Gibbs Semigroups

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Operator Theory Advances and Applications 273

Valentin A. Zagrebnov

Gibbs Semigroups

Operator Theory: Advances and Applications Volume 273 Founded in 1979 by Israel Gohberg

Series Editors: Joseph A. Ball (Blacksburg, VA, USA) Albrecht Böttcher (Chemnitz, Germany) Harry Dym (Rehovot, Israel) Heinz Langer (Wien, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan (Odessa, Ukraine) Wolfgang Arendt (Ulm, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa, IA, USA) Kenneth R. Davidson (Waterloo, ON, Canada) Fritz Gesztesy (Waco, TX, USA) Pavel Kurasov (Stockholm, Sweden) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Ilya Spitkovsky (Abu Dhabi, UAE)

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More information about this series at http://www.springer.com/series/4850

Valentin A. Zagrebnov

Gibbs Semigroups

Valentin A. Zagrebnov Université d’Aix-Marseille Institut de Mathématiques de Marseille Marseille, France

ISSN 2296-4878 (electronic) ISSN 0255-0156 Operator Theory: Advances and Applications ISBN 978-3-030-18876-4 ISBN 978-3-030-18877-1 (eBook) https://doi.org/10.1007/978-3-030-18877-1 Mathematics Subject Classification (2010): 47D03, 47D06, 46C05, 47B10 © 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of Robert A. Minlos and Hagen Neidhardt

Preface The theory of one-parameter semigroups of operators emerged as an important branch of mathematics in the forties when it was realised that this theory has direct applications to partial differential equations, random processes, infinitedimensional control theory, mathematical physics, etc. The theory of (strongly continuous) one-parameter semigroups is now generally accepted as an integral part of contemporary functional analysis. In this framework the compact strongly continuous semigroups have been for a long time an important subject for research, as in almost all applications to partial differential equations with bounded domains the corresponding solution semigroups turn out to be families of compact operators. From this point of view, in the present book the emphasis is on a special subclass of compact semigroups. In fact, we mainly focus on the strongly continuous semigroups with values in the trace-class ideal C1 pHq of bounded operators acting on a Hilbert space H. Historically, this class of semigroups is closely related to the so-called Density Matrix in Quantum Statistical Mechanics. For this reason throughout the text for strongly continuous semigroups with values in the ideal C1 pHq we follow the terminology introduced in the first papers on this subject:[Uhl71], [ANB75], and call them the Gibbs semigroups. The aim of this book is to give an accessible and complete introduction to different aspects of the theory of Gibbs semigroups. More than fifteen years has elapsed since the publication of the Leuven University Notes [Zag03b] on this topic. The present book constitutes a major expansion of those Notes with more details and new aspects, which almost doubles the number of chapters. It is halfway between a textbook and a monograph, which provides a systematic and a comprehensive up-to-date account of the Gibbs semigroup theory, addressed to students as well as to experienced researchers. To this aim the first three chapters: Chapter 1 (Semigroups and their generators), Chapter 2 (Classes of compact operators), and Chapter 3 (Trace inequalities), contain preliminary material for the reading of the remaining main chapters. These preliminaries are accessible to graduate students with prerequisites corresponding to an introductory course in functional analysis. To recall certain elements of the spectral analysis, Appendix A, Spectra of closed operators, is also included in the preliminaries. vii

viii

Preface

Each of the Chapters 1-3 concludes with Notes, which provide references to the literature, comments to discussions in each section of the corresponding chapter, and relevant historical remarks. The material of Chapters 1–3 and of Appendix A is standard and can be found in many sources. Therefore, the Notes are limited by knowledge and prejudices of the author. The core of the book is constituted by Chapter 4 (Gibbs semigroups), Chapter 5 (Product formulae for Gibbs semigroups), Chapter 6 (Symmetrically-normed ideals), and Chapter 7 (Product formulae in the Dixmier ideal ), where the principal topics are presented. Although the selection of material in these chapters is determined by the personal taste of the author, this part of the volume provides a self-consistent introduction to the theory of Gibbs semigroups for nonspecialists interested in this specific class of semigroups. The reader is expected to be familiar with basics of functional analysis and linear operator theory, but above this level, the necessary notations, definitions and propositions are provided in the text. The Appendix B, More inequalities, serves to recall some particular inequalities useful for the exposition in Chapter 3 and in Chapters 5–7 of the book. Appendix C, Kato functions, introduces and classifies different types of the Kato functions needed in the text. We also recall certain properties of these functions, that are used in Chapters 5–7. The Notes at the end of each of Chapters 4–7 and Appendix B, Appendix C contain references and some historical remarks related to the corresponding sections. Sometimes we use the space of the Notes in order to elaborate on a discussion launched in a particular section. Finally, Appendix D is dedicated to a review of results and of the literature closely related to the Gibbs semigroups. Mainly it is about evolution of our knowledge concerning convergence of the Trotter-Kato product formulae in the operators-norm and the trace-norm topologies. The exact formulation of results and applications to lifting the operator-norm estimates of rates of convergence to the trace-norm estimates are presented in Chapter 5 – Chapter 7. Recall that after the introduction of the self-adjoint Gibbs semigroups in the 1970s by [Uhl71] and [ANB75], the analysis of bounded [Uhl71] and infinitesimally Kato-small [ANB75] perturbations of generators was developed rather repidly. The analytic theory for the Kato-small perturbations of generators is more involved. It began developing in [Mai71] and then independently in [ZBT75], [Zag89]. Note that besides a purely mathematical interest, this activity was motivated by the study of the trace-class operator known in the Quantum Statistical Mechanics as a density matrix. We resume the analysis of strongly continuous semigroups with values in the trace-class ideal in Chapter 4. My interest in Gibbs semigroups in the early 1980s was renewed after the following question posed to me by Robert A. Minlos: ”Does the exponential Trotter product formula approximating these semigroups converge in the trace-norm topology?” This question was motivated by our discussion of the very new at the time infrared bounds method in the mathematical theory of phase transitions proposed

Preface

ix

by J. Fr¨ ohlich, B. Simon and T. Spencer [FSS76]. In particular, we were intrigued by its quantum version ([FrLi78], [FILS78], [DLS78]) since in this case the proof of the infrared bounds involves the limit of the Trotter product formula approximants under the trace. Finally, to obtain the infrared bound one has to interchange the trace and the limit of approximants. In fact this interchange is not problematic for quantum spin systems since the underlying Hilbert spaces are finite-dimensional, but it is so for, e.g., unbounded spins. A typical example is the problem of proving infrared bounds for the case of structural phase transitions in anharmonic quantum crystals [Fr76]. Then the arguments in the papers [DLP79], [PK87] (essentially motivated by [Fr76]) indicate that a combination of the Trotter product formula approximation and the path integral representation for the trace is sufficient for obtaining the infrared bounds and the localisation estimate [Kon94], [MVZ00]. A message delivered by Chapter 5 is that the answer to the question posed by Robert A. Minlos is affirmative. Consequently, this allows to interchange for Gibbs semigroups the limit in the Trotter product formula and the trace, which ensures the proof of the infrared bounds in general settings. Initially the convergence of the Trotter product formula in the trace-norm topology was proven in our paper [Zag88] for the Schrödinger (Gibbs) semigroups. The first results about the product formulae for abstract Gibbs semigroups in a Hilbert space, including a generalisation to convergence of the Trotter-Kato product formulae in the tracenorm, is due to our papers with Hagen Neidhardt [NZ90a], [NZ90b]. Moreover, Chapter 5 presents the trace-norm error bounds for the convergence rates of the Trotter-Kato product formulae. This stronger result is, in turn, due to the discovery that for strongly continuous semigroups the Trotter product formula converges in the operator-norm topology [Rog93]. This paper also suggested an error bound estimate for the rate of convergence. Note that between 1959 (H. Trotter [Tro59]) and 1993 (Dzh. Rogava [Rog93]) it was common knowledge that for self-adjoint strongly continuous semigroups the Trotter product formula converges only in the strong operator topology. Although they did not provide error bound estimates, our papers about self-adjoint Gibbs semigroups [NZ90a], [NZ90b], published in 1990 were the first to challenge this state of affairs. In [NZ98] the examination of the Rogava theorem allowed us to improve the estimate of the convergence rate in the operator-norm topology for the TrotterKato (i.e., non-exponential ) generalisation of the product formulae under the same conditions as in [Rog93] The first estimate of the trace-norm error bounds for the rate of convergence of the exponential Trotter-Kato product formula in the case of the self-adjoint Schr¨ odinger (Gibbs) semigroups is due to [IT98b]. In Chapter 5 the trace-norm error bounds for the rate of convergence are obtained by lifting the operatornorm error bounds. This general method was invented in [NZ99d]. It allows to lift all, including the optimal, estimates of the convergence rates for the TrotterKato product formulae in the operator-norm topology to trace-norm error bounds

x

Preface

estimates. The method is also able to cover the case of non-self-adjoint Gibbs semigroups [CZ01c] for a quite large class of Kato functions. Chapters 6 and 7 are devoted to extending the Trotter-Kato product formulae to the more abstract framework of the symmetrically-normed ideals and to the case of the Dixmier ideal. It turns out that results for the Gibbs semigroups (Chapter 5), and in general for the von Neumann-Schatten ideals, admit an extension to arbitrary symmetrically-normed ideals. Convergence of the Trotter-Kato product formulae in the norm of the symmetrically-normed ideals is proved together with estimates of the rates of convergence. These estimates correspond to the lifting of the operator-norm error bounds. To this end we essentially follow in Chapter 6 the results from [NZ99d] and our paper [Zag19]. The Trotter-Kato product formulae in the Dixmier ideal were considered for the first time in [NZ99d]. There under certain conditions the convergence of the (singular ) Dixmier trace for the Trotter-Kato product formulae was conjectured. I made these arguments explicit in order to prove in Chapter 7 the convergence of the Trotter-Kato product formulae in the Dixmier ideal topology. The estimates of the rate of convergence are similar to those in the symmetrically-normed ideals [Zag19]. The Notes at the end of each chapter attempt to complete the exposition by comments and references that should make the contents more accessible to the readers, including graduate students in mathematics and mathematical physics.

Preface

xi

Acknowledgements I would like to thank many people, who have contributed to my understanding of the Gibbs semigroups and who have made the writing of this book possible. First of all, I am grateful to my mentor Robert Adolfovich Minlos, who introduced me to functional analysis and sparked my interest in the problems surrounding the Trotter product formula. I must also mention Nicolae Angelescu, who many years ago brought to my attention the notion of the Gibbs semigroups. Never published with them on this particular topic, I benefited greatly from many discussions and collaborations with them in this and other projects. Second, I am thankful to my co-authors in publications on the Gibbs semigroups theory and the Trotter-Kato product formulae: Hagen Neidhardt, Takashi Ichinose, Pavel Exner, Hiroshi Tamura, Hideo Tamura, Vincent Cachia and Artur Stephan. It was always a pleasure collaborating with them. With Hagen Neidhardt we obtained a complete answer to the Minlos question concerning the Trotter-Kato product formulae trace-norm convergence for abstract Gibbs semigroups. In the beginning of 2000 Minlos strongly suggested that I should summarise the existing results on Gibbs semigroups as a detailed review. My first attempt to collect such results took the form the KU Leuven Notes [Zag03]. I am thankful to André Verbeure, the editor of this series, who initiated and supported this first enterprise, which was very helpful in the preparation of the present comprehensive volume. It is my pleasure to thank Michael Th. Rassias for reading the manuscript and for making numerous suggestions and corrections. Last but not least, I want to express my gratitude to my wife Galina. Her support over all these years has meant so much to me. Marseilles, February 2018 – January 2019 Valentin A. Zagrebnov Institut de Mathématiques de Marseille and Département de Mathématiques, AMU

Contents 1

2

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Semigroups and their generators 1.1 The exponential function . . . . . . . . . . 1.2 Strongly continuous semigroups . . . . . . . 1.3 Generators and quasi-bounded semigroups . 1.4 Norm and other continuity conditions . . . 1.5 Holomorphic semigroups . . . . . . . . . . . 1.6 Holomorphic semigroups on a Hilbert space 1.7 Perturbations of semigroups . . . . . . . . . 1.8 Notes . . . . . . . . . . . . . . . . . . . . .

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1 1 5 11 14 22 28 33 38

Classes of compact operators 2.1 Compact operators on a Hilbert space . . 2.2 The canonical form of a compact operator 2.3 Trace class and Cp pHq-ideals . . . . . . . . 2.4 Convergence theorems for Cp pHq . . . . . 2.5 Notes . . . . . . . . . . . . . . . . . . . .

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41 41 46 51 64 77

Trace inequalities 3.1 Singular values of compact operators . . . 3.2 Inequalities for s-numbers and eigenvalues 3.3 Trace and Cp pHq-norm estimates . . . . . 3.4 Monotonicity, convexity and inequalities . 3.5 Notes . . . . . . . . . . . . . . . . . . . .

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Gibbs semigroups 4.1 Gibbs semigroups . . . . . . . . . . . . 4.2 Norm continuity, revisited . . . . . . . 4.3 Generators . . . . . . . . . . . . . . . 4.4 P-perturbations of generators . . . . . 4.5 Holomorphic Gibbs semigroups . . . . 4.6 Pb -perturbations of Gibbs semigroups 4.7 Notes . . . . . . . . . . . . . . . . . .

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xiii

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Contents 143 . . . . . . . . 144

Product formulae for Gibbs semigroups 5.1 The Lie-Trotter product formula . . . . . . . . . . . 5.2 Trotter-Kato product formulae: operator-norm convergence, error bounds . . . . . . 5.3 Operator-norm convergence: self-adjoint and non-self-adjoint semigroups . . . . . . . . . . . . 5.4 Trotter-Kato product formulae: trace-norm convergence, error bounds . . . . . . . . 5.5 Product formulae: non-self-adjoint Gibbs semigroups 5.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . .

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Product formulae in symmetrically-normed ideals 6.1 Preliminaries . . . . . . . . . . . . . . . . . . 6.2 Symmetrically-normed ideals . . . . . . . . . 6.3 Convergence in Cφ pHq-ideals . . . . . . . . . . 6.4 Lifting for Trotter-Kato product formulae . . 6.5 Product formulae in Cφ pHq-ideals . . . . . . . 6.6 Product formulae: error bound estimates . . . 6.7 Notes . . . . . . . . . . . . . . . . . . . . . .

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Product formulae in the Dixmier 7.1 Ideals and singular traces . 7.2 Dixmier trace . . . . . . . . 7.3 Product formulae . . . . . . 7.4 Notes . . . . . . . . . . . .

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Appendix B. More inequalities B.1 The Araki inequality . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 The Araki-Lieb-Thirring inequality in symmetrically-normed ideals B.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271 271 274 274

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Appendix A. Spectra of closed operators A.1 Resolvents and spectra . . . . . . . . . . . . p A.2 Core σpAq of σpAq . . . . . . . . . . . . . . A.3 Subsets of the spectrum σpAq . . . . . . . . A.4 Approximate and essential spectra . . . . . A.5 Fredholm operators and essential spectrum A.6 Spectrum of compact operators . . . . . . . A.7 Example: the Volterra operator . . . . . . . A.8 Example: unbounded operators . . . . . . . A.9 Spectral mapping theorem for semigroups . A.10 Notes . . . . . . . . . . . . . . . . . . . . .

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Contents Appendix C. Kato functions pβ . . . . . . . . . . . . . C.1 Classes: K, Kα , K C.2 Auxiliary functions and K˚ . . . . . . . . C.3 Regularity, domination: Kr , KD , Ks-d , K1 C.4 Notes . . . . . . . . . . . . . . . . . . . .

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Appendix D. Lie-Trotter-Kato product formulae: comments bibliography D.1 From the strong to the norm convergence . . . . . D.2 Norm convergence: optimal rate . . . . . . . . . . . D.3 Norm convergence: non-self-adjoint semigroups and D.4 Trace-norm convergence . . . . . . . . . . . . . . . D.5 Unitary product formulae . . . . . . . . . . . . . .

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Index

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List of Symbols

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Chapter 1

Semigroups and their generators This chapter contains a brief account of basic results of the theory of generators of strongly continuous operator semigroups. We introduce here some notations and definitions indispensable for the following chapters and consider only some restricted type of semigroups and their perturbations. After we define of the strongly continuous exponential function, we consider contraction and quasi-bounded semigroups, emphasising the holomorphic semigroups on a complex separable Hilbert space. Preparing the introduction of the Gibbs semigroups we address the question of the continuity of semigroups in various topologies.

1.1

The exponential function

Let B be a Banach space. By LpBq :“ LpB, Bq we denote the Banach space of bounded linear transformations from B to itself. For any bounded operator A P LpBq one can define the exponential function t ÞÑ e´tA “ Ut pAq, for complex t P C, by the series 8 ÿ tn Ut pAq :“ pÁqn , (1.1) n! n“0 which is convergent in the operator norm }¨} on the Banach space LpBq. Therefore, for any A P LpBq, the mapping t ÞÑ Ut pAq is an entire } ¨ }-holomorphic operatorvalued function on the complex plane C. The (semi)group property Ut1 `t2 pAq “ Ut1 pAqUt2 pAq,

t1 , t2 P C

(1.2)

is a direct consequence of formula (1.1) as well as the equation } ¨ }- Bt Ut pAq “ pÁqUt pAq “ Ut pAqpÁq, © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1_1

(1.3) 1

Chapter 1. Semigroups and their generators

2

where } ¨ }- Bt stands for differentiation in the sense of the operator norm on B. Now, let A be an unbounded linear operator in B with domain dom A Ă B. Since now the series (1.1) is ill-defined, the existence of the exponential function Up¨q pAq : D Ñ LpBq for D Ă C is less obvious. One possibility of defining Ut pAq is an alternative to (1.1), namely, via the famous Euler formula ´ t ¯ń Ut pAq “ lim 1 ` A , nÑ8 n

(1.4)

` first for t P R` 0 :“ t0u Y R “ r0, `8q. The following proposition gives sufficient conditions for that to be feasible.

Proposition 1.1. Let A P CpHq be a closed linear operator with a dense domain dom A Ă B such that the following two conditions hold : (i) R´ :“ p´8, 0q “ RzR` 0 belongs to the resolvent set ρpAq of A; (ii) }pA ` λ1q´1 } ď λ´1 for λ ą 0. Then: (a) The operator A generates for t ě 0 an operator-valued (exponential ) function t ÞÑ Ut pAq P LpBq. (b) }Ut pAq} ď 1 for all t ě 0. (c) The mapping R` 0 Q t ÞÑ Ut pAqf is continuous for every f P B, i.e., strongly continuous in R` 0 with limtÓ0 Ut pAqf “ f . (d) For u0 P dom A the function uptq :“ Ut pAqu0 is strongly differentiable in R` and is a solution of the equation Bt uptq “ Á uptq with initial condition uptq|t“0 “ u0 . (e) Ut`s pAq “ Ut pAqUs pAq for all t, s ě 0 and U0 pAq “ 1. Proof. (a) Let us define for t ě 0 the sequence ´ t ¯ń , Ut,n pAq “ 1 ` A n

n “ 1, 2, . . .

(1.5)

Then condition (ii) implies that }Ut,n pAq} ď 1. Moreover, t ÞÑ Ut,n pAq is } ¨ }holomorphic in the right half-plane C` :“ tz P C : <e z ą 0u, inheriting this property from the resolvent Rζ pAq :“ pA ´ ζ1q´1

(1.6)

with ζ in the resolvent set ρpAq :“ tζ P C : Rζ pAq P LpBqu, see condition (i). Notice that ´ t ¯´pn`1q P LpBq, t ą 0, (1.7) } ¨ }- Bt Ut,n pAq “ pÁq 1 ` A n

3

1.1. The exponential function but Ut,n pAq is right continuous at t “ 0 only in the strong sense

(1.8)

s-lim Ut,n pAq “ U0,n pAq “ 1, tÓ0

which means that lim }pUt,n pAq ´ 1qf } “ 0, tÓ0

f P B.

Here t Ó 0 denotes t Ñ `0. By (1.8), ż t´ε dτ Bτ rUt´τ,k pAq Uτ,n pAqs f, Ut,n pAqf ´ Ut,k pAqf “ lim εÓ0

f P B.

ε

Using (1.7) we get Ut,n pAqf ´ Ut,k pAqf ż t´ε ´ τ t ´ τ ¯ 2´ t ´ τ ¯´pk`1q ´ τ ¯´pn`1q “ lim dτ ´ A 1` A 1` A f. εÓ0 ε n k k n Now let f P dom A2 . By the strong continuity of the integrand for τ P r0, ts and condition (ii) t2 ´ 1 1¯ 2 }Ut,n pAqf ´ Ut,k pAqf } ď ` }A f }, (1.9) 2 n k which means that tUt,n pAqf uně1 is a Cauchy sequence for f P dom A2 . Since for λ ą 0 dom A2 “ pA ` λ1q´1 dom A is dense in B, and since }Ut,n pAq} ď 1, it follows that the limit (1.4) exists in the strong sense Ut pAq “ s-lim Ut,n pAq, nÑ8

t ě 0.

(1.10)

(b) Thanks to the bound }Ut,n pAq} ď 1, relation (1.8) and the estimate (1.9), we have }Ut pAq} ď 1 and U0 pAq “ 1. (1.11) (c) Since t ÞÑ Ut,n pAq is strongly continuous for t ě 0 and since the estimate (1.9) is uniform in t for any finite interval 0 ď t ď T , the operator-valued function t ÞÑ Ut pAq : R` (1.12) 0 Ñ LpBq is strongly continuous in R` , including the right limit at t “ `0, with s-limtÓ0 Ut pAq “ 1. (d) From (1.7) and (1.8), it follows that żt ´ τ ¯´1 pUt,n pAq ´ 1qf “ ´ dτ 1 ` A Uτ,n pAqAf, f P dom A. (1.13) n 0 Since

´ τ ¯´1 s-lim 1 ` A Uτ,n pAq “ Uτ pAq nÑ8 n


4

uniformly for any finite interval 0 ď τ ď T , we get from (1.13) that żt pUt pAq ´ 1qf “ ´ dτ Uτ pAqAf, f P dom A.

(1.14)

0

On the other hand, due to (1.12) the vector-valued family tUτ pAqpAf quτ ě0 is continuous in τ for any f P dom A. Therefore, (1.14) shows that the vector valued function t ÞÑ Ut pAqf is differentiable with respect to t for all f P dom A Bt pUt pAqf q “ Ut pAqpÁf q,

t ą 0, f P dom A,

(1.15)

where by definition 1 Bt Ut“0 pAqf :“ lim pUt pAq ´ 1qf “ lim Ut pAqpÁf q. tÓ0 t tÓ0 In fact, (1.7) gives more. Since ´ t ¯´1 1` A B Ď dom A, n

t ą 0,

one has ´ t ¯´1 Bt Ut,n pAq “ Út,n pAqA 1 ` A n ´ t ¯´1 “ Á Ut,n pAq 1 ` A , n

(1.16)

and ´ ´ t ¯´1 t ¯´1 lim A 1 ` A u “ lim 1 ` A Au “ Au, nÑ8 nÑ8 n n

u P dom A,

because A is a closed operator. Hence, by virtue of (1.10) and the closedness of A, one gets that for f P dom A ´ t ¯´1 lim Ut,n pAqA 1 ` A f “ Ut pAqAf nÑ8 n ´ t ¯´1 “ lim A Ut,n pAq 1 ` A f nÑ8 n “ A Ut pAqf. (1.17) Therefore, Ut pAqf P dom A and the differential equation (1.15) takes the form Bt pUt pAqf q “ pÁqpUt pAqf q,

t ą 0, f P dom A.

(1.18)

Summarising, uptq “ Ut pAqu0 is a solution of the differential equation Bt uptq “ Á uptq,

(1.19)

5

1.2. Strongly continuous semigroups

provided u0 :“ up0q P dom A. (e) Let uptq be a solution of (1.19) with initial condition u0 “ up0q P dom A. This means that the vector-valued function up¨q : R0` Ñ dom A is strongly differentiable, i.e., the limit lim δ ´1 pupt ` δq ´ uptqq “ pBt uqptq

δÑ0

exists, and such that (1.19) holds for t ą 0. Then by virtue of (1.15) and (1.19), Bτ pUt´τ pAqupτ qq “ 0,

0 ă τ ă t,

with the left derivative lim Bτ pUt´τ pAqupτ qq “ 0. τ Ó0

Thus, for any 0 ď τ ď t one gets, for t ą 0, that Ut´τ pAqupτ q “ Ut pAqupτ “ 0q “ uptq P dom A.

(1.20)

This means that any solution of (1.19) can be uniquely written as uptq “ Ut pAqu0 ,

(1.21)

that is, Ut´τ pAqUτ pAqu0 “ Ut pAqu0 for any u0 P dom A. Since this holds on the dense set dom A Ă B for Ut pAq P LpBq, t ě 0, continuity yields the composition law: (1.22) Us pAqUτ pAq “ Us`τ pAq , s, τ ě 0, called the semigroup functional equation, on all elements of B. It is these observations that legitimate for Ut pAq the notation of exponential function: (1.23) Ut pAq “ e´tA P LpBq, t ě 0, generated by the operator A.

1.2

l

Strongly continuous semigroups

Proposition 1.1 motivates the following definition: Definition 1.2. An operator-valued function of t P R` 0 , t ÞÑ Ut P LpBq, or a family of bounded operators tUt utě0 , on a Banach space B is called a strongly continuous one-parameter semigroup (or C0 -semigroup) if (a) U0 “ 1, (b) Ut Us “ Ut`s for t, s P R` 0, (c) t ÞÑ Ut f P B is a continuous function of the parameter t P R` such that limtÓ0 Ut f “ f , for every f P B.

6


Therefore, in Section 1.1 we proved existence of a special class of strongly continuous semigroups with }Ut } ď 1, generated by operators satisfying the conditions of Proposition 1.1. Definition 1.3. An operator-valued function t ÞÑ Ut is called a strongly continuous contraction semigroup if it is a C0 -semigroup with the property that }Ut } ď 1 for all t ě 0. It is important that the converse of Proposition 1.1 is also true. For this purpose we first briefly discuss integration of functions taking values in the Banach spaces B and LpBq. Remark 1.4. (a) Recall that a function F : R Ñ LpBq is norm continuous (respectively norm differentiable) if it is continuous (respectively differentiable) with respect to the operator norm topology on the space LpBq, and F is strongly continuous (respectively strongly differentiable) if t ÞÑ F ptqf is continuous (respectively differentiable) in t P R with respect to the norm topology of the space B for all f P B. (b) By the uniform boundedness principle (Proposition 1.6), one gets that if the function F is strongly continuous, then it is locally uniformly bounded: MI :“ sup }F ptq} ă 8, tPI

for any compact I Ă R. The local boundedness yields that the product of two strongly continuous mappings is jointly strongly continuous. In particular, the mapping pt, sq ÞÑ F ptqupsq P B is norm continuous, for any t ÞÑ F ptq, which is LpBq-valued strongly continuous, and for s ÞÑ upsq, which is B-valued norm continuous. One also sees that the product of strongly differentiable mappings is strongly differentiable in each variable. (c) We recall that the strong continuity of t ÞÑ F ptq implies for all f P B the continuity of the function t ÞÑ }F ptqf }. Then, by the local uniform boundedness principle, the function I Q t ÞÑ }F ptq} is bounded by MI . (d) We stress the point that t ÞÑ }F ptq} is not necessarily continuous, and may not even be measurable, unless B is separable. (e) Below we systematically consider continuous one-parameter functions with values in Banach spaces B or LpBq. Since the continuous image of a separable space (in our case R), is separable, the corresponding integrals are well-defined (via the standard Darboux–Riemann scheme approximating this function by piecewise ş ş constant functions) and finite if and only if I ds}upsq} ă 8 or I dt}F ptq} ă 8. (f) Note that the space of bounded operators LpBq is nonseparable, but if B is separable it is convenient to use this fact to simplify some estimates. For example }F ptq} is then a locally bounded measurable function, and hence locally integrable for t P I. (g) If F ptq “ Ut is a strongly continuous semigroup on a (a priori nonseparable) space B, then this continuity and, hence, the local boundedness implies

7

1.2. Strongly continuous semigroups that 1 } lim tÑ0 t

żt dτ Uτ f ´ f } ď lim sup }Uτ f ´ f } “ 0, tÑ0 0ďsďt

0

(1.24)

for any f P B. Proposition 1.5. Let tUt utě0 be a contraction semigroup on a Banach space B. Then there exists a unique, closed, densely defined operator A with domain dom A, satisfying conditions (i) and (ii) of Proposition 1.1, such that Ut “ Ut pAq,

t ě 0.

(1.25)

Proof. Set Aptq :“ t´1 p1 ´ Ut q and define DpAq :“ tu P B : lim Aptq u existsu. tÓ0

Then, for u P DpAq, we can define the operator ˇ Au :“ lim Aptq u “ ´Bt Ut ˇt“`0 , tÓ0

(1.26)

see (1.15), that is, DpAq “ dom A. To verify that the closure dom A “ B, we follow Remark 1.4 and define for t ą 0 and u P B the Riemann integral żt ut :“ dτ Uτ u . (1.27) 0

Then we obtain the representation ż ż 1 s`t 1 s dτ Uτ u ´ dτ Uτ u . Apsq ut “ s 0 s t Since Ut is strongly continuous, (1.24) and (1.26) yield lim Apsq ut “ Aut “ u ´ Ut u, sÓ0

u P B.

(1.28)

Therefore, ut P dom A for each u P B and t ą 0. Consequently, for any s ą 0 the vector t´1 ut P dom A. Since (1.24) yields limsÓ0 t´1 ut “ u, the operator A is densely defined in B. To prove that A is closed we use (1.28) for u P dom A. Then by (1.27) one gets ż t

lim Apsq ut “

dτ Uτ Au “ u ´ Ut u,

sÓ0

u P dom A.

(1.29)

0

Here we used that for u P dom A the functions τ ÞÑ Uτ Apsq converge uniformly on r0, ts to the function Uτ Au when s Ó 0. Now we choose a sequence tun uně1 Ă dom A, such that un Ñ v and Aun Ñ w. Then, by (1.29) for any t ą 0, żt dτ Uτ Aun “ un ´ Ut un , 0


8

and the uniform convergence for n Ñ 8 of the functions τ ÞÑ Uτ A un on r0, ts implies that żt dτ Uτ w “ v ´ Ut v . 0

Therefore, one can take the limits: ż 1 t 1 lim dτ Uτ w “ lim pv ´ Ut vq. tÓ0 t 0 tÓ0 t Since by (1.24) the limit in the left-hand side exists and is equal to w, we deduce the existence of the limit in the right-hand side. By (1.26), this implies v P dom A and w “ Av and proves the closedness of A. To check the uniqueness of A and the properties (i) and (ii) we first note that for any τ ą 0 one obviously has Apτ q Ut “ Ut Apτ q . If u P dom A, the limτ Ó0 Ut Apτ q u “ Ut Au exists by (1.26). Then the limit limτ Ó0 Apτ q Ut u also exists. This means that Ut u P dom A, i.e., Ut : dom A Ñ dom A, as well as that AUt u “ Ut Au. Since, by the definition of Apτ ą0q , 1 lim Ut Apτ q u “ ´ lim pUt`τ u ´ Ut q “ ´Bt Ut u , u P dom A, τ Ó0 τ Ó0 τ our arguments prove also the right-differentiability of tUt uutě0 for any u P dom A. The left derivative at t ą 0 for u P dom A is calculated by using representation (1.29) and the limit (τ ă t) ż 1 1 t dσ Uσ Au “ Út Au. Bt Ut u “ lim pUt u ´ Ut´τ uq “ ´ lim τ Ó0 τ τ Ó0 τ t´τ Therefore, if u P dom A, then one gets the equation Bt Ut u “ ÁUt u “ Út Au.

(1.30)

Next we use (1.30) to verify (i) and (ii). To this end we analyse the Laplace ˆz of the semigroup Ut : transform U żT ż8 ´zt ˆ Uz “ lim dt e Ut “ dt e´zt Ut . (1.31) T Ñ8 0

0

The integral in the right-hand side of (1.31) is an improper strong Bochner integral, ˆz,T “ which is defined as the strongly convergent limit of the bounded operators tU şT ´zt dt e Ut uT ą0 for z P C` . Here, the proper Bochner integrals are well-defined 0 because the integrand is a strongly continuous function of t. Since }Ut } ď 1, (1.31) ˆz } ď p<e zq´1 . Now, let u P dom A. Then defines a bounded linear operator with }U by virtue of (1.30), Bt pe´zt Ut uq “ é´zt Ut pz1 ` Aqu,

<e z ą 0.

9

1.2. Strongly continuous semigroups Integration of this equality gives ż8 dt e´zt Ut pz1 ` Aqu, u“ 0

or, with v :“ pz1 ` Aqu, ż8 dt e´zt Ut v,

pz1 ` Aq´1 v “

<e z ą 0.

(1.32)

0

ˆz } ď p<e zq´1 , the equality (1.32), means that Since }U ˆz “ pz1 ` Aq´1 “ R´z pAq, U

(1.33)

coincides with the resolvent of the operator A at the point ζ “ ´z. Hence, the set <e ζ ă 0 belongs to ρpAq (i) and we get the estimate (ii) of Proposition 1.1. p2q p1q To prove uniqueness, let Ut and Ut be two different contraction semip1q p2q groups, i.e., Ut0 ‰ Ut0 for some t0 ą 0. Then there exists a linear functional p1q p2q ϕ P B ˚ and v P B such that ϕpUt0 vq ‰ ϕpUt0 vq, which, due to (1.32), im´1 ´1 plies that ϕppz1 ` A1 q vq ‰ ϕppz1 ` A2 q vq for some z with <e z ą 0. Thus Rζ pA1 q ‰ Rζ pA2 q, so A1 ‰ A2 . Conversely, two different generators correspond, by (1.32), to different semigroups. It remains to prove (1.25), i.e., that the semigroup tUt utě0 coincides with the one exhibited in Proposition 1.1 if we take as a generator the operator A just constructed above. Since this operator A satisfies the conditions of Proposition 1.1, the contraction semigroup Ut pAq “ e´tA is well defined for t ě 0. The Laplace transformation of this semigroup, calculated as in (1.31) and (1.32), gives ż8 dt e´zt Ut pAq “ pz1 ` Aq´1 . (1.34) 0

Therefore, (1.25) follows from the uniqueness we proved above.

l

Since in this book we consider also other types of semigroup continuity (see Sections 1.4, 4.2) as well as degenerate semigroups (Chapters 5, 6), it is instructive to examine Definition 1.2 in more details and in a wider setting. For example, although conditions (a) and (b) are important since they define the semigroup functional equation (1.22), the strong continuity (c) at t “ `0 together with the topology of the image of the semigroup determine its analytic behaviour. A similar observation in Section 4.2 concerns the case of the norm and trace-norm continuity in R` . This demonstrates the important rˆ ole playing by the condition at t “ 0 for the notion of operator semigroup and for the entire theory, thus leading to a variety of semigroups, see Notes in Section 1.8. To advance with analysis of behaviour at the point t “ 0 we recall first the uniform boundedness principle for subsets of the set of bounded operators LpBq endowed with various topologies.


10

Proposition 1.6. For a subset S Ă LpBq the following properties are equivalent: (a) The subset S is bounded in the strong operator topology, that is, the set t}Bx} : B P S, x P Bu is bounded. (b) The subset S is bounded in the operator norm (uniform operator topology), i.e., the set t}B} : B P Su is bounded. Proposition 1.7. Let tUt utě0 be a semigroup of operators on a Banach space B, see Definition 1.2 (a) and (b). Then the following assertions are equivalent. (a) The family tUt utě0 is strongly continuous. (b) The family tUt utě0 is strongly right-continuous at t “ 0. (c) There exist δ ą 0, Mδ ě 1, and a dense D Ă B such that (1) }Ut } ď Mδ for 0 ď t ď δ; (2) limtÑ`0 Ut x “ x for all x P D. Proof. Note that the implication (a)ñ(b) is trivial. Now, if (b) holds, then for any t0 ą 0 the sequence tαn pt0 q :“ sup0ďtďt0 {n }Ut }unPN is such that αn˚ pt0 q ă 8 for some n˚ . For otherwise there would exist tn P r0, t0 {ns such that }Utn } ě n. Therefore, by the uniform boundedness principle, Proposition 1.6, one gets for some x P B: lim sup }Utn x} “ 8 , nÑ8

which contradicts the strong right-continuity in (b) saying that limnÑ8 Utn x “ x for any x P B. Since αn˚ pt0 q is bounded, the semigroup property yields on the interval r0, t0 s the estimate ˚

n } ď sup }Ut{n }n ď αn˚ pt0 qn “: αpt0 q. }Ut } ď sup }Ut{n 0ďtďt0

(1.35)

0ďtďt0

Then for any x P B and for any t0 ą 0 we obtain for 0 ă τ ă t0 Ut0 `τ x ´ Ut0 x “ Ut0 pUτ x ´ xq , Ut0 ´τ x ´ Ut0 x “ Ut0 ´τ px ´ Uτ xq . These two identities together with (1.35) yield for |h| ă t0 the estimate }Ut0 `h x ´ Ut0 x} ď αpt0 q}pU|h| x ´ xq} ,

x P B,

which proves the strong continuity of the semigroup tUt utě0 for any t0 ą 0, and hence (b)ñ(a). Since D is dense, the implication (a)ñ(c)-(2) is obvious. The implication (a)ñ (c)-(1) follows by reductio ad absurdum. Indeed, suppose that for some sequence tδn ą 0unPN , such that limnÑ8 δn “ 0, one has limnÑ8 }Uδn } “ 8. Then by the uniform boundedness principle there exists x P B such that in turn one gets

11

1.3. Generators and quasi-bounded semigroups

limnÑ8 }Uδn x} “ 8, but this contradicts to the right-continuity at t “ 0, which is a part of the properties in (a). To prove that (c)ñ(b) we note that for y P B and x P D one gets }Ut y ´ y} ď }Ut x ´ x} ` }Ut }}y ´ x} ` }y ´ x} .

(1.36)

Since (c)-(2) yields limtÑ`0 }Ut x ´ x} “ 0 and (c)-(1) gives }Ut } ď Mδ for 0 ď t ď δ, the denseness of D in B ensures that the right-hand side of (1.36) can be infinitesimally small in the limit limtÑ`0 . This proves (c)ñ(b), and by the equivalence (a)ô(b), also (c)ñ(a). l Corollary 1.8. By virtue of (1.35) one readily obtains the following estimate of }Ut } for any t ě 0, since the semigroup property yields }Ut } ď pαpt0 qqrt{t0 st0 `1 ď pαpt0 qqt`1 ď M eωt , for some M ą 0 and ω ě 0. Here rt{t0 s denotes the integer part of t{t0 ě 0.

1.3 Generators and quasi-bounded semigroups ole Propositions 1.1 and 1.5 show that the operator (generator ) A plays a central rˆ in the characterisation of a strongly continuous semigroup. Definition 1.9. The infinitesimal generator A of a strongly continuous semigroup Ut on a Banach space B is defined by 1 Au “ lim p1 ´ Ut qu, tÓ0 t

u P dom A,

(1.37)

where dom A is the set of all u P B for which the limit (1.37) exists. Since Definition 1.9 is (in a certain sense) a corollary of Proposition 1.5, below we collect explicitly some other fundamental properties of generators that follow from this proposition. Corollary 1.10. If A is the generator of a strongly continuous contraction semigroup tUt pAqutě0 on a Banach space B, then : (a) The dense subspace dom A Ă B is invariant under Ut pAq in the sense that Ut pAqpdom Aq Ď dom A for all t ě 0. (b) For all u P dom A and all t ą 0 one has AUt u “ Ut Au “ ´Bt Ut u. In fact, the proof of the Proposition 1.5 shows that the existence of the closed densely defined generator for a semigroup Ut is ensured by its strong continuity alone, i.e., it does not require that Ut is a contraction. Hence, in order that an operator A generates a strongly continuous semigroup, it is not necessary that the generator satisfies condition (ii) of Proposition 1.1.


12

Let Ut be a strongly continuous semigroup and τ ą 0. Since the vectorvalued function t ÞÑ Ut f : r0, τ s Ñ B is continuous and r0, τ s is compact, the set t}Ut f } : 0 ď t ď τ u is bounded. Therefore, the uniform boundedness principle implies that there is an M such that }Ut } ď M for t P r0, τ s. Then (Corollary 1.8), by virtue of the semigroup property, one gets the estimate }Ut } “ }pUτ qn Us } ď M eωt ,

(1.38)

´1 lnpM q. for any t “ nτ ` s P R` 0 , n P t0u Y N, where N “ t1, 2, . . .u and ω “ τ Therefore, all strongly continuous semigroups are exponentially bounded.

Definition 1.11. Let ω0 be the infimum of the numbers ω so that there is M ě 1 satisfying the estimate (1.38). If ω0 “ 0, we shall call Ut a bounded semigroup. If ω0 ‰ 0, we shall call it quasi-bounded of type ω0 . We denote the corresponding class of generators by QpM, ω0 q. Proposition 1.12. Let A be a closed linear operator in a Banach space B with dom A “ B. Then, A is the generator of a strongly continuous quasi-bounded semigroup Ut pAq of type ω0 , that is, A P QpM, ω0 q, if and only if the following three conditions are satisfied : (i) There is a real ω0 such that tz P C : <e z ă ´ω0 u belongs to the resolvent set ρpAq. (ii) There exists M ě 1 such that }pz1 ` Aq´1 } ď M p<epz ´ ω0 qq´1 , z P C. (iii) One has › › ›pλ1 ` Aq´m › ď

M , pλ ´ ω0 qm

(1.39)

for all λ ą ω0 and m P N. Proof. We denote Aζ :“ A ´ ζ1, ζ P C. Then, (1.39) takes the form › › ›pξ1 ` A´ω q´m › ď M ξ ´m , ξ ą 0, m P N, 0 or, equivalently, › › ›p1 ` α A´ω q´m › ď M, 0

α ě 0, m P N.

(1.40)

Inequality (1.40) implies that the operators Ut,n pA´ω0 q defined by (1.5) are uniformly bounded by M . Consequently, using (1.40) and the same reasoning as in the derivation of (1.9), one gets }Ut,n pA´ω0 qf ´ Ut,k pA´ω0 qf } ż t`ε ´ › › ¯´pk`1q A2´ω0 τ t ´ τ ¯´ t´τ › › “ › lim dτ ´ 1` A´ω0 f ` ˘ n`1 › τ εÓ0 ε n k k 1 ` n A´ω0 ˆ ˙ 2 2 › › 1 1 › 2 t M ` A´ω0 f › , (1.41) ď 2 n k

13

1.3. Generators and quasi-bounded semigroups

for any f P dom A2 . Since dom A2 “ B and }Ut,n pA´ω0 q} ď M , (1.41) implies the existence of the limit Ut pA´ω0 q “ s-lim Ut,n pA´ω0 q, nÑ8

t ě 0.

(1.42)

Moreover, since Ut,n pA´ω0 q is strongly continuous for t ě 0, and since the estimate (1.41) is uniform in t P r0, T s, the limit (1.42) is strongly continuous for t ě 0, Ut“0 pA´ω0 q “ 1, and }Ut pA´ω0 q} ď M.

(1.43)

Now, the arguments used to establish (1.13)–(1.23) carry over verbatim to verify that (1.42) defines a bounded semigroup Ut pA´ω0 q “ e´tA´ω0 ,

t ě 0.

(1.44)

Therefore, (1.43), (1.44) and the definition of Aζ imply the existence of the strongly continuous quasi-bounded semigroup Ut pAq “ etω0 Ut pA´ω0 q “ e´tA ,

t ě 0,

(1.45)

of type ω0 , with generator A and }Ut pAq} ď M etω0 . To show the necessity of conditions (i), (ii) and (iii) we proceed as in the proof of Proposition 1.5. It follows from (1.45) that any quasi-bounded semigroup ˜t “ e´tω0 Ut . Ut of type ω0 defines a strongly continuous bounded semigroup U ´1 ˜t q, t ą 0, and the Then, we can define the family of operators A˜ptq “ t p1 ´ U ˜ ˜ ˜ ˜t is limit operator Au “ limtÓ0 Aptq u, u P dom A, see (1.37). Since the semigroup U bounded, its Laplace transform (1.31) exists for <e z ą 0 and ż8

˜t f “ pz1 ` Aq ˜ ´1 f, dt e´zt U

f P B,

(1.46)

0

˜ ´1 } ď M p<e zq´1 , or with }pz1 ` Aq › › ›pz1 ` Aq´1 › ď

M , <epz ´ ω0 q

<epz ´ ω0 q ą 0,

(1.47)

which proves (i) and (ii). If <e z “ λ ą ω0 , one gets from (1.46) that › m › › › ›pλ1 ` Aq´m f › “ ›› p´1q Bλm pA ` λ1q´1 f ›› m! ż8 M }f } 1 }f } dt tm e´λt M eω0 t “ , ď m! pλ ´ ω0 qm 0 which together with (1.47) proves (iii).

l


14

We conclude this section by a result that is partially motivated by Proposition 1.7 (c). It is useful if one needs to determine a core for generator A. Recall that in general a set D :“ core A Ă dom A is called a core for operator A if the closure of the restriction of A to this set restores A, or explicitly if pA æ Dq “ A. It is clear that core A is not unique. Proposition 1.13. Let D Ă dom A be dense in B and invariant under the semigroup tUt pAqutě0 . Then D is a core for the generator A. Proof. First we note that since the generator A of a strongly continuous semigroup is a closed operator, the space dom A is complete with respect to the graph norm }f }A :“ }f } ` }Af },

f P dom A.

(1.48)

Let D denote the closure of D in the set dom A with respect to the norm (1.48). Let f P dom A. Then since D is dense in B, there exists a sequence tfn uně1 Ă D such that limnÑ8 }f ´ fn } “ 0. Recall that the strongly continuous semigroup tUt pAqutě0 is continuous in topology (1.48). Therefore, żt fnt

dτ Uτ pAqfn P D.

:“ 0

Since Afnt “ fn ´ Ut pAqfn , definition (1.48) and the boundedness property }Ut pAq} ď M eωt yield ›ż t › › › lim }fnt ´ f t }A “ lim › dτ Uτ pAqpf ´ fn q›

nÑ8

nÑ8

0

` lim }fn ´ Ut pAqfn ´ f ` Ut pAqf } “ 0, nÑ8

for any t ą 0 and hence, f t P D. Here we used that Af t “ f ´ Ut pAqf . Now using this relation again we obtain that › ›1 › ›1 › ›1 › › › › › › lim › f t ´ f › “ lim › f t ´ f › ` lim › pf ´ Ut pAqf q ´ Af › “ 0. tÑ0 t tÑ0 t tÑ0 t A So, f P D, and therefore dom A “ D.

1.4

l

Norm and other continuity conditions

Here, we consider another continuity condition for semigroups. Suppose that instead of the strong continuity in Definition 1.2 (c), one has a semigroup Ut pAq which is norm continuous in the sense that pcq1

lim }Ut pAq ´ 1} “ 0. tÓ0

(1.49)

15

1.4. Norm and other continuity conditions

Since (1.49) implies (c) and consequently the quasi-boundedness (1.38) of Ut pAq, with generator A defined by (1.37), we also get › › lim }Ut`τ pAq ´ Ut pAq} ď lim M eωpt`|τ |q ›U|τ | pAq ´ 1› “ 0, (1.50) τ Ñ0

τ Ñ0

that is, the uniform continuity of the semigroup Ut pAq at any t ą 0. If A is a bounded generator, then the semigroup Ut pAq is well-defined by the } ¨ }-convergent series (1.1) for any t P C. It is obviously } ¨ }-continuous in the sense of (1.49) and (1.50). Proposition 1.14. A semigroup tUt pAqutě0 is norm continuous for t ě 0 if and only if its generator A P LpBq. Proof. By the remark above and Propositions 1.1 and 1.2, we only have to prove that (1.49) (c)1 implies A P LpBq. Since › ›ż τ › › (1.51) › ds pUs pAq ´ 1q› ď τ sup }Us pAq ´ 1} , 0ďsďτ

0

taking τ ą 0 small enough, τ ă τ0 , see (c)1 , one gets żτ › › › › ›τ ´1 ds Us pAq ´ 1› ă 1. 0

şτ

ds Us pAq, 0 ă τ ă τ0 , is invertible. Since żτ ż t`τ pUt pAq ´ 1qLτ “ ds Us pAq ´ ds Us pAq t 0 żt ż t`τ ds Us pAq ´ ds Us pAq “ τ 0 żt “ pUτ pAq ´ 1q ds Us pAq

Therefore, the operator Lτ :“

0

0

for t ě 0, and, since all these operators commute, we obtain żt Ut pAq ´ 1 “ ´ ds Us pAqKτ ,

(1.52)

0

with Kτ :“ ´pUτ pAq ´ 1qL´1 τ , a bounded operator. Relation (1.52) implies that the operator-valued function t ÞÑ Ut pAq is norm differentiable for t ą 0, see (1.51), › › › ›1 lim › pUt`δ pAq ´ Ut pAqq ` Ut pAqKτ › δÑ0 δ żδ ›1 › › › “ lim › Ut pAq ds pUs pAq ´ 1qKτ › δÑ0 δ 0 ´ ¯ ď }Ut pAq} ¨ }Kτ } lim sup }Us pAq ´ 1} “ 0, δÑ0 0ďsď|δ|


16 with the operator-norm derivative

} ¨ }- Bt Ut pAq “ Út pAqKτ .

(1.53)

The bounded operator Kτ generates a } ¨ }-continuous semigroup tUt pKτ q “ e´tKτ utě0 defined by the corresponding } ¨ }-convergent series. The function t ÞÑ Ut pKτ q is in fact } ¨ }-holomorphic in the complex plane C. Then, by (1.53), we get for 0 ă s ă t that ¯ ´ } ¨ }- Bs Us pAqe´pt´sqKτ “ Ús pAqKτ e´pt´sqKτ ` Us pAqKτ e´pt´sqKτ “ 0. Hence, Ut pAq “ e´tKτ , with Kτ P LpBq. On the other hand, by (1.51), (1.52), and Definition 1.9, the infinitesimal generator A of the semigroup Ut pAq can be calculated as the limit ˆ ˙ ż 1 1 t Au “ lim p1 ´ Ut pAqq u “ lim ds Us pAq Kτ u “ Kτ u, tÓ0 t tÓ0 t 0 for any u P dom Kτ “ B. So, we conclude that Ut pAq “ e´tA with A P LpBq.

l

It is relevant to mention here a special class of semigroups, which is of particular importance in the context of this chapter. It is known (Corollary 1.10) that for any strongly continuous semigroup Ut pAq on a Banach space B the subspace dom A is invariant under Ut pAq in the sense that Ut pAq maps dom A into dom A, for all t ě 0. But now we assume a bit more, namely that Ut pAq maps B into dom A, for all t ą 0. Proposition 1.15. Let tUt pAqutě0 be a strongly continuous semigroup on B such that Ut pAq B Ď dom A for t ą 0. Then R` Q t ÞÑ Ut pAq is an infinitely } ¨ }differentiable operator-valued function. Proof. Let δ ą 0. By the assumption of the proposition, the operator A Us pAq is closed with dom pA Uδ pAqq “ B, so that }A Uδ pAq} ď Cδ . By (1.14), żt Ut pAqf ´ f “ ´ dτ Uτ pAq Af, 0

for all f P dom A and t ą 0, so żt pUt`δ pAq ´ Uδ pAqq u “ ´

dτ Uτ pAq A Uδ pAq u,

(1.54)

0

for u P B. Since every strongly continuous semigroup is exponentially bounded (see (1.38)), we have }Ut pAq} ď M eωt , for some M ą 0 and ω ě 0, for all t ě 0. Then, (1.54) yields the estimate żt }Ut`δ pAq ´ Uδ pAq} ď Cδ dτ M eωτ . (1.55) 0

17


Thus, the semigroup tUt pAqutě0 is } ¨ }-continuous at t “ δ for all δ ą 0. Moreover, by (1.54), (1.55) and by the closedness of the generator A, we get ›1 › › › › pUt`δ pAq ´ Uδ pAqq ` A Uδ pAq› t ›1 ż t ` › ˘ › › “› dτ Uτ `δ{2 pAq ´ Uδ{2 pAq A Uδ{2 pAq› t 0 żt 2 ď pCδ{2 q ds M eωs , 0

which means that Ut pAq is } ¨ }-differentiable with } ¨ }- Bt Ut pAq “ A Ut pAq for all t ą 0. For 0 ă δ ă t, we can rewrite this as } ¨ }- Bt Ut pAq “ Ut´δ pAqpÁUδ pAqq, and hence differentiate (1.56) repeatedly in the operator-norm topology.

(1.56) l

Having at our disposal semigroups generated by bounded operators we can provide another construction of strongly continuous semigroups, different from the one in Sections 1.1 and 1.2. The idea is to approximate them by sequences of the norm-continuous semigroups. Since the resolvent of the generator of a strongly continuous semigroup plays a key rôle in the theory, see Proposition 1.1 and (1.34), we first recall the following general convergence property. Proposition 1.16. Let A be a closed linear operator with dense domain dom A Ď B. If there exist a real ω and an M ą 0 such that p´8, ´ωq Ă ρpAq and }λpA ` λ1q´1 } ď M for all λ ě ω, then (i) s-limλÑ8 λR´λ pAq “ 1. (ii) limλÑ8 λAR´λ pAqu “ limλÑ8 λR´λ pAqAu “ Au,

u P dom A.

Proof. If u P dom A, then λR´λ pAqu “ ´R´λ pAqAu ` u. The assumptions of the proposition imply that }R´λ pAqAu} ď λ´1 M }Au}, and we get (i) for any u P dom A. This result can be extended to all u P B, since by the same assumptions, the operators λR´λ pAq are uniformly bounded for all λ ě ω and dom A is dense in B. Assertion (ii) is then a straightforward consequence of (i). l Assertion (ii) suggests which bounded operator should be chosen to approximate the unbounded generator A. Since in Section 1.3 we showed that the results for the case of quasi-bounded semigroups can be deduced from those for contraction semigroups, the proof below is given only for the latter. Proposition 1.17. Under the assumptions of Proposition 1.1, that is, A is a closed linear operator with dense domain dom A Ď B such that p´8, 0q Ă ρpAq and }λR´λ pAq} ď 1 for λ ą 0, the operator A generates a strongly continuous contraction semigroup tUt pAqutě0 .


