A. Connes and analytic approach to the Phillips-Atiyah-Patodi-Singer spectral flow. We compare various definitions of double operator integrals in Chap. 3 and discuss problems where they are utilized in Chap. 5. Multilinear operator integrals naturally replace double operator integrals in higher order perturbation problems. For instance, for self-adjoint matrices A, B and n times differentiable functions f , we have the following extension of the representation (1.0.1): f (A)−f (B)−

n−1   1 dk f (B + t (A − B))t =0 = TfA,B,...,B (A − B, . . . , A − B), [n] k k! dt k=1

(1.0.3) where f [n] is the nth order divided difference of f and TfA,B,...,B is a n-linear Schur [n] multiplier considered in the eigen bases of the matrices A and B. On the way to the representation (1.0.3), one also derives the formula  1 dk f (B + t (A − B))t =0 = TfB,B,...,B (A − B, . . . , A − B). [k] k! dt k

(1.0.4)

The relation (1.0.3) suggests that in order to approximate a perturbed matrix function by a noncommutative analog of a Taylor polynomial with prescribed accuracy it suffices to find a suitable bound for the transformation TfA,B,...,B , which [n] is an example of a multilinear operator integral on the product of matrix spaces. The necessity to consider (1.0.3) for infinite dimensional operators A, B arose in mathematical physics when L. S. Koplienko attempted to extend the fundamental I. M. Lifshits–M. G. Krein trace formula  Tr (f (A) − f (B)) = f  (t)ξA,B (t) dt, (1.0.5) R

where ξA,B ∈ L1 (R) is the spectral shift function, to nontrace class perturbations A−B, for which the trace of f (A)−f (B) is undefined, but the trace of (1.0.3) or its modifications can possibly be defined. Using double operator integrals he succeeded to prove an analog of (1.0.5) for Hilbert-Schmidt perturbations A − B in 1984, but

4

1 Introduction

including more general perturbations required a development of a comprehensive theory of multilinear operator integration. Major steps of the development are briefly summarized below and discussed in a greater detail in Chap. 4, while applications of multilinear operator integration, including the trace formula (1.0.5) and its extensions, are discussed in Chap. 5. As in the case of the double operator integrals, there exist several constructions known under the name “multiple operator integral”, each introduced in response to a specific type of problems to be treated. The Birman–Solomyak double operator integral on S2 was extended by B. S. Pavlov to a multilinear transformation on the product space S2 × · · · × S2 in 1969, and a recent approach to the same transformation by C. Coine, C. Le Merdy, F. Sukochev opened new technical opportunities. The transformation tied to a separation of variables of the symbol ϕ was extended from B(H) to B(H) × · · · × B(H) by V. V. Peller in 2006 and to I × · · · × I, where I is a symmetrically normed ideal of a semifinite von Neumann algebra satisfying property (F), by N. A. Azamov, A. L. Carey, P. G. Dodds, F. A. Sukochev independently in 2009. The latter extension allowed to prove existence of higher order derivatives for a broad set of functions f in a general setting and justify that the trace of (1.0.4) is well defined for A − B ∈ Sn , but it did not capture nice properties of UMD ideals Sp , 1 < p < ∞, enabling sharper bounds that were essential for extending (1.0.5) to higher order Taylor remainders. L. S. Koplienko’s attempt to obtain an analog of (1.0.5) for A − B ∈ Sn was implemented in 2013 by D. Potapov, A. Skripka, F. Sukochev, who took an approach to multilinear operator transformations TfA,...,A that does not involve separation of [n] variables of the symbol. To derive bounds for the transformation on Sp1 × · · · × Spn with values in Sp , where 1 < p, pj < ∞, j = 1, . . . , n, and 0 < p1 = p11 + · · · + p1n < 1, they used an intricate recursive procedure preserving the symbol f [n] modulo change of polynomial factors in the simplex integral representation for the divided difference and, thus, keeping the smallest norm of f [n] . Apart from an estimate for TfB,...,B : Sp1 × · · · × Spn → Sp with pj , p as above, they obtained [n] the bound for the trace of the transformation   B,...,B  Tr T [n] (X, . . . , X)   cn f (n) ∞ Xn , n f

X ∈ Sn ,

(1.0.6)

leading to similar bounds for traces of higher order Taylor remainders and their representations in terms of higher order spectral shift functions. One of main objectives in the study of multiple operator integrals consists in obtaining useful estimates for their norms. The existing variety of different multiple operator integral constructions is a by-product of the search for sharp estimates that were necessary to solve particular problems. Thus, a considerable part of this book is dedicated to discussion of various bounds for multiple operator integrals. In particular, we discuss the bounds that are derived based on one of the following methods: spectral integral representation of the transformation (Sects. 3.2 and 4.2), Hilbert space factorization of symbols (Sects. 3.3.3 and 4.3.1), reduction of order of polynomial integral momenta (Sects. 3.3.4 and 4.3.2), transference to

1 Introduction

5

noncommutative Calderón-Zygmund operators (Sect. 3.3.7). The spectral integral approach supplies the best bounds, but is limited to the Hilbert input space. The approach based on factorization of the symbol is fairly universal, but inevitably leads to a larger norm of the symbol. The order reduction of polynomial integral momenta provides the best bounds when the input space enjoys the UMD property. The transference approach allows to obtain the smallest norm of the symbol for a non-UMD space S1 , but enlarges the output space to S1,∞ . From the very beginning, development of multilinear operator integration has been motivated and directed by applications. The former has supplied an indispensable part of a toolkit in perturbation theory and quantised calculus. In Chap. 5, we discuss major applications of multilinear operator integration techniques and present a number of famous problems in operator theory, functional analysis, mathematical physics, and noncommutative geometry whose resolution relied on these techniques. The best up to date results on Lipschitz and Hölder estimates for operator functions, differentiation and approximation of operator functions, trace formulas and spectral shift functions, spectral flow in the von Neumann algebra setting, quantum differentiability, differentiability of noncommutative Lp -norms are presented along with main ideas or outlines of proofs. Our exposition unfolds in order of increasing complexity. We start each section with discussion of the finite dimensional case that provides a basic insight into multiple operator integration without technicalities. We discuss properties and applications of finite dimensional multiple operator integrals or, equivalently, multilinear Schur multipliers independently of the general infinite dimensional theory. The connection between operator integrals and Schur multipliers discussed in this book allows to interchange results of both theories. We define continuous infinite dimensional versions of multiple operator integrals firstly on the Hilbert-Schmidt class, whose Hilbert space structure allows to derive the best results, secondly on Schatten-von Neumann ideals, where the UMD property, if holds, supplies second best results, and the algebra of bounded linear operators, next on noncommutative Lp -spaces and, finally, on general Banach spaces. We implement this program for double operator integrals in Chap. 3 and for general multiple operator integrals in Chap. 4. Thus, the reader looking only for multiple operator integrals in a specific setting can easily locate the respective material in the manuscript. To master multiple operator integration in its full generality, the reader should be proficient in a graduate level functional analysis, harmonic analysis, general spectral theory of linear operators, basic theory of Schatten-von Neumann classes, noncommutative Lp -spaces, as well as symmetrically normed ideals and continuous traces. Nonetheless, the reader with background in linear algebra and basic analysis of functions and matrices can appreciate the theory of finite-dimensional multiple operator integrals (multilinear Schur multipliers); the reader familiar with general theory of linear operators on a Hilbert space can appreciate multiple operator integrals on B(H) and the Schatten-von Neumann ideals. To help the reader to recover the necessary background material, we recall major concepts and give references to relevant literature in Chap. 2.

Chapter 2

Notations and Preliminaries

In this chapter we recall major concepts from function spaces, theory of linear operators, ideals in von Neumann algebras and continuous traces, noncommutative Lp -spaces, Banach space theory, and approximation theory that are involved in our discussion of multiple operator integration and its applications. We also supply references to a systematic treatment of these concepts.

2.1 Spaces of Functions We use the symbol f : D → R to denote a function f defined on a set D with values in a set R. Let I be an interval in the real line R and let f : I → C, where C is the complex plane. The Lipschitz seminorm of f is defined by f Lip(I ) := sup t,s∈I t =s

|f (t) − f (s)| . |t − s|

We say that f is Lipschitz if and only if f Lip(I ) < ∞. The class of all Lipschitz functions on I is denoted by Lip(I ). Let α (R) denote the set of Hölder functions of exponent 0 < α < 1, that is,   |f (t1 ) − f (t2 )| < ∞ . α (R) = f : R → C : f α := sup |t1 − t2 |α t1 ,t2 ∈R t1 =t2

In particular, 1 (R) = Lip(R). For I ⊂ C, the linear space of continuous complex-valued functions on I is denoted by C(I ) and its subspace of bounded functions by Cb (I ). If I ⊂ R, the © Springer Nature Switzerland AG 2019 A. Skripka, A. Tomskova, Multilinear Operator Integrals, Lecture Notes in Mathematics 2250, https://doi.org/10.1007/978-3-030-32406-3_2

7

8

2 Notations and Preliminaries

space of functions continuously differentiable n times on I is denoted by C n (I ) and its subspace of compactly supported functions by Ccn (I ). The space of continuous functions on R that decay at infinity is denoted by C0 (R). For ϕ ∈ C(T), where T is the unit circle, by its derivative at z0 ∈ T, we understand the limit ϕ  (z0 ) :=

lim

T z→z0

ϕ(z) − ϕ(z0 ) , z − z0

(2.1.1)

provided it exists. The symbol C n (T) denotes the set of functions n times continuously differentiable on T in the sense of (2.1.1). Let be a subset in Rn endowed with a Borel σ -algebra and measure μ. Let ∞ L ( , μ) denote the space of all complex-valued, essentially bounded functions on ( , μ). The ess sup norm on L∞ ( , μ) is denoted  · ∞ . When μ is the Lebesgue measure we write L∞ ( ) instead of L∞ ( , μ). For 1  p < ∞, let Lp ( , μ) denote the space of all measurable functions f : → C satisfying  1/p < ∞. The space Lp ( , μ) is equipped with the norm f p := |f |p dμ  · p . When = N, where N is the set of positive integers, and μ is a counting measure, Lp ( , μ) becomes the space of p-summable sequences, which we denote p . Let F f and F −1 f be the Fourier transform and the inverse Fourier transform, respectively, of the function f ∈ L1 (R), that is, 1 F f (t) = √ 2π

 R

f (s) e

−ist

ds, F

−1

1 f (s) = √ 2π

 R

f (t) eist dt.

Given n ∈ N, denote the Wiener space Wn (R) = {f ∈ C n (R) : f (k) , F f (k) ∈ L1 (R), k = 0, . . . , n}. By standard methods of harmonic analysis one can see that Ccn+1 (R) is a dense subset of Wn (R). Let R denote the set of bounded rational complex-valued functions on R with non-real poles, R+ (respectively, R− ) the subset of R consisting of functions with poles in the upper half-plane (respectively, in the lower half-plane). s denote a modified homogenous Besov Let 0 < p, q  ∞, s ∈ R, and let B˜ pq space either on R or on T. In the context of perturbation theory, we are particularly interested in the case p = ∞, q = 1, and s ∈ N ∪ {0}. Let w0 ∈ C ∞ (R) be such that its Fourier transform is supported in [−2, −1/2] ∪ [1/2, 2] and F w0 (y) + F w0 (y/2) = 1 for 1  y  2 and define wk (x) = 2k w0 (2k x) for x ∈ R, k ∈ Z, where Z is the set of integers. Then, for n ∈ N ∪ {0},    n (R) = f ∈ C n (R) : f (n) ∞ + 2nk f ∗ wk ∞ < ∞ . B˜ ∞1 k∈Z

2.1 Spaces of Functions

9

Alternatively, the Besov space can be characterized as follows: n f ∈ B˜ ∞1 (R) ⇐⇒ f (x) = c0 + c1 x + · · · + cn x n + f0 (x),  2nk f0 ∗ wk ∞ < ∞, supp F f0 ⊂ R \ {0}. cj ∈ C, j = 0, . . . , n, k∈Z n The space B˜ ∞1 (R) is sometimes considered with the seminorm

f B˜ n

∞1 (R)

=



2nk f ∗ wk ∞

k∈Z

and sometimes with the seminorm (n) n (R) = f ∞ + f B˜ n f B∞1

∞1 (R)

.

When the Besov space is considered with the latter seminorm, we denote it n (R). There are other deviations in definitions and notations of Besov spaces B∞1 n in different publications. In particular, initially Besov spaces B∞1 (R) were defined via summability of iterated difference operators. For a detailed exposition of Besov spaces we refer the reader to [140, 197, 207]. n Sometimes it is simpler to work with subclasses of B∞1 (R). In particular, n (R). Wn (R) ⊂ B∞1

Another known inclusion is n (R) {f ∈ C n+1 (R) : f (n) , f (n+1) ∈ L2 (R)} ⊂ B∞1

(see [157, Lemma 7]). It is also known that n {f ∈ C n (R) : f (n−1) ∈ 1− (R), f (n) ∈  (R)} ⊂ B∞1 (R)

for 0 < < 1 (see [157, Theorem 4 and Remark 5]). For a function f on the unit circle T by F f we define its Fourier series with coefficients denoted by F f (n), n ∈ Z. Let A(D) denote the disc algebra, that ¯ → C analytic on the open unit disc D and is, the algebra of functions f : D ¯ The algebra A(D) can be naturally identified continuous on the closed unit disc D. with the algebra of continuous functions on the unit circle T with vanishing Fourier coefficients F (−n), n ∈ N. Given n ∈ N, denote Wn (T) = {f : T → C : f (k) ∈ L1 (T), F f (k) ∈ 1 (Z), k = 0, . . . , n}.

10

2 Notations and Preliminaries

n (T) of functions on T can be defined as follows. Let n ∈ N. The Besov class B∞1 ∞ Let w ∈ C (R) be such that

w  0, supp w ⊂

 x 1 , 2 , and w(x) = 1 − w for x ∈ [1, 2]. 2 2

Consider the trigonometric polynomials wm and wm defined by   k  wm (z) = w m zk , m  1, w0 (z) = z + 1 + z, and 2 k∈Z

(z) = wm (z), m  0. wm

Then, for each function ϕ on T,  

ϕ= ϕ ∗ wm + ϕ ∗ wm . m0

m1

n (T) consists of functions ϕ on T such that The Besov class B∞1

{2nm ϕ ∗ wm ∞ }m  0 ∈ 1 and {2nm ϕ ∗ wm ∞ }m  1 ∈ 1

and is considered with the seminorm  

n (T) = ϕB∞1 2nk ϕ ∗ wk ∞ + 2nk ϕ ∗ wk ∞ . k0

k1

We will frequently use the fact that for n1 , n2 ∈ N, n2 n1 B∞1 (T) ⊂ B∞1 (T) if n1 < n2 .

We also have n Wn (T) ⊂ B∞1 (T).

2.2 Divided Differences We recall that the divided difference of the zeroth order f [0] is the function f itself. Let λ0 , λ1 , . . . , λn be points in R (respectively, in T) and let f ∈ C n (R) (respectively, f ∈ C n (T)). The divided difference f [n] of order n is defined recursively by f [n] (λ0 , . . . , λn ) = lim

λ→λn

f [n−1] (λ0 , . . . , λn−2 , λ) − f [n−1] (λ0 , . . . , λn−2 , λn−1 ) . λ − λn−1 (2.2.1)

2.3 Linear Operators

11

Basic properties of the divided difference of a function defined on R can be found in, for example, [65, Section 4.7]. In particular, for f ∈ C n (R) and I an interval in R, f [n] (λ0 , . . . λ0 ) =  [n]  f 

L∞ (I n+1 )



1 (n) f (λ0 ), n!

1 (n) f L∞ (I ) , n!

(2.2.2)

and the divided difference is invariant with respect to any permutation of its variables. By [191, Lemma 3.2] (which follows directly from [59, Theorem 2.1 and Lemma 2.2]), for f ∈ C n (T), we have  [n]  f  ∞ n+1  L (T )

 (n)  π (n+3)/2 f  ∞ , L (T) 2n+1 ((n + 1)/2)

(2.2.3)

where (·) is the Gamma function. For f ∈ Lip(I ) we have sup |f [1] (λ, μ)|  f Lip(I ) .

(2.2.4)

λ,μ∈I λ=μ

2.3 Linear Operators Let X be a Banach space equipped with the norm  · X and Y a Banach space equipped with the norm  · Y . Denote by B(X, Y) the Banach space of bounded linear operators mapping X to Y and equipped with the operator norm X :=

sup X(ξ )Y ,

ξ X  1

X ∈ B(X, Y).

When we need to distinguish between different norms we use more detailed symbols  · X→Y or · : X → Y instead of  · . If X = Y, we write B(X) instead of B(X, X). We identify the algebraic tensor product X∗ ⊗ Y with the space of finite rank operators in B(X, Y) via (x ∗ ⊗ y)(x) := x ∗ (x)y,

x ∈ X, x ∗ ∈ X∗ , y ∈ Y.

We will also work with multilinear transformations between Banach spaces, that is, transformations that are linear in each of the variables separately. If X1 . . . , Xn

12

2 Notations and Preliminaries

are Banach spaces, then the symbol Bn (X1 × · · · × Xn , Y) stands for the space of n-linear mappings X1 × · · · × Xn → Y equipped with the norm X :=

sup

ξ1 X1 ,...,ξn Xn  1

X(ξ1 , . . . , ξn )Y ,

X ∈ Bn (X1 × · · · × Xn , Y).

Again we can also use  · X1 ×···×Xn →Y or · : X1 × · · · × Xn → Y instead of  · . By IX ∈ B(X) we denote the identity operator on X. When there is no ambiguity, we simply write I for the identity operator.   Let H be a separable Hilbert space equipped with the inner product ·, · and the respective norm  · . We will consider both an infinite dimensional Hilbert space H and the d-dimensional space 2d , d ∈ N. In the case H = 2d , we consider the canonical inner product. Let B(H) denote the C ∗ -algebra of bounded linear operators on H equipped with the operator norm. The subset of the normal operators in B(H) is denoted by Bnorm (H) and of the self-adjoint operators by Bsa (H). The set of closed self-adjoint operators defined on a dense subset D of H is denoted Dsa . When we write A, B ∈ Dsa , we assume that A, B are densely defined on the same subset of H. We will freely apply facts from basic spectral theory of self-adjoint and, more generally, normal operators that can be found in, for example, [38, Chapters 5 and 6]. Let Pξ denote the orthogonal projection on the unit vector ξ ∈ H, that is,   Pξ (·) = ·, ξ ξ. Let σ (A) denote the spectrum of a linear operator A densely defined in H. If A ∈ Dsa , then σ (A) ⊂ R. Let EA denote the spectral measure of a normal operator A. Given a Borel function f , let f (A) denote the operator function defined by the standard functional calculus. We will work with the tensor product A⊗B of a d ×d-matrix A and m×m-matrix B, which is the dm × dm-matrix given by ⎛

⎞ a11 B · · · a1d B ⎜ .. ⎟ . A ⊗ B = ⎝ ... . ⎠ ad1 B · · · add B We will also touch upon tensor product spaces of infinite dimensional operators. For details on this account we refer the reader to [174, 202].

2.4 Schatten-von Neumann Classes Results recalled in this section and a more detailed discussion of ideals of compact operators can be found in, for instance, [38, 85, 183].

2.4 Schatten-von Neumann Classes

13

Let an infinite dimensional separable Hilbert space H be fixed. Let S∞ denote the Banach space of all compact operators on H equipped with the uniform norm. Recall that every X ∈ S∞ has a representation X=

∞ 

λn (X)Pn ,

n=1

where {λn (X)}∞ n=1 is a sequence of eigenvalues of the operator X such that the sequence {|λn (X)|}∞ n=1 is decreasing (that is, not increasing) and tends to 0 and Pn , n ∈ N, is a rank one projection on H. Denote by F the algebra of all finite rank operators on H, by Fnorm its subset of normal and by Fsa its subset of self-adjoint operators. Let 1  p < ∞. By Sp we denote the pth Schatten(-von Neumann) ideal, that is, the Banach ideal of all X ∈ S∞ such that Xp := (Tr(|X|p ))1/p < ∞, where |X| = (X∗ X)1/2 and Tr(·) is the standard trace extending the canonical matrix trace, that is, Tr(A) =

∞ 

λn (A)

n=1

for a positive semidefinite operator A. When H = 2d , we denote the respective p Schatten-von Neumann ideal (B(2d ),  · p ) by Sd . By X = X∞ we denote the operator norm of X ∈ B(H). The set of finite rank operators is dense in every Sp considered with the norm  · p , 1  p  ∞. Let {sn (X)}∞ n=1 denote the decreasing sequence of singular values of X, that is, the decreasing sequence of eigenvalues of |X|. For 1  p < ∞, X ∈ Sp if p and only if {sn (X)}∞ n=1 is an element of the Banach sequence space  , that is,  ∞ p n=1 sn (X) < ∞. In particular, we have the inclusion Sp  Sq ,

p < q,

and the inequalities X  Xq  Xp . The norm  · p satisfies the Hölder inequality XY r  Xp Y q ,

(2.4.1)

14

2 Notations and Preliminaries

where 1  p, q, r  ∞ are such that duality

1 r

=

+ q1 and X ∈ Sp , Y ∈ Sq . We have the

1 p

1 1 + = 1, p q

(Sp )∗ = Sq ,

with every functional ϕ on Sp in the form ϕ(X) = Tr (XYϕ ) for some Yϕ ∈ Sq , ϕ = Yϕ q . The ideal S1 is called the trace class ideal and S2 the Hilbert-Schmidt class. The ideal S2 with the inner product X, Y  := Tr(Y ∗ X), X, Y ∈ S2 , and corresponding norm  · 2 becomes a Hilbert space. The Hilbert-Schmidt norm of a finite-dimensional operator X ∈ B(2d ) can also be computed via a matrix  d representation of X = xj k j,k=1 as follows: X2 =

d 

|xj k |2

1/2 .

(2.4.2)

j,k=1

The weak Schatten(-von Neumann) ideal Sp,∞ , 1  p < ∞, is defined by 1   Sp,∞ = X ∈ S∞ : sup n p sn (X) < ∞ .

n∈N

The space Sp,∞ equipped with the quasi-norm 1

Xp,∞ := sup n p sn (X) n∈N

is a quasi-Banach space. If p > 1, there exists an equivalent norm 1

Xp,∞ := sup N p N∈N

−1

N 

sk (X)

k=1

with respect to which Sp,∞ is a Banach space. We have the inclusion Sp ⊂ Sp,∞ ⊂ Sq for 1  p < q < ∞.

2.5 Product of Spectral Measures

15

2.5 Product of Spectral Measures In this section we collect preliminaries on products of spectral measures crucial in the definition of a double operator integral. Let ( j , j ), j = 1, 2 be a measurable space. Let E and F be spectral measures given on 1 and 2 , respectively, with values in the set of orthogonal projections in B(H). Let us consider E(σ1 ) : X → E(σ1 )X, σ1 ∈ 1 , σ2 ∈ 2 , X ∈ S2 . F (σ2 ) : X → XF (σ2 ), It is clear that E and F are commuting spectral measures on ( 1 , 1 ) and ( 2 , 2 ), respectively, with respect to S2 . Define the product of the measures E and F G(σ1 × σ2 ) := E(σ1 )F (σ2 ), σ1 ∈ 1 , σ2 ∈ 2 ,

(2.5.1)

so that G(σ1 × σ2 )(X) = E(σ1 )XF (σ2 ), X ∈ S2 .

(2.5.2)

To verify that G is indeed a spectral measure, we need to show that G is σ -additive. We note that the product of two spectral measures can fail to be σ additive for an arbitrary Hilbert space, as it is shown in [43]. In [40, Theorem 2] it is established that G constructed above is in fact σ -additive. The proof uses the specificity of the fact that G is defined with respect to S2 , but not an arbitrary Hilbert space. A more general result for the product of n spectral measures, n  2 is proved in [138, Theorem 1]. We present the proof from [40, Theorem 2] here for the reader’s convenience. Proposition 2.5.1 Let ( 1 , 1 ) and ( 2 , 2 ) be measure spaces, H a separable Hilbert space and let E and F be spectral measures with respect to H on ( 1 , 1 ) and ( 2 , 2 ), respectively. Then, the mapping G : 1 × 2 → B(S2 ) defined by (2.5.2) extends to a spectral measure on ( , ) with respect to S2 , where = 1 × 2 and  = 1 ⊗ 2 is the minimal σ -algebra generated by the algebra 1 × 2 of “measurable rectangles”.  (i)  (k) (k) (i)  Proof If σ ∈ 1 × 2 is such that σ = nk=1 σ1 × σ2 , where σ1 × σ2 ∩  (j ) (j )  σ1 × σ2 = ∅, for i = j, then we define G(σ )X :=

n 

G(σ1(k) × σ2(k) )X.

(2.5.3)

k=1

Correctness of the definition above can be checked using standard arguments.

16

2 Notations and Preliminaries

Let us prove that G takes values in orthogonal projections on S2 . Indeed, for X, Y ∈ S2 and δ = σ1 × σ2 , σ1 ∈ 1 , σ2 ∈ 2 , we have that G(δ)2 = G(δ) and 

   G(δ)X, Y = E(σ1 )XF (σ2 ), Y = Tr(Y ∗ E(σ1 )XF (σ2 ))     = Tr((E(σ1 )Y F (σ2 ))∗ X) = X, E(σ1 )Y F (σ2 ) = X, G(δ)Y ,

that is, G(δ)∗ = G(δ). For an arbitrary set δ ∈ 1 × 2 the proof is similar using (2.5.3). Since E and F are spectral measures, it follows that G( )X = E( 1 )XF ( 2 ) = X, that is, G( ) is the identity operator on S2 . We show that G is σ -additive on 1 × 2 . For X, Y ∈ S2 consider the scalar measure   νX,Y : σ1 × σ2 → G(σ1 × σ2 )X, Y , σ1 ∈ 1 , σ2 ∈ 2 .  Let δ ∈ 1 ⊗ 2 be such that δ = ∞ n=1 δn , δn ∈ 1 × 2 and δn ∩ δm = ∅, n = m, n, m ∈ N. Recalling the inequality ∞ 

G(δn )X22  X22 , X ∈ S2 ,

n=1

(see, e.g., [123, Lemma 12.4.7]) and evaluating ∞ ∞        G(δn )X, Y  =  G(δn )X, G(δn )Y  n=1

n=1



∞ 

G(δn )X2 G(δn )Y 2

(2.5.4)

n=1



∞ 

G(δn )X22

∞

1  2

n=1

G(δn )Y 22

1 2

n=1

 X2 Y 2 ,    2 we obtain that the series ∞ n=1 G(δn )X, Y converges for any X, Y ∈ S . Next we prove that νX,Y is a σ -additive scalar measure, that is, ∞ ∞  

    G G(δn )X, Y . δn X, Y = n=1

n=1

(2.5.5)

2.5 Product of Spectral Measures

17

    If X = ·, ξ2 ξ1 and Y = ·, η2 η1 for some ξj , ηj ∈ H, and σj ∈ j , j = 1, 2, then we have that    νX,Y ∗ (σ1 × σ2 ) = Tr(Y E(σ1 )XF (σ2 )) = E(σ1 )ξ1 , η2 F (σ2 )η1 , ξ2 , which is a product of two scalar σ -additive measures. Thus, νX,Y is σ -additive for any one-dimensional operators X and Y. Therefore, any one-dimensional operators X and Y satisfy (2.5.5). If each of X and Y is represented by a finite sum of one-dimensional operators, then by linearity of the inner product, we obtain that X and Y satisfy (2.5.5). Let now X, Y ∈ S2 and {Xk }k  1 and {Yk }k  1 be sequences of finitedimensional operators such that Xk → X and Yk → Y in S2 as k → ∞. Since ∞ δ = n=1 δn ∈ 1 ⊗ 2 , it follows from (2.5.3) that 

   G(δ)Xk , Yk → G(δ)X, Y as k → ∞.

(2.5.6)

Applying (2.5.4), we have that ∞ ∞        G(δn )Xk , Yk − G(δn )X, Y   n=1

n=1

∞      | G(δn )Xk , Yk − G(δn )X, Y |



n=1



∞ ∞           | G(δn )Xk , Yk − G(δn )X, Yk | + | G(δn )X, Yk − G(δn )X, Y | n=1

=

∞ 

n=1

  | G(δn )(Xk − X), Yk | +

n=1

∞ 

  | G(δn )X, Yk − Y |

n=1

 Xk − X2 Yk 2 + X2 Yk − Y 2 . Hence, ∞   n=1

∞     G(δn )Xk , Yk → G(δn )X, Y as k → ∞, n=1

which along with (2.5.6) proves (2.5.5) for all X, Y ∈ S2 . The observation that σ -additivity of νX,Y implies σ -additivity of G completes the proof.  

18

2 Notations and Preliminaries

2.6 Classical Noncommutative Lp -Spaces and Weak Lp -Spaces In this section we recall definitions and some important properties of the classical noncommutative Lp -spaces and the weak Lp -spaces. Details can be found in, for instance, [123, 154, 206]. Noncommutative Lp -spaces generalize Schatten classes. Let M be a semifinite von Neumann algebra of bounded linear operators defined on H and let τ be a semifinite normal faithful trace on M. Note that (B(H), Tr) is one of examples of (M, τ ). We denote the subset of self-adjoint elements of M by Msa . We use the notation XηM for a closed densely defined operator X affiliated with M, and H ηMsa for a self-adjoint operator H ηM (i.e., all the spectral projections of H are elements of M). Let S(M, τ ) denote the set of τ -measurable operators affiliated with M. We recall that H ηMsa is τ -measurable if given > 0 there exists a projection P ∈ M such that P (H) ⊂ dom(H ) and τ (I − P ) < [79, Definition 1.2]. Let μt (X) denote the tth generalized s-number [79, Definition 2.1] of X ∈ S(M, τ ). By [79, Proposition 2.2], μt (X) = inf{s  0 : τ (E|X| (s, ∞))  t}. Further properties of generalized s-numbers can be found in [79]. For instance, if (M, τ ) = (B(H), Tr), then S(B(H), Tr) = B(H) and the generalized s-numbers coincide with the singular values of the operators, namely, μt (X) = sn (X),

t ∈ [n − 1, n), n ∈ N.

An operator X ∈ M is said to be τ -compact if and only if lim μt (X) = 0.

t →∞

An operator X ∈ M is said to be τ -(Breuer-)Fredholm if the projections on ker X and ker X∗ are τ -finite and there exists a τ -finite projection P ∈ M such that ran(I − P ) ⊆ ran(X). If an operator has τ -compact resolvent, then it is τ -Fredholm. The noncommutative Lp -space, 1  p < ∞, associated with (M, τ ) is   Lp (M, τ ) := XηM : Xp := τ (|X|p )1/p < ∞ . This space can also be described as Lp (M, τ ) = {X ∈ S(M, τ ) : μ(X) ∈ Lp (0, ∞)},

2.6 Classical Noncommutative Lp -Spaces and Weak Lp -Spaces

19

where (Lp (0, ∞), ·p ) is the usual Lebesgue space, and μ(X) denotes the function t → μt (X). In this case we have Xp = μ(X)p ,

X ∈ Lp (M, τ ).

An example of a noncommutative Lp -space is the Schatten ideal Sp Lp (B(H), Tr ). We denote

=

L∞ (M, τ ) := M and let  · ∞ =  ·  stand for the operator norm. The Hölder inequality (2.4.1) extends to the setting of noncommutative Lp -spaces: XY r  Xp Y q , where 1  p, q, r  ∞ are such that Lq (M, τ ). We have the duality

1 r

=

1 p

+

1 q

and X ∈ Lp (M, τ ), Y ∈

 p ∗ L (M, τ ) = Lq (M, τ ), where 1  p < ∞, p1 + q1 = 1. The weak noncommutative Lp -space, 1  p < ∞, associated with (M, τ ) is the space 1   Lp,∞ (M, τ ) := X ∈ S(M, τ ) : XLp,∞ := sup t p μt (X) < +∞ .

t 0

If 1  p < ∞, then the space Lp,∞ (M, τ ) equipped with the quasi-norm  · Lp,∞ given above becomes a quasi-Banach space (see, e.g., [123, Example 2.6.10] and [68]). For 1 < p < ∞, there exists a norm  · Lp,∞ on Lp,∞ (M, τ ) given by 1

XLp,∞ := sup t p −1 t >0



t

μs (X)ds, X ∈ Lp,∞ (M, τ ),

0

which satisfies  · Lp,∞   · Lp,∞ 

p  · Lp,∞ . p−1

(2.6.1)

In the special case when M is the commutative von Neumann algebra L∞ (0, ∞) equipped with the normal semifinite trace given by integration with respect to the Lebesgue measure, the space Lp,∞ (M, τ ) coincides with the classical commutative weak Lp -space Lp,∞ (0, ∞) (see, e.g., [68, 117, 122]). The following result is the Hölder type inequality for the quasi-norm  · Lp,∞ and the norm  · Lp,∞ .

20

2 Notations and Preliminaries

Lemma 2.6.1 Let m ∈ N and let 1  p, p1 , . . . , pm < ∞ be such that 1 pm

=

1 p.

1 p1

+ ··· +

(i) For all Xj ∈ Lpj ,∞ (M, τ ), 1  j  m, 1

X1 · . . . · Xm Lp,∞  m p

m

Xj Lpj ,∞ . j =1

(ii) If 1 < p, p1 , . . . pm < ∞, then for all Xj ∈ Lpj ,∞ (M, τ ), 1  j  m, X1 · . . . · Xm Lp,∞ 

1 p mp p−1

m

Xj Lpj ,∞ . j =1

Proof (i) It follows from [123, Corollary 2.3.16] that for all t > 0, m

μt (X1 · . . . · Xm ) 

m

μ (Xj ) j =1

=

t m

t m

− p1

j =1 m j =1

t − 1 pj Xj Lpj ,∞ m

Xj Lpj ,∞ .

Hence, 1

1

m

t p μt (X1 · . . . · Xm )  m p j =1

Xj Lpj ,∞ .

Taking the supremum over t > 0 and using (2.6.1) proves (i). (ii) It follows from (2.6.1) and (i) that X1 · . . . · Xm 

Lp,∞

1 p p  X1 · . . . · Xm Lp,∞  mp p−1 p−1

m

Xj Lpj ,∞ , j =1

completing the proof.

 

Recall that for 1  r, q  ∞, the space (Lq + Lr )(M, τ ) is defined as (Lq + Lr )(M, τ ) := {X ∈ S(M, τ ) : X(Lq +Lr )(M,τ ) 0 such that T : Lp1 (M, τ ) × . . . × Lpk (M, τ ) → Lp (M, τ )  cp .

(2.6.2)

Then, q

T ∈ Bk ((Lq + Lr )(M, τ )×k , (L k + L k )(M, τ )). r

Proof By the definition of the space Lq + Lr , for X1 , . . . , Xk ∈ (Lq + Lr )(M, τ ), there are Yj ∈ Lq (M, τ ), Zj ∈ Lr (M, τ ) such that Xj = Yj + Zj , j = 1, . . . , k. Hence, T (X1 , . . . , Xk )

q

r

L k +L k

= T (Y1 + Z1 , . . . , Yk + Zk ) q r L k +L k      = T (X1,A , . . . , Xk,A ) q 



T (X1,A , . . . , Xk,A )

where Xj,A =

j ∈A

Zj ,

j∈ /A

q

r

L k +L k

A ⊂{1,...,k}

! Yj ,

r

L k +L k

A ⊂{1,...,k}

, j = 1, . . . , k.

Fix A ⊂ {1, . . . , k}. If 1 < pA < ∞ is such that 1 |A | k − |A | + , = pA r q then r q  pA  k k

,

(2.6.3)

22

2 Notations and Preliminaries q

r

and, therefore, LpA (M, τ ) ⊂ (L k + L k )(M, τ ). Thus, T (X1,A , . . . , Xk,A )

q

r

L k +L k

 const T (X1,A , . . . , Xk,A )LpA .

(2.6.4)

Using (2.6.2), we obtain T (X1,A , . . . , Xk,A )LpA  cpA

Yj Lq j ∈A

Zj Lr j∈ /A

k

 cpA

(Yj Lq + Zj Lr ).

(2.6.5)

j =1

Combining (2.6.3)–(2.6.5) implies T (X1 , . . . , Xk )

q

r

L k +L k

 const



T (X1,A , . . . , Xk,A )LpA

A ⊂{1,...,k}

 const



k

(Yj Lq + Zj Lr ).

cpA

A ⊂{1,...,k}

j =1

Taking the infimum over all the representations Xj = Yj + Zj , Yj Lq (M, τ ), Zj ∈ Lr (M, τ ), j = 1, . . . , k, completes the proof.

∈  

2.7 The Haagerup Lp -Space In this section we recall the construction of noncommutative Lp -spaces associated with an arbitrary von Neumann algebra. We use Haagerup’s definition [87], and Terp’s exposition of the subject [206]. The basics on von Neumann algebras and Tomita’s modular theory can be found in [96]. Let M be an arbitrary von Neumann algebra with a faithful normal semifinite weight φ0 . We consider the one-parameter modular automorphism group σ φ0 = φ {σt 0 }t ∈R (associated with φ0 ) on M and obtain a semifinite crossed product von Neumann algebra N := M σ φ0 R,

(2.7.1)

which admits the canonical semifinite trace τ and a trace-scaling dual action θ = {θs }s∈R such that τ ◦ θs = e−s τ for all s ∈ R.

2.7 The Haagerup Lp -Space

23

The original von Neumann algebra M can be identified with a θ -invariant von Neumann subalgebra L∞ H aag (M) of N. For 1  p < ∞, the noncommutative p Haagerup Lp -space LH aag (M) is defined by p

LH aag (M) := {X ∈ S(N, τ ) : θs (X) = e

− ps

X for all s ∈ R}.

It is known from [206, Part II, Theorem 7] that there is a linear bijection ψ → Xψ between the predual space M∗ and L1H aag (M). Due to this correspondence we define the trace tr : L1H aag (M) → C by tr(Xψ ) := ψ(I ), Xψ ∈ L1H aag (M).

(2.7.2)

p

Given any X ∈ LH aag (M), 1  p < ∞, we have the polar decomposition X = p U |X|, where |X| is a positive operator in LH aag (M) and U is a partial isometry contained in M. It is established in [206, Proposition 12] that |X|p ∈ L1H aag (M). p Thus, we can define a Banach norm (see [206, Corollary 27]) on LH aag (M) by setting 1

XLp

H aag

p

:= tr(|X|p ) p , X ∈ LH aag (M).

(2.7.3) p

The following lemma due to [79, Lemma 1.7] shows that if X ∈ LH aag (M), then = XLp,∞

XLp

H aag

(2.7.4)

p

and, therefore, LH aag (M) is a closed linear subspace in Lp,∞ (N, τ ). p

Lemma 2.7.1 Let 1  p < ∞. If X ∈ LH aag (M), then μt (X) = XLp

H aag

·t

− p1

, t > 0.