18

Proof. With Proposition 1.16 in mind, we define a sequence of bounded operators An :“ nARń pAq “ n1 ´ n2 Rń pAq,

n P N,

(1.57)

which, by assertion (ii) therein, converges to A on dom A. Consider then the corresponding sequence of norm-continuous semigroups Ut pAn q :“ e´tAn ,

t ě 0, n P N.

(1.58)

By (1.57), each tUt pAn qutě0 is a contraction semigroup: }Ut pAn q} ď e´tn et}n

2

Rń pAq}

ď 1.

Therefore, by the uniform boundedness principle (Proposition 1.6), it is sufficient to prove the convergence of this approximating sequence of semigroups on the dense set dom A. For this purpose, we introduce the operator-valued functions s ÞÑ Ut´s pAm qUs pAn qu,

u P dom A,

(1.59)

for 0 ď s ď t and m, n P N. The elements of the semigroups tUt pAn qutě0 commute for all n P N and are norm-continuous. Then, by virtue of (1.59), one gets żt Ut pAn qu ´ Ut pAm qu “ ds Bs pUt´s pAm qUs pAn quq 0 żt “ ds Ut´s pAm qUs pAn q pAm ´ An q u, 0

which yields the estimate }Ut pAn qu ´ Ut pAm qu} ď t }pAn ´ Am q u} ,

(1.60)

for any u P B. Since, by Proposition 1.16 (ii), tAn uuně1 is a Cauchy sequence for each u P dom A, estimate (1.60) implies that tUt pAn quu converges uniformly on each interval r0, T s. Hence, by the uniform boundedness of the sequence tUt pAn quně1 , the limit Ut u :“ lim Ut pAn qu (1.61) nÑ8

exists for each u P B and for each t ě 0. By virtue of (1.61), the limit tUt utě0 inherits the semigroup property of tUt pAn qu. Hence, it is a contraction semigroup, i.e., }Ut } ď 1. Thanks to the estimate }pUt ´ 1q u} ď }pUt ´ Ut pAn qq u} ` }pUt pAn q ´ 1q u} , and the uniform convergence of tUt pAn quu on each interval r0, T s for any u P dom A, we get that (1.62) lim Ut u “ u, u P dom A. tÓ0

19


The uniform boundedness of }Ut } ď 1 allows us to extend (1.62) to B and conclude that the semigroup tUt utě0 is strongly continuous. It remains to prove that the operator A generates this semigroup. Let B be the generator of tUt utě0 . Fix u P dom A. Since the sequence tUt pAn quuně1 converges uniformly on each interval r0, T s, we get u ´ Ut u “ u ´ lim Ut pAn qu nÑ8 żt żt ds Us pAn qAn u “ ds Us Au, “ lim nÑ8 0

(1.63)

0

by virtue of (1.60), (1.61) and Proposition 1.16(ii). This implies that Au “ Bu,

u P dom A,

(1.64)

or dom A Ď dom B. Since tUt utě0 is a strongly continuous contraction semigroup, its Laplace transform (1.34) exists for the set tz : <e z ą 0u. Hence, p´8, 0q Ă ρpBq. Moreover, by (1.64), we get Rζ pAq “ Rζ pBq for any ζ P p´8, 0q, which implies A “ B. l We have already observed that inspecting the properties of semigroup one has first to focus on its behaviour at t “ 0. After the strong and operator-norm continuity hypothesis in Definition 1.2(c) it is natural to discuss semigroups that are continuous at t “ `0 in the weak operator topology. By observations in Proposition 1.7 and by (1.49) the continuity at t “ `0 is decisive for the analysis of semigroups. So, we consider, instead of pcq, or stronger pcq1 , a weak continuity condition pcq2 , see Definition 1.20. For this purpose we need some preliminaries. Recall that on a Banach space B one can define a space of bounded linear functionals LpB, Cq “ tφ : B Ñ Cu, where φ : u ÞÑ xu|φy, u P B, and the norm }φ}˚ :“ supt|xu|φy| : }u} ď 1, u P Bu is finite. Then B ˚ :“ LpB, Cq is the dual space of B. The space B ˚ itself is Banach, with the norm } ¨ }˚ . Definition 1.18. The dual space B ˚ of B allows to define on B the weak topology, σpB, B ˚ q, as the weakest topology on B in which all the functionals φ P B ˚ are continuous. This topology is generated on B by a collection of neighborhoods: tu P B : |xu|φy| ď ε, φ P B ˚ , ε ą 0u, of the vector u “ 0. Then the equality lim xun |φy “ xu|φy , for all φ P B ˚ ,

nÑ8

means that the sequence tun P Buně1 converges weakly to the vector u P B, and we write w-limnÑ8 un “ u. The weak operator topology on LpBq is the weakest topology such that the maps Φu,φ : LpBq Ñ C given by Φu,φ pBq :“ xBu|φy are all continuous for all u P B, φ P B ˚ . Then the convergence w-limnÑ8 Bn “ B of the sequence tBn uně1 in the weak operator topology means that lim xBn u|φy “ xBu|φy,

nÑ8


20 for each u P B and φ P B ˚ .

Remark 1.19. (1) Although the topology σpB, B ˚ q is evidently weaker than the vector-norm topology on B, the uniform boundedness principle is still valid for the weak topology on B. (2) One should not confuse the weak operator topology on LpBq with the weak Banach space topology σpLpBq, LpBq˚ q on LpBq generated by the dual of LpBq space LpBq˚ . (3) The weak Banach space and the operator-norm topologies are stronger than the weak operator topology on LpBq, but the uniform boundedness principle is valid in this topology on LpBq. So, we can add in Proposition 1.6 to the equivalent properties (a), (b), also: (c) The subset S is bounded in the weak operator topology, that is, the set t|xBx | φy| : B P S, x P B, φ P B ˚ u is bounded. Definition 1.20. A semigroup tUt utě0 on B is weakly continuous (or continuous in the weak operator topology) if in Definition 1.2 one substitutes (c) by the condition (cf. Definition 1.18): (c)2 t ÞÑ Ut f P B is a weakly continuous vector-valued function of the parameter t P R` for every f P B and weakly right-continuous at t “ 0, that is, w-lim Ut u “ lim xUt u|φy “ xu|φy, tÓ0

tÓ0

(1.65)

for each u P B and φ P B ˚ . Proposition 1.21. Let tUt utě0 be a semigroup on a Banach space B, see Definition 1.2 (a) and (b). Then the following assertions are equivalent: (a) The semigroup tUt utě0 is strongly continuous, Definition 1.2(c). (b) The semigroup tUt utě0 is strongly right-continuous at t “ 0. (a-w) The semigroup tUt utě0 is weakly continuous, Definition 1.20. (b-w) The semigroup tUt utě0 is weakly right-continuous at t “ 0. The main impact of this surprising result is that it rules out possible nontrivial generalisations of the notion of semigroup by highlighting the relevance of the strong continuity at t “ `0. The proof in a Banach space B needs a number of facts and tools, which are out of the scope of this book. Thus, the reader has to consult the Notes in Section 1.8 for the references. Since the main objects studied in this book are the Gibbs semigroups, which are strongly continuous on a complex separable Hilbert space H, we provide the proof of Proposition 1.21 only on such a space H. Note that the self-duality of H, which implies that the functionals, φ P H, and gives their explicit form, (they are scalar products: xu|φy “ pu, φq), makes the proof simpler.


21

Proposition 1.22. Let tUt utě0 be a contraction semigroup on a Hilbert space H. Then the following assertions are equivalent: (a) The semigroup tUt utě0 is strongly continuous, Definition 1.2(c). (b) The semigroup tUt utě0 is strongly right-continuous at t “ 0. (a-w) The semigroup tUt utě0 is weakly continuous, Definition 1.20. (b-w) The semigroup tUt utě0 is weakly right-continuous at t “ 0. Proof. The equivalence: (a)ô(b), follows from Proposition 1.6 since the strong right limit (b) implies operator-norm boundedness of semigroup in the vicinity of t “ 0. By virtue of Remark 1.19(3) about the part (c) of the weak uniform boundedness principle, the proof of the equivalence (a-w) and (b-w) (1.65) follows the same line of reasoning as for (a)ô(b). Since (b)ñ(b-w) is evident, it is sufficient to prove that (b-w)ñ(b). To this aim we note that since tUt utě0 is a contraction, lim }Ut u ´ u}2 “ lim t}Ut u}2 ` }u}2 ´ 2 <epUt u, uqu

tÑ`0

tÑ`0

ď lim t2}u}2 ´ 2 <epUt u, uqu “ 0, tÑ`0

for any u P H, so tUt utě0 is strongly right-continuous at t “ 0.

l

Corollary 1.23. If tUt utě0 is strongly continuous, then the adjoint semigroup tUt˚ utě0 is also strongly continuous. To see this, note that by the definition pUt u, vq “ pu, Ut˚ vq, u, v P H, the strong continuity of Ut yields the weak continuity of Ut˚ , and hence its strong continuity. It is known that on a Banach space the dual semigroup Ut˚ is not necessarily strongly continuous for a strongly continuous semigroup Ut . But it is always weak ˚ -continuous (in the topology σpLpBq, LpBq˚ q), which makes it still interesting to study, see comments to Section 1.4 in Notes in Section 1.8. We conclude this section by describing a situation of failure of C0 -continuity, which is of a different nature than the modification of topology of continuity with respect to the parameter t P R0` . For that we modify (relax) the conditions (a) and (c) in Definition 1.2. Definition 1.24. An operator-valued function of t P R` , t ÞÑ Ut P LpBq, is called a degenerate one-parameter semigroup on B if (a1 ) U0 ‰ 1, (b1 ) Ut Us “ Ut`s for t, s P R` , (c1 ) t ÞÑ Ut f P B is a continuous function of the parameter t P R` for every fixed f P B. Recall that for a degenerate semigroup tUt utą0 the subspace B0 :“ tf P B : lim Ut f “ f u Ď B , tÓ0

(1.66)

22


is known as the space of strong continuity. Therefore, if B0 “ B, then the degenerate semigroup is a C0 -semigroup. Otherwise, the operator P :“ limtÓ0 Ut ‰ 1 violates condition (a) in Definition 1.2. By virtue of the semigroup property (b1 ) and by the continuity (c1 ), the operator P is a projection, which is orthogonal in the case of self-adjoint semigroups. We comment here that the limits of the Trotter-Kato product formulae (away from zero) are often degenerate semigroups, see Chapter 5 and Chapter 6. Proposition 1.25. For a degenerate semigroup tUt utą0 on H the following holds true: (i) tUt utą0 is exponentially bounded. (ii) kerpP q and ranpP q are closed, Ut -invariant, and H “ kerpP q ‘ ranpP q, for self-adjoint semigroups. (iii) Ut has the representation Ut “ St P “ P St , where tSt utě0 is a C0 -semigroup on H. In Sections 5.4-5.6 and Sections 6.4-6.6 one encounters degenerate semigroups in the following context. Let P0 : H Ñ H0 be an orthogonal projection from a Hilbert space H onto H0 and te´tH utě0 be a C0 -semigroup on H0 with domain of generator dom H Ă H0 . Then by Definition 1.24, the operator family tWt :“ e´tH P0 utą0 is obviously a degenerate semigroup on H with the subspace H0 as its space of strong continuity (1.66).

1.5

Holomorphic semigroups

One can ask whether, instead of the construction of the strongly continuous semigroup tUt pAq “ e´tA utě0 described in Sections 1.1–1.4, the functional calculus in the form of the Riesz-Dunford integral ż 1 Ut pAq “ dz e´tz pz1 ´ Aq´1 , Γ Ă ρpAq , (1.67) 2πi Γ is appropriate for this purpose? This formula is obviously valid if A P LpBq and if Γ is a positively-oriented contour in the resolvent set ρpAq, which encloses the spectrum σpAq :“ CzρpAq of the operator A in its interior. In fact, (1.67) is a way to construct semigroups if one assumes slightly more about generator A than in Section 1.1 or in Section 1.3. Definition 1.26. Let θ P p0, π{2s and denote by Sθ the open sector Sθ “ tz P C` : | arg z| ă θu. A strongly continuous semigroup tUt utě0 on a Banach space B is called a bounded holomorphic semigroup of semi-angle θ if: (i) tUt utą0 is the restriction to R` of an analytic family of bounded operators tUz uzPSθ , which obeys the semigroup property (functional equation): Uz`z1 “ Uz Uz , for z, z 1 P Sθ .

23

1.5. Holomorphic semigroups (ii) For θ1 ă θ and for z P S θ1 , one has }Uz } ď M 1 and s-lim Uz “ 1. zÑ0

If A is the generator of the semigroup tUt utě0 , we write Uz pAq “ e´zA . We recall that for holomorphic families of bounded operators on LpHq there is no distinction between uniform, strong, or weak operator analyticity. From Definition 1.26, one can deduce properties of the generator A. By (i) and (ii), for each ϕ P p´θ, θq the family tUz pAquz“reiϕ is a bounded strongly continuous iϕ semigroup of parameter r P R` 0 with generator e A. Then, by Proposition 1.12, iϕ the spectrum σpe Aq of this generator lies in C` . Since this is true for all |ϕ| ă θ, the spectrum of A belongs to the closed sector S π{2´θ : ) ! π σpAq Ă S π{2´θ “ z P C` : | arg z| ď ´ θ . 2

(1.68)

By Definition 1.26(ii), for any ε ą 0 and |ϕ| ď θ1 “ θ ´ ε{2, the semigroup tUreiϕ pAqurě0 is uniformly bounded by Mε1 . Therefore, Proposition 1.12 yields the estimate }pz1 ` eiϕ Aq´1 } ď Mε1 p<e zq´1 for all z with <e z ą 0. For | arg z| ď π{2 ´ ε{2, this estimate implies › › ›pz eíϕ 1 ` Aq´1 › ď

Mε1 . |z| sin ε

This means that for ε ą 0 › › ›pζ1 ` Aq´1 › ď Mε , |ζ|

ζ P S π{2`θ´ε zt0u,

(1.69)

with Mε independent of ζ and ε ă θ. In fact, the conditions (1.68) and (1.69) are also sufficient. Proposition 1.27. A closed linear operator A in a Banach space B is the generator of a bounded holomorphic semigroup tUz pAquzPSθ of semi-angle 0 ă θ ď π{2 if and only if A satisfies conditions (1.68) and (1.69). Proof. The necessity part was already proved. To prove the sufficiency part we use the Riesz-Dunford integral representation (1.67). By virtue of (1.68) and (1.69), the integral is absolutely convergent for t ą 0 in the operator-norm topology if Γ is chosen as a contour in ρpAq running from infinity with arg z “ π{2 ´ θ ` ε and to infinity with arg z “ ´pπ{2 ´ θ ` εq inside the domain Dθ´ε “ CzS π{2´θ`ε . To verify the semigroup property, we define Us pAq by (1.67), with the contour of integration Γ1 Ă Dθ´ε similar to Γ, but slightly shifted to the left. Then


24

ż ż 1 1 Ut pAqUs pAq “ dz dz 1 e´tz´sz pz1 ´ Aq´1 pz 1 1 ´ Aq´1 p2πiq2 Γ 1 Γ ż ż “ ‰ 1 1 ´tz´sz 1 “ dz dz e pz1 ´ Aq´1 ´ pz 1 1 ´ Aq´1 pz 1 ´ zq´1 2 p2πiq Γ Γ1 ż ż 1 1 ´tz ´1 “ dz e pz1 ´ Aq dz 1 e´sz pz 1 ´ zq´1 2 p2πiq Γ Γ1 ż ż 1 1 ´sz 1 1 ´1 dz e pz 1 ´ Aq dz e´tz pz ´ z 1 q´1 ` p2πiq2 Γ1 Γ ż 1 ´pt`sqz ´1 “ dz e pz1 ´ Aq “ Ut`s pAq, 2πi Γ where we have used the resolvent equation Rz pAqRz1 pAq “ pRz pAq ´ Rz1 pAqqpz ´ z 1 q´1 , and properties of the Cauchy integral. The integral (1.67) can be made absolutely convergent in the operator norm even for complex t by deforming the contour Γ Ă Dθ´ε , if one ensures that | argptzq| ă π{2 for z P Γ and |z| Ñ 8. Note that in the domain Dθ´ε one has | arg z| ą π{2 ´ θ. Hence, the semigroup Ut pAq can be extended to complex values of t in the sector | arg t| ă θ. Since the Riesz-Dunford representation (1.67) is operator-norm differentiable under the integral sign, it follows that Ut pAq is } ¨ }-holomorphic in the open sector Sθ “ tt P C : | arg t| ă θu. Within this sector, the Cauchy theorem gives ż 1 } ¨ }- Bt Ut pAq “ dz e´tz p´zqpz1 ´ Aq´1 (1.70) 2πi Γ “

p´1q 2πi

“ pÁq

ż “ ‰ dz e´tz 1 ` Apz1 ´ Aq´1 Γ

1 2πi

ż dz e´tz pz1 ´ Aq´1 “ pÁqUt pAq P LpBq, Γ

where the closedness of the operator A allows to take A out of the integral. Hence, we get Ut pAqA Ă AUt pAq “ ´Bt Ut pAq P LpBq, t P Sθ . (1.71) Setting z 1 “ zt, formula (1.67) becomes ż ´ z1 ¯´1 1 1 Ut pAq “ dz 1 e´z 1Á , 2πit Γ1 t

(1.72)

where the contour Γ1 may be taken inside the domain Dθ independently of t. For z 1 P Γ1 , ε ą 0 and t P S θ´ε , we have }pz 1 {t ´ Aq´1 } ď Mε |t{z 1 |. Moreover, the condition Γ1 Ă Dθ ensures the convergence of the integral (1.72), see estimate (1.69).

25

1.5. Holomorphic semigroups Therefore, 1 }Ut pAq} ď Mε 2π

ż

1

|dz 1 | |z 1 |´1 |e´z | “ M 1 .

(1.73)

Γ1

Thus, the semigroup Ut pAq is uniformly bounded in the sector S θ1 for θ1 ă θ. Similarly, we get the estimate, see (1.70), › 1 ż ´ 1 ¯´1 › › › 1 1 ´z 1 z }Bt Ut pAq} “ › dz z e 1 ´ A › 2πit2 Γ1 t Mε ď |t|

ż

1

|dz 1 | |e´z | :“ Γ1

M11 , |t|

t P S θ1 ăθ .

(1.74)

To verify the strong continuity of Ut pAq when t Ñ 0 in the sector S θ´ε , we use again the representation (1.72), writing Ut pAq ´ 1 “

1 2πi

ż dz 1 Γ1

1 ¯´1 e´z ´ z 1 1 ´ A A. z1 t

Then, due to (1.69), for any u P dom A we get ż 1 }pUt pAq ´ 1qu} ď |t| Mε }Au} |dz 1 | |z 1 |´2 |e´z |,

t P S θ´ε .

(1.75)

Γ1

Since dom A “ B and }Ut pAq} ď M 1 for t P S θ´ε and ε ą 0, it follows that s-lim Ut pAq “ Ut“0 pAq “ 1, tÑ0

t P S θ´ε ,

which finishes the proof.

l

Corollary 1.28. Since tUt pAqutě0 is holomorphic, a calculation similar to the one used in (1.74) gives that for any t ą 0, }Btn Ut pAq} “ }An Ut pAq} ż › ´ 1 ¯´1 › 1 › › 1 1 n ´z 1 z dz pz q e 1 ´ A “› › 2πitn`1 Γ1 t M1 ď nn . t

(1.76)

The next corollary gives an alternative characterisation of the bounded holomorphic semigroups and their relation to the operator-norm continuity, see Section 1.4. Corollary 1.29. Let tUz pAquzPSθ be a bounded holomorphic semigroup on B in the sector Sθ with θ ă π{2. Then Uz pAq : B Ñ dom A

(1.77)


26

for all z P Sθ . Moreover, for any ε P p0, θq, there is a Cε ą 0 such that }A Uz pAq} ď

Cε , |z|

(1.78)

for all z P Sθ´ε . Proof. The mapping property (1.77) follows from (1.70) and (1.71). Estimate (1.78) follows from (1.70) and (1.74). l Proposition 1.30. Let tUt pAqutě0 be a strongly continuous semigroup on B with generator A such that for u P B, u ÞÑ Ut pAqu P dom A, }Ut pAq} ď M and }A Ut pAq} ď M11 t´1 for all t ą 0. Then there exists an angle θ ą 0, which depends on M11 , such that Ut pAq can be analytically continued to a bounded }¨}-holomorphic semigroup in the sector Sθ . Proof. Proposition 1.15 shows that Ut pAq is an infinitely } ¨ }-differentiable operator-valued function on R` with } ¨ }- Btn Ut pAq “ pÁ Ut{n pAqqn , see (1.56). Hence, by the assumptions of the proposition and by (1.76), }Btn Ut } ď pn M11 t´1 qn . This means that the function Ut pAq can be analytically continued from R` to the disc Dt :“ tz P C` : |z ´ t| ă pe M11 q´1 tu with t ą 0 and e M11 ą 1, by setting Uz pAq “ The union of these discs

Ť tą0

8 ÿ pz ´ tqn n Bt Ut pAq. n! n“0

(1.79)

Dt “ Sθ is a sector with

θpM11 q

“ arcsin pe M11 q´1 ă π{2.

The rest of the proof that the operator-valued function (1.79) satisfies conditions (i) and (ii) of Definition 1.26 can be found in references provided in Notes in Section 1.8 l Remark 1.31. For t ą 0, the holomorphic semigroups enjoy an even stronger form of continuity than norm continuity. Since the family tUt pAqutą0 can be differentiated any number of times, one gets by (1.76) Ut pAqu P

8 č

dom An

and }An Ut pAq} ď Mn1 tń ,

(1.80)

n“1

for all t ą 0. This in turn yields the estimate: ›ż t › › › n }A pUt pAq ´ Us pAqq} “ › dτ An Bτ Uτ pAq› s

M1 ´ 1 1¯ ď n`1 n ´ n , n s t for all 0 ă s ď t and n ě 1.

(1.81)

27

1.5. Holomorphic semigroups

The notion of a bounded holomorphic semigroup can be generalised in the same way as in Section 1.3 for strongly continuous quasi-bounded semigroups. This means that we can relax the boundedness condition in (ii) of Definition 1.26. Definition 1.32. A strongly continuous semigroup tUt pAqutě0 on a Banach space B is called a quasi-bounded holomorphic semigroup of semi-angle θ P p0, π{2q if tUz pAquzPSθ is as in Definition 1.26, except that it is not required to be uniformly bounded in the sector S θ1 for θ1 ă θ. Proceeding as in Section 1.3, we can obtain an exponential estimate on the growth of Uz pAq in the sector Sθ1 . If Uz pAq is a holomorphic semigroup of angle θ, then }Uz pAqf } is bounded in the domain Dθ1 ,τ “ tz P S θ1 : |z| ď τ, τ ą 0u for any f P B. Hence, by the uniform boundedness principle, }Uz pAq} ď M for z P Dθ1 ,τ . For any z P S θ1 , the semigroup property (Definition 1.26 (i)) implies that Uz pAq “ pUz1 pAqqn Uζ pAq, where z “ nz 1 ` ζ and z 1 , ζ P Dθ1 ,τ . Therefore, one concludes that there are constants M, ω ą 0 such that }Uz pAq} ď M eω|z| ,

z P S θ1 ,

(1.82)

˜z pAq :“ e´ωz Uz pAq is a bounded holowhere M and ω depend on θ1 , and that U 1 morphic semigroup of angle θ . The arguments above motivate a generalisation of Proposition 1.27 (cf. Proposition 1.12). Proposition 1.33. A closed operator A in a Banach space B is the generator of a quasi-bounded holomorphic semigroup tUz pAquzPSθ of semi-angle θ P p0, π{2s if and only if : (i) ρpAq “ Cztz P C : z ` ω0 P S π{2´θ u for some ω0 P R, and (ii) for ε ą 0, there exists a constant Mε ą 0 such that }pz1 ` ω0 1 ` Aq´1 } ď

Mε , |z|

z P S π{2`θ´ε zt0u , ε ă θ .

(1.83)

We denote this class of generators by H pθ, ω0 q, where ω0 is the type of semigroup, cf. Definition 1.11. Proof. The operator A0 :“ ω0 1 ` A satisfies the conditions of Proposition 1.27. Therefore, Uz pA0 q “ e´ω0 z Uz pAq is a bounded holomorphic semigroup of semiangle θ1 “ θ ´ ε. Therefore, }Uz pA0 q} ď M 1 (1.73), and }Uz pAq} ď M 1 eω0 |z| , which yields the proposition. Note that H pθ, ω0 ă 0q Ă H pθ, 0q, whereas A P H pθ, ω0 ą 0q if and only if A “ A0 ´ ω0 1, where A0 P H pθ, 0q. l Remark 1.34. To make evident that condition (1.83) (or (1.69) for ω0 “ 0) is stronger than (1.39), it is sufficient to note that, since t ÞÑ Ut pAq is strongly


28

continuous for t ě 0, the condition (1.83) implies the existence of the operatornorm convergent Laplace transform ˆλ pAq “ U

ż8 dt e´λt Ut pAq “ pA ` λ1q´1 P LpBq, 0

for λ ą ω0 , cf.(1.31)–(1.33). Then, using Fubini’s theorem, we get ż8 pλ1 ` Aqń “ 0

“

ż8 dt1

ż8 dtn e´λpt1 `¨¨¨`tn q Ut1 `¨¨¨`tn pAq

dt2 . . . 0

0

1 pn ´ 1q!

ż8 dτ τ n´1 e´λτ Uτ pAq.

(1.84)

0

Since by Proposition 1.33 we have }Uτ pAq} ď M 1 eω0 τ , we deduces the estimate (1.39) from the integral formula (1.84) for M “ M 1 ě 1. Note that the condition (1.69) (or (1.83)) guarantees that the semigroup Ut pAq is } ¨ }-continuous and that for u P B, u ÞÑ Ut pAqu P dom A for t ą 0, see Section 1.4 and Corollary 1.29. This is in contrast with the strongly continuous semigroups, for which one has only for u P dom A, u ÞÑ Ut pAqu P dom A for t ą 0, see Section 1.3.

1.6

Holomorphic semigroups on a Hilbert space

Since our main subject, the Gibbs semigroups, are defined on a Hilbert space H, in this section we look closer at holomorphic semigroups on a separable complex space H with a sesquilinear inner product p¨, ¨q : H ˆ H Ñ C. We note that pαu, βvq “ αβpu, vq for u, v P H and α, β P C. First, we recall the notion of numerical range NrA of an operator A. Let A be a linear operator in a Hilbert space H. Then NrA :“ tpAu, uq : u P dom A, }u} “ 1u.

(1.85)

In general, NrA is neither open nor closed, even if A is a closed operator in H. A theorem of Hausdorff states only that NrA is a convex subset in C. Therefore NrA is a closed convex set, and the complement ∆ :“ CzNrA is either a connected open set, or it consists of two components ∆1 and ∆2 . The latter possibility occurs whenever NrA is a strip bounded by two parallel straight lines, including the degenerate case when those lines coincide. Next, we denote by def T :“ dimpran T qK the deficiency (or defect) of the closed operator T in H. Proposition 1.35. Let A be a closed operator in H. Then for any ζ P ∆, the operator A ´ ζ1 is injective with a closed range ranpA ´ ζ1q. If, in addition, for

29

1.6. Holomorphic semigroups on a Hilbert space

each of those ζ, the deficiency defpA ´ ζ1q is equal to zero, then ∆ Ă ρpAq or, equivalently, σpAq Ă NrA. Moreover, }Rζ pAq} ď

1 . distpζ, NrAq

(1.86)

Proof. For any u P dom A with }u} “ 1, one has |pAu, uq ´ ζ| “ |ppA ´ ζ1qu, uq| ď }pA ´ ζ1qu},

ζ P C.

(1.87)

Then, if ζ P ∆ “ CzNrA so that distpζ, NrAq “ δ ą 0, the estimate (1.87) gives }pA ´ ζ1qu} ě δ, or }pA ´ ζ1qv} ě δ}v},

v P dom A.

(1.88)

This implies that the operator A´ζ1 is injective, i.e., kerpA´ζ1q “ t0u. Moreover, since A is closed, the set ranpA ´ ζ1q is also closed, i.e., it is a subspace. If the dimension of its orthogonal complement pranpA ´ ζ1qqK is zero, then ranpA ´ ζ1q “ H. Thus, the inverse operator pA ´ ζ1q´1 “ Rζ pAq has domain H, and }Rζ pAq} ď δ ´1 by virtue of (1.88). This proves that ζ P ρpAq, or ∆ Ă ρpAq, and the estimate (1.86). l In fact, in Proposition 1.35 it is sufficient that defpA ´ ζ1q “ 0 for only one ζ P ∆. As above, we set Aζ :“ A ´ ζ1 for ζ P C. Lemma 1.36. The deficiency def Aζ is constant for ζ P ∆, except for the above mentioned special case, in which the def Aζ is constant in both ∆1 and ∆2 . These constants are called the deficiency of a A in ∆, ∆1 and ∆2 , respectively. Proof. Let ζ P ∆ so that distpζ, NrAq “ δ ą 0. We set dζ :“ def Aζ . For a ą 1 and ζ 1 P C such that |ζ 1 ´ ζ| ď aδ, we have by (1.88) that a}Aζ u} ě }pζ ´ ζ 1 qu},

u P dom A.

(1.89)

Notice that the operator Aζ 1 is closed and Aζ 1 “ Aζ ` pζ ´ ζ 1 q1. Suppose that dζ 1 ă dζ . Then there exists a ϕ P H a ran Aζ , ϕ ‰ 0, such that ϕ K pH a ran Aζ 1 q. This means that ϕ P ran Aζ 1 , that is, ϕ “ Aζ 1 v for some v P dom A. Since ϕ K ran Aζ , one has pϕ, Aζ vq “ 0, and hence, pAζ v, Aζ vq “ ´pζ ´ ζ 1 qv, Aζ vq.

(1.90)

Suppose now that dζ 1 ą dζ . Then, similarly, one can find ψ P H a ran Aζ 1 , ψ ‰ 0, such that ψ P ran Aζ , i.e., ψ “ Aζ w for some w P dom A. Since ψ K ran Aζ 1 , we get pAζ w, Aζ wq “ ´pAζ w, pζ ´ ζ 1 qwq. (1.91) Both (1.90) and (1.91) are impossible for a ă 1, because, by (1.89) }Aζ v}2 ď }pζ ´ ζ 1 qv}}Aζ v} ď a}Aζ v}2 ,


30 and

}Aζ w}2 ď }pζ ´ ζ 1 qw}}Aζ w} ď a}Aζ w}2 . Therefore, dζ 1 “ dζ for all ζ 1 in the disc Dr“δ pζq :“ tz P C : |z ´ ζ| ă δu Ă ∆. By the same reasoning, one can extend the equality dζ 2 “ dζ to all ζ 2 in the disc Drďδ pzq Ă ∆ for any centre z P Dr“δ pζq. Covering ∆ by these intersecting discs Drďδ pzq we conclude that the deficiency of A is constant on ∆. The proof l for the case with two components ∆1 and ∆2 is similar. The next statement gives a relation between the spectrum σpAq and the numerical range NrA in the case of a bounded operator A. Proposition 1.37. If A P LpHq, then σpAq Ă NrA.

(1.92)

Proof. Since for u P H, }u} “ 1, one has |pAu, uq| ď }A}, we get NrA Ď Dr“}A} pz “ 0q. Hence ∆ is a connected open set containing the exterior of the disc Dr“}A} pz “ 0q. Since A P LpHq, this exterior belongs to the resolvent set ρpAq, i.e., the resolvent Rζ pAq P LpHq, or def Aζ “ 0 for ζ P CzD}A} pz “ 0q Ă ∆. By Lemma 1.36, the same must be true for all ζ P ∆ “ CzNrA. Since the operator A is closed, the conditions of Proposition 1.35 are satisfied, and this implies that ∆ Ă ρpAq, which is equivalent to (1.92). l Remark 1.38. For a closed unbounded linear operator A in H, one needs the additional condition def Aζ “ dimpH a ran Aζ q “ 0,

for all ζ P ∆,

(1.93)

to ensure (1.92), cf. Proposition 1.35. Recall that z is in the spectrum σpAq “ tζ P C : Rζ pAq R LpHqu if z belongs to one of the following disjoint sets (Appendix A, Section A.3): (i) The point spectrum σp pAq :“ tζ P C : ker Aζ ‰ t0uu. In the case when ker Az “ t0u, but ran Az ‰ H, i.e., z P σpAqzσp pAq, we have two other possibilities. (ii) The continuous spectrum σcont pAq :“ tζ P C : ker Aζ “ t0u, ran Aζ ‰ H, ran Aζ “ Hu. (iii) The residual spectrum σres pAq :“ tζ P C : ker Aζ “ t0u, ran Aζ ‰ Hu.

31

1.6. Holomorphic semigroups on a Hilbert space

Therefore, in the case (iii), the deficiency def Az ‰ 0, i.e., the corresponding condition in Proposition 1.35 serves to exclude σres pAq from the set ∆ “ CzNrA. Otherwise, we would get instead of (1.92) that the essential part of the spectrum, σp pAqYσcont pAq, belongs to the closure of the numerical range: σp pAqYσcont pAq Ă NrA. The notion of numerical range is very useful for the classification of generators of contraction and holomorphic semigroups on a Hilbert space. Definition 1.39. An operator A in a Hilbert space H is said to be accretive, if NrA Ă C` “ tz P C : <e z ě 0u. If, in addition, A is closed, then by the stability of the deficiency, Lemma 1.36, def Az “ dimpran Az qK is constant for <e z ă 0. If def Az “ 0 for <e z ă 0, then by Proposition 1.35, one has C` “ tz P C : <e z ą 0u Ă ρpÁq with pA ` z1q´1 P BpHq and }pA ` z1q´1 } ď

1 , <e z

<e z ą 0.

(1.94)

The operator A satisfying (1.94) is called m-accretive. An m-accretive operator A is maximally accretive in the sense that A has no proper accretive extensions. Indeed, let A1 be an accretive extension of A. Then, pA ` z1q´1 “ pA1 ` z1q´1 P BpHq for <e z ą 0, which implies A “ A1 . Remark 1.40. Note that in Definition 1.39 we did not require that dom A be dense in H. In fact, an m-accretive operator A is necessarily densely defined. Indeed, since dom A “ ranpA ` z1q´1 for <e z ą 0, to prove this we have to show that ppA ` z1q´1 u, vq “ 0 for all u P H implies v “ 0. Let u “ v and w “ pA ` z1q´1 v. Then, 0 “ |pw, pA ` z1qwq| ě <epAw, wq ` <e z}w}2 ě <e z}w}2 , with <e z ą 0, and hence, w “ 0 and v “ 0. An operator A is called quasi-accretive if there is a γ P C such that A ` γ1 is accretive. Similarly, we call A quasi-m-accretive if A ` γ1 is m-accretive for some γ P C. Definition 1.41. An accretive operator A is called sectorial with semi-angle α P p0, π{2q, if NrA Ď S α :“ tz P C` : | arg z| ď αu. (1.95) If, in addition, A is m-accretive, then it is called m-sectorial. If A is quasi-maccretive and NrA Ď S α,γ :“ tz P C` : | argpz ´ γq| ď αu (1.96) for some γ P C, then it is called quasi-m-sectorial with vertex γ and semi-angle α. Notice that γ and α are not uniquely determined.

32


Corollary 1.42. Let A be a quasi-m-sectorial operator with vertex γ and semi-angle α. Then, by Proposition 1.35, we have ρpAq Ě CzS α,γ .

(1.97)

This is clear from Lemma 1.36, since CzNrA is a connected set. Using the notions of m-accretive and of m-sectorial operators, we can identify m-accretive generators of the contraction, of the quasi-bounded, and of the holomorphic semigroups. Proposition 1.43. An operator A in H is a generator of a contraction semigroup tUt pAqutě0 on H if and only if A is m-accretive. Proof. If the operator A is m-accretive, then, by Definition 1.39 and Remark 1.40, it is densely defined, closed, and its properties (1.94) imply the conditions of Proposition 1.1. Hence, by Definition 1.3, the corresponding semigroup tUt pAqutě0 is a contraction semigroup. The converse follows from Proposition 1.1. l Corollary 1.44. An operator A is the generator of a quasi-bounded semigroup with M “ 1 and of type ω0 (i.e., A P Qp1, ω0 q, see Definition 1.11), if and only if it is quasi-m-accretive for some γ such that <ep´γq “ ω0 . Remark 1.45. Let A be the generator of a quasi-bounded holomorphic semigroup with semi-angle θ P p0, π{2s, i.e., A P H pθ, ω0 q. In Section 1.5, Propositions 1.27 and Propositions 1.33, we have shown that A P H pθ, ω0 “ 0q is equivalent to the existence, for each ε ą 0, of a constant Mε1 such that eiϕ A P QpMε1 , 0q for any angle |ϕ| ď θ ´ ε{2. This remark gives a convenient condition for A to be the generator of a contraction holomorphic semigroup on H. Proposition 1.46. Let A be an m-sectorial operator in a Hilbert space H with vertex γ “ 0 and with semi-angle α P r0, π{2q. Then A P H pθ “ π{2 ´ α, 0q and the holomorphic semigroup tUz pAquzPSθ is contraction semigroup since }Uz pAq} ď 1 for | arg z| ă θ. Proof. According to Definition 1.11 and Remark 1.45, it suffices to prove that eiϕ A P QpM “ 1, 0q for any |ϕ| ď π{2 ´ α. Since, by (1.95) and (1.96), NrA Ă S α,γ“0 , one has Nrpeiϕ Aq “ eiϕ NrA Ă C` for |ϕ| ď π{2 ´ α. Thus, ρpeiϕ Aq Ą CzNrpeiϕ Aq Ą C´ :“ tz P C : <e z ă 0u , by Proposition 1.35. Therefore, if <e ζ ă 0, then by (1.86), }p´ζ1 ` eiϕ Aq´1 } ď

1 . distpζ, C` q

(1.98)

Since distpζ, C` q is actually the distance of ζ from the imaginary axis, (1.98) together with (1.94) and Corollary 1.44 imply that eiϕ A P QpM “ 1, 0q for |ϕ| ď π{2 ´ α. Therefore, }Uz pAq} “ }Ut peiϕ Aq} ď 1 where z “ eiϕ t with t ě 0. l

33

1.7. Perturbations of semigroups

Corollary 1.47. Let A be a positive self-adjoint operator: A “ A˚ ě 0, or NrA Ă R0` “ tx P R : x ě 0u. Then A P H pπ{2, 0q, which implies that, for <e z ą 0, the semigroup Uz pAq is holomorphic and a contraction, }Uz pAq} ď 1.

1.7 Perturbations of semigroups Generally, bounded and a fortiori unbounded perturbations of quasi-bounded strongly continuous semigroups with generator A P QpM, ω0 q require a nontrivial analysis. See examples in Section 4.4. The situation is much simpler if the original semigroup has a generator A P Qp1, ω0 q. Proposition 1.48. Let A P Qp1, ω0 q and B P LpBq. Then the operator H :“ A ` B with dom A is closed, and H P Qp1, ω0 ` }B}q. Proof. Since closedness is stable with respect to bounded perturbations, by virtue of Proposition 1.12 one only has to estimate }R´λ pHqm } for m “ 1. From the second Neumann series for the resolvent R´ζ pHq one gets pH ` ζ1q´1 “

8 ÿ

pA ` ζ1q´1 r´BpA ` ζ1q´1 sk .

(1.99)

k“0

Since A P QpM “ 1, ω0 q implies that }pA ` ζ1q´1 } ď p<e ζ ´ ω0 q´1 for <e ζ ą ω0 , from (1.99) we obtain for }B}p<e ζ ´ ω0 q´1 ă 1 that }pH ` ζ1q´1 } ď p<e ζ ´ ω0 ´ }B}q´1 . By Definition 1.11, the estimate (1.100) yields H P QpM “ 1, ω0 ` }B}q.

(1.100) l

Remark 1.49. It is known that, in general, it is difficult to perturb a generator A of a strongly continuous semigroup by an unbounded operator B so that A ` B remains being a generator. For example, the operator A`B need not be a generator in QpM 1 , ω01 q even if B is a relatively bounded perturbation of A P QpM, ω0 q, that is, (1.101) }Bu} ď a}u} ` b}Au}, u P dom A Ă dom B, for A ě 0 and a ě 0 , b ą 0. The infimum of all possible constants b ą 0 in (1.101) is called the relative bound of the operator B (with respect to A), which we denote again by b. Definition 1.50. Let A be a closed operator in B. We denote by Pb pAq the class of closed operators verifying (1.101) with a relative bound b ą 0. The class Pbă1 pAq contains the so-called Kato-small perturbations of the operator A. On the other hand, if an unbounded operator B verifies (1.101) for any b ą 0, i.e., the infimum is zero, we say that b “ `0 (or b :“ 0` ) and that operator B belongs to the class P0` pAq of infinitesimally small unbounded perturbations of A.


34

Proposition 1.48 hints that one can make the class of admissible perturbations larger by enforcing conditions on the generator A. It is evident that if B P LpBq, then in (1.101) one can put a “ }B} and b “ 0. Hence, the bounded operator B P P0 for any unbounded operator A. So the class P0 of bounded perturbations considered in Proposition 1.48 is the smallest in the hierarchy: P0 Ă P0` pAq Ă Pb pAq .

(1.102)

In Section 4.4 we shall study another class of infinitesimally small unbounded perturbations P, which lies between P0 and P0` , see (4.56). Below we treat perturbations from the class Pbă1 pAq under a rather strong condition on A. Namely, we request analyticity of the original unperturbed semigroup in a sector Sθ with semi-angle θ P p0, π{2s. The measure of the impact of perturbation is expressed by decrement θ ´ ε, of the semi-angle of analyticity (0 ď ε ă θ), and by a variation of the semigroup type ω01 . Proposition 1.51. Let A be the generator of a quasi-bounded holomorphic semigroup of semi-angle θ P p0, π{2s, i.e., A P H pθ, ω0 q. Then, for any non-negative ε ă θ, there exist constants ω01 ě 0 and δ ă 1 such that, if B P Pb pAq for some a ě 0 and b ď δ, then the operator H “ A ` B P H pθ ´ ε, ω01 q. If, in particular, ω0 “ 0 and a “ 0, then H P H pθ ´ ε, 0q is the generator of a bounded holomorphic semigroup. Proof. Without loss of generality, we may assume that the type ω0 “ 0, see Proposition 1.33. Let A P H pθ, 0q. Then B P Pb pAq implies that }BpA ` ζ1q´1 } ď a}pA ` ζ1q´1 } ` b}ApA ` ζ1q´1 }.

(1.103)

If | arg ζ| ď π{2 ` θ ´ ε, then by (1.69), }pA ` ζ1q´1 } ď

Mε , |ζ|

and consequently, }ApA ` ζ1q´1 } “ }1 ´ ζpA ` ζ1q´1 } ď 1 ` Mε . Hence, (1.103) yields the estimate }BpA ` ζ1q´1 } ď a Mε |ζ|´1 ` b p1 ` Mε q,

(1.104) ´1

which ensures the convergence of the Neumann series for pH ` ζ1q provided the right-hand side of (1.104) is smaller than one. The norm estimate of this series gives the additional condition Mε |ζ|´1 1 ´ a Mε |ζ|´1 ´ b p1 ` Mε q Mε p1 ´ b p1 ` Mε qq´1 “ , |ζ| ´ a Mε p1 ´ b p1 ` Mε qq´1

}pH ` ζ1q´1 } ď

(1.105)

35


for | arg ζ| ď π{2 ` θ ´ ε. If b ă p1 ` Mε q´1 “ δ, then to satisfy both conditions ζ should belong to a shifted sector Sπ{2`θ´ε, γ with the vertex γ :“ γpa, b, Mε q ą 0 such that the disc tζ : |ζ| ă aMε p1 ´ b{δq´1 u Ă CzSπ{2`θ´ε,γ .

(1.106)

Equivalently, (1.105) and (1.106) mean that }pH ` γ1 ` ζ1q´1 } ď

M1 , |ζ|

ζ P Sπ{2`θ´ε .

(1.107)

Therefore, by Proposition 1.33, the operator H P H pθ ´ ε, ω01 q for M 1 :“ Mε p1 ´ b{δq´1 ą 0 and for the type ω01 ď γpa, b, Mε q. Note that then the vertex γH ě ´γpa, b, Mε q. From (1.105) we also deduce that one can put ω01 “ 0 if a “ 0, that is, H P H pθ ´ ε, 0q. l Another case when one can extend the class of admissible perturbations to unbounded operators is perturbations of contraction semigroups, i.e., generators from the class Qp1, 0q, cf. Proposition 1.48. Since this book is about the Gibbs semigroups, we consider only Hilbert spaces, see comments in Notes in Section 1.8. Proposition 1.52. Let A P Qp1, 0q be the generator of a contraction semigroup on H. If B P Pbă1 pAq is an accretive operator, then H “ A ` B with dom H “ dom A is the generator of a contraction semigroup on H, that is, H P Qp1, 0q. Proof. By Proposition 1.43, to prove the assertion it is sufficient to verify that H is an m-accretive operator in H. According to Definition 1.39, we have first to check that the operator H “ A ` B with dom H “ dom A is closed. Indeed, since B P Pbă1 pAq, (1.101) shows that á}u} ` p1 ´ bq}Au} ď }Hu} ď a}u} ` p1 ` bq}Au}, (1.108) for u P dom A. Then by virtue of the inequalities (1.108), the operator H is closed since A is closed. Next, by Proposition 1.43, A is an m-accretive operator because A P Qp1, 0q. Since B is an accretive operator, it follows that <epu, Huq “ <epu, Auq ` <epu, Buq ě 0,

u P dom A .

Therefore, H is a closed accretive operator with NrH Ď C` , and }pH ` ζ1qu} ě <epu, pH ` ζquq ě <e ζ}u}2 ,

u P dom A ,

for ζ P C` . This implies (Proposition 1.35) that the deficiency of the closed operator H is defpH ` ζ1q “ 0. Hence, }pH ` ζ1q´1 } ď p<e ζq´1 and, by Definition 1.39, the operator H is m-accretive. Hence, it is a generator of the contraction semigroup tUt pHqutě0 on H. l


36

Remark 1.53. Consider the operator Hpκq :“ A ` κB, dom Hpκq “ dom A, where A and B are the same as in Proposition 1.52. Then one can not guarantee the result of this proposition for complex κ P C, even if |κ| is small. However, if A is a generator of a holomorphic semigroup and B P Pb pAq, see Proposition 1.51, there is a nice theory for A perturbed by κB (κ P C) for |κ b| ă 1. Recall that the linear function: tHpκq :“ A ` κButκ: |κ|ăb´1 u , is a simplest example of a holomorphic family of type (A), see Notes in Section 5.6. Below we treat perturbations from the class Pb pAq under the assumption that the original unperturbed semigroup is generated by an m-sectorial operator A with semi-angle α P r0, π{2q. The effect of the perturbation is expressed by the increment of the semi-angle for Hpκq. In contrast to Propositions 1.51 and 1.52, it is the analytic properties of the family of holomorphic semigroups tUz pHpκqquκPC which are here the focus of our discussion. The case of a self-adjoint generator A ě 0, that is, of an m-sectorial operator A with semi-angle α “ 0, is of a particular interest, see Proposition 1.46. Then some estimates, which are due to Proposition 1.51, become more explicit. Proposition 1.54. Let A ě 0 be a densely defined self-adjoint operator in H. If B P Pb pAq for some b ě 0, then the following statements hold : (i) For any |κ b| ă 1, the operator Hpκq :“ A ` κB is m-sectorial and Hpκq P H pθpκ, bq, ω0 pκ, a, bqq for the semi-angle defined by equation |κ|b , ctg θpκ, bq “ a 1 ´ |κ|2 b2

(1.109)

and the semigroup type ω0 pκ, a, bq ď

a|κ| . 1 ´ b|κ|

(1.110)

(ii) For a fixed r ă b´1 , and for any z in the sector with vertex zero, Sθpr,bq :“

č

Sθpκ,bq ,

(1.111)

|κ|ăr

the family of holomorphic semigroups tUz pHpκqquκPDr is operator-norm holomorphic in the disc Dr “ tκ P C : |κ| ă ru. Proof. (i) By Proposition 1.46, A is the generator of a holomorphic contraction semigroup: A P H pθ “ π{2, 0q. Since κB P P|κb|ă1 pAq, Proposition 1.51 shows that Hpκq is the generator of a bounded holomorphic semigroup: Hpκq P H pθ1 , ωq, for some θ1 ă π{2 and type ω ě 0. To localise the numerical range of the operator Hpκq we proceed as follows.


37

Note that tUz pAquzPSπ{2 is a contraction holomorphic semigroup in the open sector Sθ“π{2 . Since A is self-adjoint, Proposition 1.35 yields # |ζ|´1 , if | arg ζ| ď π2 , }R´ζ pAq} ď (1.112) ´1 | =m ζ| , if π2 ă ˘ arg ζ ă π. By the spectral representation we have ›ż 8 λ ›› › ´1 }ApA ` ζq } “ › dEA pλq › λ`ζ 0 ˇ λ ˇ ˇ ˇ ď sup ˇ ˇ λ ` ζ λě0 # 1, if | arg ζ| ď π2 , “ ´1 |ζ|| =m ζ| , if π2 ă ˘ arg ζ ă π. Then (1.103), (1.112) and (1.113) yield # a|ζ|´1 ` b, if | arg ζ| ď π2 , ´1 }BpA ` ζ1q } ď a| =m ζ|´1 ` b|ζ|| =m ζ|´1 , if π2 ă ˘ arg ζ ă π.

(1.113)

(1.114)

Therefore, the condition }κ B R´ζ pAq} ă 1 for the operator-norm convergence of the Neumann series for the resolvent R´ζ pHpκqq and the estimates (1.112), (1.114) show that the resolvent set ρp´Hpκqq contains the domain ! π |κ|a ) Mκ,b :“ ζ P C : | arg ζ| ď ^ |ζ| ą (1.115) 2 1 ´ |κ|b ! π Y ζPC: ď | arg ζ| ď π ^ 2 „ 1{2 ) |κ|a |κ|b |κ|2 a2 2 | =m ζ| ą à . p<e ζq ` 1 ´ |κ|2 b2 1 ´ |κ|2 b2 1 ´ |κ|2 b2 By virtue of (1.115), for any κ P Db´1 the asymptotic slopes of the border line of Mκ,b , when |ζ| Ñ 8, are equal to ´π ¯ =m ζ |κ|b “ ˘ tg ` θpκ, bq “ ˘ a , 2 |ζ|Ñ8 | <e ζ| 1 ´ |κ|2 b2 lim

(1.116)

for, respectively, π{2 ă ˘ arg ζ ă π. Since Mκ,b Ă ρp´Hpκqq, by (1.115) and (1.116) we obtain that Hpκq is a sectorial operator (Definition 1.41) with semi-angle α “ π{2 ´ θpκ, bq (1.109). Its numerical range lies in the sector: NrpHpκqq Ă S π2 ´θpκ,bq, γpκ,a,bq ,

(1.117)


38

with vertex γpκ, a, bq :“ ´|κ|a{p1 ´ |κ|bq. Then for the vertex of m-sectorial operator Hpκq one gets γHpκq ě γpκ, a, bq. Since for the corresponding semigroup the type is ω0 “ ´ γHpκq , this proves (1.110) for ω0 pκ, a, bq ď ´ γpκ, a, bq. Therefore, Hpκq P H pθpκ, bq, ω0 pκ, a, bqq, by Proposition 1.33 and tUz pHpκqquzPSθpκ,bq is a quasi-bounded holomorphic semigroup for any κ P Db´1 Ş. (ii) For a fixed r ă b´1 , there is a contour Γ Ă κPDr Mκ,b such that, for any κ P Dr , one gets the Riesz-Dunford representation (1.67), ż 1 Uz pHpκqq “ dζ ezζ pζ1 ` Hpκqq´1 , (1.118) 2πi Γ for the family of holomorphic semigroups tUz pHpκqquzPSθpr,bq for the sector Ş Sθpr,bq “ κPDr Sθpκ,bq (1.111) and for the operator-norm convergent Bochner integral. Now we take into account that the resolvent identity gives the representation Uz pHpκ ` qq ´ Uz pHpκqq ż 1 “ dζ ezζ R´ζ pHpκ ` qqp´BqR´ζ pHpκqq 2πi Γ

(1.119)

for κ, κ ` P Dr . Therefore, for any z P Sθpr,bq , the function: κ ÞÑ Uz pHpκqq is operator-norm holomorphic in κ P Dr . l Corollary 1.55. If a “ 0, then by virtue of (1.110), we get ω0 pκ, a “ 0, bq “ 0. Then operator Hpκq ě 0 is the generator of a contraction holomorphic semigroup, see Proposition 1.46. Corollary 1.56. If b “ 0, that is, the operator B P P0 pAq, then Hpκq P H pπ{2, ωpκ, a, b “ 0qq for κ P C, see (1.109).

1.8

Notes

Notes to Section 1.1. The material of this chapter is standard. We give only some of the popular references for further reading and a few historical remarks. One of the first definitions of the exponential function in infinite-dimensional spaces appeared probably in the paper [Gra10]. Here we follow Kato’s book [Kat80], Ch. IX. A brief history of the exponential function (by T. Hahn and C. Parazzoli) can be found in a rather exhaustive treatise [EN00], Ch.VII. Notes to Section 1.2. The continuity assumption at t “ 0 is crucial in the semigroup theory. The right strong continuity for t “ 0 seems to be the most appropriate for the basic classes of semigroups, see discussion in [HP57], [Dav80], [Dav07] and [EN00]. Several other types of semigroups are studied in the book of E. Hille and R. S. Phillips [HP57], the great classic on one-parameter semigroup theory, and in the more recent encyclopedia on semigroups by K.-J. Engel and R. Nagel [EN00].