The following version of the Hölder inequality is proved, for instance, in [206, Theorem 23]. Lemma 2.7.2 Let k ∈ N and 1  p, pj  ∞, j = 1, . . . , k, be such that ···+

1 pn

=

1 p.

For every Xj ∈

X1 · . . . · Xk Lp

H aag

p LHjaag (M),

 X1 Lp1

1 p1

+

1  j  k,

H aag

· . . . · Xk Lpk .

(2.7.5)

H aag

Let now M be a semifinite von Neumann algebra and τ0 a faithful normal ¯ ∞ (R) be the von Neumann algebra tensor product semifinite trace on M. Let M⊗L

24

2 Notations and Preliminaries

¯ 2 (R) equipped of M and L∞ (R) acting on the Hilbert space tensor product H ⊗L  with the tensor product trace τ := τ0 ⊗ ν, where ν is the trace on L∞ (R), given by  ν(f ) =

R

f (s)e−s ds, 0  f ∈ L∞ (R).

¯ ∞ (R) satisfying Recall that τ  is the unique faithful normal semifinite trace on M⊗L  ∞ τ (X ⊗ f ) = τ0 (X)ν(f ), X ∈ M, f ∈ L (R). It is known that in the case when M is a semifinite von Neumann algebra there exists a trace preserving ∗-isomorphism between the crossed product von ¯ ∞ (R), τ  ) (see [209, Part II, Neumann algebra (N, τ ) defined in (2.7.1) and (M⊗L ∞ ¯ Proposition 4.2]). We identify (N, τ ) with (M⊗L (R), τ  ). It is also known that ∞ for all X ∈ M, f ∈ L (R), θs (X ⊗ f ) = X ⊗ ls (f ), s ∈ R, where ls is the left translation by s (see, e.g., [209, Part II, Proposition 4.2]). The following result is well-known (see, e.g., [87, Theorem 2.1], [206, p. 62]). t

Theorem 2.7.3 Let 1  p  ∞ and ζp (t) = e p , t ∈ R. Let M be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ0 . The following assertions hold. ¯ ∞ (R) is τ  -measurable for all X ∈ (i) The operator X ⊗ ζp affiliated with M⊗L Lp (M, τ0 ). p (ii) X ⊗ ζp ∈ LH aag (M), for all X ∈ Lp (M, τ0 ). (iii) The mapping X → X ⊗ ζp , X ∈ Lp (M, τ0 ) p

is an isometry from Lp (M, τ0 ) into LH aag (M).

2.8 Symmetrically Normed Ideals The definitions of objects recalled in this section can be found in, for instance, [123]. For a more detailed treatment of the subject we refer the reader to [57, 85, 183]. Let M be a semifinite von Neumann algebra. Let I be a symmetrically normed ideal of M with norm  · I , that is, I is a two sided Banach ideal with respect to the norm  · I , which satisfies the properties (i) A ∈ M, B ∈ I, 0  A  B implies A ∈ I and AI  BI , (ii) there is a constant c1 > 0 such that B  c1 BI for every B ∈ I, (iii) for all A, C ∈ M and B ∈ I, we have ABCI  A BI C.

2.8 Symmetrically Normed Ideals

25

A simple example of I is the ideal Lp (M, τ ) := Lp (M, τ ) ∩ M equipped with the norm  · I = max{ · ,  · p }. Another example is the dual Macaev (also called classical Dixmier-Macaev) ideal L

(1,∞)

 := A ∈ S∞ : A(1,∞) := sup

 n  1 sk (A) < ∞ . n∈N log(1 + n) k=1

To get another class of examples we let ψ be a concave function satisfying lim ψ(t) = 0,

t →0+

lim ψ(t) = ∞.

t →∞

Then the Marcinkiewicz (also called Lorentz) ideal Iψ associated with a σ -finite, semifinite von Neumann algebra factor M and its ideal norm  · Iψ are defined by     t 1 Iψ = A ∈ M : AIψ := max A, sup μs (A) ds < ∞ . t >0 ψ(t) 0 Let τI be a trace on a symmetrically normed ideal I, that is, τI is a linear functional on I satisfying the unitary invariance τI (U AU ∗ ) = τI (A) for all A ∈ I, unitary U ∈ M. The unitary invariance of the trace τI implies its cyclicity property τI (AB) = τI (BA) for all A ∈ M, B ∈ I (see, e.g., [96, Proposition 8.1.1]). We also assume that τI is positive, that is, τI (A)  0 for all A ∈ I, A  0 and that τI is bounded (equivalently, continuous) with respect to the ideal norm  · I , that is, there is a constant c2 > 0 such that |τI (A)|  c2 AI for all A ∈ I. Examples of (I, τI ) include (S1 , Tr), (L1 (M, τ ), τ ), (L(1,∞) , Trωd ), and (Iψ , τωc ), where Trωd is the Dixmier trace corresponding to a generalized limit ωd on ∞ and τωc is the Dixmier trace corresponding to a dilation invariant Banach limit ωc on L∞ (0, ∞).

26

2 Notations and Preliminaries

Given a symmetrically normed ideal I equipped with a continuous trace τI , we consider the root ideals   I1/p = A ∈ M : |A|p ∈ I , p ∈ N, whose elements satisfy the Hölder inequality ABI  AI1/p BI1/q ,

1 1 + = 1, p q

where AI1/p := ( |A|p I )1/p (see [77, 200]).

2.9 Traces on L1,∞ (M, τ ) Let M be a semifinite von Neumann algebra equipped with a normal faithful semifinite trace τ . In this section we adopt some terminology from the theory of singular traces on symmetric operator spaces (see [123]) to traces on the noncommutative weak L1 -space L1,∞ (M, τ ) (see also [73, Section 6]). A trace φ on L1,∞ (M, τ ) is a linear functional, which is unitarily invariant, that is, φ : L1,∞ (M, τ ) → C satisfies φ(U XU ∗ ) = φ(X) for all X ∈ L1,∞ (M, τ ) and all unitaries U ∈ M. A trace φ on L1,∞ (M, τ ) is said to be normalized if φ(X) = 1 for every 0  X ∈ L1,∞ (M, τ ) with μt (X) = t −1 , t > 0. Fix a free ultrafilter ω on N. The limit with respect to the ultrafilter ω is denoted by limn→ω . In the following lemma we introduce a particular trace on L1,∞ (M, τ ), which is essential in the proof of the main result in Sect. 5.8. Lemma 2.9.1 The functional φ : (L1,∞ (M, τ ))+ → R+ , given by φ(X) := lim

n→ω

1 log(1 + n)



n

μt (X) dt, 0  X ∈ L1,∞ (M, τ ),

(2.9.1)

1

extends to a positive normalized trace on L1,∞ (M, τ ). Proof First we show that φ is a linear functional on all positive elements of L1,∞ (M, τ ). The equality φ(αX) = αφ(X), α > 0, 0  X ∈ L1,∞ (M, τ ), is obvious. Next we show that φ(X + Y ) = φ(X) + φ(Y ) for 0  X, Y ∈ L1,∞ (M, τ ).

2.9 Traces on L1,∞ (M, τ )

27

Let n > 1 be fixed. By [97, Lemma 8.4], there exists λ ∈ N satisfying 

n



n

μt (X + Y )dt 

λa

(μt (X) + μt (Y ))dt,

a

for all λa < n. Taking a = 1/λ we infer that 

n



n

μt (X + Y ) dt 

1

(μt (X) + μt (Y )) dt

1/λ



n

=



1

(μt (X) + μt (Y )) dt +

1

(μt (X) + μt (Y )) dt.

1/λ

Dividing both parts of the latter inequality by log(1 + n) and taking limn→ω imply φ(X + Y )  φ(X) + φ(Y ). Conversely, by [123, Lemma 3.4.4], there exists λ ∈ N such that 

n



n

(μt (X) + μt (Y )) dt  2

λa

μ2t (X + Y ) dt, λa < n.

a

Without loss of generality, we may assume that λ > 2. Setting a = 1/λ implies 

n

 (μt (X)+μt (Y )) dt  2

1

 =



n

2n

μ2t (X+Y ) dt= 1/λ



1

n

μt (X+Y ) dt+ 2/λ

(2.9.2)

μt (X+Y ) dt 2/λ



μt (X+Y ) dt+ 1

2n

μt (X+Y ) dt. n

Observe that 

2n n

μt (X + Y ) dt  X

+ Y L1,∞



2n n

t −1 dt = X + Y L1,∞ log 2.

Dividing (2.9.2) by log(1 + n) and taking limn→ω imply φ(X) + φ(Y )  φ(X + Y ). Thus, φ is additive and positively homogeneous on (L1,∞ (M, τ ))+ := {X ∈ L1,∞ (M, τ ) : X  0}. It extends to L1,∞ (M, τ ) by linearity, and we denote this extension by φ. Since μ(U XU ∗ ) = μ(X) for all X ∈ L1,∞ (M, τ ) and all unitaries U ∈ M, it follows that

28

2 Notations and Preliminaries

φ is unitarily invariant. The fact that φ is positive and normalized follows directly from definition (2.9.1).   Lemma 2.9.2 The extension of φ defined in (2.9.1) is a bounded linear functional on L1,∞ (M, τ ). Proof If X ∈ (L1,∞ (M, τ ))+ , then |φ(X)|  XL1,∞ lim

n→ω

1 log(1 + n)



n 1

t −1 dt = XL1,∞ .

Let now X ∈ L1,∞ (M, τ ). Then, X = (Re X)+ − (Re X)− + i(Im X)+ − i(Im X)− . By [98, Chapter 1], (Re X)± L1,∞ , (Im X)± L1,∞  c0 XL1,∞ , where c0 is the modulus of concavity of the quasi-norm  · L1,∞ . Therefore,

1 2 |φ(X)|  max{φ((Re X)+ )2 , φ((Re X)− )2 }+ max{φ((ImX)+ )2 , φ((Im X)− )2 } √  2 c0 XL1,∞ . Lemma 2.9.3 If X ∈ L1,∞ (M, τ ) is such that τ (E|X| (0, ∞)) < ∞, then φ(X) = 0, where φ is defined in (2.9.1). Proof Let c := τ (E|X| (0, ∞)) for some c > 0. Since μ(X) is the right inverse function of t → τ (E|X| (t, ∞)), it follows that μc (X) = 0. Therefore, 

n 1



c

μt (X) dt =

μt (X) dt, for all n  c,

1

implying φ(X) = 0.

 

The proof of the following lemma immediately follows from Lemma 2.7.1 and existence of the Jordan decomposition of elements from L1H aag (M) (see [15, Theorem 6]). Lemma 2.9.4 Let (N, τ ) be the crossed product von Neumann algebra defined in (2.7.1) and let tr be the trace on L1H aag (M) defined by (2.7.2). Then, for any normalized trace φ on L1,∞ (N, τ ), φ(X) = tr(X), X ∈ L1H aag (M).

2.10 Banach Spaces and Spectral Operators

29

2.10 Banach Spaces and Spectral Operators In this section we collect facts on operators on Banach spaces for which double operator integrals can be constructed and admit results similar to those on a Hilbert space. Let μ be a complex Borel measure on a measurable space ( , ) and X a Banach space. A function f : → X is called μ-measurable if there exists a sequence of Xvalued simple functions converging to f μ-almost everywhere. For Banach spaces X and Y, we say that a function f : → B(X, Y) is strongly measurable if ω → f (ω)x is a μ-measurable mapping → Y for each x ∈ X. A Banach space X is said to have a bounded approximation property if there exists M  1 such that for every compact subset K ⊆ X and > 0, there exists X ∈ X∗ ⊗X with XX→X  M and supx∈K Xx−xX < . If X and Y are Banach spaces such that X is separable and either X or Y has a bounded approximation property, then every T ∈ B(X, Y) is a limit in the strong operator topology of a norm bounded sequence of finite rank operators [173, Lemma 3.2]. Let (Z,  · Z ) be a Banach space which is continuously embedded in B(X, Y). Following [210], we say that Z has the strong convex compactness property if for every finite measure space ( , , μ) and every strongly measurable bounded f : → Z, the operator T ∈ B(X, Y) defined by  T x :=

f (ω)x dμ(ω),

x ∈ X,



belongs to Z with  T Z  inf

g(ω) dμ(ω),

where the infimum is taken over all measurable functions g : → [0, ∞] such that f (ω)Z  g(ω) for ω ∈ . Any separable Z has the strong convex compactness property. The subspaces of compact and weakly compact operators in B(X, Y) have the strong convex compactness property, but not all subspaces of B(X, Y) do. If N is a semifinite von Neumann algebra on a separable Hilbert space H with a faithful normal semifinite trace τ and F is a rearrangement invariant Banach function space with the Fatou property, then N ∩ F (N, τ ) has the strong convex compactness property (see [173] for more details). Let X and Y be Banach spaces and I a Banach space which is continuously embedded in B(X, Y). We say that (I,  · I ) is a Banach ideal in B(X, Y) if • R ∈ B(Y), X ∈ I, and T ∈ B(X) imply RXT ∈ I and RXT I  RY→Y XI T X→X ; • X∗ ⊗ Y ⊆ I and x ∗ ⊗ yI = x ∗ X∗yY for all x ∗ ∈ X∗ and y ∈ Y.

30

2 Notations and Preliminaries

For separable X and Y, any maximal Banach ideal (for the definition see, e.g., [152]) in B(X, Y) has the strong convex compactness property. This includes a large class of operator ideals, such as the ideal of absolutely p-summing operators, the ideal of integral operators, etc. Below we summarize some of the basics of scalar type operators, as taken from [72]. Let X be a Banach space and B a Borel σ -algebra. A spectral measure on X is a map E : B → B(X) such that the following hold: • • • •

E(∅) = 0 and E(C) = IX ; E(ς1 ∩ ς2 ) = E(ς1 )E(ς2 ) for all ς1 , ς2 ∈ B; E(ς1 ∪ ς2 ) = E(ς1 ) + E(ς2 ) − E(ς1 )E(ς2 ) for all ς1 , ς2 ∈ B; E is σ -additive in the strong operator topology.

Note that these conditions imply that E is projection-valued. Moreover, by [72, Corollary XV.2.4] there exists a constant K such that E(ς )B(X)  K,

ς ∈ B.

(2.10.1)

An operator A ∈ B(X) is a spectral operator if there exists a spectral measure E on X such that AE(ς ) = E(ς )A and σ (A, E(ς )X) ⊆ ς for all ς ∈ B, where σ (A, E(ς )X) denotes the spectrum of A in the space E(ς )X. For a spectral operator A, we let spec(A) denote the minimal constant K occurring in (2.10.1) and call spec(A) the spectral constant of A, which is well-defined since the spectral measure E associated with A is unique (see [72, Corollary XV.3.8]). By [72, Corollary XV.3.5], E(σ (A)) = IX . Let B(σ (A), E) denote the space of all bounded E-measurable complex-valued functions on σ (A). The integral of a function f ∈ B(σ (A), E) is defined as follows. For f = nj=1 αj χςj a finite simple function with αj ∈ C and ςj ⊆ σ (A) mutually disjoint Borel sets for j = 1, . . . , n, we let  f dE := σ (A)

n 

αj E(ςj ).

j =1

This definition is independent of the representation of f , and   

σ (A)

  f dE 

X→X

 4 spec(A)f ∞ .

Since the simple functions are dense in B(σ (A), E), for a general f ∈ B(σ (A), E) set   f dE := lim fn dE ∈ B(X), σ (A)

n→∞ σ (A)

2.10 Banach Spaces and Spectral Operators

31

where {fn }∞ n=1 ⊆ B(σ (A), E) is a sequence of simple functions satisfying fn − f ∞ → 0 as n → ∞. This definition is independent of the choice of approximating sequence and   

σ (A)

  f dE 

 4 spec(A)f ∞ .

X→X

(2.10.2)

It is straightforward to check that 





(αf + g) dE = α σ (A)

f dE +





σ (A)

fg dE =

g dE, σ (A)

 f dE

σ (A)

σ (A)

 g dE

σ (A)

for all α ∈ C and simple f, g ∈ B(σ (A), E), and approximation then extends these identities to general f, g ∈ B(σ (A), E). Moreover, σ (A) χσ (A) dE = E(σ (A)) = IX . Hence the map f → σ (A) f dE is a continuous morphism B(σ (A), E) → B(X) of unital Banach algebras. Since the spectrum of A is compact, the identity function is bounded on σ (A) and σ (A) λ dE(λ) ∈ B(X) is well defined. Definition 2.10.1 A spectral operator A ∈ B(X) with spectral measure E is a scalar type operator if  A=

λ dE(λ). σ (A)

The class of scalar type operators on X is denoted by Bs (X). For A ∈ Bs (X) with spectral measure E and f ∈ B(σ (A), E) we define  f (A) :=

f dE. σ (A)

As remarked above, f → f (A) is a continuous morphism B(σ (A), E) → B(X) of unital Banach algebras with norm bounded by 4 spec(A). Finally, we note that a normal operator A on a Hilbert space H is a scalar type operator with spec(A) = 1, and in this case (2.10.2) improves to   

σ (A)

  f dE   f ∞ ,

as is known from the Borel functional calculus for normal operators. Let now X possess an unconditional Schauder basis {ej }∞ j =1 ⊆ X (see, e.g., [121]). For j ∈ N, let P ∈ B(X) be the projection given by P j j (x) = xj ej for all  x= ∞ k=1 xk ek ∈ X. An operator A ∈ B(X) is called diagonalizable (with respect

32

2 Notations and Preliminaries

∞ ∞ of to {ej }∞ j =1 ) if there exist invertible U ∈ B(X) and a sequence {λj }j =1 ∈  complex numbers such that

U AU −1 x =

∞ 

λj Pj x,

x ∈ X,

j =1

where the series converges since {ek }∞ k=1 is unconditional. Observe that any diagonalizable operator is a scalar type operator.

2.11 Differentiability of Maps on Banach Spaces The first order Fréchet and Gâteaux differentiability as well as higher order Fréchet differentiability are standard concepts (see, e.g., [181, Chapter I, Sections B and F]). A map F between two Banach spaces (X,  · X ) and (Y,  · Y ) is called Fréchet differentiable at a point x ∈ X if there exists DF (x) ∈ B(X, Y) such that for every ε > 0 there is δ > 0, so that F (x + h) − F (x) − DF (x)(h)Y  εhX for every h ∈ X with hX < δ. Note that if the map F is a bounded linear operator, then obviously DF (x)(h) = F (h), that is, DF (x) = F for every x ∈ X. To define the higher order differentiability of a map between Banach spaces, we recall that there is an isometric isomorphism between the spaces B2 (X × X, Y) and B(X, B(X, Y)); more generally, between Bn (X×n , Y) and B(X, B(X, . . . , B(X, Y)) . . . ). The latter identification of spaces is a standard fact of functional analysis (see, e.g., [181, Lemma 1.41]). The map F is called twice Fréchet differentiable at a point x ∈ X if it is Fréchet differentiable in a neighborhood of x and the respective differential DF (x) ∈ B(X, Y) is Fréchet differentiable at x. In this case, the second Fréchet differential D 2 F (x) is an element of B(X, B(X, Y)) = B2 (X × X, Y). More generally, we have the following definition. Definition 2.11.1 Let n ∈ N. The map F between Banach spaces (X,  · X ) and (Y,  · Y ) is said to be n times Fréchet differentiable at x ∈ X if it is n − 1 times Fréchet differentiable in a neighborhood of x and there is D n F (x) ∈ Bn (X×n , Y) satisfying   n−1 (D F (x + h) − D n−1 F (x))(h1 , . . . , hn−1 ) − D n F (x)(h1 , . . . , hn−1 , h) = o(hX )h1 X · · · hn−1 X as hX → 0 for all h1 , . . . , hn−1 ∈ X.

Y

2.11 Differentiability of Maps on Banach Spaces

33

We further say that f is n times continuously Fréchet differentiable at x if it is n times Fréchet differentiable in a neighborhood of x and  n  (D F (x + h) − D n F (x))(h1 , . . . , hn ) = o(1)h1 X · · · hn X X as hX → 0 for all h1 , . . . , hn ∈ X. Another definition of the higher order Fréchet differentiability is given in [126, Subsection 8.62]. To distinguish it from the differentiability defined above, we call it Taylor-Fréchet differentiability. Definition 2.11.2 The map F between Banach spaces (X,  · X ) and (Y,  · Y ) is called n-times Taylor-Fréchet differentiable at x ∈ X if there exist Fx(k) ∈ Bk (X×k , Y), k = 1, . . . , n, such that for every ε > 0 there is δ > 0 so that for every h ∈ X with hX < δ, n    1 (k)   Fx (h, . . . , h)  εhnX . F (x + h) − F (x) − Y k! k=1

It is proved in the proposition on pages 311–312 of [126] that Fréchet differentiability in the sense of Definition 2.11.1 implies the Taylor-Fréchet differentiability in the sense of Definition 2.11.2 and the respective equality D n F (x) = Fx(n) . The map F between Banach spaces (X,  · X ) and (Y,  · Y ) is called Gâteaux differentiable at x if it is differentiable along every direction in X, that is, there exists a map DG F (x) from X to Y defined as F (x + th) − F (x) t →0 t

DG F (x)(h) :=  · Y - lim

(2.11.1)

for every h ∈ X. We note that DG F (x) is assumed to be neither bounded nor linear, but it will possess such properties in the problems that we consider. The Fréchet differentiability implies the Gâteaux differentiability and equality DF (x) = DG F (x)

(2.11.2)

of the Fréchet and Gâteaux derivatives, but the converse holds only under an additional assumption on F . The following property is standard (see, e.g., [181, Lemma 1.15]). Proposition 2.11.3 Let F be a map between Banach spaces (X,  · X ) and (Y,  · Y ). If F is Gâteaux differentiable in a neighborhood U of x ∈ X, DG F (ξ ) given by (2.11.1) is an element of B(X, Y) for every ξ ∈ U , and ξ → DG F (ξ ) is continuous at ξ = x, then F is Fréchet differentiable at x. We will understand higher order Gâteaux differentiability in the following sense.

34

2 Notations and Preliminaries

Definition 2.11.4 Let n ∈ N. The map F between Banach spaces (X,  · X ) and (Y,  · Y ) is said to be n times Gâteaux differentiable at x ∈ X if it is n − 1 times n F (x) between Gâteaux differentiable in a neighborhood of x and there is a map DG Banach spaces (X,  · X ) and (Y,  · Y ) satisfying n−1 n−1 DG F (x + th)(h) − DG F (x)(h) n F (x)(h) = DG t →0 t

 · Y - lim for every h ∈ X.

In applications to differentiation of operator functions considered in Sect. 5.3, the n map DG F (x) turns out to be bounded and homogeneous of order n. When we discuss differentiability of operator functions we deal with modified concepts of Gâteaux and Fréchet derivatives. Such derivatives can be defined at points that do not belong to a Banach space, but are self-adjoint operators densely defined in a separable Hilbert space H. Another modification consists in that the operator derivative can be calculated at an element of the Banach space B(H) while the direction of differentiation can be an element of a different Banach space, for instance, the Schatten ideal Sp or the noncommutative Lp space Lp (M, τ ). To indicate that the operator derivative is calculated in the Sp -, n Lp (M, τ )- or Lp (M, τ )-norm, we introduce the notations DG,p F (x) and Dpn F (x) for the respective nth order Gâteaux derivative and Fréchet differential of a function F at a point x along a direction in Sp , Lp , or Lp . If F is defined at self-adjoint operators, then directions of differentiation are automatically assumed to be selfadjoint.

Chapter 3

Double Operator Integrals

The concept of a double operator integral on B(H) was first introduced by Yu. L. Daletskii and S. G. Krein in [61]. They launched this theory in order to compute the derivative of the function t → f (A(t)), where {A(t)}t is a family of bounded self-adjoint operators depending on the parameter t. Although the initial construction allowed to handle a limited class of functions f and produced bounds that depended on the spectra of operators, it created new conceptual and technical opportunities. Further development of perturbation theory and its applications stimulated extension and refinement of double operator integral constructions and methods, with ground breaking contributions made in [34–36, 49, 141, 159]. There are also generalizations of double operator integrals to multilinear transformations, which are considered in the next section. In this chapter we discuss the main constructions and properties of double operator integrals that have found important applications.

3.1 Double Operator Integrals on Finite Matrices Let A, B ∈ Bsa (2d ). Let {ξj }dj=1 , {ηk }dk=1 be complete systems of orthonormal eigenvectors and {λj }dj=1 , {μk }dk=1 sequences of the eigenvalues of A and B, respectively. This notation is assumed throughout the section.

© Springer Nature Switzerland AG 2019 A. Skripka, A. Tomskova, Multilinear Operator Integrals, Lecture Notes in Mathematics 2250, https://doi.org/10.1007/978-3-030-32406-3_3

35

36

3 Double Operator Integrals

3.1.1 Definition Self-adjoint Case Let ϕ : R2 → C. We define TϕA,B : B(2d ) → B(2d ) by TϕA,B (X) :=

d d  

ϕ(λj , μk )Pξj XPηk ,

(3.1.1)

j =1 k=1

for X ∈ B(2d ). The operator TϕA,B is a discrete version of a double operator integral. The function ϕ is usually called the symbol of the operator TϕA,B . d

1 Let {λj }j 0=1 and {μk }dk=1 denote the sets of distinct eigenvalues of A and B, respectively. By properties of the spectral measure,   Pξk , i = 1, . . . , d0 , EB ({μj }) = Pηk , j = 1, . . . , d1 . EA ({λi }) =

1k d λk =λi

1k d μk =μj

Thus, (3.1.1) can be rewritten in terms of the spectral measures of A and B as TϕA,B (X) =

d0  d1 

ϕ(λj , μk )EA ({λj })XEB ({μk }).

(3.1.2)

j =1 k=1

Unitary Case In the case when A, B are unitary operators the definition (3.1.1) is similar. The only difference is that the function ϕ is defined on the torus T2 . Further in this section we distinguish the self-adjoint and unitary cases only if it is stated explicitly.

3.1.2 Relation to Finite-Dimensional Schur Multipliers  Let c ∈ 2d have the representation c = dl=1 cl ηl . Then the action of the operator TϕA,B (X) on the element c ∈ 2d can be written as follows: TϕA,B (X)c =

d  d  d 

  cl ϕ(λj , μk )Pξj X ηl , ηk ηk

j =1 k=1 l=1

=

d d   j =1 k=1

  ck ϕ(λj , μk ) Xηk , ξj ξj .

(3.1.3)

3.1 Double Operator Integrals on Finite Matrices

37

Every linear operator X ∈ B(2d ) can be identified with the d × d complex matrix  d X = xj k j,k=1 , where   xj k = Xηk , ξj . Note that, since the systems {ξj }dj=1 , {ηk }dk=1 are orthonormal, the (k0 , j0 )-entry for 1  j0 , k0  d of the matrix corresponding to the operator TϕA,B (X) given by (3.1.1) can be calculated from (3.1.3) as follows: 

d      TϕA,B (X)ηk0 , ξj0 = ϕ(λj , μk0 ) Xηk0 , ξj ξj , ξj0 j =1

  = ϕ(λj0 , μk0 ) Xηk0 , ξj0 = ϕ(λj0 , μk0 )xj0 k0 .

(3.1.4)

d  Thus, the matrix TϕA,B (X) is simply the matrix ϕ(λj , μk )xj k j,k=1 , that is, an entrywise product (usually called a Schur product or Hadamard product) of matrices d  d  ϕ(λj , μk ) j,k=1 and xj k j,k=1 . In other words, TϕA,B acts as a Schur multiplier on B(2d ). For other properties of finite dimensional Schur matrices we refer the reader to [32, 90]. Infinite dimensional Schur multipliers and their generalizations are discussed in [153, 203].

3.1.3 Properties of Finite Dimensional Double Operator Integrals Algebraic Properties The following properties of the operator integral TϕA,B are direct consequences of the Schur multiplication properties and the representation (3.1.4) (for brevity, below we use the notation Tϕ ): Tαϕ+βψ = αTϕ + βTψ , α, β ∈ C, Tϕψ = Tϕ Tψ , Tϕ = I, provided ϕ ≡ 1. For future use, we make the next simple observation. Assume that the function ϕ has the representation ϕ(λ, μ) = a1 (λ)a2 (μ)

38

3 Double Operator Integrals

for some complex valued functions a1 and a2 . The definition of the double operator integral (3.1.1) and the spectral resolution of operator functions a1 (A) and a2 (B) imply that TaA,B (X) = a1 (A) · X · a2 (B), X ∈ B(2d ). 1 ·a2

(3.1.5)

The formula (3.1.5) lies at the foundation of the definition of a continuous variant of an operator integral (see Sect. 3.3).

Norm Estimates An estimate for the Hilbert–Schmidt norm of TϕA,B (X) is easy to obtain, as we see below. Proposition 3.1.1 Let A, B ∈ B(2d ) be self-adjoint (or unitary) operators with corresponding d-tuples of the eigenvalues {λj }dj=1 and {μk }dk=1 . Then, for a function ϕ : R2 → C (or ϕ : T2 → C), TϕA,B (X)2  max |ϕ(λj , μk )| X2 . j,k

(3.1.6)

Proof Combining (3.1.4) and (2.4.2) gives d   A,B 2 T  = (X) |ϕ(λj , μk )|2 |xj k |2  max |ϕ(λj , μk )|2 X22 , ϕ 2 j,k

j,k=1

 

completing the proof.

The estimate (3.1.6) has a straightforward dimension-dependent analog (3.1.7) for the operator norm. More sophisticated, dimension-independent estimates for the operator norm of a double operator integral are discussed in Sect. 3.3. Corollary 3.1.2 Let A, B ∈ B(2d ) be self-adjoint (or unitary) operators with corresponding d-tuples of the eigenvalues {λj }dj=1 and {μk }dk=1 . Then, for a function ϕ : R2 → C (or ϕ : T2 → C), TϕA,B (X) 

√ d max |ϕ(λj , μk )| X. j,k

(3.1.7)

Proof By standard connections between different matrix norms (see, e.g., [32, Chapter IV, Section 2]), TϕA,B (X)  TϕA,B (X)2 and X2  which along with (3.1.6) proves the result.

√ d X,  

3.1 Double Operator Integrals on Finite Matrices

39

An estimate of type (3.1.6) is the best possible. We will strive to obtain analogous estimates for double (multiple) operator integrals for other norms on B(2d ) and its infinite dimensional analogs.

Perturbation Formula The discrete symbol f [1] and the corresponding double operator integral was first studied by K. Löwner in [125]. Proposition 3.1.3 Let A, B ∈ Bsa (2d ) and σ (A)∪σ (B) ⊂ [a, b]. Let f : [a, b] → C and let ψ : [a, b] × [a, b] → R be any function such that ψ(λ, μ) = f [1] (λ, μ) for all λ ∈ σ (A), μ ∈ σ (B), λ = μ. Then, f (A) − f (B) = TψA,B (A − B).

(3.1.8)

Proof By the spectral theorem, A=

d  j =1

λj Pξj , B =

d 

μk Pηk

k=1

and f (A) =

d  j =1

f (λj )Pξj , f (B) =

d 

f (μk )Pηk .

k=1

Since f (A)ξj = f (λj )ξj , f (B)ηk = f (μk )ηk , we have 

   (f (A) − f (B))ηk , ξj = ψ(λj , μk ) (A − B)ηk , ξj ,

(3.1.9)

Rewriting Löwner’s formula (3.1.9) in terms of the double operator integral and using (3.1.4), we obtain (3.1.8).   A completely analogous result holds in the case of unitary operators. Proposition 3.1.4 Let U, V ∈ B(2d ) be unitary operators. If f is differentiable on T, then f (U ) − f (V ) = TfU,V [1] (U − V ).

(3.1.10)

40

3 Double Operator Integrals

Lipschitz Estimates The Lipschitz estimate for the Hilbert–Schmidt norm is a well-known result and is given in the following proposition. Similar estimate in the infinite-dimensional case is established in [36] and discussed in Sect. 5.1.1 (see Theorem 5.1.3). Proposition 3.1.5 Let A, B ∈ Bsa (2d ) and σ (A)∪σ (B) ⊂ [a, b]. If f ∈ Lip[a, b], then f (A) − f (B)2  f Lip[a,b] A − B2 . Proof Combining (3.1.8), (3.1.6), and (2.2.4) yields the result.

(3.1.11)  

The estimate above is given in the case of self-adjoint operators A, B; however, similar argument shows that it also works for unitary operators with the constant f  ∞ . The estimate (3.1.11) is the most desirable Lipschitz estimate. Estimates in other Schatten norms and in the operator norm require much more care since a straightforward application of (3.1.7), (2.2.4), and (3.1.8) gives the dimensiondependent estimate f (A) − f (B) 

√ d f Lip[a,b] A − B.

(3.1.12)

The following result √ due to [8, Theorem 11.4] shows that the dimensiondependent component d in the inequality above can be improved. Theorem 3.1.6 Let A, B ∈ Bsa (2d ) and σ (A) ∪ σ (B) ⊂ [a, b]. If f ∈ Lip[a, b], then f (A) − f (B)  C(1 + log d)f Lip[a,b] A − B, for some absolute constant C. We will come back to this question in Sect. 5.1, where we study the Lipschitz estimate problem in the infinite-dimensional case. We will see, in particular, that in order to make the bounds for the operator and trace class norms of f (A) − f (B) in terms of the same norm of A − B independent of the dimension, we have to restrict the class of functions f .

Continuity In the next lemma we state continuity of a double operator integral with respect to auxiliary self-adjoint (or unitary) matrix parameters. The proof is postponed to Proposition 4.1.6 applicable in the higher order case.

3.2 Double Operator Integrals on S2

41

Proposition 3.1.7 Let Am , A, Bm , B ∈ B(2d ), m ∈ N be self-adjoint (respectively, unitary) operators such that Am → A and Bm → B as m → ∞. Let ψ ∈ Cb (R2 ) (respectively, ψ ∈ C(T2 ) ). Then, TψAm ,Bm (X) → TψA,B (X), m → ∞, for all X ∈ B(2d ).

3.2 Double Operator Integrals on S2 There are several approaches to double operator integrals on S2 . In this section we define the continuous version of the double operator integral (3.1.1) as an integral with respect to the spectral measure constructed as a product of two other spectral measures. Such an approach was firstly suggested by M. S. Birman and M. Z. Solomyak in [34] (see also [35, 36]), where the authors have built the foundation of the theory of double operator integrals. We will also discuss an alternative approach due to B. S. Pavlov [138].

3.2.1 Definition We start with the construction due to M. S. Birman and M. Z. Solomyak.

Birman–Solomyak’s Approach Let = 1 × 2 and ϕ ∈ L∞ ( , G), where G is defined in (2.5.1). Recall that L∞ ( , G) consists of all G-measurable functions ϕ on for which ϕ∞ := G-sup ϕ = inf{α ∈ R+ : |ϕ(·)|  α, G-a.e.} Definition 3.2.1 The Birman–Solomyak double operator integral TϕG : S2 → S2 is defined as the integral of the symbol ϕ with respect to the spectral measure G, that is,  TϕG (X) := ϕ(ω) dG(ω)(X), X ∈ S2 . (3.2.1)

42

3 Double Operator Integrals

The notation  TϕG (X) =:

1 × 2

ϕ(ω1 , ω2 )dE(ω1 )XdF (ω2 ), X ∈ S2 ,

is frequently used.

The property  ϕ(ω) dG(ω) = ϕL∞ ( ,G) of the functional calculus implies the following important result. Proposition 3.2.2 The operator integral TϕG is bounded on S2 if and only if ϕ ∈ L∞ ( , G). In this case, TϕG : S2 → S2  = ϕL∞ ( ,G) .

(3.2.2)

Let A, B ∈ Dsa and let E and F be spectral measures of A and B, respectively. Let 1 = 2 = R and let 1 = 2 be the σ -algebra of all Borel sets on R. In this case, we use the notation TϕA,B := TϕG ,

(3.2.3)

for ϕ ∈ L∞ (R2 , G).

Pavlov’s Approach If Y ∈ S2 and ϕ ∈ L∞ ( , G), where G is given by (2.5.1), then σ → Tr(G(σ )Y ) is a scalar measure on  = 1 ⊗ 2 and ϕ is integrable with respect to Tr(G(·)Y ) (for the construction of G see Proposition 2.5.1). Thus,  θG : Y →

ϕ(ω1 , ω2 ) d Tr(G(ω1 × ω2 )Y )

(3.2.4)



is a linear bounded functional on S2 . Since (S2 )∗ = S2 , it follows that there exists AG ∈ S2 such that θG (Y ) = Tr(AG Y ) for all Y ∈ S2 . We set  ϕ(ω1 , ω2 )dG(ω1 × ω2 ) := AG .

The transformation AG is the Pavlov’s version of the double operator integral. Applying the double operator integral (3.2.1) to Y ∈ S2 and taking the trace, we obtain θG (Y ), where θG is defined in (3.2.4). Therefore, Pavlov’s definition and Birman–Solomyak’s one coincide (see also [34, (2.12)]).

3.2 Double Operator Integrals on S2

43

We will discuss Pavlov’s approach [138] in full generality of arbitrary order multiple operator integrals in Sect. 4.2.

3.2.2 Relation to Schur Multipliers on S2 ∞ Fix two orthonormal bases {ξj }∞ j =1 and {ηk }k=1 in H. Then, every operator X ∈ S2 can be represented as an infinite matrix X = (xj k )∞ j,k=1 , where xj k =   ∞ X(ηk ), ξj , j, k ∈ N. For a matrix Y = (yj k )j,k=1 let X ◦ Y := (yj k xj k )∞ j,k=1 denote the Schur product of the matrices X and Y. The matrix Y is called a Schur multiplier if the mapping X → Y ◦ X is a bounded operator on S2 . We denote by M(S2 , S2 ) the space of all multipliers on S2 with the norm

Y 2,2 := sup{Y ◦ X2 : X2  1}. Suppose that each of A, B ∈ Bsa (H) has a discrete spectrum. Consider ∞ sequences λ = {λj }∞ j =1 and μ = {μk }k=1 consisting of the points of the ∞ spectrum of A and B, respectively, counted with multiplicities. Let {ξj }∞ j =1 , {ηk }k=1 be corresponding orthonormal bases of eigenvectors of the operators A and B, respectively, and let ϕ ∈ L∞ (R2 , G). Then, we have the formulas TϕA,B (X) =



ϕ(λj , μk )Pξj XPηk ,

j,k



 TϕA,B (X)ηk , ξj = ϕ(λj , μk )xj k .

Thus, TϕA,B (X) is represented as the Schur product of the matrices (ϕ(λj , μk ))j,k and X = (xj k )j k (compare with finite versions (3.1.1) and (3.1.4)). The double operator integral can be viewed as a continuous version of a Schur multiplier and, therefore, sometimes the operator integral TϕA,B is called a Schur multiplier. Observe also that since the operator TϕA,B is bounded on S2 if and only if ϕ ∈ L∞ (R2 , G) (see Proposition 3.2.2), it follows that the matrix Y = (yj k )j k is the Schur multiplier on S2 if and only if supj,k |yj k | < ∞. In this case, Y 2,2 = sup |yj k | j,k

(for an alternative proof of the latter result see, e.g., [17, Proposition 2.1]). When A and B have arbitrary spectra, the transformation TϕA,B can be viewed as a Schur multiplier acting by pointwise multiplication on direct integral decompositions of operators (see, e.g., [41, Subsection 3.2] or [203]).

44

3 Double Operator Integrals

3.2.3 Basic Properties of Double Operator Integrals on S2 Algebraic Properties The following properties of the operator integral TϕA,B = TϕG on S2 defined in (3.2.1) and (3.2.3) hold (for brevity, we use the notation Tϕ ): Tαϕ+βψ = αTϕ + βTψ , α, β ∈ C; Tϕψ = Tϕ Tψ ; Tϕ = I, provided ϕ ≡ 1; Tϕ = Tϕ∗ ; where ϕ denotes the complex conjugate of ϕ. Proposition 3.2.3 Assume that ϕ ∈ L∞ ( , G), where G is defined in (2.5.1), has a representation ϕ(ω1 , ω2 ) = a1 (ω1 )a2 (ω2 ) for some a1 , a2 ∈ L∞ (Ω, G). Then, the formula  TϕG (X) =

 a1 (ω1 )dE(ω1 ) · X ·

(3.2.5)

a2 (ω2 )dF (ω2 )

1

2

holds for all X ∈ S2 . Proof Indeed, observing that TaG1 (X) = TaE1 (X) =



 a1 (ω1 )dE(ω1 )(X) =

1

a1 (ω1 )dE(ω1 ) · X,

(3.2.6)

1

and, similarly, TaG2 (X)

=

TaF2 (X)

 =X·

(3.2.7)

a2 (ω2 )dF (ω2 ), 2

and using multiplicativity of the spectral integral, we obtain  TϕG (X) = TaG1 a2 (X) = (TaG1 TaG2 )(X) = TaG1 X · 

 a1 (ω1 )dE(ω1 ) · X ·

= 1

a2 (ω2 )dF (ω2 ) 2

a2 (ω2 )dF (ω2 ). 2

3.3 Double Operator Integrals on Schatten Classes and B(H)

45

Corollary 3.2.4 Let ϕ ∈ L∞ (R2 , G) have the representation ϕ(λ, μ) a1 (λ)a2 (μ) for some a1 , a2 ∈ L∞ (Ω, G). Then,

=

TϕG (X) = TϕA,B (X) = a1 (A) · X · a2 (B).

(3.2.8)

Proof The result follows from (3.2.5) upon applying the spectral resolutions of a1 (A) and a2 (B).   The resolution (3.2.8) extends its finite-dimensional version (3.1.5); it plays a crucial role in the further presentation. In particular, one of definitions of the double operator integral on B(H) and Schatten ideals is based on this formula (see Sect. 3.3). Further properties of double operator integrals on S2 are collected in Sect. 3.3.5.

3.3 Double Operator Integrals on Schatten Classes and B(H) We start with a brief discussion of the original construction due to Yu. L. Daletskii and S. G. Krein [61] and then turn to a detailed discussion of constructions frequently used in contemporary analysis.

3.3.1 Daletskii-Krein’s Approach The double operator integral in [61] was introduced as an iterated integral on B(H). We present the definition below. Let A ∈ Bsa (H), let [a, b] contain σ (A), and let {Eμ }a  μ  b be the spectral family of A. Let F (μ) be a bounded operator on H depending on the parameter μ. Consider the abstract Stieltjes integral 

b

F (μ) dEμ ,

(3.3.1)

a

which is defined as the limit of the operators Fδ =

n 

F (μk )E(δk )

k=1

in the operator norm as the maximal length of the intervals δk = [λk−1 , λk ] tends to zero, where δ = {δk } is a partition of [a, b] by the points a = λ0 < λ1 < · · · < λn−1 < λn = b and μk ∈ δk . If the integral in (3.3.1) exists, it is called an operator integral. The following estimate is obtained in [61, Theorem 1.4].

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3 Double Operator Integrals

Theorem 3.3.1 Let ϕ(λ, μ) be a continuous operator function with partial deriva (λ, μ) continuous in the rectangle a  λ, μ  b and tives ϕλ (λ, μ), ϕμ (λ, μ), ϕλμ let X be a bounded operator which depends on neither λ nor μ. Then the iterated integral  TϕA,A (X) :=

b



a

b

 ϕ(λ, μ) dEλ X dEμ

a

exists and    

a

b





b

ϕ(λ, μ) dEλ a

  X dEμ  

  X max |ϕ(λ, μ)| + (b − a) max |ϕλ (λ, μ)| λ,μ

λ,μ

(3.3.2)

  + (b − a) max |ϕμ (λ, μ)| + (b − a)2 max |ϕλμ (λ, μ)| . λ,μ

λ,μ

The operator TϕA,A is the first version of the double operator integral on B(H) and if the Hilbert space H is finite-dimensional, then the definition of the double operator integral above coincides with (3.1.1). The estimate (3.3.2) is the best one attained for the operator integral defined in [61], but it depends on the size of the spectrum of A (in particular, does not apply to unbounded A) and imposes unnecessary restrictions on the symbol ϕ. There exist other constructions of double operator integrals, which are presented further in the text, that provide better estimates under weaker restrictions on the operators and symbols.

3.3.2 Extension from the Double Operator Integral on S2 Another approach of double operator integral in B(H) is suggested by M. S. Birman and M. Z. Solomyak in [34] (see also [41, Section 4]), where the authors extend the double operator integral from S2 to B(H), with an additional assumption on the symbol ϕ. Recall that S1 ⊂ S2 ⊂ B(H) with continuous inclusions. Moreover, (S1 )∗ = B(H). For any L∞ -function ϕ, we have that the operator Tϕ maps S1 into S2 . If ϕ is such that the image Tϕ (S1 ) lies in S1 and the restriction Tϕ |S1 is a bounded operator on

3.3 Double Operator Integrals on Schatten Classes and B(H)

47

S1 , then we say that the double operator integral Tϕ is bounded on S1 . One can show that in this case the operator Tϕ is also bounded on S1 and has the same norm. Then the adjoint operator Tϕ |∗S1 acts on the space B(H). Hence, we naturally define Tϕ (X) := (Tϕ |∗S1 )(X),

X ∈ B(H).

If X ∈ S∞ , then also Tϕ (X) ∈ S∞ . Indeed, for any finite rank operator X, we have Tϕ (X) ∈ S1 ⊂ S∞ and the set of all finite rank operators is dense in S∞ . So the operator Tϕ defined above acts from S∞ to S∞ . Moreover, Tϕ : S∞ → S∞  = Tϕ : B(H) → B(H) = Tϕ : S1 → S1  and Tϕ : B(H) → B(H)  Tϕ : S2 → S2  = ϕ∞ . We discuss the existence of this double operator integral and its estimate later, in Sect. 3.3.6.

3.3.3 Approach via Separation of Variables This approach was introduced by M. S. Birman and M. Z. Solomyak in [34, Section 6] and developed further [35, 36, 141]. It is now a particular case of the theory of multilinear operator integrals constructed in [24, 146].

Demonstration in a Simple Case The simplest approach to the definition of a double operator integral on Sp is to use the property (3.2.8). Let A, B ∈ Bsa (H) be self-adjoint operators and let ϕ ∈ L∞ (R2 ) be given by the formula ϕ(λ, μ) =

n 

a1 (j, λ)a2 (j, μ), λ, μ ∈ R,

j =1

where a1 (j, ·), a2 (j, ·) ∈ L∞ (R), j = 1, . . . , n. Define the operator TϕA,B : Sp → Sp by TϕA,B (X) :=

n  j =1

a1 (j, A)Xa2 (j, B).

48

3 Double Operator Integrals

From the Hölder inequality (2.4.1) it follows that TϕA,B is a bounded linear operator on Sp for 1  p  ∞ and TϕA,B : Sp → Sp  

n 

a1 (j, ·)∞ a2 (j, ·)∞ .

j =1

Similarly, the operator TϕA,B is defined on B(H). General Case Below we present the largest class of functions to which the above approach applies. This class is the integral projective tensor product of L∞ -spaces introduced in [141]. Let A1 be the class of functions ϕ : R2 → C admitting the representation  ϕ(λ, μ) =

a1 (λ, ω) a2 (μ, ω)dν(ω),

(3.3.3)



for some finite measure space ( , ν) and bounded measurable functions ai (·, ·) : R × → C, i = 1, 2, where on R we consider the Borel σ -algebra, such that 

a1 (·, ω)∞ a2 (·, ω)∞ d |ν| (ω) < ∞.

The class A1 is an algebra with respect to the operations of pointwise addition and multiplication [24, Proposition 4.10]. The formula  ϕA1 = inf



a1 (·, ω)∞ a2 (·, ω)∞ d |ν| (ω),

where the infimum is taken over all possible representations (3.3.3), defines a norm on A1 (see [63, Lemma 4.6]). The class A1 can also be defined as the class of functions ϕ : R2 → C admitting the representation  ϕ(λ, μ) =

b1 (λ, ω) b2 (μ, ω) dν2 (ω),

(3.3.4)



for some (not necessarily finite) measure space ( , ν2 ) and bounded measurable functions bi (·, ·) : R × → C, i = 1, 2,

3.3 Double Operator Integrals on Schatten Classes and B(H)

49

such that 

b1 (·, ω)∞ b2 (·, ω)∞ d |ν2 | (ω) < ∞

(see, e.g., [63, 158]). These definitions coincide since the representation (3.3.3) of the function ϕ can be obtained from (3.3.4) with a1 (λ, ω) =

b1 (λ, ω) b2 (λ, ω) , a2 (λ, ω) = b1 (·, ω)∞ b2 (·, ω)∞

and the finite measure ν defined by ν = b1 (·, ω)∞ b2 (·, ω)∞ ν2 . Another important description of the class A1 is given in [141, Theorem 1, (3)⇔(4)] (for (3)⇐(4) see [36]). Namely, A1 can also be defined as the class of functions ϕ : R2 → C admitting the representation  ϕ(λ, μ) =

c1 (λ, ω) c2 (μ, ω) dν1 (ω), 1

for some (not necessarily finite) measure space ( 1 , ν1 ) and measurable functions ci (·, ·) : R × 1 → C, i = 1, 2, such that    

1

  |c1 (·, ω)|2 dν1 (ω) 

∞

   

1

  |c2 (·, ω)|2 dν1 (ω) 

∞

< ∞.

Definition The definition of a double operator integral based on a separation of variables is given below. Definition 3.3.2 Let 1  p  ∞. For every ϕ ∈ A1 , and a fixed couple A, B ∈ Dsa , the double operator integral TϕA,B : Sp → Sp (respectively, TϕA,B : B(H) → B(H)) is defined by  TϕA,B

(X) :=

a1 (A, ω) X a2 (B, ω) dν(ω),

50

3 Double Operator Integrals

for X ∈ Sp (respectively, X ∈ B(H)), where aj and ( , ν) are taken from the representation (3.3.3) and the integral is understood in the sense of the Bochner integral 



  a1 (A, ω) Xa2 (B, ω) dν(ω) (y) = a1 (A, ω) Xa2 (B, ω) (y) dν(ω), y ∈ H



(see, e.g., [141] and also [24, Definition 4.1]). It follows directly from the definition that if A, B ∈ Dsa and ϕ ∈ A1 , then TϕA,B is a bounded linear operator on Sp , 1  p  ∞, and on B(H). One of important results of this theory is that the value TϕA,B (X) does not depend on the particular representation on the right-hand side of (3.3.3) (see [71, Lemma 7.2 and Theorem 7.5]). Similarly to the proof of (3.2.8), one can show that if ϕ ∈ A1 , then TϕA,B on S2 given by Definition 3.3.2 coincides with TϕA,B given by Definition 3.2.1 of Sect. 3.2, and so these definitions also agree with the definition of double operator integral on S1 , S∞ , and B(H) given in Sect. 3.3.2.

3.3.4 Approach Without Separation of Variables An approach without separation of variables that enables strong estimates of Schatten norms of double operator integrals was introduced by D. Potapov and F. Sukochev in [159]. This double operator integral is a particular case of a general order multiple operator integral introduced in [163], which is discussed in Chap. 4. In the present subsection we follow [159] and consider only the special case of operators with spectra consisting of a finite number of rational  points. N  Let A and B be operators with spectra contained in − N m , . . . , m for some m, N ∈ N and let ϕ : R2 → C be a bounded Borel function. The double operator integral associated with A, B, ϕ is defined on B(H) by TϕA,B (X)

:=

 |l0 |,|l1 |  N

 ϕ

l0 l1 , m m



 EA

l0 l0 , +1 m m





XEB

l1 l1 , +1 m m

 ,

(3.3.5) for X ∈ B(H). In the case when A, B are finite dimensional matrices, the transformation defined in (3.3.5) coincides with the one in (3.1.2). It follows from the spectral theorem that the transformations given by Definition 3.3.2 and by the formula (3.3.5) coincide for all ϕ ∈ A1 and all A, B with  N for some m, N ∈ N. spectra contained in − N , . . . , m m

3.3 Double Operator Integrals on Schatten Classes and B(H)

51

3.3.5 Properties of Double Operator Integrals on Sp and B(H) Algebraic Properties It is established in [24, Proposition 4.10] that the double operator integral given by Definition 3.3.2 satisfies Tα1 ϕ1 +α2 ϕ2 = α1 Tϕ1 + α2 Tϕ2 , Tϕ1 ϕ2 = Tϕ1 Tϕ2 , for ϕ1 , ϕ2 ∈ A1 , α1 , α2 ∈ C.

Norm Estimates The following estimate for a double operator integral is an immediate consequence of Definition 3.3.2 and the ideal property of Schatten classes. Theorem 3.3.3 Let A, B ∈ Dsa , ϕ ∈ A1 , and 1  p  ∞. Then,    A,B  Tϕ (X)  ϕA1 Xp p

(3.3.6)

for every X ∈ Sp if 1  p < ∞ or X ∈ B(H) if p = ∞. The double operator integral (3.3.5) as well as its continuous version with symbol ϕ equal to the divided difference of a Lipschitz function discussed in Sect. 3.3.2 admits extension and improvement of the estimate (3.3.6) on the Schatten classes Sp , p > 1. The following result is established in [159, Theorem 7]. Theorem 3.3.4 Let A, B ∈ Dsa , f ∈ Lip(R), and 1 < p < ∞. Then, there exists cp > 0 such that TfA,B [1] (X)p  cp f Lip(R) Xp for every X ∈ Sp , where f [1] is defined in (2.2.1). The original version of the proof of Theorem 3.3.4 in [159, Theorem 7] is based on decomposition of the symbol f [1] that realizes TfA,B [1] as a suitable combination of triangular truncations and multipliers on UMD spaces handled by MarcinkiewiczMihlin multiplier theory. Results on UMD spaces can be found in, for instance, [91]; in particular, the mentioned multiplier theory is applied in [91, Proposition 5.48] and discussed in references cited therein. In addition, the result of Theorem 3.3.4 is derived in the proof of the basis of induction for Theorem 4.3.10 with involvement of a vector-valued harmonic analysis and interpolation. It can also be derived from the boundedness of the double operator integral S1 → S1,∞ (discussed in Sect. 3.3.7)

52

3 Double Operator Integrals

and the double operator integral S2 → S2 for 1 < p < 2 by interpolation and then extended to 2 < p < ∞ by duality. The sharp estimate for the constant cp from Theorem 3.3.4 in terms of p is established in [48] based on harmonic analysis of UMD spaces Sp , 1 < p < ∞. Theorem 3.3.5 The constant cp from Theorem 3.3.4 satisfies cp ∼

p2 , 1 < p < ∞. p−1

Perturbation Formula The following nice property of the double operator integral TfA,B [1] is obtained in [39, Theorem 2.1]. It also admits an extension to the case of unbounded operators A and B [39, Theorem 2.2]. Theorem 3.3.6 Let A, B ∈ Bsa (H) with σ (A) ∪ σ (B) ⊆ [a, b] and X ∈ S2 . If f ∈ Lip[a, b], then f (A)X − Xf (B) = TfA,B [1] (AX − XB).

(3.3.7)

Proof Let p1 (λ, μ) = λ and p2 (λ, μ) = μ, for λ, μ ∈ R. Since f is Lipschitz, it follows that f is continuous on [a, b] and, therefore, f ◦ p1 , f ◦ p2 ∈ L∞ ([a, b] × [a, b], G), where G is defined in (2.5.1). Moreover, since (f ◦ p1 )(λ, μ) = f (p1 (λ, μ)) = f (λ) and (f ◦ p2 )(λ, μ) = f (μ), by (3.2.6) and (3.2.7), we have A,B TfA,B ◦p1 (X) = f (A)X, Tf ◦p2 (X) = Xf (B),

(X) = AX, TpA,B (X) = XB, TpA,B 1 2 for X ∈ S2 . Thus, using basic properties of double operator integrals, we obtain A,B A,B f (A)X − Xf (B) = TfA,B ◦p1 (X) − Tf ◦p2 (X) = Tf ◦p1 −f ◦p2 (X) A,B = T(p −p 1

=

2 )f

[1]

(X) = TpA,B (X) − TpA,B (X) f [1] f [1]

A,B TfA,B [1] (Tp1 (X))

1

2

A,B − TfA,B [1] (Tp2 (X))

= TfA,B [1] (AX − XB).

Although Theorem 3.3.6 is not applicable to the identity operator X = I , an analog of (3.1.8) remains valid. This result is the Consequence of [36, Theorem 4.5].

3.3 Double Operator Integrals on Schatten Classes and B(H)

53

Theorem 3.3.7 Let A, B ∈ Dsa be such that A − B ∈ S2 and let f ∈ Lip(R). Then, f (A) − f (B) ∈ S2 and f (A) − f (B) = TfA,B [1] (A − B).

(3.3.8)

We omit the proof of Theorem 3.3.7, but demonstrate its major ideas later, in the proof of Theorem 4.4.8. The above representation for an increment of an operator function extends to the case of non-Hilbert–Schmidt perturbations under an additional assumption on the scalar function. The next result is due to [36, Consequence of Theorem 4.5]. Theorem 3.3.8 Let A, B ∈ Dsa be such that A − B ∈ Sp , 1  p  ∞ (B(H), respectively). If f ∈ Lip(R) such that f [1] ∈ A1 , then f (A) − f (B) ∈ Sp (B(H), respectively) and f (A) − f (B) = TfA,B [1] (A − B).

(3.3.9)

Continuity Let C1 denote the subset of A1 of functions admitting the representation (3.3.3), ∞ where ∪∞ k=1 k = for a growing sequence { k }k=1 of measurable subsets of such that the families {aj (·, ω)}ω∈ k , j = 1, 2, are uniformly bounded and uniformly continuous. A norm on C1 is defined by  ϕC1 = inf



a1 (·, ω)∞ a2 (·, ω)∞ d |ν| (ω),

where the infimum is taken over all possible representations (3.3.3) with {aj (·, ω)}ω∈ k uniformly bounded and uniformly continuous for j = 1, 2, k ∈ N. Recall that {An }∞ n=1 ⊂ Dsa converges to A ∈ Dsa in the strong resolvent sense −1 in the strong operator topology for if {(λI − An )−1 }∞ n=1 converges to (λI − A) all λ such that Im λ = 0. We have the following continuity properties of double operator integrals. Proposition 3.3.9 Let 1  p  ∞. Let {Ai,n }∞ n=1 ⊂ Dsa converge to Ai ∈ Dsa , p i = 1, 2, in the strong resolvent sense and let {Xn }∞ n=1 ⊂ S (B(H), respectively) p converge to X ∈ S (B(H), respectively). The following assertions hold. (i) Let ϕ ∈ C1 . If X ∈ Sp , 1  p < ∞, then A1,n ,A2,n

lim Tϕ

n→∞

(X) − TϕA1 ,A2 (X)p = 0;

(3.3.10)

54

3 Double Operator Integrals

if X ∈ B(H), then A1,n ,A2,n

sot- lim (Tϕ n→∞

(X) − TϕA1 ,A2 (X)) = 0.

(3.3.11)

Moreover, if {Ai,n }∞ n=1 converge to Ai , i = 1, 2, in the operator norm, then the strong operator topology convergence in (3.3.11) can be replaced with the operator norm convergence. (ii) For every ϕ ∈ A1 , lim TϕA1 ,A2 (Xn ) − TϕA1 ,A2 (X)p = 0.

n→∞

(iii) Let {ϕn }∞ n=1 ⊂ A1 converge to ϕ in the norm  · A1 . Then, lim TϕAn1 ,A2 (X) − TϕA1 ,A2 (X)p = 0.

n→∞

Proof (i) The functions aj (·, ω) from the decomposition (3.3.3) are continuous and bounded (uniformly in ω on every k ), so the sequence {ai (An,i , ω)}n converges to ai (Ai , ω) in the strong operator topology for every ω ∈ , i = 1, 2 [171, Theorem VIII.20 (b)]. By [85, Theorem 6.3], Sp - lim a1 (An,1 , ω)Xa2 (An,2 , ω) = a1 (A0 , ω)Xa2 (A2 , ω), n→∞

for every ω ∈ if 1  p < ∞. We also have sup a1 (An,1 , ·)Xa2 (An,2 , ·)p ∈ L1 ( , μ). n

Therefore, by the Lebesgue dominated convergence theorem for Bochner integrals, we have the convergence of double operator integrals in the Schatten norm  · p , 1  p < ∞. A completely analogous argument works for convergence in the strong operator topology and operator norm. The continuity in (ii) and (iii) follows immediately from the estimate (3.3.6).   The result of Proposition 3.3.9(i) holds for a broader set of symbols ϕ when p = 2. The following fact is established in [55, Proposition 3.1]. Proposition 3.3.10 Let {Ai,m }∞ m=1 ⊂ Dsa converge to Ai ∈ Dsa , i = 1, 2, in the strong resolvent sense and let X ∈ S2 . Then, (3.3.10) with p = 2 holds for every ϕ ∈ Cb (R2 ).

3.3.6 Symbols of Bounded Double Operator Integrals There have been many attempts to find an appropriate class of symbols for which the operator Tϕ defined in Sect. 3.3.2 is bounded on B(H). As noted in Sect. 3.3.5,

3.3 Double Operator Integrals on Schatten Classes and B(H)

55

Tϕ is bounded if ϕ ∈ A1 . It is straightforward to see that all functions in A1 are bounded; therefore, the class A1 does not contain polynomials. Further results on boundedness of Tϕ are stated below. Let 1  p  ∞ and define Mp = Mp (EA , EB ) := {ϕ ∈ L∞ (R2 ) : TϕA,B ∈ B(Sp )}, MB(H) = MB(H) (EA , EB ) := {ϕ ∈ L∞ (R2 ) : TϕA,B ∈ B(B(H))}. It follows from Proposition 3.2.2 that M2 (EA , EB ) = L∞ (R2 , G), where G is given by (2.5.1) with E = EA , F = EB . We also have the following characterization of MB(H) , M1 , and M∞ , where the first two equalities can be found in [41, Section 4.1] and the third equality is due to [141, Theorem 1]. Theorem 3.3.11 MB(H) = M1 = M∞ = A1 . The description of bounded double operator integrals on B(H) given by Theorem 3.3.11 is analogous to the celebrated Grothendieck’s characterization of bounded Schur multipliers on B(2 ) that can be found in [153, Theorem 5.1]. The class Mp , 1 < p = 2 < ∞, has not been described yet. However, the following inclusions are known to be strict (see, e.g., [41, (5.7)]): M∞  Mp  M2 .

(3.3.12)

For instance, it is proved in [159] that for the absolute value function f (t) = |t|, its divided difference f [1] belongs to Mp whenever 1 < p < ∞. However, it follows from [80] and [99] that f [1] ∈ / A1 , so the left inclusion is strict. Below we demonstrate that the right inclusion in (3.3.12) is also strict. Let 2 < α < p, Md := (e−2πij k/d )dj,k=1 ,

Xd := d −1/2−1/p−1/α · (e2πij k/d )dj,k=1 .

Since d −1/2 (e2πij k/d )dj,k=1 is a unitary matrix, Xd p = d −1/p−1/α (Tr(I ))1/p = d −1/α . Note that the rescaled Schur product d −1/2+1/p+1/α  (Md ◦ Xd ) is the orthogonal projection onto the unit vector d −1/2 1, 1, . . . , 1 ∈ 2d . Thus, Md ◦ Xd p = d p/2−1−p/α > d −1 . p

56

3 Double Operator Integrals

Consider X := ⊕∞ d=1 Xd ,

M := ⊕∞ d=1 Md ,

the elements of the ∞ -direct sum of the matrix algebras B(2d ), which embeds in B(H), ∞ where H is a separable infinite-dimensional Hilbert space. Since Tr(X) = d=1 Tr(Xd ), we have p

Xp =

∞  d=1

p

Xd p =

∞ 

d −p/α < ∞,

M ◦ Xp = ∞.

d=1

Let A, B be self-adjoint operators such that σ (A) consists of the eigenvalues ∞ ∞ 2 {λj }∞ j =1 and σ (B) consists of the eigenvalues {μk }k=1 and let ϕ ∈ L (R ) be such that M = (ϕ(λj , μk ))∞ j,k=1 . Then, M ◦ X = TϕA,B (X) is the double operator integral defined in Sect. 3.2.2. We conclude from the argument above that TϕA,B ∈ / Mp for p > 2. By the duality argument (see, e.g., [41, (5.6)] or Theorem 4.1.9(iii)), Mp = Mp , where 1 < p, p < ∞, 1/p + 1/p = 1. / Mp for 1 < p < 2 as well. Since, ϕ ∈ L∞ (R2 ), it follows that Hence, TϕA,B ∈ TϕA,B ∈ M2 . One can also generalize the ideas demonstrated above to the case of double operator integrals TϕA,B , where A, B have arbitrary spectra. Problem 3.3.12 Describe the class Mp for 1 < p = 2 < ∞. Now we address the question for which functions f : R → C the divided difference f [1] belongs to the class A1 . Recall that W1 (R) is the Wiener class defined in Sect. 2.1. Proposition 3.3.13 If f ∈ W1 (R), then f [1] ∈ A1 . Proof Recalling that F −1 F f = f, we have f [1] (λ, μ) =

  iλt  1 f (λ) − f (μ) 1 = √ e − eiμt F f (t) dt λ−μ 2π λ − μ R   t

i = √ ei(λs+μ(t −s)) ds F f (t) dt. 2π R 0

3.3 Double Operator Integrals on Schatten Classes and B(H)

57

Making the substitution u = s, v = t − s in the latter integral, we obtain i f [1] (λ, μ) = √ 2π Taking = R2 , dν(u, v) =

√i 2π

 R2

eiλu eiμv F f (u + v) du dv.

F f (u + v) du dv, a1 ((u, v), λ) = eiλu , and

a2 ((u, v), μ) = eiμv , u, v ∈ R, we obtain the representation (3.3.3) for the function f [1] . We also have f [1] ∈ A1 because  |t F f (t)| dt = F f  L1 < ∞. R

A more delicate harmonic analysis leads to the following result obtained in [141, Theorem 2]: 1 (R) ⇒ f [1] ∈ A1 . f ∈ B∞1

The above result is sharpened by the theorem below, which is proved in [161, Theorem 5(ii)]. Recall that C1 is the class of symbols defined in Sect. 3.3.5, “Continuity”. 1 (R), then f [1] ∈ C and Theorem 3.3.14 If f ∈ B∞1 1

f [1] C1  const f B 1

∞1 (R)

.

Let 1  p  ∞ and define Fp := {f : R → C : f [1] ∈ Mp }. 1 (R) In the following theorem B11 loc denotes the space of functions that belong to 1 the Besov class B11 (R) locally.

Theorem 3.3.15 1 (R) 1 (i) B11 loc  F1 = F∞  B∞1 (R); (ii) Fp = Lip(R), 1 < p < ∞.

Proof The assertion (i) is proved in [141]. Proposition 3.2.2 implies (ii) for p = 2. The assertion (ii) for an arbitrary 1 < p < ∞ is proved in [159] (see Theorem 5.1.8).   The result of Theorem 3.3.15(i) was improved in [19], where the authors introduced a new class of functions D such that 1 ⊂ D ⊂ F1 . B∞1

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3 Double Operator Integrals

The fact that the left inclusion above is strict was proved later and can be found in [9, Remark at p. 90]. The original approach of [19] was for the Besov classes on the unit circle. Its analogy for the real line R can be found in [9, Subsection 3.13]. The class F1 coincides with the class of operator Lipschitz functions (see [9]). However, the structural properties of the class F1 in pure scalar terms are not fully understood. Problem 3.3.16 Describe the class F1 in terms of familiar function spaces.

3.3.7 Transference Principle The estimate for the double operator integral Tf [1] on Sp , 1 < p < ∞, given by Theorem 3.3.4 is derived in [159] by transferring boundedness of certain operators on L2 (R, X) to operators on the UMD space X = Sp , 1 < p < ∞. The related transference principle and history of transference ideas in general are discussed in [91, Chapter 5]. As noted in Sect. 3.3.6, the double operator integral T|tA,B is not |[1] bounded on S1 for some choice of A, B with A − B ∈ S1 . However, as we see below, T|t |[1] is a bounded map from S1 to S1,∞ and an estimate similar to the one obtained in Theorem 3.3.4 holds. This estimate for Tf [1] in B(S1 , S1,∞ ) for every f in the largest set of admissible functions Lip(R) is established in [49, Theorem 1.2]. It follows from deep results in noncommutative analysis and is based on transferring boundedness of singular integral operators on the tensor product space L2 (T2 ) ⊗ H [137] to operators on H. The following result is due to [49, Theorem 1.2], and it proves the conjecture of [133]. In fact, the result is established in a more general setting for perturbations in the noncommutative L1 -space of a semifinite von Neumann algebra.

Theorem 3.3.17 Let A ∈ Dsa and f ∈ Lip(R). Then, there exists an absolute constant c > 0 such that TfA,A [1] (X)1,∞  c f Lip(R) X1

(3.3.13)

for every X ∈ S1 . Proof Since the proof of this result is technical, we demonstrate the main idea of the transference method under additional assumptions on A, X, f . Assume that A ∈ B(H) and σ (A) is a finite subset of Z, which we denote by J. Let {Pj }j ∈J be a sequence of mutually orthogonal projections such that A=

 j ∈J

j Pj , I =

 j ∈J

Pj .

3.3 Double Operator Integrals on Schatten Classes and B(H)

59

Assume that X ∈ S1 satisfies Pj XPj = 0 for every j ∈ J. Finally, assume that f : Z → Z. Define a unitary operator U=



Me(j,f (j)) ⊗ Pj ∈ B(L2 (T2 ) ⊗ H),

j ∈J

where Me(m,n) , m, n ∈ Z, is a multiplication by the function e(m,n) (t1 , t2 ) := exp(−imt1 − int2 ) defined on the torus T2 . Then,



 U (I ⊗ X)U ∗ = Me(j3 ,f (j3 )) ⊗ Pj3 I ⊗ Pj1 XPj2 j3 ∈J

×

 j4 ∈J

=



j1 ,j2 ∈J

j1 ,j2 ∈J

Me(−j4 ,−f (j4 )) ⊗ Pj4

Me(j1 −j2 ,f (j1 )−f (j2 )) ⊗ Pj1 XPj2 .

(3.3.14)

Similarly,

 f (j1 ) − f (j2 )   ∗ U I ⊗ TfA,A Pj1 XPj2 U ∗ [1] (X) U = U I ⊗ j1 − j2 j1 ,j2 ∈J

 f (j1 ) − f (j2 ) = Me(j1 −j2 ,f (j1 )−f (j2 )) ⊗ Pj1 XPj2 . j1 − j2 j1 ,j2 ∈J

(3.3.15) Let g ∈ C 1 (C \ {0}) be a homogeneous function such that g(t1 + it2 ) =

t2 , |t2 |  |t1 | = 0. t1

Such function g exists and is defined in the proof of [49, Theorem 4.4]. Then, g(∇T2 ) acts as the Fourier multiplier g(∇T2 )e(m−n,f (m)−f (n)) =

f (m) − f (n) e(m−n,f (m)−f (n)) , m = n ∈ Z. m−n (3.3.16)

Combining (3.3.14)–(3.3.16) implies   ∗ ∗ U I ⊗ TfA,A [1] (X) U = (g(∇T2 ) ⊗ I )(U (I ⊗ X)U ).

60

3 Double Operator Integrals

Hence, 1 1,∞ TfA,A  X1 . [1] (X)1,∞  g(∇T2 ) ⊗ I : S → S

By a particular result of [137] stated in [49, Theorem 2.1] and by [49, Lemma 4.3], g(∇T2 ) ⊗ I ∈ B(S1 , S1,∞ ), implying (3.3.13) under the additional assumptions on A, X, f . For the complete proof of (3.3.13) in the general case we refer the reader to [49].  

3.4 Nonself-adjoint Case By analogy with double operator integrals for self-adjoint operators defined in Sects. 3.3.3 and 3.3.4, one can define double operator integrals for unitary A and B. The following properties of the transformations discussed in Sect. 3.3.5 extend to the unitary case: the algebraic properties, estimate (3.3.6), perturbation formulas (3.3.7) and (3.3.8), continuity. A unitary analog of the estimate obtained in Theorem 3.3.4 is stated in Theorem 4.3.19 in the general higher order case. Double operator integrals were introduced for contractions A, B in [142, 147] on the space S2 similarly to (3.2.1) and on B(H) similarly to Definition 3.3.2. In the definition of a double operator integral for contractions spectral measures of unitary operators are replaced with semi-spectral measures. Analogous extensions of double operator integrals from self-adjoint to maximal dissipative operators were defined in [6]. When a problem for contractions (respectively, maximal dissipative operators) can be reduced to a problem for unitaries (respectively, self-adjoints) by means of dilations (see, e.g., [132]), one can apply double operator integrals for unitaries without involving the double operator integration theory for contractions (see, e.g., [164]).

3.5 Double Operator Integrals on Noncommutative Lp -Spaces Double operator integrals were introduced for perturbations in a semi-finite von Neumann algebra M equipped with a normal semi-finite trace τ . The three constructions analogous to those in the setting of B(H) are due to [24, 64, 159].

3.5 Double Operator Integrals on Noncommutative Lp -Spaces

61

3.5.1 Extension from the Double Operator Integral on L2 (M, τ ) The double operator integral TϕG on the Hilbert space L2 (M, τ ) can be defined by adjusting Definition 3.2.1. If TϕG extends from a dense subset L2 (M, τ )∩Lp (M, τ ) of Lp (M, τ ), 1  p < ∞, to a bounded transformation on Lp (M, τ ), then this extension is called a double operator integral on Lp (M, τ ). This approach to double operator integration was implemented in [64].

3.5.2 Approach via Separation of Variables The double operator integral given by Definition 3.3.2 was extended in [24] to perturbations in M and, in particular, in a symmetrically normed ideal I of M with property (F ). A symmetrically normed ideal I of a semifinite von Neumann algebra M is said to have property (F ) if for every net {Aα } in I satisfying supα Aα I  1 and converging to some A ∈ M in the strong operator topology such that {A∗α } converges to A∗ in the strong operator topology, it follows that A ∈ I and AI  1. Examples of such ideals include Lp (M, τ ) = Lp (M, τ ) ∩ M, 1 ≤ p < ∞, and Lp,∞ (M, τ ) = Lp,∞ (M, τ ) ∩ M, 1 < p < ∞. It is proved in [24, Lemma 4.6] that the transformation given by Definition 3.3.2 is a bounded map on I.

3.5.3 Approach Without Separation of Variables The definition (3.3.5) extends to perturbations X in the noncommutative Lp space Lp (M, τ ), 1  p < ∞. A more general definition that does not impose restrictions on the spectra of A and B is given in the multilinear setting in Chap. 4. It is proved in [163, Lemma 3.5] that the multiple operator integrals given by Definition 3.3.2 and formula (3.3.5) coincide for ϕ ∈ C1 on M. They also coincide with the extension of TϕG from L2 (M, τ ) ∩ Lp (M, τ ) to Lp (M, τ ), 1  p < ∞, discussed in Sect. 3.5.1.

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3 Double Operator Integrals

3.5.4 Properties of Double Operator Integrals on Lp (M, τ ) Algebraic Properties Each of the three constructions of a double operator integral considered in this section satisfies the linearity Tα1 ϕ1 +α2 ϕ2 = α1 Tϕ1 + α2 Tϕ2 . Further algebraic properties are discussed in Sect. 4.4.3 for general order multiple operator integrals on Lp (M, τ ).

Norm Estimates The following bound is a consequence of [24, Lemma 4.6]. Theorem 3.5.1 Let A, BηMsa , ϕ ∈ A1 , and let I be a symmetrically normed ideal of M with property (F ). Then, the double operator integral given by Definition 3.3.2 satisfies    A1 ,A2  (X)  ϕA1 XI Tϕ I

for all X ∈ I. The following result is established in [159, Theorem 7]. Theorem 3.5.2 Let A, BηMsa , f ∈ Lip(R), and 1 < p < ∞. Then, there exists cp > 0 such that TfA,B [1] (X)p  cp f Lip(R) Xp for every X ∈ Lp (M, τ ). The following bound is proved in [49, Theorem 1.2]. Theorem 3.5.3 Let AηMsa and f ∈ Lip(R). Then, there exists an absolute constant c > 0 such that TfA,A [1] (X)1,∞  c f Lip(R) X1 for every X ∈ L1 (M, τ ) ∩ L2 (M, τ ).

3.6 Double Operator Integrals on Banach Spaces

63

Perturbation Formula The following analog of Theorem 3.3.8 is obtained in [24, Theorem 5.3]. Theorem 3.5.4 Let A, BηMsa be such that A − B ∈ M and let f ∈ W1 (R). Then, f (A) − f (B) = TfA,B [1] (A − B).

(3.5.1)

By the method discussed in Theorem 4.4.8 below, one can extend (3.5.1) to A, B ∈ (Lq + Lr )(M, τ ), 1 < r < q < ∞, and f, f  ∈ Cb (R).

3.6 Double Operator Integrals on Banach Spaces One more interesting direction of development of double operator integration theory is to consider a double operator integral on the space B(X, Y), where X and Y are Banach spaces. Such an attempt was firstly made in [64], where the theory of double operator integration was extended in various directions, including the Banach space setting. However, the results in the general setting were much weaker than in the Hilbert space setting. In this section we present results of [173] for scalar type operators on Banach spaces that match analogous results on Hilbert spaces. Fix Banach spaces X and Y, scalar type operators A ∈ Bs (X) and B ∈ Bs (Y) with spectral measures E and F , respectively, and let ϕ ∈ A1 . Let a representation as in (3.3.3) for ϕ be given, with corresponding ( , ν) and a1 , a2 . For ω ∈ , let a1 (A, ω) := a1 (·, ω)(A) ∈ B(X) and a2 (B, ω) := a2 (·, ω)(B) ∈ B(Y) be defined by the functional calculus for A and B, respectively. The next property is due to [173, Lemma 4.1]. Lemma 3.6.1 Let X ∈ B(X, Y) have the separable range. Then, for each x ∈ X, ω → a2 (B, ω)Xa1 (A, ω)x is a weakly measurable map → Y. Now suppose that Y is separable, I is a Banach ideal in B(X, Y) and let X ∈ B(X, Y). By (2.10.2), a2 (B, ω)Xa1 (A, ω)I 16 spec(A) spec(B)XI a1 (·, ω)∞a2 (·, ω)∞ (3.6.1) for ω ∈ . Since I is continuously embedded in B(X, Y), by the Pettis measurability theorem, Lemma 3.6.1, and (3.6.1) we can define the double operator integral  TϕA,B (X)x

:=

a2 (B, ω)Xa1 (A, ω)x dν(ω) ∈ Y,

The next property is due to [173, Proposition 4.2].

x ∈ X.