1.8. Notes

39

Notes to Section 1.3. The material of this section is a part of standard courses on strongly continuous one-parameter semigroups. To complete the references quoted above, we only add here the books [Yos65], [Kre71] as well as [Paz83], [Gol85] and [Ves96]. Proposition 1.12 for ω0 “ 0 and M “ 1 is the famous Hille-Yosida theorem for contraction semigroups, whereas the general case is often called the HilleYosida-Phillips theorem. Our proof of Proposition 1.12 is based on the classical Euler formula (or limit). Another popular way to prove this proposition resorts to the so-called Yosida approximants (1.57), that we consider in Section 1.4. The criterion proved in Proposition 1.13 is due to E. Nelson. Here we followed [Dav07], Ch.6.1. Notes to Section 1.4. Proposition 1.14 states that if a semigroup is right normcontinuous at t “ 0, then it is in some sense trivial. Recall that it is not the case for the so-called immediately norm-continuous or eventually norm-continuous semigroups, that we discuss in Section 4.2, see also [EN00] Ch.II, Section 4. In the last case, the semigroup is norm-continuous for all t ą t0 and some t0 ą 0. For the immediately norm-continuous semigroup t0 “ 0. The both are strongly right-continuous at t “ 0. Although such semigroups are known since [HP57], the complete characterisation of generators of these semigroups is (as far as I know) still an open problem, see [You92]. Proposition 1.17 is again the Hille-Yosida theorem. For the construction of the strongly continuous contraction semigroup, we used the Yosida approximants (1.57). The main importance of the ”no-go” Propositions 1.21 and 1.22 is that they rule out nontrivial extensions to weakly continuous semigroups. In a Hilbert space, Corollary 1.23 easily settles the question of continuity for the adjoint semigroup. In a Banach space the dual Ut˚ (to a strongly continuous on B semigroup Ut ) is ˚ not, in general, strongly continuous on B ˚ . But R` 0 Q t ÞÑ xu|Ut φy is continuous for any u P B, φ P B ˚ . The topology σpB ˚ , Bq generated on B ˚ by the set of functionals from B is the weak˚ -topology. This observation allows one to develop on the dual space B ˚ a consistent theory of dual weak˚ -continuous semigroups, see [EN00] Ch.II, Section 2. Notes to Section 1.5. Holomorphic semigroups arise from parabolic partial differential equations and represent an important subclass of the strongly continuous semigroups, which are also immediately norm continuous in the above sense. In this section we followed essentially [Dav80], [Kat80] and [RS75]. A complete proof of the Proposition 1.30 one can find in [Dav07], Chapter 8.4, Theorem 8.4.9. Notes to Section 1.6. The numerical range of a generator seems a very natural way to characterise the holomorphic semigroups on a Hilbert space. Our exposition follows [Kat80] and [RS75].

40


Notes to Section 1.7. We recall that there are essentially two ways to study perturbations of semigroups. They are the semigroup based methods and the resolvent based methods, Chapters 11.4 and 11.5 in [Dav07]. Since for contraction as well as for holomorphic semigroups one has to control only the resolvent (and not all its powers), we consider in this section only the resolvent based methods. Proposition 1.51 (Hille theorem, [HP57], p.418) and Proposition 1.52 give examples of such a method. We note that the proof of Proposition 1.51 in a Banach space is more complicated. The standard Kato perturbation theory yields straightforwardly the proof for the relative bound b ă 1{2, but the extension to b ă 1 needs additional, although not very complicated arguments, see for example, 2.7 Theorem in [EN00], Ch.III, Section 2. If the Banach space is reflexive, the proof for b ă 1 is simpler (Corollary III.2.9, [EN00]), and thus so is the proof of Proposition 1.51 for a Hilbert space. We return to the perturbation theory in Chapter 4, where the semigroup based methods will be also presented. The results of this section are contained in the monographs [HP57], [Kat80] and [Dav80]. In our proof of Propositions 1.54, we followed the line of reasoning of [Mai71] and [Zag89].

Chapter 2

Classes of compact operators In this chapter, we deal with norm ideals (classes) of compact operators on a Hilbert space. After a rather standard presentation of compact operators, we introduce the von Neumann-Schatten ideals and discuss their properties making essential use of the notion of singular values. The following section is devoted to a detailed discussion of norm convergence theorems in these ideals. More inequalities involving the singular values of compact operators are featured in the last section, as they are indispensable for the study of Gibbs semigroups.

2.1

Compact operators on a Hilbert space

Let H be a complex, separable, infinite-dimensional Hilbert space with inner product p¨, ¨q as in Section 1.6. For u, v P H and α, β P C, pαu, βvq “ αβ pu, vq. Recall that a bounded operator A P LpHq is called non-negative (or positive), if pAu, uq ě 0 for all u P H. We then write A ě 0, and correspondingly A ě B if A ´ B ě 0. Remark 2.1. Every non-negative operator A P LpHq on a complex Hilbert space H is self-adjoint: A˚ “ A. This is a consequence of 0 ď pAu, uq “ pu, A˚ uq “ pu, Auq “ pu, Auq and the polarisation identity 1 tp}u ` v}2 ´ }u ´ v}2 q ´ ip}u ` iv}2 ´ }u ´ iv}2 qu, 4 a which leads to pAu, vq “ pu, Avq for u, v P H. Here }f } :“ pf, f q denotes the usual vector norm on H. pu, vq “

For any A P LpHq, one has A˚ A ě 0 since pA˚ Au, uq “ }Au}2 ě 0. Let EK pλq be the spectral measure corresponding to the self-adjoint operator K “ A˚ A. Then © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1_2

41

42

Chapter 2. Classes of compact operators

(the square-root lemma) the bounded operator ż ? ? K“ dEK pλq λ

(2.1)

σpKq

is well defined, self-adjoint and non-negative, cf. Section 3.1. Therefore,?for any A P LpHq, one can define the absolute value of A by (2.1), i.e., |A| :“ A˚ A “ |A|˚ . Remark 2.2. In spite of the fact that |αA| “ |α| |A| for α P C, relations such as |AB| “ |A| |B|, |A| “ |A˚ | or |A ` B| ď |A| ` |B| are false in general. Recall that an operator U P LpHq such that }U u} “ }u} for all u P H is called an isometry. The operator U is a partial isometry if U æ ker U K , i.e., the restriction of U to the orthogonal complement of ker U :“ tf P H : U f “ 0u, is an isometry. Hence, in this case H “ ker U ‘ ker U K “ ran U ‘ ran U K and U : ker U K Ñ ran U is unitary. Note that U ˚ : ran U Ñ ker U K acts as the inverse of U . Therefore, P1 “ U ˚ U and P2 “ U U ˚ are the orthogonal projections onto ker U K and ran U , which are also called the initial and final spaces of U . Proposition 2.3 (polar decomposition). Let A P LpHq. There exists a unique partial isometry U such that A “ U |A| and ker U “ ker A. Moreover, ran U “ ran A. Proof. Define U : ran |A| Ñ ran A by U p|A|uq :“ Au. Since }|A|u} “ }Au}, one has ker |A| “ ker A. Therefore, U is well defined, that is, if U f “ g, where f “ |A|u and g “ Au, then f “ 0 implies that u P ker |A| and hence g “ 0. Moreover, we get also that ran |A| “ H a ker |A| “ H a ker A “ ran A˚ . Hence, as }f } “ }g}, the operator U is isometric, i.e. it extends to an isometry from ran |A| “ ran A˚ to ran A. To extend U to all of H, we define it to be zero on ran |A|K . Therefore, U is a partial isometry with initial space ran A˚ and final space ran U “ ran A. Since |A| is self-adjoint, ran |A|K “ ker |A| “ ker A, thus ker U “ ker A. l Now we pass to the main subject of this section: the compact operators and norm ideals (classes) in this ring of operators on a Hilbert space H. Definition 2.4. A bounded operator A on H is compact if the image tAun uně1 of any bounded sequence tun uně1 Ă H contains a Cauchy sequence. This is equivalent to the statement that A maps bounded subsets of H into precompact sets, i.e., into subsets with a compact closure. We shall denote the set of compact operators on H by C8 pHq. Recall that a sequence tun uně1 on H converges weakly to u if for any v P H, pun , vq Ñ pu, vq when n Ñ 8. This will be denoted by w-limnÑ8 un “ u. The corresponding topology is weaker than the topology defined by the vector norm on H, and used in Definition 2.4. Note that there is no difference between compact C8 pHq and completely continuous operators L8 pHq on H, see Notes to Section 2.1 in Section 2.5.

2.1. Compact operators on a Hilbert space

43

Proposition 2.5. A compact operator A maps any weakly convergent sequence tun uně1 into a norm convergent sequence tAun uně1 . Proof. Suppose that w-limnÑ8 un “ u. Then by the uniform boundedness principle, supn }un } ă 8 and Au “ w-limnÑ8 Aun . If }Aun ´ Au} Ñ 0, for n Ñ 8, then there exists a subsequence tun1 un1 ě1 such that inf n1 }Apun1 ´ uq} “ ε ą 0. As A P C8 pHq, the set tApun1 ´ uqu is precompact. Therefore, one can find a subsequence tun2 un2 ě1 such that limn2 Ñ8 Apun2 ´ uq “ g with }g} ě ε. However, w-limnÑ8 Aun “ Au implies that g “ 0, which is impossible if ε ą 0. Thus, l inf ně1 }Apun ´ uq} “ 0, or limnÑ8 Aun “ Au. Remark 2.6. Since any ball in a separable Hilbert space H is weakly compact, the converse of Proposition 2.5 also holds. Indeed, suppose that A maps any weakly convergent sequence tun uně1 into a vector-norm convergent sequence tAun uně1 . Since by the uniform boundedness principle supně1 }un } ă 8, the sequence tun uně1 belongs to a ball in H. Then there is a subsequence tun1 un1 ě1 so that u “ w-limn1 Ñ8 un1 , and consequently Aun1 Ñ Au. Hence, the operator A is compact by Definition 2.4. This is not true in a Banach space if the space is not reflexive, and that makes there a difference between completely continuous and compact operators, see Notes to Section 2.1 in Section 2.5 Proposition 2.7. The set of compact operators C8 pHq is a two-sided ˚-ideal in the algebra of bounded operators LpHq: (a) C8 pHq is a linear subspace of LpHq. (b) If A P C8 pHq and B P LpHq, then AB P C8 pHq and BA P C8 pHq. (c) If A P C8 pHq, then A˚ P C8 pHq. Proof. (a) Let tun uně1 be a weakly convergent sequence in H. For A, B P C8 pHq and α, β P C one has pαA ` βBqun “ αAun ` βBun Ñ αAu ` βBu. So, by Remark 2.6, pαA ` βBq P C8 pHq. (b) Since a bounded operator B preserves the norm and weak topologies on H, the assertion is a direct consequence of Proposition 2.5 and Remark 2.6. (c) Since }A} “ }A˚ }, the operator A˚ belongs to LpHq and, therefore, ˚ AA P C8 pHq. Let u “ w-limnÑ8 un . Then AA˚ un Ñ AA˚ u, and since }A˚ un ´ A˚ u}2 “ pAA˚ pun ´ uq, pun ´ uqq ď }AA˚ pun ´ uq}}pun ´ uq}, we conclude that A˚ un Ñ A˚ u, that is, A˚ P C8 pHq.

l

We caution the reader that AB P C8 pHq does not imply that one of these operators is compact. The next statement says that the ideal C8 pHq is closed in the operator norm topology. This type of convergence is denoted by } ¨ }-lim An “ A nÑ8

meaning

lim sup }pA ´ An qϕ} “ 0.

nÑ8 ϕPH }ϕ}“1

44


Proposition 2.8. If a sequence tAn uně1 of compact operators converges to A in the operator norm topology, then A P C8 pHq. Proof. Suppose that tum umě1 is bounded. Then supmě1 }um } “ δ ă 8 and for any subsequence tumk ukě1 and u P H, }Apumk ´ uq} ď }pA ´ An qumk } ` }An pumk ´ uq} ` }pAn ´ Aqu}.

(2.2)

For any n2 ą n1 ě 1 there are subsequences tun2 k ukě1 Ă tun1 k ukě1 such that tAn1 un1 k ukě1 and tAn2 un2 k ukě1 are Cauchy sequences. Then, by a diagonal trick, one finds a subsequence tukk ukě1 such that tAn ukk ukě1 is Cauchy for every n ě 1. Hence, for any ε ą 0 one can find N pεq and Kpεq so that in (2.2) }pA ´ An qukk } ď }A ´ An }δ ă ε, also }pAn ´ Aqukk `p } ď }An ´ A}δ ă ε, for all n ą N pεq, and at the same time }An pukk úkk `p q} ă ε for kk , kk `p ą Kpεq. Thus tAukk ukě1 is a Cauchy sequence. Hence, A is compact by Definition 2.4. l An important subclass of compact operators is that of the finite-rank operators. Definition 2.9. A bounded operator A is of finite rank if dim ran A ă 8 or, equivalently, there exists an orthonormal basis tej ujě1 Ă H such that Au “

r ÿ

pAu, ej qej ,

uPH

(2.3)

j“1

for some finite r. The set of all finite-rank operators will Ť be denoted by KpHq. Let Kr pHq :“ tA P LpHq : dim ran A “ ru. Then KpHq “ rě0 Kr pHq. Remark 2.10. The following properties of KpHq are a direct consequence of Definition 2.9: (a) KpHq Ă C8 pHq because for any A P KpHq the set tAϕ : }ϕ} ď 1u is precompact in H. (b) By (2.3) one has pAu, vq “

r ÿ

pu, A˚ ej qpej , vq “ pu, A˚ vq.

j“1

Denoting e˜j :“ A˚ ej , we get the representations A“

r ÿ j“1

p¨, e˜j qej , A˚ “

r ÿ

p¨, ej q˜ ej

j“1

for A and A˚ . Hence, if A P Kr pHq, then A˚ P Kr pHq.

(2.4)

45

2.1. Compact operators on a Hilbert space

(c) KpHq is a linear subspace of C8 pHq. Moreover, if A P KpHq and B P C8 pHq, then AB P KpHq and BA P KpHq by (2.4). Hence, KpHq is a two-sided ˚-ideal in C8 pHq and in LpHq. The following property is typical for compact operators, cf. Proposition 2.8. Proposition 2.11. Every compact operator A P C8 pHq is the operator norm limit of a sequence tAn uně1 Ă KpHq. Proof. Let tej ujě1 be an orthonormal basis in H and let HnK be the orthogonal complement of the subspace Hn :“ re1 , . . . , en s spanned by tej unj“1 . We set K Hn“0 “ H and define sn`1 :“ sup }Au}. (2.5) uPHK n }u}“1

Then the sequence tsn uně1 is monotonically decreasing, so it converges to a limit K s˚ ě 0. Choose a normalised sequence tun P Hn´1 uně1 such that }Aun } ě s˚ {2. Since for any f P H lim pf, un q “ lim

nÑ8

8 ÿ

nÑ8

pf, ej qpej , un q “ 0,

j“n`1

w-lim un “ 0, and,řby Proposition 2.5, one gets limnÑ8 Aun “ 0. Thus s˚ “ 0. n Define now An :“ j“1 p¨, ej qAej P Kn pHq. Then › 8 › › › ÿ pv, ej qAej › }A ´ An } “ sup }pA ´ An qv} “ sup › vPH }v}“1

vPH }v}“1 j“n`1

“ sup }Au} “ sn`1 . K uPHn }u}“1

Hence, } ¨ }-limnÑ8 An “ A, as claimed.

l

Remark 2.12. The above statement means that C8 pHq is the closure of KpHq in the operator norm. Let Pn : H Ñ Hn be the orthogonal projection onto the subspace spanned by tej : j “ 1, 2, . . . , nu, where tej ujě1 is an orthonormal basis in H. Then, taking for any B P LpHq the operator Bn :“ Pn B P Kn pHq, we have lim }pB ´ Bn qv} ď }B} lim }p1 ´ Pn qv} “ 0,

nÑ8

nÑ8

v P H.

We say that B is the strong limit of the Bn : B “ s-lim Bn “ B. Therefore, the strong closure of the class KpHq coincides with LpHq. For example, s-limnÑ8 Pn “ 1 R C8 pHq.

46


2.2 The canonical form of a compact operator Definition 2.13. Let A be a closed linear operator in H with dom A. The resolvent set ρpAq of A is defined by ρpAq :“ tλ P C : ker Aλ “ t0u and ran Aλ “ Hu. Thus, the resolvent Rλ pAq :“ Aλ´1 has dom Rλ pAq “ H and is a bounded operator with ran Rλ pAq “ dom A for any λ P ρpAq. The set σpAq “ CzρpAq is called the spectrum of A. Here Aλ “ A ´ λ1. We need to distinguish three subsets of the spectrum. For convenience they are recalled below, cf. Appendix A, Chapter 7.4. Definition 2.14. Let A be a closed linear operator in H. Then (a) the set σp pAq :“ tλ P C : ker Aλ ‰ t0uu is called the point spectrum of A; (b) the set σcont pAq :“ tλ P C : ker Aλ “ t0u ^ ran Aλ ‰ ran Aλ “ Hu is called the continuous spectrum of A. For λ P σcont pAq, the domain of the resolvent is a dense subspace of H; (c) the set σres pAq :“ tλ P C : ker Aλ “ t0u ^ ran Aλ Ă Hu is called the residual spectrum of A. Note that the range of Aλ is not dense for λ P σres pAq. Remark 2.15. If λ “ 0 would belong to the resolvent set ρpAq of a compact operator A, then A´1 P LpHq and, by Proposition 2.7, one would get 1 “ A´1 A P C8 pHq, which is impossible if H is infinite-dimensional. Hence, λ “ 0 always belongs to σpAq for any A P C8 pHq. Remark 2.16. According to Definition 2.14, σpAq “ σp pAq Y σcont pAq Y σres pAq,

(2.6)

and these three subsets do not intersect. Remark 2.17. Let λ0 P ρpAq. By Definition 2.13, there is a Cpλ0 q ą 0 such that }Aλ0 u} ě Cpλ0 q}u} for all u P dom A. This implies that for |λ0 ´ λ| ă Cpλ0 q }rpλ0 ´ λq1 ` Aλ0 s´1 u} “ }pAλ0 q´1

8 ÿ

pλ ´ λ0 qn pAλ0 qń u}

n“0

ď C ´1 pλ0 qp1 ´ |λ0 ´ λ|C ´1 pλ0 qq}u}. Therefore, the resolvent set ρpAq is open, i.e., the open neighbourhood tλ P C : |λ ´ λ0 | ă Cpλ0 qu of a regular point λ0 P ρpAq belongs to the resolvent set. Hence, the spectrum σpAq is a closed subset of C.

47

2.2. The canonical form of a compact operator

For a bounded operator A P LpHq, we can easily localise its spectrum σpAq in the complex plane. Let |λ| ą }A}, then › › 8 › ›ÿ }Rλ pAqu} “ |λ|´1 › pA{λqn u› ď |λ|´1 t1 ´ }A}{|λ|u´1 }u},

u P H,

n“0

which means that σpAq Ă tλ : |λ| ď }A}u. For compact operators C8 pHq we can say more. Proposition 2.18. Let A be a compact operator on H. Then σpAq is a discrete set with no other accumulation point than zero. Furthermore, each non-zero λ P σpAq is an eigenvalue of A of finite multiplicity, i.e., the corresponding space of eigenvectors is finite-dimensional, and λ is an eigenvalue of A˚ with the same multiplicity as λ. Lemma 2.19. The eigenvalues of a compact operator A have no other accumulation points than zero. Proof. Suppose that the statement of the lemma is false. Then there exists a sequence tλn uně1 of eigenvalues of A with corresponding eigenvectors un such that |λn | ě δ ą 0. Suppose first that tλn u contains only a finite number of distinct eigenvalues. Then there is a subsequence tλnj unj ě1 such that all λnj are equal: λnj “ λ˚ . Let Hλ˚ Ă H be the subspace spanned by the corresponding eigenvectors tunj u. It is obviously invariant under A. In particular, the operator A maps the bounded set tv P Hλ˚ : }v} ď pλ˚ q´1 u into the unit ball B “ tu P Hλ˚ : }u} ď 1u, which should be compact due to the compactness of A. This is possible only if dim Hλ˚ ă 8. Hence, the subsequence tλnj unj ě1 is finite and λ˚ is not a limit point, i.e., each eigenvalue has a finite multiplicity. Next, suppose that tλn u contains an infinite number tλα u of distinct eigenvalues with |λα | ě δ ą 0. Then, for any m “ 1, 2, . . ., the eigenvectors tuα : α “ řm´1 1, 2, . . . , mu are linearly independent. If this is not the case, then um “ α“1 Cα uα řm´1 řm´1 and λm um “ α“1 Cα λα uα , which implies that α“1 p1 ´ λα {λm quα “ 0. Let Hm Ă H now be the subspace spanned by tuα : α “ 1, 2, . . . , mu; then Hn is invariant under A. Since the vectors tuα : α “ 1, 2, . . . , mu are linearly independent, Hm´1 is a proper subspace of Hm and, by the Gram-Schmidt orthogonalisation procedure, there is a vm P Hm with }vm } “ 1 such that vm K Hm´1 . Hence, the sequence tvα uαě1 is orthonormal and the set tλ´1 α vα u Ă H is bounded. Con´1 ´1 ´1 sider Apλ´1 n vn ´ λm vm q “ vn ´ rλm Avm ´ λn pA ´ λn qvn s with m ă n. Then Avm P Hn´1 because vm P Hn´1 and Hn´1 is invariant under A. By the construction of Hn the operator pA´λn 1q maps Hn into Hn´1 . Hence, ´1 pA ´ λn 1qvn P Hn´1 and rλ´1 m Avm ´ λn pA ´ λn 1qvn s P Hn´1 K vn . Therefore, ´1 ´1 }Apλn vn ´ λm vm q} ě }vn } “ 1 for any m ă n, showing that the set tApλ´1 α vα qu is not compact. This contradicts to compactness of A. l Lemma 2.20. Suppose that λ ‰ 0 is not an eigenvalue of the compact operator A. Then ran Aλ is closed.

48


Proof. Suppose that a sequence tAλ un uně1 Ă ran Aλ converges to v. We need to show that v P ranpλ1 ´ Aq. If tun u is bounded, then by the compactness of A, there is a subsequence tun1 un1 ě1 such that Aun1 Ñ w. Hence, λun1 “ pλ1 ´ Aqun1 ` Aun1 Ñ v ` w and λAun1 Ñ Apv ` wq. Thus, λw “ Apv ` wq, or v “ λ´1 pλ1 ´ Aqpv ` wq P ran Aλ . Now, if tun u is unbounded, then there is a subsequence tun1 un1 ě1 such that un1 un1 ě1 is a bounded sequence and }un1 } Ñ 8. Set u ñ1 :“ un1 {}un1 }; then t˜ limn1 Ñ8 Aλ un1 {}un1 } “ 0. By the same argument as above, there is a subseun2 ` A˜ un2 un2 ě1 such that A˜ ˜ λ˜ quence t˜ un2 “ pλ1 ´ Aq˜ un2 Ñ w, ˜ and un2 Ñ w ˜ “ limn2 Ñ8 }λ˜ un2 Ñ Aw, ˜ “ 0, where }w} ˜ ´ Aw un2 } “ |λ| ą 0. So, w λA˜ ˜ or λw ˜ is an eigenvector of A corresponding to an eigenvalue λ ‰ 0, which is contrary to the assumption of the lemma. l Lemma 2.21. Let λ ‰ 0. If A is a compact operator and ker Aλ “ t0u, then ran Aλ “ H. Proof. Define a sequence of subspaces of H by Hpnq :“ ranpRλń pAqq for n “ 0, 1, 2, . . ., where Hp0q :“ H. Then H Ě Hp1q Ě ¨ ¨ ¨ and, by Lemma 2.20, each Hpn`1q “ Aλ Hpnq is a closed subspace of Hpnq . Moreover, there is an n˚ so that Hpn`1q “ Hpnq for all n ě n˚ . Indeed, suppose to the contrary that all tHpnq u are distinct. Then by the Gram-Schmidt orthogonalisation procedure there exists an orthonormal sequence tvn uně1 such that vn`1 P Hpn`1q and vn`1 K Hpnq . Then Avn ´ Avm “ ´λvm ` pλvn `Rλ´1 pAqvn ´Rλ´1 pAqvm q and the vector λvn `Rλ´1 pAqvn ´Rλ´1 pAqvm belongs to Hpm`1q for m ă n. Therefore, }Avn ´ Avm } ě |λ|}vm } “ |λ|, showing that the set tAvn u is not compact. This contradicts the assumption that A is compact. Assume that ran Aλ Ă H, i.e., H Ą Hp1q . Since ker Aλ “ t0u, the mapping Aλ : H Ñ Hp1q is a bijection and consequently Aλ : Hp1q Ñ Hp2q Ă Hp1q . If this would not be the case, then Hp2q “ Hp1q but, as Aλ´1 Hp2q “ Hp1q and Aλ´1 Hp1q “ H, it would follow that Hp1q “ H. The same reasoning applies to all n ě 1: Aλ : Hpnq Ñ Hpn`1q Ă Hpnq , which contradicts to the existence of n˚ . Hence, l ran Aλ “ H. Lemma 2.22. Let A P LpHq. Then σpA˚ q “ tλ P C : λ P σpAqu and Rλ pA˚ q “ Rλ pAq˚ . Proof. If λ P ρpAq, then Rλ pAqAλ “ 1 “ Aλ Rλ pAq. Therefore, Rλ pAq˚ A˚λ “ 1 “ A˚λ Rλ pAq˚ , which proves the lemma. l Proof of Proposition 2.18. By Remark 2.15, σpAq ‰ H and contains at least the point λ “ 0. Lemma 2.20 excludes that the continuous spectrum of A contains any λ ‰ 0, and the same holds for the residual spectrum by Lemma 2.21. Hence, according to Lemma 2.19 and (2.6), we have σpAqzt0u “ σp pAqzt0u, and the point spectrum σp pAqzt0u is a discrete set of eigenvalues of finite multiplicities. By Remark 2.15 and Lemma 2.22, 0 P σpA˚ q and σpA˚ qzt0u “ σp pA˚ qzt0u “ tλ ‰ 0 : λ P σp pAqu. l

49

2.2. The canonical form of a compact operator

Remark 2.23. The only point in the spectrum σpAq that we did not yet classify is λ “ 0. In general, it may belong to σcont pAq Y σres pAq as well as to σp pAq, see Appendix A (Section A.6). In the latter case it can be an eigenvalue of finite or infinite multiplicity. By Remark 2.15 the point λ “ 0 always belongs to σpAq, if the Hilbert space H is infinite-dimensional. Compact operators for which σpAq “ t0u are known as abstract Volterra operators. For more details about the classification of the point λ “ 0 and for examples of compact operators, see Sections A.6 and A.7 of Appendix A. A particularly interesting case is that of the compact self-adjoint operators, Section A.6 of Appendix A. Proposition 2.24. Let A “ A˚ P C8 pHq, then (a) σpAq Ă R and σres pAq “ H; (b) eigenvectors corresponding to distinct eigenvalues of A are orthogonal; (c) there is an orthonormal basis ten uně1 in H such that Aen “ λn en . The set tλn ‰ 0u coincides with σp pAqzt0u, and each eigenvalue λn has ř a finite multiplicity. Finally, any u P H has a unique representation u “ ně1 un en ` ξ, where ξ P ker A. Proof. (a) Notice that in this item we do not use the compactness of the operator A. Let λ “ α ` iβ. By self-adjointness, }Aλ u}2 “ }pAλ q˚ u}2 ě |β|2 }u}2 ,

u P H.

Therefore, ker Aλ “ kerpAλ q˚ “ t0u and H “ ran Aλ ‘ kerpAλ q˚ “ ran Aλ ,

=m λ ‰ 0.

Then by Definition 2.13 we obtain that σpAq Ă R. Now, since H “ ran Aα ‘ kerpAα q˚ “ ran Aα ‘ ker Aα , Definition 2.14 implies that σres pAq “ H, cf. Remark 2.23. (b) Let λm , λm1 P σp pAq with λm ‰ λm1 . By Proposition 2.18, there exist two eigenvectors um and um1 corresponding to λm and λm1 . But then, pum , Aum1 q “ λm1 pum , um1 q “ pAum , um1 q “ λm pum , um1 q, which is only possible if um K um1 . (c) For each non-zero eigenvalue λm one can choose an orthonormal basis in the finite-dimensional subspace Hλm spanned by eigenvectors corresponding to λm . Since, by (a), eigenvectors corresponding to distinct tλn uně1 are mutually orthogonal, the collection of all these vectors ten uně1 is an orthonormal set in H. Let M be the closure of the span of ten uně1 . Since A “ A˚ and M is invariant under A by construction, one has also that A : MK Ñ MK . If AK :“ A æ MK denotes the restriction of A to MK , then AK “ A˚K P C8 pHq since A “ A˚ P C8 pHq. Therefore, by Proposition 2.18, any non-zero λ P σpAK q belongs

50


to σp pAK q, and thus to σp pAq. However, since all eigenvectors corresponding to non-zero eigenvalues of A are in M, the spectral radius r “ supλPσpAK q |λ| “ 0. Since σpAK q is a closed set, ş σpAK q “ t0u. By the spectral theorem for self-adjoint operators, one gets AK “ σpAK q“t0u dEAK pλq λ “ 0, that is, AK is the zero operator on MK or, equivalently, MK “ ker A. Decomposing H as M ‘ MK , each vector u P H has the unique representation ÿ u“ un en ` ξ, (2.7) ně1

with un “ pu, en q and ξ P ker A.

l

Corollary 2.25. We know that the subspace ker A, at least if it is not reduced to t0u, corresponds to the point 0 P σpAq, see Remark 2.23. Let te1m řumě1 be an K ker Then orthonormal in u basis A. (2.7) M us to write allows “ “ ně1 un en ` ř 1 1 1 H basis in is an orthonormal u e u u te , i.e., corresponding Y te ně1 mě1 n m m m m to the compact self-adjoint operator A. Therefore, 0 P σp pAq is an eigenvalue of finite or infinite multiplicity according to whether dim MK ă 8 or dim MK “ 8. Corollary 2.26 (Canonical form for compact operators). If A is a compact operator on H, then A˚ A is compact and self-adjoint, see Proposition 2.7. By Proposition 2.24 and Corollary 2.25 there is an, not necessarily complete, orthonormal set tϕn uně1 such that A˚ Aϕn “ λn pA˚ Aqϕn with λn pA˚ Aq ‰ 0. The numbers λn pA˚řAq are the non-zero eigenvalues of A˚ A, and for any u P H one has A˚ Au “ ně1 λn pA˚ Aqun ϕn , with un “ pu, ϕn q. Moreover, kerpA˚ Aq “ a rϕ1 , ϕ2 , . . .sK and, since A˚ A ě 0, each λn pA˚ Aq ą 0. Let sn pAq :“ λn pA˚ Aq and set ψn “ sn pAq´1 Aϕn . Then the system of vectors tψn uně1 is orthonormal and we obtain the canonical representation of the operator A: ÿ ÿ Au “ sn pAqpu, ϕn qψn , A˚ u “ sn pAqpu, ψn qϕn , u P H. (2.8) ně1

ně1

Remark 2.27. The canonical representation (2.8) entails (a) that the orthonormal sets tϕn uně1 and tψn uně1 are related by ψn “ sn pAq´1 Aϕn , and that they are complete in ran A and in ran A˚ respectively; (b) that the numbers tsn pAquně1 , called the singular values?of A, are the eigenvalues tλn p|A|quně1 of the self-adjoint operator |A| “ A˚ A, see Proposition 2.3), with corresponding eigenvectors tϕn uně1 . Therefore, if A “ U |A| is the polar decomposition of A, then λn p|A|q “ sn pAq and ψn “ U ϕn ; (c) that although |A| ‰ |A˚ |, one has tsn pAq “ sn pA˚ quně1 . For details, see Section 3.1. The representation (2.8) of a compact operator is called canonical because the inverse to Corollary 2.26 is also true.

2.3. Trace class and Cp pHq-ideals

51

Proposition 2.28. Let tϕn uně1 be an orthonormal (not necessarily complete) set and let U be a partial isometry on H. Let tsn uně1 be a non-increasing sequence of positive real numbers such that sn Ñ 0 when n Ñ 8. Then (2.8), with ψn “ U ϕn , defines a compact operator A “ U |A|, such that tsn pAq “ sn uně1 . Proof. Consider the rank-r operator Ar :“

r ÿ

sn p¨, ϕn qU ϕn .

(2.9)

n“1

Since }pAr`N ´ Ar qu}2 “

r`N ÿ

sn2 |pu, ϕn q|2

n“r`1 2 ď sr`1

r`N ÿ

2 }u}2 , |pu, ϕn q|2 ď sr`1

n“r`1

the sequence tAr urě1 is norm convergent when r Ñ 8. By Proposition 2.11, the limit operator (2.10) } ¨ }-lim Ar “ A rÑ8

is compact with ker A “ ker U and sn “ sn pAq.

l

Remark 2.29. In general the operators ÿ ÿ sn pAqp¨, ϕn qA˚ ψn “ sn pAq2 p¨, ϕn qϕn , A˚ A “ ně1

ně1

and AA˚ “

ÿ ně1

sn pAqp¨, ψn qAϕn “

ÿ

sn pAq2 p¨, ψn qψn ,

ně1

are different. Since |A| ‰ |A˚ |, their eigenfunctions tϕn ‰ ψn uně1 , although the singular values coincide tsn pAq “ sn pA˚ quně1 . If the operator A is normal, i.e., A˚ A “ AA˚ , then tϕn “ ψn uně1 . Therefore, operators A and A˚ have a common set of eigenfunctions, but with different eigenvalues: λpA˚ q “ λpAq, cf. Section 3.1.

2.3

Trace class and Cp pHq-ideals

Above, we associated with any compact operator A its singular values tsn pAq “ a λn pA˚ Aquně1 , where tλn pA˚ Aquně1 is the set of decreasing eigenvalues of ˚ the a positive self-adjoint compact operator A A ą 0. It is clear that sn pAq “ 2 λn p|A| q “ sn p|A|q and that a sn pAq “ λn pAA˚ q “ sn pA˚ q, (2.11)

52


see Remark 2.29. With singular values at our disposal, we can describe the von Neumann-Schatten classes Cp pHq (1 ď p ă 8) of linear operators on a Hilbert space H. We start with the trace class C1 pHq. Definition 2.30. A compact operator A P C8 pHq belongs to the trace class C1 pHq if ÿ (2.12) }A}1 :“ sn pAq ă 8. ně1

By (2.8), Proposition 2.28 and Definition 2.12, }A} “ s1 pAq ď }A}1 ,

(2.13)

where equality holds only for rank-one operators: A P K1 pHq. Moreover, }A}1 “ }|A|}1

and }A}1 “ }A˚ }1 ,

(2.14)

by virtue of (2.11). Proposition 2.31. C1 pHq is a two-sided ˚-ideal in the ring of bounded operators LpHq: (a) If A P C1 pHq, then A˚ P C1 pHq. (b) If A P C1 pHq and B P LpHq, then AB P C1 pHq and BA P C1 pHq. (c) C1 pHq is a vector space. Proof. (a) Follows from (2.11), see (2.14). (b) By the canonical representation (2.8) of a compact operator with nonincreasing singular values tsn pAquně1 , one readily gets s1 pAq “ supuPH:}u}“1 }Au} and (2.15) sup sn pAq “ }Au}, for n ě 2. }u}“1 , uPrϕ1 ,...,ϕn´1 sK

If B P LpHq, then }BAu} ď }B}}Au}. Together with (2.15), this gives sn`1 pBAq ď }B}sn`1 pAq.

(2.16)

Since by (2.11) sn`1 pABq “ sn`1 pB ˚ A˚ q˚ q “ sn`1 pB ˚ A˚ q, we also get sn`1 pABq ď }B ˚ }sn`1 pA˚ q “ }B}sn`1 pAq.

(2.17)

Then (2.12) and the estimates (2.16) and (2.17) give }BA}1 ď }B}}A}1

and }AB}1 ď }B}}A}1 .

(2.18)

(c) Notice that by (2.8), (2.9) and (2.15) we obtain: }A ´ Ar } “

sup }u}“1 ^ uPrϕ1 ,...,ϕn´1 sK

}Au} “ sr`1 pAq.

(2.19)


53

On the other hand, there is a characterisation of singular values, which generalises (2.15) or (2.19), namely, the minimax principle. For the case of unbounded operators, see Proposition 4.23. Remark 2.32 (Minimax principle). Let s1 pAq “ supuPH:}u}“1 }Au} and sn`1 pAq “

inf

sup }Au},

Mn ĂH }u}“1 ^

for n P N .

(2.20)

K uPMn

Here Mn Ă H are linear subspaces of dimension n and MK n are their orthogonal Ăn such that }Au} ă complements. Indeed, suppose that there exists a subspace M K Ă . Let Fn`1 “ rϕ1 , . . . , ϕn`1 s be the subspace spanned by sn`1 pAq for all u P M n ĂK ‰ t0u, since dim M Ăn ă the eigenvectors tϕk u of the operator A. Then Fn`1 X M n K Ă u} ă sn`1 pAq. dim Fn`1 and there is a unit vector u ˜ P Fn`1 X Mn such that }A˜ řn`1 řn`1 2 u, ϕk q|2 sk2 pAq ě sn`1 u}2 “ }A k“1 p˜ pAq since However, }A˜ u, ϕk qϕk }2 “ k“1 |p˜ the tsk pAqukě1 are non-increasing. This contradiction shows that (2.21)

sup }Au} ě sn`1 pAq. }u}“1 ^ K uPMn

It remains only to observe that the equality in (2.21) is attained for Mn “ Fn . Now, let A, B P C1 pHq and let re1 , . . . , en`m s be a subspace spanned by linearly independent vectors tej : j “ 1, 2, . . . , n ` mu. Then }pA ` Bqu} ď }Au} ` }Bu} and sup

}pA ` Bqu} ď

}u}“1 , uPre1 ,...,en`m sK

sup

}Au} `

}u}“1 , uPre1 ,...,en`m sK

sup

}Bu}.

(2.22)

}u}“1 , uPre1 ,...,en`m sK

Minimising the left- and right-hand sides of (2.22) over tej : j “ 1, 2, . . . , n ` mu, one gets by (2.20) that sn`m`1 pA ` Bq ď sn`1 pAq ` sm`1 pBq, which shows that A ` B P C1 pHq.

(2.23) l

Corollary 2.33. (a) Let U1 and U2 be unitary operators. By (2.16) and (2.17) we get sn pAq “ sn pU1´1 U1 AU2 U2´1 q ď sn pU1 AU2 q ď sn pAq for any A P C1 pAq. Therefore, sn pAq “ sn pU1 AU2 q

and

}A}1 “ }U1 AU2 }1 .

(2.24)

54


(b) Let C P Kn pHq be a rank-n operator, see Definition 2.9. Then combining (2.19) and (2.20), we obtain another characterisation of the singular values of an operator A P C8 pHq : sup

sn`1 pAq ď

|pAu, uq|

}u}“1 , uPranpCqK

|ppA ´ Cqu, uq| ď sup |ppA ´ Cqu, uq|

sup

“

sup

}Au} “

}u}“1 , uPranpCqK

}u}“1 , uPranpCqK

}u}“1

(2.25)

“ }A ´ C}.

On the other hand, we know that the equality in (2.25) is attained for C “ An , see (2.19). Therefore sn`1 pAq “

inf tC: CPKn pHqu

}A ´ C}.

(2.26)

Corollary 2.34. Suppose that the compact operators A and B satisfy 0 ď A ď B. Then A “ |A|, B “ |B|, and }A1{2 u} ď }B 1{2 u}. The minimax principle (2.20) 1{2 1{2 and sn pA1{2 q “ sn pAq, sn pB 1{2 q “ sn pBq imply that sn pAq ď sn pBq,

n ě 1.

(2.27)

See Section 3.1 (e) for details. Lemma 2.35. Let A P LpHq and suppose that basis tej ujě1 in H. Then A P C8 pHq.

ř8 j“1

}Aej }2 ă 8 for an orthonormal

Proof. Consider a bounded sequence tun uně1 with }un } ď a. Then }A˚ un }2 “

8 ÿ

|pA˚ un , ej q|2 “

j“1

8 ÿ

|pun , Aej q|2 .

j“1

Now, let u “ w-lim un . Then pun ´ u, Aej q Ñ 0, and one has }A˚ pun ´ uq}2 “

8 ÿ

|pun ´ u, Aej q|2

j“1

ď }pun ´ uq}2

8 ÿ

}Aej }2 ă 8.

j“1

This means that }A˚ pun ´ uq} Ñ 0, or A˚ P C8 pHq, see Proposition 2.5. Hence, l A P C8 pHq, by Proposition 2.7. ř8 Lemma 2.36. Let A P LpHq and A ě 0. If j“1 pej , Aej q ă 8 for an orthonormal basis tej ujě1 in H, then A P C1 pHq and 8 ÿ

ej q “ ej , A˜ p˜

j“1

8 ÿ n“1

sn pAq “ }A}1


55

for any orthonormal basis t˜ ej ujě1 in H. Proof. Recall that A ě 0 implies A “ A˚ , see Remark 2.1. Therefore, the operator ř8 A1{2 is well defined and, by the assumption of the lemma, j“1 }A1{2 ej }2 ă 8. By Lemma 2.35, A1{2 P C8 pHq, and by Proposition 2.7, A P C8 pHq. Notice that for A ě 0 the polar decomposition has the form |A| “ A. Hence, in the canonical representation of A, see (2.8), ψn “ ϕn and we get 8 ÿ

ej q “ p˜ ej , A˜

8 ÿ 8 ÿ

ej , ϕn q|2 “ sn pAq|p˜

j“1 n“1

j“1

8 ÿ

sn pAq}ϕn }2 “

n“1

8 ÿ

sn pAq.

l

n“1

ř8 Proposition 2.37. Let A P LpHq and suppose that j“1 pej , A˜ ej q ă 8 for any two ej ujě1 . Then A P C1 pHq. orthonormal bases tej ujě1 and t˜ Proof. By the polar decomposition, A “ U |A|, see Proposition 2.3. Let t˜ ej ujě1 be an orthonormal basis in ran |A| and ej :“ U e˜j . Since the restriction of U to ran |A| is an isometry, the system tej u is also orthonormal and 8 ÿ

p˜ ej , |A|˜ ej q “

j“1

8 ÿ

8 ÿ

pU e˜j , U |A|˜ ej q “

j“1

pej , A˜ ej q ă 8,

(2.28)

j“1

by the assumption of the proposition. We can complete the system t˜ ej ujě1 to an orthonormal basis for H without altering the sum (2.28) because H a ran |A| “ ker |A|. Then by Lemma 2.36 we get that |A| P C1 pHq, and hence, A “ U |A| P C1 pHq. l The next statement says that the converse is also true. Proposition 2.38. Let A P C1 pHq and tej u, t˜ ej u be two arbitrary orthonormal bases for H. Then 8 ÿ |pej , A˜ ej q| ď }A}1 . (2.29) j“1

Equality is reached for ej “ ψj and e˜j “ ϕj , where tψj u and tϕj u are defined by the canonical representation (2.8). Proof. Since A P C8 pHq, one can use the canonical representation (2.8) to get the following estimate: 8 ÿ

|pej , A˜ ej q| ď

j“1

8 ÿ

sn pAq|p˜ ej , ϕn qpej , ψn q|

j,n“1 8 ÿ

ď n“1 8 ÿ

“ n“1

sn pAq

8 ”ÿ

|p˜ ej , ϕn q|2

8 ı1{2 ” ÿ

j“1

sn pAq}ϕn }}ψn } “

|pej , ψn q|2

ı1{2

j“1 8 ÿ n“1

sn pAq “ }A}1 .

(2.30)

56


Again by (2.8) we get that equality in (2.30) is attained for ej “ ψj and e˜j “ ϕj . l Instead of Proposition 2.37, the following property may be useful: ř8 Proposition 2.39. If A P LpHq and j“1 }Aej } ă 8 for an orthonormal basis tej ujě1 of H, then A P C1 pHq. Proof. By the polar decomposition, we have }Au} “ }|A|u} and 8 ÿ

pej , |A|ej q ď

8 ÿ

8 ÿ

}|A|ej } “ j“1

j“1

}Aej } ă 8. j“1

Therefore, by Lemma 2.36, |A| P C1 pHq and, consequently, A P C1 pHq.

l

ej ujě1 be two orthonormal bases and A P C1 pHq. Remark 2.40. Let tej ujě1 and t˜ Then 8 ÿ

ej q “ ej , A˜ p˜

8 8 ÿ ÿ

ej , en qpen , Aen1 qpen1 , ej q p˜

j“1 n,n1 “1

j“1

8 ÿ

“

pen , Aen1 q

n,n1 “1

8 ÿ

8 ÿ

ej , en qpen1 , e˜j q “ p˜

j“1

pen , Aen q,

n“1

where the interchanging of sums is allowed by absolute convergence, see Proposition 2.38. This motivates the following definition. Definition 2.41. The map Tr : C1 pHq Ñ C given by A ÞÑ Tr A :“

8 ÿ

pej , Aej q, A P C1 pHq,

j“1

where tej ujě1 is any orthonormal basis in H, is called the trace. To bolster this concept we calculate the result of this map explicitly. Proposition 2.42. Let A P C1 pHq. Then for any orthonormal basis tej ujě1 Ă H the ř8 sum j“1 |pej , Aej q| ă 8 and the matrix trace Tr A is independent of the choice of a basis: 8 8 ÿ ÿ sn pAqpϕn , ψn q . (2.31) pej , Aej q “ j“1

n“1

The map A ÞÑ Tr A is a bounded linear functional on the space C1 pHq with | Tr A| ď }A}1 .


57

Proof. By the canonical representation (2.8) of compact operators and the absolute convergence of the double sum, which justifies their interchange, one obtains the representation (2.31): 8 ÿ

pej , Aej q “

j“1

8 ÿ

sn pAqpϕn , ej qpej , ψn q “

8 ÿ

sn pAqpϕn , ψn q .

n“1

j,n“1

Moreover, since sn pAq ě 0 and |pϕn , ψn q| ď 1, this representation yields the } ¨ }1 estimate | Tr A| ď }A}1 for the linear functional Tr p¨q. l Corollary 2.43. The (matrix) trace Tr p¨q is } ¨ }1 -continuous. A P C1 pHq. By the estimate in Corollary 3.4, the sum of eigenRemark 2.44. Let ř8 values ΛpAq :“ n“1 λn pAq (spectral trace) converges absolutely and defines on C1 pHq another bounded linear functional. Whether the spectral and the matrix traces coincide is a subtle question, the proof of which is out of the scope of this book. See Notes in Section 2.5 for references. Proposition 2.45 (Lidskiˇı trace theorem). If the operator A P C1 pHq, then Tr A “ ΛpAq. This positive answer for A P C1 pHq allows us to establish a number of trace and } ¨ }1 -norm inequalities, Section 3.3. To appreciate the nontriviality of equality Tr A “ ΛpAq we discuss in Appendix A (Section A.7) the Volterra operator. The next assertion confirms that the domain C1 pHq for the functional ΛpAq is also reasonable, because it is closed under the mapping A ÞÑ |A|. Proposition 2.46. An operator A P LpHq belongs to the trace class C1 pHq if and only if Tr |A| ă 8, and in this case Tr |A| “ }A}1 .

(2.32)

by (2.14) |A| P C1 pHq. By (2.8), the canonical represenProof. If A P C1 pHq, then ř tation of |A| reads |A| “ ně1 sn pAqp¨, ϕn qϕn , where the eigenvalues λn p|A|q of |A| coincide with the singular values sn pAq of A and tϕn uně1 is the orthonormal set of the corresponding eigenvectors. By the Hilbert-Schmidt theorem (Corollary 2.25) we can complete tϕn uně1 to an orthonormal basis tej ujě1 of H, such that 8 8 ÿ ÿ ÿ (2.33) pej , |A|ej q “ pϕn , |A|ϕn q “ sn p|A|q. j“1

ně1

ně1

Since |A| P C1 pHq, the sum (2.33) converges. Then by Lemma 2.36 it converges for any orthonormal basis of H, i.e., by Definitions 2.30 and 2.41 one gets ÿ (2.34) Tr |A| “ sn p|A|q “ }A}1 ă 8 . ně1

58


|A| ă 8. Then for any orthonormal Now, let A P LpHq ř8be such that Trř 8 basis t˜ ej ujě1 one has j“1 p˜ ej q “ j“1 }|A|1{2 e˜j }2 ă 8. Therefore, by ej , |A|˜ Lemma 2.35, |A|1{2 P C8 pHq. Hence, |A| P C8 pHq. Then, calculating the trace in the complete orthonormal basis tej ujě1 corresponding to the compact self-adjoint operator |A|, see Corollaries 2.25 and 2.26, we get Tr |A| “

8 ÿ

8 ÿ

pej , |A|ej q “

j“1

λj p|A|q “

j“1

ÿ

sn pAq “ }A}1 .

l

ně1

ř8 Remark 2.47. It is not true that if j“1 |p˜ ej q| ă 8 for some orthonormal ej , A˜ basis t˜ ej ujě1 , then A P C1 pHq, cf. Propositions 2.37–2.39 and Remark 2.40. Some basic properties of the trace are: Proposition 2.48. Let A, B P C1 pHq. Then: (a) TrpA ` Bq “ Tr A ` Tr B. (b) TrpλAq “ λ Tr A, λ P C. (c) TrpU AU ´1 q “ Tr A for any unitary operator U . (d) TrpACq “ TrpCAq for any C P LpHq. (e) If 0 ď A ď B, then Tr A ď Tr B. (f) If a sequence tAk ukě1 from the cone C1,` pHq of positive trace-class operators converges weakly to a trace-class operator A: w-limkÑ8 Ak “ A, then Tr A ď lim inf kÑ8 Tr Ak , that is, the functional Tr p¨q is weakly lower semicontinuous. Proof. Properties (a) and (b) follow directly from Definition 2.41. ej 1 uj 1 ě1 “ (c) Note that if tej ujě1 is an orthonormal basis, then so is t˜ ř8 ej , U AU ´1 e˜j q “ Tr A. tU ej uj 1 ě1 . Thus TrpU AU ´1 q “ j“1 p˜ (d) Choose an orthonormal basis tej ujě1 Ą tψn uně1 , where tψn uně1 is the orthonormal system corresponding to the canonical form of A˚ , (2.8), then TrpACq “

8 ÿ

pej , ACej q “

ÿ

pA˚ ψn , Cψn q “

sn pAqpϕn , Cψn q.

ně1

ně1

j“1

ÿ

Similarly, using the orthonormal system tϕn uně1 corresponding to A (2.8), one gets ÿ ÿ sn pAqpϕn , Cψn q “ TrpACq. pϕn , CAϕn q “ TrpCAq “ ně1

ně1

(e) Since A ď B is equivalent to pu, Auq ď pu, Buq for any u P H, the from the definition of trace. inequality Tr A ď Tr B follows ř 8 (f) Note that Tr Ak “ n“1 pen , Ak en q for any k ě 1 independently of the orthonormal basis ten uně1 , where each positive term pen , Ak en q has limit:


59

limkÑ8 pen , Ak en q “ pen , Aen q. Then using a discrete version of Fatou’s lemma we get inequality Tr A “

8 ÿ n“1

lim pen , Ak en q ď lim inf kÑ8

kÑ8

8 ÿ

pen , Ak en q,

(2.35)

n“1

which proves the weak lower semi-continuity of the trace when the weak limit A P C1 pHq. l Note that the weak operator convergence of the trace-class operators tAk ukě1 above does not guarantee that the positive bounded operator A P C1,` pHq, since both sides of inequality (2.35) could be infinite. We can summarise the properties of the functional } ¨ }1 :“ Trp| ¨ |q on C1 pHq as Proposition 2.49. The functional } ¨ }1 is a norm on the two-sided ˚-ideal C1 pHq: (a) If A P C1 pHq, then }A}1 ě 0, and }A}1 “ 0 implies A “ 0. (b) }αA}1 “ |α|}A}1 for α P C and A P C1 pHq. (c) }A ` B}1 ď }A}1 ` }B}1 for A, B P C1 pHq. (d) }AB}1 ď }A}1 }B}1 for A, B P C1 pHq. (e) The functional } ¨ }1 is weakly lower semi-continuous. Proof. (a) This follows from (2.12) by the positivity of the singular values and by the canonical representation (2.8) of compact operators. (b) This results from the polar decomposition |αA| “ |α||A| and Proposition 2.53: }αA}1 “ Tr |αA| “ |α| Tr |A| “ |α|}A}1 . (c) Let tej ujě1 and t˜ ej ujě1 be two orthonormal bases in H. Then 8 ÿ j“1

|pej , pA ` Bq˜ ej q| ď

8 ÿ

|pej , A˜ ej q| `

j“1

ď }A}1 ` }B}1 ,

8 ÿ

ej q| |pej , B˜

j“1

(2.36)

by (2.29). Applying Proposition 2.38 to the left-hand side of (2.36), we get }A ` B}1 ď }A}1 ` }B}1 . (d) This follows directly from (2.13) and (2.18): }AB}1 ď }A}1 }B} ď }A}1 }B}1 . (e) The proof is similar to (f) in Proposition 2.48. l In fact, the ˚-ideal C1 pHq equipped with the trace-norm } ¨ }1 is a Banach space. This is similar to the ˚-ideal C8 pHq, which is a Banach space with respect to the operator norm } ¨ }. We shall consider topological properties of the ˚-ideals Cp pHq, 1 ď p ă 8, in the next section.