(3.6.2)

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3 Double Operator Integrals

Proposition 3.6.2 Let X and Y be separable Banach spaces such that X or Y has the bounded approximation property, A ∈ Bs (X), B ∈ Bs (Y), and ϕ ∈ A1 . Let I be a Banach ideal in B(X, Y) with the strong convex compactness property. Then (3.6.2) defines an operator TϕA,B ∈ B(I) which is independent of the choice of representation of ϕ in (3.3.3), with TϕA,B I→I  16 spec(A) spec(B)ϕA1 .

(3.6.3)

If A and B are self-adjoint operators on separable Hilbert spaces X and Y, then (3.6.3) improves to TϕA,B I→I  ϕA1 . If X = Y = H is an infinite-dimensional separable Hilbert space and I = S2 (H), then the definition of a double operator integral on S2 given in Sect. 3.2 coincides with the definition above for all ϕ ∈ A1 and self-adjoint operators A, B ∈ B(H). Remark 3.6.3 Proposition 3.6.2 cannot in general be extended to a class of functions larger than A1 . Indeed, as we saw in Theorem 3.3.11, TϕA,B is a bounded operator on I = B(H) if and only if ϕ ∈ A1 . However, for specific Banach ideals, for instance, ideals with the UMD property, results were obtained for larger classes of functions [64, 159] (see Theorem 3.3.15(ii)). Let us present the following fundamental fact, which is proved in [173, Lemma 4.5], extending the result of [158, Lemma 3] for self-adjoint operators on a separable Hilbert space with difference in a Schatten class. Let p1 , p2 : R2 → R be the coordinate projections given by p1 (λ, μ) := λ, p2 (λ, μ) := μ for (λ, μ) ∈ R2 . Note that f ◦ p1 , f ◦ p2 ∈ A1 for all bounded Borel functions f . Lemma 3.6.4 Under the assumptions of Proposition 3.6.2, the following properties hold: (i) The map ϕ → TϕA,B is a morphism A1 → B(I) of unital Banach algebras. (ii) Let f be a bounded Borel function and X ∈ B(X, Y). Then, A,B TfA,B ◦p1 (X) = Xf (A) and Tf ◦p2 (X) = f (B)X. A,B In particular, TpA,B 1 (X) = XA and Tp2 (X) = BX.

This approach to double operator integration theory in the setting of Banach spaces allows to solve a series of interesting problems concerning Lipschitz type estimates in different spaces of operators (see Sect. 5.1.5).

Chapter 4

Multiple Operator Integrals

Theory of multiple operator integrals arose as an extension of the double operator integration theory to the settings that could not be encompassed by the latter constructions. In particular, multilinear transformations naturally arise in finding summable approximations to operator functions in the case of nontrace class perturbations, as we will see in the next chapter. The first attempts to construct suitable multilinear extensions of double operator integrals were made in [138, 196, 199]; the more recent approaches important for applications are due to [24, 56, 146, 163]. In this chapter we discuss the main constructions and properties of multiple operator integrals suitable for applications.

4.1 Multiple Operator Integrals on Finite Matrices 4.1.1 Definition (j )

Let A0 , . . . , An ∈ Bsa (2d ), let g(j ) = {gi }di=1 be an orthonormal basis of (j ) eigenvectors of Aj and let {λi }di=1 be the corresponding d-tuple of eigenvalues, j = 0, . . . , n. Let ϕ : Rn+1 → C. We define TϕA0 ,...,An : B(2d ) × · · · × B(2d ) → B(2d ) by " #$ % n times

TϕA0 ,...,An (X1 , . . . , Xn ) =

d  r0 ,...,rn =1

  (1) (n) ϕ λ(0) r0 , λr1 . . . , λrn Pg(0) X1 Pg(1) . . . Xn Pg(n) , r0

r1

rn

(4.1.1)

© Springer Nature Switzerland AG 2019 A. Skripka, A. Tomskova, Multilinear Operator Integrals, Lecture Notes in Mathematics 2250, https://doi.org/10.1007/978-3-030-32406-3_4

65

66

4 Multiple Operator Integrals A ,...,A

n for X1 , . . . , Xn ∈ B(2d ). The operator Tϕ 0 is a discrete version of a multiple operator integral. The function ϕ is usually called the symbol of the operator A ,...,An . Tϕ 0

(j ) d

j Assume that {λi }i=1 is the set of pairwise distinct eigenvalues of the operator Aj , where dj ∈ N, dj  d. Then, (4.1.1) can be rewritten in terms of the spectral measures of Aj as

TϕA0 ,...,An (X1 , . . . , Xn ) =

d0  r0 =1

···

dn 

(4.1.2)

(1) (n) (0) (1) (n) ϕ(λ(0) r0 , λr1 . . . , λrn )EA0 ({λr0 })X1 EA1 ({λr1 }) . . . Xn EAn ({λrn }).

rn =1

4.1.2 Relation to Multilinear Schur Multipliers Let n, d ∈ N and let m(n) := {mr0 ,...,rn }dr0 ,...,rn =1 ⊂ C. An n-linear Schur multiplier (or a linear Schur multiplier in case n = 1) Mn = Mm(n) : B(2d ) × · · · × B(2d ) → B(2d ) #$ % " n times

associated with symbol m(n) is defined via Mn (X1 , . . . , Xn ) :=

d 

mr0 ,...,rn · xr(1) x (2) . . . xr(n) · Er 0 r n , 0 r1 r1 r2 n−1 rn

(4.1.3)

r0 ,...,rn =1

where  d Xk = xij(k) i,j =1 is a d × d matrix, 1  k  n. The latter can be rewritten in the form Mn (X1 , . . . , Xn ) =

d 

mr0 ,...,rn · Er0 X1 Er1 X2 Er2 . . . Xn Ern .

(4.1.4)

r0 ,...,rn =1

Multilinear Schur multipliers whose symbols are divided differences were considered in [73] in the context of perturbation theory. General multilinear Schur multipliers were introduced in [78].

4.1 Multiple Operator Integrals on Finite Matrices

67

The norm of a multilinear Schur multiplier is estimated via the norm of a linear Schur multiplier in [169, Theorem 2.3]. Theorem 4.1.1 Let 1  p1 , . . . , pn , p  ∞ be such that Mn defined in (4.1.3) or (4.1.4) and M1,k˜ given by M1,k˜ (C) =

d  r,s=1

1 p1

+ ··· +

1 pn

=

1 p.

For

mr,k, . . . , k ,s · Er CEs , " #$ % n−1

we have p

p

p

Mn : Sd 1 × · · · × Sd n → Sd  

p

max M1,k˜ : S1d → Sd .

1k d

The following result relates multilinear Schur multipliers to multiple operator integrals. Proposition 4.1.2 Let n ∈ N and ϕ : Rn+1 → C. Let A0 , . . . , An be selfadjoint (or unitary) elements in B(2d ) and X1 , . . . , Xn ∈ B(2d ). Let g(k) be an orthonormal basis of eigenvectors of Ak with the corresponding d-tuple of (k) (k) eigenvalues {λl }dl=1 and suppose that Xk has the matrix representation (xij )di,j =1 in the bases {g(k), g(k−1) }, k = 1, . . . , n. The matrix representation of the multiple operator integral with symbol ϕ given by (4.1.1) in the bases {g(n) , g(0) } coincides with the value of the Schur multiplier associated with the matrix (n) d m(n) = {ϕ(λ(0) r0 , . . . , λrn )}r0 ,...,rn =1

  (1) (n) given by (4.1.3) on the n-tuple (xij )di,j =1 , . . . , (xij )di,j =1 , that is, Mm(n) (X1 , . . . , Xn ) = TϕA0 ,...,An (X1 , . . . , Xn ).

4.1.3 Properties of Finite Dimensional Multiple Operator Integrals Algebraic Properties The following properties of the operator integral TϕA0 ,...,An are direct consequences of (4.1.4) and Proposition 4.1.2 (for brevity, below we use the notation Tϕ ): Tαϕ+βψ = αTϕ + βTψ , α, β ∈ C, Tϕ = I, provided ϕ ≡ 1, where I is the identity operator on B(2d ).

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4 Multiple Operator Integrals

Norm Estimates In this subsection we extend the estimates (3.1.6) and (3.1.7) to the higher order case. The first estimate requires a more delicate care while the second one is obtained by a very minor adjustment of the argument. Proposition 4.1.3 Let n ∈ N and let A0 , . . . , An ∈ B(2d ) be self-adjoint (or (j ) unitary) operators with corresponding d-tuples of the eigenvalues {λi }di=1 , j = n+1 n+1 0, . . . , n. Then, for ϕ : R → C (or ϕ : T → C), TϕA0 ,...,An : S2d × · · · × S2d → S2d  =

max

1  r0 ,...,rn  d

  (0) (1)  ϕ λ , λ . . . , λ(n) . r0

r1

rn

(4.1.5) (n) d Proof Let m(n) := {ϕ(λ(0) r0 , . . . , λrn )}r0 ,...,rn =1 . In view of Proposition 4.1.2, in order to prove (4.1.5) it suffices to prove that the Schur multiplier defined in (4.1.3) satisfies

Mn : S2d × · · · × S2d → S2d  =

max

1  r0 ,...,rn  d

|mr0 ,...,rn |.

(4.1.6)

Alternatively, one could adjust the method demonstrated below and derive (4.1.5) directly from (4.1.1). Below we include the proof of the equality (4.1.6) due to [169, Lemma 2.1]. For a fixed (n + 1)-tuple (r0 , . . . , rn ), by (4.1.3), we have that Mn (Er0 r1 , Er1 r2 , . . . , Ern−1 rn ) = mr0 ,...,rn Er0 rn and thus, Mn : S2d × · · · × S2d → S2d   |mr0 ,...,rn | for all 1  r0 , . . . , rn  d. Hence, trivially we have Mn : S2d × · · · × S2d → S2d  

max

1  r0 ,...,rn  d

|mr0 ,...,rn |.

Conversely, by (4.1.3), we have that Mn (X1 , . . . , Xn )22 =

d    

d 

r0 ,rn =1 r1 ,...,rn−1 =1



max

1r0 ,...,rn d

|mr0 ,...,rn |

2 (n)  mr0 ,...,rn · xr(1) . . . x rn−1 rn  0 r1

2

d 

d 

(4.1.7)

2 (n) |xr(1) . . . x | , r r r 0 1 n−1 n

r0 ,rn =1 r1 ,...,rn−1 =1

4.1 Multiple Operator Integrals on Finite Matrices

69

for all X1 , . . . , Xn ∈ S2d . Using the Cauchy-Schwartz inequality n-times, we obtain that d  r0 ,rn =1

=

d 

2

r1 ,...,rn−1 =1

d d   |xr(1) |· 0 r1 r0 ,rn =1



|xr(1) . . . xr(n) | 0 r1 n−1 rn

r1 =1

d d  

= X1 22

d  r1 ,rn =1

|xr(2) . . . xr(n) | 1 r2 n−1 rn

2

r2 ,...,rn−1 =1

|xr(1) |2 · 0 r1

r0 ,rn =1 r1 =1

d 

d  r1 =1

d  r2 ,...,rn−1 =1

d 

|xr(2) . . . xr(n) | 1 r2 n−1 rn

2

r2 ,...,rn−1 =1

|xr(2) . . . xr(n) | 1 r2 n−1 rn

2

 . . .  X1 22 . . . Xn 22 . Combining the latter with (4.1.7) proves the converse inequality, which completes the proof of (4.1.6) and, hence, of the proposition.   Corollary 4.1.4 Let n ∈ N and let A0 , . . . , An ∈ B(2d ) be self-adjoint (or unitary) (j ) operators with corresponding d-tuples of the eigenvalues {λi }di=1 . Then, for ϕ : n+1 n+1 2 → C (or ϕ : T → C) and X1 , . . . , Xn ∈ B(d ), R TϕA0 ,...,An (X1 , . . . , Xn )    (1) (n)  X1  . . . Xn .  d n/2 max ϕ λ(0) r0 , λr1 . . . , λrn 1  r0 ,...,rn  d

(4.1.8)

We note that the bound (4.1.8) with d n instead of d n/2 trivially follows from the definition of the multilinear Schur multiplier. In some cases such coarse bound is sufficient; see [88, Theorem 2.3.1] or a remark at the end of Sect. 5.3.1.

Perturbation Formula In the next proposition we establish a standard perturbation formula for a multiple operator integral with respect to auxiliary self-adjoint matrix parameters.

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4 Multiple Operator Integrals

Proposition 4.1.5 Let 2  n ∈ N, f be n times differentiable on R (or on T), and A, B, H1 , . . . , Hn−1 ∈ B(2d ) be self-adjoint (or unitary) operators. Then, H ,...,Hj−1 ,A,Hj ,...,Hk

Tf [k]1

H ,...,Hj−1 ,B,Hj ,...,Hk

(X1 , . . . , Xk ) − Tf [k]1

(X1 , . . . , Xk ) (4.1.9)

=

H1 ,...,Hj−1 ,A,B,Hj ,...,Hk Tf [k+1] (X1 , . . . , Xj −1 , A − B, Xj , . . . , Xk )

for all X1 , . . . , Xk ∈ B(2d ), 0  k  n − 1, 1  j  k + 1. Proof Fix n  2 and 0  k  n − 1. For 1  j  k + 1, denote ψj (x0 , . . . , xk+1 ) := xj f [k+1] (x0 , . . . , xk+1 ), φj (x0 , . . . , xk+1 ) := f [k] (x0 , . . . , xj −1 , xj +1 , . . . , xk+1 ). From the definition of the divided difference, we see that ψj −1 − ψj = φj − φj −1 . For brevity, we set H˜ j −1 := (H1 , . . . , Hj −1 ), j H˜ k

X˜ j −1 := (X1 , . . . , Xj −1 ), ˜k jX

:= (Hj , Hj +1 , . . . , Hk ),

:= (Xj , Xj +1 , . . . , Xk ).

By the definition of the multiple operator integral (4.1.1), H˜

,A,B, j H˜ k

(X˜ j −1 , A − B, j X˜ k )

H˜

,A,B, j H˜ k

Hj−1 ,A,B, j Hk (X˜ j −1 , A, j X˜ k ) − Tf [k+1] (X˜ j −1 , B, j X˜ k )

H˜

,A,B, j H˜ k

H ,A,B, j Hk (X˜ j −1 , I, j X˜ k ) − Tψjj−1 (X˜ j −1 , I, j X˜ k )

H˜

,A,B, j H˜ k

(X˜ j −1 , I, j X˜ k )

H˜

,A,B, j H˜ k

(X˜ j −1 , I, j X˜ k )

H˜

,A,B, j H˜ k

Hj−1 ,A,B, j Hk (X˜ j −1 , I, j X˜ k ) − Tφj−1 (X˜ j −1 , I, j X˜ k )

H˜

,A, j H˜ k

j−1 RHS of (4.1.9) = Tf [k+1] j−1 = Tf [k+1] j−1 = Tψj−1

j−1 = Tψj−1 −ψj

= Tφj j−1 −φj−1 = Tφj j−1 = Tf [k]j−1 proving the assertion.

˜

H˜

(X˜ k ) − Tf [k]j−1

,B, j H˜ k

˜

˜

˜

˜

˜

(X˜ k ) = LHS of (4.1.9),  

4.1 Multiple Operator Integrals on Finite Matrices

71

We note that the formula (4.1.9) is proved in [53, Theorem 15] for matrices in the special case n = 2.

Continuity In the next proposition we establish continuity of a multiple operator integral with respect to self-adjoint matrix parameters. (m)

Proposition 4.1.6 Let k ∈ N, Aj , Aj ∈ B(2d ), j = 1, . . . , k + 1, m ∈ N, be self-adjoint (respectively, unitary) operators such that A(m) → A, as m → ∞. Let k ∈ N and let f ∈ C k (R) (respectively, f ∈ C k (T)). Then, (m) ,...,A(m)

TfA[k]

(X1 , . . . , Xk ) → TfA,...,A (X1 , . . . , Xk ), [k]

m → ∞,

for all X1 , . . . , Xk ∈ B(2d ). Proof We prove the assertion for self-adjoint operators. The case of unitary operators is similar. Assume first that f ∈ C k+1 (R). Denote ˜ k+1 jB

:= (Bj , Bj +1 , . . . , Bk+1 ).

Applying a telescoping summation and Proposition 4.1.5 gives  A(m) ,...,A(m)   1  A ,...,Ak+1 k+1 (X1 , . . . , Xk ) − Tf [k]1 (X1 , . . . , Xk ) Tf [k]

2

k+1   ˜ j−1 , j A˜ (m) ˜ j , j+1 A˜ (m) 1A 1A   k+1 k+1 = Tf [k] (X1 , . . . , Xk ) − Tf [k] (X1 , . . . , Xk )

2

j =1

k+1   ˜ j−1 , A(m) ,Aj , j A˜ (m) 1A   k+1 = Tf [k+1] j (X1 , . . . , Xj −1 , A(m) − A , X , . . . X )  . j j k j 2

j =1

Applying the triangle inequality, Proposition 4.1.3, and the inequality (2.2.2) (or (2.2.3) for the unitary case), gives  A(m) ,...,A(m)   1  A1 ,...,Ak+1 k+1 (X1 , . . . , Xk ) − Tf [k] (X1 , . . . , Xk ) Tf [k]

2

 f (k+1) ∞ ·

k+1  j =1

which approaches 0 as m → ∞.

(m)

Aj

− Aj 2 · X1 2 · · · · · Xk 2 ,

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4 Multiple Operator Integrals

Assume now that f ∈ C k (R). Given > 0 there exists f ∈ C k+1 (R) such that f (k) − f (k) ∞ X1 2 · · · Xn 2
0, lim EAm ((λj − , λj + )) = EA ((λj − , λj + ))

m→∞

∈ σ (Am ), m ∈ N, such that and there exist λ(m) j (m) lim λ m→∞ j

= λj

(see, e.g., [171, Theorem VIII.24]). However, this property extends to the infinite dimensional case only under the restriction that λj − , λj + are not eigenvalues of A. On the other hand, the ideas used to prove Proposition 4.1.6 generalize to the infinite dimensional setting.

4.1 Multiple Operator Integrals on Finite Matrices

73

Reduction to Identical Spectral Measures Below we prove a useful property of multiple operator integrals, which shows that instead of working with TϕB,A,...,A defined in (4.1.1) for A = B, it suffices to work with TϕA,A,...,A . Proposition 4.1.8 Let A, B ∈ B(2d ) be self-adjoint operators and X1 , . . . , Xn ∈ B(2d ). Let Eij denote the elementary 2 × 2 matrix whose only nonzero entry equals 1 and has indices (i, j ) and denote C = E11 ⊗A+E22 ⊗B,

X˜ 1 = E12 ⊗X1 ,

X˜ j = E22 ⊗Xj , j = 2, . . . , n.

and

Then, for a bounded Borel function ϕ : Rn+1 → C, TϕC,...,C (X˜ 1 , . . . , X˜ n ) = E12 ⊗ TϕA,B,...,B (X1 , . . . , Xn ). 

 m  Proof Let σ (A) = {λi }m i=1 and σ (B) = {λj }j =1 for m, m  d. The spectral projection of the operator C associated with λ ∈ σ (A) ∪ σ (B) is given by

EC (λ) = E11 ⊗ EA (λ) + E22 ⊗ EB (λ). By (4.1.2), TϕC,...,C (X˜ 1 , . . . , X˜ n )=



ϕ(λr0 , . . . , λrn )EC (λr0 )X˜ 1 . . .X˜ n EC (λrn ).

λr0 ,...,λrn ∈σ (A)∪σ (B)

Moreover, EC (λr0 )X˜ 1 . . . X˜ n EC (λrn )   = E11 ⊗ EA (λr0 ) + E22 ⊗ EB (λr0 ) (E12 ⊗ X1 )EC (λr1 )X˜ 2 . . . X˜ n EC (λrn )   = E12 ⊗ EA (λr0 )X1 EC (λr1 )(E22 ⊗ X2 )EC (λr2 )X˜ 3 . . . X˜ n EC (λrn )   = E12 ⊗ EA (λr0 )X1 EB (λr1 )X2 EC (λr2 )X˜ 3 . . . X˜ n EC (λrn )   = · · · = E12 ⊗ EA (λr0 )X1 EB (λr1 )X2 . . . Xn EB (λrn ) . Therefore, TϕC,...,C (X˜ 1 , . . . , X˜ n )    = ϕ(λr0 , . . . , λrn )E12 ⊗ EA (λr0 )X1 EB (λr1 )X2 . . .Xn EB (λrn ) λr0 ∈σ (A),λr1 ...,λrn ∈σ (B)

= E12 ⊗ TϕA,B,...,B (X1 , . . . , Xn ).

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4 Multiple Operator Integrals

4.1.4 Estimates of Multiple Operator Integrals via Double Operator Integrals The theorem below reduces estimates for multiple operator integrals to estimates for double operator integrals. Theorem 4.1.9 Let 1  p1 , . . . , pn , p  ∞ be such that p11 + · · · + p1n = p1 and let ψ : Cn+1 → C be a bounded Borel function. Let A, B ∈ B(2d ). The following assertions hold. (i) If A and B are self-adjoint, 0 ∈ σ (A), and ϕ(x0 , xn ) := ψ(x0 , 0, . . . , 0, xn ),

for x0 , xn ∈ R,

then p

p

p

p

TψA+B,A,...,A : Sd 1 × · · · × Sd n → Sd   TϕA+B,A : S1d → Sd . (ii) If A and A + B are unitary, 1 ∈ σ (A), and ϕ(z0 , zn ) := ψ(z0 , 1, . . . , 1, zn ),

for z0 , zn ∈ T,

then p

p

p

p

TψA+B,A,...,A : Sd 1 × · · · × Sd n → Sd   TϕA+B,A : S1d → Sd . (iii) If n = 1, 1  p, p  ∞, and

1 p

+

1 p

= 1, then

 A+B,A  p p p p  T : Sd → Sd  = TψA+B,A : Sd → Sd . ψ Proof The properties (i) and (ii) are immediate consequences of Theorem 4.1.1 and Proposition 4.1.2. The property (iii) follows from the representation (4.1.4) and Proposition 4.1.2.  

4.2 Multiple Operator Integrals on S2 4.2.1 Pavlov’s Approach Let A1 , . . . , An+1 ∈ Dsa and X1 . . . , Xn ∈ S2 . Let = σ (A1 ) × · · · × σ (An+1 ) and δk be a Borel subset of σ (Ak ), k = 1, . . . , n + 1. Consider the finitely additive S2 -valued measure m(δ1 × · · · × δn+1 ) = EA1 (δ1 )X1 . . . Xn EAn+1 (δn+1 )

4.2 Multiple Operator Integrals on S2

75

defined on finite unions of the sets δ1 ×· · ·×δn+1 , which we call rectangular subsets of σ (A1 ) × · · · × σ (An+1 ). It is proved in [138, Theorem 1] that m has a bounded semivariation m  X1 2 . . . Xn 2 and is σ -additive. Since the space S2 is reflexive, it follows from [66, Chapter 1, Section 5, Theorem 2] that m has a unique σ -additive extension m ˜ :  → S2 , where  is the σ -algebra generated by the rectangular subsets of σ (A1 ) × · · · × σ (An+1 ). The measure m ˜ has a bounded semivariation and is absolutely continuous with respect to the direct product λA1 × · · · × λAn+1 of scalar-valued spectral measures of A1 , . . . , An+1 . Hence, L∞ ( , λA1 × · · · × λAn+1 ) ⊂ L∞ ( , m) ˜ and, by [66, Chapter 1, Section 1, Theorem 13], the S2 -valued integral  ϕ(ω) d m(ω) ˜

(4.2.1)



is well defined for ϕ ∈ L∞ ( , λA1 × · · · × λAn+1 ). This integral is called in [138] a multiple operator integral associated with ϕ, A1 , . . . , An+1 , and X1 , . . . , Xn . If n = 1, then m(E ˜ A1 (δ1 )XEA2 (δ2 )) = G(δ1 × δ2 ) for every X ∈ S2 , where G is given by (2.5.1). Thus, the multiple operator integral given by (4.2.1) extends the double operator integral given by (3.2.4).

4.2.2 Coine-Le Merdy-Sukochev’s Approach The following multiple operator integral is defined in [56]. Given a self-adjoint operator A ∈ Dsa , let λA denote its scalar-valued spectral measure. For A1 , . . . , An+1 ∈ Dsa , let = σ (A1 ) × · · · × σ (An+1 ). Definition 4.2.1 Let  be the unique linear map from the tensor product space L∞ (σ (A1 ), λ1 ) ⊗ · · · ⊗ L∞ (σ (An+1 ), λn+1 ) into Bn (S2 × · · · × S2 , S2 ) such that (f1 ⊗ · · · ⊗ fn+1 )(X1 , . . . , Xn ) = f1 (A1 )X1 f2 (A2 )X2 · · · fn (An )Xn fn+1 (An+1 ) for all fj ∈ L∞ (σ (Aj ), λj ), j = 1, . . . , n, and all X1 , . . . , Xn ∈ S2 . According to [56, Proposition 6],  uniquely extends to a w∗ -continuous and contractive map  A1 ,...,An+1 : L∞ ( , λA1 × · · · × λAn+1 ) −→ Bn (S2 × · · · × S2 , S2 ). ϕ ∈ L∞ ( , λA1 × · · · × Let ϕ : Rn+1 → C be a bounded Borel function and let & λAn+1 ) be the class of its restriction to . Denote the n-linear map  A1 ,...,An+1 (& ϕ) by  A1 ,...,An+1 (ϕ) : S2 × · · · × S2 −→ S2 .

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4 Multiple Operator Integrals

We note that Bn (S2 × · · · × S2 , S2 ) is a dual space, and the predual is given by ∧

∧

S2 ⊗ · · · ⊗ S2 , ∗ the projective tensor product of n copies of S2 [66]. The

w -continuity of '  A1 ,...,An+1 means that if a net {ϕi }i∈I in L∞ , n+1 i=1 λAi converges to ϕ ∈ '

∗ 2 L∞ , n+1 i=1 λAi in the w -topology, then for all X1 , . . . , Xn ∈ S , the net

 A ,...,A  n+1 (ϕ )(X , . . . , X )  1 i 1 n i∈I converges to  A1 ,...,An+1 (ϕ)(X1 , . . . , Xn ) weakly in S2 . The following connection between different multiple operator integrals is established in [56, Remark 8]. Proposition 4.2.2 The multiple operator integral (ϕ) given by Definition 4.2.1 coincides with Pavlov’s multiple operator integral given by (4.2.1). The crucial point is the construction leading to Definition 4.2.1 is the w∗ continuity of  A1 ,...,An+1 (·), which allows to reduce various computations to elementary tensor product manipulations. These ideas are illustrated in [55]. The symbols ϕ for which the triple operator integral  A1 ,A2 ,A3 (ϕ) maps S2 × S2 to S1 are characterized in [56, Theorem 23]. In fact, this characterization is obtained for a natural generalization of  A1 ,A2 ,A3 (ϕ) to normal operators A1 , A2 , A3 . Theorem 4.2.3 Let A, B and C be normal operators densely defined in H and let φ ∈ L∞ (σ (A) × σ (B) × σ (C), λA × λB × λC ). The following are equivalent: (i)  A,B,C (φ) ∈ B2 (S2 × S2 , S1 ). (ii) There exist a separable Hilbert space H and two functions a ∈ L∞ (σ (A)×σ (B), λA ×λB ; H ) and b ∈ L∞ (σ (B)×σ (C), λB ×λC ; H ) such that φ(t1 , t2 , t3 ) = a(t1 , t2 ), b(t2 , t3 ) for a.e. (t1 , t2 , t3 ) ∈ σ (A) × σ (B) × σ (C). In this case,    A,B,C (φ) : S2 × S2 −→ S1  = inf a∞ b∞ , where the infimum runs over all pairs (a, b) satisfying (ii).

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

77

4.3 Multiple Operator Integrals on Schatten Classes and B(H) 4.3.1 Approach via Separation of Variables In this subsection we extend the definition of Sect. 3.3.3 to the higher order case. This approach to multiple operator integral was developed in [146], [24] and summarized in [149]. Let n ∈ N and let An be the class of functions ϕ : Rn+1 → C admitting the representation  ϕ(λ1 , . . . , λn+1 ) =

a1 (λ1 , ω) . . . an+1 (λn+1 , ω) dν(ω),

(4.3.1)



for some finite measure space ( , ν) and bounded measurable functions ai (·, ·) : R × → C, i = 1, . . . , n + 1, where on R we consider the Borel σ -algebra, such that 

a1 (·, ω)∞ . . . an+1 (·, ω)∞ d |ν| (ω) < ∞.

The class An is an algebra with respect to the operations of pointwise addition and multiplication [24, Proposition 4.10]. The formula  ϕAn = inf



a1 (·, ω)∞ . . . an+1 (·, ω)∞ d |ν| (ω),

where the infimum is taken over all possible representations (4.3.1), defines a norm on An (see [63, Lemma 4.6]). The class An can also be defined as the class of functions ϕ : Rn+1 → C admitting the representation  ϕ(λ1 , . . . , λn+1 ) =

b1 (λ1 , ω) . . . bn+1 (λn+1 , ω) dν2 (ω),

(4.3.2)



for some (not necessarily finite) measure space ( , ν2 ) and bounded measurable functions bi (·, ·) : R × → C, i = 1, . . . , n + 1, such that 

b1 (·, ω)∞ . . . bn+1 (·, ω)∞ d |ν2 | (ω) < ∞

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4 Multiple Operator Integrals

(see, e.g., [63, 158]). These definitions coincide since, as in the case of double operator integrals, the representation (4.3.1) of the function ϕ can be obtained from (4.3.2) with a1 (λ, ω) =

b1 (λ, ω) bn+1 (λ, ω) , . . . , an+1 (λ, ω) = b1 (·, ω)∞ bn+1 (·, ω)∞

and the finite measure ν defined by ν = b1 (·, ω)∞ . . . bn+1 (·, ω)∞ ν2 . The following definition was introduced independently in [24, Definition 4.1] and [146, (3.9)]. Definition 4.3.1 Let 1  p, p1 , . . . , pn  ∞. For every ϕ ∈ An , and a fixed tuple A ,...,An+1 : Sp1 × · · · × Spn → A1 , . . . , An+1 ∈ Dsa , the double operator integral Tϕ 1 A ,...,A n+1 Sp , where p11 + · · · + p1n = p1 (respectively, Tϕ 1 : B(H) × · · · × B(H) → B(H)) is defined by A1 ,...,An+1

Tϕ

 (X1 , . . . , Xn ) :=

a1 (A1 , ω) X1 . . . Xn an+1 (An+1 , ω) dν(ω),

for Xj ∈ Spj (respectively, Xj ∈ B(H) if pj = ∞), where aj and ( , ν) are taken from the representation (4.3.1) and the integral is understood in the sense of the Bochner integral 

a1 (A1 , ω) X1 . . . Xn an+1 (An+1 , ω) dν(ω) (y)





=

 a1 (A1 , ω) X1 . . . Xn an+1 (An+1 , ω) (y) dν(ω), y ∈ H.



It is worth noting that the multiple operator integral is introduced above for the symbol ϕ from the integral projective tensor product of L∞ -spaces which coincides with the extended Haagerup tensor product in the case n = 2. The definition and study of the multiple operator integral with respect to the symbol from the extended Haagerup tensor product for n > 2 can be found in [95] and [11]; see also [14]. A ,...,An+1 The value Tϕ 1 (X1 , . . . , Xn ) does not depend on the particular representation on the right-hand side of (4.3.1). This fact is stated in [146, Lemma 3.1] and proved under the assumption ai (·, ω) be bounded and continuous for every ω ∈ , i = 1, . . . , n + 1, in [24, Lemma 4.3]. A ,...,An+1 It follows directly from Definition 4.3.1 that if ϕ ∈ An , then Tϕ 1 p is a bounded n-linear transformation that maps S 1 × · · · × Spn → Sp , 1  p, p1 , . . . , pn  ∞, and B(H) × · · · × B(H) → B(H). The following connection between different multiple operator integrals is established in [56, Proposition 28].

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

79

Proposition 4.3.2 Let A1 , . . . , An+1 ∈ Dsa and ϕ ∈ An . Then, the multiple operator integral  A1 ,...,An+1 (ϕ) given by Definition 4.2.1 coincides on S2 ×· · ·×S2 A ,...,An+1 with the multiple operator integral Tϕ 1 given by Definition 4.3.1.

4.3.2 Approach Without Separation of Variables In this section we extend the definition of Sect. 4.1.1 to the case of infinitedimensional operators. The multiple operator integral that we introduce below is different from the one introduced in Sect. 4.3.1; however, the two constructions coincide for a broad set of symbols used in applications. Let Aj ∈ Dsa , j = 0, . . . , n, and denote 

j El,m

= EAj

l l+1 , m m

 ,

for every m ∈ N, l ∈ Z, and j = 0, . . . , n. Let n ∈ N and 1  p, pj  ∞, j = 1, . . . , n, be such that 0  p1 = p11 + · · · + p1n  1. Let Xj ∈ Spj and let ϕ : Rn+1 → C be a bounded Borel function. The following definition is due to [163, Definition 3.1]. Definition 4.3.3 Suppose that for every m ∈ N, the series Sϕ,m (X1 , . . . , Xn ) : =





ϕ

l0 ,...,ln ∈Z

= lim

N→∞

ln l0 ,..., m m

 |lj |  N 0j n

 ϕ

 El00 ,m X1 El11 ,m X2 . . . Xn Elnn ,m

ln l0 ,..., m m

 El00 ,m X1 El11 ,m X2 . . . Xn Elnn ,m

converges in the norm of Sp and (X1 , . . . , Xn ) → Sϕ,m (X1 , . . . , Xn ), m ∈ N, is a sequence of bounded multilinear operators that map Sp1 × · · · × Spn → Sp .  ∞ If the sequence of operators Sϕ,m m=1 converges strongly to some multilinear ∞  operator Tϕ , then, according to the Banach-Steinhaus theorem, Sϕ,m m=1 is uniformly bounded and the operator Tϕ is also bounded. The transformation Tϕ is called the multiple operator integral associated with ϕ and the operators A0 , . . . , An (or the spectral measures EA0 , . . . , EAn ). Let Cn denote the subset of An of functions admitting the representation (4.3.1), ∞ where ∪∞ k=1 k = for a growing sequence { k }k=1 of measurable subsets of such that the families {aj (·, ω)}ω∈ k , j = 1, . . . , n + 1, are uniformly bounded and

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4 Multiple Operator Integrals

uniformly continuous. A norm on Cn is defined by  ϕCn = inf



a1 (·, ω)∞ . . . an+1 (·, ω)∞ d |ν| (ω),

where the infimum is taken over all possible representations (4.3.1) with aj , j = 1, . . . , n + 1, as above. Multiple operator integrals with symbols in Cn are known to enjoy nicer properties than multiple operator integrals with symbols in An . The following sufficient condition for ϕ ∈ Cn is established in [161, Theorem 5]. A more general set of symbols in Cn will be considered in Sect. 4.3.3. n (R), then f [n] ∈ C and Theorem 4.3.4 If f ∈ B∞1 n n (R) . f [n] Cn  constn f B∞1

The following result is established in [163, Lemma 3.5]. Proposition 4.3.5 The multiple operator integrals Tϕ given by Definitions 4.3.1 and 4.3.3 coincide for ϕ ∈ Cn . The multiple operator integral on a Cartesian product of Hilbert-Schmidt ideals defined in this subsection coincides with the one given by Definition 4.2.1, as obtained in [119, Lemma 2.6 and Remark 2.7(i)]. Proposition 4.3.6 The multiple operator integrals Tϕ and (ϕ) given by Definitions 4.3.3 and 4.2.1 coincide on S2 × · · · × S2 for every ϕ ∈ Cb (Rn+1 ). Proof Let n ∈ N, let A1 , . . . , An+1 be self-adjoint operators, X1 , . . . , Xn ∈ S2 , A ,...,An+1 : S2 × · · · × S2 → S2 is well and ϕ ∈ Cb (Rn+1 ). It can be shown that Tϕ 1 defined. (  For any r ∈ N, l ∈ Z, set Jl,r = rl , l+1 and for N ∈ N, consider r ϕr,N =



 ϕr,N

|lj |  N 1  j  n+1

ln+1 l1 ,..., r r

 χJl1 ,r ⊗ · · · ⊗ χJln+1 ,r .

Since ϕ : Rn+1 → C is continuous, ϕ = w∗ - lim lim ϕr,N r→∞ N→∞

in L∞ ( , λA1 × · · · × λAn+1 ). Hence for all X1 , . . . , Xn ∈ S2 ,  A1 ,...,An+1 (ϕ)(X1 , . . . , Xn ) = lim lim  A1 ,...,An+1 (ϕr,N )(X1 , . . . , Xn ) r→∞ N→∞

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

81

in S2 . Comparing the latter with Definition 4.3.3 implies A1 ,...,An+1

 A1 ,...,An+1 (ϕ)(X1 , . . . , Xn ) = Tϕ

(X1 , . . . , Xn ).

(4.3.3)

4.3.3 Properties of Multiple Operator Integrals on Sp and B(H) Algebraic Properties The multiple operator integral given by Definition 4.3.1 satisfies Tα1 ϕ1 +α2 ϕ2 = α1 Tϕ1 + α2 Tϕ2 ,

(4.3.4)

for ϕ1 , ϕ2 ∈ An , α1 , α2 ∈ C (see, e.g., [24, Proposition 4.10]). The multiple operator integral given by Definition 4.3.3 satisfies (4.3.4) for all bounded Borel functions ϕ1 , ϕ2 : Rn+1 → C for which it is defined, as it follows immediately from the definition. The following result is established in [163, Lemma 3.2]. Proposition 4.3.7 Let n ∈ N and 1  p, pj  ∞, j = 1, . . . , n, be such that 0  p1 = p11 + · · · + p1n  1. Let Xj ∈ Spj . Let A ∈ Dsa , let ϕ : Rn+1 → C be a bounded Borel function, and let Tϕ be the transformation associated with A0 = . . . = An = A and ϕ according to Definition 4.3.3. The following assertions hold. (i) Let Tϕ : Sp1 × · · · × Spn → Sp be bounded. If ϕ(λ0 , λ1 , . . . , λn ) := ϕ(λn , λn−1 , . . . , λ0 ), then Tϕ : Spn × · · · × Sp1 → Sp is bounded and     Tϕ  = Tϕ  . (ii) Assume that 1  p0  ∞ and · · · × Spn → S

p0 p0 −1

1 p0

+···+

1 pn

= 1. Assume also that Tϕ : Sp1 ×

exists and is bounded. Define

ϕ ∗ (λn , λ0 , . . . , λn−1 ) := ϕ (λ0 , . . . , λn−1 , λn ) . pn

If Tϕ ∗ : Sp0 × · · · × Spn−1 → S pn −1 exists and is bounded, then     Tr X0 Tϕ (X1 , . . . , Xn ) = Tr Tϕ ∗ (X0 , . . . , Xn−1 ) Xn , for Xj ∈ Spj , j = 0, . . . , n.