60


Definition 2.50. An operator A P C8 pHq is said to belong to the von NeumannSchatten class Cp pHq, 1 ď p ă 8, if +1{p # ÿ p }A}p :“ sn pAq ă 8. (2.37) ně1

Since by (2.37) p ÞÑ }A}p is a non-increasing function of p ě 1, one immediately gets for 1 ă p ă q ă 8 that C1 pHq Ă Cp pHq Ă Cq pHq Ă C8 pHq,

(2.38)

which corresponds to the estimates (2.39)

}A}1 ě }A}p ě }A}q ě }A}.

The norms in (2.39) coincide only for rank-one operators A P K1 pHq. Another immediate consequence of (2.37) is an analogue of Proposition 2.46. If A P Cp pHq, then by (2.38) A P C8 pHq and |A| P C8 pHq. By the spectral representation of the self-adjoint operator |A|, we can define, for any 1 ď p ă 8, the operator ż ÿ λnp p|A|qp¨, ϕn qϕn , |A|p “ (2.40) dE|A| pλq λp “ σp|A|q

ně1

where tϕn u and tλn p|A|q “ sn pAqu are the eigenvectors and eigenvalues of |A|. Therefore, |A|p P C1 pHq and (2.37) reads }A}p “ pTr |A|p q1{p .

(2.41)

The class Cp pHq is a linear space. If A P Cp pHq, then obviously }αA}p “ |α|}A}p , i.e., αA P Cp pHq. Now let A, B P Cp pHq. Then by (2.38) we get pA ` Bq P C8 pHq. For the singular values of this compact operator we have inequalities s2n pA ` Bq ď s2n´1 pA ` Bq ď sn pAq ` sn pBq, see (2.23). Then by using the elementary estimates ” ıp psn pAq ` sn pBqqp ď maxt2sn pAq, 2sn pBqu ď p2sn pAqqp ` p2sn pBqqp , we obtain }A ` B}p “

ÿ

tps2n´1 pA ` Bqqp ` ps2n pA ` Bqqp u

ně1

ď 2p`1

”ÿ

psn pAqqp `

ně1 p`1

“2

p}A}p ` }B}p q,

ÿ

psn pBqqp

ı

ně1

(2.42)


61

that is, pA ` Bq P Cp pHq. As sn pA˚ q “ sn pAq, see (2.11), we get that A˚ P Cp pHq whenever A P Cp pHq, and that }A}p “ }A˚ }p . (2.43) Inequality (2.17) implies }AB}p ď }B}}A}p

and }BA}p ď }B}}A}p ,

(2.44)

which means that Cp pHq is a two-sided ˚-ideal in LpHq. Notice that the inequalities (2.39) and (2.44) imply }AB}p ď }A}p }B}p ,

(2.45)

and that }A}p “ 0 implies A “ 0, see (2.37) and (2.40). Therefore, we can summarise the above observations about } ¨ }p as Proposition 2.51. For p ě 1 the functional } ¨ }p is a norm on the ˚-ideal Cp pHq. To prove this statement it remains to verify the triangle inequality }A ` B}p ď }A}p ` }B}p

(2.46)

for A, B P Cp pHq. Note that A ` B P Cp pHq by (2.42) and that we have already established (2.46) for p “ 1 in Proposition 2.49 (c). We shall first establish some auxiliary results before proving (2.46) in Proposition 2.55. Lemma 2.52. Let W “ rwnm s be a square matrix of dimension r satisfying r ÿ

|wnm | ď 1

and

r ÿ

|wnm | ď 1.

(2.47)

m“1

n“1

If x “ px1 , x2 , . . . , xr q and y “ py1 , y2 , . . . , yr q are two vectors with ordered components 0 ď xr ď xr´1 ď ¨ ¨ ¨ ď x1 and 0 ď yr ď yr´1 ď ¨ ¨ ¨ ď y1 , then |pW x, yq| ď px, yq.

(2.48)

Proof. Let fk “ p1, . . . , 1, 0, . . . , 0q be the vector with the first k components equal to 1. Then according to (2.47), ˇ ÿ ˇ ˇ ˇ |pW fn , fm q| “ ˇ wij ˇ ď mintn, mu “ pfn , fm q, (2.49) iďn,jďm

for 1řď n, m ď r. It is ř clear that the vectors x and y can be represented as r r x “ n“1 αn fn and y “ m“1 βm fm , where αn ě 0 and βm ě 0. By (2.49) we get r r ˇ ÿ ˇ ÿ ˇ ˇ αn βm pfn , fm q “ px, yq. αn βm pW fn , fm qˇ ď |pW x, yq| “ ˇ n,m“1

n,m“1

l

62


Lemma 2.53. Let A, B P C8 pHq. The singular values of A, B and AB satisfy the inequalities r r ÿ ÿ sn pAqsn pBq, r P N. sn pABq ď (2.50) n“1

n“1

Proof. By the polar decomposition, AB “ U |AB|. Hence, |AB| “ U ˚ AB. Let Pr be a projection onto the subspace rϕ1 , ϕ2 , . . . , ϕr s spanned by the eigenvectors tϕj : j “ 1, 2, . . . , ru of the compact operator |AB|, then r ÿ

sn pABq “

n“1

r ÿ

sn p|AB|q “ Trp|AB|Pr q

n“1

˜ “ TrpPr U ˚ ABPr q “ TrpA˜Bq,

(2.51)

˜ :“ BPr P Kr pHq. Using the canonical decomwhere A˜ :“ Pr U ˚ A P Kr pHq and B position (2.8) one gets r ÿ

A˜ “

˜ ˜ n pAq, ˜ ϕn pAqψ sn pAqp¨,

r ÿ

˜“ B

˜ ˜ ˜ n pBq. ϕn pBqψ sn pBqp¨,

(2.52)

n“1

n“1

Therefore, (2.51) and (2.52) yield r ÿ

sn pABq “

n“1

r ÿ

˜ ϕn pAqq ˜ ˜ Bψ ˜ n pAq, sn pAqp

n“1 r ÿ

“

˜ m pBqpψ ˜ ˜ n pAq, ˜ ˜ ϕm pBqqpψ ˜ sn pAqs m pBq, ϕn pAqq

n,m“1

“: pW x, yq,

(2.53)

where matrix W :“ rwnm s with ˜ ˜ ϕm pBqqpψ ˜ ˜ wnm :“ pψn pAq, m pBq, ϕn pAqq, and ˜ s2 pBq, ˜ ˜ . . . , sr pAqq, ˜ s2 pAq, ˜ ˜ . . . , sr pBqq. y :“ ps1 pBq, x :“ ps1 pAq, The matrix W satisfies the conditions of Lemma 2.52 r r r ”ÿ ı1{2 ” ÿ ı1{2 ÿ ˜ 2 ˜ 2 ˜ ϕn pAqq| ˜ ϕm pBqq| |pψm pBq, |pψn pAq, |wnm | ď n“1

n“1

n“1

› ›› › ˜ ››ψm pBq ˜ › “ 1, ď ›ϕm pBq and similarly for the row sums r ÿ

r ”ÿ

|wnm | ď m“1

˜ 2 ˜ ϕm pBqq| |pψn pAq,

m“1

› › ›› ˜ › “ 1. ˜ ››ϕn pAq ď ›ψn pAq

r ı1{2 ” ÿ m“1

˜ 2 ˜ ϕn pAq| |pψm pBq,

ı1{2


63

Therefore, by the estimate (2.48) and (2.53) one gets r ÿ

r ÿ

sn pABq “ pW x, yq ď px, yq “

˜ n pBq. ˜ sn pAqs

(2.54)

n“1

n“1

Then (2.50) follows from the inequalities (2.16), (2.17) and (2.54), since ˜ ď }Pr }sn pBq “ sn pBq. ˜ ď }Pr }}U ˚ }sn pAq “ sn pAq and sn pBq sn pAq

l

Proposition 2.54 (Cp -Hölder inequality). Let A P Cp pHq and B P Cq pHq, where p ą 1 and p´1 ` q ´1 “ 1. Then AB P C1 pHq and }AB}1 ď }A}p }B}q .

(2.55)

ÿ

(2.56)

Proof. By Definition 2.30 }AB}1 “

sn pABq.

ně1

Applying the classical Hölder inequality to the estimate (2.50) for the truncated sum corresponding to (2.56), one gets r ÿ

sn pABq ď

n“1

r ÿ

sn pAqsn pBq

n“1 r ”ÿ

ď

spn pAq

r ı1{p ” ÿ

n“1

sqn pBq

ı1{q .

(2.57)

n“1

Taking the limit r Ñ 8, we obtain (2.55).

l

Proposition 2.55 (} ¨ }p -triangle inequality). Let A P Cp pHq and B P Cp pHq, where 1 ă p ă 8. Then inequality (2.46) holds: }A ` B}p ď }A}p ` }B}p . Proof. Let C P C1 pHq. Then, by Proposition 2.38, | Tr C| ď }C}1 .

(2.58)

For C “ T Q, with T P Cp pHq, Q P Cq pHq, p ą 1 and p´1 ` q ´1 “ 1, the estimate (2.58) implies (2.59) | TrpT Qq| ď }T }p }Q}q , see Proposition 2.54. For a fixed T P Cp pHq the inequality (2.59) is saturated for Q “ Qmax :“ |T |p´1 U ˚ , where we used the polar decomposition T “ U |T |. Indeed, TrpT Qmax q “ TrpU |T |p U ˚ q “ p}T }p qp ,

64


see (2.32), (2.37) and Proposition 2.48 (d)) and ”ÿ }Qmax }q “

psn pT qqpp´1qq

ı1{q

ně1

”ÿ “

ı1{q “ p}T }p qp´1 . spn pT q

ně1

Therefore, }T }p “

| TrpT Qq| “ sup | TrpT Qq|, }Q}q QPCq pHq }Q}q “1 sup

(2.60)

where p ą 1 and p´1 ` q ´1 “ 1. For A, B P Cp pHq, Q P Cq pHq, p ą 1 and p´1 ` q ´1 “ 1 we have the obvious inequality | TrpA ` BqQ| ď | TrpAQq| ` | TrpBQq|. (2.61) Taking here the supremum over tQ : }Q}q “ 1u, we get by (2.60) the } ¨ }p -triangle inequality }A ` B}p ď }A}p ` }B}p . (2.62) l Proof of Proposition 2.51. We have already checked all necessary properties of the functional } ¨ }p , including the triangle inequality for p “ 1, see Proposition 2.49 and for p ą 1, see Proposition 2.55. l In fact, the Cp pHq-ideals are Banach spaces with respect to the norms } ¨ }p , p ě 1. We shall discuss this in the next section.

2.4

Convergence theorems for Cp pHq

This section is devoted to topological properties of the space Cp pHq. We begin with the following lemma. Lemma 2.56. If C1 , C2 P C8 pHq, then |s` pC1 q ´ s` pC2 q| ď }C1 ´ C2 },

` ě 1.

(2.63)

Proof. Put in the estimate (2.23) A ` B “ C1 , A “ C2 and m “ 0. Then, taking into account (2.8), sn`1 pC1 q ´ sn`1 pC2 q ď s1 pC1 ´ C2 q “ }C1 ´ C2 },

n ě 0.

(2.64)

Now, let A ` B “ C2 and A “ C1 . Then for m “ 0 the estimate (2.23) gives sn`1 pC2 q ´ sn`1 pC1 q ď }C1 ´ C1 }, which together with (2.64) proves (2.63).

n ě 0, l

Lemma 2.57. The space LpHq of bounded operators equipped with the operatornorm topology is a Banach space.

2.4. Convergence theorems for Cp pHq

65

Proof. We only have to prove the completeness of LpHq. Let tAk ukě1 be a Cauchy sequence of elements of LpHq. Then, as }Ak`n u ´ Ak u} ď }Ak`n ´ Ak }}u}, tAk uukě1 is a Cauchy sequence in H for each fixed u P H. Since the space H is complete, there is a v P H such that limkÑ8 }Ak u ´ v} “ 0. Let A be an operator defined by Au :“ v. Then A is obviously linear and bounded, with }A} ď supkě1 }Ak } ď M . The latter holds because tAk ukě1 is a Cauchy sequence, which implies that t}Ak }ukě1 is also a Cauchy sequence. Hence, }pAk ´ Aqu} ď lim }Ak ´ Ak`n }}u}, nÑ8

or lim }Ak ´ A} ď lim }Ak ´ Ak`n } “ 0.

kÑ8

k,nÑ8

l

Corollary 2.58. The space C8 pHq is a ˚-ideal in LpHq, see Proposition 2.7. By Proposition 2.8, it is a closed manifold of the Banach space LpHq. Thus C8 pHq is itself a Banach space with respect to the operator norm topology. Moreover, since the finite-rank operators KpHq are dense in C8 pHq in the operator norm topology, see Proposition 2.11, the Banach space C8 pHq is separable. Similar properties hold for the von Neumann-Schatten classes Cp pHq, 1 ď p ă 8. Proposition 2.59. For any 1 ď p ă 8, the space Cp pHq has the following properties: (a) Cp pHq is closed in the } ¨ }p -norm. (b) The finite-rank operators are } ¨ }p -dense in Cp pHq. (c) Cp pHq is a separable Banach space with respect to the norm } ¨ }p . Proof. (a) Suppose that a sequence tAk P Cp pHqukě1 converges in the } ¨ }p -norm to operator A, i.e., }AÁk }p ă ε for k ą kε . By the }¨}p -triangle inequality (2.62), }pA ´ Ak q ` Ak }p ď }A ´ Ak }p ` }Ak }p . Hence, A P Cp pHq. (b) Using the canonical representation (2.8) one can construct for any A P řN Cp pHq a finite-rank operator ApN q “ n“1 sn pAqp¨, ϕn qψn such that }AÁpN q }p “ ´ř ¯1{p p ă ε for N ą Nε . něN `1 sn pAq (c) Let tAk ukě1 be a Cauchy sequence of elements of Cp pHq, i.e., }Ak`n ´ Ak }p ă ε for k ą kε and n ě 1 arbitrary. Since Cp pHq Ă C8 pHq and } ¨ } ă } ¨ }p , tAk ukě1 is a Cauchy sequence in C8 pHq. Then by Corollary 2.58 there exists an operator A P C8 pHq such that A “ } ¨ }-limkÑ8 Ak . Because tAk ukě1 is a Cauchy sequence in the } ¨ }p -norm ÿ s` pAk`n ´ Ak q ď ε, k ą kε , n ě 1. (2.65) `ě1

By Lemma 2.56, the singular values are } ¨ }-continuous functions on the compact operators. Therefore, we can let n Ñ 8 in (2.65) to obtain ÿ (2.66) s` pA ´ Ak q ă ε, k ą kε . `ě1

66


The estimate (2.66) means that limkÑ8 }A ´ Ak }p “ 0, see (2.37). Hence, by (a), A P Cp pHq, that is, Cp pHq is a Banach space. Since finite-rank operators KpHq are l } ¨ }p -dense in Cp pHq, the space Cp pHq is separable. Remark 2.60. The above arguments show that, for 1 ď p ă 8, the space Cp pHq is not closed in the operator norm topology. By Remark 2.12, the } ¨ }-closure of Cp pHq coincides with C8 pHq. Next we collect and discuss a number of convergence theorems in Cp pHq, which are indispensable for the ensuing exposition of the theory of Gibbs semigroups. – The first type of the convergence theorems is a direct analogue in Cp pHq of the classical dominated convergence theorem in Lp . – The second type of them is a kind of lifting propositions, stating that convergence of norms of operators in Cp pHq bolstered by a weak convergence of the operators themselves yields their norm convergence. – The third type concerns a local uniformity of convergence in Cp pHq of oneparameter operator-valued functions tt ÞÑ Φn ptq P Cp pHqunPN for t P ra, bs Ă R. The following proposition provides analogue in Cp pHq of the dominated convergence theorem for Lp -spaces. Note that analogue of the pointwise convergence here is convergence in the weak operator topology. Proposition 2.61. Let 0 ď B P LpHq and suppose that tAk ukě1 is a sequence of bounded operators such that |Ak | ď B, |Ak˚ | ď B for all k ě 1. Suppose also that the weak operator limit exists: A “ w-limkÑ8 Ak , with |A| ď B, |A˚ | ď B. If B P Cp pHq for 1 ď p ă 8, then A “ } ¨ }p -limkÑ8 Ak P Cp pHq. Proof. First, note that 0 ď B P C8 pHq implies that B 1{2 P C8 pHq, see Corol1{2 lary 2.26. Moreover, sn pB 1{2 q “ sn pBq and B 1{2 has the same eigenvectors as B. Since |Ak | ď B is equivalent to }|Ak |1{2 u} ď }B 1{2 u}, the operator |Ak |1{2 maps any weakly convergent sequence tun uně1 into a convergent sequence t|Ak |1{2 un uně1 . Therefore, |Ak |1{2 P C8 pHq and, consequently, Ak “ Uk |Ak | P C8 pHq for each k ě 1. By the same argument, A P C8 pHq. Now, for any fixed ε ą 0, we can find a finite-rank projection P such that }QBQ} ă ε, where Q “ 1 ´ P and p ď 8, see Proposition 2.28. Note that |Ak | ď B implies Q|Ak |Q ď QBQ. Hence, by Corollary 2.34, sn pQ|Ak |Qq ď sn pQBQq and, consequently, }Q|Ak |Q}p ď }QBQ}p ď ε, or }|Aj |1{2 Q}2p ď ε1{2 . Note that } ¨ }8 “ } ¨ }. The estimate }P Ak Q}p ď }Ak Q}p “ }U |Ak |Q}p ď }|Ak |Q}p follows from (2.44), }P } ď 1, }U } “ 1, the Hölder inequality (2.55) and the fact that |Ak | ď B: }|Ak |1{2 |Ak |1{2 Q}p ď }|Ak |1{2 }2p }|Ak |1{2 Q}2p ď }B}p ε1{2 .


67

Similar estimates for A˚k , A and A˚ hold by the assumptions of the proposition. Next, we write }A ´ Ak }p “ }pP ` QqpA ´ Ak qpP ` Qq}p ď }P pA ´ Ak qP }p ` }P pA ´ Ak qQ}p ` }QpA ´ Ak qP }p ` }QpA ´ Ak qQ}p .

(2.67)

We then estimate the different terms in (2.67). For the first term we get, }QpA ´ Ak qQ}p ď }|A|Q}p ` }|Ak |Q}p ď 2ε1{2 }B}.

(2.68)

The second and the third terms can be estimated in a similar way, see (2.43), }P pA ´ Ak qQ}p ď }pA ´ Ak qQ}p ď 2ε1{2 }B}p , }QpA ´ Ak qP }p ď }pA ´ Ak q˚ Q}p ď 2ε1{2 }B}p .

(2.69)

Therefore, (2.67)–(2.69) yield }A ´ Ak }p ď 6ε1{2 }B}p ` }P pA ´ Ak qP }p .

(2.70)

Since P is a finite-rank operator, }P pAÁk qP }p Ñ 0 as k Ñ 8 as w-limkÑ8 Ak “ l A. This proves the proposition. Remark 2.62. The condition |A˚k | ď B cannot be dropped. Indeed, let tek ukě1 be an orthonormal set and Ak :“ p¨, e1 qek P C1 pHq. Then w-limkÑ8 Ak “ 0 and |Ak | ď p¨, e1 qe1 “: B P C1 pHq, but limkÑ8 }Ak } ‰ 0. In this example, A˚k “ p¨, ek qe1 and |Ak˚ | “ p¨, ek qek P C1 pHq, but the estimate |A˚k | ď B is not valid. If in the assumption of Proposition 2.61 we replace the weak convergence by the strong convergence, then we get the following statement for the case when B P C8 pHq. Proposition 2.63. Let B be a positive compact operator and let tAk ukě1 be a sequence of bounded self-adjoint operators such that ´B ď Ak ď B

and

s-lim Ak “ A. kÑ8

(2.71)

Then Ak P C8 pHq, A “ A˚ P C8 pHq, ´B ď A ď B, and also limkÑ8 }A ´ Ak } “ 0. Proof. The two-side bound (2.71) by compact operator B imply that Ak P C8 pHq for any k ě 1. Since A˚k “ Ak and since the strong convergence (2.71) yields w-limkÑ8 A˚k “ A˚ , we get A˚ “ A. Moreover, this limit and (2.71) give ´pBu, uq ď limkÑ8 pAk u, uq ď pBu, uq, or ´B ď A ď B. For the rest of the proof, we can restrict ourselves to the case A “ 0. Suppose that for a subsequence tA` u`ě1 one has }A` } ě a ą 0. Since A` P C8 pHq, there is a sequence tu` u`ě1 of normalised vectors such that A` u` “ b` u` with |b` | ě a.

68


Then (2.71) implies |pu` , A` u` q| ď pu` , Bu` q, or }B 1{2 u` } ě a1{2 . As B 1{2 P C8 pHq, there exists a subspace M of H, with dim M ă 8 such that }B 1{2 v} ď δa1{2 }v},

v P M K,

(2.72)

for some 0 ă δ ă 1, see (2.20) and (2.26). Let PM be the projection onto M . Then (2.72) and the estimate }B 1{2 u` } ď }B 1{2 PM u` } ` }B 1{2 p1 ´ PM qu` } show that }B 1{2 }}PM u` } ě }B 1{2 PM u` } ě p1 ´ δqa1{2 . Hence, }PM u` } ě p1 ´ δqa1{2 }B 1{2 }´1 , }u` } “ 1 and }PM A` u` } ě p1 ´ δqa3{2 }B 1{2 }´1 ,

}u` } “ 1.

(2.73)

On the other hand, s-lim`Ñ8 A` “ 0 implies }A` PM } Ñ 0 and hence lim` }PM A` } “ 0, which contradicts the estimate from below in (2.73). l Corollary 2.64. Propositions 2.61 and 2.63 are valid assuming that B P Cp pHq with 1 ď p ă 8 and Ak “ A˚k . The conclusion is that Ak P Cp pHq and } ¨ }p -limkÑ8 Ak “ A P Cp pHq. The proofs carry through verbatim. In applications the following version of the lifting proposition for convergence of compact operators C8 pHq is useful. Proposition 2.65. Let B be a positive compact operator and let tAk ukě1 be a sequence of bounded self-adjoint operators such that 0 ď Ak ď B

and

w-lim Ak “ A. kÑ8

(2.74)

Then Ak P C8 pHq, A “ A˚ P C8 pHq, 0 ď A ď B, and limkÑ8 }A ´ Ak } “ 0. Proof. The first part of this assertion including the inequalities: 0 ď A ď B, is a corollary of the arguments of the proof for tAk ukě1 in Propositions 2.63. To prove the result on the lifting of the weak operator convergence (2.74) to the operator-norm convergence we first note that operators B 1{2 P C8 pHq, A1{2 P C8 pHq, Ak 1{2 P C8 pHq, k P N, and that 0 ď A1{2 ď B 1{2

and 0 ď Ak 1{2 ď B 1{2 , k P N.

(2.75)

Now, we use the canonical representation for the compact operators A1{2 , (2.8) (Corollary 2.26) and the minimax principle (Corollary 2.34), which B gives sn pA1{2 q ď sn pB 1{2 q, n ě 1. They allow to construct a contraction ΓA,B such that (2.76) ΓA,B : ran B 1{2 Ñ H and A1{2 “ ΓA,B B 1{2 . 1{2


69

Similarly one constructs the contractions tΓA,B pkqukě1 such that ΓA,B pkq : ran B 1{2 Ñ H

1{2

and Ak

“ ΓA,B pkqB 1{2 .

(2.77)

Since A “ pA1{2 q˚ A1{2 “ B 1{2 Γ˚A,B ΓA,B B 1{2 , and similarly Ak 1{2 1{2 pAk q˚ Ak

1{2

˚

“ B ΓA,B pkq ΓA,B pkqB for any u, v P H that

1{2

“

, the weak operator limit in (2.74) yields

lim ppA ´ Ak qu, vq

kÑ8

(2.78)

“ lim ppΓ˚A,B ΓA,B ´ ΓA,B pkq˚ ΓA,B pkqqB 1{2 u, B 1{2 vq “ 0. kÑ8

Note that by virtue of the canonical form of self-adjoint compact operators (2.8) (Corollary 2.26), ran B 1{2 “ H. Therefore, (2.78) implies w-limkÑ8 ΓA,B pkq˚ ΓA,B pkq “ Γ˚A,B ΓA,B , and thus lim }B 1{2 pΓ˚A,B ΓA,B ´ ΓA,B pkq˚ ΓA,B pkqqB 1{2 } “ 0,

kÑ8

since B 1{2 P C8 pHq.

(2.79) l

The use of Proposition 2.61 may pose a problem because the triangle inequality fails for absolute values of operators, see Remark 2.2. For this reason, the following notion of the operator psq-order and the next statement are helpful. psq

Definition 2.66. We write A ă B, if sn pAq ď sn pBq for n ě 1. We say that psq- limkÑ8 Ak “ A, if limkÑ8 sn pAk q “ sn pAq for all n ě 1. Remark 2.67. Since s1 pAq “ }A} ě sn pAq, psq- limkÑ8 Ak “ 0 if and only if limkÑ8 }Ak } Ñ 0. psq

Proposition 2.68. Let tAk ukě1 be a sequence of bounded operators such that tAk ă Bukě1 for an operator B P Cp pHq. If } ¨ }-limkÑ8 Ak “ A, then } ¨ }p -limkÑ8 Ak “ A. Proof. Since by Remark 2.67 }Ak ´ A} Ñ 0 implies sn pAk ´ Aq Ñ 0, it is sufficient to show that sn pAk ´ Aq ď αn with tαn uně1 P lp , because then we can appeal to the dominated convergence theorem in the lp -space. By inequality (2.23) and sn pAk q ď sn pBq, one gets s2n`1 pAk ´ Aq ď sn`1 pAk q ` sn`1 pAq ď 2sn`1 pBq, s2n pAk ´ A ď sn`1 pAk q ` sn pAq ď sn pAk q ` sn pAq ď 2sn pBq. Hence, we can take tαn uně1 “ 2tsn pBquně1 P lp .

l

70


Therefore, Proposition 2.68 is an alternative device to Proposition 2.61 and Corollary 2.64. The next result is of the second type. Here the lifting of the convergence of norms of operators is bolstered by conditions of strong convergence of operators and their adjoints. Proposition 2.69. Let tAk ukě1 be a sequence of bounded operators such that s-limkÑ8 Ak “ A and s-limkÑ8 Ak˚ “ A˚ . If there is 1 ď p ă 8 such that Ak , A P Cp pHq and limkÑ8 }Ak }p “ }A}p , then } ¨ }p -lim Ak “ A.

(2.80)

kÑ8

To carry on we first have to establish two lemmata. Lemma 2.70 (Strong continuity of multiplication). Let s-limkÑ8 Ak “ A P LpHq and s-limkÑ8 Bk “ B P LpHq. Then s-limkÑ8 Ak Bk “ AB. Proof. For any u P H one has Ak Bk u ´ ABu “ Ak pBk ´ Bqu ` pAk ´ AqBu.

(2.81)

Then the first term in (2.81) converges to zero because }Ak pBk ´ Bqu} ď sup }Ak }}pBk ´ Bqu}, kě1

where supkě1 }Ak } ă 8 by the uniform boundedness principle (Proposition 1.6). l The second term in (2.81) converges to zero because s-limkÑ8 Ak “ A. Lemma 2.71. Suppose that s-limkÑ8 Ak “ A P LpHq and that also s-limkÑ8 Ak˚ “ A˚ . Then s-limkÑ8 |Ak | “ |A| and s-limkÑ8 |Ak˚ | “ |A˚ |. Proof. First, by Lemma 2.70, s-lim |Ak |2 “ s-lim A˚k Ak “ |A|2 , kÑ8

kÑ8

and similarly s-limkÑ8 |Ak˚ |2 “ |A˚ |2 . Next, let Ck :“ Ak {M and C :“ A{M , where M “ supkě1 }Ak }. Then }Ck }, }C} ď 1. We now observe that |Ck | can be computed using the series |Ck | “

8 b ÿ 1 ´ p1 ´ Ck˚ Ck q “ 1 ` γm p1 ´ Ck˚ Ck qm ,

(2.82)

m“1

? which converges in the operator norm. Indeed, 1 ´ z is analytic for |z| ă 1, and as all γmě1 are negative, we have for all N the estimate N ÿ m“0

|γm | “ 2 ´

N ÿ m“0

γm “ 2 ´ lim xÒ1

N ÿ m“0

? γm xm ď 2 ´ lim 1 ´ x “ 2. xÒ1

2.4. Convergence theorems for Cp pHq This implies that the power series for Since 0 ď Ck˚ Ck ď 1, we have

71 ? 1 ´ z is absolutely convergent for |z| ď 1.

}1 ´ Ck˚ Ck } “ sup pu, p1 ´ Ck˚ Ck quq ď 1, uPH }u}“1

and consequently the series (2.82) yields |Ck |. Since the convergence in (2.82) is uniform, we can use the strong continuity of multiplication, see Lemma 2.70, to take the limit in each term of the sum, therefore |C| “ s-lim |Ck | and similarly kÑ8

|C ˚ | “ s-lim |Ck˚ |. kÑ8

l

Proof of Proposition 2.69. Since A ‰ 0, then by virtue of the estimate › › A A › › k }Ak }p ´ }A}p › }Ak ´ A}p “ › }Ak }p }A}p p › A A ›› › k ď |}Ak }p ´ }A}p | ` › ´ › }A}p , }Ak }p }A}p p

(2.83)

we can assume with no loss of generality that }Ak }p “ }A}p “ 1. Fix ε ą 0. Then we can find a projection P P KpHq such that }P |A|P }p ě 1 ´ ε and }P |A˚ |P }p ě 1 ´ ε, see Remark 2.12 and Proposition 2.59 (b). For example, P can be the projection onto the span of the first few eigenvectors of |A| and |A˚ |. Since s-lim |Ak | “ |A| kÑ8

and

s-lim |A˚k | “ |A|, kÑ8

(2.84)

there exists an N pεq such that for k ě N pεq we have }P |Ak |P }p ě 1 ´ 2ε and }P |A˚k |P }p ě 1 ´ 2ε. Put Q :“ 1 ´ P . By Definition 2.50 of } ¨ }p and by the canonical representation of |Ak |, 1“

dim ÿP n“1

spn pAk q `

8 ÿ

spn pAk q “ }P |Ak |P }pp ` }Q|Ak |Q}pp ,

n“1`dim P

i.e., we get }Q|Ak |Q}p ď t1 ´ p1 ´ 2εqp u1{p “: δp2εq,

(2.85)

and the analogous for }Q|A˚k |Q}. Hence, }|Ak |1{2 Q}2p ď δ 1{2 pεq and }|Ak |Q}p ď }|Ak |1{2 }2p }|Ak |1{2 Q}2p ď δ 1{2 p2εq,

(2.86)

1{2

where we used }|Ak |1{2 }2p “ }Ak }p “ 1 and the Hölder inequality (2.55). Similarly, we derive the inequalities }|A˚k |Q}p ď δ 1{2 p2εq and }|A|Q}p ď δ 1{2 pεq and }|A˚ |Q}p ď δ 1{2 pεq.

(2.87)

72


Now we can present (2.83) using inequality (2.67) }Ak ´ A}p ď }P pAk ´ AqP }p ` }P pAk ´ AqQ}p ` }QpAk ´ AqP }p ` }QpAk ´ AqQ}p . The different terms can be estimated as follows: the last as }QpAk ´ AqQ}p ď }|Ak |Q}p ` }|A|Q}p ď δ 1{2 p2εq ` δ 1{2 pεq,

(2.88)

and the second and third ones in a similar manner as }P pAk ´ AqQ}p ď }pAk ´ AqQ}p ď δ 1{2 p2εq ` δ 1{2 pεq, }QpAk ´ AqP }p ď }pA˚k ´ A˚ qQ}p ď δ 1{2 p2εq ` δ 1{2 pεq.

(2.89)

Therefore, (2.88) and (2.89) yield }Ak ´ A}p ď 3δ 1{2 p2εq ` 3δ 1{2 pεq ` }P pAk ´ AqP }p .

(2.90)

Since P is a finite-rank operator, limkÑ8 }P pAk ´ AqP }p “ 0, because A “ s-limkÑ8 Ak . This proves the proposition. l Corollary 2.72. Examining the proof given above one finds that taking into account results and arguments of dominated convergence Proposition 2.61, the conditions of the strong operator convergence in Proposition 2.69 can be relaxed to the weak operator convergence. Consequently, the assertion of Proposition 2.69 follows in fact from the next statement. Proposition 2.73. Fix 1 ď p ă 8. Suppose that w-limkÑ8 Ak “ A, w-limkÑ8 |Ak | “ |A|, w-limkÑ8 |Ak˚ | “ |A˚ |. If operators Ak , A P Cp pHq and limkÑ8 }Ak }p “ }A}p , then } ¨ }p -limkÑ8 Ak “ A. Remark 2.74. We note that if 1 ă p ă 8, then due to the uniform convexity of the Banach spaces C1ăpă8 pHq, Proposition 2.73 can be strengthened to a “perfect” assertion about lifting the weak operator convergence to the } ¨ }p -convergence, namely: – Suppose that w-limkÑ8 Ak “ A and that limkÑ8 }Ak }p “ }A}p . Then } ¨ }p -limkÑ8 Ak “ A. Moreover, this statement is valid even for the more difficult case p “ 1. The reader is addressed to the Notes in Section 2.5 for a discussion and for references. Therefore, in the case p “ 1 it is instructive to consider first the positive cone of trace-class operators C1,` pHq Ă C1 pHq, when }A}1 “ Tr A ě 0. Then weak conditions of Proposition 2.73 reduce to only one condition: w-limkÑ8 Ak “ A. On the cone of positive trace-class operators C1,` pHq Proposition 2.73 takes the following ”imperfect”, cf. Remark 2.74, yet useful form.


73

Proposition 2.75. Let the sequence tAk P C1,` pHqukě1 converge in the weak operator topology to A P C1,` pHq. If the sequence tTr Ak ukě1 is uniformly bounded, then Tr A ď lim inf kÑ8 Tr Ak . Moreover, limkÑ8 }Ak ´ A}1 “ 0 if and only if limkÑ8 Tr Ak “ Tr A. Proof. Note that the first part of the proposition is simply the weak lower semicontinuity of the trace, Proposition 2.48(f). The second part needs arguments similar to those for of Proposition 2.73. On the positive cone C1,` pHq they are straightforward. The proof in one direction is evident. Thus, let w-limkÑ8 Ak “ A and limkÑ8 }Ak }1 “ }A}1 . Since for A “ 0 the statement is trivial, let A ‰ 0. Then by virtue of the inequality › A › A › › k }Ak ´ A}1 “ › }Ak }1 ´ }A}1 › }Ak }1 }A}1 1 › A A ›› › k ď |}Ak }1 ´ }A}1 | ` › ´ › }A}1 , }Ak }1 }A}1 1 we can consider the case }Ak }1 “ }A}1 “ 1. Then for any finite-rank orthogonal projection P one gets }Ak ´ A}1 ď }P pAk ´ AqP }1 ` 2}P Ak p1 ´ P q}1

(2.91)

` 2}P Ap1 ´ P q}1 ` }p1 ´ P qAk p1 ´ P q}1 ` }p1 ´ P qAp1 ´ P q}1 . Note that on the positive cone C1,` pHq one gets }p1 ´ P qAk p1 ´ P q}1 “ Trp1 ´ P qAk “ 1 ´ Tr P A ` Tr P pA ´ Ak q, }p1 ´ P qAp1 ´ P q}1 “ 1 ´ Tr P A, }P pAk ´ AqP }1 “ | Tr P pA ´ Ak q|. If P Ak p1 ´ P q “ Uk |P Ak p1 ´ P q| is the polar decomposition then }P Ak p1 ´ 1{2 1{2 P q}1 “ Tr Uk˚ P Ak Ak p1 ´ P q and the Cauchy-Schwarz inequality yields the estimate }P Ak p1 ´ P q}1 ď pTr P Ak q1{2 p1 ´ Tr P Ak q1{2 . (2.92) One obviously obtains the similar estimate for the term with A: }P Ap1 ´ P q}1 ď pTr P Aq1{2 p1 ´ Tr P Aq1{2 .

(2.93)

Now, by weak convergence: w-limkÑ8 Ak “ A of trace-class operators, for any (small) ε ą 0 there exist a projection Pε such that 1 ´ Tr Pε A ă ε2 , and a number Nε such that | Tr Pε pA ´ Ak q| ă ε2 for all k ě Nε . Since Tr P Ak ď 1, Tr P A ď 1, then collecting (2.92), (2.93) and the identities above, we deduce from (2.91) the estimate }Ak ´ A}1 ď ε2 ` 2 p2ε2 q1{2 ` 2ε ` 2ε2 ` ε2 ă 10 ε, for all k ě Nε , which proves the assertion.

l

74


Summarising these facts let us list conditions that are sufficient for the application of Proposition 2.69 and Propositions 2.73. Corollary 2.76. Let 1 ď p ă 8. Suppose that limkÑ8 }Ak ´ A} “ 0 and limkÑ8 }Ak }p “ }A}p . Then limkÑ8 }Ak ´ A}p “ 0. Indeed, since }Ak ´ A} “ }A˚k ´ A˚ }, one gets s-limkÑ8 Ak “ A and s-limkÑ8 A˚k “ A˚ . Hence, } ¨ }p -limkÑ8 Ak “ A by Proposition 2.69. We know that the absolute value | ¨ | is strongly continuous in the sense of Lemma 2.71. It follows that in fact, Proposition 2.69 implies } ¨ }p -continuity of | ¨ |. Proposition 2.77. Let 1 ď p ă 8. Suppose that tAk ukě1 Ă Cp pHq and } ¨ }p -limkÑ8 Ak “ A, then } ¨ }p -limkÑ8 |Ak | “ |A|. Proof. So far as limkÑ8 }Ak ´ A}p “ 0, we have both s-limkÑ8 Ak “ A and s-limkÑ8 Ak˚ “ A˚ . Then by Lemma 2.71 we get s-limkÑ8 |Ak | “ |A| and s-limkÑ8 |A˚k | “ |A˚ |. Next, by the triangle inequality (2.46), |}Ak }p ´ }A}p | ď }Ak ´ A}p .

(2.94)

Since }Ak }p “ }|Ak |}p and }A}p “ }|A|}p , the estimate (2.90) implies } ¨ }p -lim }|Ak |} “ }|A|}.

(2.95)

kÑ8

Therefore, the proof follows from Proposition 2.69.

l

The next proposition shows that one can lift the strong continuity of multiplication in Lemma 2.70, up to } ¨ }p -continuity for 1 ď p ă 8 if one of the factors converges in the } ¨ }p -norm. Proposition 2.78. Suppose that s-limkÑ8 Ak “ A P LpHq and that } ¨ }p -limkÑ8 Bk “ B P Cp pHq for a 1 ď p ă 8. Then } ¨ }p -lim Ak Bk “ AB.

(2.96)

kÑ8

If, in addition, s-limkÑ8 Ak˚ “ A˚ , then also } ¨ }p -lim Bk Ak “ BA.

(2.97)

kÑ8

Proof. Writing }Ak Bk ´ AB}p ď }Ak }}Bk ´ B}p ` }pAk ´ AqB}p , and using the uniform boundedness principle sup }Ak } ď M ă 8, kě1

(2.98)


75

for a strongly convergent sequence tAk ukě1 , we see that it suffices to show that in (2.98) limkÑ8 }pAk ´ AqB}p “ 0. Given ε ą 0, we can find a projection P P KpHq such that }p1 ´ P qB}p ă ε. Then the last term in (2.98) can be estimated as }pAk ´ AqB}p ď }pAk ´ AqP }}B}p ` p}Ak } ` }A}q}p1 ´ P qB}p .

(2.99)

Then, since the strong convergence of tAk ukě1 implies limkÑ8 }pAk ´ AqP } “ 0, (2.99) gives lim sup }pAk ´ AqB}p ď pM ` }A}qε. kÑ8

Since ε is arbitrary, the result (2.96) is proven. The proof of (2.97) follows along the same line of reasoning, with the specil ficity of the norm } ¨ }p taken into account. Remark 2.79. This result does not hold if in the statement of the proposition the strong convergence is replaced by the weak convergence w-limkÑ8 Ak “ A. Take for example Ak “ p¨, e1 qek and Bk “ p¨, e1 qe1 P C1 pHq. Then } ¨ }p -limkÑ8 Ak Bk “ } ¨ }p -limkÑ8 p¨, e1 qek ‰ 0, while w-limkÑ8 Ak “ 0. Here, tek ukě1 is an orthonormal basis in H. Remark 2.80. This result also fails for the sequence tBk Ak ukě1 instead of tAk Bk ukě1 . Take Ak “ p¨, ek qe1 . Then s-lim Ak “ 0 and Bk “ p¨, e1 qe1 P C1 pHq. However, } ¨ }p -lim Ak Bk “ } ¨ }p -limpe1 , ek qp¨, e1 qe1 “ 0 , whereas, kÑ8

kÑ8

} ¨ }p -lim Bk Ak “ } ¨ }p -limp¨, ek qe1 ‰ 0. kÑ8

kÑ8

We conclude this section by extending the convergence results of Proposition 2.61 and Proposition 2.69 to sequences of operator-valued in Cp pHq functions. This is the third type of convergence theorems that we need, in particular, for the Trotter-Kato product formulae in Chapters 5–7. Definition 2.81. Let tt ÞÑ Φn ptqunPN be a sequence of operator-valued functions in Cp pHq for t P R. We say that the sequence tΦn p¨qunPN , converges in } ¨ }p -topology to the function Φp¨q P Cp pHq locally uniformly in D Ď R if lim sup }Φn ptq ´ Φptq}p “ 0,

nÑ8 tPra,bs

for any compact ra, bs Ă D. Lemma 2.82. Let tXn p¨qunPN and tYn p¨qunPN be sequences of operator-valued functions respectively in LpHq and in Cp pHq for t P ra, bs Ă R, and let suptPra,bs,ně1 }Xn ptq} ă 8.

76

Chapter 2. Classes of compact operators Let respectively, Xp¨q : ra, bs Ñ LpHq and Y p¨q : ra, bs Ñ Cp pHq be such that sup }Y ptq} ă 8, and } ¨ }p -lim Yn ptq “ Y ptq, nÑ8

tPra,bs

uniformly in t P ra, bs. (i) If s-limnÑ8 Xn ptq “ Xptq uniformly on ra, bs and if for some sequence of finite-rank orthogonal projections tPk ukě1 such that s-limkÑ8 Pk “ 1 one has lim sup }p1 ´ Pk qY ptq}p “ 0, kÑ8 tPra,bs

then } ¨ }p -limnÑ8 Xn ptqYn ptq “ XptqY ptq uniformly on ra, bs. (ii) If s-limnÑ8 Xn ptq˚ “ Xptq˚ uniformly in t P ra, bs and if for some sequence of finite-rank orthogonal projections tQk ukě1 such that s-limkÑ8 Qk “ 1 one has lim sup }Y ptqp1 ´ Qk q}p “ 0 , kÑ8 tPra,bs

then } ¨ }p -limnÑ8 Yn ptqXn ptq “ Y ptqXptq uniformly on ra, bs. Proof. Modulo the evident t-uniformity, the proof essentially mimics the line of reasoning of the proof of Proposition 2.78. l Proposition 2.83. Let tXn ptqunPN be a sequence of operator-valued functions in Cp pHq for t P ra, bs and 1 ď p ă 8, such that suptPra,bs,ně1 }Xn ptq} ă 8. Let Xp¨q : ra, bs Ñ Cp pHq be such that there are two sequences of finite-rank orthogonal projections tPk ukě1 and tQk ukě1 obeying s-limkÑ8 Pk “ s-limkÑ8 Qk “ 1, with the property: lim sup }p1 ´ Pk qXptq} “ lim sup }Xptqp1 ´ Qk q} “ 0, kÑ8 tPra,bs

kÑ8 tPra,bs

(2.100)

and in addition lim sup }p1 ´ Pk qXptq}p “ 0 _ lim sup }Xptqp1 ´ Qk q}p “ 0.

kÑ8 tPra,bs

kÑ8 tPra,bs

(2.101)

If s-limnÑ8 Xn ptq “ Xptq, s-limnÑ8 Xn ptq˚ “ Xptq˚ , and lim }Xn ptq}p “ }Xptq}p , 1 ď p ă 8,

nÑ8

(2.102)

uniformly on ra, bs, then } ¨ }p -limnÑ8 Xn ptq “ Xptq uniformly in t P ra, bs for p P N. Proof. Note that the conditions of this assertion for t P ra, bs coincide with the conditions of Proposition 2.69 pointwise, and so does the proof. The extension to uniformity follows from the conditions (2.100)–(2.102) that should be incorporated in the argument. However, the latter is straightforward. l

77

2.5. Notes

Corollary 2.84. Let tXn p¨qunPN and tYn p¨qunPN be sequences of operator-valued functions ra, bs Ă R with values in Cp pHq, 1 ď p ă 8, such that suptPra,bs,ně1 }Xn ptq} ă 8. Let Xp¨q : ra, bs Ñ Cp pHq be a function obeying (2.101) and (2.102). If s-lim Xn ptq “ Xptq, nÑ8

s-lim Xn ptq˚ “ Xptq˚ , nÑ8

} ¨ }p -lim Yn ptq “ Y ptq, nÑ8

uniformly on ra, bs and }Xn ptq}p ď }Yn ptq}p for t P ra, bs, then } ¨ }p -lim Xn ptq “ Xptq, 1 ď p ă 8 , nÑ8

uniformly on ra, bs. Furthermore, we shall use a certain generalisation of the Lebesgue dominated convergence theorem for one-parameter operator-valued families in Cp pHq-ideals. First, we need the following lemma. Lemma 2.85. Let tXn unPN and tYn unPN be sequences of non-negative self-adjoint operators strongly converging, respectively, to the operators X and Y . Suppose that tYn unPN Ă Cp pHq, Y P Cp pHq. If Xn ď Yn for all n P N, then tXn unPN Ă Cp pHq and s-limnÑ8 Xn “ X P Cp pHq, 1 ď p ă 8. The proof is straightforward and essentially the same as in Lp spaces. Proposition 2.86. Let tXn p¨qunPN , Xp¨q and tY p¨qu be operator-valued functions on ra, bs Ă R such that for each t P ra, bs the operators satisfy the conditions of Lemma 2.85. If in addition s-limnÑ8 Xn ptq “ Xptq, } ¨ }p -limnÑ8 Yn ptq “ Y ptq uniformly on ra, bs, suptPra,bs,ně1 }Xn ptq} ă 8 and Y ptq satisfy conditions (2.100): lim sup }p1 ´ Pk qY ptq} “ lim sup }Y ptqp1 ´ Qk q} “ 0,

kÑ8 tPra,bs

kÑ8 tPra,bs

then } ¨ }p -limnÑ8 Xn ptq “ Xptq uniformly on ra, bs. Proof. Pointwise the conditions of this assertion for t P ra, bs coincide with the conditions of Lemma 2.85. The extension to uniformity follows from the uniform boundedness suptPra,bs,ně1 }Xn ptq} ă 8 and conditions (2.100). l

2.5

Notes

Notes to Section 2.1. The basic definitions and properties of compact operators are standard and can be found in many books on operators on Hilbert spaces. We were inspired by [BS87], [KF99], [Sim05], [BEH94], [RS80], and [HN01]. Note that Definition 2.4 proposes a general definition for compact operators. One can repeat it verbatim for a Banach space B.

78


From Proposition 2.5 and Remark 2.6 it follows that the more general concept of the completely continuous operators L8 pBq is useful, see [KF99](Ch.IV, §6), [Horn50], [Fan51]. Definition. A bounded operator A on a Banach space B is called completely continuous if it maps any weakly convergent sequence txn uně1 Ă B into a normconvergent sequence tAxn uně1 . Note that if A is a compact operator on B, then A is completely continuous, that is, C8 pBq Ď L8 pBq. If A is completely continuous on B and the Banach space B is reflexive, then A is compact. In particular, C8 pHq “ L8 pHq when this concept is applied for operators in Hilbert spaces. The following example shows that in general for Banach spaces the difference L8 pBqzC8 pBq is nontrivial. Example. Recall that every bounded operator on the Banach space B“ l1 is completely continuous, but among them there are many, that are noncompact, for example, the identity operator. Notes to Section 2.2. There are many good references on the material of this section. It is only question of the place in the spectrum of the accumulation point 0 which is a rather subtle matter. See also the comments in Appendix A. Here we followed essentially [KF99], [BS87], [RS80] and [Sad91]. Notes to Section 2.3. The great classics on the Cp pHq-ideals are [GK69], [Sch70] and [Sim05]. H. Weyl and R. Courant understood the importance of the minimax principle for the study of singular values. Most textbooks on the theory of linear operators deal with this topic, the treatments in [BS87] and in [BEH94] are especially well written. Proposition 2.45 is the celebrated Lidskiˇı Trace Theorem, see [Lid59] and [Sim05] for more comments. Notes to Section 2.4. This section contains the most important convergence theorems in Cp pHq-ideals that we need. The proof of the C8 pHq-dominated convergence theorem, Proposition 2.61, and Remark 2.62 are due to [Sim05], Theorem 2.16. Proposition 2.63 and Corollary 2.64 are proved in [Zag80]. Proposition 2.65 can be found in [NZ99b]. The device in Definition 2.66 as well as Proposition 2.68 were proposed by B. Simon [Sim05], Theorem 2.17. Propositions 2.69 and 2.78 are proved in [Gr¨ u73], but Remarks 2.79 and 2.80 are again due to [Sim05]. The observation that Gr¨ umm’s convergence theorem (Propositions 2.69) for 1 ď p ă 8 can be strengthened to Proposition 2.73 is in fact Theorem 2.20 in [Sim05]. Note that for 1 ă p ă 8 there exists an improved version of these observations that we formulated in Remark 2.74 as a ”perfect” lifting assertion. It can be stated as follows:

2.5. Notes

79

– The weak convergence of operators enhanced by the convergence of their operator norms implies the norm convergence of these operators. For 1 ă p ă 8 this follows from the uniform convexity of Banach spaces Cp pHq, see [Sim05], Theorem 2.21. The long standing question: whether this assertion holds true also for the ”flat” trace-class ideal C1 pHq, was solved affirmatively in the general setting of symmetrically normed ideals by the Arazy-Simon theorem, see e.g., [Sim05] (Addendum H), Theorem A6. The affirmative answer (Proposition 2.75) for the special case of the positive cone C1,` pHq is due to G. F. dell’Antonio [dellA67] and E. B. Davis [Dav69]. For the extension of convergence theorems to sequences of operator-valued functions in Cp pHq (Lemmata 2.82, 2.85 and Propositions 2.83, 2.86) we followed here [NZ90b].

Chapter 3

Trace inequalities To study the Gibbs semigroups (including the semigroups in symmetrically-normed ideals) we need more Cp -norm estimates than those provided in Section 2.3. Here we present inequalities involving eigenvalues and singular values of operators from the ideals tCp pHqupě1 .

3.1

Singular values of compact operators

Here we collect some properties of singular values of compact operators that we systematically use throughout the book. Some of them were already mentioned in Chapter 2. (a) Recall that, by the square-root lemma, any positive (hence self-adjoint) ? square root K bounded operator P has a unique positive LpHq K P LpHq, such ? that K commutes with all the operators C P LpHq verifying rC, Ks “ 0, and that ? p Kq2 “ K, cf. Section 2.1. Since the operator K “ A˚ A ě 0, this observation ? motivates the definition of the absolute value of A by |A| :“ A˚ A ě 0. Then }|A|u} “ }Au} since |A|2 “ A˚ A. For some other properties of the operator absolute value, see Remark 2.2. By the polar decomposition theorem (Proposition 2.3), for any A P LpHq there exists a unique partial isometry U : pker U qK Ñ ran U , such that one has the representation A “ U |A|. Here ran U “ ran U , the absolute value is |A| “ U ˚ A, and U is uniquely determined by the condition ker U “ ker A. Then U ˚ A “ A˚ U , but also by definition |A| ‰ |A˚ |, if A is not normal. If A ě 0, then A “ |A|, i.e., U “ 1. (b) Let A P C8 pHq be a compact operator. Then by Proposition 2.7(b) its absolute value |A| “ U ˚ A is also a compact operator. Proposition 2.24 states that: - The operator |A| has an orthonormal set of eigenfunctions tϕn uně1 with eigenvalues tλn p|A|quně1 |A| ϕn “ λn p|A|qϕn , λn p|A|q “: sn pAq, n P N. © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1_3

(3.1) 81

82

Chapter 3. Trace inequalities – The positive eigenvalues tλn p|A|quně1 of |A| convergent to zero and have finite multiplicity. They are called the singular values (or s-numbers) spAq :“ tsn pAquně1 of the operator A. – The eigenvectors tφn uně1 of the operator |A| form an orthonormal basis in H.

The importance of the s-numbers as compared with the eigenvalues follows from the fact that it is the s-numbers that express the canonical form of compact operators as well as define the norms on the ideals Cp pHq, p ě 1, of bounded operators in LpHq. (c) By virtue of (3.1) tϕn uně1 are also eigenfunctions of the operator |A|2 “ ˚ A A with eigenvalues tλn p|A|2 q “ s2n pAquně1 and similarly for |A˚ |2 “ AA˚ . Therefore λn pA˚ Aq “ s2n pAq and λn pAA˚ q “ sn2 pA˚ q, n P N.

(3.2)

Since the identity λpλ1 ´ A˚ Aq´1 “ 1 ` A˚ pλ1 ´ AA˚ q´1 A implies that σp pA Aqzt0u “ σp pAA˚ qzt0u, we infer that sn pAq “ sn pA˚ q in (3.2). So, if A is not normal, then |A| ‰ |A˚ |, yet λn p|A|q “ λn p|A˚ |q. (d ) If A is a normal compact operator, then its singular values are tsn pAq “ |λn pAq|uně1 , cf. (3.1). To see this we note that by normality, i.e., AA˚ “ A˚ A, the operators A and A˚ have a common set of eigenfunctions tun uně1 and λn pA˚ q “ λn pAq. Indeed, A˚ Aun “ λn pA˚ Aqun yields tλn pA˚ Aq “ |λn pAq|2 “ s2n pAquně1 for n P N. (e) If A ě 0 is a compact operator, then |A| “ A and the singular values 1{2 sn pAq “ λn pAq ě 0. By (3.1) this yields sn pA1{2 q “ sn pAq, cf. Corollary 2.34, for n P N. (f ) We recall the following Weyl-Horn chain of inequalities that involves snumbers, which will be needed in the sequel. Let A P C8 pHq. Then for any system of vectors tuj ukj“1 one has inequalities: ˚

detrpAui , Auj qsk1 ď

k ź

s2n pAq detrpui , uj qsk1 , k P N .

n“1

Here rpξi , ηj qsk1 denotes the k ˆ k matrix with entries mij :“ pξi , ηj q, for vectors tξi : i “ 1, 2, . . . , ku and tηj : j “ 1, 2, . . . , ku in H.

3.2

Inequalities for s-numbers and eigenvalues

First we consider inequalities relating the s-numbers and eigenvalues of compact operators.

83

3.2. Inequalities for s-numbers and eigenvalues Lemma 3.1. Let A be a compact operator. Then for any k P N k k ˇ ź ˇź ˇ ˇ λj pAqˇ ď sj pAq , ˇ j“1

j“1

and equalities hold if and only if A is normal. The proof is based on the Weyl-Horn inequalities and on the relations between eigenvalues and singular values for normal operators, Section 3.1 (d ) and the Notes to Section 3.2. To continue we need the following abstract lemma. Lemma 3.2. Let Φ : R Ñ R be a convex function that vanishes at ´8 (which implies Φpxq ě 0). Let taj ujě1 and tbj ujě1 be non-increasing sequences of real numbers such that k k ÿ ÿ aj ď bj k “ 1, 2, . . . . (3.3) j“1

j“1

Then k ÿ

Φpaj q ď

k ÿ

Φpbj q,

k “ 1, 2, . . . .

(3.4)

j“1

j“1

If in addition Φ is strictly convex, then the equalities k ÿ j“1

Φpaj q “

k ÿ

Φpbj q,

k “ 1, 2, . . . ,

(3.5)

j“1

imply aj “ bj . Proof. Since Φ is convex, the left derivative Φl1 pxq :“ lim Ó0

Φpxq ´ Φpx ´ q

exists for any x P R and it is a non-negative nondecreasing function. Let us now consider the Stieltjes integral ż8 żx 1 dΦl puq px ´ uq` “ dΦ1l puq px ´ uq Ń Ń żx 1 “ ´Φl pŃ qpx ` N q ` du Φ1l puq, (3.6) Ń

where y` :“ maxty, 0u and N ą 0. Since the integral (3.6) is positive, we obtain żx 1 du Φ1l puq “ Φpxq ´ ΦpŃ q ď Φpxq, (3.7) px ` N q Φl pŃ q ď Ń

84

Chapter 3. Trace inequalities

and hence lim sup N Φl1 pŃ q ă 8 and N Ñ8

lim Φ1l pŃ q “ 0.

N Ñ8

(3.8)

By assumption, limxÑ´8 Φpxq “ 0. Therefore, (3.7) and (3.8) imply that lim px ` N q Φ1l pŃ q “ lim N Φ1l pŃ q “ 0. N Ñ8

N Ñ8

(3.9)

Now, taking the limit N Ñ 8 in (3.6) and using (3.9), we get the representation ż8 dΦ1l puq px ´ uq` ,

Φpxq “

x P R.

(3.10)

´8

Let k ÿ

Ak puq :“

paj ´ uq`

and Bk puq :“

j“1

k ÿ

pbj ´ uq` .

(3.11)

j“1

Then the representation (3.10) yields k ÿ

ż8 dΦ1l puq Ak puq,

Φpaj q “ ´8

j“1 k ÿ

(3.12) ż8 dΦl1 puq Bk puq.

Φpbj q “ ´8

j“1

We now derive inequalities between the functions Ak and Bk defined in (3.11). First, let u ě b1 . Then, obviously Ak puq “ Bk puq “ 0. If u ď mintak , bk u, then by (3.3) one gets Ak puq “

k ÿ

aj ´ ku ď

j“1

k ÿ

bj ´ ku “ Bk puq.

j“1

Now let aq`1 ď u ă aq

and bp`1 ď u ă bp

for some q, p ď k. Then for p ě q we have Ak puq “

q ÿ

aj ´ qu ď

q ÿ

bj ´ qu ` pbq`1 ´ uq ` ¨ ¨ ¨ ` pbp ´ uq “ Bk puq.

j“1

j“1

Similarly, if p ă q, then Ak puq “

q ÿ j“1

aj ´ qu ď

q ÿ j“1

aj ´ qu ´ pbq ´ uq ´ ¨ ¨ ¨ ´ pbp`1 ´ uq ď Bk puq.

85

3.2. Inequalities for s-numbers and eigenvalues These estimates yield Ak puq ď Bk puq,

k “ 1, 2, . . . , u P R.

(3.13)

The representations (3.12) show that (3.4) follows from the inequality (3.13). In the case of a strictly convex function Φpxq the equality (3.5) and representation (3.12) imply Ak puq “ Bk puq for k “ 1, 2, . . . and for any u P R, which in turn is only possible if aj “ bj for j “ 1, 2, . . . l Proposition 3.3. Let A P C8 pHq and let f : R` 0 Ñ R be a function such that f p0q “ 0 and R Q t ÞÑ f pet q is convex. Then k ÿ

f p|λj pAq|q ď

j“1

k ÿ

f psj pAqq,

k “ 1, 2, . . . .

(3.14)

j“1

Proof. We use Lemmata 3.1 and 3.2. Notice that the function t ÞÑ f pet q verifies the assumptions on the function Φptq in Lemma 3.2. If we define aj :“ ln |λj pAq|

and bj :“ ln sj pAq,

then the inequalities (3.3) follow from the estimate given by Lemma 3.1. Consequently, (3.14) results from (3.4) for Φptq “ f pet q. l Corollary 3.4. For any compact operator A P C8 pHq and p ą 0, k ÿ

|λj pAq|p ď

k ÿ

sj pAqp ,

k “ 1, 2, . . . .

(3.15)

j“1

j“1

If the right-hand side has a finite limit for p ě 1 when k Ñ 8, then A P Cp pHq. Now we consider relations between singular values when more than one operator A is involved. They complement the inequalities provided by Lemma 2.53 older and by (2.57) and will unable us to establish in Section 3.3 generalised Cp -H¨ inequalities. Lemma 3.5. Let A, B P C8 pHq. The singular values sn pAq and sn pBq satisfy the relations k k k ź ź ź (3.16) sj pAq sj pBq, k “ 1, 2, . . . , sj pABq ď j“1

j“1

j“1

and k ÿ j“1

sj pA ` Bq ď

k ÿ j“1

sj pAq `

k ÿ j“1

sj pBq,

k “ 1, 2, . . . .

(3.17)

86


Proof. Recall that for any C P C8 pHq and any system of vectors tφl : l “ 1, 2, . . . , ku Ă H one has the Weyl-Horn inequalities in the form detrpCφi , Cφj qs ď

k ź

s2l pCq detrpφi , φj qs.

(3.18)

l“1

Here rpξi , ηj qs denote the k ˆ k matrix with elements mij :“ pξi , ηj q, for vectors tξi : i “ 1, 2, . . . , ku and tηj : j “ 1, 2, . . . , ku. Applying the inequalities (3.18) to the product C “ AB one gets detrpABei , ABej qs ď

k ź

s2l pAq detrpBei , Bej qs

l“1 k ź

ď

s2l pAq

k ź

s2l pBq.

(3.19)

l“1

l“1

It follows from the canonical representation of the compact self-adjoint operator B ˚ A˚ AB that there exists a complete system of eigenvectors tej : j “ 1, 2, . . . , ku such that, by Remark 2.29, we have detrpABei , ABej qs “

k ź

s2l pABq.

(3.20)

l“1

The inequalities (3.16) then follow from (3.19) and (3.20). By Propositions 2.28 and 2.38, we can choose an orthonormal system of vectors tψj : j “ 1, 2, . . . , ku and a partial isometry U such that k ÿ

|pU pA ` Bqψj , ψj q| “

j“1

k ÿ

sj pA ` Bq.

(3.21)

j“1

Since, again by Proposition 2.38, we have k ÿ

|pU pA ` Bqψj , ψj q| ď

k ÿ

|pU Aψj , ψj q| `

j“1

j“1

k ÿ

ď j“1

k ÿ

|pU Bψj , ψj q|

j“1

sj pAq `

k ÿ

sj pBq,

(3.22)

j“1

the inequalities (3.17) follow from (3.21) and (3.22).

l

With the help of the lemmata above, we can establish a generalisation of Proposition 3.3 for s-numbers of a pair of compact operators.

3.3. Trace and Cp pHq-norm estimates

87

Proposition 3.6. Let A, B P C8 pHq and let f : R` 0 Ñ R be a function such that f p0q “ 0 and for t P R, t ÞÑ f pet q is convex. Then k ÿ

f psj pABqq ď

k ÿ

f psj pAqsj pBqq,

k “ 1, 2, . . . .

(3.23)

j“1

j“1

Proof. Notice that the function t ÞÑ f pet q verifies the assumptions on the function Φptq in Lemma 3.2. Setting aj :“ ln sj pABq and bj :“ lnpsj pAqsj pBqq,

(3.24)

the inequalities (3.3) follow from (3.16). Consequently, (3.23) results from (3.4) with Φptq “ f pet q. l Corollary 3.7. Choosing f pxq “ x for x ě 0, (3.23) reproduces the result of Lemma 2.53. On the other hand, since the inequality (3.16) can be extended to a product of operators tAs P C8 pHq : s “ 1, . . . , ru, we get the following generalisation of (2.50): k ÿ

sj pA1 A2 ¨ ¨ ¨ Ar q ď

k ÿ

sj pA1 q ¨ ¨ ¨ sj pAr q.

(3.25)

j“1

j“1

Corollary 3.8. Let A P C8 pHq. For any n “ 1, 2, . . . and any α ą 0 we get k ÿ

α{n

sj

j“1

pAn q ď

k ÿ

sjα pAq,

k “ 1, 2, . . . .

(3.26)

j“1

Indeed, by (3.23) we have k k r r ´ `ź ´ź ¯ ÿ ˘¯ ÿ As ď f f sj sj pAs q , j“1

s“1

j“1

k “ 1, 2, . . . .

(3.27)

s“1

Then for As “ A and r “ n in (3.27) and with the function f pxq “ xα{n , for x ě 0, the inequalities (3.27) imply (3.26).

3.3

Trace and Cp pHq-norm estimates

The inequalities for singular values that we established in Section 3.2 yield a number of important relations involving traces and } ¨ }p -norms on Cp pHq, p ě 1. Here we present some of them. To this aim we consider first trace and norm estimates motivated by upper bounds from Proposition 3.3 and Corollary 3.4. ř Recall that for a tracek class operator X P C1 pHq its spectral trace ΛpXq “ j“1 λj pXq (Proposition

88


2.45) and its matrix trace ΛpXq “ TrpXq coincide. Since λj pX p q “ λj pXqp and sj pXq “ λj p|X|q, we can rewrite (3.15) for p ě 1 as follows: | TrpAp q| “ |

8 ÿ

λj pAqp | ď

j“1

8 ÿ

|λj pAq|p

(3.28)

j“1 8 ÿ

ď

sj pAqp “

8 ÿ

λj p|A|qp “ Trp|A|p q .

j“1

j“1

Remark 3.9. Note that a direct corollary of (3.28) for the trace-class operators (p “ 1) is the estimate | TrpAq| ď Trp|A|q “ }A}1 . It implies the } ¨ }1 -norm continuity of the map Tr : C1 pHq Ñ C given by A ÞÑ TrpAq “

8 ÿ

pej , Aej q,

j“1

where tej ujě1 is any orthonormal basis in H, cf. Corollary 2.43. Corollary 3.10. Let X ě 0 and Y ě 0 be self-adjoint trace-class operators. Then TrpXY q “ TrpX 1{2 Y X 1{2 q ě 0 and TrppXY qp q “ TrppX 1{2 Y X 1{2 qp q ě 0. Since |XY |2 “ pXY q˚ pXY q “ Y XXY , one also gets Trpp|XY |q2 q “ TrpX 2 Y 2 q. These observations together with inequality (3.28) for the trace-class operator A :“ XY and for integer p “ 2q yield 0 ď TrppXY q2q q ď TrppX 2 Y 2 qq q,

q P N.

(3.29)

If q “ 2n´1 , n P N, then iterating the estimate (3.29) we get n

n´1

0 ď TrppXY q2 q ď TrppX 2 Y 2 q2

n

n

q ď . . . ď TrpX 2 Y 2 q.

(3.30)

An important applications of the estimates (3.30) concerns the operatorvalued exponential functions: X “ eÁ , Y “ e´B . Proposition 3.11. Let A and B be self-adjoint non-negative operators such that B P Pbă1 pAq is a Kato-small perturbation of A (Definition 1.50). Let eÁ P Cs pHq, e´B P Cr pHq for 1{s ` 1{r “ 1, s, r P N, including the case s “ 1, when we set e´B P LpHq. Then the operator sum A ` B is a well-defined non-negative selfadjoint operator on dom pA ` Bq “ dom A, and Trpe´pA`Bq q ď TrpeÁ e´B q.

(3.31)

Proof. The first part of the assertion follows from the Kato-smallness of the perturbation B, see Proposition 1.51, which gives a C0 -semigroup with generator A ` B.

3.3. Trace and Cp pHq-norm estimates

89

By the assumptions made on the operator exponentials the product eÁ e´B belongs to C1 pHq. Hence, the trace in the left-hand side of (3.31) exists. Then for n n X :“ eÁ{2 and Y :“ e´B{2 the inequalities (3.30) yield n

n

n

0 ď TrppeÁ{2 e´B{2 q2 q ď TrpeÁ e´B q.

(3.32)

Note that the generators A and B satisfy the conditions of Proposition 5.8. Then by the Lie-Trotter product formula, there exists the strong limit n n n s-limnÑ8 peÁ{2 e´B{2 q2 “ e´pA`Bq . Since the trace (as well as the trace norm) is weakly lower semi-continuous (Proposition 2.48 (f) and Corollary 2.75) and the left-hand side of (3.32) is bounded from above, this strong limit gives the estimate n

n

n

Trpe´pA`Bq q ď lim inf TrpeÁ{2 e´B{2 q2 . nÑ8

(3.33)

Therefore, taking the limit n Ñ 8 in the left-hand side of (3.32) and using the l lower semi-continuity (3.33) we obtain the inequality (3.31). Since by the definition of the } ¨ }p -norm one has }A}p “ pTrp|A|p qq1{p , p ě 1, the trace inequality (3.31) gives the inequality for norms }e´pA`Bq }1 ď }eÁ{2 e´B eÁ{2 }1 .

(3.34)

In the next assertion we relax the conditions of Proposition 3.11 on the generators A and B. Since in (3.33) the estimate from below comes from the weak lower semi-continuity of the trace, the topology of convergence for the Lie-Trotter product formula was not important for the proof of inequality (3.31). However, at least one of the generators A or B has to produce a Gibbs semigroup, see Chapter 4. Proposition 3.12. Let A ě 0 be generator of the self-adjoint Gibbs semigroup tGt pAq “ e´tA utě0 . Then Trotter-Kato product formula converges away from zero in the trace-norm topology for Kato functions: f pxq “ gpxq “ e´x , and for any self-adjoint operator B ě 0 to a degenerate Gibbs semigroup ` ˘n . } ¨ }1 - lim e´tA{n e´tB{n “ e´tH P0 , H “ A ` B, (3.35) nÑ8

where orthogonal projection P0 : H Ñ dom H and t ą 0. For the proof one has to consult Section 5.4, Proposition 5.53. n

n

Corollary 3.13. For the operators X “ eÁ{2 , Y “ e´B{2 and for inequalities (3.30) we obtain the same estimate (3.32). However, because of (3.35), the limit in the left-hand side of (3.33) takes now the different form n

n

n

Trpe´tH P0 q ď lim inf TrpeÁ{2 e´B{2 q2 , nÑ8

(3.36)

which gives instead of (3.31) the estimate Trpe´tH P0 q ď TrpeÁ e´B q,

t ą 0.

This inequality involves the degenerate Gibbs semigroup te´tH P0 utą0 .

(3.37)

90


Proposition 3.14. Let tAs P Cps pHq : s “ 1, 2, . . . , ru and suppose that 1. Then Aprq :“

r ź

with p´1 :“

As P Cp pHq

s“1

r ÿ

p´1 s

řr s“1

ps´1 ď

(3.38)

s“1

and r r › ›ź ź › › }As }ps . As › ď › p

s“1

(3.39)

s“1

Proof. Take f pxq “ xp , x ě 0, in (3.27). By the definition (2.37) of the } ¨ }p -norm, we obtain 8 ź r ”ÿ ı1{p (3.40) }Aprq }p ď . sjp pAs q j“1 s“1

Since by the H¨ older inequality }Aprq }p ď

r 8 ź ”ÿ

sjp pAs q

ı1{p

j“1 s“1 8 ”ÿ

ď

spj 1 pA1 q

8 ”ÿ

ı1{p1 ¨¨¨

j“1

“ }A1 }p1 ¨ ¨ ¨ }Ar }pr ,

ı1{pr sjpr pAr q

(3.41)

j“1

for p´1 “

r ÿ

p´1 s ,

s“1

(3.40) and (3.41) imply (3.38) and (3.39).

l

Remark 3.15. This result is in fact a generalisation of the Cp -H¨ older inequality proved in Proposition 2.54. Corollary 3.16. If A P Cp pHq for some p ą 0, then Aα P Cp{α pHq, for any α ą 0, and }Aα }p{α ď p}A}p qα . (3.42) Proof. This result follows directly from the Definition 2.50 of }¨}p and from (2.41), (3.28). We note that the quantity } ¨ }p defined for 0 ă p ă 1 by (2.37) has no l longer the properties of a norm. Now we consider non-self-adjoint operator exponentials that belong to the trace-class C1 pHq. In fact they are related to the non-self-adjoint Gibbs semigroups that we study in Section 5.5. Proposition 3.17. For any trace-class operator X and p P N one has inequalities TrppX ˚ qp X p q ď TrpX ˚ Xqp .

(3.43)

91

3.4. Monotonicity, convexity and inequalities

Proof. For matrices the result (3.43) follows directly from the Weyl estimate (3.15) in Corollary 3.4. The extension to C1 pHq is carried out by trace-norm finite-rank approximation of X. l We apply (3.43) for the Gibbs exponential X “ eÁ P C1 pHq generated by m-sectorial operator A with vertex γ “ 0 and semi-angle α, Definition 4.26. Note that by Corollary 4.33 (or Corollary 5.79) we obtain ˚

} ¨ }1 - lim pe´tA nÑ8

˚

{n ´tA{n n

e

q “ e´tpA

.

Àq

,

t P Sθ ,

(3.44)

.

where θ “ π{2 ´ α and A˚ ` A is the form-sum of the operators A˚ and A. Since K ÞÑ Tr K for K P C1 pHq is continuous in the the trace-norm topology (Remark 3.9), the inequality (3.43) and the Trotter formula (3.44) yield the following assertion. Proposition 3.18. For the Gibbs exponential X “ eÁ generated by the m-sectorial operator A one has the inequality ˚

˚

TrpeÁ eÁ q ď Tr e´pA

.

Àq

.

(3.45)

Comparing this result with Proposition 3.11 we see that the inequality (3.45) is opposite in sign to the inequality (3.31) for self-adjoint exponentials. We also note that (3.45) becomes equality if and only if operator A is normal.

3.4

Monotonicity, convexity and inequalities

Similarly to the Gibbs semigroups, the inequalities that we study in this section have significant applications in quantum statistical mechanics. For this reason we essentially focus here on the Gibbs exponential function and leave the discussion regarding general convex functions for the end. In order to be able to provide proofs we shall need to refer to some results of Chapter 4 (Sections 4.4 and 4.5). Proposition 3.19 (Peierls-Bogoliubov inequality). The Gibbs exponential X “ eÁ P C1,` pHq generated by a self-adjoint non-negative operator A ě 0 satisfies the inequality (3.46) peÁ u, uq ě e´pAu,uq , for any unit vector u P dom A. Equality is attained for eigenvectors of the operator A. Proof. Since X ˚ “ X belongs to the positive cone C1,` pHq, there exists a complete orthonormal set of eigenvectors tϕj ujě1 of the operator X ě 0 that forms an orthonormal basis (ONB) in H. By the spectral mapping theorem (Section A.9), one gets that also Aϕj “ λj pAqϕj for non-negative eigenvalues tλj pAqujě1 . Then

92


ř ř u “ jě1 upjq ϕj and since jě1 |upjq |2 “ 1, the Jensen inequality for (convex) exponential function yields peÁ u, uq “

ÿ

|upjq |2 e´λj pAq ě e´

ř jě1

|upjq |2 λj pAq

“ e´pAu,uq ,

(3.47)

jě1

which proves (3.46). Note that equality is attainded in (3.47) when u “ ϕj , for l any j ě 1. Corollary 3.20. Let tej ujě1 be an ONB in the Hilbert space H. Then (3.46) yields for the Gibbs exponential X “ eÁ ě 0 the inequality ÿ ÿ peÁ ej , ej q ě e´pAej ,ej q . iě1

jě1

Since the left-hand side is the trace (Definition 2.41), which is independent of the choice of ONB, for any basis tej ujě1 one has the inequality Tr eÁ ě

ÿ

e´pAej ,ej q ,

(3.48)

jě1

also known as the Peierls-Bogoliubov inequality. Moreover, taking into account the last assertion of Proposition 3.19 and inequality (3.48) we obtain that Tr eÁ “

ÿ

e´pAϕj ,ϕj q “ sup

ÿ

e´pAej ,ej q .

(3.49)

tej ujě1 jě1

jě1

Proposition 3.21. Let Op1,` :“ tA ě 0 : X “ eÁ P C1,` pHqu. Then the realvalued function A ÞÑ ln Tr eÁ

(3.50)

is convex and monotonically decreasing on the cone of positive operators Op1,` . older inequality Proof. Let A1 , A2 P Op1,` . Taking into account (3.49) and the H¨ one gets ÿ e´ppαA1 `p1´αqA2 qej ,ej q Tr e´pαA1 `p1´αqA2 q “ sup tej ujě1 jě1

“ sup

ÿ

pe´pA1 ej ,ej q qα pe´pA2 ej ,ej q q1´α

tej ujě1 jě1

ď sup tej ujě1

”´ ÿ jě1

e´pA1 ej ,ej q

¯α ´ ÿ jě1

e´pA2 ej ,ej q

¯1´α ı ,

(3.51)

93

3.4. Monotonicity, convexity and inequalities for α P r0, 1s. Then elementary estimate and again (3.49) yield ”´ ÿ ¯α ´ ÿ ¯1´α ı e´pA2 ej ,ej q e´pA1 ej ,ej q sup tej ujě1

ď sup

jě1

´ÿ

(3.52)

jě1

e´pA1 ej ,ej q

¯α

tej ujě1 jě1

sup

´ÿ

e´pA2 ej ,ej q

¯1´α “

tej ujě1 jě1

pTr eÁ1 qα pTr eÁ2 q1´α . The inequalities (3.51) and (3.52) imply ln Tr e´pαA1 `p1´αqA2 q ď α ln Tr eÁ1 ` p1 ´ αq ln Tr eÁ2 ,

(3.53)

which proves the convexity of the mapping (3.50). Let A1 , A2 P Op1,` and A1 ď A2 . Then by (3.49) ÿ ÿ e´pA2 ej ,ej q “ Tr eÁ2 , e´pA1 ej ,ej q ě sup Tr eÁ1 “ sup tej ujě1 jě1

tej ujě1 jě1

and consequently ln Tr eÁ1 ě ln Tr eÁ2 , A1 ď A2 .

(3.54)

That is the mapping (3.50) is monotonically decreasing on the positive cone Op1,` . l Corollary 3.22. We set A1 :“ A and A2 :“ A`B , where B ě 0 is also self-adjoint. Then Ωpαq :“ ln Tr e´pαA1 `p1´αqA2 q “ ln Tr e´pA`p1´αqBq (3.55) is a convex monotonically increasing function Ω : r0, 1s Ñ rΩp0q, Ωp1qs. Recall that convex functions are almost everywhere differentiable and possess everywhere right- and left-derivatives. Hence, by convexity one gets the inequalities Bα Ωp`0q ď Ωp1q ´ Ωp0q ď Bα Ωp1 ´ 0q .

(3.56)

Remark 3.23. The calculation in (3.56) of the right-derivative at α “ `0 and the left-derivative at α “ 1 ´ 0 needs an additional analysis of differentiability of the mapping (3.50) on the cone of positive operators Op1,` . This analysis is presented in full generality in Sections 4.5 and 4.6. Here we recall only some necessary facts concerning the calculation of derivatives. Heuristically one gets for the derivative Bα Ωpαq of the function (3.55) the following expression Bα Ωpαq “

1 Tr B e´pA`p1´αqBq “: xByA`p1´αqB . Tr e´pA`p1´αqBq

(3.57)

Then the convexity inequality (3.56) takes on the form xByA`B ď ln Tr eÁ ´ ln Tr e´pA`Bq ď xByA , which is known as the two-sided Bogoliubov (convexity) inequality.

(3.58)

94


Remark 3.24. A key step towards justification of (3.57) is including the Gibbs exponential under the trace, into a C0 -semigroup family, which for t ą 0 takes values in the trace class C1 pHq for any α P r0, 1s. This yields tGt pHpκqq “ e´t Hpκq utě0 , which is a Gibbs semigroup with self-adjoint generator Hpκq “ A ` κB in the sense of Definition 4.1. Here we set κ :“ 1 ´ α. Then to proceed further we use that by Corollary 4.32, the function: t ÞÑ e´t Hpκq , is trace-norm differentiable for t ą 0 and any fixed α P r0, 1s. Then for the derivative of the function Wt,s pκ, q :“ Gt´s pHpκqqGs pHpκ ` qq we obtain } ¨ }1 - Bs Wt,s pκ, q “ Gt´s pHpκqqp´BqGs pHpκ ` qq P C1 pHq.

(3.59)

Thus, definition of Wt,s pκ, q and (3.59) yield żt ds Gt´s pHpκqqp´BqGs pHpκ ` qq,

Gt pHpκ ` qq ´ Gt pHpκqq “

(3.60)

0

where the integral is well defined and belongs to C1 pHq. Note that if there exists ξ ą 0, such that operator BpA ` ξ 1q´1 P LpHq, then (3.60) gives limÑ0 }Gt pHpκ ` qq ´ Gt pHpκqq}1 “ 0. Consequently, by (3.60) we find that the } ¨ }1 -derivative żt ds Gt´s pHpκqq B Gs pHpκqq P C1 pHq,

Bκ Gt pHpκqq “ ´

(3.61)

0

exists and is also an operator in C1 pHq, Proposition 4.52. Since Trp¨q is continuous in the } ¨ }1 -topology, TrpBκ Gt pHpκqqq “ Bκ Tr Gt pHpκqq. Similarly, since the integrals in (3.60) and (3.61) are constructed as Bochner integrals in C1 pHq, one can interchange Trp¨q and the integration. Therefore, application of the Trp¨q to derivative (3.61) yields żt ds Tr Gt´s pHpκqq B Gs pHpκqq “

Bκ Tr Gt pHpκqq “ ´ 0

´ t Tr B Gt pHpκqq,

(3.62)

by the cyclicity of the trace. Recall that Gt pHpκ “ 1 ´ αqq “ e´t pA`p1´αqBq . Then (3.62) gives for the derivative of Ωpαq the heuristical value (3.57), provided that B is relatively small with respect to the generator A in the sense that BpA ` ξ 1q´1 P LpHq for some ξ ą 0. This proves the two-sided Bogoliubov (convexity) inequality (3.58). We conclude this section by some remarks on trace functions and operator trace inequalities, which are similar to what we obtained for the Gibbs exponential f pxq “ e´x denoted by fppAq “ eÁ . Let f : R Ñ R be a monotonically decreasing convex function. Recall that Op1,` denotes cone of positive self-adjoint operators, see Proposition 3.21. By the

95

3.4. Monotonicity, convexity and inequalities

spectral functional calculus, the mapping Op1,` Q A ÞÑ fppAq, corresponding to the function f , is well defined and we assume that fp : Op1,` Ñ C1,` pHq. Then by the spectral mapping theorem (Section A.9) σpfppAqq “ f pσpAqq, and the non-negative spectrum σpAq “ tλj pAqujě1 is pure point with accumulation at `8. Proposition 3.25 (general Peierls-Bogoliubov inequality). Let f : R Ñ R` 0 be a monotonically decreasing convex function. Then for A P Op1,` and for the corresponding operator fppAq P C1,` pHq one has the inequalities pfppAqu, uq ě f ppAu, uqq

and

Tr fppAq ě

ÿ

f ppAej , ej qq,

(3.63)

jě1

for any unit vector u P dom A and for any ONB tej ujě1 Ă dom A. Equalities are attained on eigenvectors tϕj ujě1 of the operator A. Proof. The proof closely mimics that of Proposition 3.19 and Corollary 3.20. Since inequality in (3.63), we note that for fppAqϕj “ it is sufficient to checkřonly the firstř pjq f pλj pAqqϕj and u “ jě1 u ϕj , jě1 |upjq |2 “ 1, the Jensen inequality for the convex function f yields pfppAqu, uq “

ÿ

|upjq |2 f pλj pAqq ě f p

ÿ

|upjq |2 λj pAqq “ f ppAu, uqq,

jě1

jě1

which proves (3.63). Note that the inequalities (3.63) become equalities, when l u “ ϕj , for any j ě 1, and for tej “ ϕj ujě1 . Corollary 3.26. (a) The monotonicity of the function f implies the monotonicity of the trace function A ÞÑ Tr fppAq. (b) The convexity of the function f implies the convexity of the trace function A ÞÑ Tr fppAq. Proof. (a) Let A1 , A2 P Op1,` and A1 ď A2 , i.e., dom A1 Ą dom A2 . Then by Proposition 3.25 and the monotonicity of f , Tr fppA1 q “ sup

ÿ

f ppA1 ej , ej qq

tej ujě1 jě1

ě sup

ÿ

f ppA2 ej , ej qq “ Tr fppA2 q,

tej ujě1 jě1

that is, the trace function is monotone. (b) Using Proposition 3.25 and the convexity of f we estimate the trace function for a convex combination of operators A1 , A2 P Op1,` and α P r0, 1s on

96


the common domain D “ dom A2 : Tr fppαA1 ` p1 ´ αqA2 q “ sup

ÿ

f pppαA1 ` p1 ´ αqA2 qej , ej qq

tej ujě1 jě1

ď sup

ÿ

pαf ppA1 ej , ej qq ` p1 ´ αqf ppA2 ej , ej qqq

tej ujě1 jě1

ď α sup

ÿ

f ppA1 ej , ej qq ` p1 ´ αq sup

tej ujě1 jě1

ÿ

f ppA2 ej , ej qq

tej ujě1 jě1

“ α Tr fppA1 q ` p1 ´ αq Tr fppA2 q . Hence, the trace function A ÞÑ Tr fppAq is convex.

l

Corollary 3.27 (Klein inequality). Let the function α ÞÑ ωpαq :“ Tr fppαA1 ` p1 ´ αqA2 q be convex for α P r0, 1s, i.e., we get that ωp1q ´ ωp0q ě

ωpαq ´ ωp0q . α

(3.64)

So, the right derivative Bα ωpαq|α“`0 exists and we obtain the Klein inequality in the form: TrtfppA1 q ´ fppA2 qu ě Bα Tr fppA2 ` αpA1 ´ A2 qq α“`0 , (3.65) regardless of what the explicit expression in the right-hand side is. Let Apαq :“ A2 `αpA1 Á2 q. To proceed with the calculation of derivative in (3.65) for small α, the function α ÞÑ fppApαqq should be trace-norm differentiable. Since for α ă 1{2 the operator αpA1 ´ A2 q is Kato-small with respect to A2 , with a relative bound less than one, the linear function α ÞÑ Apαq can be extended to a holomorphic family tApzquzPD1{2 of type (A) in the disc of radius 1{2, Section 5.5, (subsection 5.5.1). Now we can use the Riesz-Dunford functional calculus (1.67) to express the operator fppApαqq as ż 1 f pζq p f pApαqq “ dζ . (3.66) 2πi Γ ζ1 ´ Apαq Here Γ Ă ρpApαqq is a positively oriented contour in the resolvent set of Apαq that encircles the spectrum σpApαqq. Note that ρpApαqq Ă CzR` 0 for any α P r0, 1s. To ensure that the representation (3.66) exists and gives trace-class operators, we require that – the resolvent Rλ pA2 q “ pA2 ´ λ1q´1 P C1 pHq for λ ă 0; – for <e ζ Ñ `8, the function f pζq Ñ 0 fast enough to guarantee that the integral is } ¨ }1 -convergent.

97

3.5. Notes

Then the left-hand side of (3.66) is } ¨ }1 -differentiable, which implies that the derivative Bα now commutes with the integral and also with Trp¨q. Consequently, Bα Tr fppA2 ` αpA1 ´ A2 qq α“`0 “ TrpA1 ´ A2 qfp1 pA2 q , where

1 fp1 pA2 q “ 2πi

ż dζ Γ

f pζq . pζ1 ´ A2 q2

Hence, we obtain the Klein inequality (3.65) in its well-known canonical form: TrrfppA1 q ´ fppA2 q ´ pA1 ´ A2 qfp1 pA2 qs ě 0 .

3.5

(3.67)

Notes

Notes to Section 3.1. The description of singular values in (a)–(e) is standard and can be found in [GK69] (Ch.II, §1–§2) and in [Sim05], Ch.1. For the square-root lemma in (a), see [RS80] Ch.VI.4. The usual way to prove that singular values satisfy tsn pAq “ sn pA˚ quně1 is a reference to the canonical form of the compact operators. In (c) we outline another way, based on the observation that the compact operators AB and BA have the same non-zero eigenvalues, with the same multiplicities, see [Sim05], Ch.1. Section 3.1 contains only preliminaries about s-numbers; other properties of singular values of compact operators are explained in other parts of the book, where they are indispensable. The Weyl-Horn chain of inequalities in (f) is due to H. Weyl [Wey49] and A. Horn [Horn50]. For a short proof, see [GK69] (Ch.II, §3) or [Sad91] (Ch.5.2.3). Notes to Section 3.2. Lemma 3.1 is due to H. Weyl [Wey49] and A. Horn [Horn50], for the proof see, e.g., [Sim05] (Theorem 1.13 and Theorem 1.14). The important Lemma 3.2 was proved by H. Weyl [Wey49], here we followed [GK69] (Ch.II, §3.2). Lemma 3.5 is due to A. Horn [Horn50] and K. Fan [Fan51]. Our proof here is based on the Weyl-Horn chain of inequalities. Proposition 3.6 is due to A. Horn [Horn50]. Notes to Section 3.3. For additional information about spectral and matrix traces, including historical remarks, see [Piet14]. Proposition 3.11 proves the Golden-Thompson inequality (3.31) for selfadjoint exponentials with values in the trace ideals for a Hilbert space. For matrices it was proved in [Gold65] and [Thom65]. Comprehensive surveys of related trace inequalities with historical remarks can be found in [Sim05] (Chapter 8) and in [Petz83], [Car09]. Corollary 3.13 shows the Golden-Thompson inequality for the case when the Trotter product limit is a degenerate Gibbs semigroup. Similar to the case of C0 semigroups the weak lower semi-continuity of the trace is sufficient to infer the result in this case.

98


Proposition 3.6, together with Corollaries 3.7 and 3.8, yield fundamental estimates for Cp pHq-ideals, see [GK69], [Sch70]. These estimates are collected in Proposition 3.14 and Corollary 3.16. They are useful in applications to non-selfadjoint Gibbs semigroups. Proposition 3.17 is a generalisation to the Hilbert space case of the Ky Fan inequalities for matrices [Fan49]. Then Proposition 3.18 is an extension to non-selfadjoint Gibbs exponentials of the matrix Bernstein inequality [Bern88]. Note that inequality (3.45) is opposite in sign to the Golden-Thompson inequality (3.31) for self-adjoint exponentials. Notes to Section 3.4. The Peierls-Bogoliubov inequality appeared originally in [Peie38]. Apparently it was rediscovered in fifties and intensively exploited by Bogoliubov’s school as a quantum variational principle, especially in the theory of magnetism [Tya67]. To keep a contact with the main topic of this book, we provide here the proof of Proposition 3.19 and Corollary 3.20 in a Hilbert space for Gibbs exponentials. It is a direct proof with reference to the Jensen inequality, cf. [Sim05] and compare with e.g., [Car09]. Proposition 3.21 is also proved in a Hilbert space and for Gibbs exponentials, cf. [Rue69], Proposition 2.5.5. The proof avoids derivatives, which are easy to treat for matrices (see [Car09], [Petz83]) but difficult in infinite-dimensional spaces. Corollary 3.22 (3.55) serves as a preparation of the two-side Bogoliubov convexity inequality (3.58). Note that the right-hand side of (3.58) appeared first as a new variational principle in the paper of one of Bogoliubov’s student [Kva56]. Then it became popular first in the theory of magnetism [Tya67]. Although almost evident, the two-side estimate (3.58) is much more efficient. It is a key for the Approximating Hamitonian Method (see [BBZKT81] and [BBZKT84]) in quantum statistical mechanics. The proof of the Bogoliubov convexity inequality (3.58) for the case of a bounded operator B, that is, without the issues with derivatives, is straightforward. Remark 3.24 explains a strategy for the general case. It was realised in [ZBT75], [BBZKT81]. The generalisation of the Peierls-Bogoliubov inequality in Proposition 3.25 as well as Corollary 3.26 for non-exponential convex functions are standard. They are independent of Klein’s inequality [Kle31] that has stated at the end in Corollary 3.27. Again the main problem here is dealing with the derivative involved in the Klein inequality (3.65), or (3.67), cf. [Rue69], [Car09].

Chapter 4

Gibbs semigroups This chapter contains notations and definitions concerning the main subject of the book. Here we introduce the Gibbs semigroups. They are strongly continuous (or degenerate) semigroups with values in the trace-class ideal of bounded operators on a Hilbert space. Although they constitute a subclass of the more general class of compact semigroups, the Gibbs semigroups have many special properties of their own. We present here both eventually and immediately Gibbs semigroups. The immediately Gibbs semigroups may be in turn degenerate or not. The immediately Gibbs semigroups play important rôle in Quantum Statistical Mechanics, see the Notes in Section 4.7. In our discussion of the non-self-adjoint Gibbs semigroups, we define a class of p-generators. The perturbation theory is developed first for a restricted class of the P0` -perturbations of quasi-bounded semigroups. Then we consider the holomorphic Gibbs semigroups, for which this class may be extended to Pbă1 -perturbations with b ą 0.

4.1

Gibbs semigroups

The following is a key definition in the theory of the Gibbs semigroup. Definition 4.1. A strongly continuous semigroup tGt utě0 on a Hilbert space H is called an (immediately) Gibbs semigroup if for t P R` , t ÞÑ Gt P C1 pHq. We call tGt utě0 a self-adjoint Gibbs semigroup if Gt˚ “ Gt , for t P R` 0. Although we study both self-adjoint and non-self-adjoint semigroups, the book places a certain emphases on the self-adjoint Gibbs semigroups. The first straightforward corollary of Definition 4.1 and the } ¨ }p -continuity of the multiplication for 1 ď p ă 8, is the following statement. Proposition 4.2. Any immediate Gibbs semigroup tGt utě0 is trace-norm continuous for t ą 0. © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1_4

99

Chapter 4. Gibbs semigroups

100

Proof. For any t ą 0, there is a δ P R such that t{2 ` δ ą 0. From the semigroup property it follows that Gt`δ “ Gt{2`δ Gt{2 , where Gt{2`δ P LpHq and Gt{2 P C1 pHq. Since the semigroup tGt utě0 is strongly continuous, s-limδÑ0 Gt{2`δ “ Gt{2 . Hence, for t ą 0 } ¨ }1 -lim Gt`δ “ } ¨ }1 -limpGt{2`δ Gt{2 q δÑ0

(4.1)

δÑ0

“ ps-lim Gt{2`δ q Gt{2 “ Gt , δÑ0

by the } ¨ }1 -continuity of multiplication on C1 pHq, Proposition 2.78.

l

According to the next statement, the semigroup property of tGt utě0 allows one to relax in Definition 4.1 the trace-class condition Gt P C1 pHq for any t ą 0, to the condition Gt P Cp pHq for any t ą 0, and an ideal Cp pHq Ą C1 pHq with p ą 1. Proposition 4.3. A semigroup tGt utě0 such that Gt P Cp pHq for all t ą 0 and a number p ą 1 is an immediate Gibbs semigroup. Proof. Let t ą 0. Then by assumption one has Gt{α P Cp pHq for any α ą 1, and in particular for α “ p. Therefore, by Corollary 3.16, Gt “ pGt{p qp P C1 pHq,

(4.2)

that is, Gt P C1 pHq for all t ą 0.

l

Remark 4.4. Let ten uně1 be an orthonormal basis in the Hilbert space H. Then the one-parameter family of operators tTt utě0 defined for any u P H by the mappings u ÞÑ Tt u “

8 ÿ

e´t lnpn`1q pu, en qen ,

t ě 0,

(4.3)

n“1

is a self-adjoint strongly continuous contraction semigroup on H. The semigroup tTt utě0 is obviously compact for t ą 0 with the singular values tsn pTt q “ pn ` 1q´t uně1 , but it has the Gibbs semigroup property only for t ą t0 “ 1. Similarly, Tt P Cp pHq only for t ą t0 ppq “ p´1 . Definition 4.5. If there exists a threshold t0 ą 0 such that Ttąt0 P C1 pHq, then the semigroup tTt utě0 is called an eventually Gibbs semigroup away from t0 . If t0 “ 0 and tTt P C1 pHqutą0 , then tTt utě0 is called an immediately Gibbs semigroup, or briefly the Gibbs semigroup, cf. Definition 4.1. Corollary 4.6. By Proposition 4.2 and Definition 4.5 we conclude that an immediately Gibbs semigroup is immediately } ¨ }1 -continuous for t ą 0, whereas an eventually Gibbs semigroup (with threshold t0 ą 0) is } ¨ }1 -continuous only for t ą t0 .

4.1. Gibbs semigroups

101

Since by definition any Gibbs semigroup tGt utě0 is strongly continuous, it is quasi-bounded and it has a generator A which satisfies the conditions of Proposition 1.12. Hence, we can write Gt pAq :“ e´tA ,

t ě 0,

(4.4)

and }Gt pAq} ď M eω0 t ,

M ą 0, ω0 ě 0,

where p´8, ´ω0 q P ρpAq, the resolvent set of the closed linear operator A. Moreover, if <e ζ ą ω0 , the Laplace transform of (4.4) exists and it is given by the operator-norm convergent integral ż8 ˆ ζ pAq “ G dt e´ζt Gt pAq “ pζ1 ` Aq´1 . (4.5) 0

Note that (4.5) is related to the resolvent, Rζ pAq “ Aζ´1 , of the operator A by ˆ ´ζ pAq, Rζ pAq “ G see (1.33). We used here the notation Aζ :“ A ´ ζ1 introduced in Chapter 1. Remark 4.7. The inclusion C1 pHq Ă C8 pHq implies that every Gibbs semigroup tGt pAqutě0 is also a C0 -semigroup of compact operators. Since the integral (4.5) converges in the operator-norm topology and the integrand is a compact operator, ˆ ζ pAq P C8 pHq. This implies that the resolvent Rζ pAq is compact. the operator G Hence, the generator A is an unbounded operator, cf. Proposition 1.14. Similarly to Sections 1.3 and 1.5, the Laplace transform (4.5) is a key formula for understanding the properties of generator of Gibbs semigroup. We start with statements that are valid also in the general case of Banach spaces, cf. Appendix A, Section A.4. Proposition 4.8. Let A be a closed linear operator in a Hilbert space H. If λ P C is such that there exists a sequence txn uně1 with the properties lim inf }xn } ě c ą 0 nÑ8

and

lim }Aλ xn } “ 0,

nÑ8

then λ P σpAq. Proof. Suppose, by contradiction, that λ P CzσpAq “ ρpAq. Then the resolvent Rλ pAq is a bounded operator on H, i.e., there exists a constant a ą 0 such that }Rλ pAqu} ď a}u} for all u P H. This means that }Aλ x} ě a´1 }x} for any x P dom Aλ , which contradicts the hypothesis of the proposition. l We use this proposition to study spectral properties of generators of Gibbs semigroups.


102

Lemma 4.9. The spectrum of the resolvent Rζ pAq of an unbounded, closed, densely defined operator A in H admits for any ζ R σpAq the representation σpRζ pAqq “ t0u Y tpλ ´ ζq´1 : λ P σpAqu .

(4.6)

Proof. Since A is not bounded, there is a sequence tfn uně1 Ă dom A, with }fn } “ 1, such that lim supnÑ8 }Aζ fn } “ `8. This implies that there exists a sequence txn uně1 such that }xn } ě c ą 0 and limnÑ8 }Rζ pAqxn } “ 0: for example, one can take xn :“ Aζ fn {}Aζ fn }. Therefore, by virtue of Proposition 4.8, we obtain that t0u Ă σpRζ pAqq. Next, since ran Rζ pAq “ dom A, the operator L :“ pz ´ ζqAζ Rz pAq,

z R σpAq,

is bounded, and it commutes with Rζ pAq for any ζ P C. Moreover, we have L rpz ´ ζq´1 1 ´ Rζ pAqs “ Aζ Rz pAq ´ pz ´ ζqRz pAq “ 1. Hence, pz ´ ζq´1 R σpRζ pAqq for z R σpAq. Conversely, suppose that pλ ´ ζq´1 R σpRζ pAqq and consider the operator M defined by M :“ r1 ´ pλ ´ ζqRζ pAqs´1 Rζ pAq “ Rζ pAqr1 ´ pλ ´ ζqRζ pAqs´1 . If f P H, then Aλ M f “ rpζ ´ λq1 ` A ´ ζ1sRζ pAqr1 ´ pλ ´ ζqRζ pAqs´1 f “ r1 ´ pλ ´ ζqRζ pAqsr1 ´ pλ ´ ζqRζ pAqs´1 f “ f. On the other hand, if f P dom A, then M Aλ f “ r1 ´ pλ ´ ζqRζ pAqs´1 Rζ pAqrpζ ´ λq1 ` A ´ ζ1sf “ r1 ´ pλ ´ ζqRζ pAqs´1 r1 ´ pλ ´ ζqRζ pAqsf “ f. Hence, λ R σpAq.

l

Corollary 4.10. If the operator Rζ pAq is compact for some ζ R σpAq, then Rz pAq P C8 pHq,

z R σpAq,

and the spectrum σpAq is either empty, or consists of at most countably many eigenvalues tλn uně1 of finite multiplicity, such that lim |λn | “ `8.

nÑ8

Moreover, we have that the point spectrum σp pRζ pAqq “ tpλn ´ ζq´1 : λn P σpAqu, whereas σess pRζ pAqq “ t0u.

4.1. Gibbs semigroups

103

Proof. The first part of the statement follows from the resolvent identity Rz pAq “ Rζ pAq ` Rz pAqpz ´ ζqRζ pAq, and from Proposition 2.7, which states that C8 pHq is a ˚-ideal in LpHq. Note that by Lemma 4.9, t0u Ă σpRζ pAqq for unbounded operator A even if σpAq “ H. Then the second part of the statement is a consequence of Lemma 4.9, Proposition 2.18, and Remark 2.23. In order to prove the third part, recall that Rζ pAq P C8 pHq implies 0 P σpRζ pAqq, Remark 2.15. By virtue of Remark 2.23, this point may belong to any of the three disjoint components of the spectrum σpRζ pAqq, Appendix A (Section A.6). Suppose that u P H is an eigenvector of Rζ pAq with eigenvalue equals to zero: Rζ pAqu “ 0. Since Rζ pAqu P dom A, we get u “ pA ´ ζ1qRζ pAqu “ 0 . This means that ker Rζ pAq is trivial, and we infer that the point 0 R σp pRζ pAqq. Next, following Definition 2.14, we consider the case when ran Rζ pAq “ dom A. Since A is unbounded, closed, and densely defined, ran Rζ pAq ‰ H, but ran Rζ pAq “ H, i.e., t0u Ď σess pRζ pAqq. Therefore, σpRζ pAqqzt0u “ σp pRζ pAqq and the equality σp pRζ pAqq “ tpλn ´ ζq´1 : λn P σpAqu follows from (4.6).

l

Now, let A be generator of a strongly continuous semigroup tUt pAqutě0 on H, see Chapter 1. In this general setup, the relationship between σpAq and σpUt pAqq is not straightforward, cf. Section A.9. Proposition 4.11. Suppose that tUt pAqutě0 is a strongly continuous semigroup on H with generator A. Then te´λt : λ P σpAqu Ď σpUt pAqq. Proof. Let A be the maximal Abelian subalgebra of LpHq containing Ut pAq for all t ě 0. This maximal subalgebra A obviously exists by Zorn’s lemma, and it is closed under the strong limit. Representation (4.5) implies that Rζ pAq P A ,

ζ R σpAq.

Moreover, if X P A is invertible within LpHq, then X ´1 P A. Therefore, the spectrum of X as an operator coincides with its spectrum as an element of the ˜ Banach algebra A and hence, it coincides with tXpmq : m P M u, where M is the ˜ maximal ideal space of A, and X is the Gel’fand transform of X. Let λ P σpAq and ζ R σpAq. Then pλ ´ ζq´1 P σpRζ pAqq, so there exists ˜ ζ pmq “ pλ ´ ζq´1 ‰ 0. If γptq “ U ˜t pmq, then γp0q “ 1 and m P M with R


104

γpsqγptq “ γps ` tq for s, t ě 0. Since for any u P H, Rζ pAqu P dom A, the operator-valued function R` Q t ÞÑ Ut pAqRζ pAq P LpHq is continuous in the operator norm, cf. Proposition 1.15. Hence, γptqpλ ´ ζq´1 depends continuously on t P R` . Consequently, t ÞÑ γptq “ e´βt for some β P C. If <e z ą ω0 , the integral ż8 dt e´zt Ut pAqRζ pAq “ R´z pAqRζ pAq 0

is operator-norm convergent, so its Gel’fand transform is ż8 ˜ ´z pmq. dt e´zt e´βt pλ ´ ζq´1 “ pλ ´ ζq´1 R 0

˜ z pmq “ pβ ´ zq´1 . On the other hand, from the Gel’fand transform Therefore, R of the resolvent identity, ˜ ζ pmqR ˜ ζ pmq ´ R ˜ z pmq “ pζ ´ zqR ˜ z pmq, R we obtain ˜ z pmq “ pλ ´ ζq´1 t1 ` pζ ´ zqpλ ´ ζq´1 u´1 “ pλ ´ zq´1 . R ˜t pmq “ e´λt , so e´λt P σpUt pAqq. We conclude that λ “ β, that is, U

l

In general, the converse of Proposition 4.11 is false. However, if some further conditions are imposed on the semigroup tUt utě0 , see Section 1.4, one gets the following statement. Proposition 4.12. Let tUt pAqutěτ be a norm-continuous semigroup on H for t P rτ, `8q, τ ą 0. Then µt ‰ 0 is a non-zero element of the spectrum, µt P σpUt pAqq, if and only if it can be written as µt “ e´tλ for some λ P σpAq. Proof. For a non-zero element µt P σpUt pAqq, there exists an element m P M , where M is the maximal ideal space, such that the Gel’fand transform γptq “ ˜t pmq satisfies γptq “ µt . Moreover, γps1 ` s2 q “ γps1 qγps2 q for s1 , s2 P p0, 8q, U and γpsq is continuous for s P rτ, `8q. This implies that γpsq “ e´sλ for some λ and s P R` . Since for the operator-norm continuous semigroup tUt pAqutěτ the integral in representation ż 8

dt e´zt Uτ `t pAq

Uτ pAqpz1 ` Aq´1 “ 0

is convergent in the operator norm, the corresponding Gel’fand transform can be written as ż8 ´1 „ ´λτ dt e´zt e´pτ `tqλ e tpz1 ` Aq u pmq “ 0

˜ ´z pmq. “ e´λτ pz ` λq´1 “ e´λτ R This means that pλ ´ zq´1 P σpRz pAqq and λ P σpAq, by Lemma 4.9.

l

4.2. Norm continuity, revisited

105

If we assume that tUt pAqutě0 is a compact operator-valued semigroup for some τ ą 0, then even a stronger statement about the spectral properties of tUt pAqutěτ can be proven. We return to this problem in the next section, Proposition 4.21.

4.2 Norm continuity, revisited In this section we revise the operator-norm continuous semigroups. They were introduced in Section 1.4 and we found (Proposition 1.14) that operator-norm continuity of the semigroup tUt utě0 at t “ `0 implies the boundedness of its generator. So, first we relax this condition to the usual C0 -continuity at t “ `0. Then we study the impact on the semigroup continuity away from zero of the topology of the image ImpUtą0 q of the map t ÞÑ Ut . For example, the strongly continuous immediately Gibbs semigroups are defined by ImpGtą0 q Ď C1 pHq. We also observed (Corollary 4.6) that away from zero the semigroup continuity inherits the tracenorm topology of the image. Below we elucidate a relation between compactness of the image ImpUtą0 q Ď C8 pHq, and topology of continuity of semigroup tUt utě0 away from t “ 0. It turns out that the semigroup tUt utą0 inherits for its continuity the topology of the ideal of compact operators C8 pHq, that is, the operator norm topology. To continue we first introduce few notations and definitions similar to the ones used for the Gibbs semigroups. Definition 4.13. A strongly continuous semigroup tUt utě0 on a Banach space B is said to be eventually norm-continuous if there exists a threshold t0 ą 0 such that the mapping t ÞÑ Ut is operator-norm continuous for t P pt0 , 8q. If t0 “ 0, then a strongly continuous semigroup tUt utě0 is called immediately norm-continuous, or briefly a norm-continuous semigroup, cf. Section 1.4. Next we introduce the compact semigroups on a Hilbert space. The related results we shall use for their spectral analysis. Definition 4.14. A strongly continuous semigroup tUt utě0 on a Hilbert space H is said to be eventually compact if there exists t0 ą 0 such that t ÞÑ Ut P C8 pHq for t ą t0 .