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4 Multiple Operator Integrals

(iii) Let ϕ1 : Rk+1 → C and ϕ2 : Rn−k+1 → C be bounded Borel functions such that the operators Tϕ1 : Sp1 × · · · × Spk → Sq and Tϕ2 : Spk+1 × · · · × Spn → 1 + · · · + p1n . Sr exist and are bounded, where q1 = p11 + · · · + p1k and 1r = pk+1 If ψ(λ0 , . . . , λn ) := ϕ1 (λ0 , . . . , λk ) · ϕ2 (λk , . . . , λn ) , then the operator Tψ exists and is bounded on Sp1 × · · · × Spn and Tψ (X1 , . . . , Xn ) = Tϕ1 (X1 , . . . , Xk ) · Tϕ2 (Xk+1 , . . . , Xn ) , for Xj ∈ Spj , j = 1, . . . , n. (iv) Let ϕ1 : Rk+1 → C and ϕ2 : Rn−k+2 → C be bounded Borel functions such that Tϕ1 : Sp1 × · · · × Spk → Sq and Tϕ2 : Sq × Spk+1 × · · · × Spn → Sr 1 exist and are bounded, where q1 = p11 + · · · + p1k and 1r = q1 + pk+1 + · · · + p1n . If ψ(λ0 , . . . , λn ) := ϕ1 (λ0 , . . . , λk ) · ϕ2 (λ0 , λk , . . . , λn ) , then the operator Tψ exists and is bounded on Sp1 × · · · × Spn and   Tψ (X1 , . . . , Xn ) = Tϕ2 Tϕ1 (X1 , . . . , Xk ), Xk+1 , . . . , Xn , for Xj ∈ Spj , j = 1, . . . , n. Proof Given m, N ∈ N and a bounded Borel function ϕ : Rn+1 → C, we denote Tϕm,N (X1 , . . . , Xn ) (4.3.5)  

   

 ln l0 l0 l0 ln ln := ,..., E , +1 X1 . . . Xn E , +1 , ϕ m m m m m m |lj |  N 0j n

where Xj ∈ Spj , j = 0, . . . , n. Note that by the definition of Tϕ , Tϕ (X1 , . . . , Xn ) = lim

lim T m,N (X1 , . . . , Xn ). m→∞ N→∞ ϕ

(4.3.6)

(i) By taking the adjoint in (4.3.5) we obtain  m,N ∗ Tϕ (X1 , . . . , Xn ) = Tϕm,N (Xn∗ , . . . , X1∗ ).

(4.3.7)

By (4.3.6), Tϕ (Xn∗ , . . . , X1∗ ) = lim

lim T m,N (Xn∗ , . . . , X1∗ ) m→∞ N→∞ ϕ

(4.3.8)

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

83

Combining (4.3.7)–(4.3.8) implies 

Tϕ (X1 , . . . , Xn ))∗ = Tϕ (Xn∗ , . . . , X1∗ ).

Hence, Tϕ : Sp1 × · · · × Spn → Sp  = Tϕ : Spn × · · · × Sp1 → Sp . (ii) By cyclicity of the trace,     l0 ln ϕ ,..., X0 El0 ,m X1 El1 ,m X2 . . . Eln−1 ,m Xn Eln ,m m m     l0 ln = Tr ϕ ,..., Eln ,m X0 El0 ,m X1 El1 ,m X2 . . . Eln−1 ,m Xn . m m

Tr

Hence,



Tr X0 Tϕm,N (X1 , . . . , Xn ) = Tr Tϕm,N (X0 , . . . , Xn−1 ) Xn . ∗

(4.3.9)

Combining (4.3.6) and (4.3.9) completes the proof of the claim. (iii) Immediately from (4.3.5) we obtain Tψm,N (X1 , . . . , Xn ) = Tϕm,N (X1 , . . . , Xk ) · Tϕm,N (Xk+1 , . . . , Xn ) . 1 2 (4.3.10) Passing to the limit in (4.3.10) and applying (4.3.6) proves the claim. (iv) Immediately from (4.3.5) we obtain

m,N Tψm,N (X1 , . . . , Xn ) = Tϕm,N T (X , . . . , X ), X , . . . , X 1 k k+1 n . ϕ1 2 (4.3.11) Passing to the limit in (4.3.11) and applying (4.3.6) proves the claim.

 

The proof of Proposition 4.3.7 is an example of a general approach to deriving results on multiple operator integrals given by Definition 4.3.3: prove results for an operator A whose spectrum is a finite subset of Z (by working with a finite sum Tϕm,N defined in (4.3.5)) and then transfer the results to an operator A with an arbitrary spectrum by approximations. It is also convenient to establish results for a multiple operator integral associated with A0 = · · · = An and then transfer the results to a multiple operator integral associated with distinct A0 , . . . , An by adopting a trick with tensor products illustrated in Proposition 4.1.8. This approach is utilized in [163, 164, 191] and demonstrated in the proof of Theorem 4.3.10 below.

Norm Estimates Immediately from Definition 4.3.1 and Hölder’s inequality, we have the following estimate for a multiple operator integral.

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4 Multiple Operator Integrals

Theorem 4.3.8 Let n ∈ N, A1 , . . . , An+1 ∈ Dsa , ϕ ∈ An , and 0  p1 = 1 pn

 1. Then,

1 p1

    A1 ,...,An+1 (X1 , . . . , Xn )  ϕAn X1 p1 . . . Xn pn Tϕ p

+···+

(4.3.12)

for all Xj ∈ Spj (or Xj ∈ B(H) if pj = ∞), j = 1, . . . , n. An estimate with a norm of the symbol ϕ smaller than ϕAn is obtained in [163] for the multiple operator integral given by (4.3.3) via a subtle analysis. We state the result and outline the scheme of its proof below. The largest set of symbols for which the aforementioned result holds is described via polynomial integral momenta introduced in [163]. Let Sn be the simplex   n  sj = 1, sj  0, j = 0, . . . , n Sn = (s0 , . . . , sn ) ∈ Rn+1 :

(4.3.13)

j =0

equipped with the finite measure dσn defined by  n−1   φ s0 , . . . , sn−1 , 1 − sj dvn .



 φ(s0 , . . . , sn ) dσn = Sn

Rn

j =0

for every continuous function φ : Rn+1 → C, where  Rn = (s0 , . . . , sn−1 ) ∈ R : n

n−1 

 sj  1, sj  0, j = 0, . . . , n − 1

j =0

and dvn is the Lebesgue measure on Rn . Let Pn denote the set of polynomials with real coefficients of n + 1 variables. Given h ∈ Cb (R) and ℘ ∈ Pn , we call the function φn,h,℘ defined by   n ℘ (s0 , . . . , sn ) h sj λj dσn

 φn,h,℘ (λ0 , . . . , λn ) = Sn

(4.3.14)

j =0

a polynomial integral momentum. The function φ is continuous because h is. The divided difference is an example of a polynomial integral momentum, as it follows from standard properties of divided differences (see, e.g., [163, Lemma 5.1], where a detailed comparison of these definitions is presented). Proposition 4.3.9 For n ∈ N, f ∈ C n (R), f [n] = φn,f (n) ,1 , where φn,f (n) ,1 is given by (4.3.14).

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

85

The following estimate is established in [163, Theorem 5.6]. Theorem 4.3.10 Let n ∈ N, ℘ ∈ Pn , h ∈ Cb (R), and let φn,h,℘ be the polynomial integral momentum defined by (4.3.14). Let 1 < p, pj < ∞, j = 1, . . . , n, be such 0 ,...,An be that 0 < p1 = p11 + · · · + p1n < 1. Let Aj ∈ Dsa , j = 0, . . . , n and let TφAn,h,℘ the transformation given by Definition 4.3.3. Then, Tφn,h,℘ ∈ Bn (Sp1 × · · · × Spn , Sp ) and there exists c℘,p1 ,...,pn > 0 such that 0 ,...,An   c℘,p1 ,...,pn h∞ . TφAn,h,℘

(4.3.15)

Proof of Theorem 4.3.10 The result is proved by induction on n. Assume first that A0 = · · · = An =: A and σ (A) is a finite subset of Z. In this case Tφn,h,℘ := TφA,...,A is a finite sum. n,h,℘ (1) Base of induction: n = 1. Let 2 < p1 < ∞ be sufficiently large. It is proved in [163, Theorem 5.6] (see also [163, Lemma 4.6]) that if 2 < q < p1 satisfies p11 + q1 = 12 , then there exists c˜℘,p1 > 0 such that      p1 Tφ → Sp1   c˜℘,p1 1 + Tφ1,h,℘ : Sq → Sq  . 1,h,℘ : S

(4.3.16)

We present two key technical ideas of the proof, which are also used in the proof of the induction step below. They consist in decomposing the whole multiple operator integral into simpler components that can be handled by means of harmonic analysis. Note that  Tφ (X) = Tφ (Tu (X)) + Tφ (Xd ) + Tφ (Tl (X)) = φ(λ0 , λ1 )E({λ0 })XE({λ1 }) λ0 λ1

where Tu and Tl are respectively upper and lower triangular truncations, which are bounded operators on Sp , 1 < p < ∞ [85, 91], and Xd is the part of X diagonal with respect to the spectral measure E. Thus, estimating the whole multiple operator integral can be reduced to estimating it on truncated perturbations.

86

4 Multiple Operator Integrals

It is proved in [159, Lemma 6] using Fourier inversion that there exists a function g : R → C satisfying  R

|s|n |g(s)| ds < ∞, n ∈ N ∪ {0},

and 1 l1 − l2 = √ l1 − l0 2π

 R

g(s)(l1 − l2 )is (l1 − l0 )−is ds

(4.3.17)

for |l1 − l2 |  1. |l1 − l0 | Define Xs,1 =

 l0 0 such that   p1 Tφ → Sp1   c℘,p1 h∞ , 1,h,℘ : S By duality and Proposition 4.3.7(i) and then by interpolation, the latter bound implies (4.3.15) with n = 1 for all values of p1 ∈ (1, ∞).

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

87

(2) Induction step: show that Tφn−1,h,℘   c℘,p1 ,...,pn−1 h∞ implies (4.3.15). The proof involves reduction of order of the polynomial integral momentum, which exists only for a certain subset of the variables (λ0 , . . . , λn ) considered below. Let λ0  λ2  λ1 with λ0 = λ1 . One of the key technical steps in the proof is the decomposition  t  κ t m dt s k h(κλ2 + (λ0 − λ2 )t + (λ1 − λ0 )s) ds 0

0

 λ0 − λ2 = ℘˜ 1 , κ, θ h(κλ2 + (λ0 − λ2 )θ ) dθ λ0 − λ1 0    κ λ0 − λ2 ℘˜ 2 , κ, σ h(κλ2 + (λ1 − λ2 )σ ) dσ + λ0 − λ1 0 

κ



(4.3.21)

derived in [163, Lemma 5.9] for κ > 0, where ℘˜ 1 and ℘˜ 2 are the polynomials depending on m and k but not on h and given by  κ (t − θ )k t m dt, ℘˜ 1 (ζ, κ, θ ) = ζ k+1 θ



℘˜ 2 (ζ, κ, σ ) = (1 − ζ )

κ

 k (1 − ζ )σ + ζ t t m dt.

σ

The decomposition (4.3.21) is used in [163, Lemma 5.8] to obtain the following order reduction of the polynomial integral momentum:   λ0 − λ2 φn,h,℘ (λ0 , λ1 , λ2 , λ3 , . . . , λn ) = ψn−1,h,℘1 , λ0 , λ2 , λ3 , . . . , λn λ0 − λ1   λ0 − λ2 +ψn−1,h,℘2 , λ1 , λ2 , λ3 , . . . , λn , λ0 − λ1 (4.3.22) where ℘1 , ℘2 ∈ Pn depend only on ℘ and ψn−1,h,℘i (λ, λi−1 , λ2 , . . . , λn )  = Sn−1

  n−1 ℘i (λ, s0 , . . . , sn−1 ) h sj λj dσn−1 , i = 1, 2, j =0

with Sn defined in (4.3.13). The polynomial ℘i (ζ, s0 , . . . , sn−1 ), i = 1, 2, is a linear combination of the polynomials ℘(ζ, ˘ s0 , . . . , sn−1 ) = (1 − ζ )m ℘˘ r (s0 , . . . , sn−1 ), where ℘˘ r ∈ Pn−1 and m ∈ N ∪ {0}.

88

4 Multiple Operator Integrals

Consider the transformations  (0,2,1) Tφn,h,℘ (X1 , . . . , Xn ) :=

φn,h,℘ (l0 , . . . , ln )El0 X1 El1 X2 . . . Xn Eln ,

l0  l2 lj +1 ,

(4.3.26)

where the decrement and increment of the indices j − 1 and j + 1 are understood modulo n, that is, if j = 0, then j − 1 = n and if j = n, then j + 1 = 0. By fixing j we further split K into subspaces K ,i , i = 0, 1, where K ,0 = {(l0 , . . . , ln ) ∈ K : lj −1  lj +1 } and K ,1 = {(l0 , . . . , ln ) ∈ K : lj −1 > lj +1 }. The space Zn+1 \ D splits into the disjoint union of 2n+1 sets K ,i , ∈ {−1, 1}n , i = 0, 1. Hence,   ,i Tφn,h,℘ = TφDn,h,℘ + Tφn,h,℘ , (4.3.27) ∈{−1,1}n i=0,1

where TφDn,h,℘ =



φn,h,℘ (l0 , . . . , ln )El0 X1 . . . Xn Eln

(l0 ,...,ln )∈D

and Tφ ,i = n,h,℘

 (l0 ,...,ln )∈K ,i

φn,h,℘ (l0 , . . . , ln )El0 X1 . . . Xn Eln .

90

4 Multiple Operator Integrals

We fix ∈ {−1, 1}n and j ∈ {0, 1, . . . , n} as in (4.3.26) and obtain ! (l0 , . . . , ln ) ∈ K ,i ⇒

lj −1  lj +1 < lj

if i = 0

lj +1 < lj −1  lj

if i = 1.

If j = 1 and i = 0, then Tφ ,i = Tφ(0,2,1) , which boundedness is proved n,h,℘ n,h,℘ in (4.3.23). By shifting and reversing the enumeration of variables as in Proposi(0,2,1) tion 4.3.7(i) and (ii) we reduce estimating Tφ ,i to estimating Tφn,h,℘ . Therefore, n,h,℘ Tφ ,i   c℘,p1 ,...,pn h∞ . n,h,℘

(4.3.28)

We note that  TφDn,h,℘ (X1 , . . . , Xn ) =

℘ (s0 , . . . , sn ) dσn Sn



h(l)El

l∈σ (A)



Ek X1 . . . Xn Ek ,

k∈σ (A)

so TφDn,h,℘ (X1 , . . . , Xn )p

       c℘ h∞  Ek X1 . . . Xn Ek   . k∈σ (A)

(4.3.29)

p

Consider the family of unitaries Ut =



e2πilt El , t ∈ [0, 1].

l∈σ (A)

Due to the orthogonality of the trigonometric functions,  k∈σ (A)





1

Ek X1 . . . Xn Ek =

1 n

... 0

0 j =1

Ut∗j Xj Utj dt1 . . . dtn .

(4.3.30)

Applying the Hölder inequality in (4.3.30) and combining the outcome with (4.3.29) gives TφDn,h,℘   c℘,p1 ,...,pn h∞ .

(4.3.31)

Combining (4.3.27), (4.3.28), and (4.3.31) proves (4.3.15) under the additional assumption A0 = · · · = An = A and σ (A) is a finite subset of Z. A careful investigation of the proofs of [163, Lemmas 3.3 and 5.5] shows that their results also hold for distinct A0 , . . . , An . This immediately implies that to get (4.3.15) in the general case it suffices to prove (4.3.15) for A0 , . . . , An whose spectra are finite subsets of Z. Below we reduce the case of distinct A0 , . . . , An to the case of identical A0 = · · · = An that was handled above.

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

91

Assume that the spectra of A0 , . . . , An are finite subsets of Z. Let 0  k  n − 1 and assume that the (n−k) last self-adjoint operators Ak+1 , Ak+2 , . . . , An are equal. Let Eij denote the elementary 2 × 2 matrix whose nonzero entry has indices (i, j ). Let Xl = Xl ⊗ E22 for l = 1, . . . , k − 1, Xk = Xk ⊗ E21 , and Xl = Xl ⊗ E11 for l = k + 1, . . . , n. Then for every l = 0, . . . , k, let Al be the self-adjoint operator with the spectral measure EAk+1 ⊗ E11 + EAl ⊗ E22 and for every l = k + 1, . . . , n,  let Al = Ak . Consider X ∈ Sp , with p1 + p1 = 1, and let X = X ⊗ E12 . A straightforward calculation (see, e.g., the proof of [191, Theorem 3.3]) implies that   A0 ,...,An   A0 ,...,An (X1 , . . . , Xn )X = Tr Tφn,h,℘ (X1 , . . . , Xn )X . Tr Tφn,h,℘ Note that by construction, the (n − k + 1) self-adjoint operators Ak , Ak+1 , . . . , An are equal. Further, Xl p = Xl p for every l = 1, . . . , n and Xp = Xp . Using this process inductively for k = n − 1, n − 2, . . . , 0, we obtain that to prove (4.3.15), it suffices to have it when the self-adjoint operators are all equal. This concludes the proof of the theorem.   In view of its importance, we state a particular case of Theorem 4.3.10 for the polynomial integral momentum φn,f (n) ,1 = f [n] (see Proposition 4.3.9). Theorem 4.3.11 Let n ∈ N, f ∈ C n (R), f (n) ∈ Cb (R). Let 1 < p, pj < ∞, j = 1, . . . , n, be such that 0 < p1 = p11 +· · ·+ p1n < 1. Let Aj ∈ Dsa , j = 0, . . . , n 0 ,...,An and let TfA[n] be the transformation given by Definition 4.3.3. Then, 0 ,...,An TfA[n] ∈ Bn (Sp1 × · · · × Spn , Sp )

and there exists cp1 ,...,pn > 0 such that A ,...,An

0 Tf [n]

  cp1 ,...,pn f (n) ∞ .

Problem 4.3.12 Find an estimate for the constant c℘,p1 ,...,pn in (4.3.15). Perturbation Formula The following extension of (4.1.9) to the infinite dimensional case is obtained in [119, Lemma 3.10]. Theorem 4.3.13 Let 1 < p < ∞, n ∈ N, n  2, and f ∈ C n (R), f (n−1) , f (n) ∈ Cb (R). Let A1 , . . . , An−1 , A, B ∈ Dsa with B − A ∈ Sp , let X1 , . . . , Xn−1 ∈ Sp . Then, for every i = 1, . . . , n, A ,...,Ai−1 ,A,Ai ,...,An−1

1 Tf [n−1]

A ,...,Ai−1 ,A,B,Ai ,...,An−1

1 = Tf [n]

A ,...,Ai−1 ,B,Ai ,...,An−1

1 (X1 , . . . , Xn−1 ) − Tf [n−1]

(X1 , . . . , Xn−1 )

(X1 , . . . , Xi−1 , A − B, Xi , . . . , Xn−1 ).

92

4 Multiple Operator Integrals

The assumption of f (n) be continuous in Theorem 4.3.13 is eliminated in [52] for the transformation  given by Definition 4.2.1. Theorem 4.3.13 has an analog for perturbations in S1 (or B(H)) and a smaller set of functions f . Below we state the result of [119, Lemma 4.2] extending [146, Lemma 5.4] from the case n = 2 to a general n. n−1 n (R). Let Theorem 4.3.14 Let n ∈ N, n  2, and f ∈ B∞1 (R) ∩ B∞1 1 A1 , . . . , An−1 , A, B ∈ Bsa (H) with B − A ∈ S (or B(H)), let X1 , . . . , Xn−1 ∈ S1 (or B(H)). Then, for every i = 1, . . . , n, A ,...,Ai−1 ,A,Ai ,...,An−1

1 Tf [n−1]

A ,...,Ai−1 ,B,Ai ,...,An−1

1 (X1 , . . . , Xn−1 ) − Tf [n−1]

A ,...,Ai−1 ,A,B,Ai ,...,An−1

1 = Tf [n]

(X1 , . . . , Xn−1 )

(X1 , . . . , Xi−1 , A − B, Xi , . . . , Xn−1 ).

Continuity The following continuity properties of multiple operator integrals can be established completely analogously to the properties established in Proposition 3.3.9. Recall that the class Cn is defined in Sect. 4.3.2. Proposition 4.3.15 Let 1  p, p1 , . . . , pn  ∞ satisfy p1 = p11 + · · · + p1n . Let {Ai,m }∞ m=1 ⊂ Dsa converge to Ai ∈ Dsa , i = 1, . . . , n + 1, in the strong resolvent pi pi sense and let {Xi,m }∞ m=1 ⊂ S (B(H), respectively) converge to Xi ∈ S (B(H), respectively), i = 1, . . . , n. The following assertions hold for multiple operator integrals given by Definition 4.3.1. (i) Let ϕ ∈ Cn . If Xi ∈ Spi , 1  p < ∞, then A1,m ,...,An+1,m

lim Tϕ

m→∞

A1 ,...,An+1

(X1 , . . . , Xn ) − Tϕ

(X1 , . . . , Xn )p = 0; (4.3.32)

if Xi ∈ B(H), then A1,m ,...,An+1,m

sot- lim (Tϕ m→∞

A1 ,...,An+1

(X1 , . . . , Xn ) − Tϕ

(X1 , . . . , Xn )) = 0. (4.3.33)

Moreover, if {Ai,m }∞ m=1 converges to Ai , i = 1, . . . , n+1, in the operator norm, then the strong operator topology convergence in (4.3.33) can be replaced with the operator norm convergence. (ii) For every ϕ ∈ An , A1 ,...,An+1

lim Tϕ

m→∞

A1 ,...,An+1

(X1,m , . . . , Xn,m ) − Tϕ

(X1 , . . . , Xn )p = 0.

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

93

(iii) Let {ϕk }∞ k=1 ⊂ An converge to ϕ in the norm  · An . Then, A ,...,An+1

lim Tϕk 1

k→∞

A1 ,...,An+1

(X1 , . . . , Xn ) − Tϕ

(X1 , . . . , Xn )p = 0.

The result of Proposition 4.3.15(i) holds for a broader set of symbols ϕ when p = 2. The following fact is established in [55, Proposition 3.1]. Proposition 4.3.16 Let {Ai,m }∞ m=1 ⊂ Dsa converge to Ai ∈ Dsa , i = 1, . . . , n + 1, in the strong resolvent sense and let X1 , . . . , Xn ∈ S2 . Then, (4.3.32) with p = p1 = · · · = pn = 2 holds for every ϕ ∈ Cb (Rn+1 ). With help of Proposition 4.3.15 one can transfer results from simpler A1 , . . . , An+1 or X1 , . . . , Xn or else ϕ to a general case by approximations.

4.3.4 Nonself-adjoint Case By analogy with multiple operator integrals for self-adjoint operators defined in Sects. 4.3.1 and 4.3.2, one can define multiple operator integrals for unitary operators. The following properties of the transformations discussed in Sect. 3.3.5 extend to the unitary case: the algebraic properties, estimate (4.3.12) (see (4.3.34) below), perturbation formulas (3.3.7) and (3.3.8) (see [168, Lemma 2.4(i)]), continuity. A unitary analog of the estimate obtained in Theorem 3.3.4 is stated in Theorem 4.3.19 below. Immediately from the definition of a multiple operator integral for unitary operators and Hölder inequality, we have the following estimate. Theorem 4.3.17 Let A1 , . . . , An+1 be unitary operators, ϕ ∈ An , and let 0  p1 = 1 p1

+ ··· +

1 pn

 1. Then,

   A1 ,...,An+1  (X1 , . . . , Xn )  ϕAn X1 p1 . . . Xn pn . Tϕ p

(4.3.34)

for all Xj ∈ Spj (or Xj ∈ B(H) if pj = ∞), j = 1, . . . , n. It can be shown based on the results of [146] that    [n]  n (T) f   constn f B∞1 An

(4.3.35)

n (T). for f ∈ B∞1 We start with an adjustment of Definition 4.3.3 to the case of unitary operators, which is due to [164, Definition 2.5].

Definition 4.3.18 Let n, m ∈ N and let A0 , . . . , An be unitary operators. Denote zj,m := e2πij/m ,

(l)

Ej,m := EAl ([zj,m , zj +1,m )),

94

4 Multiple Operator Integrals

for j = 0, . . . , m − 1, l = 0, . . . , n. Let 1  α, αi  ∞ for i = 1, . . . , n be such that 1/α1 + · · · + 1/αn = 1/α. For ψ a bounded Borel function on Tn+1 , the mapping A ,...,An

: Sα1 × · · · × Sαn → Sα

Tψ 0 defined by A ,...,An

Tψ 0

(X1 , . . . , Xn ) m−1 

:= lim

m→∞

j0 ,...,jn =0

ψ(zj0 ,m , . . . , zjn ,m )Ej(0) X1 Ej(1) X2 . . . Xn Ej(n) , n ,m 0 ,m 1 ,m

provided the limit exists for all Xi ∈ Sαi , i = 1, . . . , n, is called a multiple operator integral with symbol ψ. Below we define algebraic analogs of polynomial integral momenta (4.3.14) considered in [191, Definition 2.3]. Given n ∈ N, l ∈ {1, . . . , n}, d ∈ {0, . . . , n − l}, consider the polynomial 

pl,d (t0 , . . . , tl−1 ) :=

d

l−1 ad,d0,...,dl−1 t0d0 . . . tl−1 .

d0 ,...,dl−1  0 d0 +···+dl−1 =d

and define the function ϕn−d,f (n) ,pl,d (λ0 , . . . , λl−1 , λl+d , . . . , λn ) on Tn−d+1 recursively as follows. For d0 ∈ {0, . . . , n − 1}, set ϕ

d

n−d0 ,f (n) ,t0 0

(λ0 , λd0 +1 , . . . , λn ) := d0 ! f [n] (λ0 , . . . , λ0 , λd0 +1 , . . . , λn ); " #$ % d0 +1

for d0 , . . . , dl−1  0 satisfying d0 + · · · + dl−1 = d, set ϕ

d

d

l−1 n−d,f (n) ,t0 0 ... tl−1

(λ0 , . . . , λl−1 , λl+d , . . . , λn )

:= d0 ! . . . dl−1 ! f [n] (λ0 , . . . , λ0 , . . . , λl−1 , . . . , λl−1 , λl+d , . . . , λn ) " #$ % " #$ % d0 +1

+

 (i1 ,...,il−1 )∈Il

×ϕ

dl−1 +1

    dl−1 d−d0 −i1 −···−il−1 +1 d1 (−1) ... i1 il−1 d−i1 −···−il−1 i1 il−1 t1 ... tl−1

n−d,f (n) ,t0

(λ0 , . . . , λl−1 , λl+d , . . . , λn ),

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

95

  where Il = {0, . . . , d1 } × · · · × {0, . . . , dl−1 } \ {(d1 , . . . , dl−1 )}, and set ϕn−d,f (n) ,pl,d (λ0 , . . . , λl−1 , λl+d , . . . , λn )  := ad,d0 ,...,dl−1 ϕ (n) d0 n−d,f

d0 ,...,dl−1  0 d0 +···+dl−1 =d

d

l−1 ,t0 ... tl−1

(λ0 , . . . , λl−1 , λl+d , . . . , λn ).

The result of Theorem 4.3.19 below is obtained in [164, Theorems 2.8 and 2.17] for a polynomial f and extended to functions with derivatives representable by absolutely convergent Fourier series in [191, Theorem 3.3]. Theorem 4.3.19 Let n ∈ N, let 1 < p, pj < ∞, j = 1, . . . , n, satisfy 0
0 such that  A0 ,...,An  T [n]   cp,n f (n) ∞ . f Remark 4.3.20 The bound of Theorem 4.3.19 extends to the bound   A ,...,A n  0 (f [n] ) : Sp1 × · · · × Spn → Sp   cp,n f (n) ∞ for all functions f ∈ C n (T), where  A0 ,...,An is the extension of the transformation given by Definition 4.2.1 from the set (S2 ∩Sp1 )×· · ·×(S2 ∩Spn ) to Sp1 ×· · ·×Spn , by adjusting the methods of [52]. Proof of Theorem 4.3.19 The strategy of the proof is the same as for Theorem 4.3.10, but the technical realization is more subtle than in the self-adjoint case. We briefly outline the technical distinctions below. Similarly to the self-adjoint case it suffices to prove the result for A 0 = · · · = An = A with spectrum contained in e2πij/m : j = 0, . . . , m − 1 with fixed m ∈ N. Denote Tf [n] := TfA,...,A , Ej := EA ([zj,m , zj +1,m )), [n]

96

4 Multiple Operator Integrals

where zj,m = e2πij/m . Given a Borel subset B ⊂ Tn+1 and a Borel function φ on Tn+1 , denote  φ(zj0 , . . . , zjn )Ej0 X1 Ej1 X2 . . . Xn Ejn , TφB (X1 , . . . , Xn ) := (zj0 ,...,zjn )∈B

for every (X1 , . . . , Xn ) ∈ Sp1 × · · · × Spn . By a modification of [159, Lemma 6], given δ > 0, there exists a function gδ : R → C such that  |s|n |gδ (s)| ds < ∞, n ∈ N ∪ {0}, R

and λ2 − λ1 1 =√ λ1 − λ0 2π



|λ1 − λ0 | λ1 − λ0



λ2 − λ1 |λ2 − λ1 |



 R

gδ (s)

|λ2 − λ1 |is |λ1 − λ0 |is

 ds,

whenever |λ2 − λ1 |  δ, |λ1 − λ0 | which replaces (4.3.17) in the unitary case. The following analogs of (4.3.18)  2πij/m are established in [110, Theorem 3.4]. Let B, C be subsets of e : j =  0, . . . , m − 1 , let r ∈ N, and denote 

ϒr (X) :=



z∈B,w∈C

ϒ−r (X) :=



 z∈B,w∈C

s (X) :=



z−w |z − w|

r

|z − w| z−w

E({z}) X E({w}) r E({z}) X E({w})

|z − w|is E({z}) X E({w}).

z∈B,w∈C

Then, there are constants cα , cα,r > 0 such that ϒr (X)α  cα,r Xα , ϒ−r (X)α  cα,r Xα , s (X)α  cα (1 + |s| + |s|2 )Xα .

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

97

If f is analytic on the closed unit desk, then the decomposition (4.3.21) extends from the portion of the simplex on the real line determined by 0  λ  ξ  μ  1 to all λ, ξ, μ ∈ T with λ = μ by uniqueness of meromorphic functions. Subsequently, we obtain (4.3.22) for λ, ξ, μ ∈ T with λ = μ. For a general function f ∈ ∞  C n (T) satisfying |j |n |F f (j )| < ∞, the representation (4.3.22) is replaced j =−∞

with algebraic counterparts given below. The algebraic approach is taken to avoid integration in (4.3.14) over a region containing the point 0, for which a trigonometric polynomial with negative powers of the variable is undefined. Let n ∈ N, l ∈ {1, . . . , n}, d ∈ {0, . . . , n − l}. Let pl,d be a polynomial of l variables  dl−1 ad,d0,...,dl−1 t0d0 . . . tl−1 . pl,d (t0 , . . . , tl−1 ) := d0 ,...,dl−1  0 d0 +···+dl−1 =d

If n = 1, then  ξ −μ n ϕ1,f (n) ,t d (ξ, μ) 0 λ−μ       n−1  n−1 λ − ξ j +1 ξ − μ n−1−j + ϕ1,f (n) ,t d (λ, ξ ) 0 λ−μ λ−μ j 

ϕ1,f (n) ,t d (λ, μ) = 0

j =0

thanks to [191, Lemma 2.10]. If n  2, then it is established in [191, Lemma 2.7] that there exist polynomials ql,d,t0,...,tl−1 (t0 , . . . , tl−1 )=



k

l−1 bpl,d ,k1 ,...,kl−1 q˜pl,d ,k1 ,...,kl−1 (t0 ) t1k1 . . . tl−1

k1 ,...,kl−1  0 k1 +···+kl−1 =d+1

and rl,d,t0,...,tl−1 (t0 , . . . , tl−1 )=



k

l−1 cpl,d ,k1 ,...,kl−1 r˜pl,d ,k1 ,...,kl−1 (t0 ) t1k1 . . . tl−1 ,

k1 ,...,kl−1  0 k1 +···+kl−1 =d+1

98

4 Multiple Operator Integrals

where q˜pl,d ,k1 ,...,kl−1 and r˜pl,d ,k1 ,...,kl−1 are polynomials of one variable, such that for all λ0 , . . . , λl−1 , λl+d , . . . , λn ∈ T with λ0 = λ1 , (4.3.36)

ϕn−d,f (n) ,pl,d (λ0 , . . . , λl−1 , λl+d , . . . , λn ) 

=



bpl,d ,k1 ,...,kl−1 q˜pl,d ,k1 ,...,kl−1

k1 ,...,kl−1  0 k1 +···+kl−1 =d+1

×ϕ

k



+

k

l−1 n−d−d1 −1,f (n) ,t1 1 ...tl−1

(λ0 , λ2 , . . . , λl−1 , λl+d , . . . , λn ) 

cpl,d ,k1 ,...,kl−1 r˜pl,d ,k1 ,...,kl−1

k1 ,...,kl−1  0 k1 +···+kl−1 =d+1

×ϕ

k

k

l−1 n−d−d0 −1,f (n) ,t1 1 ...tl−1

λ0 − λ2 λ0 − λ1



λ0 − λ2 λ0 − λ1



(λ1 , λ2 , . . . , λl−1 , λl+d , . . . , λn ).

The analog of (4.3.27) requires more care in the unitary case. Denote Q(n) k

 

2πk 2π(k + 1) , , := z ∈ T : arg(z) ∈ n+2 n+2

k = 0, . . . , n + 1,

where arg(z) denotes the principal value of the argument of the complex number z. By additivity of the multiple operator integral, Tϕ =



(n) 0

(n) 1

(n)

Qk ×Qk ×···×Qkn

Tϕ

.

k0 ,k1 ,...,kn ∈{0,1,...,n+1} (n) (n) There are two principal types of the set Q(n) k0 × Qk1 × · · · × Qkn . One is when there exists an index i ∈ {0, 1, . . . , n} such that1 |ki+1 −ki |  2 and, hence, |z−w|  cn > (n) 0 for z ∈ Q(n) ki+1 , w ∈ Qki , and the other is when |ki+1 − ki |  1 for all i. In the (n)

latter case,2 there is a ∈ (0, π] such that arg(z) ⊆ [a, a + π] whenever z ∈ Qki , for each i. Thus, in this case we have the inequality |zj1 − zj0 | > |zj2 − zj1 | whenever (n) (n) zj0 , zj1 , zj2 ∈ Q(n) k0 ∪ Qk1 ∪ · · · ∪ Qkn and j0  j2 < j1 . The subcase k0 = k1 = (n) · · · = kn is treated by decomposing the off-diagonal part of Q(n) 0 × · · · × Q0 into the disjoint union of 2n+1 sets K ,i , ∈ {−1, 1}n , i = 0, 1, analogously to how it

1 Here the increment and decrement of the index i is understood modulo n, that is, if i = n, then i + 1 = 0 and if i = 0, then i − 1 = n.! i  n2 2 If n is even, then a typical example is ki = i, ki = n − i + 1, i > n2 .

4.3 Multiple Operator Integrals on Schatten Classes and B(H)

99

was done in the proof of Theorem 4.3.15. The other subcases are treated by a similar analysis.   We note that multiple operator integrals analogous to those in Definition 4.3.1 were introduced for contractions in [147] and maximal dissipative operators in [6].

4.3.5 Change of Variables In this section we give an example of the change of operator variables in the multiple operator integral. The respective transition from unbounded self-adjoint operators with non-Schatten-class difference to unitary operators with Schattenclass difference is applied in perturbation theory, for instance, in the proof of Theorem 5.5.5. The following result is established in [166, Theorem 3.2] and [191, Theorem 5.2]. Theorem 4.3.21 Let n ≥ 2. Let H ∈ Dsa and V ∈ Bsa (H). Assume the condition (H + V − iI )−1 − (H − iI )−1 ∈ Sn . Define the unitary operators U0 := (H + iI )(H − iI )−1 and U1 := (H + V + iI )(H + V − iI )−1 . If ∞ * )  |F f (j )||j |n < ∞ ϕ ∈ f ∈ C n (T) : j =−∞

and  ψ(λ) = ϕ

λ+i λ−i

 , λ ∈ R,

then U ,...,U0 Tϕ [n]0 (U1 −U0 , . . . , U1 −U0 )

=

n 



k=1 0=i0 0 such that the multiple operator integral given by Definition 4.3.1 satisfies     A1 ,...,An+1 (X1 , . . . , Xn )  cI ϕAn X1 I . . . Xn I Tϕ I

for all Xj ∈ I, j = 1, . . . , n. Below we inherit the notation of Theorem 4.3.10 and state the result of Theorem [163, Theorem 5.3] in the full generality. Theorem 4.4.7 Let n ∈ N, ℘ ∈ Pn , h ∈ Cb (R), and let φn,h,℘ be the polynomial integral momentum defined by (4.3.14). Let 1 < p, pj < ∞, j = 1, . . . , n, be 0 ,...,An be the such that p1 = p11 + · · · + p1n . Let Aj ηM, j = 0, . . . , n, and let TφAn,h,℘ transformation given by Definition 4.4.1. Then, A ,...,An

0 Tφn,h,℘

∈ Bn (Lp1 (M, τ ) × · · · × Lpn (M, τ ), Lp (M, τ ))

and there exists c℘,p1 ,...,pn > 0 such that    A0 ,...,An  Tφn,h,℘   c℘,p1 ,...,pn h∞ .

Perturbation Formula Denote the polynomial integral momentum ϕn,h,1 defined in (4.3.14) by ϕn,h , that is,   n h sj λj dσn .