(4.7)

If tUt P C8 pHqutą0 , then the semigroup tUt utě0 is called immediately compact, or briefly a compact semigroup. Lemma 4.15. Any immediately compact C0 -semigroup tUt utě0 is norm-continuous for t ą 0. The same is true for any eventually compact C0 -semigroup tUt utě0 with a threshold t0 ą 0, when t ą t0 .


106

Proof. We consider only the case of eventually compact C0 -semigroups, since in the case of immediately compact semigroups it is sufficient to set t0 “ 0. Let 0 ă t0 ă τ . Then for any ε ą 0, there is a finite-rank operator UτF P KpHq such that }Uτ ´ UτF } ă ε, see Proposition 2.11. Hence, for τ ď t ď t1 and f P H, we have the estimate }Ut1 f ´ Ut f } “ }pUt1 ´τ ´ Ut´τ qUτ f } ď }pUt1 ´τ ´ Ut´τ qUτF f } ` }Ut1 ´τ ´ Ut´τ }}Uτ ´ UτF }}f }.

(4.8)

Since UτF P KpHq and s-limt1 Ñt Ut1 ´τ “ Ut´τ , we get that lim }Ut1 ´τ ´ Ut´τ qUτF } “ 0.

t1 Ñt

(4.9)

Note that }Ut1 ´τ ´ Ut´τ } ď 2 maxt}Ut1 ´τ }, }Ut´τ }u. Then the proof follows from (4.8) and (4.9) due to the estimate }Uτ ´ UτF } ă ε, for any ε ą 0. l Although essentially we are going to deal with semigroups in a Hilbert space, certain results are also relevant in Banach spaces. In fact, Definition 4.14 and Lemma 4.15 hold in the general case of a Banach space B. Indeed, since Uτ maps the unit ball in B into a precompact set C Ă B, the strong operator convergence s-limt1 Ñt Ut1 ´τ “ Ut´τ , implies the operator-norm convergence on C, see (4.8). An example of strongly continuous, immediately compact, and eventually Gibbs semigroup is given in Remark 4.4. A simple example illustrating that eventual does not imply immediate comes from nilpotent semigroups. Example 4.16. Define a C0 -semigroup tUt utě0 on the Hilbert space H “ L2 pr0, 1sq by # f px ` tq for x ` t ď 1 , pUt f qpxq “ 0 for x ` t ą 1 . Then tUt utě0 is nilpotent since Utą1 f “ 0 for any f P H. Note that this strongly continuous semigroup is eventually norm-continuous and eventually compact (with threshold t0 “ 1), but it is not immediately norm-continuous or compact. Proposition 4.17. A quasi-bounded strongly continuous semigroup tUt pAqutě0 on H with generator A P QpM, ω0 q is immediately compact if and only if the map R` 0 Q t ÞÑ Ut pAq P LpHq is immediately norm-continuous and the resolvent Rz pAq is compact for z P ρpAq. Proof. Let the semigroup tUt pAqutě0 be compact. Then by Lemma 4.15, it is normcontinuous for all t ą 0. Now, since the generator A P QpM, ω0 q, it follows from Definition 1.11 for semigroups that }e´tλ Ut pAq} ď M etpω0 ´λq . Hence, for λ ą ω0 the resolvent of A can be represented by the Laplace transform (Proposition 1.12): ż8 R´λ pAq “ dt e´λt Ut pAq. 0


107

Note that the norm-continuity of the semigroup tUt pAqutě0 away from zero implies that the (Bohner) integral ż8 pεq R´λ pAq “ (4.10) dt e´λt Ut pAq, εą0

for some ε ą 0, is the limit of the corresponding operator-norm convergent Darboux-Riemann sums. Since the semigroup tUt pAqutě0 is compact away from zero, each of these sums is in turn a compact operator. Note that the set of compact operators is closed in the operator-norm topology (Proposition 2.8), which yields pεq R´λ pAq P C8 pHq for any ε ą 0 and λ ą ω0 . Since the semigroup is quasi-bounded and the integrand in (4.10) is norm-continuous, pεq

lim }R´λ pAq ´ R´λ pAq} “ 0, εÓ0

λ ą ω0 ,

and again by Proposition 2.8, we conclude that R´λ pAq is compact. To conclude this part of the proof, we remark that p´8, ´ω0 q Ă ρpAq, see Proposition 1.12, and that by the resolvent identity, the fact that Rζ pAq P C8 pHq for some ζ P p´8, ´ω0 q implies that Rz pAq P C8 pHq for any z P ρpAq. To prove the converse, we assume now that the map R` 0 Q t ÞÑ Ut pAq P LpHq is immediately norm-continuous and that Rz pAq P C8 pHq for some z P ρpAq. Then by the remark just above, the resolvent Rz pAq is compact for every z P ρpAq. Since tUt pAqutě0 is strongly continuous and quasi-bounded, it follows from Proposition 1.12 that the set p´8, ´ω0 q Ă ρpAq. We fix a constant λ ą ω0 and define a family of operators: żt Ct :“ dτ e´λτ Uτ pAq, t ě 0. (4.11) 0

Then integrating the equation Bt pe´λt Ut pAquq “ é´λt Ut pAqpλ1 ` Aqu, we obtain

u P dom A ,

żt p1 ´ e

´λt

dτ e´λτ Uτ pAqpλ1 ` Aqu,

Ut pAqqu “ 0

or with v :“ pλ1 ` Aqu, the equation żt p1 ´ e

´λt

Ut pAqqpA ` λ1q

´1

dτ e´λτ Uτ pAqv.

v“

(4.12)

0

Since ´λ P ρpAq, the range ranpλ1 ` Aq “ H. Therefore, equation (4.12) implies that operator (4.11) is compact: Ct “ p1 ´ e´λt Ut pAqqpA ` λ1q´1 P C8 pHq,

(4.13)


108

since R´λ pAq P C8 pHq. By the assumption of norm-continuity of the semigroup tUt pAqutą0 , it follows that for any t ą 0 ż 1 t`δ Ut pAq “ eλt } ¨ }-lim dτ e´λτ Uτ pAq δ t δÓ0 1 “ eλt } ¨ }-lim pCt`δ ´ Ct q. (4.14) δ δÓ0 By virtue of (4.12) and (4.13), the operator-norm limit of compact operators in the right-hand side of (4.14) exists which implies that Ut pAq P C8 pHq for t ą 0. l Now we consider eventually norm-continuous and eventually compact C0 semigroups. A sufficient condition guaranteeing that the eventual property does imply the immediate property for norm-continuity or for compactness of semigroups is their analyticity. Remark 4.18. It is straightforward that if a C0 -semigroup tUt pAqutě0 is holomorphic and eventually norm-continuous for threshold t0 ą 0, then it is immediately norm-continuous. This follows directly from the construction of tUt pAqutě0 in Proposition 1.27. It ensures that a strongly continuous holomorphic semigroup (Definition 1.26) is, in fact, immediately norm-continuous and even normholomorphic for t in some open sector Sθ Ă C` . The next statement together with Example 4.16 settle also the problem of existence of immediately, and of elimination of eventually compact semigroups. Proposition 4.19. Let tUt pAqutě0 be a holomorphic semigroup with generator A P H pθ, ω0 q. If tUt pAqutě0 is eventually compact with threshold t0 ą 0, then it is immediately compact. Proof. Let p t0 ą t0 such that p t0 ´ t0 ă p t0 sin θ. Since by Corollary 1.29 and Proposition 1.33 the semigroup tUt pAqutě0 is norm-holomorphic in the open sector Sθ Ă C` , one gets the operator-norm convergent power series expansion Uz pAq “

8 ÿ pz ´ p t0 qn n Bt Ut pAq|t“tp0 n! n“0

(4.15)

in the disc Dtp0 :“ tz P C` : |z ´ p t0 | ă peM11 q´1 p t0 u, where eM11 ą 1 and θ “ 1 ´1 arcsin peM1 q , see (1.79). Therefore, t0 P Dtp0 . Note that the analyticity of the map z ÞÑ Uz pAq allows us to calculate all derivatives in (4.15). For |δ| ă p t0 ´ t0 the operators Utp0 pAq and Utp0 `δ pAq are compact. Then the derivative ˇ 1 Bt Ut pAqˇt“tp0 “ } ¨ }-lim rUtp0 `δ pAq ´ Utp0 pAqs P C8 pHq, δ δÑ0

(4.16)

being a ˇlimit in the operator-norm topology, is also a compact operator with }Bt Ut pAqˇt“tp0 } ă M11 {p t0 , see (1.76).


109

one can iterate the argument Since the map z ÞÑ Uz pAq is norm-analytic, ˇ in (4.16) for higher-order derivatives: Btn Ut pAqˇt“tp0 P C8 pHq, for n P N, with esˇ timates }Btn Ut pAqˇt“tp } ă pnM11 {p t0 qn . Then the operator-norm convergent power 0 series (4.15) defines compact operators tUz pAquzPDtp in the disc Dtp0 and in par0 ticular, Ut0 pAq P C8 pHq. Now we can take Ut0 pAq P C8 pHq and consider the operator-norm convergent in the disc Dt0 :“ tz P C` : |z ´ t0 | ă t0 sin θu power series expansion (4.15) at t “ t0 . Then by the same argument as above the operators tUz pAquzPDt0 are compact in the disc Dt0 , and so Ut1 pAq P C8 pHq, for example, at t1 “ t0 p1 ´ psin θq{2q. If one considers expansion (4.15) at t1 , then Ut2 pAq P C8 pHq for t2 “ t1 p1 ´ psin θq{2q, and so on for n steps. Therefore, Utn pAq P C8 pHq for tn “ t0 p1 ´ l psin θq{2qn Ñ 0, when n Ñ 8, establishes the assertion. Remark 4.20. By Definitions 4.1 and 4.5, the Gibbs semigroups are obviously compact. Remark 4.4 gives an example of a self-adjoint eventually Gibbs semigroup. On the other hand, since this semigroup is holomorphic, it is immediately compact by Proposition 4.19. The lack of analyticity in Example 4.16 of the nilpotent semigroup leads to an eventually compact and eventually Gibbs semigroup, which has zero trace-norm for t ě t0 “ 1, see Appendix A, Section A.7. Now we are in position to analyse spectra of compact semigroups versus spectra of their generators. Proposition 4.21. Let A be generator of a compact semigroup tUt pAqutě0 . Then the following assertions hold for dim H “ 8. (a) The spectrum σpAq is either empty, or consists of a finite or countable set of eigenvalues tλn uně1 of finite multiplicity, which have no limit point in C. (b) The spectrum of the semigroup has the representation σpUt pAqq “ t0u Y te´tσpAq u,

t ą 0,

(4.17)

where σpUt pAqqzt0u “ σp pUt pAqqzt0u “ te´tσpAq u, that is, Aϕn “ λn ϕn if and only if Ut pAqϕn “ e´tλn ϕn . Proof. (a) Since compact semigroups are norm-continuous, A is the generator of a strongly continuous semigroup. Then, by Proposition 1.12, the resolvent set ρpAq is nonempty, and by virtue of Proposition 4.17, the resolvent Rz pAq is compact for any z P ρpAq. Therefore, the assertion follows from Corollary 4.10. pbq Note that in contrast to Corollary 4.10, we cannot claim that 0 R σp pUt pAqq. We assume that the spectrum of A is non-empty, otherwise σpUt pAqq “ t0u in (4.17). It then follows from (a) that σpAq “ tλn uně1 . For an eigenvector ϕn P dom A corresponding to the eigenvalue λn , the identity pA ´ z1qRz pAqϕn “ ϕn “ Rz pAqpλn ´ zqϕn ,

z P ρpAq,


110 and the fact that A P QpM, ω0 q imply (see (1.5))

´ ¯´m t Ut,m pA´ω0 qϕn “ 1 ` A´ω0 ϕn m ´ ¯ ´m t “ 1 ` pλn ` ω0 q ϕn , m

(4.18)

where A´ω0 “ A ` ω0 1. By virtue of Proposition 1.12 and taking in (4.18) the limit m Ñ 8, we get Ut pAqϕn “ etω0 Ut pA´ω0 qϕn “ e´tλn ϕn ,

(4.19)

for t ě 0. Conversely, since Ut pAq is compact for t ą 0, there exists an eigenvector ψ P H with an eigenvalue of the form e´tµ , i.e., Ut pAqψ “ e´tµ ψ. Calculating the limit 1 1 lim p1 ´ Ut pAqqψ “ lim p1 ´ e´tµ qψ “ µψ, tÓ0 t tÓ0 t it follows from the definition of the generator A (Definition 1.9) that ψ P dom A and Aψ “ µψ, i.e., µ P tλn uně1 . In the general case, fix t0 ą 0 and let ηp‰ 0q P σpUt0 pAqq; since Ut0 pAq is compact, this η must be an eigenvalue. We denote the corresponding eigenvector by v. Then for any t ě 0, we have Ut0 pAqUt pAqv “ Ut pAqUt0 pAqv “ η Ut pAqv , which implies that the subspace Hη :“ kerpUt0 pAq ´ η1q is invariant under the semigroup. Since Ut0 pAq is compact, any proper subspace Hη must be finitepηq dimensional. Let tUt :“ Ut pAq æ Hη utě0 be the restriction of the semigroup pηq tUt pAqutě0 to the subspace Hη . Then tUt utě0 is a strongly continuous semigroup acting on a finite-dimensional space, and hence, its generator K is a bounded linear operator with dom K “ Hη . Since tUtη “ e´tK utě0 acts on the finite-dimensional pηq space Hη , the spectra σpUt q and σpKq are non-empty, discrete, and are related pηq by σpUt q “ e´tσpKq . Let λ P σpKq. There exists an eigenvector fλ P Hη such that pηq Ut fλ “ e´tλ fλ , t ě 0. (4.20) pηq

Since Ut

“ Ut pAq æ Hη , (4.20) implies Ut pAqfλ “ e´tλ fλ ,

t ě 0.

(4.21)

Thus, the eigenvalues of tUt pAqutě0 indeed have an exponential form, in particular, η “ e´t0 λ . From the arguments developed for this case, we get that λ P σpAq “ tλn uně1 , with fλ being the corresponding eigenvector. Since when dim H “ 8, the point t0u always belongs to the spectrum of a compact operator, this completes the proof of the representation (4.17). l


111

Although in general for compact operators one has t0u Ď σess , (Appendix A, Section A.6), in the next statement the essential spectrum shrinks to its continuous part. Proposition 4.22. Let A be a densely defined generator of a strongly continuous semigroup tUt pAqutě0 on a Hilbert space H. (a) tUt pAqutą0 is self-adjoint and compact if and only if the operator A is selfadjoint and has a compact resolvent Rζ pAq for some ζ P ρpAq. (b) In this case, there is an orthonormal basis tϕn uně1 in H consisting of eigenvectors of A with real eigenvalues tλn uně1 satisfying minně1 λn “ ´ω0 , such that we have the representation ÿ e´tλn pu, ϕn qϕn u P H, t ě 0. Ut pAqu “ (4.22) ně1

(c) For dim H “ 8 and t ą 0, the spectrum of Ut pAq can be written as σpUt pAqq “ t0u Y te´tσpAq u, where t0u “ σcont pUt pAqq and te´tσpAq u “ σp pUt pAqq. In this case the self-adjoint operators Ut pAq are compact for t ą 0. Proof. (a) Since the semigroup tUt pAqutě0 is strongly continuous, it is quasibounded with generator A P QpM, ω0 q, see Section 1.3. Suppose that Ut pAq is compact for t ą 0 and self-adjoint. If ω0 ă λ, then by Proposition 4.17, the resolvent ż 8

R´λ pAq “ pA ` λ1q´1 “

dt e´λt Ut pAq,

(4.23)

0

is compact and the integral converges in the operator-norm topology. Since Ut pAq is self-adjoint for all t ě 0, the latter implies that R´λ pAq is also self-adjoint. For real λ we have R´λ pAq˚ “ R´λ pA˚ q, and therefore A “ A˚ . Then the resolvent identity implies that Rz pAq is compact for any z P ρpAq. Conversely, assume that A ě ´ω0 1 is self-adjoint and has a compact resolvent. Since the numerical range NrpA ` ω0 1q Ď R` 0 , by Proposition 1.33 and Corollary 1.47, the operator A P H pπ{2, ω0 q, i.e., it is the generator of a holomorphic semigroup. Holomorphic semigroups are norm-continuous for t ą 0, see Corollary 1.29, so Proposition 4.17 implies that Ut pAq is compact for t ą 0. (b) Since R´λ pAq P C8 pHq for ω0 ă λ, by Corollary 4.10, we have ker R´λ pAq “ H, i.e., R´λ pAq ‰ 0 and t0u Ă σess pR´λ pAqq. Since the operator R´λ pAq is non-zero and self-adjoint, the set σpR´λ pAqqzt0u is nonempty, by consequence, also σpAq is nonempty. Furthermore, σpR´λ pAqqzt0u “ σp pR´λ pAqq “ tpλn ` λq´1 : λn P σpAqu,

(4.24)

see Corollary 4.10. By Corollary 2.25 and Corollary 2.26, the normalised eigenvectors tϕn uně1 corresponding to the point spectrum (4.24) of the self-adjoint


112

compact operator R´λ pAq form an orthonormal basis in H, i.e., ÿ u“ pu, ϕn qϕn , ně1

R´λ pAqu “

ÿ

(4.25)

pλn ` λq´1 pu, ϕn qϕn ,

ně1

for any u P H, where R´λ pAqϕn “ pλn ` λq´1 ϕn . Since ϕn P dom A, the latter implies that tϕn u are eigenvectors of the self-adjoint operator A: Aϕn “ λn ϕn , with real eigenvalues tλn uně1 “ σpAq. By virtue of Proposition 4.21(b), tϕn u are also eigenvectors of the elements of the semigroup tUt pAqutě0 with eigenvalues te´tλn uně1 . Since tϕn uně1 is a basis in H, we get the representation (4.22). By Proposition 4.21(a), the spectrum σpAq has no accumulation point in R, and denoting λ´ :“ minně1 λn , expression (4.22) becomes ÿ e´tpλn ´λ´ q pu, ϕn qϕn . Ut pAqu “ e´tλ´ ně1

Hence, }Ut pAqu} ď e´tλ´ }u},

u P H.

(4.26)

Since A P QpM, ω0 q, the estimate (4.26) implies ´λ´ ď ω0 . On the other hand, Ut pAqϕn “ e´tλn ϕn , i.e., Ut pAqϕ´ “ e´tλ´ ϕ´ , so that }Ut pAq} ě e´tλ´ ,

ln }Ut pAq} ě ´λ´ , tÑ`8 t lim

and we infer that, λ´ “ ´ω0 , by virtue of (4.26). (c) If dim H “ 8, the point t0u belongs to the spectrum of Ut pAq P C8 pHq for t ą 0. The structure of σpUt pAqq has already been established in Proposition 4.21(b). It only remains to classify the point t0u. By virtue of representation (4.22), the relation 0 “ }Ut pAqu}2 “

8 ÿ

e´2tλn |pu, ϕn q|2 ,

t ě 0,

u P H,

n“1

implies u “ 0. Therefore, ker Ut pAq “ H, that is, 0 R σp pUtą0 pAqq. Since Ut pAq is self-adjoint, we have H “ ker Ut pAq ‘ ran Ut pAq, and by the remark just above, H “ ran Ut pAq. Hence, by Definition 2.14 (cf. Appendix A, Section A.1), we see that 0 R σres pUt pAqq. Consequently, the nonisolated point zero belongs to the continuous spectrum and we infer that in fact t0u “ σcont pUt pAqq. l Since C1 pHq Ă C8 pHq, our observations concerning compact semigroups, i.e. Propositions 4.17–4.22, are obviously applicable to Gibbs semigroups. More details on the spectral mapping theorem for semigroups one finds in Section A.9.


113

We conclude this section with a version of the minimax principle (2.20) for unbounded self-adjoint operators. It is useful for the identification of generators of compact self-adjoint semigroup. Proposition 4.23. For an unbounded self-adjoint operator A with a dense domain dom A Ă H, we define µpMn q :“

sup

pAu, uq,

(4.27)

uPMn Ădom A }u}“1

where Mn is a finite-dimensional subspace with dim Mn “ n ě 1. If we set µn pAq :“

inf

Mn Ădom A

(4.28)

µpMn q,

then (a) µ1 pAq ą ´8 if and only if A is semibounded from below. (b) If µ1 pAq ą ´8 and µn pAq Ñ `8, for n Ñ 8, then σpAq is a point spectrum with finite multiplicities, that is, σpAq “ σp pAq. (c) If µ1 pAq ą ´8 and σpAq “ σp pAq with finite multiplicities, then µn pAq “ λn pAq, where tλn pAquně1 is the set of eigenvalues in nondecreasing order and repeated according to the multiplicity. Proof. (a) If A ě ´γA 1, then µ1 pAq ě ´γA , by definitions (4.27) and (4.28). Conversely, ´γA ď µ1 pAq “ “

sup

inf

M1 Ădom A

pAu, uq

uPM1 Ădom A }u}“1

inf pAu, uq.

uPdom A }u}“1

(b) If σpAq “ σp pAq and some λs pAq P σp pAq has infinite multiplicity, the corresponding subspace Hs Ă dom A of eigenvectors is infinite-dimensional. Hence, psq psq there is a sequence of subspaces Mm Ă Hs , with dim Mm “ m, such that psq µpMm q “ λs pAq ă `8,

for m Ñ 8,

which contradicts the hypothesis. Since for a self-adjoint operator A, the residual spectrum σres pAq “ H, it only remains to show that σcont pAq “ H (Appendix A). Suppose that there is a λ0 P σcont pAq. Then λ0 would be a non-isolated point of continuous nonconstancy of the spectral measure EA pλq, that is, EA p∆0 pεqq :“ EA pλ0 ` εq ´ EA pλ0 ´ εq ‰ 0,

∆0 pεq “ pλ0 ´ ε, λ0 ` εq,

such that dim EA p∆0 pεqqH “ 8,

for any ε ą 0.

(4.29)


114

Now, considering the subspaces Mm Ă EA p∆0 pεqqH Ă dom A, we get a contradiction, since ż λ0 `ε µpMm q “ sup pdEA pλqu, uq, λ ď |λ0 ` ε| , uPMm Ădom A }u}“1

λ0 ´ε

is bounded for m Ñ 8, whereas µm pAq Ñ 8 (4.28) for m Ñ 8 by conditions in (b). (c) Let tλn pAquně1 be the nondecreasing sequence of the eigenvalues of A repeated according to multiplicity, and let tϕn u be the corresponding family of orthonormal eigenvectors. We denote by Fn :“ rϕ1 , . . . , ϕn s the space spanned by the first n eigenvectors. From Definitions (4.27) and (4.28), we obviously get µn pAq “

inf

Mn Ădom A

µpMn q ď µpFn q “ λn pAq.

(4.30)

˜ n such that To prove the equality in (4.30), suppose that there is a subspace M the strict inequality holds: ˜ n q ă µpFn q “ λn pAq . µpM

(4.31)

˜ n is finite-dimensional, there exists a number N large Since the subspace M ˜ n Ă FN . Counting then the dimensions, we see that enough such that M ˜ n X rϕn , ϕn`1 , . . . , ϕN s ‰ H, i.e., M ˜ n and the space spanned by tϕj uN M j“n ˜. Therefore, contain a common vector u ˜ n q ě pA˜ µpM u, u ˜q “

N ÿ

u, ϕj q|2 ě λn pAq, λj pAq|p˜

j“n

which contradicts the strict inequality (4.31).

l

Corollary 4.24. If A and B are densely defined self-adjoint operators in H such that A ď B, i.e., pAu, uq ď pBu, uq for u P dom B Ď dom A, then Proposition 4.23 (see (4.27) and (4.28)) implies that λn pAq ď λn pBq for n ě 1.

4.3 Generators We start by observing that an operator A P QpM, ω0 q with a compact resolvent Rζ pAq, ζ P ρpAq, is not necessarily the generator of a compact semigroup, Proposition 4.17. On the other hand, Proposition 4.22 (a) implies that the compactness of the resolvent Rζ pAq, ζ P ρpAq, of a self-adjoint operator A ě ´ω0 1 is a sufficient and necessary condition for the compactness of the semigroup tUt pAqutě0 generated by A. Hence, the generators of self-adjoint compact semigroups are exhaustively characterised by the property Rz pAq P C8 pHq for some z P ρpAq. Based on Proposition 4.22, we can formulate a sufficient condition on the generator of a self-adjoint strongly continuous semigroup ensuring that this semigroup is a Gibbs semigroup.

4.3. Generators

115

Proposition 4.25. A self-adjoint operator A ě ´ω0 1 is the generator of a Gibbs semigroup tGt pAqutě0 if the resolvent Rζ pAq P Cp pHq for some ζ P ρpAq and 1 ď p ă 8. Proof. Since A is the generator of a strongly continuous semigroup, that is, A P QpM, ω0 q, by hypothesis and by Proposition 4.22, we get the representation (4.22). Hence, e´tλn “ sn pe´tA q are the singular values of the compact semigroup te´tA utě0 and, by Lemma 2.36, we have }e´tA }1 “

8 ÿ

e´tλn ,

t ą 0.

(4.32)

n“1

By expression (4.25), we get that pλn ` λq´1 “ sn pR´λ pAqq are the singular values of the self-adjoint operator R´λ pAq P Cp pHq, λ ą ω0 . For the resolvent R´λ pAq P Cp pHq, its } ¨ }p -norm is well defined, see Definition 2.50, and the series 8 ”ÿ ı1{p “ }R´λ pAq}p pλn ` λq´p n“1

is convergent. This implies that the sum in (4.32) is convergent for any t ą 0. l This observation motivates the introduction of the following class of generators of Gibbs semigroups. Definition 4.26. A quasi-m-sectorial operator A with vertex γ “ ´ω0 and with semi-angle α is called a p-generator if for some ζ0 P ρpAq, the resolvent Rζ0 pAq belongs to the von Neumann-Schatten class Cp pHq for a finite p ě 1. A self-adjoint operator A satisfying the criterion of Proposition 4.25 is a p-generator. Proposition 4.27. Any p-generator A is the generator of a holomorphic Gibbs semigroup tGz pAquzPSπ{2´α . Proof. Proposition 1.46 implies that the operator A´ω0 “ A`ω0 1 is the generator of a holomorphic semigroup, that is, A´ω0 P H pπ{2´α, 0q and tUz pA´ω0 quzPSπ{2´α is a contraction holomorphic semigroup in the open sector Sπ{2´α “ tz P C : | arg z| ă π{2 ´ αu. Hence, for this semigroup we can use representation (1.67) ż 1 Uz pA´ω0 q “ dζ e´zζ pζ1 ´ A´ω0 q´1 , (4.33) 2πi Γ where Γ is a positively oriented contour enclosing the sector Sα Ą NrpA´ω0 q, i.e., Γ Ă ρpAq. Since the resolvent identity implies that Rζ pA´ω0 q P Cp pHq for any ζ P ρpAq, whenever Rζ0 pA´ω0 q P Cp pHq, we assume that ζ0 P Γ. Then formula (4.33) yields ż 1 }Uz pA´ω0 q}p ď |dζ| e´ <epzζq }Rζ0 pA´ω0 q}p 2π Γ ˆ p1 ` |ζ0 ´ ζ|}Rζ pA´ω0 q}q , z P Sπ{2´α . (4.34)


116 By estimate (1.83), }Rζ pA´ω0 q} ď

Mε , |ζ|

(4.35)

for ζ P CzSα`ε and ε ą 0. Therefore, we can choose the contour Γ Ă CzSα`ε in such a way that }RζPΓ pA´ω0 q} ď L, and there is δ ą 0 ensuring that <epzζq ě δ|z||ζ|,

for z P Sπ{2´α , ζ P Γ X C` .

(4.36)

The estimates (4.34)–(4.36) yield ˜ }Rζ pA´ω q}p , }Uz pA´ω0 q}p ď L 0 0

z P Sπ{2´α .

By Proposition 4.3, this estimate implies that Uz pA´ω0 q “: Gz pA´ω0 q P C1 pHq is a Gibbs semigroup for z P Sπ{2´α . Since by Corollary 1.28, the holomorphic semigroup Uz pA´ω0 q is } ¨ }differentiable in the sector Sπ{2´α , and since C1 pHq is the ˚-ideal in LpHq, for z, z ` ∆z P Sπ{2´α the derivative ˘ 1 ` Gz`∆z pA´ω0 q ´ Gz pA´ω0 q (4.37) ∆zÑ0 ∆z ˘ 1 ` Gz{2`∆z pA´ω0 q ´ Gz{2 pA´ω0 q Gz{2 pA´ω0 q “ lim ∆zÑ0 ∆z “ Á Gz pAq P C1 pHq ,

Bz Gz pA´ω0 q “ lim

exists in the trace-norm by the } ¨ }1 -continuity of multiplication, Proposition 2.78. Hence, the Gibbs semigroup tGz pA´ω0 quzPSπ{2´α is } ¨ }1 -holomorphic in the sector Sπ{2´α . This implies that the same is true for the Gibbs semigroup with generator A, tGz pAquzPSπ{2´α :“ teω0 z Gz pA´ω0 quzPSπ{2´α , and completes the proof.

l

Remark 4.28. The example in Remark 4.4 shows that the conditions of Proposition 4.27 are suitable for strongly continuous and immediate Gibbs semigroups. Indeed, the self-adjoint operator A with σpAq “ tlnpn`1quně1 is not a p-generator. The C0 -semigroup tTt utě0 defined by (4.3) is only an eventually Gibbs semigroup with the threshold t0 “ 1. On the other hand, the semigroup tGt pAqutě0 defined as: tGt“0 pAq “ 1u _ tGt pAq :“ Tt`1 utą0 , is an immediate, but degenerate Gibbs semigroup since limtÓ0 Gt pAq ‰ 1. To continue we mention a contact with construction of generators by sesquiliniar forms. We develop this approach below (see, e.g., Section 5.5) in a framework of perturbations and in analytic continuations of forms and the associated operators.

4.3. Generators

117

Remark 4.29. Let A be an m-sectorial operator with vertex γ “ 0 and semi-angle α P r0, π{2q, see Definition 1.41. We recall that there is a one-to-one correspondence between the set of all m-sectorial operators and the set of all densely defined, closed sesquilinear sectorial forms. We denote the form corresponding to the operator A by aru, vs “ <e aru, vs ` i =m aru, vs , u, v P dom a . 1 Here <e a :“ 21 paà˚ q and =m a :“ 2i paá˚ q are the real and the imaginary parts ˚ of a. The adjoint form a ru, vs :“ arv, us, dom a˚ “ dom a, which corresponds to the operator A˚ , is in turn densely defined, closed and sectorial. The form a is called symmetric if aru, vs “ a˚ ru, vs. Since by definition the symmetric form <e aru, vs is closed and γ “ 0, the representation theorem for non-negative forms shows that it defines a non-negative self-adjoint operator AR :“ <e A ě 0 with dom AR Ă dom a, which is called the real part of A. Recall that AR “ 12 pA ` A˚ q, if A P LpHq, but this is not true in general for unbounded operators. Since the sum pa ` bqru, vs of the closed forms corresponding to two m-sectorial operators . A and B is also closed, one calls the m-sectorial operator pA ` Bq associated via representation theorem with pa ` bq, the form-sum of the operators A and B. This definition yields for unbounded case a non-negative self-adjoint operator ? . <e A :“ 12 pA ` A˚ q, with dom <e A “ dom a.

We refer to Section 5.5 for more details about sesquilinear sectorial forms and their properties that we use below in this chapter. By Proposition 1.46, any m-sectorial operator A with vertex γ “ 0 and semi-angle α P r0, π{2q is the generator of a contraction holomorphic semigroup: A P H pπ{2 ´ α, 0q. Then one gets a relation between generators A and <e A in the case of the Gibbs semigroups. Proposition 4.30. Let A P H pπ{2 ´ α, 0q be an m-sectorial operator with real part <e A ě 0. If e´t <e A P C1 pHq for t ą 0, then e´tA P C1 pHq for t P Sθ , θ “ π{2 ´ α, and p

p

p`1

}e´tA }1 ď ¨ ¨ ¨ ď }e´tA{2 }22p ď }e´tA{2

2p`1

}222p`1 p

p

ď ¨ ¨ ¨ ď lim }e´tA{2 }22p “ }e´t <e A }1 . (4.38) pÑ8

Proof. First we note that since <e A ě 0 and e´t <e A P C1 pHq, ż8 dt e´tp1`<e Aq P C8 pHq,

p1 ` <e Aq´1 “

(4.39)

0

because the right-hand side operator can be arbitrary well approximated in the operator norm by finite-rank operators. By Remark 5.65, this implies that msectorial generators A and A˚ also have compact resolvents. Since dom a “ H for the sesquilinear form a corresponding to A, this ensures that the Trotter product


118 formula

1 ˚

} ¨ }- lim pe´tA

˚

{2n ´tA{2n n

q “ e´tpA

e

nÑ8

.

Àq{2

,

t P Sθ ,

(4.40)

converges in the operator-norm topology to the semigroup generated by half of the form-sum of A and A˚ . By virtue. of Remark 4.29, it coincides with non-negative self-adjoint generator <e A “ pA ` A˚ q{2 ě 0. Secondly, by Lemma 2.56, the operator-norm convergence of the Trotter product formula (4.40) implies the convergence of all singular values of the compact operators involved in expression (4.40), i.e., for j “ 1, 2, 3, . . . , we have ˚

sj ppe´tA

q q ÝÝÝÑ sj pe´t <e A q .

{2n ´tA{2n n

e

nÑ8

(4.41)

Furthermore, by the inequalities (3.26) in Corollary 3.8, Sk ppq :“

k ÿ

p

p

sj pe´tA{2 q2 ď

j“1

k ÿ

p`1

sj pe´tA{2

q2

p`1

“: Sk pp ` 1q .

(4.42)

j“1

The limit (4.41) yields the convergence of any finite sum in (4.42), Sk :“ lim Sk ppq “ pÑ8

k ÿ

sj pe´t <e A q ,

(4.43)

j“1

thus, (4.42) and (4.43) give k “ 1, 2, . . . .

Sk ppq ď Sk pp ` 1q ď ¨ ¨ ¨ ď Sk ,

Hence, the series tSk ppqukě1,pě0 is monotonically increasing in k and p. Since e´t <e A P C1 pHq for t ą 0, this series is bounded by supkě1 Sk “ }e´t <e A }1 , see (4.43). In particular, lim Sk p0q “ }e´tA }1 (4.44) kÑ8

and

p

p

lim Sk ppq “ }e´tA{2 }22p .

kÑ8

(4.45)

Therefore, taking the limit k Ñ 8 in the inequalities Sk p0q ď ¨ ¨ ¨ ď Sk ppq ď Sk pp ` 1q ď ¨ ¨ ¨ ď Sk ď }e´t <e A }1 , we get in view of the limits (4.44) and (4.45) the announced inequalities (4.38) and p p (4.46) lim }e´tA{2 }22p ď }e´t <e A }1 . pÑ8

1 See

Chapter 5 (Section 5.5, Propositon 5.75).

4.3. Generators

119

On the other hand, by (4.42), we also have p

p

}e´tA{2 }22p ě Sk ppq, or, see (4.43), p

p

lim }e´tA{2 }22p ě sup Sk “ }e´t <e A }1 ,

pÑ8

kě1

which together with the opposite inequality (4.46) imply the equality in (4.38). l Corollary 4.31. If the generator A satisfying the conditions of Proposition 4.30 is normal, all the inequalities in (4.38) become equalities. Proof. If the generator A is normal, then the operator e´tA is also normal and thus sj pe´tA q “ |λj pe´tA q|. Since the corresponding spectral representations take the form A“

8 ÿ

e´tA “

λj pAqPj ,

j“1

8 ÿ

e´tλj pAq Pj ,

<e A “

j“1

8 ÿ

<e λj pAqPj ,

j“1

where Pj are the orthogonal spectral projectors associated with the eigenvalues λj pAq, we obtain: sj pe´tA q “ |λj pe´tA q| “ |e´tλj pAq | “ e´t <e λj pAq “ e´tλj p<e Aq “ sj pe´t <e A q, which implies }e´tA }1 “ }e´t <e A }1 in (4.38).

l

Corollary 4.32. An m-sectorial operator A with positive real part <e A ě 0 is the generator of a contraction holomorphic Gibbs semigroup tGz pAquzPSπ{2´α if e´t <e A P C1 pHq, for t ą 0. Proof. By Proposition 1.46, the operator A P H pπ{2 ´ α, 0q, that is, the semigroup te´zA uzPSπ{2´α is holomorphic, and it is a contraction. By Proposition 4.30, we have e´tA P C1 pHq, t ą 0. Since C1 pHq is a ˚-ideal in LpHq, in view of the semigroup property, we get e´zA P C1 pHq for z P Sπ{2´α . By the same reason, the } ¨ }1 -derivative of e´zA exists in this sector. Hence, te´zA “ Gz pAquzPSπ{2´α is a contraction } ¨ }1 -holomorphic Gibbs semigroup. l Corollary 4.33. Let A be an m-sectorial operator with real part <e A ě 0. If e´t <e A P C1 pHq for t ą 0, then the Trotter product formula (4.40) converges in the trace-norm topology: ˚

} ¨ }1 - lim pe´tA pÑ8

{2p ´tA{2p 2p´1

e

q

“ e´t <e A ,

t P Sθ .

(4.47)


120

Proof. Let t ą 0. By the definition of singular values (2.11) and Definition 2.50 one gets p p´1 p p ˚ p }e´tA{2 }22p “ }pe´tA {2 e´tA{2 q2 }1 . Since by (4.38) the left-hand side converges, we obtain ˚

lim }pe´tA

{2p ´tA{2p 2p´1

pÑ8

e

q

}1 “ }e´t <e A }1 .

(4.48)

On the other hand, (4.40) yields the operator-norm convergence ˚

lim }pe´tA

pÑ8

{2p ´tA{2p 2p´1

e

q

´ e´t <e A } “ 0.

(4.49)

Then by virtue of Proposition 2.69 (or Corollary 2.76) the limits (4.48) and (4.49) imply assertion (4.47). l This result is also a corollary of a more general Proposition 5.78 about Trotter product formula for Gibbs semigroups.

4.4 P-perturbations of generators In this section we study a class of infinitesimally small unbounded perturbations, called P-perturbations. Definition 4.34. Recall that a closed operator B belongs to the class of Pperturbations of the C0 -semigroup generator A if ď (4.50) Ut pAqH , dom B Ě piq tą0

ż1 dt }B Ut pAq} ă 8 .

piiq

(4.51)

0

We denote this property as B P PpAq, or abbreviate to B P P, if there is no danger of confusion. Remark 4.35. For a closed B the condition (i) implies that the product BUt pAq P LpHq, for all t ą 0 by the closed graph theorem. Note that then the function t ÞÑ B Ut pAq is strongly continuous away from zero. This implies that the function t ÞÑ }B Ut pAq} is measurable, locally bounded, and, hence, strongly integrable on any finite interval I Ă R` . The integral in (ii) is understood in the sense Ñ 0` for the lower limit. Finally, since I is a separable set, the range of any strongly continuous operator-valued function is a separable subset of LpHq. Hence, the corresponding integral exists in the sense of Bochner. Remark 4.36. In general, for a quasi-bounded semigroup tUt pAqutě0 the image Ut pAqH is not included in dom A, for t ě 0, see discussion in Chapter 1. However,

4.4. P-perturbations of generators

121

it is true that Ut pAq : dom A Ñ dom A, for t ą 0 by Propositions 1.1 and 1.5. If instead of (4.50), we consider the domain ď (4.52) Ut pAqdom A , D :“ tą0

then D Ě dom A, and for B P PpAq one also has dom B Ě D. Since the domain D is by construction invariant under the semigroup tUt pAqutě0 , for any u P dom A there exists a sequence tun uně1 Ă D such that limnÑ8 }u ´ un } “ 0, i.e., D is a core of A, see Proposition 1.13. Therefore, dom B contains a core A “ D. The next assertion says more. Lemma 4.37. Let B P PpAq for generator A P QpM, ω0 q of C0 -continuous quasibounded semigroup tUt pAqutě0 . Then (4.53)

dom B Ě dom A, and for any ε ą 0 there exists λε ą ω0 such that }BR´λ pAq} ă ε ,

(4.54)

for all λ ą λε . Proof. By virtue of Proposition 1.12 and condition (4.50) of Definition 4.34, we have: }BUt pAq} ď }BUt“1 pAq } M eω0 pt´1q , t ą 1. Hence, by condition (4.51) of Definition 4.34, ż8 dt }BUt pAqe´λt } ă 8, 0

for all λ ą ω0 . Moreover, for any ε ą 0, there is a constant λε ą ω0 such that ›ż 8 › › › (4.55) › dt BUt pAqe´λt u› ă ε }u}, u P H, 0

for all λ ą λε . Since ż8 dt Ut pAqe´λt u “ pA ` λ1q´1 u,

u P H,

0

cf. (1.46), the closedness of B and the bound (4.55) yield R´λ pAqu P dom B. Thus, l they imply (4.53) and (4.54). Corollary 4.38. By virtue of (4.54) }Bu} ď ελε }u} ` ε}Au}, u P dom A Ă dom B,


122

for any ε ą 0. This means that B P PpAq is an infinitesimally Kato-small perturbation with the relative A-bound equals to b “ 0` , Section 1.7. By Remark 1.49 and Definition 1.50 we infer that B P P0` pAq, or that the class of unbounded perturbations PpAq Ă P0` pAq. Taking into account (1.102) one gets the hierarchy P0 Ă PpAq Ă P0` pAq Ă Pb pAq .

(4.56)

Here P0 denotes the class of bounded perturbations Pb“0 pAq. If B P PpAq, then the operator H :“ A ` B has dom H “ dom A and it is closed, by the stability of closeness under relatively small perturbations Pbă1 pAq. In Section 5 we consider other classes of infinitesimally small unbounded perturbations of generators such that the relative bound b “ 0` . One of this classes consists of the relatively compact perturbations (Remark 5.31), while another one consists of perturbations verifying certain fractional power conditions, Remark 5.48. Before proceeding with the construction of C0 -semigroups for perturbations of class P we need a preparatory lemma. Lemma 4.39. Suppose the densely defined operator A is the generator of a C0 semigroup tUt pAqutě0 on a Hilbert space H and H “ A ` B is the generator of the C0 -semigroup tUt pA ` Bqutě0 with a bounded perturbation B P P0 . Then this semigroup satisfies the following two integral equations: żt Ut pA ` Bqu “ Ut pAqu ´

ds Ut´s pA ` Bq B Us pAq u,

(4.57)

ds Ut´s pAq B Us pA ` Bq u,

(4.58)

0 żt

Ut pA ` Bqu “ Ut pAqu ´ 0

for every t ě 0 and u P H. Taking advantage of the uniform boundedness of integrands the integrals are understood in the operator-norm sense. Proof. Since dom A “ dom pA`Bq is invariant under the perturbed tUt pA`Bqutě0 and nonperturbed tUt pAqutě0 semigroups, the function r0, ts Q s ÞÑ Wsp1q v :“ Ut´s pA ` Bq Us pAqv ,

(4.59)

is continuously differentiable for any v P dom A. By the fundamental semigroup equation (1.30) one obtains Bs Wsp1q v :“ Ut´s pA ` Bq B Us pAqv .

(4.60)

Then after integration for any v P dom A, definitions (4.59) and (4.60) yield żt Ut pA ` Bqv ´ Ut pAqv “ ´

ds Ut´s pA ` Bq B Us pAq v . 0

(4.61)


123

Since dom A is dense in H, the boundedness of operators in the left- and righthand sides of (4.61) ensures that the integral equation (4.57) holds on H in the operator-norm sense. The same arguments, applied to the family r0, ts Q s ÞÑ Wsp2q v :“ Ut´s pAq Us pA ` Bqv P H, yield the integral equation (4.58).

(4.62) l

The integral formulae (4.57), (4.58) are a source of the semigroup-based method for the construction of perturbed semigroups. In contrast to the resolventbased method, it involves the Dyson-Phillips series, see Notes in Section 1.8 and in Section 4.7. In the next Proposition 4.41 we use the semigroup-based method to treat the P-perturbations of quasi-bounded semigroups. According to Lemma 4.39, in order to construct the perturbed semigroup tUt pA ` Bqutě0 one can use the solutions of either of the integral equations (4.57), or (4.58), and then check that they verify all the properties of C0 -semigroups. Remark 4.40. To proceed, we rewrite (4.57) in the operator-valued function space Ft :“ Cpr0, ts, LpHqq,

(4.63)

such that for each element Φ P Ft and any u P H the function r0, ts Q s ÞÑ Φs u is continuous in the topology of H. This space endowed with the norm topology }Φ}8,t :“ sup }Φs },

Φ P Ft ,

sPr0,ts

is a Banach space. Motivated by (4.57) we define on (4.63) the operator Vp1q : Φ ÞÑ Vp1q Φ such that for each r P r0, ts we have żr p1q pV Φqprq :“ ds Φr´s B Us pAq. (4.64) 0

By virtue of condition (4.51), the Volterra-type operator (4.64) on the space Ftď1 is a bounded operator: żt p1q p1q }V }Ft “ sup (4.65) }V Φ}8,t ď ds }BUs pAq}. tΦPFt :}Φ}8,t “1u

0

Moreover, there exists τ ą 0 such that in the space Fτ ă1 the norm (4.65) obeys the estimate żτ }Vp1q }Fτ ď ds }BUs pAq} “: ξ ă 1. (4.66) 0

Therefore, p1`Vp1q q P LpFτ q is invertible and the solution of the integral equation (4.57) for t P r0, τ s can be defined in the space Fτ as Up¨q pA ` Bq “ p1 ` Vp1q q´1 Up¨q pAq.

(4.67)


124

Similarly, we define on Ft the operator Vp2q : Φ ÞÑ Vp2q Φ such that for each r P r0, ts żr pVp2q Φqprq :“ ds Ur´s pAq B Φs . (4.68) 0

Then by condition (4.51), the operator (4.68) in the space Ftď1 is bounded żt }Vp2q }Ft “

}Vp2q Φ}8,t ď

sup tΦPFt :}Φ}8,t “1u

ds }Us pAqB},

(4.69)

0

and there exists τ ą 0 such that in the space Fτ ă1 the norm (4.69) obeys the estimate żτ p2q }V }Fτ ď (4.70) ds }BUs pAq} “: η ă 1. 0 p2q

Consequently, p1 ` V q P LpFτ q is invertible and we can define the solution of the integral equation (4.58) in the space Fτ by Up¨q pA ` Bq “ p1 ` Vp2q q´1 Up¨q pAq .

(4.71)

Proposition 4.41. Let B P PpAq for a generator A P QpM, ω0 q. Then dom A Ă dom B and the operator sum H :“ A ` B, with dom H “ dom A, is the generator of a strongly continuous quasi-bounded semigroup on H. Proof. Let t P r0, τ s, where τ is defined by condition (4.66). Then by definition (4.64) the right-hand side of representation (4.67) is the convergent in Fτ infinite series 8 ÿ (4.72) Sn ptq, pp1 ` Vp1q q´1 Up¨q pAqqptq “ n“0

where the terms in the sum (4.72) are defined by the recurrence relation S0 ptq “ Ut pAq, żt Sn ptq “ ´ ds Sn´1 pt ´ sqBUs pAq,

n ě 1.

(4.73)

0

Indeed, since by (4.73) the operators Sně1 ptq are the n-fold operator-norm convergent Bochner integrals ż s1

żt Sn ptq “

ds2 . . .

ds1 0

ż sn´1

0

dsn 0

Ut´s1 pAqp´BqUs1 ´s2 pAq ¨ ¨ ¨ Usn´1 ´sn pAqp´BqUsn pAq, (4.74) the estimate (4.66) implies that for 0 ď t ď τ }Sn ptq} ď M eω0 t ξ n ,

n ě 1.

(4.75)


125

Therefore, (4.72) converges in the LpHq operator norm, and }Ft } ď M eω0 t p1 ´ ξq´1 ,

0ďtďτ,

(4.76)

where we denote the series in (4.72) by Ft :“

8 ÿ

Sn ptq .

(4.77)

n“0

Now we proceed with the verification of the (semigroup) properties of the solution tFt utďτ , which is the operator-norm convergent Dyson-Phillips series (4.77). (1) For any u P H one gets the immediate estimate }pFt ´ 1qu} ď }pUt pAq ´ 1qu} ` }

8 ÿ

Sn ptq}}u}.

n“1

For the first term of this estimate we obtain limtÑ`0 }pUt pAq ´ 1qu} “ 0. To make evident the time dependence of the sum, we use (4.74) to refine the upper bound of (4.75) as follows: n „ż t 8 8 ÿ ÿ (4.78) Sn ptq} ď } M eω0 τ ds}B Us pAq} . n“1

0

n“1

Then by Definition (4.34)(ii) and a comment in Remark 4.35, we have żt lim ds}B Us pAq} “ 0 . tÑ`0 0

So, the solution (4.77) is strongly continuous at t “ `0: lim }pFt ´ 1qu} “ 0, tÓ0

u P H.

(4.79)

(2) Let 0 ď t ď τ . By virtue of (4.73), for any u P H the sum (4.77) can be rearranged as follows: żt 8 ÿ Sn´1 pt ´ sq B Us pAqu Ft u “ Ut pAqu ´ ds 0

n“1

żt ds Ft´s B Us pAqu ,

“ Ut pAqu ´

0ďtďτ.

(4.80)

0

Thus, the family tFt utďτ satisfies the integral equation (4.57). Similarly, with the help of (4.74) for any u P H the sum (4.77) can be rearranged differently: żt 8 ÿ Ft u “ Ut pAqu ´ ds1 Ut´s1 pAq B Sn´1 ps1 qu 0

n“1

żt ds Ut´s pAq B Fs u ,

“ Ut pAqu ´ 0

0ďtďτ.

(4.81)


126

Therefore, the family tFt utďτ satisfies also the integral equation (4.58). (3) Both equations (4.80) and (4.81) can be extended to any t ě 0. Suppose for example that (4.80) is valid for 0 ď t ď nτ . Let nτ ă t1 ď pn ` 1qτ , then Ft1 “ Fτ Ft1 ´τ ż t1 ´τ ´ ¯ “ Fτ Ut1 ´τ pAq ´ ds Ft1 ´τ ´s BUs pAq 0

ż t1 ´τ

żτ ds Fτ ´s BUs pAqUt1 ´τ pAq ´

“ Uτ Ut1 ´τ ´ 0

ds Ft1 ´s BUs pAq 0

ż t1 ds Ft1 ´s BUs pAq,

“ Ut1 ´ 0

and by induction, (4.80) holds for all t ě 0. The same is true for (4.81). (4) Although the above arguments bring forward the semigroup functional equation (Definition 1.2(b)) for the family tFt utě0 , it is necessary to check this property for any choice of t1 , t2 ě 0. Taking into account (4.74) and the representation (4.77), one can verify by a direct multiplication of the series Ft1 and Ft2 for t1 , t2 ě 0 with t1 ` t2 ď τ , that Ft1 Ft2 “ Ft2 Ft1 “ Ft1 `t2 .

(4.82)

Then due to (3) this property can be extended to t ě τ by putting Ft “ pFτ qn Ftńτ ,

nτ ă t ď pn ` 1qτ .

(4.83)

Summarising (1)–(4), or (4.76)–(4.83), we conclude that the family tFt utě0 is in fact a quasi-bounded C0 -semigroup with M 1 “ M p1 ´ ξq´1 and ω01 “ ω0 . (5) Finally, with the help of equation (4.80) we shall determine the Ť generator of the semigroup tFt utě0 . To this end, we consider the domain D “ tą0 Ut pAqdom A, (4.52). If v P D, there exist a vector u P dom A and a constant τ ą 0 such that v “ Uτ pAqu. By virtue of (4.80) and by the definition of the semigroup generator, we get for B P P0` 1 1 lim p1 ´ Ft qv “ lim p1 ´ Ut pAqqv tÓ0 t tÓ0 t ż 1 t ` lim ds Ft´s pB Uτ pAqqUs pAqu tÓ0 t 0 “ Av ` BUτ pAqu “ pA ` Bqv , v P D.

(4.84)

This means that the generator H of the semigroup tFt utě0 satisfies dom H Ě D, and that H “ A ` B on D. By Remark 4.36, we get that dom A Ď dom H and Hu “ pA ` Bqu,

u P dom A.

(4.85)


127

To finish the proof, we have to verify that dom H is not larger than dom A. Note that dom A Ď dom B and that the bound (4.54) holds by Lemma 4.37. Then the Laplace transform of (4.80) for λ ą ω0 and λ large enough (such that ε ă 1 in (4.54)), gives pH ` λ1q´1 “ pA ` λ1q´1 ´ pH ` λ1q´1 BpA ` λ1q´1 “ pA ` λ1q´1 t1 ` BpA ` λ1q´1 u´1 .

(4.86)

Since }BpA`λ1q´1 } ă ε ă 1, equation (4.86) implies that dom H “ ran R´λ pHq Ď ran R´λ pAq “ dom A. Together with expression (4.85), this implies dom H “ l dom A. Corollary 4.42. The sum of a bounded operator B P LpHq and a generator A P QpM, ω0 q, H “ A ` B, is the generator of a strongly continuous quasi-bounded semigroup tUt pHqutě0 , that is, H P QpM 1 , ω01 q. Note also that in this case the operator (4.64) is indeed an abstract Volterra operator. Then iterations of (4.64), (4.65) yield the estimate }Vn }Ft “

}Vn Φ}8,t ď

sup tΦPFt :}Φ}8,t “1u

1 pt }B}M eω0 t qn , n!

which shows that the spectral radius of this operator is zero, see Section A.7. Then the proof of Proposition 4.41 is straightforward. In Proposition 4.41, the perturbed semigroup tUt pHqutě0 with perturbation B P P0` pAq is constructed via an operator-norm convergent Dyson-Phillips series (4.77). The extension of this perturbation theory to Gibbs semigroups needs the following preparatory lemma. Lemma 4.43. Let the m-sectorial operator A be such that e´t <e A P C1 pHq for t ą 0, and let V1 , V2 , . . . , Vn be bounded operators on H. For any set of positive numbers t1 , t2 , . . . , tn , n n ›ź › ź › › Vj e´tj A › ď }Vj }}e´pt1 `t2 `...`tn q <e A{4 }1 . › j“1

1

(4.87)

j“1

Proof. Firstly, let Vj P C8 pHq for j “ 1, 2, . . . , n. We set tm :“ minttj unj“1 ą 0 řn and T :“ j“1 tj ą 0. For any 1 ď j ď n, we define an integer `j P N by 2`j tm ď tj ď 2`j `1 tm . We then set

řn j“1

2`j tm ą T {2 and n ź j“1

Vj e´tj A “

n ź j“1

`j

Vj e´ptj ´2

tm qA

`j

pe´tm A q2 .

(4.88)


128

By the definition of the } ¨ }1 -norm and by the inequality (3.25) for singular values, see Corollary 3.7, we get 8 n n ›ź › ÿ `ź `j `j ˘ › › Vj e´tj A › “ sk Vj e´ptj ´2 tm qA pe´tm A q2 › 1

j“1

j“1

k“1 n 8 ź ÿ

ď

¯“ ´ ‰2`j `j sk e´ptj ´2 tm qA sk pe´tm A q sk pVj q

k“1 j“1 8 ÿ

ď

řn

sk pe´tm A q

j“1

2`j

n ź

}Vj } .

(4.89)

j“1

k“1

`j

`j

tm qA q ď }e´ptj ´2 tm qA } ď 1 and that sk pVj q ď }Vj }, Here we used that sk pe´ptj ´2 řn `j see Section 4.2. Let N :“ j“1 2 and Tm :“ N tm ą T {2; then inequality (4.89) yields n n ›ź › ´› ¯N ź › › › (4.90) Vj e´tj A › ď ›e´Tm A{N ›q“N }Vj }. › j“1

j“1

In order to apply Proposition 4.30, we consider an integer p P N such that 2p ď N ă 2p`1 . It then follows that T {4 ă Tm {2 ă 2p Tm {N , and hence we obtain 8 ¯N ´› ÿ › ›e´Tm A{N › skN pe´Tm A{N q “ q“N k“1 8 ÿ

ď k“1 8 ÿ

ď

p

p

s2k pe´2 p

Tm A{2p N

q

p`2

s2k pe´T A{2

q,

(4.91)

k“1 p

p

where we used that sk pe´Tm A{N q “ sk pe´2 Tm A{2 N q ď }e´Tm A{N } ď 1, and that sk pe´pt`τ qA q ď }e´tA }sk pe´τ A q ď sk pe´τ A q for any t, τ ą 0. Therefore, the estimates (4.90), (4.91) and inequalities (4.38) give the bound (4.87), see (2.32). Secondly, let Vj P LpHq, j “ 1, 2, . . . , n, and set V˜j :“ Vj e´εA for 0 ă ε ă tm . Hence, V˜j P C1 pHq and sk pV˜j q ď }V˜j } ď }Vj }. If we set t˜j :“ tj ´ ε, then n › n › ź › ›ź ˜ ˜ ˜ Vj e´tj A › ď }Vj }}e´pt1 `t2 `¨¨¨`tn q <e A{4 }1 . › j“1

1

(4.92)

j“1

Since the semigroup te´t <e A utě0 is } ¨ }1 -continuous for t ą 0, we can now take in (4.92) the limit ε Ó 0, which gives the result (4.87) in the general case. l Proposition 4.44. Let the m-sectorial operator A be such that e´t <e A P C1 pHq for t ą 0, and let the perturbation B P PpAq define the generator Hpκq :“ A ` κB of a Gibbs semigroup tGt pHpκqqutě0 , for κ P C. Then the function C Q κ ÞÑ Gt pHpκqq P C1 pHq is } ¨ }1 -holomorphic for any fixed t ą 0.


129

Proof. By virtue of Definition 1.41, Proposition 1.43 and Proposition 1.46, A P QpM “ 1, ω0 “ 0q. Hence, by Proposition 4.41, we can define a perturbation of the Gibbs semigroup tGt pAqutě0 , see Proposition 4.30, by the norm-convergent Dyson-Phillips series (4.77) Ft pκq :“

8 ÿ

Sn pt, κq.

(4.93)

n“0

Here, as, e.g., in (4.81), we define S0 pt, κq “ Gt pAq, żt Sn pt, κq “ ds Gt´s pAqp´κBqSn´1 ps, κq,

n ě 1.

(4.94)

0

Since B P P0` pAq implies that the operator BGt pAq P LpHq for t ą 0, we get that Gt´s pAqp´κBqSn´1 ps, κq P C1 pHq for s ą 0 and Sn pt, κq is an n-fold } ¨ }1 convergent Bochner integral, which can be estimated as }Sn pt, κq}1 ď żt żt dτ0 . . . dτn χnt pτ0 , . . . , τn q}Gτ0 pAqκBGτ1 pAq ¨ ¨ ¨ κBGτn pAq}1 . (4.95) 0

Here,

0

χtn pτ0 , . . . , τn q

is the characteristic function of the set

n n ! ) ą ÿ τi “ t Ă τi ě 0, i “ 0, 1, . . . , n : R` 0 . i“0

i“0

Now, we can use inequality (4.87) to estimate the integrand in (4.95). To this end, we set V0 :“ Gτ0 {2 pAq and Vj :“ BGτj {2 pAq, j “ 1, . . . , n, and introduce the functions t ÞÑ qptq :“ }Gt pAq} ď 1 and t ÞÑ pptq :“ }BGt pAq}. By Lemma 4.43, see (4.87), we then get from (4.95) that p ˚ ¨ ¨ ¨ ˚ pqpt{2q Tr Gt{8 pAq, }Sně1 pt, κq}1 ď |κ|n 2n pq ˚ loooomoooon

(4.96)

n

where ˚ ¨ ¨ ¨ ˚ pqpt{2q pq ˚ p loooomoooon n

ż t{2

ż t{2 dτn χnt{2 pτ0 , . . . , τn qqpτ0 qppτ1 q ¨ ¨ ¨ ppτn q. (4.97)

dτ0 . . .

“

0

0

Since B P PpAq, (4.51) implies that for any R ą 0 there is an ε ą 0 small enough such that żε dτ ppτ q :“ γε ă p2Rq´1 . (4.98) 0


130 The estimates (4.96)–(4.98) then yield

}Sn pt, κq}1 ď |κ|n 2n γεn Tr Gt{8 pAq,

(4.99)

for 0 ă t ď 2ε. Therefore, the series (4.93) converges uniformly in the } ¨ }1 topology in the disk DR “ tκ P C : |κ| ă Ru and for t in any compact K Ă p0, 2εs. Therefore, Ft pκq P C1 pHq. Finally, similarly to Proposition 4.41, we can use the semigroup integral equation (4.80), or (4.81), to extend this statement to any compact K Ă R` . Proposition 4.41, (4.84), enables us to identify the generator of the Gibbs semigroup l (4.93) with Hpκq “ A ` κB, that is, tFt pκqutě0 “ tGt pHpκqqutě0 .

4.5 Holomorphic Gibbs semigroups In this section, we enlarge the class of perturbations discussed in Section 4.4 to the class Pb with relative bound b ą 0. First, we consider a rather restricted set of selfadjoint generators A and study perturbations of the self-adjoint Gibbs semigroups. Then we pass to the theory of holomorphic non-self-adjoint Gibbs semigroups. Note that, by Proposition 4.27 any p-generator A is the generator of a holomorphic Gibbs semigroup. Therefore, below we focus, in particular, on developing of the corresponding perturbation theory. Proposition 4.45. The generator A ě ´ω0 1 of a self-adjoint Gibbs semigroup tGt pAqutě0 on H and a symmetric operator B P Pbă1 pAq, define the operator H “ A`B, with dom H “ dom A, as the generator of a quasi-bounded self-adjoint Gibbs semigroup tGt pHqutě0 . Proof. Recall that B P Pbă1 pAq means that the operator B is Kato-small with respect to A with the relative bound b ă 1, see Definition 1.50. Since B is symmetric, by the Kato-Rellich theorem, the operator sum H “ A ` B with dom H “ dom A, see (1.101), is a self-adjoint operator and H ě ´ω01 1, with ω01 :“ ω0 ` maxta{p1 ´ bq, a ` b|ω0 |u. Hence, H is an m-sectorial operator with vertex γ “ ´ω01 and semi-angle α “ 0, i.e., H P H pπ{2, ω01 q. Since Gt pAq P C1 pHq for t ą 0 and A “ A˚ , by Proposition 4.21, the spectrum σpAq “ σp pAq is pure point with real eigenvalues λn pAq Ñ `8, for n Ñ 8. Applying the minimax principle to the perturbed operator H “ A ` B, or more precisely to its resolvent Rγă´ω01 pHq, see Section 2.3, we then get that σpHq “ σp pHq and λn pHq ě a ` p1 ´ bqλn pAq ą 0 for all n greater than some n0 ą 1. This implies that 8 ÿ }e´tH }1 “ e´tλn pHq ď c1 }eć2 tA }1 n“1

for t ą 0 and some c1 , c2 ą 0. Hence, H is the generator of a quasi-bounded self-adjoint Gibbs semigroup. l

4.5. Holomorphic Gibbs semigroups

131

Remark 4.46. Any self-adjoint Gibbs semigroup tGt pAqutě0 can be extended to a } ¨ }1 -holomorphic Gibbs semigroup tGz pAquzPSπ{2 . Indeed, since the operator A “ A˚ ě ´ω0 1, it is quasi-m-sectorial with vertex γ “ ´ω0 and semi-angle α “ 0. Thus, A P H pθ “ π{2, ω0 q is the generator of the quasi-bounded holomorphic semigroup tUz pAquzPSπ{2 , see Corollary 1.47. By the semigroup property we get that Gz pAq :“ Uz“tìτ pAq “ Ut pAqUiτ pAq P C1 pHq, z P Sπ{2 , since Ut pAq “ Gt pAq P C1 pHq for t ą 0 and Uiτ pAq P LpHq. Recall that by Proposition 1.27 and Corollary 1.29, the semigroup tGz pAquzPSπ{2 is } ¨ }-holomorphic in the open sector Sπ{2 . Therefore, it is } ¨ }1 differentiable, cf. (4.37): Bz Gz pAq “ } ¨ }1 -lim ∆zÑ0

˘ 1 ` Gz{2`∆z pAq ´ Gz{2 pAq Gz{2 pAq ∆z

“ ÁGz pAq P C1 pHq,

(4.100)

for z P Sπ{2 by the } ¨ }1 -continuity of multiplication, Proposition 2.78. Equation (4.100) now implies that the complex operator-valued function of z P Sπ{2 , z ÞÑ Gz pAq P C1 pHq is } ¨ }1 -holomorphic. Remark 4.47. (a) The condition that the symmetric perturbation is Kato small, B P Pbă1 pAq, can be relaxed if one assumes that B ě 0 and D :“ dom A X dom B is dense in H. Let us suppose for simplicity that the generator A of a self-adjoint semigroup tGt pAqutě0 satisfies A “ A˚ ě α1 with α ą 0. Since the densely defined symmetric operator T :“ A ` B, dom T “ D, is semi-bounded from below by α1, it has a self-adjoint extension T˜ ě α1, and T˜ ě A by B ě 0. Since α ą 0, the inverse operators T˜´1 and A´1 exist and 0 ď T˜´1 ď A´1 , where A´1 P C8 pHq. Then T˜´1 P C8 pHq and λn pT˜q ě λn pAq for eigenvalues, n ě 1 (cf. Proposition 2.63). Hence, ˜

}e´tT }1 ď }e´tA }1 . Therefore tGt pT˜qutě0 is a self-adjoint Gibbs semigroup, which can be extended to z P Sπ{2 by Remark 4.46. (b) Let A ě 0 be generator of a self-adjoint Gibbs semigroup tGt pAq “ e´tA utě0 and B ě 0 be generator of a self-adjoint contraction semigroup. It may happen that domain dom pAq X dom pBq is not dense in H, or even is trivial, that is, reduces to t0u. Then the construction of semigroup corresponding to the pair A, B via a perturbation theory is difficult/impossible. On the other hand, in Section 5.4 (Proposition 5.53) we shall show that a construction is possible via the Trotter-Kato product formulae approximation. We call it the product formulabased construction of semigroups. (c) For example, let dom A X dom B “ t0u, whereas dom A1{2 X dom B 1{2 is nontrivial. Then the closure dom A1{2 X dom B 1{2 “: H0 is a subspace H0 Ď H.


132

.

The non-negative self-adjoint form-sum operator H :“ A ` B is densely defined in the Hilbert subspace H0 , see Remark 4.29. The exponential Trotter-Kato product formula converges in the trace-norm topology away from zero for any self-adjoint operator B ě 0 to a degenerate Gibbs semigroup: ´ ¯n . } ¨ }1 - lim e´tA{n e´tB{n “ e´tH P0 , H “ A ` B . nÑ8

Here t ą 0 and P0 : H Ñ H0 is the orthogonal projection. In order to generalise this observation to any A ě ´γA 1 and B ě ´γB 1, ˜ “ γA , γB P R, one can consider the operators A˜ “ A ` pγA ` εq1 ą 0 and B B ` γB 1 ě 0. Remark 4.48. According to Remark 4.46, the perturbed self-adjoint semigroup defined in Remark 4.47 can be extended to a holomorphic Gibbs semigroup tGz pA ` BquzPSπ{2 . Similarly to Section 1.7 and Section 4.4, we would like to study here the stability of generators with respect to perturbations, or the analytic properties of the perturbed semigroups with respect to the parameter κ of the family of perturbations tκBuκPC . However, in general, even for κ ă 0 this is not possible if there is no subordination between A and B, cf. Proposition 1.52. Note that the construction of perturbed semigroups in Proposition 4.45 and in Remarks 4.46, 4.47 is not semigroup- or resolvent-based, see notes in Section 1.8 and in Section 4.7. In fact, only the operator values of the one-parametric family were enough to identify that the corresponding semigroups are Gibbs. A particular case one finds in Remark 4.47(c). There the construction of a perturbed semigroup is entirely based on the product formula, which is an example of the product formula-based construction of semigroups. Below we consider the case, when A is a p-generator, Definition 4.26, and κB P Pbă1 pAq for κ P C and b ě 0. Recall that, by Proposition 4.27, any pgenerator A is the generator of a holomorphic Gibbs semigroup. For the construction of the perturbed semigroup we use the resolvent-based method to lift the result of Proposition 1.54 for the operator-norm topology to the trace-norm topology. Proposition 4.49. Let the positive self-adjoint operator A ě 0 in H be a p-generator and let the perturbation operator B P Pb pAq for b ă 1. Then the operator-valued map into the trace-class: D Q pz, κq ÞÑ e´zpA`κBq P C1 pHq,

(4.101)

is holomorphic in the trace-norm topology on the domain D :“ tz P Sθp|κ|,bq Ă Cu ˆ tκ P Dră1{b Ă Cu (4.102) a 2 2 2 “ tpz, κq P C : |argpzq| ă arctgp 1 ´ |κ| b {|κ|bq ^ |κ| ă 1{bu.

4.5. Holomorphic Gibbs semigroups

133

Proof. By Proposition 1.54, the quasi-sectorial for each semi-angle αpκ, bq “ π{2´θp|κ|, bq operators tHpκq :“ A`κBuκPD1{b in the unit disc D1{b are generators of the operator-norm holomorphic family tUz pHpκqq “ e´zHpκq uκPD1{b of holomorphic semigroups tUz pHpκqquzPSθp|κ|,bq in sectors with semi-angles θp|κ|, bq “ a arctgp 1 ´ |κ|2 b2 {|κ|bq. Then for B P Pb pAq the } ¨ }-convergent Neumann series and the Riesz-Dunford formula (1.118) give the representation Uz pHpκqq “

1 2πi

ż dζ ezζ R´ζ pAq Γ

8 ÿ

pκBR´ζ pAqqn .

(4.103)

n“0

Here the } ¨ }-convergent integral is taken along the contour Γ Ă p´Mκ, b q, see (1.115). Since p´Mκ, b q Ă ρpAq and since A is a p-generator, we obtain that R´ζ pAq “ Rζ0 pAqr1 ´ pζ0 ` ζqR´ζ pAqs P Cp pHq , for any ζ P Γ. Hence, the integral is in fact } ¨ }p -convergent and the representation (4.103) leads to the estimate ż 1 ´1 }Uz pHpκqq}p ď |dζ| eRepzζq }Rζ pAq}p t1 ´ |κ|}BR´ζ pAq}u 2π Γ ż 1 1 ` |ζ0 ` ζ|}R´ζ pAq} ď |dζ| eRepzζq }Rζ0 pAq}p , (4.104) 2π Γ 1 ´ |κ|}BR´ζ pAq} where the last integral converges for z P Sθp|κ|,bq and κ P D1{b . Therefore, the estimate (4.104) implies Uz pHpκqq P Cp pHq, and by Proposition 4.3 we obtain that tUz pHpκqquzPSθp|κ|,bq is a Gibbs semigroup: Uz pHpκqq “: Gz pHpκqq P C1 pHq. Since tUz pHpκqquzPSθp|κ|,bq is } ¨ }-holomorphic in the open sector Sθp|κ|,bq , the map z ÞÑ Gz pHpκqq is } ¨ }1 -holomorphic in the same sector by (4.100). We note also that for a fixed r ă b´1 the family of Gibbs semigroups tGz pHpκqquκPDr is } ¨ }1 -uniformly bounded: }Gz pHpκqq}1 ď Mr pzq,

(4.105)

for any z P Sθpr,bq . Now we recall that, by Proposition 1.54 (ii), the series in (4.103) is a uniformly } ¨ }-convergent Taylor series for |κ| ď r ă b´1 . Therefore, we can rewrite (4.103) as 8 ÿ κn n B Gz pHp0qq, z P Sθpr,bq . (4.106) Gz pHpκqq “ n! κ n“0 Here, the } ¨ }-derivatives have the representation ż n! Bκn Gz pHp0qq “ dκ κ´pn`1q Gz pHpκqq, 2πi C

(4.107)


134

where the contour C Ă D1{b encircles the point κ “ 0. If the contour Cr1 is the circle of radius r ă r1 ă b´1 , then by (4.107) and by the } ¨ }1 -estimate (4.105) in the disc D1 one gets the bounds ż n! 2π n }Bκ Gz pHp0qq}1 ď dϕpr1 qń sup }Gz pHpκqq}1 2π 0 |κ|ďr 1 “ n! pr1 qń Mr1 pzq,

(4.108)

for any z P Sθpr,bq . Consequently, (4.106) and (4.108) yield for any z P Sθpr1 ,bq the estimate N 8 › › ÿ ÿ κn n › › Bκ Gz pHpκ “ 0qq› ď p|κ|{r1 qn Mr1 pzq, ›Gz pHpκqq ´ n! 1 n“0 n“N `1 which implies the } ¨ }1 -convergence of the series (4.106), uniformly in the closed disc |κ| ď r. Therefore, the function κ ÞÑ Gz pHpκqq is } ¨ }1 -holomorphic in disc D1{b . l Corollary 4.50. Let the closed operators tHpκquκPD1{c ĂC form in a Hilbert space H, a holomorphic family of type (A), Definition 5.67, and let D1{c Q 0. If Hp0q P H pθ, ωq is the generator of a quasi-bounded holomorphic semigroup, then for any u P dom Hp0q by the criterion in Proposition 5.68 one can estimate the difference }pHpκq ´ Hp0qqu} ď

|κ| pa}u} ` b}Hp0qu}q. 1 ´ |κ|c

(4.109)

For |κ| ă pb ` cq´1 the difference in (4.109) is relatively Hp0q-small. Thus by Propositions 1.51 the operator Hpκq P H pθ1 , ω 1 q generates a holomorphic semigroup and one can use the Riesz-Dunford representation ż 1 Uz pHpκqq “ dζ ezζ pζ1 ` Hpκqq´1 , (4.110) 2πi Γ for z P Sθ1 , κ P D1{pb`cq and with the operator-norm convergent Bochner integral taken along an appropriate contour Γ. Note that due to the resolvent identity we get the representation Uz pHpκqq ´ Uz pHp0qq ż 1 “ dζ ezζ pζ1 ` Hpκqq´1 ppHp0q ´ Hpκqqqpζ1 ` Hp0qq´1 2πi Γ for κ P D1{pb`cq . Then the smallness (4.109) ensures the existence of the operatornorm derivative at κ “ 0. ż 1 Bκ Uz pHp0qq “ dζ ezζ Bκ pζ1 ` Hp0qq´1 2πi Γ ż ż 1 zζ 1 “ dζ e dκ1 pκ1 q´2 pζ1 ` Hpκ1 qq´1 . 2πi Γ 2πi C

4.6. Pb -perturbations of Gibbs semigroups

135

In the last line we use that the type (A) analyticity of Hpκq implies the operator-norm κ-analyticity of the resolvent Rζ pHpκqq. So, we calculate the above derivative as the integral for a small circle C Ă D1{pb`cq around zero in the κ-plane. Then for derivatives of any order at non-zero points κ one gets the expression ż n! Bκn pζ1 ` Hpκqq´1 “ dκ1 pκ1 ´ κqń´1 pζ1 ` Hpκ1 qq´1 , (4.111) 2πi C for a small circle C around κ. Due to fundamental inequality (1.69), which is uniform in a small disc D1{I with the centre at κ, (4.111) yields the estimate }Bκn pζ1 ` Hpκqq´1 } ď

Mε I n n! π , |argζ| ď ` θ ´ ε . |ζ| 2

(4.112)

The estimate (4.112) ensures for small |κ| the uniform in ζ P ρpHpκqq operator-norm convergence of the Taylor series pζ1 ` Hpκqq´1 “

8 ˇ ÿ κn n ˇ Bκ pζ1 ` Hpκqq´1 ˇ . n! κ“0 n“0

(4.113)

Integration of expansion (4.113) in (4.110) gives the } ¨ }-convergent Taylor series for holomorphic semigroup tUz pHpκqquzPSθp|κ|,bq , cf. (4.106), and also proves that the family tUz pHpκqquκPD is norm holomorphic in a small disc D. We note that most of the above arguments can be used verbatim in the proof for the case of the } ¨ }p -norm topology. Corollary 4.51. Let closed operators tHpκquκPDĂC form a holomorphic family of type (A). If Hp0q P H pθ, ωq is a p-generator, then the Riesz-Dunford formula (4.110) allows to proceed with the same arguments as above using systematically the } ¨ }p -topology. This also includes the integration. Finally, the semigroup property and Proposition 4.3 allow to lift the holomorphic properties of the map D Q pz, κq ÞÑ e´z Hpκq P Cp pHq , to the trace-norm topology for p “ 1.

4.6 Pb -perturbations of Gibbs semigroups In this section we return to the Pb -perturbations of Gibbs semigroups constructed in the previous section using the Riesz-Dunford formula. Here we use for analysis the semigroup-based method. In contrast to the resolvent-based, used in the previous section, it makes the perturbation series (cf. Section 4.4) more explicit. The semigroup-based method is motivated by the abstract Volterra-type integral equations (Lemma 4.39) and their iterative solutions by the convergent


136

series, Propositions 4.41 and 4.44. But within this direct method we forfeit the full power of advantages deriving from the analyticity which is essential for the resolvent-based method. The Proposition 4.49 and Corollaries 4.50, 4.51, show the power of the resolvent-based method for Gibbs semigroups. We learned a similar arguments before in Sections 1.6, 1.7 for the case of the strongly continuous holomorphic semigroups. A discrepancy between these methods gets even more sound, when one moves from P0` -perturbations to the Pb -perturbations, or to stronger topologies of continuity. To illustrate the limitations of the semigroup-based method for Pb -perturbations we consider here a concrete example of perturbations from the holomorphic family of type (A), tHpκq :“ A ` κBuκPDĂC , which is a linear function. If conditions of Proposition 4.49 are satisfied, then the function r0, 1s Q α ÞÑ Vαp1q :“ Gp1´αqz pHpκ0 ` κqq Gαz pHpκ0 qq P C1 pHq

(4.114)

in the sector St|κ0 |,|κ0 `κ|u :“ Sθp|κ0 |,bq X Sθp|κ0 `κ|,bq , is continuously } ¨ }1 differentiable for s P p0, 1q and for the fixed parameters κ0 , pκ0 ` κq P D1 , with a given z P C. The derivative of (4.114) has the form Bα Vαp1q “ z Gp1´αqz pHpκ0 ` κqq κ B Gαz pHpκ0 qq P C1 pHq.

(4.115)

Applying to (4.115) the } ¨ }1 -convergent Bochner integral one gets the } ¨ }1 valued relation Gz pHpκ0 ` κqq ´ Gz pHpκ0 qq ż1 “ ´z dα Gp1´αqz pHpκ0 ` κqq κ B Gαz pHpκ0 qq.

(4.116)

0

For κ0 “ 0 the relation (4.116) is the integral equation ż dζ Gz´ζ pHpκqq κ B Gζ pAq, Gz pHpκqq “ Gz pAq ´

(4.117)

r0,zs

for the construction of a perturbed holomorphic Gibbs semigroup tGz pHpκqquzPSθp|κ|,bq by the semigroup-based method. Here the integral is taken along the ray r0, zs in the sector Sθp|κ|,bq Y t0u. After a change of variables the equations (4.116) and (4.117) are reduced to the standard integral equations with integration along the ray r0, ts Ă R0` , Section 4.4 (4.61).


137

Proposition 4.52. Let the self-adjoint operator A ě 0 be a p-generator. Let the linear holomorphic family of type (A): tHpzq :“ A ` zBuzPD1 , for unit disc D1 Ă C, be generated by B P Pbă1 pAq. Suppose also that the adjoint operator B ˚ P Pb˚ ă1 pAq. Then the family of Gibbs semigroups tGt pHpκqqutκ:|κ|ă1u , where t P R` , is } ¨ }1 -differentiable with respect to κ P p´1, 1q. Proof. Since the conditions for Proposition 4.49 are satisfied, tGt pHpκqqutě0 is a Gibbs semigroup for any |κ| ď 1. Using the equation (4.116) with the integral along the ray r0, ts Ă R` 0 , we obtain żt dsGt´s pHpκ ` κ1 qq B Gs pHpκqq,

Gt pHpκ ` κ1 qq ´ Gt pHpκqq “ ´κ1

(4.118)

0

where κ P r´1, 1s and pκ ` κ1 q P r´1, 1s. Then to prove the existence of the } ¨ }1 -derivative Bk Gt pHpκqq one has to check (at least) that the } ¨ }1 -limit of the right-hand side of (4.118) exists and is equal to zero, when κ1 Ñ 0. For this purpose we split the integral in (4.118) into two parts: ż t{2 ds Gt´s pHpκ ` κ1 qq B Gs pHpκqq 0 żt ds Gt´s pHpκ ` κ1 qq B Gs pHpκqq, `

(4.119)

t{2

to estimate the integrand in the } ¨ }1 -norm on two intervals. First we consider the integrand for s P rt{2, ts. Then }Gt´s pHpκ ` κ1 qq B Gs pHpκqq}1

(4.120)

1

ď }Gt´s pHpκ ` κ qq}}BGs{2 pHpκqq}}Gs{2 pHpκqq}1 . Since tGz pHpκqquzPSθp|κ|,bq is a quasi-bounded holomorphic semigroup (Proposition 1.54), we have }Gt´s pHpκ ` κ1 qq} ď M eps´tq γp1,a,bq , where its type ω0 ď ω :“ ´γp1, a, bq. The last factor }Gs{2 pHpκqq}1 is bounded, since tGt pHpκqqutě0 is a Gibbs semigroup and since s ě t{2 ą 0. Let us set Hω pκq :“ Hpκq ` ω 1 ě 0. Then for the factor in the middle we obtain for µ ą 0 that }BGs{2 pHpκqq} ď }BpHω pκq ` µ1q´1 }}pHω pκq ` µ1qGs{2 pHpκqq}.

(4.121)

By the Definition 1.50 of the Kato-small perturbations Pbă1 pAq (1.101) one gets for u P H that }BpHω pκq ` µ1q´1 u} ď a}pHω pκq ` µ1q

´1

(4.122) ´1

u} ` b}ApHω pκq ` µ1q ´1

ď b}u} ` pa ` bpω ` µqq}pHω pκq ` µ1q

u}

u} ` b|κ|}BpHω pκq ` µ1q´1 u}.


138

Since Hω pκq ě 0, these inequalities imply the estimate }BpHω pκq ` µ1q´1 } ď

1 p2b ` pa ` ωbq{µq . 1 ´ b|κ|

(4.123)

For the bound of the norm of the last factor in (4.121) we use that operator Hω pκq P H pθp|κ|, bq, 0q is the generator of a holomorphic contraction semigroup. By the same reason as in (1.76) we obtain the upper bound }pHω pκq ` µ1qGs{2 pHpκqq} ď

2 M11 ` pω ` µqM es ω{2 . s

(4.124)

Combining the estimates (4.121)–(4.124) proves the } ¨ }1 -norm uniform boundedness of (4.120): }Gt´s pHpκ ` κ1 qq B Gs pHpκqq}1 ď Mrt{2,ts ,

(4.125)

and consequently the boundedness of the integral (4.119) on the interval rt{2, ts. To estimate the integral (4.119) on the interval s P r0, t{2s we use for µ ą 0 the inequality: }Gt´s pHpκ ` κ1 qq B Gs pHpκqq}1

(4.126)

1

ď }Gpt´sq{2 pHpκ ` κ qq}1 }Gpt´sq{2 pHpκ ` ˆ }pHω pκ `

κ1 q

`

µ1q´1 B}}G

κ1 qqpHω pκ

`

κ1 q

` µ1q}

s pHpκqq} .

Here the overline stands for the closure of the indicated operators. Note that the both of these closed operators are bounded. We prove this only for the less evident case of the second operator since the line of reasoning for the first one is similar. To this aim we introduce the auxiliary operator O :“ B ˚ pHω˚ pκ ` κ1 q ` µ1q´1 .

(4.127)

Since B ˚ P Pb˚ ă1 pAq and pκ ` κ1 q P r´1, 1s, we have H ˚ pκ ` κ1 q “ A ` pκ ` κ1 qB ˚ . Then by the same arguments as for B (see (4.122), (4.123)) we deduce that the operator (4.127) is bounded and its norm is estimated as }O} ď

1 p2b˚ ` pa ` ωb˚ q{µq. 1 ´ b˚ |κ ` κ1 |

Consequently, for all u P H and any v P dom B the scalar product pOu, vq “ pu, O˚ vq “ pu, pHω pκ ` κ1 q ` µ1q´1 B vq,

(4.128)

is bounded. Since the adjoint of a bounded operator is also bounded, we have O˚ P LpHq with }O˚ } “ }O}. Moreover, by (4.128) we obtain pHω pκ ` κ1 q ` µ1q´1 B Ď O˚ .


139

This means that the operator pHω pκ ` κ1 q ` µ1q´1 B with a dense domain dom B is closable and its closure is the bounded operator pHω pκ ` κ1 q ` µ1q´1 B “ ppHω pκ ` κ1 q ` µ1q´1 Bq˚˚ “ O˚ ,

(4.129)

with the norm }pHω pκ ` κ1 q ` µ1q´1 B} “ }pB ˚ pHω˚ pκ ` κ1 q ` µ1q´1 q˚ } “ }O}. Since the semigroup tGz pHpκqquzPSθp|κ|,bq is holomorphic, the operator pHω pκ ` κ1 q ` µ1qGpt´sq{2 pHpκ ` κ1 qq is bounded. Then by similar arguments as above, the operator Gpt´sq{2 pHpκ ` κ1 qqpHω pκ ` κ1 q ` µ1q is also bounded. This yields the uniform } ¨ }1 -boundedness of (4.126): }Gt´s pHpκ ` κ1 qq B Gs pHpκqq}1 ď Mr0,t{2s ,

(4.130)

and, hence of the integral (4.119) on the interval s P r0, t{2s. Together with the estimate on the interval s P rt{2, ts our arguments yield the }¨}1 -norm boundedness of the } ¨ }1 -integral in (4.118). Thanks to the uniform boundedness of integral in the right-hand side of (4.118), one first gets the } ¨ }1 -continuity of the family tGt pHpκqqutκ:|κ|ă1u : lim Gt pHpκ ` κ1 qq “ Gt pHpκqq.

κ1 Ñ0

(4.131)

Using the limit (4.131) in the integrand and the uniform } ¨ }1 -convergence of the integral in (4.118) we obtain for the } ¨ }1 -derivative the explicit formula żt Bκ Gt pHpκqq “ ´ ds Gt´s pHpκqq B Gs pHpκqq. (4.132) 0

This proves the assertion about the } ¨ }1 -differentiability of the family of Gibbs semigroups tGt pHpκqqutκ:|κ|ă1u , for t ą 0. l Corollary 4.53. Using the estimates (4.125) and (4.130) one can iterate (4.132) to calculation high-order derivatives. This leads to the Dyson-Phillips representation (4.106), in correspondence with the semigroup-based method. A difference is that the resolvent-based method gives simultaneously a control of convergence the series (4.106), as in (4.108). Corollary 4.54. The same arguments are applicable to the case of integration along the ray ζ P r0, zs (4.117) in the sector Sθp|κ|,bq Y t0u. This gives for the } ¨ }1 derivative the formula żz Bκ Gz pHpκqq “ ´ dζ Gz´ζ pHpκqq B Gζ pHpκqq , (4.133) 0

for Gibbs semigroups, when z P Sθp|κ|,bq .

140


4.7 Notes Notes to Section 4.1. Compact semigroups are well-known for already a long time. They appear naturally, for example, in the study of parabolic partial differential equations in bounded spatial domains [RR93]. The extension to semigroups with values in von Neumann-Schatten Cp pHq-ideals is conceptually straightforward, see for example, [Bal76], [Dav07]. However, it seems that it was D. A. Uhlenbrock [Uhl71] who proposed for the first time the notion of Gibbs semigroups (Definition 4.1), motivated by Quantum Statistical Mechanics. Then this concept was supported in [ANB75]. Indeed, the canonical quantum Gibbs state of a finite system is defined by the ´1 density matrix operator ZΛ pβq expp´βHΛ q, where β ´1 ě 0 is the temperature of the system, HΛ is its Hamiltonian, and the canonical partition function ZΛ pβq “ Tr expp´βHΛ q, see e.g., [BR96]. Remark 4.4 says that, according to our Definition 4.1, the semigroup corresponding to the Quantum Statistical Mechanics is immediately Gibbs [Zag80], [Zag89]. In contrast, the semigroup defined by expression (4.3) is an example of an eventually Gibbs semigroup, i.e., tTt utě0 becomes a Gibbs semigroup only after some threshold t0 ą 0. Then, by Corollary 4.6, the semigroup tTt utě0 is also eventually } ¨ }1 -continuous, that is, continuous in the trace-norm topology for t ą t0 . The terminology immediately/eventually, is the same as for compact semigroups, see for example, Section 4.2 and [EN00]. We resume this section by discussing the relation between spectra σpAq and σpUt pAqq. For details concerning the application of the Gel’fand transform, see [Dav80], Chapter 2.2, and [BR79], Section 2.3.5. Notes to Section 4.2. For strongly continuous semigroups tUt utě0 with values in C1ďpď8 pHq-ideals for t away from zero there is a curious relation between their topology of continuity and topology of image Cp pHq as the Banach space. Here we continue to discuss this observation (started in Section 4.1 for the Gibbs semigroups), but now for the case of compact semigroups, see Proposition 4.17. Note that in Chapter 5 we shall show that this relation also concerns the topology of continuity of semigroups away from zero and the topology of convergence of the Trotter-Kato product formulae. The first result was due to V. A. Zagrebnov [Zag88]. In [NZ90a, NZ90b] the existence of such relation was formulated for abstract Gibbs semigroups. Then it was confirmed for symmetrically-normed ideals in [NZ99d]. For the case of compact semigroups this relation with the operator-norm convergence of the Trotter-Kato product formulae was established in [NZ99b]. Definitions 4.13 and 4.14 of immediately and eventually compact C0 semigroups as well as Proposition 4.19 for holomorphic semigroups are standard, see [EN00]. Since (in contrast to the Gibbs semigroups, see Remark 4.4) they are immediately norm-continuous for t ą 0 (Lemma 4.15), the relation between the spectral properties of these semigroups and their generators, Proposition 4.21 and Proposition 4.22, is more explicit than in the general case, see Proposition 4.11.

4.7. Notes

141

Different formulations of the minimax principle (Proposition 4.23), including formulations for unbounded operators, as well as its history can be found in, e.g., [RS78] and [BEH94]. Notes to Section 4.3. The notion of p-generator was introduced for the particular case of self-adjoint semigroups by V. A. Zagrebnov [Zag89]. Proposition 4.27 is a generalisation of Theorem 4.1 from [Zag89]. Proposition 4.30 and Corollaries 4.31 and 4.32 are due to V. Cachia and V. A. Zagrebnov [CZ01c]. Note that Proposition 4.30 is an extension of the result [BG72] for self-adjoint generators to the case of m-sectorial generators. Notes to Section 4.4. In [Uhl71] D. A. Uhlenbrock considered the perturbation theory of Gibbs semigroups generators for bounded perturbations from the class P0 . The class of P-perturbations was introduced in [HP57], see also [Dav07], Chapter 11.4. The perturbation theory of quasi-bounded strongly continuous semigroups QpM, ω0 q for the case of P-perturbations (Proposition 4.41) is standard, [Dav80], Chapter 2 and [Dav07], Chapter 11. For self-adjoint Gibbs semigroups, the theory concerned with P-perturbations was developed by N. Angelescu, G. Nenciu, and M. Bundaru in [ANB75]. Both results, [Uhl71] and [ANB75], for Gibbs semigroups are based on the GinibreGruber inequality [GG69]. A generalisation of this inequality to m-sectorial generators (Lemma 4.43) is due to V. Cachia and V. A. Zagrebnov [CZ01c]. This generalisation allows to prove the trace-norm convergence for the P-perturbation series of Gibbs semigroups generated by m-sectorial operators, see Proposition 4.44. Notes to Section 4.5. Proposition 4.45 on Pbă1 -perturbations for self-adjoint Gibbs semigroups is due to V. A. Zagrebnov [Zag89]. The analytic theory for the case of Pbă1 -perturbations of Gibbs semigroups with p-generators was developed for the first time by D. Maison [Mai71]. In our exposition, Proposition 4.49 and Corollary 4.50, we follow essentially [Mai71] and [Zag89]. Notes to Section 4.6. The names resolvent-based and semigroup-based methods are due to [Dav07], Chapters 11.4 and 11.5. They indicate clearly the difference between two ways for construction of semigroups. The product formula-based method mentioned in Remark 4.47(b),(c), was advocated in [Che74]. The illustrative example and Proposition 4.52 are motivated by [Mai71] and [Zag89]. Condition B ˚ P Pb˚ ă1 pAq does not follow from B P Pbă1 pAq. It is indispensable for estimating products (like (4.126)) including unbounded operators. The line of reasoning is standard, see [NZ98] Section 1, or Remark 5.14. The question whether there exists a modification of Definition 4.34 such that it gives non-infinitesimally small perturbations, was positively solved in [Voi77]. Modifying the integral condition (4.51) he proposed a new (Miyadera-Voigt) class of unbounded perturbations PM V Ă Pbą0 , such that P0` Ă PM V , cf. (4.56). For

142


details see, e.g., [EN00], Chapter III, Section 3c. Similarly to perturbations of class Pb (Proposition 1.51), the holomorphic semigroups are stable with respect to Miyadera-Voigt perturbations, [EN00], 3.17 Exercises.

Chapter 5

Product formulae for Gibbs semigroups A wealth of results in the literature deal with the Lie-Trotter and Trotter-Kato product approximations for strongly continuous semigroups in the strong operator topology. However, it has been known since a long time that for the Gibbs semigroups the Trotter-Kato product formulae converges also in the trace-norm topology, see Notes in Section 5.6 and comments in Appendix D.4. On the other hand, the operator-norm convergence of the Trotter-Kato product formulae with error bound estimates is a relatively recent result for self-adjoint strongly continuous semigroups. We give a detailed proof of this fact in Section 5.2. In Section 5.3 some general conditions ensuring the operator-norm TrotterKato product formulae convergence of the degenerate self-adjoint semigroup without error bound estimates are considered. We also present there some results about the non-self-adjoint strongly continuous semigroups. In Section 5.4, it is shown how to lift the error bound estimates from Section 5.2 to establish the trace-norm convergence of the Trotter-Kato product formulae for the self-adjoint Gibbs semigroups. This lifting method allows also to prove that the estimates of rate of convergence for the operator-norm and the tracenorm cases coincide. We complete this chapter by Section 5.5, where we prove the convergence of the Trotter-Kato product formulae in the trace norm for non-self-adjoint Gibbs semigroups. First, we develop for our purpose the analytic continuation method for associated with generators sesquilinear forms. It allows us to establish the convergence, but it does not yield results on the error bound estimates. Then similarly to Section 5.4 we show how to lift the error bound estimates obtained for the operator-norm convergence of the Trotter-Kato product formulae for strongly continuous non-self-adjoint semigroups, to trace-norm estimates for non-self-adjoint © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1_5

143

144

Chapter 5. Product formulae for Gibbs semigroups

Gibbs semigroups. In the next Section 5.1 we start by recalling the first known result about the product formulae, which is due to Sophus Lie (1875). We also present there a general programme that one follows developing the analysis of the product formulae convergence in various topologies.

5.1

The Lie-Trotter product formula

Since the observation by S. Lie for finite matrices, it is known that the semigroup tUt pA ` Bqutě0 generated by A ` B, the sum of two bounded operators A and B on a Banach space B, can be approximated in terms of the semigroups tUt pAqutě0 and tUt pBqutě0 . Proposition 5.1 (Lie product formula). Let A, B P LpBq. Then for any t ě 0 and n P N, we have (a) the product approximation ›` › ˘ › Ut{n pAqUt{n pBq n ´ Ut pA ` Bq› ď c , n

(5.1)

where c “ cpt, }A}, }B}q ą 0; (b) the symmetrised product approximation ›` › ˘ › Ut{2n pAqUt{n pBqUt{2n pAq n ´ Ut pA ` Bq› ď csym , n2

(5.2)

where csym “ csym pt, }A}, }B}q ą 0. Proof. (a) Since A, B and A ` B are bounded, we can use the norm convergent exponential series (1.1). Let P ptq :“ Ut pAqUt pBq. Then the difference n P pt{nqn ´ Ut{n pA ` Bq n´1 ÿ

“

` ˘ m P pt{nqn´m´1 P pt{nq ´ Ut{n pA ` Bq Ut{n pA ` Bq

(5.3)

m“0

can be estimated as follows › › › › n pA ` Bq› ›P pt{nqn ´ Ut{n ` ˘n ď n maxt}P pt{nq}, }Ut{n pA ` Bq}u }P pt{nq ´ Ut{n pA ` Bq}. Using exponential series (1.1) one gets the estimates }P pt{nq} ď etp}A}`}B}q{n

and }Ut{n pA ` Bq} ď etp}A}`}B}q{n .

(5.4)

5.1. The Lie-Trotter product formula

145

Then (5.4) yields › › › › n pA ` Bq› ď n etp}A}`}B}q }P pt{nq ´ Ut{n pA ` Bq}. ›P pt{nqn ´ Ut{n

(5.5)

Since › › › › ›P pt{nq ´ Ut{n pA ` Bq› 8 8 ›8 ´ t ¯k ÿ ıs › 1 ´ t ¯` ÿ 1 ” t ›ÿ 1 › “› ´ A ´ B ´ ´ pA ` Bq › k! n `! n s! n s“0 k“0 `“0 8 8 8 ¯k ÿ ¯` ÿ ¯k ÿ 1´t 1´t 1´t }A} }B} ` }A} k! n `! n k! n k“2 `“2 k“3

ď

`

8 8 ¯` ¯` ÿ ÿ 1´t t 1´t }B} ` }A} }B} `! n n `! n `“3 `“2

`

8 8 ¯k ÿ ıs ÿ 1´t 1”t t }B} }A} ` p}A} ` }B}q n k! n s! n s“3 k“2

`

t2 1 }AB ´ BA} “: 2 Mn pt, }A}, }B}q, 2n2 n

(5.6)

the inequality (5.5) yields the estimate (5.1), where c is defined as the maximum over n ě 1 of the right-hand side of (5.6) multiplied by n2 etp}A}`}B}q . Then c “ etp}A}`}B}q M pt, }A}, }B}q, where M pt, }A}, }B}q :“ maxně1 Mn pt, }A}, }B}q. 1{2 1{2 (b) Let T ptq :“ Ut pAqUt pBqUt pAq be a symmetrised product approximation. Then arguing as in (a) we obtain › › › › › › › › n pA ` Bq› ď n etp}A}`}B}q ›T pt{nq ´ Ut{n pA ` Bq›. (5.7) ›T pt{nqn ´ Ut{n Since the symmetrisation eliminates the last term in (5.6), we get › › › › ›T pt{nq ´ Ut{n pA ` Bq› 8 8 t ¯k1 ÿ 1 ´ t ¯` ÿ 1 ´ t ¯k2 1 ´ ´ A ´ B ´ A k ! 2n `! n k ! 2n “0 1 `“0 k “0 2

8 › ÿ › “› k1

2

8 ıs › ÿ 1” t › ´ ´ pA ` Bq › s! n s“0

ď

1 Lpt, }A}, }B}q. n3

(5.8)

where L is determined in a similar way as the right-hand side of (5.6). The estimate (5.8) gives a better than (5.1) rate of convergence (5.2) for csym “ etp}A}`}B}q Lpt, }A}, }B}q. l

146


Remark 5.2. (1) In Section 5.6 and Appendix D.4 we present a brief history of the product formulae since the work of S. Lie. Here we would like to notice only that in Hilbert and Banach spaces the (exponential ) Trotter product formula ¯n ´ γ-lim e´tA{n e´tB{n “ e´tC , (5.9) nÑ8

has permeated throughout operator and probability theory, for various topologies of convergence γ. The challenge is to find general hypotheses under which the formula holds in infinite-dimensional settings, including the strongest topology of convergence γ. (2) A realisation of this programme requires at least the following steps: (a) to give a description of the set of pairs of operators tA, Bu for which the limit (5.9) exists (in some sense) and yields a semigroup; (b) to find the corresponding optimal (or the strongest) topology γ, for which this limit exists; (c) to reconstruct the generator C from the pair of operators tA, Bu; (d) to generalise, if possible, the exponential functions in (5.9) to Borel measurable functions f, g : R0` Ñ r0, 1s, with the properties: f p0q “ 1, f 1 p`0q “ ´1 and gp0q “ 1, g 1 p`0q “ ´1; (e) to consider product formulae for different product approximants, for example, nonsymmetrised tP pt{nqn uně1 or symmetrised tT pt{nqn uně1 , including various order of factors, to optimise the rate of convergence.

5.2 Trotter-Kato product formulae: operator-norm convergence, error bounds For a long time, the Lie-Trotter product formula (5.9) for strongly continuous contraction semigroups was known only for exponential functions f, g and solely in the strong operator topology γ “ s, with a limit that in turn is a contraction C0 -semigroup. In the present section we show that the arguments used in the proof of Proposition 5.1 can be refined to yield a remarkable and rather nontrivial statement about convergence in the operator-norm topology γ “ } ¨ }, of various TrotterKato product formulae corresponding to different types of non-exponential approximants generated by a set of Borel measurable functions f, g, Proposition 5.8. We note at this point the following important fact, which explains why we undertake the study below: – It is the operator-norm continuity away from zero, which holds for the strongly continuous self-adjoint semigroups generated by A and B, that yields γ “ } ¨ }.

5.2. Trotter-Kato product formulae: operator-norm convergence

147

– For strongly continuous Gibbs semigroups, which are trace-norm continuous away from zero, the Trotter-Kato product formulae were established in the trace-norm topology γ “ } ¨ }1 . – These observations confirm a conventional wisdom that a natural topology γ for the convergence of the product formulae inherits the (strongest) topology in which the semigroup is continuous away from zero. – The γ-limit of the product formulae approximants gives one-parameter family for t P R` 0 , which is either a C0 -semigroup, or a degenerate semigroup in the sense of Definition 1.24, and is in turn γ-continuous away from zero. Before proceeding, we first specify in subsection 5.2.1 (Preliminaries and Proposition 5.8) the conditions on the pair of generators tA, Bu, including a hierarchy between the operators A and B. Below this is a Kato-smallness condition of one operator with respect to the other. These conditions allow us to obtain in Proposition 5.8 an operator-norm error bound estimate for the convergence rate of the Trotter-Kato product formulae for a certain subclass of generic Kato functions f, g P K. For details about these functions, see Appendix C. Subsection 5.2.2, Auxiliary lemmata, is collection of statements indispensable for subsection 5.2.3, Proof of Proposition 5.8. The exponential Kato functions f pxq “ gpxq “ e´x trivially satisfy conditions formulated in these subsections. We conclude Section 5.2 by a discussion of a quite subtle optimality property of the convergence rate. In subsection 5.2.4, Optimal rate of the operator-norm convergence, we study sufficient conditions on the pairs of tA, Bu and on the corresponding admissible functions tf, gu that ensure in Proposition 5.19 and in Proposition 5.25 the optimality of the error bound estimate for the convergence rate. Note that a choice of the suitable Kato functions from K is decisive for the convergence of Trotter-Kato product formulae as well as for the error bound optimality. In subsection 5.2.5, Optimal rate: fractional conditions, we show how subtle is the tuning of the class of admissible Kato functions tf, gu, when the operators tA, Bu are related by some fractional smallness conditions, see Proposition 5.27.

5.2.1 Preliminaries and Proposition 5.8 (i) Let A and B be densely defined, semibounded from below, self-adjoint operators in a Hilbert space H. Without loss of generality we can assume that A ě 1,

B ě 1.

(5.10)

(ii) Let B P Pb pAq. Again for simplicity, we assume that a “ 0 in (1.101), that is, dom A Ă dom B and }Bu} ď b}Au}, u P dom A, (5.11) for 0 ď b ă 1.

148


Hence, the operator H “ A ` B, with dom H “ dom A, is self-adjoint thanks to the b ă 1 A-smallness of B (5.11). Note that by (5.10) and Corollary 1.46 the operators A, B and H are generators of holomorphic contraction semigroups on H. Remark 5.3. We note that there are others hierarchy of conditions on the pair of generators tA, Bu that ensure convergence in the topology γ “ } ¨ }, without as well as with error bound estimate. We consider some of them in Section 5.3, see Propositions 5.36, 5.45 for the operator-norm convergence without estimate and Propositions 5.47, 5.49 for proofs with error bound estimates. In Section 5.4 we show that for the Gibbs semigroups the results of Section 5.3 can be lifted to convergence of the Trotter-Kato product formulae in the trace-norm topology, correspondingly without or with the error bound estimates, inherited from the operator-norm estimates. Definition 5.4. A pair of the Borel measurable functions f, g , which is defined on R` 0 “ r0, 8q and satisfies conditions 0 ď f pxq ď 1, 0 ď gpxq ď 1,

f p0q “ 1, gp0q “ 1,

f 1 p`0q “ ´1 1

g p`0q “ ´1.

(5.12) (5.13)

is said to belong to the class of generic Kato functions. We denote the algebra of monomials generated by f, g , including their fractional powers, by K. (iii) For the Trotter-Kato product formulae we consider the functions f, g which belong to generic Kato functions from the class K. Note that for the operators (5.10) and for any α ě 0, conditions (5.12) and (5.13) yield s-lim f ptAqα “ 1 and s-lim gptBqα “ 1 , tÑ`0

tÑ`0

in the strong operator topology on LpHq. Next, we formulate conditions on the pair of Kato functions from K under which one can lift the Trotter product formula (5.9) to the operator-norm convergent Trotter-Kato product formula: ` ˘n } ¨ }-lim f ptA{nqgptB{nq “ Ut pHq :“ e´tH . (5.14) nÑ8

(iv) To this aim we assume more about the admissible Kato functions: a x f pxq ă 8, C1{2 :“ ess sup 1 ´ f pxq xą0 1 ´ f pxq C1 :“ ess sup ă 8, x xą0 ˇ´ 1 ¯ 1 ˇˇ ˇ C2 :“ ess supˇ f pxq ´ ˇ ă 8, 1 ` x x2 xą0

(5.15) (5.16) (5.17)

5.2. Trotter-Kato product formulae: operator-norm convergence 1 ´ gpxq S1 :“ ess sup ă 8, x xą0 ˇ´ 1 ¯ 1 ˇˇ ˇ S2 :“ ess supˇ gpxq ´ ˇ ă 8. 1 ` x x2 xą0

149 (5.18) (5.19)

(v) Our last condition reads as b C1{2 S1 ă 1.

(5.20)

Remark 5.5. Besides the standard (exponential) Kato functions f pxq “ gpxq “ e´x , the class of Borel functions satisfying conditions (5.12)–(5.19) contains, for example, f pxq “ p1 ` x{2q´2 with C1{2 “ 2, but not f pxq “ p1 ` xq´1 , since in this case C1{2 “ 8. On the other hand gpxq “ p1 ` xq´1 belongs to the class described by (iii) and (iv), since there is no condition on gpxq similar to (5.15). This asymmetry in conditions is related to the hierarchy (ii) of operators tA, Bu. Remark 5.6. Note that by (5.12) and (5.13), we always have C1{2 ě 1 and S1 ě 1. Therefore, condition (v) implies 0 ď b ă pC1{2 S1 q´1 ď 1.