 ϕn,h (λ0 , . . . , λn ) = Sn

(4.4.4)

j =0

The following algebraic property extends [161, Theorem 11] to the case of τ measurable operators; it is proved in [170, Theorem 28]. Theorem 4.4.8 Let k ∈ N, k + 1 < r < q < ∞. Let A, B ∈ (Lq + Lr )(M, τ ) and A1 , . . . , Ak be self-adjoint operators. Let h ∈ Cb (R) be such that h ∈ Cb (R). Then 1 ,...,Ak (X , . . . , X ) − T B,A1 ,...,Ak (X , . . . , X ) TϕA,A 1 k 1 k ϕk,h k,h 1 ,...,Ak (A − B, X1 , . . . , Xk ) = TϕA,B,A k+1,h

holds for all X1 , . . . , Xk ∈ (Lq + Lr )(M, τ ), where ϕk,h , ϕk+1,h are given by (4.4.4). If, in addition, f is such that h = f (k) , then A,A1 ,...,Ak

Tf [k]

A,B,A1 ,...,Ak

= Tf [k+1]

B,A1 ,...,Ak

(X1 , . . . , Xk ) − Tf [k]

(A − B, X1 , . . . , Xk ).

(X1 , . . . , Xk )

106

4 Multiple Operator Integrals

Proof In the proof we will frequently use [51, Proposition 2.5], implying that if a sequence of projections {Pn }∞ n=1 ⊂ M decreases to 0 in the strong operator topology, M is atomless, and X ∈ Lq (M, τ ) + Lr (M, τ ), then XPn Lq +Lr → 0. By considering N ⊗ L∞ (0, 1) instead of N, we can assume without loss of generality that N is atomless. By Theorem 4.4.7 and Lemma 2.6.2, q

r

1 ,...,Ak , T B,A1 ,...,Ak ∈ B ((Lq + Lr )(M, τ )×k , (L k + L k )(M, τ )) TϕA,A k ϕk,h k,h

(4.4.5)

and q

r

q r ×(k+1) 1 ,...,Ak ∈ B , (L k+1 + L k+1 )(M, τ )). TϕA,B,A k+1 ((L + L )(M, τ ) k+1,h

(4.4.6)

Denote ψ0 (x0 , . . . , xk+1 ) := x0 ϕk+1,h (x0 , . . . , xk+1 ), ψ1 (x0 , . . . , xk+1 ) := x1 ϕk+1,h (x0 , . . . , xk+1 ). Let Q be a projection such that τ (Q) < ∞. Since A, B ∈ (Lq + Lr )(M, τ ), it follows that AQ, QB, AQ − QB ∈ (Lq + Lr )(M, τ ). Let A(m) = AEA ([−m, m]) and B (m) = BEB ([−m, m]). By multilinearity of the multiple operator integral, property (4.4.1), and Lemma 4.4.4, (m)

TϕAk+1,h,B

(m) ,A ,...,A 1 k

(m)

= TϕAk+1,h,B

((A(m) Q − QB (m) ), X1 , . . . , Xk )

(m) ,A ,...,A 1 k

(m)

− TϕAk+1,h,B

(A(m) Q, X1 , . . . , Xk )

(m) ,A ,...,A 1 k

(m) ,B (m) ,A ,...,A 1 k

= TψA0

A(m) ,B (m) ,A1 ,...,Ak

= Tψ0 −ψ1

(QB (m) , X1 , . . . , Xk ) (m) ,B (m) ,A ,...,A 1 k

(Q, X1 , . . . , Xk ) − TψA1 (Q, X1 , . . . , Xk ).

(Q, X1 , . . . , Xk ) (4.4.7)

By Proposition 4.3.9, ϕk+1,h (x0 , . . . , xk+1 ) = f [k+1] (x0 , . . . , xk+1 ) [1] = (f [k] )[1] (x0 , . . . , xk+1 ) = ϕk,h (x0 , . . . , xk+1 ).

Hence, (ψ0 − ψ1 )(x0 , x1 , . . . , xk+1 ) = ϕk,h (x0 , x2 , . . . , xk+1 ) − ϕk,h (x1 , x2 , . . . , xk+1 ).

4.4 Multiple Operator Integrals on Noncommutative and Weak Lp -Spaces

107

Therefore, by Lemma 4.4.5, (m)

TψA0 −ψ,B1 =T

(m) ,A ,...,A 1 k

A(m) ,A1 ,...,Ak ϕk,h

(Q, X1 , . . . , Xk ) (m) ,A ,...,A 1 k

(QX1 , . . . , Xk ) − QTϕBk,h

(X1 , . . . , Xk ).

(4.4.8)

Combining (4.4.7) and (4.4.8) implies (m)

TϕAk+1,h,B

(m) ,A ,...,A 1 k

(m) ,A ,...,A 1 k

= TϕAk,h

((A(m) Q − QB (m) ), X1 , . . . , Xk ) (m) ,A ,...,A 1 k

(QX1 , . . . , Xk ) − QTϕBk,h

(X1 , . . . , Xk ).

(4.4.9)

By Definition 4.4.1, (m)

TϕAk+1,h,B

(m) ,A ,...,A 1 k

((A(m) Q − QB (m) ), X1 , . . . , Xk )

1 ,...,Ak (E ([−m, m])(A(m) Q − QB (m) )E ([−m, m]), X , . . . , X ). = TϕA,B,A A B 1 k k+1,h

Clearly, EA ([−m, m])(A(m)Q − QB (m) )EB ([−m, m]) = EA ([−m, m])(AQ − QB)EB ([−m, m]) → AQ − QB in (Lq + Lr )(M, τ ) as m → ∞. Hence, by (4.4.6), (m)

TϕAk+1,h,B

(m) ,A ,...,A 1 k

((A(m) Q − QB (m) ), X1 , . . . , Xk )

1 ,...,Ak ((AQ − QB), X1 , . . . , Xk ) → TϕA,B,A k+1,h q

(4.4.10)

r

in (L k+1 + L k+1 )(M, τ ) as m → ∞. We also have (m) ,A ,...,A 1 k

TϕAk,h

1 ,...,Ak (QX1 , . . . , Xk ) = EA ([−m, m])TϕA,A (QX1 , . . . , Xk ). k,h

Hence, by (4.4.5), (m) ,A ,...,A 1 k

TϕAk,h q

1 ,...,Ak (QX , . . . , X ) (QX1 , . . . , Xk ) → TϕA,A 1 k k,h

(4.4.11)

r

in (L k + L k )(M, τ ) as m → ∞. Similarly, (m) ,A ,...,A 1 k

QTϕBk,h

1 ,...,Ak (X , . . . , X ) (X1 , . . . , Xk ) → QTϕB,A 1 k k,h

(4.4.12)

108

4 Multiple Operator Integrals q

r

in (L k + L k )(M, τ ) as m → ∞. Combining (4.4.10)–(4.4.12) and (4.4.9) gives 1 ,...,Ak ((AQ − QB), X , . . . , X ) TϕA,B,A 1 k k+1,h 1 ,...,Ak (QX , . . . , X ) − QT B,A1 ,...,Ak (X , . . . , X ). = TϕA,A 1 k 1 k ϕk,h k,h

(4.4.13)

Since τ is a semifinite trace, it follows that there exists a sequence {Qn }n∈N of projections satisfying Qn ↑ I and τ (Qn ) < ∞. By (4.4.13), 1 ,...,Ak ((AQ − Q B), X , . . . , X ) TϕA,B,A n n 1 k k+1,h 1 ,...,Ak (Q X , . . . , X ) − Q T B,A1 ,...,Ak (X , . . . , X ), n ∈ N. = TϕA,A n 1 k n ϕk,h 1 k k,h (4.4.14)

Since AQn − Qn B → A − B as n → ∞, in (Lq + Lr )(M, τ ), it follows that 1 ,...,Ak ((AQ −Q B), X , . . . , X ) → T A,B,A1 ,...,Ak ((A −B), X , . . . , X ) TϕA,B,A n n 1 k 1 k ϕk+1,h k+1,h q

r

in (L k+1 +L k+1 )(M, τ ), and so also with respect to the measure topology. Similarly, since Qn X1 → X1 in (Lq + Lr )(M, τ ) and 1 ,...,Ak (X , . . . , X ) → T B,A1 ,...,Ak (X , . . . , X ) Qn TϕB,A 1 k 1 k ϕk,h k,h q

r

in (L k + L k )(M, τ ), it follows that 1 ,...,Ak 1 ,...,Ak TϕA,A (Qn X1 , . . . , Xk ) − Qn TϕB,A (X1 , . . . , Xk ) k,h k,h 1 ,...,Ak (X , . . . , X ) − T B,A1 ,...,Ak (X , . . . , X ) → TϕA,A 1 k 1 k ϕk,h k,h q

r

in (L k + L k )(M, τ ), and so also with respect to the measure topology. Taking the limit in (4.4.14) with respect to the measure topology we complete the proof of the theorem.  

Hölder-Type Estimates The main result of this section is the Hölder-type estimate given in Theorem 4.4.11, which is established in [170, Theorem 34] and extends [161, Corollary 13 and Theorem 14]. In the proof of Theorem 4.4.11, the interpolation argument of [161, Theorem 14] is replaced with a different technique based on the Calderón-type operator Pq,r defined in (4.4.15) below. Such operators provide a useful technical tool in many questions of interpolation theory (see, e.g., [28, Chapter 3, Section 5]).

4.4 Multiple Operator Integrals on Noncommutative and Weak Lp -Spaces

109

For 1 < q, r < ∞ and X ∈ S(M, τ ), consider the operator Pq,r : S(M, τ ) → S(0, ∞), given by (Pq,r (X))(t) :=

1  t

t 0

μrs (X) ds

1/r

+

1  t

∞

1/q

q

μs (X) ds

,

t > 0.

t

(4.4.15) Observe that if X ∈ (Lr + Lq )(M, τ ), r < q, then the value Pq,r (X)(t) is a finite number for all t > 0. Observe also that there is t > 0 such that Pq,r (X)(t) = 0 if and only if X = 0. Lemma 4.4.9 If 1 < r < q < ∞, then for p ∈ (r, q) there exists a constant c(p, q, r) > 0 such that Pq,r (X)Lp,∞  c(p, q, r)XLp,∞ , X ∈ Lp,∞ (M, τ ). Proof Let X ∈ Lp,∞ (M, τ ), t > 0. Then, (Pq,r (X))(t) =

1  t

t

0

μrs (X)ds

1 p

 sup s μs (X) · s>0

= XLp,∞

+

1 

1  t

1/r

t t

s

1 

t

s

∞

t

t

− pr

ds

1/r

0 − pr

1/r ds

1/q

q

μs (X)ds

+

+

1 

1  t

∞

s

− pq

1/q ds

t

∞

s

t t 0 1

p q1 p r −1 + . = t p XLp,∞ p−r q −p

− pq

1/q ds

 

Appealing to (2.6.1) completes the proof.

The main technical tool in the proof of Theorem 4.4.11 is the estimate obtained in the next result due to [170, Theorem 33]. Let σu , u ∈ (0, ∞), denote the dilation operator s . σu (f )(s) = f u We note that σu is a bounded linear operator on the Banach space Lp,∞ (0, ∞) for 1 < p < ∞ with the norm 1

σu Lp,∞ →Lp,∞ = u p .

(4.4.16)

Theorem 4.4.10 Let k ∈ N and let k + 1 < r < q < ∞. Then, there exists a constant C(k, q, r) > 0 such that for a pair of self-adjoint operators A, B ∈ (Lr + Lq )(M, τ ), for all self-adjoint operators A1 , . . . , Ak affiliated with M and

110

4 Multiple Operator Integrals

every compactly supported function h ∈ α , α ∈ [0, 1],   1 ,...,Ak 1 ,...,Ak μ TϕA,A (X1 , . . . , Xk ) − TϕB,A (X1 , . . . , Xk ) k,h k,h k

 C(k, q, r)hα σ3 (Pr,q (A − B))α

σ3 (Pr,q (Xj )) j =1

holds for all X1 , . . . , Xk ∈ (Lr + Lq )(M, τ ), where ϕk,h is given by (4.4.4). The proof of Theorem 4.4.10 involves Theorems 4.4.7, 4.4.8, and Lemma 2.6.2. We refer the reader for details to [170, Theorem 33]. The following Hölder-type estimate for multiple operator integrals is due to [170, Theorem 35]. Theorem 4.4.11 Let k ∈ N and let k + 1 < p < ∞. Then, there exists a constant C(k, p) > 0 such that for every compactly supported function h ∈ α , α ∈ [0, 1], and all self-adjoint operators A, B ∈ Lp,∞ (M, τ )  A,A ,...,A  1 k T (X1 , . . . , Xk ) − T B,A1 ,...,Ak (X1 , . . . , Xk ) ϕk,h

ϕk,h

 C(k, p) · hα A − BαLp,∞

p

L k+α



k

  Xj 

,∞

Lp,∞

j =1

holds for all X1 , . . . , Xk ∈ Lp,∞ (M, τ ), where ϕk,h is given by (4.4.4). Proof Let k ∈ N, k + 1 < p < ∞ and α ∈ [0, 1] be fixed. Observe that A, B, X1 , . . . , Xk ∈ (Lr + Lq )(M, τ ) for r = 12 (p + k + 1) and q = 2p. Denote for brevity 1 ,...,Ak 1 ,...,Ak D := TϕA,A (X1 , . . . , Xk ) − TϕB,A (X1 , . . . , Xk ). k,h k,h

By Theorem 4.4.10, k

μ(D)  C(k, q, r)hα · σ3 (Pr,q (A − B))α ·

(4.4.17)

σ3 (Pr,q (Xj )). j =1

Taking the norm  ·  p ,∞ on both sides of (4.4.17) and applying Lemma 2.6.1(ii) L k+α gives D

p ,∞ L k+α

  , C(k, q, r) C  (k, p) hα σ3 (Pr,q (A − B))α  k

· j =1

  σ3 (Pr,q (Xj )) p,∞ , L

p

L α ,∞

4.4 Multiple Operator Integrals on Noncommutative and Weak Lp -Spaces

 p  and c2 (m, q) = max c2 k + 1, k+α

where C  (k, p) =

0α 1

111 1 q q q−1 m .

Recall-

ing (4.4.16), we infer that D

p ,∞ L k+α

k+α p

3

  C(k, q, r)C  (k, p)hα (Pr,q (A − B))α    Pr,q (Xj )

k

·

Lp,∞

j =1 k+1 p

3

p

L α ,∞

α  C(k, q, r)C  (k, p)hα Pr,q (A − B)Lp,∞   Pr,q (Xj )

k

· j =1

Lp,∞

.

Hence, by Lemma 4.4.9, D

p

L k+α

,∞

3

k+1 p

C(k, q, r)C  (k, p)c(p, q, r)k+α hα A − BαLp,∞

k

·

Xj Lp,∞ j =1

 C  (k, p)hα A − BαLp,∞ ·

k

Xj Lp,∞ , j =1

where C  (k, p) = 3

k+1 p

C(k, q, r)C  (k, p) max c(p, q, r)k+α .

 

0α1

The following assertion due to [170, Theorem 35] generalizes Theorem 4.4.11 and also extends the result of [4, Theorem 5.8] (see Sect. 5.2) for Schatten class perturbations to a multilinear setting. Its proof is completely analogous to the proof of Theorem 4.4.11 and, therefore, it is omitted. Theorem 4.4.12 Let α ∈ [0, 1], k ∈ N, and let 1 < p, pj < ∞, j = 0, . . . , k be such that p1 = pα0 + p11 + · · · + p1k . Then, there are constants c(k, p1 , . . . , pk ), c(k, ˜ p1 , . . . , pk ) > 0 such that for every compactly supported function h ∈ α and (i) for all A = A∗ , B = B ∗ ∈ Lp0 (M, τ ), Xj ∈ Lpj (M, τ ), j = 1, . . . , k,    A,A1 ,...,Ak  1 ,...,Ak (X1 , . . . , Xk ) − TϕB,A (X , . . . , X ) Tϕk,h 1 k  k,h  c(k, p1 , . . . , pk ) · hα A − BαLp0



k j =1

Lp

  Xj  pj ; L

112

4 Multiple Operator Integrals

(ii) for all A = A∗ , B = B ∗ ∈ Lp0 ,∞ (M, τ ), Xj ∈ Lpj ,∞ (M, τ ), j = 1, . . . , k,    A,A1 ,...,Ak 1 ,...,Ak (X , . . . , X ) (X1 , . . . , Xk ) − TϕB,A Tϕk,h  1 k k,h

Lp,∞

 c(k, ˜ p1 , . . . , pk ) · hα A − BαLp0 ,∞



k

  Xj 

j =1

L

pj ,∞

,

where ϕk,h is given by (4.4.4). The following strong technical result is established in [170, Theorem 36] by utilizing a two-dimensional induction and multiple operator integration techniques, including Theorems 4.4.8 and 4.4.11. Theorem 4.4.13 Let m ∈ N, m  2 and p ∈ (m, m + 1]. Then, there exists a constant c(p) > 0 such that for every compactly supported function g on R satisfying g (j ) (0) = 0, j = 0, . . . , m − 1, and g (m−1) ∈ p−m ,  A0 ,...,Ak   T [k] g

p ,∞ (Lp,∞ )×k →L p−1

 c(p) · g (m−1) p−m

k  j =0

holds for all self-adjoint elements A0 . . . , Ak ∈ Lp,∞ (M, τ ).

Aj Lp,∞

p−k−1

Chapter 5

Applications

In this chapter we discuss various results of operator theory, functional analysis, mathematical physics, and noncommutative geometry that rely on methods of multiple operator integration.

5.1 Operator Lipschitz Functions The study of operator Lipschitz functions emerged from the now answered question of M. G. Krein [115]. A detailed exposition on operator Lipschitzness with respect to the operator and trace class norms is given in [9]; for a brief summary see [149]. In this section we briefly discuss major results on operator Lipschitzness with respect to the operator and Schatten norms. Initially the operator Lipschitz functions were introduced in the case of selfadjoint operators, but the definition naturally generalizes to the nonself-adjoint case. Definition 5.1.1 Let I be an interval in R, f : I → C a continuous function, and 1  p  ∞. We say that f is operator Lipschitz on I with respect to the norm  · p if there exists a constant cf,p > 0 such that f (A) − f (B)p  cf,p A − Bp for all A, B ∈ Dsa with σ (A) ∪ σ (B) ⊂ I and every separable Hilbert space H. We will briefly call the operator Lipschitz functions with respect to the Sp -norm “operator Sp -Lipschitz functions” and operator Lipschitz functions with respect to the B(H)-norm “operator Lipschitz functions”.

© Springer Nature Switzerland AG 2019 A. Skripka, A. Tomskova, Multilinear Operator Integrals, Lecture Notes in Mathematics 2250, https://doi.org/10.1007/978-3-030-32406-3_5

113

114

5 Applications

5.1.1 Commutator and Lipschitz Estimates in S2 Proposition 5.1.2 Let A, B ∈ Bsa (H) and σ (A)∪σ (B) ⊆ [a, b]. If f ∈ Lip[a, b], then f (A)X − Xf (B)2  f Lip[a,b] AX − XB2 , X ∈ S2 . Proof The result is an immediate consequence of Theorem 3.3.6 and the property 2 2 [1] ∞ = f Lip[a,b] TfA,B [1] : S → S  = f

 

following from (3.2.2) and (3.2.3).

The following result is an immediate consequence of Proposition 3.2.2 and Theorem 3.3.7. Nonetheless, we demonstrate an independent proof of it based on a similar result for finite matrices (see [83, Theorem 4.1]). Theorem 5.1.3 For every A, B ∈ Dsa such that A − B ∈ S2 and f ∈ Lip(R), the estimate f (A) − f (B)2  f Lip(R) A − B2

(5.1.1)

holds, that is, f is operator Lipschitz on R with respect to the S2 -norm. Proof (Proof in the Case A, B ∈ Bsa (H)) Let {ξj }∞ j =1 be an orthonormal basis in H and PN the orthogonal projection onto the linear span of {ξj }N j =1 , N ∈ N. Firstly we observe that PN (A − B)PN 22 =

∞     PN (A − B)PN ξj , ξk 2 j,k=1

∞     (A − B)PN ξj , PN ξk 2 = j,k=1

=

N     (A − B)ξj , ξk 2  A − B2 . 2

j,k=1

Hence, from the estimate (3.1.11) obtained for finite matrices, we have f (PN APN ) − f (PN BPN )2  f Lip(R) PN APN − PN BPN 2  f Lip(R) A − B2 .

5.1 Operator Lipschitz Functions

115

Thus, for any n ∈ N, it follows that n     (f (PN APN ) − f (PN BPN ))ξj , ξk 2  f 2

2 Lip(R) A − B2 .

j,k=1

Since PN APN → A and PN BPN → B in the strong operator topology as N → ∞, by [171, Theorem VIII.20(b)] we infer that f (PN APN ) → f (A) and f (PN BPN ) → f (B) in the strong operator topology. Thus, n     (f (A) − f (B))ξj , ξk 2  f 2

Lip(R) A

− B22 , for any n ∈ N.

j,k=1

Taking n → ∞, we obtain f (A) − f (B) ∈ S2 and the inequality (5.1.1).

 

5.1.2 Commutator and Lipschitz Estimates in Sp and B(H) The class of operator Lipschitz functions in Sp , 1 < p < ∞, coincides with the set of scalar Lipschitz functions, while the set of operator Lipschitz functions in S1 and B(H) is smaller. The details are discussed below. The next result is a consequence of [41, Theorem 8.2] and the bound for the double operator integral (3.3.6). In the particular case of A, B ∈ Bsa (H) with σ (A) ∪ σ (B) ⊆ [a, b], the proof of this result goes along the lines of the proof of Theorem 3.3.6 and applies properties of the double operator integral discussed in Sect. 3.3.5. Theorem 5.1.4 Let A, B ∈ Dsa be such that A − B ∈ B(H) and let X ∈ B(H). If f ∈ Lip(R) is such that f [1] ∈ A1 , then f (A)X − Xf (B) = TfA,B [1] (AX − XB)

(5.1.2)

f (A)X − Xf (B)  f [1] A1 AX − XB.

(5.1.3)

and

We note that applying (5.1.2) to X = I recovers the representation (3.3.9) and applying (5.1.3) to X = I gives the estimate below. Theorem 5.1.5 Let 1  p  ∞ and A, B ∈ Dsa be such that A − B ∈ Sp (or A − B ∈ B(H) if p = ∞). If f ∈ Lip(R) such that f [1] ∈ A1 , then f (A) − f (B)p  f [1] A1 A − Bp .

(5.1.4)

116

5 Applications

We have the following necessary and sufficient conditions for operator Lipschitzness described in terms of harmonic analysis. Theorem 5.1.6 1 (R) is operator Lipschitz on R with respect to the (i) Every function f ∈ B∞1 operator and Schatten norms. (ii) If f is operator Lipschitz on R with respect to the operator and trace class 1 (R) . norms, then f ∈ B11 loc

Proof The property (i), which was established in [143], is an immediate consequence of Theorems 3.3.14 and 3.3.8. The property (ii) is established in [141].   Remark 5.1.7 If f is as in Theorem 3.3.8, then f Lip(R)  f [1] A1 by the straightforward estimate  |f [1] (λ, μ)|  |a1(λ, ω)| · |a2 (λ, ω)| d|ν|(ω)

 



a1 (·, ω)∞ a2 (·, ω)∞ d|ν|(ω).

Therefore, the constant in the estimate (5.1.4) is worse than in the case p = 2 (established in Theorem 5.1.3). The restriction f [1] ∈ A1 in Theorem 5.1.5 is removed in [159, Theorem 1] in the case p = 1, implying that every Lipschitz function is operator Lipschitz with respect to the Schatten Sp -norm, p > 1. Theorem 5.1.8 Let 1 < p < ∞ and I be a (bounded or unbounded) interval in R. A continuous function f : I → C is operator Lipschitz with respect to the Sp -norm on I if and only if f ∈ Lip(I ). Moreover, there is cp > 0 such that f (A) − f (B)p  cp f Lip(I ) A − Bp for all A, B ∈ Dsa with σ (A) ∪ σ (B) ⊆ I and all f ∈ Lip(I ). Proof (Proof Outline) If f is operator Lipschitz in Sp , then it is scalar Lipschitz. This immediately follows from considering operators on a one dimensional Hilbert space. Assume now that f ∈ Lip(I ). Adjusting the estimate for the double operator integral in Theorem 3.3.4 to the interval I and combining it with the result of [156, Theorem 5.3] implies f (U )V − Vf (U )p  cp U V − V U p

(5.1.5)

5.1 Operator Lipschitz Functions

117

for every self-adjoint operator U and every bounded operator V . Applying (5.1.5) to     A 0 0I U= and V = 0 B I 0  

implies the result.

Remark 5.1.9 It was proved in [62] that the space of all operator Lipschitz functions with respect to the Schatten norm  · p , 1 < p < ∞, contains nondifferentiable functions, for example, f (t) = |t|. The fact that the absolute value function f (t) = |t| is not operator Lipschitz on R with respect to  · 1 and  ·  was proved earlier in [62, 99]. More generally, it follows from the results in [93] and from [104, Corollary 3.7] that all functions that are operator Lipschitz with respect to the operator norm  ·  are differentiable. There are also continuously differentiable Lipschitz functions that are not operator Lipschitz with respect to ·1 [80, 131, 208]. These functions are also not operator Lipschitz with respect to  ·  because operator Lipschitzness with respect to the operator norm is equivalent to the operator Lipschitzness with respect to the trace class norm [9, Theorem 3.6.5]. The converse problem whether there exist operator Lipschitz functions with respect to · that are not continuously differentiable was posed in [213] and found affirmative answers in [104, 109] as detailed below. One of the ways to construct functions that are not operator Lipschitz is to find a sequence of finite matrices of increasing dimension so that the respective Lipschitz bounds grow logarithmically with the dimension and then take appropriate direct sums of such matrices. This strategy is at the heart of the counterexample for the function f (t) = |t| outlined in [62]. The following finite-dimensional result for the trace class norm is obtained in [62, Theore 13] and for the operator norm in [8, Remark after Theorem 11.4] with involvement of [62, Lemma 15]. Theorem 5.1.10 (i) For every d ∈ N there exist Ad , Bd ∈ Bsa (22d ) such that Ad = Bd and  |Ad | − |Bd | 1  const · log d · Ad − Bd 1 . (ii) For every d ∈ N there exist Ad , Bd ∈ Bsa (2d ) such that Ad = Bd and  |Ad | − |Bd |   const · (1 + log d) · Ad − Bd . Below we provide another finite-dimensional construction that is suitable to build higher order counterexamples for Taylor remainders with bounded operators in Sect. 5.4. The result is derived in [169, Theorem 5.2] from [54, Theorem 7], while the latter is based on results of [208].

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5 Applications

Theorem 5.1.11 For every d ∈ N, d  2, there exist non-zero self-adjoint operators Ad , Bd ∈ B(22d ) such that σ (Ad + Bd ) = σ (Ad ) ⊂ [−e−1 , e−1 ], 0 ∈ σ (Ad ) has multiplicity 2, every λ ∈ σ (Ad ) \ {0} has multiplicity 1, and 1

h(Ad + Bd ) − h(Ad )  const (log d) 2 Bd , where ! h=

− 1   |x| log  log |x| − 1 2 , x ∈ [−e−1 , e−1 ] \ {0} 0, x=0

(5.1.6)

is a function in C 1 (R). The following conditions for f to be operator Lipschitz on [a, b] are due to [109, Corollary 4.6]. Theorem 5.1.12 Suppose that f ∈ C[a, b] and there are xn $ a, x0 = b, such that f is operator Lipschitz on each segment In = [xn , xn−1 ], that is, f (A) − f (B)  cn A − B

(5.1.7)

for all A, B ∈ Bsa (H) such that σ (A) ∪ σ (B) ⊆ In and some cn > 0. Then, f is operator Lipschitz on [a, b] if and only if ∞   n=1

fn xn − xn−1

2 < ∞, where fn = sup{|f (x) − f (a)| : x ∈ [a, xn−1 ]},

and sup cn < ∞, where cn satisfy (5.1.7). n∈N

Based on Theorem 5.1.12 one can construct a large variety of operator Lipschitz functions that are not continuously differentiable. The first example of such function is given in [104, Theorem 3.8]: ⎧ ⎨t 2 sin 1 if t =  0 t f (t) = ⎩0 if t = 0. The following result is obtained in [109, Corollary 5.2]. Theorem 5.1.13 Let ϕ be an infinitely many times differentiable, nonnegative function on R such that supp(ϕ) = [−1, 1],

max ϕ(t) = 1, t ∈R

ϕ  (−1/2) = 1,

ϕ  (1/2) = −1.

5.1 Operator Lipschitz Functions

119

 3 2 Let {σn }∞ 0, σ1 < 14 and ∞ n=1 σn < ∞. Set dn = 2n+1 , n=1 in R+ be such that σn $

∞ n . Then, the function g(t) := an = σ2nn and ϕn (t) = an ϕ t −d n=1 ϕn (t) is an infinitely many times differentiable on R \ {0}, differentiable but not continuously at t = 0, and operator Lipschitz on supp(g). Although not every perturbation in S1 produces an increment of an operator function in S1 , this increment belongs to the larger ideal S1,∞ , provided the respective scalar function is Lipschitz. The following analog of Theorem 5.1.8 is a consequence of Theorem 3.3.17. Theorem 5.1.14 Let f ∈ Lip(R). Then, there exists an absolute constant c > 0 such that f (A) − f (B)1,∞  c f Lip(R) A − B1 for all A, B ∈ Dsa .

5.1.3 Commutator and Lipschitz Estimates: Nonself-adjoint Case Operator Lipschitzness of functions of unitary operators is completely analogous to the one of self-adjoint operators. The following analog of Theorem 5.1.8 for unitaries is established in [21, Theorem 2]. Theorem 5.1.15 Let A, B be unitaries, 1 < p < ∞. Then, there is cp > 0 such that f (A) − f (B)p  cp f Lip(T) A − Bp ,

(5.1.8)

where |f (x) − f (y)| . |x − y| x=y∈T

f Lip(T) = sup

An analog of Theorem 5.1.6 for functions of unitary operators is established in [141]. Theorem 5.1.16 1 (T) is operator Lipschitz on T with respect to the (i) Every function f ∈ B∞1 operator and Schatten norms. (ii) If f is operator Lipschitz on T with respect to the operator and trace class 1 (T). norms, then f ∈ B11

An example of a function in C 1 (T) that is not operator Lipschitz in S1 on T is given in [141]. Similarly to a self-adjoint case, such example can also

120

5 Applications

be constructed based on dimension dependent bounds for matrix functions. The following dimension dependent bound is derived in [54, Theorem 8]. Theorem 5.1.17 For every integer d  3, there exist unitary operators Hd , Kd ∈ B(22d+1) such that σ (Hd ) = σ (Kd ), 1 ∈ σ (Hd ), and 1

u(Kd ) − u(Hd )  const (log d) 2 Kd − Hd , where u is given by ˜ ) u(eiθ ) := h(θ

(5.1.9)

and h˜ is a 2π-periodic function in C 1 (R) ∩ C n (R \ {0}) extending the function h defined by (5.1.6). Below we state results on Lipschitzness of functions of normal operators. The first one is proved in [110, Corollary 6.1]. Theorem 5.1.18 Let 1 < p < ∞ and I be a compact set in C. A continuous function f on C is operator Lipschitz with respect to the Sp -norm on the set Bnorm (H)(I ) := {A ∈ B(H) : A is normal, σ (A) ⊆ I }

(5.1.10)

if and only if f ∈ Lip(I ). Moreover, there is cp > 0 such that f (A) − f (B)p  cp f Lip(I ) A − Bp

(5.1.11)

for all A, B ∈ Bnorm (H)(I ) and all f ∈ Lip(I ), where f Lip(I ) = sup

x=y∈I

|f (x) − f (y)| . x − y1

A bound similar to (5.1.11) is also obtained for the commutator f (A)X −Xf (A) in [110, Corollary 6.1]. The following operator norm bound for a quasicommutator f (A)X − Xf (B) is due to [13, Theorem 10.3]. Theorem 5.1.19 Let A, B ∈ B(H) be normal operators and X ∈ B(H). Then, f (A)X − Xf (B)  c f B 1

∞1 (R

2)

max{AX − XB, A∗ X − XB ∗ }.

The appropriate adjustment of the latter result also holds for unbounded normal operators A, B. We note that the aforementioned results are based on theory of double operator integrals, each one employing different aspects of this theory. To achieve their result, the authors of [13] studied operator Lipschitzness of functions of two variables.

5.1 Operator Lipschitz Functions

121

The investigation of the operator Lipschitzness of functions of two variables in the operator and Schatten-von Neumann norms was continued in [11, 14]. Operator Lipschitzness has also been studied for contractions and dissipative operators. The following analog of (5.1.8) for contractions is proved in [110, Theorem 6.4]. Theorem 5.1.20 Let A, B be contractions, 1 < p < ∞, and f ∈ A(D). Then, f (A) − f (B)p  cp f Lip(D) A − Bp , where |f (x) − f (y)| . x − y1 x=y∈D

f Lip(D) = sup

It is established in [107, Theorem 3.4] that every function f ∈ A(D) that is operator Lipschitz on T (for all pairs of unitaries) with respect to the operator norm is also operator Lipschitz on D (for all pairs of contractions) and, moreover, f (A)X − Xf (B)  f OL(T) AX − XB where A, B are contractions, X ∈ B(H), and  f OL(T) = sup

 f (A) − f (B) : A = B are unitaries . B − A

It is proved in [12, Theorem 5.3] that f (A)X − Xf (B)  f OL(C+ ) AX − XB where X ∈ B(H), A, B are maximal dissipative operators with bounded quasicommutator AX − XB, and   f (A)−f (B) ¯ : A = B are normal with σ (A) ∪ σ (B) ⊂ C+ . f OL(C+ ) = sup B − A

5.1.4 Lipschitz Type Estimates in Noncommutative Lp -Spaces The results of Theorems 5.1.8 and 3.3.17 extend to the setting of noncommutative Lp spaces, as it is done in [159, Theorem 1] and [49, Theorem 5.3]. Theorem 5.1.21 Let (M, τ ) be a semifinite von Neumann algebra and AηMsa , BηMsa . Let f ∈ Lip(R) and 1 < p < ∞. Then, there exists cp > 0 such that f (A) − f (B)Lp (M,τ )  cp f Lip(R) A − BLp (M,τ ) .

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5 Applications

Theorem 5.1.22 Let (M, τ ) be a semifinite von Neumann algebra and AηMsa , BηMsa . If f ∈ Lip(R), then there exists c > 0 such that f (A) − f (B)L1,∞ (M,τ )  c f Lip(R) A − BL1 (M,τ ) .

5.1.5 Lipschitz Type Estimates in Banach Spaces Let X, Y be Banach spaces. We are interested in the Lipschitz type estimates f (B) − f (A)B(X)  cA,B,f B − AB(X)

(5.1.12)

for A, B ∈ B(X) and, more generally, commutator estimates f (B)X − Xf (A)B(X,Y)  cA,B,f BX − XAB(X,Y)

(5.1.13)

for A ∈ B(X), B ∈ B(Y), X ∈ B(X, Y). As it was mentioned above, this problem is well-known in the special case when X = Y is a separable Hilbert space, such as 2 , and A and B are normal operators on X. Here we present such estimates in the Banach space setting, and specifically for X = p and Y = q with p, q ∈ [1, ∞]. For all relevant definitions see Sect. 2.10. The following result is due to [173, Theorem 4.6]. Theorem 5.1.23 Let X, Y be separable Banach spaces such that either X or Y has a bounded approximation property and let I be a Banach ideal in B(X, Y) with a strong convex compactness property. If A, B ∈ Bs (X) and f [1] ∈ A1 , then f (B)X − Xf (A)I  16 spec(A) spec(B) f [1] A1 BX − XAI and, in particular, f (B) − f (A)I  16 spec(A) spec(B) f [1] A1 B − AI . It is immediate from the definition of a scalar type operator that every normal operator on H is of scalar type, which extends Theorem 3.3.8 for p = ∞ to the Banach space setting. If A and B are diagonalizable operators, then the class of functions in the above theorem is extended in [173, Theorem 7.3]. Given Banach spaces X, Y and 1  p < ∞, let !p denote the ideal in B(X, Y) consisting of all S : X → Y such that for every n ∈ N and every collection {xj }nj=1 ⊂ X, n  j =1

p 1/p

S(xj )Y

C

sup

x ∗ X∗  1

n   1/p  | x ∗ , x j |p j =1

5.1 Operator Lipschitz Functions

123

The infimum of C as above gives a norm on !p , which we denote by πp . The ideal (!p , πp ) is called the ideal of p-summing operators from X to Y; it is the Banach ideal by [67, Propositions 2.3, 2.4, 2.6]. ∗

Theorem 5.1.24 Let 1 < p < ∞ and p1 + p1∗ = 1. Let A ∈ B(p ) (respectively, A ∈ B(c0 )) and B ∈ B(p ) (respectively, B ∈ B(1 )) be diagonalizable operators. ∗ Let (I, ·I ) be the ideal of p-summing operators from p to p (respectively, from c0 to 1 ). Then, every f ∈ Lip(C) satisfies (5.1.13) with cA,B,f = cA,B f Lip(C) . Analogous results for different pairs of spaces (p , q ) are derived in [173, Theorems 6.8 and 6.9]. Theorem 5.1.25 Let 1  p < q < ∞. Let A ∈ B(p ) and B ∈ B(q ) (respectively, B ∈ B(c0 )) be diagonalizable operators with real spectra. Then, (5.1.13) holds with B(X, Y) = B(p , q ) (respectively, B(X, Y) = B(p , c0 )) and f (t) = |t|, where cA,B,f = cA,B f Lip(R) . Theorem 5.1.26 Let 1  p < q < ∞. Let A ∈ B(1 ) and B ∈ B(q ) (respectively, B ∈ B(c0 )) be diagonalizable operators. Then, (5.1.13) holds with B(X, Y) = B(1 , q ) (respectively, B(X, Y) = B(p , c0 )) for every Lipschitz function f , where cA,B,f = cA,B f Lip(C) . In addition, (5.1.12) holds with X = 1 and X = c0 . The proofs in [173] rely on the theory of Schur multipliers on the space B(p , q ) developed by G. Bennett [26, 27] and double operator integrals discussed in Sect. 3.6. Commutator estimates for f (t) = |t| and different Banach ideals in B(H) were also studied in [62, 70], where the proofs are based on Macaev’s celebrated theorem (see [85]) or on the UMD-property of the reflexive Schatten-von Neumann ideals. However, the spaces B(X, Y) are not UMD-spaces and, therefore, the techniques used in [62, 70] do not apply to them.

5.1.6 Operator I-Lipschitz Functions Operator I-Lipschitz and Commutator I-Bounded Functions Let I be a symmetrically normed (s. n.) ideal of B(H) equipped with the norm  · I . Denote by Inorm the set of all normal operators in I and by I a compact subset of C. Set Inorm (I ) := {A ∈ Inorm : σ (A) ⊆ I }. Definition 5.1.27 (i) f ∈ C(C) is called an I-Lipschitz function on I ⊂ C if there is D > 0 such that f (A) − f (B) ∈ I and f (A) − f (B)I  D A − BI ,

A, B ∈ Inorm (I ).

(5.1.14)

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5 Applications

(ii) f is a commutator I-bounded function on I ⊂ C if there is D > 0 such that, for all A ∈ Inorm (I ) and X ∈ B(H), we have f (A)X − Xf (A) ∈ I and f (A)X − Xf (A)I  D AX − XAI .

(5.1.15)

We note that if I = B(H), then f is an operator Lipschitz function. The spaces of all I-Lipschitz and commutator I-bounded functions on the interval I are denoted I-Lip(I ) and I-CB(I ), respectively. The result below is due to [104, Theorem 3.5]. Theorem 5.1.28 Let f ∈ C(C), I ⊂ C and let I be an s. n. ideal. The following properties are equivalent. (i) (ii) (iii) (iv)

f is an I-Lipschitz function on I . (5.1.14) holds for all A, B ∈ Fnorm (I ). (5.1.15) holds for all A ∈ Inorm (I ), all X ∈ Bsa (H). (5.1.15) holds for all A ∈ Fnorm (I ), all X ∈ Fsa .

Condition (5.1.15) is equivalent to the following stronger condition (see [104, Proposition 4.1]): f is commutator I-bounded on I if and only if there is D > 0 such that, for all A, B ∈ Inorm (I ) and X ∈ B(H), we have f (A)X − Xf (B) ∈ I and f (A)X − Xf (B)I  D AX − XBI .

(5.1.16)

For all s. n. ideals, including I = B(H), [104, Corollaries 3.6 and 5.4] and (5.1.16) yield the following result. Corollary 5.1.29 (i) I-CB(I ) ⊆ I-Lip(I ) for I ⊂ C and I-CB(I ) = I-Lip(I ) for I ⊂ R. (ii) B(H)-CB(I ) = S∞ -CB(I ) = S1 -CB(I ) for all I ⊂ C. For I = Sp , 1 < p < ∞, the above results were noticed in [62]. By Corollary 5.1.29(i), the condition that f ∈ I-CB(I ) is stronger than the condition that f ∈ I-Lip(I ) for I ⊂ C. If I ⊂ R these conditions are equivalent. The possibility to reduce the study of I-Lipschitz functions to the study of commutator I-bounded functions is important since it enables us to use interpolation theory techniques. Definition 5.1.30 (i) A compact set I ⊂ C is called I-Fuglede if I-CB(I ) = I-Lip(I ). (ii) A s. n. ideal I is called a Fuglede ideal if all compacts I in C are I-Fuglede. Proposition 5.1.31 ([104, Proposition 4.5]) A compact I ⊂ C is I-Fuglede if and only if the function h(z) = z is commutator I-bounded on I, that is, there is D > 0 such that  ∗  A X − XA∗   D AX − XAI , I

A ∈ Inorm (I ) and X ∈ B(H).

5.1 Operator Lipschitz Functions

125

The following sufficient conditions for an ideal to be Fuglede are obtained in [105, Corollary 3.8]; for the definition of Boyd indices see, for instance, [16]. Theorem 5.1.32 Let I be a separable s. n. ideal and (pI , qI ) be its Boyd indices. If 1 < pI , qI < ∞, then I is a Fuglede ideal. The theorem above, in particular, implies that all Sp , 1 < p < ∞, are Fuglede ideals (see also [1, 182, 211]). The following results are obtained in [105, Theorem 4.3 and Corollary 4.6]. Theorem 5.1.33 Let I be a separable s. n. ideal. (i) If f ∈ I-CB(I ) for I ⊂ C, then there exists D > 0 such that (5.1.16) holds for all A, B ∈ Bnorm (H)(I ) satisfying AX − XB ∈ I for all X ∈ B(H), where Bnorm (H)(I ) is defined in (5.1.10). (ii) If f ∈ I-Lip(I ) and I ⊂ C is I-Fuglede compact, then (5.1.16) holds for all A, B ∈ Bnorm (H)(I ) satisfying AX−XB ∈ I for all X ∈ B(H). In particular, A − B ∈ I implies f (A) − f (B) ∈ I and f (A) − f (B)I  D A − BI . Apart from separable ideals, Theorem 5.1.33 holds for a large variety of other s. n. ideals (see [105, Theorem 4.5 and Corollary 4.6]). It is established in [105, Corollary 3.8] that the ideals S1 , S∞ , B(H) are not Fuglede by testing the properties of f (z) = z and using the result of [92] that there are A ∈ Bnorm (H) and X ∈ B(H) such that AX − XA ∈ S1 , but A∗ X − XA∗ ∈ / S1 .

(5.1.17)

It is established in [211] that the operator X in (5.1.17) can be chosen compact. Further, it is proved in [102, Corollary 4.3] that X can be chosen in any Sp for p > 1. It is shown in [105, Corollary 5.9] that both A and X can be chosen compact. Initially, it was thought that all spaces Sp -Lip(R), p ∈ (1, ∞), are different. However, it is shown in [159] that they are all the same and coincide with the space Lip(R). In [110] this result is extended to the spaces Sp -Lip(C), p ∈ (1, ∞), which is shown to coincide with the space Lip(C). Problem 5.1.34 Let I be a separable s. n. ideal. Does I-Lip(C) coincide with Lip(C)? I-Stable and Commutator I-Stable Functions The notions of I-stable and commutator I-stable functions were introduced in [105]. They are close to the notions of I-Lipschitz and commutator I-bounded functions. Definition 5.1.35 (i) f ∈ C(C) is called I-stable on a compact I ⊂ C if, for A, B ∈ Bnorm (H)(I ), the condition A − B ∈ I implies f (A) − f (B) ∈ I.

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5 Applications

(ii) f ∈ C(C) is called commutator I-stable on I if AX − XA ∈ I for A ∈ Bnorm (H)(I ) and all X ∈ B(H) implies A∗ X − XA∗ ∈ I. (iii) A compact I is called weakly I-Fuglede if h(z) = z is a commutator I-stable function on I . (iv) A s. n. ideal I is called weakly Fuglede if all compacts I in C are weakly I-Fuglede. In general, the condition of commutator I-stability is stronger than I-stability. In fact, f ∈ C(C) is I-stable on I if and only if the implication in Definition 5.1.35(ii) holds for all X = X∗ ∈ B(H) (see [105, Proposition 5.2]). If, however, I is weakly I-Fuglede, then commutator I-stability coincides with I-stability (see [105, Proposition 5.5]). Clearly, all I ⊂ R are weakly I-Fuglede for all ideals I. We also have the following result, which proof is left as an exercise. Proposition 5.1.36 Let I be a separable s. n. ideal. (i) If I is a I-Fuglede compact, then I is weakly I-Fuglede. (ii) If I is Fuglede, then I is weakly Fuglede. Stability and Fuglede properties of the ideals Sp , 1  p  ∞, and B(H) are summarized below. Theorem 5.1.37 (i) Let 1 < p < ∞. For each compact I ⊂ C, Sp -Lip(I ) = Sp -CB(I ) = Lip(I ). (ii) S∞ , B(H) are weakly Fuglede, but not Fuglede ideals. S1 is not weakly Fuglede. (iii) For each compact I ⊂ C, S1 -CB(I ) = S∞ -CB(I ) = B(H)-CB(I ) ⊆ Lip(I ). (iv) The ideals S1 , S∞ , B(H) have the same Fuglede compacts. If I is one of them, then S1 -Lip(I ) = S∞ -Lip(I ) = B(H)-Lip(I ) = S1 -CB(I ) = S∞ -CB(I ) = B(H)-CB(I ) = {g ∈ C(I ): g is commutator S1 -stable on I }. If I ⊂ R then the above spaces coincide with {g ∈ C(I ): g is S1 -stable on I }. A-Lipschitz Functions on Semisimple Hermitian Banach *-Algebras A Let A be a semisimple Hermitian Banach *-algebra and Asa be the set of all selfadjoint elements in A. Denote by C*(A) the C*-algebra completion of A with respect to Ptak-Rajkov C*-norm: Ar = rA (A∗ A)1/2 for A ∈ A, where rA (A) is the spectral radius of A in A. For all relevant algebraic notions we refer the reader to [60].

5.1 Operator Lipschitz Functions

127

Each f ∈ C(R) acts on C*(A)sa , that is, for each A ∈ C*(A)sa , there is f (A) ∈ C*(A). For I ⊆ R, set Asa (I ) := {A ∈ Asa : σ (A) ⊆ I }. A function f ∈ C(R) acts on Asa (I ) if f (A)∈A (instead of f ∈C∗ (A)) for all A ∈ Asa (I ). For example, all Sp , 1  p < ∞, are semisimple Hermitian Banach p ∞ p *-algebras and C*(S  ) = S . A function f ∈ C(R) acts on all S if and only if  f (t )  f (0) = 0 and  t  < C for some C > 0 and t in a neighbourhood of 0 (see [108, Theorem 2.2]). If δ is a weakly closed *-derivation on Sp , 1  p < ∞, its domain D(δ) is a semisimple Hermitian Banach *-algebra with norm Aδ = A + δ(A) , A ∈ D(δ), and C*(D(δ)) = S∞ . If f is Lipschitz on each compact in R and f (0) = 0 then, by [108, Theorem 3.8], f acts on D(δ). If δ is a weakly closed *-derivation on a C*-algebra A, then D(δ) is a semisimple Hermitian Banach *-algebra and C*(D(δ)) = A. It is shown in [103, Theorem 8.4] that f ∈ C(R) acts on D(δ) if and only if f is operator Lipschitz. Definition 5.1.38 A function f ∈ C(R) is called A-Lipschitz on I0 ⊆ R if it acts on Asa (I0 ) and, for each compact I ⊆ I0 , there is DI > 0 such that f (A) − f (B)A  DI A − BA for all A, B ∈ Asa (I ). Denote by  an open subset of R containing 0 and by A-Lip() the space of all functions A-Lipschitz on each compact in . The presence of non-trivial ALipschitz functions reflects the structure of the algebra A. The following result is due to [106, Corollary 2.5 and Theorem 4.4]. Theorem 5.1.39 Let A be a semisimple Hermitian Banach *-algebra. (i) Let A be unital. If all infinitely differentiable functions on R act on Asa (R), then A is C*-equivalent, that is, it is a unital C*-algebra in some equivalent norm (C*(A) = A). (ii) Let A be unital. If there is a non-linear A-Lipschitz function on some I = [a, b] which extends to a function on C analytic in a neighbourhood of I in C, then A is C*-equivalent. (iii) Let A be not unital. If f (t) = t 2 is an A-Lipschitz function on some 0 ∈  ⊆ R, then A is a dense symmetrically normed Jordan ideal of C*(A), that is, there is K > 0 such that AX + XA ∈ A and AX + XAA  K A XA for all A ∈C*(A), X ∈ A.

128

5 Applications

It is proved in [82] that if C*(A) is S∞ or a properly infinite W*-algebra, then all s. n. Jordan ideals are two-sided ideals of S∞ . Various conditions when A is a s. n. two-sided ideal of S∞ are given in [106]. Let !(A) be the set of all irreducible *-representations of A. For π ∈ !(A), let Hπ be the representation space. The following result is proved in [106, Theorem 3.3]. Theorem 5.1.40 Let A be a C*-algebra and 0 ∈  ⊆ R. (i) Let dim Hπ  N for some N ∈ N and all π ∈ !(A). Then, the space A-Lip() coincides with the space of all functions g on  such that g|I ∈ Lip(I ) for all compacts I ⊂ . (ii) If N in (i) does not exist, then A-Lip() = B(H)-Lip(). If A is a s. n. Jordan ideal of C*(A), the space A-Lip() is described in [106, Theorem 4.20]. Theorem 5.1.41 Let A be a s. n. Jordan ideal of a separable C*-algebra C*(A). Let 0 ∈  ⊆ R. (i) If C*(A) is not a CCR-algebra, then A-Lip() = B(H)-Lip(). (ii) Let C*(A) be a CCR-algebra. Then, for each π ∈ !(A), there is a s. n. ideal J π of S∞ such that J π = S1 , J π = S∞ and A-Lip() ⊆ J π -Lip().

5.2 Operator Hölder Functions Operator functions inherit the Hölder property from the respective scalar Hölder functions, which is quite different from the operator Lipschitzness discussed in the previous section. Below we discuss several Hölder-type inequalities for operator functions. The following results are established in [5, Theorem 4.1]; see also [9]. Theorem 5.2.1 Let A, B ∈ Dsa and f ∈ α (R) for α ∈ (0, 1). Then, there exists a constant c > 0 such that f (A) − f (B)  c(1 − α)−1 f α (R) A − Bα . Operator Hölderness of functions of self-adjoint operators with perturbations in Schatten classes is established in [4, Theorem 5.8]. Theorem 5.2.2 Let A, B ∈ Dsa and f ∈ α (R) for α ∈ (0, 1), and 1 < p < ∞. Then, there exists a constant c > 0 such that f (A) − f (B)p/α  cf α (R) A − Bαp .

5.3 Differentiation of Operator Functions

129

The following operator Hölder property in the unitary case is due to [5, Theorem 5.1]. Theorem 5.2.3 Let A, B be unitary operators and f ∈ α (T) for α ∈ (0, 1). Then, there exists a constant c > 0 such that f (A) − f (B)  c(1 − α)−1 f α (T) A − Bα . A result completely analogous to the one of Theorem 5.2.3 for contractions A, B and functions f ∈ α (T) analytic in the unit disc is derived in [5, Section 6]. The following result is established in [13, Theorem 9.1]. Theorem 5.2.4 Let A, B be normal operators with the same domain in H, let f ∈ α (R2 ) for α ∈ (0, 1), and 1 < p < ∞. Then, there exists a constant cp,α > 0 such that f (A) − f (B)p/α  cp,α f α (R2 ) A − Bαp . Modifications of Theorem 5.2.4 for p = 1 and for a quasinormed ideal with upper Boyd index less than 1 are discussed in [13, Section 9].

5.3 Differentiation of Operator Functions Differentiability is one of natural properties studied in theory of functions, in particular, operator functions. Pioneering results on differentiability of operator functions were obtained in [61], with restrictive assumptions on functions and operators in the infinite-dimensional setting. These results were substantially refined and extended in the series of papers [18, 19, 24, 36, 55, 63, 102, 110, 119, 141, 143, 146, 147, 163, 199] in response to development of perturbation theory and also influenced by the question published in [212]. In this section, we prove the best known results on differentiability of matrix functions and outline the proof of differentiability of operator functions in the infinite-dimensional case. We also outline the proof of best estimates for Schatten norms of operator derivatives obtained in [163].

5.3.1 Differentiation of Matrix Functions Following the proof of [61, Theorem 1] and supplementing omitted details, we demonstrate below that t → f (X(t)) is differentiable in the operator norm and the respective derivative can be written as a double operator integral (3.1.1).

130

5 Applications

Proposition 5.3.1 Let t → X(t) be a C 1 -function with values in Bsa (2d ). If f ∈ C 1 (R), then the function t → f (X(t)) is differentiable in the operator norm and

df (X(t)) ),X(t ) dX(t) = TfX(t . [1] dt dt

(5.3.1)

Proof Let {λj (t)}dj=1 be a complete list of eigenvalues and {ξj (t)}dj=1 a respective orthonormal basis of eigenvectors of X(t). Let f (λ) = λm , m ∈ N. Then, differentiating by the product rule gives m−1  dXm (t) df (X(t)) dX(t) m−i−1 = = X Xi (t) (t). dt dt dt i=0

By the spectral theorem, the latter equals m−1 d  i=0 j =1

=

k=1

d  d m−1   j =1 k=1

! =

dX(t)  m−i−1 λk Pξk dt d

λij Pξj

i=0

d

dX(t) Pξj Pξk λij λm−i−1 k dt

m λm d dX(t ) j −λk j =1 k=1 λj −λk Pξj dt Pξk , d d ) m−1 Pξj dX(t j =1 k=1 mλj dt Pξk ,

λj = λk λj = λk

,

where we suppressed the parameter t from the notation of the spectral projection and eigenvalues. Applying (3.1.1) completes the proof of (5.3.1) for a monomial. By linearity, (5.3.1) extends from monomials to general polynomials. Now we extend (5.3.1) to a general f ∈ C 1 (R) via approximations. Let [a, b] contain all the points λj (t), j = 1, 2, . . . , d. Let ϕl ⇒ f  on [a, b] as l → ∞, λ where ϕl is a polynomial, l ∈ N. Then, fl (λ) := f (a) + a ϕl (t) dt ⇒ f (λ) on [a, b]. Applying the estimate (3.1.12) gives    f (X(t + s)) − f (X(t)) fl (X(t + s)) − fl (X(t))    −   s s   √  X(t + s) − X(t)  .  d f  − fl ∞    s By the estimate (3.1.7),    X(t ),X(t )  √ X(t ),X(t ) T [1]   d f  − f  ∞ X(t). (X(t)) − T (X(t)) [1] l  f  fl

5.3 Differentiation of Operator Functions

131

Combining the latter two inequalities with (5.3.1) for polynomials completes the proof.   The formula (5.3.1) is an excellent instrument for studying the derivatives of operator functions. In particular, by (3.1.6) and the formula (5.3.1), we obtain the following estimate for the derivative:  df (X(t))       2 dt

 dX(t)    max |f  (λ)|   . 2 a λ b dt

(5.3.2)

In this section we present more delicate estimates for operator derivatives which extend and complement (5.3.2) to the case of other norms. Below we prove extension of (5.3.1) to the higher order case. Theorem 5.3.2 Let H, V ∈ Bsa (2d ). If f ∈ C k (R), k ∈ N, then the function t → f (H + tV ) is differentiable k times in the operator norm and   d k f (H + tV ) = k! TfH[k]+t V ,...,H +t V V , . . . , V . k " #$ % dt

(5.3.3)

k

Proof We proceed by induction and, for simplicity of exposition, calculate only the derivative at t = 0. The base of induction is proved in Proposition 5.3.1, so we only need to confirm the inductive step. Assume that the function t → f (H +tV ) is differentiable k −1 times and (5.3.3) holds for k − 1. Thus, we have  dk f (H + tV )t =0 dt k  (k − 1)!  H +t V ,...,H +t V = lim (V , . . . , V ) − TfH,...,H Tf [k−1] [k−1] (V , . . . , V ) t →0 t H + tV , . . . , H + tV ,H, . . . , H #$ % " #$ % k−1  " (k − 1)!  k−j j (V , . . . , V ) = lim Tf [k−1] t →0 t j =0

H + tV , . . . , H + tV ,H, . . . , H " #$ % " #$ % − Tf [k−1]

k−j−1

j+1

 (V , . . . , V ) .

(5.3.4)

132

5 Applications

By the perturbation formula (4.1.9), H + tV , . . . , H + tV ,H, . . . , H H + tV , . . . , H + tV ,H, . . . , H " " #$ % " #$ % #$ % " #$ % 1 k−j j k−j−1 j+1 Tf [k−1] − Tf [k−1] t × (V , . . . , V ) H + tV , . . . , H + tV ,H, . . . , H " #$ % " #$ % k−j

= Tf [k]

j+1

(V , . . . , V ).

(5.3.5)

By Proposition 4.1.6, lim T H[k]+t V ,...,H +t V ,H,...,H (V , . . . , V ) t →0 f

= TfH,...,H (V , . . . , V ). [k]

(5.3.6)

Combining (5.3.4)–(5.3.6) completes the proof of (5.3.3).

 

The function in Theorem 5.3.2 is, in fact, k times continuously Fréchet differentiable at every H ∈ Bsa (2d ). This fact is proved in [88, Theorem 2.3.1] by direct computation for polynomials and induction along with performing an approximation of general functions by polynomials that is based on coarse Hilbert-Schmidt bounds for multilinear Schur multipliers, where the constant grows polynomially with the dimension of the matrix (see Sect. 4.1.3 for details about the bound). Ideologically this is similar to but technically different from the proof of Theorem 5.3.2. In Theorem 5.3.2, we use a general approximation method of Proposition 4.1.6 and perform the induction step based on purely algebraic Proposition 4.1.5. More general results on Fréchet differentiability of operator functions in the infinite dimensional setting along with delicate bounds for Fréchet derivatives are discussed in Sect. 5.3.4.

5.3.2 Differentiation in B(H) Along Multiplicative Paths of Unitaries To prove an analog of (5.3.3) in the infinite-dimensional unitary case, we need the following lemma obtained in [168, Lemma 2.6 and (3.2)]. n (T), let A and U ∈ B(H) be self-adjoint and unitary Lemma 5.3.3 Let f ∈ B∞1 operator, respectively. Denote U (t) = eit A U, t ∈ R. Then, for all 1  k  n − 1 and all j1 , . . . , jk ∈ N,

 d U˜ k+1 (t ) j1 Tf [k] (A U (t), . . . , Ajk U (t))t =s dt =i

k+1  m=1

U˜

(s)

k+2 Tf [k+1] (Aj1 U (s), . . . , Ajm−1 U (s), AU (s), Ajm U (s), . . . , Ajk U (s))

5.3 Differentiation of Operator Functions k 

+i

U˜

k+1 Tf [k]

133

(s)

m=1

× (Aj1 U (s), . . . , Ajm−1 U (s), Ajm +1 U (s), Ajm+1 U (s), . . . , Ajk U (s)), where U˜ r (t) is the tuple consisting of r copies of U (t). The following theorem is proved for n = 1 in [141, (5)] and for a general n in [168, Theorem 3.1], with precise coefficients found in [205]. n (T), let A ∈ B (H) and let U ∈ B(H) be a unitary Theorem 5.3.4 Let f ∈ B∞1 sa it A operator. Set U (t) = e U , t ∈ R. Then, k   1 dk k  f (U (t)) = i t =s k! dt k



l=1 j1 ,...,jl  1 j1 +···+jl =k

1 U˜ (s) T l+1 (Aj1 U (s), . . . , Ajl U (s)), j1 ! . . . jl ! f [l] (5.3.7)

for all 1  k  n − 1. Moreover,  n    d n    n (T) A .  dt n f (U (t)) t =s   cn f B∞1

(5.3.8)

Proof The formula (5.3.7) in the case k = 1 was established in [141, (5)]. The formula (5.3.7) for an arbitrary 1  k  n−1 can be established by induction. We assume that it holds for k = p < n − 1 and verify below that it holds for k = p + 1. Denote Vj,t := Aj U (t). Applying Lemma 5.3.3 gives   d p+1 p  f (U (t)) = i t =s dt p+1 p



l=1 j1 ,...,jl  1 j1 +···+jl =p

= i p+1

p 



l=1 j1 ,...,jl  1 j1 +···+jl =p

×

l 

U˜

 d U˜ l+1 (t ) p! Tf [l] (Vj1 ,t , . . . , Vjl ,t )t =s j1 ! . . . jl ! dt

p! j1 ! . . . jl !

(t )

Tf [l]l+1 (Vj1 ,t , . . . , Vjm−1 ,t , Vjm +1,t , Vjm+1 ,t , . . . , Vjl ,t )

m=1

+ i p+1

p 



l=1 j1 ,...,jl  1 j1 +···+jl =p

×

l+1 

U˜

(t )

l+2 Tf [l+1] (Vj1 ,t , . . . , Vjm−1 ,t , V1,t , Vjm ,t , . . . , Vjl ,t )

m=1

=: i

p+1

p! j1 ! . . . jl !

(S1 + S2 ),

134

5 Applications

where U˜ k+1 (t) is the tuple consisting of k+1 copies of U (t). Making the substitution ir = jr , 1  r = m  l and im = jm + 1 in S1 , we obtain S1 =

p  l 



l=1 m=1 ir  1,r=m,im  2 i1 +···+il =p+1 U˜

p! i1 ! . . . im−1 !(im − 1)!im+1 ! . . . il !

(t )

× Tf [l]l+1 (Vi1 ,t , . . . , Vil ,t ). Relabeling the summands of S2 via the mapping l → l − 1 and making the substitution ir = jr , 1  r  m − 1, im = 1 and ir = jr−1 , m + 1  r  l, we obtain p+1 

S2 =



l=2 j1 ,...,jl−1  1 j1 +···+jl−1 =p

×

l 

U˜

p! j1 ! . . . jl−1 !

(t )

Tf [l]l+1 (Vj1 ,t , . . . , Vjm−1 ,t , V1,t , Vjm ,t , . . . , Vjl−1 ,t )

m=1

=

p+1 

l 



l=2 m=1 i1 ,...,il  1,im =1 i1 +···+il =p+1

p! U˜ (t ) T l+1 (Vi1 ,t , . . . , Vil ,t ). i1 ! . . . im−1 !im+1 ! . . . il ! f [l]

Thus, ˜

2 (t ) S1 + S2 = TfU[1] (Vp+1,t )

+

p  l  l=2 m=1 U˜





+

ir  1,r=m,im  2 i1 +···+il =p+1

i1 ,...,il  1,im =1 i1 +···+il =p+1

p! i1 ! . . . im−1 !(im − 1)!im+1 ! . . . il !

(t )

× Tf [l]l+1 (Vi1 ,t , . . . , Vil ,t ) +

p+1 



m=1 i1 ,...,ip+1  1,im =1 i1 +···+ip+1 =p+1 U˜

(t )

p! i1 ! . . . im−1 !(im − 1)!im+1 ! . . . ip+1 !

p+2 × Tf [p+1] (Vi1 ,t , . . . , Vip+1 ,t ),

5.3 Differentiation of Operator Functions

135

so ˜

2 (t ) S1 + S2 = TfU[1] (Vp+1,t )

+

p  l 



l=2 m=1

i1 ,...,il  1 i1 +···+il =p+1 U˜

p! U˜ (t ) Tf [l]l+1 (Vi1 ,t , . . . , Vil ,t ) i1 ! . . . im−1 !(im − 1)!im+1 ! . . . il !

(t )

p+2 (V1,t , . . . , V1,t ). + (p + 1)! Tf [p+1]

Since l  m=1

i1 + · · · + il (p + 1)! p! = p! = , i1 ! . . . im−1 !(im − 1)!im+1 ! . . . il ! i1 ! . . . il ! i1 ! . . . il !

it follows that U˜ (t )

2 (Vp+1,t ) + S1 + S2 = Tf [1]

U˜

p 



l=2

i1 ,...,il  1 i1 +···+il =p+1

(p + 1)! U˜ l+1 (t ) T (Vi1 ,t , . . . , Vil ,t ) i1 ! . . . il ! f [l]

(t )

p+2 (V1,t , . . . , V1,t ) + (p + 1)! Tf [p+1]

=

p+1 



l=1

i1 ,...,il  1 i1 +···+il =p+1

(p + 1)! U˜ l+1 (t ) T (Vi1 ,t , . . . , Vil ,t ), i1 ! . . . il ! f [l]

proving (5.3.7). The estimate (5.3.8) follows from the estimates (4.3.34) for the multiple operator integral and (4.3.35) for its symbol.  

5.3.3 Differentiation in B(H) and S1 Along Linear Paths of Self-adjoints Differentiability in the infinite-dimensional setting imposes stronger restrictions on the respective class of scalar functions than in the finite-dimensional setting. In particular, the condition f ∈ C 1 (R) is not sufficient for differentiability of operator functions in the infinite-dimensional case even when the involved operators are bounded [141, Theorem 8] (see also [80, 81]). It is proved in [143, Theorem 2] 1 (R), then f is Gâteaux differentiable that if f is an element of the Besov space B∞1 p with respect to the S -norm, 1  p < ∞, and the operator norm at every self-

136

5 Applications

adjoint operator. This restriction on f is relaxed in [19] by methods of complex analysis (see [9, Subsection 3.13] for details). It also follows (see theorem below) 1 (R) is Fréchet Sp -differentiable, 1  p < ∞, at every self-adjoint that f ∈ B∞1 operator and Fréchet differentiable with respect to the operator norm at every bounded self-adjoint operator, and in the case p = 1 the restriction on f is substantially relaxed in the next section. A necessary condition for differentiability of f in the S1 -norm, which creates a small gap with the sufficient condition, is obtained in [143, Theorem 3] via nuclearity criterion for Hankel operators. Higher order Gâteaux differentiability of operator functions with respect to the operator and trace class norms is established in [146] and higher order Fréchet differentiability in [119, Theorem 4.1]. The aforementioned sufficient conditions for S1 - and B(H)differentiability are summarised and proved in the theorem below. 1 (R)∩ Theorem 5.3.5 Let A ∈ Dsa , X = X∗ ∈ S1 (or B(H)), n ∈ N, and f ∈ B∞1 n B∞1 (R). Then, the function f is differentiable n times at A along the direction X in the sense of Gâteaux in the S1 -norm (or operator norm),

 dn f (A + tX)t =0 = n! TfA,...,A (X, . . . , X), [n] n dt

(5.3.9)

and   n d       cn f B n (R)Xn f (A + tX) p  dt n t =0  ∞1 p

(5.3.10)

for 1  p  ∞. Moreover, the function f is n times continuously Fréchet differentiable at A in the S1 -norm (or, if A is bounded, in the operator norm). Proof We note that by (4.3.12) and Theorem 4.3.4, TfA,...,A (X, . . . , X) is well [k] 1 (R) ∩ B n (R) ⊂ ∩n B k (R). The defined for every k = 1, . . . , n because B∞1 ∞1 k=1 ∞1 proof goes by induction on n. By Theorems 3.3.8 and 3.3.14,

f (A + tX) − f (A) X,A = TfA+t (X). [1] t Applying Proposition 3.3.9(i), the estimate (3.3.6), and Theorem 3.3.8 completes the proof in the case n = 1. The proof of existence of the higher order derivatives follows the same steps as the proof of (5.3.2), with replacement of the perturbation formula and continuity of a multiple operator integral by their infinite dimensional counterparts stated in Propositions 4.3.14 and 4.3.15(i). We note that Proposition 4.3.15 is applicable thanks to Theorem 4.3.4. Now we justify the continuous Fréchet differentiability. The first order Fréchet differentiability follows from the representation (5.3.9) and Propositions 2.11.3

5.3 Differentiation of Operator Functions

137

and 3.3.9(i). Applying the connection between the Fréchet differential and Gâteaux derivative (2.11.2) along with (5.3.9) gives D11 f (A + X)(X1 ) − D11 f (A)(X)     = TfA+X,A+X (X1 ) − TfA+X,A (X1 ) + TfA+X,A (X1 ) − TfA,A [1] [1] [1] [1] (X1 ) . Applying Proposition 3.3.9(i) confirms continuity of the Fréchet differential D11 f (A). The proof of a higher order continuous Fréchet differentiability is technically more involved, and we refer the reader to [119, Theorem 4.1] for details.   Every operator differentiable function arises from a sufficiently smooth scalar function. Theorem 5.3.6 Let f ∈ C(R). If t → f (A + tX) − f (A) is differentiable as a function on R to the space B(H) equipped with the strong operator topology for all A ∈ Dsa and X ∈ Bsa (H) for every separable Hilbert space H, then f is operator 1 (R) . Lipschitz on R and, in particular, f ∈ B11 loc Proof The result follows from combination of [9, Theorem 1.2.4] and Theorem 5.1.6(ii).   Remark 5.3.7 Although existence of derivatives of operator functions for all operators A and X requires smoothness of the respective scalar function, operator derivatives at a particular operator point A along a particular direction X can exist for nonsmooth scalar functions [18]. We also refer the reader to Sect. 5.3.7, where the connection between operator Lipschitzness and operator differentiability is discussed.

5.3.4 Differentiation in Sp Along Linear Paths of Self-adjoints In this section we will see that the set of differentiable functions with respect to the Sp -norm, 1 < p < ∞, is larger than the set of S1 -differentiable functions. The following result is due to [110, Theorems 7.15, 7.17, 7.18]. Theorem 5.3.8 Let 1 < p < ∞. (i) A function f : R → R is Gâteaux Sp -differentiable at all A ∈ Bsa (H) if and only if f is differentiable on R and has bounded derivative on all compact subsets of R. Moreover, every differentiable function f : R → R with f  ∈ p Cb (R) is Gâteaux Sp -differentiable at every A ∈ Dsa along Ssa . The respective Gâteaux derivative is given by DG,p f (A) = TfA,A [1] . (ii) A function f : R → R is Fréchet Sp -differentiable at all A ∈ Bsa (H) if and only if f ∈ C 1 (R). The respective Fréchet differential is given by Dp f (A) = TfA,A [1] .

138

5 Applications

It is proved in [24, Theorem 5.5] that a function f is n times Fréchet Sp differentiable at every self-adjoint operator if f is in the Wiener class Wn+1 (R), which is improved in Theorem 5.3.5. It is proved in [55, Theorem 4.1] that a function f in C n (R) is n times Gâteaux S2 -differentiable at every bounded self-adjoint operator and, under the additional assumption “f (j ) is bounded, j = 0, . . . , n”, at every self-adjoint operator. We state below strengthening of these results due to [119, Theorems 3.4, 3.6, 3.7(ii)]. Theorem 5.3.9 Let 1 < p < ∞, n ∈ N. The following assertions hold. (i) Let f ∈ C n (R) satisfy f  , . . . , f (n−1) ∈ Cb (R) and f (n) ∈ C0 (R). Then f is n times continuously Fréchet Sp -differentiable at every A ∈ Dsa and Dpk f (A)(X1 , . . . , Xk ) =

 σ ∈ Symk

TfA,...,A (Xσ (1), . . . , Xσ (k) ), [k]

(5.3.11)

is given by for every k = 1, . . . , n, and all X1 , . . . , Xk ∈ Sp , where TfA,...,A [k] Definition 4.3.3 and Symk denotes the group of all permutations of the set {1, . . . , k}. (ii) Let f : R → C be a locally Lipschitz function. Then f is n times continuously Fréchet Sp -differentiable at every A ∈ Bsa (H) and (5.3.11) holds if and only if f ∈ C n (R). (iii) Let f ∈ C n (R) satisfy f  , . . . , f (n) ∈ Cb (R). Then, f is n − 1 times continuously Fréchet Sp -differentiable and n times Gâteaux Sp -differentiable at every A ∈ Dsa , with n f (A)(X) = n! TfA,...,A (X, . . . , X) DG,p [n]

for all X = X∗ ∈ Sp . We note that the results of Theorems 4.3.10 and 4.3.13 play a crucial role in the proof of Theorem 5.3.9, but omit the proof. We also note that the assumption “f ∈ C 1 (R) and f  is bounded” is not sufficient for Fréchet differentiability of f at an arbitrary unbounded operator, as demonstrated in [110, Example 7.20]. We also have the following characterization of n times Gâteaux Sp -differentiable functions extending the respective result of Theorem 5.3.8. The sufficient conditions are established [52] and the necessary conditions in [119, Proposition 3.9]. Theorem 5.3.10 Let 1 < p < ∞ and n ∈ N. (i) Let f : R → C be a locally Lipschitz function. Then, f is n times Gâteaux Sp -differentiable at every A ∈ Bsa (H) if and only if f is n times differentiable on R and f (n) is bounded on all compact subsets of R. (ii) Let f : R → C be a Lipschitz function. Then, f is n times Gâteaux Sp differentiable at every A ∈ Dsa if and only if f is n times differentiable on R and f  , . . . , f (n) are bounded.

5.3 Differentiation of Operator Functions

139

Problem 5.3.11 Find criteria for higher order Fréchet Sp -differentiability, 1  p  ∞, of an operator function f (A), A ∈ Dsa , in terms of smoothness properties of the respective scalar function f : R → C. The results below improve the bound (5.3.10) in the case 1 < p < ∞ and, when the norm  · 1 = Tr (| · |) is replaced with the smaller seminorm |Tr (·)|, in the case p = 1. Theorem 5.3.12 Let A ∈ Dsa , X ∈ Bsa (H), n ∈ N and f ∈ C n (R). If A is unbounded assume also that f  , . . . , f (n−1) ∈ Cb (R) and f (n) ∈ C0 (R). The following assertions hold. (i) If X ∈ Sp , 1 < p < ∞, then  n  d   (n) n (n) n     dt n f (A + tX) t =0   cp,n f ∞ Xpn  cp,n f ∞ Xp . p (5.3.12) (ii) If X ∈ Sn , then   n     Tr d f (A + tX)   cn f (n) ∞ Xn . n   t =0 n dt

(5.3.13)

Moreover, if A is bounded, then f (n) ∞ in (5.3.12) and (5.3.13) can be replaced with max |f (n) (t)|. t ∈σ (A)

Proof (i) It follows from Theorem 4.3.11 and (5.3.11) that  n  d   (n) n     dt n f (A + tX) t =0   cp,n f ∞ Xpn . p Since  · pn   · p , the inequality (5.3.12) follows. (ii) We note that  Tr

 dn  f (A + tX) t =0 dt n



  = Tr TψA,...,A (X, . . . , X )X , " #$ %

(5.3.14)

n−1

where ψ(λ0 , λ1 , . . . , λn−1 ) = f [n] (λ0 , λ0 , λ1 , . . . , λn−1 ). The result follows upon applying Hölder’s inequality and Theorem 4.3.10 in (5.3.14). More details can be found in [163, p. 536].  

140

5 Applications

5.3.5 Differentiation of Functions of Contractive and Dissipative Operators n (T), that is, Let n ∈ N and let P+ be the Riesz projection on B∞1

P+ (f ) : z →

 n0

n F f (n)zn , f ∈ B∞1 (T).

n (T). Define The operator P+ is bounded on B∞1



n (T) B∞1

 +

n := P+ B∞1 (T).

 n  ¯ and The functions in B∞1 (T) + admit a natural extension on the closed unit disc D they are analytic in D. The following result is established in [147, Theorem 5.3]. Theorem 5.3.13 Let A be a contraction on H and let X  ∈ B(H) be such that n (T) + . Then, the function f A + X is a contraction. Let n ∈ N and f ∈ B∞1 is differentiable n times at A along the direction X in the sense of Gâteaux in the operator norm,  dn f (A + tX)t =0 = n! TfA,...,A (X, . . . , X), [n] n dt and  n  d   n    n (T) X .  dt n f (A + tX) t =0   cn f B∞1 The next estimate follows from Theorems 4.3.19 and 5.3.13. Theorem 5.3.14 Let A be a contraction on H and let X ∈ Sp , 1 < p < ∞, be such ∞  |j |n |F f (j )| < that A+X is a contraction. Let n ∈ N and let f ∈ C n (T) satisfy j =0

∞. Then,

 n    d    cp,n f (n) ∞ Xn .  f (A + tX) p  dt n t =0  p

The next result is due to [147, Theorem 6.1]. Theorem 5.3.15 Let A be a contraction on H and let X ∈ B(H) be such that A + X is a contraction. Let f be a function analytic in D such that f  ∈ A(D). Then, the function f is Gâteaux differentiable at A along the direction X in the Hilbert-Schmidt norm and DG,2 f (A) = TfA,A [1] .

5.3 Differentiation of Operator Functions

141

 n  n (R) for which supp F f ⊂ Let B∞1 (R) + denote the set of those f ∈ B∞1 [0, ∞). The next result is obtained in [6, Theorem 9.1]. Theorem 5.3.16 Let A be a maximal dissipative operator densely defined in H and let X∈ B(H)be such that A + X is a maximal dissipative operator. Let n ∈ N and n (R) . Then, the function f is n times Gâteaux differentiable at A along f ∈ B∞1 + the direction X in the operator norm,  dn f (A + tX)t =0 = n! TfA,...,A (X, . . . , X), [n] n dt and  n  d   n    n (R) X .  dt n f (A + tX) t =0   cn f B∞1

5.3.6 Differentiation in Noncommutative Lp -Spaces Differentiability of operator functions in noncommutative Lp -spaces and general noncommutative symmetric spaces is studied in [24, 63]. The following results are obtained in [63, Theorem 5.16, Corollary 7.3] for noncommutative symmetric spaces E of a semifinite von Neumann algebra, including the noncommutative Lp -spaces, 1  p  ∞. Theorem 5.3.17 Let (M, τ ) be a semifinite von Neumann algebra, let E be a separable symmetric Banach function space on (0, ∞) and E = E(M, τ ) the respective noncommutative symmetric space. If f is such that f [1] ∈ A1 , then f is Gâteaux operator differentiable along Esa at every point AηMsa and the Gâteaux derivative with respect to the E-norm equals DG,E f (A) = TfA,A [1] . 1 (R), then f is Gâteaux operator differentiable along Corollary 5.3.18 If f ∈ B∞1 Esa at every point AηMsa for every semifinite von Neumann algebra (M, τ ) and noncommutative symmetric space E = E(M, τ ).

A sufficient condition for differentiability in noncommutative Lp -spaces, 1 < p < ∞, that does not involve a separation of variables of f [1] is established in [63, Corollary 6.10]. Theorem 5.3.19 If f : R → R is continuously differentiable and f  is of bounded variation, then f is Gâteaux operator differentiable along Lp (M, τ )sa , 1 < p < ∞, at every point AηMsa and the Gâteaux derivative with respect to the Lp (M, τ )norm equals DG,p f (A) = TfA,A [1] . A necessary and sufficient condition for Gâteaux differentiability in the context of an arbitrary semifinite von Neumann algebra is known only for noncommutative L2 -spaces, and it is due to [63, Proposition 6.11].

142

5 Applications

Theorem 5.3.20 A Lipschitz function f : R → R is Gâteaux differentiable along every self-adjoint direction in L2 (M, τ ) at every point AηMsa for every semifinite von Neumann algebra (M, τ ) if and only if f is continuously differentiable. In this case the Gâteaux derivative with respect to the L2 (M, τ )-norm equals DG,2 f (A) = TfA,A [1] . Higher order Fréchet differentiability in the ideal Lp (M, τ ) = Lp (M, τ ) ∩ M, 1  p < ∞, of a semifinite von Neumann algebra (M, τ ) is obtained in [24, Theorem 5.7] for a more restrictive class of functions than the one found for the first order differentiability. In fact, the latter result holds for more general symmetrically normed ideals. We state this result below. Theorem 5.3.21 Let M be a semifinite von Neumann algebra acting on a separable Hilbert space H and I a symmetrically normed ideal of M with property (F ). Let n ∈ N. Then, every f ∈ Wn+1 (R) is n times continuously Fréchet differentiable at every AηMsa and DIn f (A)(X1 , . . . , Xn ) =

 σ ∈ Symn

TfA,...,A (Xσ (1) , . . . , Xσ (n) ). [n]

The following problem generalizes Problem 5.3.11. Problem 5.3.22 Find criteria for an arbitrary order Gâteaux and Fréchet differentiability of operator functions in noncommutative symmetric spaces.

5.3.7 Gâteaux and Fréchet I-Differentiable Functions Let I be a symmetrically normed (s. n.) ideal of B(H), including the case I = B(H). Denote by Isa the set of self-adjoint elements in I. A function f ∈ C(R) is called Gâteaux I-differentiable at A ∈ Bsa (H) along I Isa if there is a bounded linear operator DG f (A) from Isa to I such that  1   I f (A)(X) = 0 f (A + tX) − f (A) − tDG I t →0 t lim

I for each X ∈ Isa and t ∈ R. The operator DG f (A) is called the Gâteaux II derivative of f at A. If f is real-valued then DG f (A): Isa → Isa . A function f ∈ C(R) is called Fréchet I-differentiable at A ∈ Bsa (H) along Isa if

 1    I f (A)(X) = 0 f (A + X) − f (A) − DG X →0 XI I I lim

for each X ∈ Isa .

5.3 Differentiation of Operator Functions

143

Clearly, Fréchet I-differentiable functions are also Gâteaux I-differentiable. Let  be an open set in R containing 0 and set Bsa (H)() := {A ∈ Bsa (H) : σ (A) ⊂ }. Denote by GDI () the space of all Gâteaux I-differentiable functions at all A ∈ Bsa (H)() along Isa and by FDI () the space of all Fréchet I-differentiable functions at all A ∈ Bsa (H)() along Isa . I-Lipschitz and Gâteaux I-Differentiable Functions In many cases the difference between I-Lipschitz and Gâteaux I-differentiable functions is not that big. It is proved in [103, Corollary 3.3] that if I = S1 is separable and an operator A ∈ Bsa (H) has no eigenvalues, then each I-Lipschitz function g on R is Gâteaux I-differentiable at A along Isa . In particular, all Lipschitz functions in the usual sense are Gâteaux Sp -differentiable, 1 < p  ∞, p at all A ∈ Bsa (H) without eigenvalues along Ssa . However, for Gâteaux Idifferentiability of f at all A ∈ Bsa (H), we need differentiability of f. The following result is established in [103, Theorem 3.6]. Theorem 5.3.23 Let  be an open set in R containing 0. Let I be a separable s. n. ideal. Then, the following properties are equivalent. (i) f ∈ I-Lip() and f is differentiable on . (ii) f ∈ GDI () (at all diagonalizable A ∈ Bsa (H)() if I = S1 ). (iii) f is Gâteaux I-Differentiable at all A ∈ Bsa (H)() along Fsa (at all diagonalizable A ∈ Bsa (H)() if I = S1 ). (iv) f is Gâteaux I-Differentiable at all A ∈ Fsa () along Fsa and, for each I compact I ⊂ , there is KI > 0 such that DG f (A)I  KI for all A ∈ Fsa (I ). For ideals Sp , 1 < p < ∞, the result of Theorem 5.3.23 is improved in [110, Theorem 7.15] (a refinement of Theorem 5.3.8): f ∈ GDSp () if and only if f is differentiable on  and has bounded derivative on all compact subsets of . Unlike the spaces GDSp (), 1 < p < ∞, there is no full description of the spaces GDS1 (), GDS∞ (), GDB(H) (). However, the theorem below links the spaces GDS∞ () and S∞ -Lip(). It follows from [103, Theorem 3.7, Corollary 3.10]. Theorem 5.3.24 Let  ⊆ R be an open set. Then, GDS1 () ⊆ GDS∞ () = S1 -Lip() = S∞ -Lip() = B(H)-Lip(). Problem 5.3.25 Let I = Sp , 1  p  ∞, be a separable s. n. ideal and f :  → R a continuous function. Does f ∈ GDI () if and only if f has bounded derivative in all compact subsets of ?

144

5 Applications

Fréchet I-Differentiable Functions Let I be a separable s. n. ideal or B(H) and  ⊆ R an open set. Similarly to Theorem 5.3.23, a function f ∈ C(R) is Fréchet I-differentiable at all A ∈ Bsa (H)() along Isa if and only if f is Fréchet I-differentiable at all A ∈ Bsa (H)() along Fsa (see [103, Theorem 4.1]). It is proved in [110, Theorem 7.17] (a refinement of Theorem 5.3.8) that FDSp () = C 1 () for 1 < p < ∞. We summarize some results on S∞ - and B(H)-differentiability below. Theorem 5.3.26 Let  ⊆ R be an open set. Then, C 2 (R)  FDB(H) () = FDS∞ () = GDB(H) () ⊆ GDS∞ () ∩ C 1 (). (5.3.15) Proof By, for instance, [73], C 2 (R) ⊂ FDB(H) (). Moreover, by [103, Proposition 2.4], FDB(H) () ⊆ C 1 (). By [103, Theorem 4.3], FDB(H) () = FDS∞ () = GDB(H) (). The latter implies GDB(H) () ⊆ B(H)-Lip() ∩ C 1(). Recalling GDS∞ () = B(H)-Lip() (see Theorem 5.3.24) completes the proof of (5.3.15). We also recall that B(H)-Lip() contains functions not in C 1 () (see Theorem 5.1.13) and there are functions in C 1 () that do not belong to B(H)-Lip() (see Remark after Theorem 5.1.8).   The relation between the spaces of Gâteaux differentiable and operator Lipschitz functions on I is studied in [103] and summarized in Theorem 5.3.28 below. We need the following auxiliary result due to [103, Proposition 7.2 and Theorem 7.3]. Theorem 5.3.27 Let I be a compact subset of R. Let ·I denote the norm on B(H)-Lip(I ) given by f I := sup |f (t)| + k(f, I ), t ∈I

where k(f, I ) is the minimal constant D satisfying (5.1.14). Then, the following assertions hold. (i) (B(H)-Lip(I ), ·I ) is a commutative Banach *-algebra. (ii) If I has a non-empty interior, then (B(H)-Lip(I ), ·I ) is not separable. Let  be an open set in R. The space B(H)-Lip() is a commutative *-algebra. Endowed with the family of seminorms {·I }I , where I are compacts in , the algebra (B(H)-Lip(), {·I }I ) becomes a complete space which is not separable by the theorem above. The following result is derived in [103, Theorem 7.9 and Theorem 7.12]. Theorem 5.3.28 Let  be an open set in R. The space GDB(H) () is a closed subalgebra of B(H)-Lip(). Moreover, the polynomials are dense in GDB(H) (), so GDB(H) () is a separable *-subalgebra of nonseparable B(H)-Lip().

5.4 Taylor Approximation of Operator Functions

145

5.4 Taylor Approximation of Operator Functions Let A, B ∈ Dsa and U be unitary. Suppose f is a C n function on R or T as applicable and consider the Taylor remainders n−1   1 dk  Rn,f,A (B) := f (A + B) − f (A + tB) t =0 k! dt k

(5.4.1)

k=0

in the self-adjoint case and Qn,f,U (B) := f (eiB U ) −

n−1   1 dk f (eit B U )t =0 k k! dt

(5.4.2)

k=0

in the unitary case. The Taylor remainders R1,f,A (B) = f (A + B) − f (A) and Q1,f,U (B) = f (eiB U ) − f (U ) of order n = 1 are studied in Sect. 5.1. In this section we discuss existence and estimates for remainders of order n  2.

5.4.1 Taylor Remainders of Matrix Functions The theorem below demonstrates that the study of matrix Taylor remainders can be reduced to the study of multilinear Schur multipliers. In the particular case n = 2 this result is justified in [53, Theorem 16] and [54, Theorem 6]. The extension to an arbitrary n  2 in the self-adjoint case is relatively standard; the extension in the unitary case is proved in [205]. Theorem 5.4.1 (i) Let f be n times differentiable on R, A, B ∈ Bsa (2d ). Then, for the remainder defined in (5.4.1), Rn,f,A (B) = TfA+B,A,...,A (B, . . . , B). [n]

(5.4.3)

(ii) Suppose f ∈ C n (T), A is a self-adjoint and U unitary operators in B(2d ). Then, for the remainder defined in (5.4.2), Qn,f,U (A) =

n 



l=1 j1 ,...,jl  1 j1 +···+jl =n

iA

Tfe[l] U,U,...,U

∞  (iA)j2 (iA)jl (iA)k U, U, . . . , U . k! j2 ! jl ! k=j1

(5.4.4)

146

5 Applications

Proof (i) Using (3.1.8) and Theorem 5.3.2, we obtain (B) − Rn,f,A (B) = TfA+B,A [1]

n−1 

TfA,A,...,A (B, . . . , B). [k]

k=1

Applying Proposition 4.1.5 with m = 1 to the difference TfA+B,A (B) − [1] TfA,A [1] (B) on the right hand-side of the equality above, we obtain (B, B) − Rn,f,A (B) = TfA+B,A,A [2]

n−1 

TfA,A,...,A (B, . . . , B). [k]

k=2

Applying the representation (4.1.9) (n − 1)-times yields (5.4.3). (ii) We will prove that for all 1  m  n the formula Qm,f,U (A) =

m 



l=1 j1 ,...,jl  1 j1 +···+jl =m

iA

Tfe[l] U,U,...,U

∞  (iA)j2 (iA)jl (iA)k U, U, . . . , U k! j2 ! jl ! k=j1

(5.4.5) holds, using induction. The formula (5.4.4) will follow from (5.4.5) for m = n. For m = 1 by (3.1.10), we have iA

Q1,f,U (A) = f (eiA U ) − f (U ) = Tfe[1]U,U (eiA U − U ) iA

= Tfe[1]U,U

∞  (iA)k k=1

k!

U ,

that is, (5.4.5) holds for m = 1. Assume that (5.4.5) holds for m = p < n, that is, we have Qp+1,f,U (A) =

p 



l=1 j1 ,...,jl  1 j1 +···+jl =p

−

iA

Tfe[l] U,U,...,U

 1 dp f (eit A U )t =0 . p p! dt

∞  (iA)j2 (iA)jl (iA)k U, U, . . . , U k! j2 ! jl ! k=j1

5.4 Taylor Approximation of Operator Functions

147

Applying Theorem 5.3.4 and decomposing i p = i j1 . . . i jl , we obtain Qp+1,f,U (A) =

p 



iA

l=1 j1 ,...,jl  1 j1 +···+jl =p p 

−

Tfe[l] U,U,...,U



l=1 j1 ,...,jl  1 j1 +···+jl =p

∞  (iA)j2 (iA)jl (iA)k U, U, . . . , U k! j2 ! jl ! k=j1

TfU,...,U [l]

(iA)j1 j1 !

U, . . . ,

(iA)jl U . jl !

The latter can be rewritten as Qp+1,f,U (A) =

p 



l=1 j1 ,...,jl  1 j1 +···+jl =p

∞ iA  (iA)j2 (iA)jl (iA)k Tfe[l] U,U,...,U U, U, . . . , U k! j2 ! jl ! k=j1

(iA)j1

(iA)jl U j1 ! jl ! (iA)j1 (iA)j1 iA (iA)jl (iA)jl

U, . . . , U − TfU,...,U U, . . . , U . + Tfe[l] U,...,U [l] j1 ! jl ! j1 ! jl ! iA

− Tfe[l] U,...,U

U, . . . ,

Applying multilinearity of the multiple operator integral and the property (4.1.9), we obtain Qp+1,f,U (A) =

p 



l=1 j1 ,...,jl  1 j1 +···+jl =p

+

p 

iA Tfe[l] U,U,...,U



l=1 j1 ,...,jl  1 j1 +···+jl =p

iA

∞  (iA)j2 (iA)jl (iA)k U, U, . . . , U k! j2 ! jl ! k=j1 +1

U,U,...,U Tfe[l+1]

∞  (iA)k k=1

=: S1 + S2 .

k!

U,

(iA)j1 (iA)jl U, . . . , U j1 ! jl ! (5.4.6)

Making the substitution i1 = j1 + 1, ir = jr for 2  r  l in the first sum on the right-hand side of (5.4.6), we obtain S1 =

p 



l=1 i1  2,i2 ,...,il  1 i1 +···+il =p+1

iA

Tfe[l] U,U,...,U

∞  (iA)i2 (iA)il (iA)k U, U, . . . , U . k! i2 ! il ! k=i1

148

5 Applications

Relabeling the summands of S2 by the mapping l → l − 1, and then making the substitution i1 = 1, ir = jr−1 for 2  r  l, we obtain S2 =

p+1 



iA

l=2 j1 ,...,jl−1  1 j1 +···+jl−1 =p

=

p+1 

Tfe[l] U,U,...,U



∞  (iA)k

iA

l=2 i1 =1,i2 ,...,il  1 i1 +···+il =p+1

k!

k=1

Tfe[l] U,U,...,U

U,

(iA)j1 (iA)jl−1 U, . . . , U j1 ! jl−1 !

∞  (iA)i2 (iA)il (iA)k U, U, . . . , U . k! i2 ! il ! k=i1

Combining the two previous equalities with (5.4.6), we arrive at iA

Qp+1,f,U (A) = Tfe[1]U,U

∞  (iA)k k! k=p+1

+

p  l=2

 i1 =1,i2 ,...,il  1 i1 +···+il =p+1

iA

× Tfe[l] U,U,...,U



+

i1  2,i2 ,...,il  1 i1 +···+il =p+1

∞  (iA)i2 (iA)il (iA)k U, U, . . . , U k! i2 ! il ! k=i1

iA

U,U,...,U + Tfe[p+1]

∞  (iA)k k=1

=

p+1 



k! iA

l=1 i1 ,i2 ,...,il  1 i1 +···+il =p+1

U, iAU, . . . , iAU

Tfe[l] U,U,...,U

∞  (iA)i2 (iA)il (iA)k U, U, . . . , U , k! i2 ! il ! k=i1

which proves (5.4.5).

 

A coarse Hilbert-Schmidt bound for Taylor remainders of matrix functions can be found in [88, Theorem 2.3.1 (4)]. More delicate estimates for Taylor remainders can be derived from estimates for multiple operator integrals discussed in this book. In particular, the following estimate holds. Theorem 5.4.2 Let A, B ∈ Bsa (2d ) and [a, b] be the convex hull containing σ (A) ∪ σ (A + B). Let 1 < p < ∞, n ∈ N, and let f be n times differentiable on [a, b] such that f (n) is bounded. Then, there exists cp,n > 0 such that Rn,f,A (B)p  cp,n f (n) L∞ [a,b] Bnpn .

5.4 Taylor Approximation of Operator Functions

149

The result of Theorem 5.4.2 is extended to the general infinite-dimensional case in the next subsection, where the bounds in the unitary case are also discussed.

5.4.2 Taylor Remainders for Perturbations in Sp and B(H) In this subsection we collect estimates for Taylor remainders of operator functions with respect to different norms. As in the finite dimensional case, the study of operator Taylor remainders can be reduced to the study of multilinear operator integrals.

Self-adjoint Case 1 (R) ∩ B n (R). Theorem 5.4.3 Let A ∈ Dsa , B ∈ Bsa (H), n ∈ N, and f ∈ B∞1 ∞1 Then, the Taylor remainder admits the representation

Rn,f,A (B) =

1 (n − 1)!



1 0

(1 − t)n−1

 dn f (A + sB)s=t dt, n ds

(5.4.7)

where the integral converges in the strong operator topology. Moreover, if B ∈ Sp , n  p < ∞, then the integral converges in Sp/n -norm and Rn,f,A (B) ∈ Sp/n .

(5.4.8)

Proof It follows  from (5.3.9) and Proposition 4.3.15(i) that the derivative t → dn  f (A + sB) is continuous on [0, 1] in the strong operator topology and in n ds s=t p/n S . Hence, we have the integral representation for the remainder (5.4.7), which can be proved as in [181, Theorem 1.43] by applying functionals in the dual space  dn  (B(H))∗ and reducing the problem to the scalar case. Since t → ds n f (A + sB) s=t is continuous in Sp/n , (5.4.8) holds.   Combining the estimate (5.3.10) and representation (5.4.7) implies the following estimate for the remainder. 1 (R) ∩ B n (R), and Theorem 5.4.4 Let A ∈ Dsa , B ∈ Bsa (H), n ∈ N, f ∈ B∞1 ∞1 1  p  ∞. Then,

  Rn,f,A (B)  cn f B n (R) Bn . pn p ∞1

(5.4.9)

The estimate (5.4.9) can be improved and the set of functions f for which it holds can be substantially enlarged when 1 < p < ∞. The next result for operator Taylor remainders centered at a bounded self-adjoint operator A is obtained in [169, Theorem 4.1].

150

5 Applications

Theorem 5.4.5 Let 1 < p < ∞, n ∈ N. Then, there exists cn,p > 0 such that for f ∈ C n (R), A = A∗ ∈ B(H), and B = B ∗ ∈ Sp , Rn,f,A (B)p  cp,n f (n) ∞ Bnpn , where Rn,f,A (B) is defined in (5.4.1). Proof Since we work with bounded operators, we assume without loss of generality that f is supported in some compact set containing the spectra of operators, that is, f ∈ Ccn (R). Let Pr , 0  r < 1, be the Poisson kernel. For k ∈ N define the convolution fk := P1/ k ∗ f . Then, fk(m) − f (m) ∞ → 0, k → ∞, for all 0  m  n, since fk = P1/ k ∗ f (m) . Moreover, {fk }∞ k=1 ⊂ Wn (R), (see, e.g., [198]). By (5.4.7) and (5.3.12) we obtain (m)

(n)

Rn,fk ,A (B)p  cn,p fk ∞ Bnpn . Again using fk(n) = P1/ k ∗ f (n) and Young’s inequality, since P1/ k 1  1, for all k ∈ N, we obtain fk(n) ∞  P1/ k 1 f (n) ∞  f (n) ∞ and so Rn,fk ,A (B)p  cn,p f (n) ∞ Bnpn .

(5.4.10)

We now prove that     Tr Rn,fk ,A (B)C → Tr Rn,f,A (B)C as k → ∞ for every C ∈ S1 . Indeed, fk (A + B) − f (A + B)  fk − f ∞ → 0 and, similarly, fk (A) − f (A)  fk − f ∞ → 0.

(5.4.11)

5.4 Taylor Approximation of Operator Functions

151

Moreover, for all 1  q  n − 1, using subsequently the representation (5.3.9), Theorem 4.3.11, and the fact that f ∈ ∩n−1 m=1 Wm (R), we obtain  dq    dq    q fk (A + tB)t =0 − q f (A + tB)t =0  p dt dt  A,...,A  A,...,A = T [q] (B, . . . , B) − Tf [q] (B, . . . , B)p fk

 A,...,A   = T(f [q] (B, . . . , B) p −f ) k  (q)  q  cp,q fk − f (q) ∞ Bpq → 0, k → ∞. Along with Hölder’s inequality, the latter implies (5.4.11) for every C ∈ S1 ⊂ (Sp )∗ . Finally, by the properties (5.4.10), (5.4.11), Hölder’s inequality, and denseness of S1 in (Sp )∗ = Sp/(p−1), we conclude Rn,f,A (B)p =

sup C∈S1 , Cp/(p−1)  1

  Tr Rn,f

k ,A

 (B)C   cn,p · f (n) ∞ · Bnpn .

As a consequence of Theorem 5.3.10 along with extension of Theorem 4.3.13 to functions that are differentiable, but not continuously, and a version of Theorem 4.3.10 for the transformation  given by Definition 4.2.1 that are derived in [52], we obtain the following estimate for operator Taylor remainders centered at an unbounded self-adjoint operator A. Theorem 5.4.6 Let 1 < p < ∞, n ∈ N and A ∈ Dsa . Let f be n times differentiable on R such that f  , . . . , f (n) are bounded and let X = X∗ ∈ Spn . Then,   n−1    1 k   DG,p f (A)(X, . . . , X)  cp,n f (n) ∞ Xnpn . f (A + X) − f (A) −   k! k=1

p

When p = 1, the condition f ∈ C n (R) is not sufficient for the remainder to be in S1 . A counterexample with unbounded A in the case n = 2 is constructed in [53]. The following result for a bounded A and general n is derived in [169, Theorem 5.1]. Theorem 5.4.7 Let n ∈ N. There exist a separable Hilbert space H and selfadjoint operators A ∈ B(H) and B ∈ Sn (H) such that Rn,fn ,A (B) ∈ / S1 (H), for the function fn ∈ C n (R) given by ! fn (x) := x

n−1

− 1   |x| log  log |x| − 1 2 , x = 0 0, x = 0.

(5.4.12)

152

5 Applications

The proof of Theorem 5.4.7 builds on the following dimension dependent bound due to [169, Theorem 5.5]. Theorem 5.4.8 Let n, d ∈ N, d  2. For Ad , Bd ∈ Bsa (22d ) satisfying Theorem 5.1.11, there are operators X1 , . . . Xn ∈ B(22d ) with Xj n = 1, 1  j  n ∈ N, such that  Ad +Bd ,Ad ,...,Ad  1 T [n] (X1 , . . . , Xn )1  const (log d) 2 fn

for a function fn : Rn+1 → R given by (5.4.12). Proof By (3.1.8) and Theorem 5.1.11, ThA[1]d +Bd ,Ad (Bd ) = h(Ad + Bd ) − h(Ad )  const (log d) 2 Bd ∞ . 1

Hence, by Theorem 4.1.9((iii)),     Ad +Bd ,Ad 1 T [1] : S12d → S12d  = ThA[1]d +Bd ,Ad : B(22d ) → B(22d )  const (log d) 2 . h It is easy to check that fn[n] (x0 , 0 . . . , 0, xn ) = h[1] (x0 , xn ). Combining the latter with Theorem 4.1.9(i) (where p1 = · · · = pn = n and p = 1) implies   A +B ,A   Ad +Bd ,Ad ,...,Ad T [n] : Sn2d × · · · × Sn2d → S12d   Th[1]d d d : S12d → S12d  fn

1

 const (log d) 2 ,  

completing the proof.

Unitary Case The following result for functions of unitary operators is established in [169, Theorem 4.2]. Theorem 5.4.9 Let 1 < p < ∞, n ∈ N. Then, there exists cn,p > 0 such that for ϕ ∈ C n (T), A = A∗ ∈ Sp , and a unitary U ∈ B(H), Qn,ϕ,U (A)p  cp,n

n  l=1

where Qn,ϕ,U (A) is defined in (5.4.2).

ϕ (l) ∞ Anpn ,

5.4 Taylor Approximation of Operator Functions

153

The aforementioned result cannot be extended to the case p = 1. A counterexample for n = 2 is obtained in [54] and in the case of a general n in [169, Theorem 5.11]. Theorem 5.4.10 Let n ∈ N. There exist a separable Hilbert space H, a unitary operator U ∈ B(H), and a self-adjoint operator B ∈ Sn (H) such that / S1 (H), Qn,ϕn ,U (B) ∈ for the function ϕn ∈ C n (T) given by ϕn (z) := (z − 1)n−1 u(z),

(5.4.13)

where u is given by (5.1.9). The proof of Theorem 5.4.10 is derived from the dimension-dependent bound of the next consequence of Theorem 5.1.17 obtained in [169, Theorem 5.15]. Theorem 5.4.11 Let d ∈ N, d  3. Then, there exist unitary operators Hd , Kd ∈ B(22d+1) satisfying Theorem 5.1.17 and such that  K ,H ,...,H  1 n 1 d : Sn T d d  2 ς 2d+1 × · · · × S2d+1 → S2d+1  const (log d) for ς : Tn+1 → C given by ς (z0 , . . . , zn ) := z1 · z2 · · · · · zn−1 · ϕn[n] (z0 , . . . , zn ) with ϕn defined in (5.4.13).

5.4.3 Taylor Remainders for Unsummable Perturbations Unsummable perturbations naturally arise in the study of differential operators, as discussed in Sect. 5.5.1. Summability of the respective approximation remainders in such cases is ensured by suitable summability restrictions on resolvents of operators as well as by choice of sufficiently nice scalar functions. When unsummable perturbations produce unsummable Taylor remainders, non-Taylor approximations can be used, as demonstrated in Sect. 5.5.1. Let (M, τ ) be a semifinite von Neumann algebra equipped with a normal faithful semifinite trace. It follows from [23, Lemmas 1.4 and 1.7] that if AηMsa and A has τ -compact resolvent, then f (A) ∈ L1 (M, τ ) for every f ∈ Cc (R). More generally, it is proved in [193, Theorem 3.1] that multiple operator integrals built over H , f [n] and applied to tuples of operators in M attain their values in L1 (M, τ ) for f ∈ Ccn+1 (R). A consequence of this result and a specific estimate for the operator Taylor remainder are established in [193, Theorem 4.1] stated below.

154

5 Applications

Theorem 5.4.12 Let AηMsa have τ -compact resolvent, B ∈ Msa , n ∈ N, and > 0. Then, for f ∈ Ccn+1 ((a, b)), Rn,f,A (B) ∈ L1 (M, τ ) and   |τ (Rn,f,A (B))|  f (n) ∞ Bn (2n (n + 1) + cn ) (b − a + 1)n 1 + a,b,A,B   √ 3 ϕ (k+1) ∞ , × a ,b ,A,B + 2 (b − a + 1) 2 max k! 1k n where cn > 0, a,b,A,B is such that   sup τ EA+t B ((a, b))  a,b,A,B

t ∈[0,1]

and   μ a,b,A,B (I + A2 )−1 

1 (1 + max{a 2 , b 2 })(1 + B

+ B2 )

hold and ϕ is a smoothening of the indicator function of (a, b) satisfying Cc∞ ((a , b )), ϕ |(a,b) ≡ 1, and 0  ϕ  1, where a = a − , b = b + .

√ 4 ϕ ∈

Theorem 5.4.12 is proved by creating weights that reduce the problem to the case of summable perturbations.

5.5 Spectral Shift A spectral shift function originates from I. M. Lifshits’ work on quantum theory of crystals summarized in [120] that was followed by M. G. Krein’s seminal paper [113] starting a mathematical theory of this object. The spectral shift function has evolved into a fundamental object in perturbation theory. It enjoyed several breakthroughs in recent years thanks to development of a new approach to multiple operator integration. There are several surveys dedicated to theory of spectral shift functions of general operators and many papers on spectral shift functions for specific models. The earlier development of the subject and applications can be found in [42, 183, 214], its development up to 2013 is surveyed in [188]. Connections of the spectral shift function to such important objects of perturbation theory as scattering phase and perturbation determinant are briefly surveyed in [33]; connection of the spectral shift function to a spectral flow is discussed in Sect. 5.6. In this subsection, we state major results on the spectral shift function for general operators, emphasizing results obtained beginning 2014.

5.5 Spectral Shift

155

5.5.1 Spectral Shift Function for Self-adjoint Operators Summable Perturbations The following fundamental result is established in [111, 113], and [163, Theorem 1.1] in the cases n = 1, n = 2, and n  3, respectively. Theorem 5.5.1 If H ∈ Dsa and V ∈ Bsa (H) are such that V ∈ Sn ,

(5.5.1)

n ∈ N, then there exist a unique real-valued function ηn = ηn,H,V ∈ L1 (R), called the n-th order spectral shift function associated with the pair of operators H and H + V , and a constant cn > 0 such that ηn 1  cn V nn and    n−1   1 dk  f (H + sV ) s=0 = f (n) (t) ηn (t) dt, Tr f (H + V ) − k! ds k R

(5.5.2)

k=0

for f ∈ Wn (R). Remark 5.5.2 (i) The condition (5.5.1) arises in the study of perturbations of discrete Laplacians on a lattice. n (ii) The formula (5.5.2) is extended from Wn (R) to the Besov class B∞1 (R) ∩ 1 (R) in [143, 145], and [7] for n = 1, n = 2, and n  3, respectively. It B∞1 is proved in [148] that (5.5.2) with n = 1 holds if and only if f is operator Lipschitz, which solves the problem posed in [113], and in [55] that (5.5.2) with n = 2 holds for f for which the divided difference f [2] admits a certain Hilbert space factorization. We recall that differentiability of operator functions is discussed in Sect. 5.3. The trace on the left hand side of (5.5.2) is well defined by (5.4.8). (iii) When H = 2d , the formula (5.5.2) extends to f ∈ C n [a, b], where [a, b] is the convex hull of the set σ (H ) ∪ σ (H + V ) [194, Theorem 2.1]. The original Krein’s proof of (5.5.2) for n = 1 is complex analytic in essence. The result is derived explicitly for rank one V , then continued to the case of finite rank V , and by approximations induced for an arbitrary V ∈ S1 . A purely real analytic proof is constructed in [165] via several stages of approximations. Existence of ηn , n  2, is established implicitly apart from some particular cases.

156

5 Applications

Proof (Proof of Theorem 5.5.1 for n  2) Evaluating the trace in (5.4.7) gives   Tr Rn,f,H (V ) =

1 (n − 1)!



1

  dn  Tr f (H + sV ) s=t dt, ds n 

(1 − t)

n−1

0

where the remainder Rn,f,H (V ) is given by (5.4.1). By (5.3.13), there exists cn > 0 such that |Tr(Rn,f,H (V ))|  cn V nn f (n) ∞ .

(5.5.3)

By the Riesz representation theorem for functionals in (Ccn+1 (R))∗ , there exists a Borel measure νn such that νn   cn V nn

(5.5.4)

and  Tr(Rn,f,H (V )) =

R

f (n) (t) dνn (t),

(5.5.5)

for every f ∈ Ccn+1 (R). By approximations, (5.5.5) extends to the space Wn (R). We prove below that the measure νn in (5.5.5), n  2, is absolutely continuous. Assume first that V ∈ S1 . Then, for every f ∈ Ccn (R), integration by parts in (5.5.5) with n − 1 gives  Tr(Rn−1,f,H (V )) = −

R

f (n) (t)νn−1 ((−∞, t)) dt.

(5.5.6)

By (5.3.13) and the Riesz representation theorem, there exists a Borel measure μn−1 such that μn−1   cn−1 V n−1 n−1 and 

 d n−1 Tr f (H + sV )s=0 n−1 ds



 =

R

f (n−1) (t) dμn−1 (t)



=−

R

f (n) (t)μn−1 ((−∞, t)) dt,

for every f ∈ Ccn (R). Combining (5.5.6) and (5.5.7) implies  Tr(Rn,f,H (V )) =

R

  f (n) (t) μn−1 ((−∞, t)) − νn−1 ((−∞, t)) dt.

(5.5.7)

5.5 Spectral Shift

157

Thus, (5.5.2) in the case V ∈ S1 holds with ηn (t) = μn−1 ((−∞, t)) − νn−1 ((−∞, t)). It follows from (5.5.4) that ηn 1  cn V nn .

(5.5.8)

1 Let V ∈ Sn and let {Vk }∞ k=1 ⊂ S be such that limk→∞ V −Vk n = 0. Consider ∞ the sequence {ηn,H,Vk }k=1 satisfying

 Tr(Rn,f,H (Vk )) =

R

f (n) (t) ηn,H,Vk (t) dt,

k ∈ N.

(5.5.9)

By duality, we obtain  R

  ηn,H,V (t) − ηn,H,V (t) dt j k =

sup f ∈Ccn+1 , f (n) ∞  1

      (n)  ηn,H,Vj (t) − ηn,H,Vk (t) f (t) dt  .  R

(5.5.10)

By the result of the previous paragraph,       (n)  ηn,H,Vj (t) − ηn,H,Vk (t) f (t) dt  = |Tr(Rn,f,H (Vj ) − Rn,f,H (Vk ))|.  R

(5.5.11) By the uniform continuity of V → Rn,f,H (V ), which follows from (5.5.3), and the representations (5.5.10) and (5.5.11),  lim

j,k→∞ R

  ηn,H,V (t) − ηn,H,V (t) dt = 0. j k

Thus, the sequence {ηn,H,Vk }∞ k=1 converges to an integrable function, which we denote by ηn,H,V . Applying (5.5.8) to ηn,H,Vk for every k ∈ N, we deduce the bound ηn,H,V 1  cn V nn . Passing to the limit in (5.5.9) as k → ∞ completes the proof of the theorem.

 

158

5 Applications

The name “spectral shift function” was given to η1 by M. G. Krein. A reason for this name can be understood from the formula     η1 (λ) = Tr EH ((−∞, λ)) − Tr EH +V ((−∞, λ)) = card{λ ∈ σ (H ) : λ < t} − card{λ ∈ σ (H + V ) : λ < t} holding for H and V finite matrices. For n  2, not only shift of eigenvalues, but also displacement of eigenvectors comes into play. The spectral shift nature of ηn in the case of commuting H and V is demonstrated in [187]. In the case of noncommuting H and V , the larger the value of n, the more intricate ηn is. The following representation for ηn is established in the case V ∈ S2 for bounded H in [74, Theorem 5.1 (ii)] and for an unbounded H in [186, Theorem 4.1]. Theorem 5.5.3 Let H ∈ Dsa , let V = V ∗ ∈ S2 , and let n ∈ N, n  2. Then, ηn (t) =

Tr (V n−1 ) − (n − 1)!



t −∞

(5.5.12)

ηn−1 (s) ds

1 (n − 1)!   [n−2]   (λ − t)n−2 (λ1 , . . . , λn−1 ) d Tr EH (λ1 )V . . . EH (λn−1 )V , × + −

Rn−1

for a.e. t ∈ R. k appearing in (5.5.12) is defined by The function x+

! k x+

:=

xk

if x > 0

0

if x  0

, for k ∈ N ∪ {0},

where we use the convention x 0 = 1 for x > 0. For the divided difference appearing [n−2]  (λ1 , . . . , λn−1 ) = 0 if λ1 = · · · = in (5.5.12) we use the convention (λ−t)n−2 + λn−1 = t. We note that the representation (5.5.12) does not hold for V ∈ Sn \ S2 because the set function   A1 × · · · × An−1 → Tr EH (A1 )V . . . EH (An−1 )V defined on rectangular sets of Rn−1 can fail to extend to a countably additive measure of bounded variation on the Borel σ -algebra of Rn−1 [74, Section 4]. Discussion of further properties of the spectral shift functions and related open questions can be found in [192, 194, 195].

5.5 Spectral Shift

159

Unsummable Perturbations The condition V ∈ Sn is not satisfied by typical perturbations of differential operators, so different restrictions, including those appearing below, are imposed in the setting of differential operators. The next result is obtained in [50, Theorem 4.6] (see also [50, Remark 4.8]). Theorem 5.5.4 Let H ∈ Dsa and V ∈ Bsa (H) be such that (H − iI )−1 V ∈ Sn ,

(5.5.13)

  dλ n ∈ N, n  2. Then there exist ηn = ηn,H,V ∈ L1 R, 1+λ 2 and a constant cn > 0 such that ηn L1 R,

dλ 1+λ2

  cn (1 + V )n−1 (H − iI )−1 V n n

and   n−1   1 dk  f (H + sV ) s=0 Tr f (H + V ) − f (H ) − k! ds k k=1

 =



2n  (t − i) f (t) ηn (t) dt, n−1

d n−1 R

dt

 for every f ∈ span λ →

1 (λ−z)j

 : j ∈ N, j  2n, Im (z) > 0 .

The following result is established in [114] and [166, Theorem 3.5] in the cases n = 1 and n  2, respectively. Theorem 5.5.5 Let H ∈ Dsa and V ∈ Bsa (H) be such that (H + V − iI )−1 − (H − iI )−1 ∈ Sn ,  n ∈ N. Then, there exist ηn = ηn,H,V ∈ L1 R,

dλ n (1+λ2 ) 2



and a constant cn > 0 such

that ηn L1 R,

dλ n (1+λ2 ) 2

(5.5.14)

  cn (H + V − iI )−1 − (H − iI )−1 n n

160

5 Applications

and  n−1  Tr f (H + V ) − f (H ) −  =

R



 Tf [k] (Vj1 , Vj2 −j1 , . . . , Vjk −jk−1 )

k=1 j1 ,...,jk ∈{1,...,n−1} j1