(5.21)

This means that for B P Pbă1 pAq, the condition (v) is not superfluous. Remark 5.7. If f pxq “ gpxq “ e´x , then C1{2 “ C1 “ C2 “ S1 “ S2 “ 1.

(5.22)

This yields 0 ď b ă 1 by condition (v), which is already ensured by (ii). We note that the case b “ 1 (see subsection 5.2.4) demands another approach to prove the operator-norm convergence in (5.14). To formulate the main statement of this section, we define for t P R` 0 the bounded operator-valued functions: t ÞÑ P ptq and t ÞÑ T ptq, by P ptq :“ f ptAqgptBq and T ptq :“ f 1{2 ptAqgptBqf 1{2 ptAq.

(5.23)

They produce two types of approximants: nonsymmetrised tP pt{nqn uně1 and symmetrised (self-adjoint) tT pt{nqn uně1 , for two different Trotter-Kato product formulae. Proposition 5.8. Suppose the operators A , B satisfy conditions (i), (ii), and the functions f, g satisfy conditions (iii), (iv). If also the condition (v) is satisfied, then there are positive constants LP and LT such that › › ›P pt{nqn ´ e´tH › ď

LP lnpnq 2 p1 ´ C1{2 S1 bq p1 ´ bq n

(5.24)

› › ›T pt{nqn ´ e´tH › ď

lnpnq LT , p1 ´ C1{2 S1 bqp1 ´ bq n

(5.25)

and

150


for n “ 3, 4, . . . , uniformly in t ě 0. Here te´tH utě0 is the self-adjoint contraction C0 -semigroup with generator H “ A ` B, dom H “ dom A. Remark 5.9. We see that under conditions (i)–(v), Proposition 5.8 provides a solution of the problems (a)–(e) of Remark 5.2 concerning the product formulae for unbounded generators. This solution includes the identification of the topology γ, which is the same as in Proposition 5.1. It inherits the topology γ “ } ¨ } in which the semigroup te´tH utě0 is continuous away from zero. The differences with Lie-Trotter product formulae are the following: – before it was known only for exponential functions f , g and only in the strong operator topology γ “ s (Trotter product formula (5.9)); – the error bound estimates of the rate of convergence (5.1) and (5.2) are optimal, i.e. the best possible, whereas those in (5.24) and (5.25) are not, see Section 5.4.

5.2.2

Auxiliary lemmata

We prove Proposition 5.8 with the help of the following series of lemmata. Lemma 5.10. Let X P LpHq be a self-adjoint operator such that 0 ď X ď 1. Then }X k p1 ´ Xq} ď p1 ` kq´1 ,

k “ 0, 1, 2, . . .

(5.26)

Proof. This estimate follows from the inequality λk p1 ´ λq ď p1 ` kq´1 ,

k “ 0, 1, 2, . . .

p0 ď λ ď 1q,

and the spectral representation ż1 X k p1 ´ Xq “

dEX pλq λk p1 ´ λq , 0

which yields (5.26).

l

Lemma 5.11. For any self-adjoint operator X ě 0 we have the estimate }Xe´αX } ď pαeq´1 ,

α ą 0.

(5.27)

Proof. Since λe´αλ ď pαeq´1 for λ ě 0, (5.27) is again a consequence of the l spectral representation for self-adjoint operators. Lemma 5.12. Let A ě 1 be a self-adjoint operator on H. If the Borel function f obeys (5.12) and (5.15), then the operator 1 ´ f pτ Aq is boundedly invertible, and }p1 ´ f pτ Aqq´1 u} ď }u} ` for each τ ą 0.

C1{2 ´1 }A u}, τ

u P H,

(5.28)


151

Proof. By virtue of f pxq ď 1 and (5.15), 0ď

xf pxq ď C1{2 , 1 ´ f pxq

or 1 ´ f pxq ě

x ě 0.

x . pC1{2 ` xq

(5.29)

Since A ě 1, the spectral theorem yields ż8 τ τ ě 1. 1 ´ f pτ Aq ě dEA pλq ´1 C1{2 λ ` τ C1{2 ` τ 1 Therefore, the operator 1 ´ f pτ Aq is invertible for τ ą 0. On the other hand, since (5.29) implies C1{2 p1 ´ f pxqq´1 ď 1 ` , x ą 0, x the spectral representation gives the estimate ˆ ˙ › ›ż 8 C1{2 › › ´1 }p1 ´ f pτ Aqq u} ď › dEA pλq 1 ` u›, τ λ 1

u P H,

which proves the assertion (5.28) for τ ą 0.

l

Lemma 5.13. Let A and B be self-adjoint operators satisfying conditions (i), (ii). If the Borel functions f and g satisfy (iii), (iv), and condition (v) is fulfilled, then the operator 1 ´ T pτ q is boundedly invertible for each τ ą 0, and }p1 ´ T pτ qq´1 u} ď

´ C1{2 ´1 ¯ 1 }u} ` }A u} , 1 ´ C1{2 S1 b τ

u P H.

(5.30)

1 ´ T pτ q “ 1 ´ f pτ Aq ` f 1{2 pτ Aqp1 ´ gpτ Bqqf 1{2 pτ Aq.

(5.31)

Proof. By definition (5.23), we get the representation

Since by Lemma 5.12 the operator 1 ´ f pτ Aq, τ ą 0, is invertible, identity (5.31) gives 1 ´ T pτ q “ p1 ´ f pτ Aqqr1 ` f 1{2 pτ Aqp1 ´ f pτ Aqq´1 p1 ´ gpτ Bqqf 1{2 pτ Aqs. (5.32) Note that for τ ą 0, we get f 1{2 pτ Aqp1 ´ f pτ Aqq´1 p1 ´ gpτ Bqqf 1{2 pτ Aq 1 “ τ Af 1{2 pτ Aqp1 ´ f pτ Aqq´1 A´1 B p1 ´ gpτ BqqB ´1 f 1{2 pτ Aq. τ

(5.33) (5.34)

152


Remark 5.14. Recall that since by (5.11) we have BA´1 P LpHq, and since pBA´1 u, vq “ pu, A´1 Bvq,

u P H, v P dom B,

the bounded operator pBA´1 q˚ Ě A´1 B is an extension of A´1 B. This implies that the operator A´1 B is closable and the closure A´1 B “ pBA´1 q˚ . Since }pBA´1 q˚ } “ }BA´1 }, the norm }A´1 B} “ }BA´1 } ď b.

(5.35)

Now, we use (5.35) to estimate (5.34) as follows: › › › 1{2 › ›f pτ Aqp1 ´ f pτ Aqq´1 p1 ´ gpτ Bqqf 1{2 pτ Aq› ď

1 }BA´1 }}p1 ´ gpτ BqqB ´1 }}f 1{2 pτ Aq} τ› › › › ˆ ›τ Af 1{2 pτ Aqp1 ´ f pτ Aqq´1 › .

(5.36)

Then by virtue of (5.10), (5.11), (5.15), (5.18) and (5.20) we get › › › 1{2 › ›f pτ Aqp1 ´ f pτ Aqq´1 p1 ´ gpτ Bqqf 1{2 pτ Aq› ď C1{2 S1 b ă 1. Hence, the operator 1 ` f 1{2 pτ Aqp1 ´ f pτ Aqq´1 p1 ´ gpτ Bqqf 1{2 pτ Aq is invertible with norm ›” ı´1 › › › › 1 ` f 1{2 pτ Aqp1 ´ f pτ Aqq´1 p1 ´ gpτ Bqqf 1{2 pτ Aq › ď p1 ´ C1{2 S1 bq´1 . The last estimate and Lemma 5.12 imply that the operator (5.31) is invertible, with estimate › › › › ›p1 ´ T pτ qq´1 u› ď p1 ´ C1{2 S1 bq´1 ›p1 ´ f pτ Aqq´1 u› ” C1{2 ´1 ı ď p1 ´ C1{2 S1 bq´1 }u} ` }A u} , u P H, τ for any τ ą 0.

l

Lemma 5.15. Let conditions (i)–(v) be satisfied. Then there exists a constant L1 ą 0 such that › 1 ›› ´1 A pT pτ q ´ Uτ pHqq› ď L1 (5.37) τ for any τ ą 0. Proof. By the definition (5.23) we get T pτ q ´ Uτ pHq “ ´ f 1{2 pτ Aqp1 ´ gpτ Bqqf 1{2 pτ Aq ´ p1 ´ f pτ Aqq ` p1 ´ Uτ pHqq,

(5.38)


153

which yields A´1 pT pτ q ´ Uτ pHqq “ ´f 1{2 pτ AqA´1 Bp1 ´ gpτ BqqB ´1 f 1{2 pτ Aq ´ p1 ´ f pτ AqqA

´1

À

´1

Hp1 ´ Uτ pHqqH

(5.39) ´1

.

(5.40)

Then, using the same argument as in (5.35), we obtain the estimate 1 ´1 1 }A pT pτ q ´ Uτ pHqq} ď }BA´1 } }p1 ´ gpτ BqqB ´1 } τ τ 1 1 ` }p1 ´ f pτ AqqA´1 } ` }HA´1 } }p1 ´ Uτ pHqqH ´1 }. (5.41) τ τ Therefore, by (5.16), (5.18) and the spectral theorem we get 1 ´1 }A pT pτ q ´ Uτ pHqq} ď }BA´1 }S1 ` C1 ` }HA´1 }. τ

(5.42)

Since by (5.11) we have }HA´1 } ď }1 ` BA´1 } ď 1 ` }BA´1 } ď 1 ` b, we obtain (5.37) from (5.42) by setting L1 :“ bS1 ` C1 ` 1 ` b. l Lemma 5.16. Let conditions (i)–(v) be satisfied. Then there exists a constant L2 ą 0 such that 1 }A´1 pT pτ q ´ Uτ pHqqA´1 } ď L2 (5.43) τ2 for any τ ą 0. Proof. From (5.38) we obtain the representation T pτ q ´ Uτ pHq “ p1 ´ f 1{2 pτ Aqqp1 ´ gpτ Bqqf 1{2 pτ Aq ` p1 ´ gpτ Bqqp1 ´ f 1{2 pτ Aqq ` pf pτ Aq ´ p1 ` τ Aq´1 q ` pgpτ Bq ´ p1 ` τ Bq´1 q ` pp1 ` τ Hq´1 ´ Uτ pHqq ` τ Hp1 ` τ Hq´1 ´ τ Ap1 ` τ Aq´1 ´ τ Bp1 ` τ Bq´1 .

(5.44)

Since A´1 tτ Hp1 ` τ Hq´1 ´ τ Ap1 ` τ Aq´1 ´ τ Bp1 ` τ Bq´1 uA´1 “ τ 2 tp1 ` τ Aq´1 ` A´1 Bp1 ` τ Bq´1 BA´1 ´ A´1 Hp1 ` τ Hq´1 HA´1 u,

(5.45)

154


we get from (5.44) the identity A´1 tT pτ q ´ Uτ pHquA´1 “ p1 ´ f 1{2 pτ AqqA´1 p1 ´ gpτ BqqB ´1 BA´1 f 1{2 pτ Aq ` A´1 Bp1 ´ gpτ BqqB ´1 p1 ´ f 1{2 pτ AqqA´1 ` pf pτ Aq ´ p1 ` τ Aq´1 qA´2 ` A´1 Bpgpτ Bq ´ p1 ` τ Bq´1 qB ´2 BA´1 ` A´1 Hpp1 ` τ Hq´1 ´ Uτ pHqqH ´2 HA´1 ` τ 2 p1 ` τ Aq´1 ` τ 2 A´1 Bp1 ` τ Bq´1 BA´1 ´ τ 2 A´1 Hp1 ` τ Hq´1 HA´1 , which yields the estimate › 1 ›› ´1 A tT pτ q ´ Uτ pHqu A´1 › 2 τ ›1› › 1 ›› › ď ›p1 ´ f 1{2 pτ AqqA´1 › ›p1 ´ gpτ BqqB ´1 › }BA´1 } τ τ › › › 1 ›› › ´1 1 › ` }BA } 1 ´ gpτ BqB ´1 › ›p1 ´ f 1{2 pτ AqqA´1 › τ τ › 1 › ` 2 ›tf pτ Aq ´ p1 ` τ Aq´1 uA´2 › τ › 1 › ` 2 ›tgpτ Bq ´ p1 ` τ Bq´1 uB ´2 › }BA´1 }2 ›τ › ` ›tp1 ` τ Hq´1 ´ Uτ pHquH ´2 › }HA´1 }2 ` 1 ` }BA´1 }2 ` }HA´1 }2 .

(5.46)

Note that by (iii) and (iv) we obtain the inequalities 1 ´ f 1{2 pxq 1 ´ f pxq ď ď C1 , x x which by the spectral representation for A give the estimate › 1 ›› › ›p1 ´ f 1{2 pτ AqqA´1 › ď C1 . τ The spectral representation gives also › 1 ( 1 ›› p1 ` τ Hq´1 ´ Uτ pHq H ´2 › ď , 2 τ 2

(5.47)

(5.48)

where 1{2 “ supxą0 rp1 ` xq´1 ´ e´x sx´2 . Now, using estimates (5.47) and (5.48), we obtain from (5.46) that › 1 ›› ´1 A tT pτ q ´ Uτ pHquA´1 › ď 2C1 S1 }BA´1 } ` C2 ` S2 }BA´1 }2 2 τ 1 ` }HA´1 }2 ` 1 ` }BA´1 }2 ` }HA´1 }2 . (5.49) 2


155

Then since }BA´1 } ď b and }HA´1 } ď 1 ` b, by setting 1 L2 :“ 2 ` C2 ` p3 ` 2C1 S1 qb ` p5{2 ` S2 qb2 , 2 we get from (5.49) the estimate (5.43).

l

5.2.3 Proof of Proposition 5.8 Proof of Proposition 5.8. First we consider the symmetric case (5.25), which is simpler to treat since the operator T pτ q is self-adjoint. To this end we set τ “ t{n, Uτ “ Uτ pHq, and we use the identity (5.3) T n pτ q ´ Uτn “ T pτ qn´1 pT pτ q ´ Uτ q `

n´1 ÿ

T pτ qn´m´1 pT pτ q ´ Uτ qUτm .

n“1

Since by Lemma 5.13 the operator p1 ´ T pτ qq´1 P LpHq for τ ą 0, we find that T n pτ q ´ Uτn “ T pτ qn´1 pT pτ q ´ Uτ q n´1 ÿ

`

T pτ qn´m´1 p1 ´ T pτ qqtp1 ´ T pτ qq´1 pT pτ q ´ Uτ qUτm u,

m“1

which in turn leads to the estimate }T pτ qn ´ Uτn } ď }T pτ qn´1 pT pτ q ´ Uτ q} n´1 ÿ

`

}T pτ qn´m´1 p1 ´ T pτ qq}}p1 ´ T pτ qq´1 pT pτ q ´ Uτ qUτm }. (5.50)

m“1

Taking into account (5.30) we get for the last factor in the sum (5.50) the estimate }p1 ´ T pτ qq´1 pT pτ q ´ Uτ qUτm } ď p1 ´ C1{2 S1 bq´1 t}pT pτ q ´ Uτ qUτm } C1{2 ´1 }A pT pτ q ´ Uτ qUτm }u. ` τ

(5.51) (5.52)

Note that the identity pT pτ q ´ Uτ qUτm “ pT pτ q ´ Uτ qA´1 AH ´1 HUτm implies the estimate (cf. (5.35)): }pT pτ q ´ Uτ qUτm } ď }pT pτ q ´ Uτ qA´1 }}AH ´1 }}HUτm }.

(5.53)

By virtue of Lemma 5.11 we have }HUτm } ď pe τ mq´1 . Then (5.53) yields }pT pτ q ´ Uτ qUτm } ď }A´1 pT pτ q ´ Uτ q}pep1 ´ bqτ mq´1 ,

(5.54)

156


where we used that }AH ´1 } “ }p1 ` BA´1 q´1 } ď p1 ´ bq´1 . Similarly, }A´1 pT pτ q ´ Uτ qUτm } ď }A´1 pT pτ q ´ Uτ qA´1 }pep1 ´ bqτ mq´1 .

(5.55)

Inserting (5.54) and (5.55) into (5.52), we obtain }p1 ´ T pτ qq´1 pT pτ q ´ Uτ qUτm } ”1 ı C1{2 ď }A´1 pT pτ q ´ Uτ q} ` 2 }A´1 pT pτ q ´ Uτ qA´1 } τ τ ˆ pep1 ´ C1{2 S1 bqp1 ´ bqmq´1 .

(5.56)

Now taking into account Lemmata 5.15 and 5.16 we deduce from (5.56) the following estimate of the last factor in in the sum (5.50) }p1 ´ T pτ qq´1 pT pτ q ´ Uτ qUτm } ď

L3 1 , p1 ´ C1{2 S1 bqp1 ´ bq m

(5.57)

where we set L3 :“ pL1 ` C1{2 L2 qe´1 . To estimate in the sum (5.50) the factor entirely dependent on the self-adjoint operator T pτ q it is sufficient to apply Lemma 5.10: }T pτ qn´m´1 p1 ´ T pτ qq} ď

1 , n´m

(5.58)

where n “ 2, 3, . . . and m “ 0, 1, . . . , n ´ 1. Inserting the estimates (5.57) and (5.58) in (5.50) we get }T pτ qn ´ Uτn } ď }T pτ qn´1 pT pτ q ´ Uτ q} ` Since

(5.59) n´1 ÿ

1 L3 . p1 ´ C1{2 S1 bqp1 ´ bq m“1 pn ´ mqm

(5.60)

n´1 ÿ

n´1 1 2 ÿ 1 2 lnpnq “ ď r1 ` lnpn ´ 1qs ď 4 pn ´ mqm n m n n m“1 m“1

for n “ 3, 4, . . ., we get for (5.60) the inequality }T pτ qn ´ Uτn } ď }T pτ qn´1 pT pτ q ´ Uτ q} `

4L3 lnpnq . p1 ´ C1{2 S1 bqp1 ´ bq n

(5.61)

It remains to estimate }T pτ qn´1 pT pτ q ´ Uτ q}. Again we refer to the existence of p1 ´ T pτ qq´1 , see Lemma 5.13, to get the estimate }p1 ´ T pτ qq´1 pT pτ q ´ Uτ q} ˆ ˙ C1{2 ´1 1 ď }T pτ q ´ Uτ } ` }A pT pτ q ´ Uτ q} 1 ´ C1{2 S1 b τ ˆ ˙ C 1 1{2 ď 2` }A´1 pT pτ q ´ Uτ q} . 1 ´ C1{2 S1 b τ

(5.62)


157

Then by Lemma 5.15 we obtain from (5.62) that }p1 ´ T pτ qq´1 pT pτ q ´ Uτ q} ď

1 p2 ` C1{2 L1 q. 1 ´ C1{2 S1 b

(5.63)

Inserting (5.63) in the inequality }T pτ qn´1 pT pτ q ´ Uτ q} ď }T pτ qn´1 p1 ´ T pτ qq}}p1 ´ T pτ qq´1 pT pτ q ´ Uτ q} we get }T pτ qn´1 pT pτ q ´ Uτ q} ď

2 ` C1{2 L1 }T pτ qn´1 p1 ´ T pτ qq}. 1 ´ C1{2 S1 b

(5.64)

Since T pτ q is self-adjoint, we can use Lemma 5.10 to estimate the right-hand side of (5.64), which gives }T pτ qn´1 pT pτ q ´ Uτ q} ď

2 ` C1{2 L1 1 . 1 ´ C1{2 S1 b n

(5.65)

Inserting (5.65) in (5.61) we get assertion (5.25): }T pτ qn ´ Uτn } ď

LT lnpnq , p1 ´ C1{2 S1 bqp1 ´ bq n

n “ 3, 4, . . . ,

(5.66)

for LT :“ p2 ` C1{2 L1 qp1 ´ bq ` 4L3 . Now we deal with the estimate (5.24). To this aim we use the identity P pτ qn ´ T pτ qn “

n´1 ÿ

P pτ qn´m´1 pP pτ q ´ T pτ qqT pτ q.

m“0

Note that P pτ q ´ T pτ q “ f 1{2 pτ Aqrf 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aqs. Then (5.67) can be rewritten as P pτ qn ´ T pτ qn “ f 1{2 pτ Aq

n´1 ÿ

T pτ qn´m´1 rf 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ AqsT pτ qm .

m“0

Since by Lemma 5.13 we have 1 “ p1 ´ T pτ qqp1 ´ T pτ qq´1 “ p1 ´ T pτ qq´1 p1 ´ T pτ qq,

(5.67)

158


the last identity may be recast as P pτ qn ´ T pτ qn “ f 1{2 pτ Aq

n´1 ÿ

T pτ qn´m´1 p1 ´ T pτ qqp1 ´ T pτ qq´1

m“0

ı ” ˆ f 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aq ˆ p1 ´ T pτ qq´1 p1 ´ T pτ qqT pτ qm . This yields the estimate }P pτ qn ´ T pτ qn } n´1 ÿ

ď

(5.68)

}T pτ qn´m´1 p1 ´ T pτ qq}

m“0

› › › › ˆ ›p1 ´ T pτ qq´1 rf 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aqsp1 ´ T pτ qq´1 › ˆ }p1 ´ T pτ qqT pτ qm }. Again by Lemma 5.13, see (5.30), we find that › › › › ›p1 ´ T pτ qq´1 rf 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aqsp1 ´ T pτ qq´1 › „› › 1 › 1{2 › 1{2 ď pτ Aqgpτ Bq ´ gpτ Bqf pτ Aq ›f › p1 ´ C1{2 S1 bq2 ¯› 2C1{2 ›› ´1 ´ 1{2 › ` f pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aq › ›A τ 2 › › ´ ¯ C1{2 › › ` 2 ›A´1 f 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aq A´1 › . τ

(5.69)

Since f 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aq “ p1 ´ gpτ Bqqf

1{2

pτ Aq ´ f

1{2

pτ Aqp1 ´ gpτ Bqq,

(5.70) (5.71)

we get A´1 rf 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aqs

(5.72)

“ A´1 Bp1 ´ gpτ BqqB ´1 f 1{2 pτ Aq ´ f 1{2 pτ AqA´1 Bp1 ´ gpτ BqqB ´1 .

(5.73)

On the other hand, since (5.71) can also be written as f 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aq “ p1 ´ f 1{2 pτ Aqqp1 ´ gpτ Bqq ´ p1 ´ gpτ Bqqp1 ´ f 1{2 pτ Aqq,


159

we get A´1 rf 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ AqsA´1 “ p1 ´ f 1{2 pτ AqqA´1 p1 ´ gpτ BqqB ´1 BA´1 ´ A´1 Bp1 ´ gpτ Bqq ˆ B ´1 p1 ´ f 1{2 pτ AqqA´1 .

(5.74)

Now we can estimate the right-hand side of (5.69). For the first term, by condition (iii), we simply get › › › 1{2 › (5.75) ›f pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aq› ď 2. The estimate of the second term needs (5.11), (5.18) and (5.73), and we obtain › 1 ›› ´1 1{2 › ›A rf pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aqs› τ 1 1 ď }BA´1 } }p1 ´ gpτ BqqB ´1 } ` }BA´1 } }p1 ´ gpτ BqqB ´1 } τ τ ď 2bS1 .

(5.76)

To estimate the third term in (5.69) we use (5.74) and (5.11), (5.16), (5.18), and (5.47), which yields › 1 ›› ´1 1{2 1{2 ´1 › rf pτ Aqgpτ Bq ´ gpτ Bqf pτ AqsA ›A › τ2 1 1 ď 2}BA´1 } }p1 ´ f 1{2 pτ AqqA´1 } }p1 ´ gpτ BqqB ´1 } τ τ ď 2bC1 S1 .

(5.77)

Inserting (5.75)–(5.77) in (5.69), we obtain L4 , p1 ´ C1{2 S1 bq2 (5.78) 2 where L4 :“ 2 ` 2bp2C1{2 S1 ` C1{2 C1 S1 q. By virtue of (5.68) and (5.78) we get › › › › ›p1 ´ T pτ qq´1 rf 1{2 pτ Aqgpτ Bq ´ gpτ Bqf 1{2 pτ Aqsp1 ´ T pτ qq´1 › ď

}P pτ qn ´ T pτ qn } ď

n´1 ÿ L4 }T pτ qn´m´1 p1 ´ T pτ qq}}p1 ´ T pτ qqT pτ qm }. (5.79) p1 ´ C1{2 S1 bq2 m“0

Applying Lemma 5.10 to the estimate (5.79) we find that }P pτ qn ´ T n pτ q} ď

n´1 ÿ L4 1 . 2 p1 ´ C1{2 S1 bq m“0 pn ´ mqpm ` 1q

(5.80)

160


Since n´1 ÿ

n ÿ 1 2 1 lnpnq “ ď4 , pn ´ mqpm ` 1q n ` 1 m n m“0 m“1

n “ 3, 4, . . . ,

estimate (5.80) implies that }P pτ qn ´ T pτ qn } ď

4L4 lnpnq , p1 ´ C1{2 S1 bq2 n

n “ 3, 4, . . .

(5.81)

By (5.66), (5.81), and the inequality }P pτ qn ´ Uτn } ď }P pτ qn ´ T pτ qn } ` }T pτ qn ´ Uτn } we finally obtain the assertion (5.24): }P pτ qn ´ Uτn } ď

LP lnpnq , p1 ´ C1{2 S1 bq2 p1 ´ bq n

where LP :“ LT p1 ´ C1{2 S1 bq ` 4L4 p1 ´ bq.

(5.82) l

Remark 5.17. In contrast to the Lie product formulae (5.1) and (5.2), the error bound estimates in Proposition 5.8 are (up to coefficients defined by the Kato functions K and the powers of p1 ´ C1{2 S1 bq) the same for symmetrised and nonsymmetrised versions of them. Remark 5.18. Note that condition (5.10) is not essential. If A and B are semibounded from below by, respectively, ´γA 1 and ´γB 1, then the operators A˜ “ ˜ “ B`pγB `1q1 satisfy (5.10). The proof goes through verbatim A`pγA `1q1 and B and gives (5.24) and (5.25) uniformly on any interval r0, t0 s (that is, locally uniformly), with LP pt0 q “ LP epγA `1qt0 epγB `1qt0 and LT pt0 q “ LT epγA `1qt0 epγB `1qt0 . Similarly, if a ‰ 0 in (1.101), one can eliminate a by modifying b, i.e., by taking b1 ą b and shifting A to obtain (5.11).

5.2.4 Optimal rate of convergence in the operator norm We conclude this section by a discussion of the optimality of the rate of convergence rate for the self-adjoint Trotter-Kato product formulae, since the rate Oplnpnq{nq in the error bound estimates (5.24) and (5.25) from Proposition 5.8 is not the best possible. Proposition 5.19. Let A and B be non-negative self-adjoint operators such that the operator sum C :“ A ` B on dom C “ dom A X dom B is also self-adjoint. Then: (a) the exponential Trotter-Kato product formula }pe´tA{n e´tB{n qn ´ e´tC } “ Opn´1 q,

n Ñ 8,

(5.83)


161

converges with the same rate Opn´1 q as its symmetrised version }pe´tB{2n e´tA{n e´tB{2n qn ´ e´tC } “ Opn´1 q,

n Ñ 8,

(5.84)

uniformly on each compact t-interval in r0, 8q. If the operator C is strictly positive, then this holds uniformly on r0, 8q; (b) the condition of self-adjointness and the error bound Opn´1 q are ultimate optimal. The proof of this proposition requires a method that is more refined than that in Proposition 5.8. This aspect of the Trotter-Kato product formulae theory is out of the scope of the present book, see Notes in Section 5.6. In a few remarks that follow we comment only the optimality claimed in Proposition 5.19 (b). Remark 5.20. One cannot relax the self-adjointness assumption on the operator sum C :“ A ` B to essential self-adjointness (even enhanced by the additional condition of Kato A-form smallness of B), since there is an example of a couple of non-negative self-adjoint operators A and B such that C on dom A X dom B is essentially self-adjoint and that lim inf }pe´tA{n e´tB{n qn ´ e´tC } ě Dptq, nÑ8

where Dptq ą 0 is continuous for t ą 0. Note that in this case we still have the well-known strong convergent Trotter product formula lim pe´tA{n e´tB{n qn u “ e´tC u,

nÑ8

for any u P H. Here self-adjoint operator C is the closure of the operator sum C :“ A ` B. Remark 5.21. In general, for noncommuting operators A and B the error bound (5.83) cannot be improved, i.e., to be made better than for matrices in the Lie product formula (5.1). Hence, the estimate Opn´1 q is ultimate optimal. This is much less evident for symmetrised Trotter product formula, since for matrices or bounded generators one obtains the rate of convergence Opn´2 q (5.2). Proposition 5.22. There exists an example of a couple of unbounded operators A and B verifying conditions of Proposition 5.19, such that }pe´tB{2n e´tA{n e´tB{2n qn ´ e´tC } ě Lptqn´1 , where Lptq ą 0 is a continuous function for t ą 0. This proposition proves optimality of the error bound Opn´1 q for the rate of convergence of symmetric Trotter product formula (5.84).

162


Remark 5.23. We advise against a conclusion that for any couple of noncommuting unbounded operators A and B verifying conditions of Proposition 5.19 the rate of convergence Opn´1 q is the best possible for the symmetric Trotter product formula. There are examples of Schrödinger C0 -semigroups that manifest for symmetric Trotter product formula a better rate of convergence than Opn´1 q. Whereas for the nonsymmetric case the error bound (5.83) is optimal, see Notes in Section 5.6 and Appendix D. Now we consider a sub-class of generic Kato functions K, which we denote p β , Appendix C (Section C.1). by K p β if: Definition 5.24. We say that h P K (i) It is a Borel measurable function h : r0, 8q Ñ r0, 1s such that hp0q “ 1, and h1 p`0q “ ´1. (ii) There exist ε ą 0 and δpεq ă 1, such that hpsq ď 1 ´ δpεq for s ě ε, and rhsβ :“ sup są0

|hpsq ´ 1 ` s| ă 8, sβ

for 1 ă β ď 2.

The standard examples are f pxq “ e´x and f pxq “ p1 ` a´1 xqá , for a ą 0, p β with Kα . see the next subsection 5.2.5 and Appendix C to compare K Then the following assertion extends Proposition 5.19 to the Trotter-Kato product formulae. p β with β “ 2, and let A, B be positive self-adjoint Proposition 5.25. Let f, g P K operators in H such that the operator sum C :“ A`B is self-adjoint on the domain dom C :“ dom A X dom B. Then the Trotter-Kato product formulae converge in the operator norm: }rgptB{2nqf ptA{nqgptB{2nqsn ´ e´tC } “ Opn´1 q , }rf ptA{nqgptB{nqsn ´ e´tC } “ Opn´1 q,

n Ñ 8.

and the error bound Opn´1 q is optimal. We note that in fact the Propositions 5.19 and 5.25 (when properly extended) include the majority of known related results. In all these cases, the operator sums of the two self-adjoint operators under study are self-adjoint, see Proposition 5.8 and Notes in Section 5.6. Nevertheless, these propositions do not cover all known (optimal) results even in the case of strongly continuous self-adjoint semigroups. For example, this is the case when the sum of operators A and B (Remark 5.20) exists only in the sense of quadratic forms, see results in Sections 5.3 and 5.4. In the next subsection we study optimality for the case when the operators A and B are related by a fractional smallness conditions.


163

5.2.5 Optimal rate: fractional conditions The fractional smallness conditions for an operator B with respect to an operator A that we consider below are the following: Let the self-adjoint operators A ě 1 and B ě 0 in a Hilbert space H be such that for α P p1{2, 1s and b P p0, 1q they satisfy (i) dom Aα Ď dom B α and }B α u} ď b}Aα u}, u P dom Aα , 9 (ii) dom H α Ď dom Aα , H :“ A`B. Definition 5.26. For each α P p1{2, 1s we denote by Kα the class of Kato functions f P K defined by the following conditions: xf pxq1{2α ă `8, xą0 1 ´ f pxq ˇ ˇ ˇ 1ˇ :“ sup ˇˇp1 ´ f pxqq ˇˇ ă `8, x xą0 ˇˆ ˙ ˇ ˇ 1 1 ˇˇ :“ sup ˇˇ f pxq ´ ă `8, 1 ` x x2 ˇ xą0 ˇ ˇ ˇ 1ˇ :“ sup ˇˇp1 ´ gpxqq ˇˇ ă `8, x xą0 ˇˆ ˙ ˇ ˇ 1 1 ˇˇ :“ sup ˇˇ gpxq ´ ă `8 . 1 ` x x2 ˇ xą0

C1{2α :“ sup C1 C2 S1 S2

(5.85) (5.86) (5.87) (5.88) (5.89)

To link the fractional smallness with properties of Kato functions f P Kα we assume also the condition b1{α C1{2α S1 ă 1 .

(5.90)

Note that the class Kα and condition (5.90) are tuned according to the fractional conditions (i), (ii), and that for α “ 1 they coincide with the sub-class of Kato functions: (5.15)–(5.19), and the smallness conditions (5.11), (5.20), for Proposition 5.8. Proposition 5.27. Let the self-adjoint operators A ě 1 and B ě 0 in a Hilbert space H be such that for α P p1{2, 1q and b P p0, 1qthe fractional smallness conditions 9 is (i) and (ii) are satisfied. Then the semibounded from below operator H :“ A`B densely defined self-adjoint form-sum of A and B and for Kato functions f, g P Kα that satisfy (5.90) one gets for some c ą 0 the operator-norm estimate: }pf ptA{nqgptB{nqqn ´ e´tH } ď uniformly for t ě 0.

c , n2α´1

n ě 2,

(5.91)

164


The same estimate for asymptotic Opn1´2α q is also true for symmetrised Trotter-Kato approximants, cf. (5.23): pP pt{2nqP ˚ pt{2nqqn “ pf ptA{2nqgptB{nqf ptA{2nqqn , T pt{nqn “ pf 1{2 ptA{nqgptB{nqf 1{2 ptA{nqqn , F pt{nqn “ pg 1{2 ptB{nqf ptA{nqg 1{2 ptB{nqqn . Proof. Recall that if positive self-adjoint operators C, D verify conditions: dom C Ď dom D and }Du} ď b}Cu}, for b P p0, 1q, and u P dom C, then the Heinz-Kato inequality yields: dom C β Ď dom Dβ and }Dβ u} ď bβ }C β u}, for any u P dom C β and β P p0, 1q. Applying this statement to the operators C “ Aα and D “ B α verifying condition (i), we obtain for β “ θ{α and u P dom Aθ that dom Aθ Ď dom B θ ,

}B θ u} ď bθ{α }Aθ u},

for θ P p0, αs.

(5.92)

If one takes θ “ 1{2, then (5.92) implies dom A1{2 X dom B 1{2 “ dom A1{2 Ď dom B 1{2 .

(5.93)

Since A and consequently A1{2 are densely defined, by (5.93) we obtain that dom A1{2 X dom B 1{2 “ dom H 1{2 if the operator H with dense domain is defined as the form-sum of A and B. By the construction of the form-sum, the 9 is self-adjoint and H ě 1. operator H “ A`B Due to condition (ii) the arguments in the proof of Propositions 5.8 can be refined. This yields for the operator-norm convergence of the Trotter-Kato product formulae the error bound estimate (5.91) uniformly for t ě 0. The corresponding argument is quite involved, see Notes in Section 5.6. We comment that if one keeps only condition (i) and skips (ii), then arguing as in the proof of Propositions 5.8 one obtains the error bound estimate, which (similar to (5.24)) has the lnpnq correction: }pf ptA{nqgptB{nqqn ´ e´tH } ď c1 for some c1 ą 0 and uniformly for t ě 0.

lnpnq , n2α´1

n ě 2,

(5.94) l

On the other hand, for exponential Kato function, which evidently belongs to Kα , the following assertion is valid. Proposition 5.28. For each α P p0, 1q and b P p0, 1q, there exist operators A and B in a Hilbert space H such that they satisfy the conditions of Propositions 5.27 but also the estimates: ˇˇ ` ˘n ˇˇˇˇ ct ˇˇ ´tpA`Bq 9 ´ e´tA{n e´tB{n ˇˇ ě 2α´1 , for α P p1{2, 1q , ˇˇe n


165

and for large n P N. Moreover, there are A and B such that ˇˇ ˘n ˇˇˇˇ ` ˇˇ 9 lim inf ˇˇe´tpA`Bq ´ e´tA{n e´tB{n ˇˇ ě dt , for α P p0, 1{2s, nÑ8

where ct and dt are positive and continuous for t ą 0. These estimates from below are also valid for self-adjoint symmetrised Trotter approximants, for example: ` ˘n F pt{nqn “ e´tB{2n e´tA{n e´tB{2n , n P N. Remark 5.29. First, we note that Proposition 5.28 implies that for α P p1{2, 1q the error bound (5.91) in the Proposition 5.27 is optimal. This means that it is the best possible in the abstract setting for asymptotics in the power of n P N. Note that it could be refined for symmetrised approximates since symmetrisation improves the convergence in the Lie product formula (5.2). Second, for α “ 1 the condition (i) yields dom H “ dom A. This case is the subject of Propositions 5.8 and further improvements in subsection 5.2.4. See Proposition 5.25 about optimality of the rate Opn´1 q for Kato functions of class pβ . K Third, the case α P p0, 1{2q is an example for which the Trotter product formula does not hold in the operator-norm convergence sense, but it holds in the strong operator topology by the standard Trotter theorem. This means that the operator-norm convergence of the product formula cannot be extended to the cases α ă 1{2 in the abstract setting. As (5.90) and Propositions 5.27, 5.28 make clear, the case α “ 1{2 needs a special analysis, see Notes in Section 5.6. Proposition 5.30. Let self-adjoint operators A ě 1 and B ě 0 in a Hilbert space H be such that (5.95) dom A1{2 Ď dom B 1{2 . Let the Kato functions f, g P K. If in addition to (5.95) the operator B 1{2 is relatively A1{2 -compact, that is, B 1{2 A´1{2 P C8 pHq,

(5.96)

and the Kato function f satisfies the condition C :“ sup xą0

xf pxq ă `8, 1 ´ f pxq

(5.97)

then the Trotter-Kato product formulae converge in the operator norm locally uni9 formly away from zero to the C0 -semigroup with self-adjoint generator H “ A`B. Note that condition (5.97) coincides with (5.85), for α “ 1{2, and with (5.146). The corresponding sub-class of the Kato functions includes for example, f pxq “ e´x and f pxq “ p1 ` xq´1 , Appendix C (Section C.3).

166


Remark 5.31. Suppose the densely defined in a Hilbert space H operators K and L with dom K Ď dom L are such that L pK ´ z1q´1 P C8 pHq for some (and then all) z P ρpKq, where ρpKq is the resolvent set of K. Then we say that L is relatively compact with respect to K, or simply that it is K-compact. We note that, if L is K-compact and closable, then L P P0` pKq. This means that L is K-bounded with the relative bound b “ 0` , Definition 1.50. Another class of P0` -perturbations will be considered in Remark 5.48. For other classes of infinitesimally small unbounded perturbations, see Section 4.4. Since B 1{2 is closed, one gets that under the condition of relative compactness (5.96) the operator B 1{2 is relatively A1{2 -bounded with relative bound b “ 0` , that is, infinitesimally small. Then there is no difference between the case α “ 1{2 and α P p0, 1{2q with conditions (i),(ii). We conclude this section by a statement, that extends Proposition 5.27 from p β , Definition 5.24. Moreover, it relaxes the Kα to a larger class of Kato functions K smallness condition in (i). For this purpose we modify the fractional conditions in (i) and (ii), which still ensure the optimal rate of convergence of the Trotter-Kato product formulae. Proposition 5.32. Let the self-adjoint operators A ě 1 and B ě 0 in a Hilbert 9 is the densely defined space H be such that non-negative operator H :“ A`B self-adjoint form-sum of A and B. Let us assume that for some α P p1{2, 1q: (i) dom A1{2 Ď dom B 1{2 , (ii) dom H α Ď dom Aα X dom B α , p β , where β “ 2α, (iii) the Kato functions f, g P K (iv) supyěx f pyq ă 1 for x ą 0. Then for some c ą 0 one gets the operator-norm estimate }pf ptA{nqgptB{nqqn ´ e´tH } ď

c , n2α´1

n ě 2,

(5.98)

uniformly for t ě 0. Remark 5.33. (a) The same example as the one that is the source of Proposition 5.28 shows that the error bound estimate (5.98) cannot be improved under the conditions of the proposition, i.e., it is ultimate optimal. (b) The application of the Heinz-Kato inequality to condition (i) of Proposition 5.27 yields (5.92), and finally the inclusion (5.93), which coincides with (i) Proposition 5.32. Hence, the smallness condition is relaxed. (c) Note that by Definition 5.26 of the class Kα , the Kato functions f, g P Kα verify (5.87) and (5.89) for any α P p1{2, 1s. This corresponds to the conditions: rf sβ“2 ă 8 and rgsβ“2 ă 8, which are stronger than condition (iii) in Proposition 5.32. Moreover, besides f pxq ă 1 the condition (5.85) demands that f decays at

5.3. Operator-norm convergence: self-adjointand non-self-adjoint semigroups167 infinity: f pxq “ Op1{x2α q. By contrast, condition (iv) of Proposition 5.32 requires only the first. For further details and references see Notes in Section 5.6.

5.3 Operator-norm convergence: self-adjoint and non-self-adjoint semigroups The conditions imposed in the previous Section 5.2 on the pair of self-adjoint generators tA, Bu, namely that operator A ` B is self-adjoint, and on the coup β , seem to be too restrictive. However, they ple of Kato functions, that f, g P K ensure in particular the optimal estimate of the rate of convergence in the operator norm for the Kato-Trotter product formulae. If one does not care about the rate of convergence, then for strongly continuous self-adjoint semigroups (or for non-exponential Kato functions f, g from K) there is an alternative approach to establish the operator-norm convergence of the Trotter-Kato product formulae. This approach covers a fairly large family of pairs of self-adjoint generators A and B since self-adjointness of the algebraic sum A`B is not required, but it gives no estimate for the rate of convergence, see comments in Remark 5.3. We present the main statement in subsection 5.3.1 Preliminaries and Proposition 5.36. The proof of the Proposition 5.36 is quite involved and so it needs certain a la preparations. First in subsection 5.3.2 Operator-norm approximation theorem ` Chernoff, we prove the lifting to the operator-norm convergence of some results that are known in the strong operator topology. This allows us to prove in subsection 5.3.3 Key preliminary statement, a central for our approach Proposition 5.40, which in the case of self-adjoint generators A and B is applicable for the operator-norm topology. The conditions on A and B are mild and demand only . existence of the (perhaps not densely) defined in H form-sum A ` B. Then for arbitrary Kato-functions f, g P K we prove in subsection 5.3.4 Proof of Proposition 5.36 a result about the operator-norm convergence of Trotter-Kato product formulae to a degenerate semigroup without estimates of the rate of convergence. It is curious that Proposition 5.36 allows in some sense a converse statement: see Proposition 5.45 for the case when f is a regular Kato-function (Appendix C). In subsection 5.3.5 Rate of convergence: non-self-adjoint semigroups, we consider extensions of the convergence in the operator norm of the Trotter product formula (i.e., f, g are exponentials) to non-self-adjoint C0 -semigroups. These extensions include the error bound estimates for the rates of convergence. The estimates are possible due to a hierarchy condition: a relative smallness of B with respect to A and of B ˚ with respect to A˚ . In Proposition 5.47 the generator A is supposed to be an m-sectorial operator and the small B to be m-accretive. To relax the smallness condition and to improve the rate of convergence the condition on

168


the generator A in Proposition 5.49 is stricter: A is assumed to be a non-negative self-adjoint operator. We lift these results to the trace-norm convergence of the product formulae for non-self-adjoint Gibbs semigroups in Section 5.5. Two lifting methods that we use there allow to prove the trace-norm convergence without and with error bound estimates only for exponential function, that is, for the Trotter product formula.

5.3.1 Preliminaries and Proposition 5.36 Let A ě 0 and B ě 0 be non-negative self-adjoint operators in a Hilbert space H, which are not hierarchically ordered. In contrast to the conditions of Propositions 5.19, here it may happen that dom A X dom B “ t0u whereas dom A1{2 X dom B 1{2 is nontrivial. If one denotes the closure dom A1{2 X dom B 1{2 by H0 Ď H, then the . non-negative self-adjoint form-sum operator H :“ A ` B is densely defined in the Hilbert subspace H0 , see Remark 4.29. Recall that under these conditions the Trotter product formula converges in the strong operator topology: ` ˘n s-lim e´tA{n e´tB{n “ e´tH P0 , t ą 0, (5.99) nÑ8

to the degenerate semigroup te´tH P0 utą0 , see Definition 1.24. Here P0 denotes the orthogonal projection of H onto H0 . Moreover, the product formula (5.99) is valid not only for strongly continuous self-adjoint semigroups, that is, for exponentials corresponding to generators A ě 0 and B ě 0, but also for Kato functions f, g P K (or P Kβ ): n

s-lim pf ptA{nqgptB{nqq “ e´tH P0 , nÑ8

t ą 0,

(5.100)

that is, for Trotter-Kato product formula. Similar to the programme outlined at the end of Section 5.1, in Remark 5.2 a natural problem is to find conditions such that the strong convergence of the Trotter (5.99) and the Trotter-Kato (5.100) product formulae can be lifted to the limit in the operator-norm topology. The aim of the Section 5.3 is to collect results concerning sufficient (and necessary) conditions, which guarantee the operator-norm convergence of the TrotterKato product formulae for a given pair of non-negative self-adjoint operators A, B and for admissible Kato functions f , g. However, in contrast to Section 5.2 these results do not yield estimates for the corresponding error bounds. Although the result in Proposition 5.36 is valid for arbitrary Kato functions f, g P K, to formulate the conditions for this assertion one needs certain functions associated with them. Note that a specific subclass of K selected by regularity will be needed only for a converse statement in Proposition 5.45. Definition 5.34. With any pair of Kato functions f, g P K we associate Borel functions f0 , g0 on R` 0 constructed as follows:

5.3. Operator-norm convergence

169

(1) First we define two auxiliary functions ˆ ˙ 1 ϕpxq :“ x´1 ´ 1 and ψpxq :“ x´1 p1 ´ gpxqq . f pxq

(5.101)

Here it is supposed that f pxq ą 0 for (5.101) to be correct. (2) Using (5.101) we define a couple of new auxiliary functions ˆ ˙ 1 0 ď ϕ0 pxq :“ inf s´1 ´1 , 0ăsďx f psq 0 ď ψ0 pxq :“ inf s´1 p1 ´ gpsqq . 0ăsďx

(5.102) (5.103)

Here f pxq “ 0 is allowed and we make the convention that 0´1 :“ `8. (3) Finally for any x P R` 0 we set " 1, if x “ 0, f0 pxq :“ p1 ` xϕ0 pxqq´1 , if x ą 0, and

" g0 pxq :“

1, if x “ 0, p1 ´ xψ0 pxqq , if x ą 0.

(5.104)

(5.105)

We note that 0 ď f pxq ď f0 pxq ď 1 and 0 ď gpxq ď g0 pxq ď 1, x P R` 0 , and that in turn the functions f0 , g0 belong to K. More details and proofs of properties concerning these functions one can found in Appendix C (Section C.2). For example, if f pxq “ e´x , x ě 0, then ϕ0 pxq “ 1, which yields f0 pxq “ p1 ` xq´1 , whereas if f pxq “ 1{p1 ` xq, x ě 0, then ϕ0 pxq “ 1, and one gets again f0 pxq “ p1 ` xq´1 . To formulate Proposition 5.36 and similar results about convergence of the Trotter-Kato product formulae we use the following terminology, cf. Proposition 5.30. Definition 5.35. We say that a sequence of families of bounded operators: tΦn ptqutě0,nPN , converges to the family tΦptqutě0 in the norm topology } ¨ }τ locally uniformly away from t0 ě 0 if lim

sup

nÑ8 ra,bsĂpt ,8q 0

}Φn ptq ´ Φptq}τ “ 0 ,

for any compact ra, bs Ă pt0 , 8q. Proposition 5.36. Let A .and B be non-negative densely defined self-adjoint operators in H and H “ A ` B be the form-sum self-adjoint operator defined in the subspace H0 . Let f and g be arbitrary Kato functions from K. If the condition f0 pt0 Aq P C8 pHq

(5.106)

170


is satisfied for some t0 ą 0, then the Trotter-Kato product formula (5.100) and all other formulae generated by f and g converge in the operator-norm topology locally uniformly away from zero (t0 “ 0) to degenerate semigroup te´tH P0 utą0 , and p10 ` Hq´1 P C8 pH0 q, 10 “ 1 æ H0 . (5.107) Here 10 :“ 1æH0 denotes restriction of the unit operator to the subspace H0 “ P0 pHq. Remark 5.37. Note that the statements below in subsections 5.3.3 and 5.3.4 are valid for general pairs f, g P K. We shall let the reader know when our arguments are restricted in subsection 5.3.4 to regular Kato functions, or in subsection 5.3.5 to exponential Kato function.

5.3.2

Operator-norm approximation theorem à la Chernoff

As the first step toward the proof of Proposition 5.36 we lift to the operator-norm convergence some results that are well-known in the strong operator topology. Let tXpsqusą0 be a family of self-adjoint operators in H and let X0 be a self-adjoint operator in the subspace H0 Ď H. Recall that tXpsqusą0 converges in the uniform resolvent sense (or in the operator-norm resolvent sense) to X0 as s Ñ `0 if lim }pXpsq ´ z1q´1 ´ pX0 ´ z10 q´1 P0 } “ 0 , (5.108) sÑ`0

for one (and hence for all) nonreal z P C. Then the following operator-norm analogue of the Trotter-Kato strong convergence theorem holds for semigroups. Lemma 5.38. Let tXpsqusą0 be a family of non-negative self-adjoint operators in a Hilbert space H and let X0 be a non-negative self-adjoint operator in the closed subspace H0 Ď H. Then the following conditions are equivalent: (a) limsÑ`0 }pλ1 ` Xpsqq´1 ´ pλ10 ` X0 q´1 P0 } “ 0, for all λ ą 0 , (b) limrÑ`8 suptPra,bs }e´tXpt{rq ´ e´tX0 P0 } “ 0, for all ra, bs Ă p0, 8q. Proof. (a) ñ (b). We note that condition (a) implies the uniform resolvent convergence of tXpsqusą0 to X0 as s Ñ `0. Let φp¨q be a real continuous function defined on R which tends to zero as x Ñ ˘8. Then by a standard argument the uniform resolvent convergence yields lim }φpXpsqq ´ φpX0 q} “ 0 ,

sÑ`0

provided X0 is defined on H, i.e., H “ H0 . However, if H0 Ă H, then one sees that the proof still remains valid. So we find that lim }φpXpsqq ´ φpX0 qP0 } “ 0 .

sÑ`0

(5.109)


171

In particular, if φpxq “ e´t|x| , x P R, t ą 0, then lim }e´tXpsq ´ e´tX0 P0 } “ 0 .

(5.110)

sÑ`0

The set of elements te´t|x| utPra,bs , ra, bs Ď p0, 8q, is compact in C0 pRq, which is the Banach space of all real continuous functions such that: limxÑ˘8 φpxq “ 0, endowed with the supremum norm. Therefore, lim sup }e´tXpsq ´ e´tX0 P0 } “ 0 ,

sÑ`0 tPra,bs

(5.111)

for all intervals ra, bs Ď p0, 8q. Since for s “ t{r one has s ď b{r, from (5.111) we immediately obtain (b). (b) ñ (a). From (b) it follows that lim }e´Xpsq ´ e´X0 P0 } “ 0.

(5.112)

sÑ`0

Setting Y psq “ e´Xpsq and Y0 “ e´X0 one gets from (5.112) that the sequence tY psqusą0 converges in the uniform resolvent sense to Y0 as s Ñ `0. Let φpxq “ pλ ` | lnp|x|q|q´1 , x P R, λ ą 0. Then again by a standard argument, applied now to tY psqusą0 and Y0 , we obtain (a). l Lemma 5.38 enables us to prove the operator-norm generalisation of the Chernoff approximation theorem. Proposition 5.39. Let tΦpsqusě0 be a family of self-adjoint non-negative contractions on a Hilbert space H and let X0 be a self-adjoint operator on the subspace H0 Ď H. Define Xpsq :“ s´1 p1 ´ Φpsqq, s ą 0. Then the family tXpsqusą0 converges in the uniform resolvent sense to X0 as s Ñ `0 if and only if the sequence tΦpt{rqr urě1 , t ą 0, converges in the operator norm to e´tX0 P0 as r Ñ `8, uniformly in t on any compact interval in R` . Proof. Since supxPr0,1s |xr ´ e´rp1´xq | ď 1{r, for r ě 1, the spectral theorem yields }Φpt{rqr ´ e´tXpt{rq } ď

1 , r

r ě 1,

t ě 0.

(5.113)

Assume that tXpsqusą0 converges in the uniform resolvent sense to X0 for s Ñ `0. Then the representation Φpt{rqr ´ e´tX0 P0 “ Φpt{rqr ´ e´tXpt{rq ` e´tXpt{rq ´ e´tX0 P0 , yields the estimate }Φpt{rqr ´ e´tX0 P0 } ď }Φpt{rqr ´ e´tXpt{rq } ` }e´tXpt{rq ´ e´tX0 P0 }. Using (5.113) one obtains that sup }Φpt{rqr ´ e´tX0 P0 } ď tPra,bs

1 ` sup }e´tXpt{rq ´ e´tX0 P0 } , r tPra,bs

(5.114)

172


for any compact interval ra, bs Ď R` . Applying now Lemma 5.38 to (5.114) we complete this part of the proof. Conversely, assume that tΦpt{rqr urě1 converges in the operator norm to e´tX0 P0 , uniformly for any compact t-interval of R` . From e´tXpt{rq ´ e´tX0 P0 “ e´tXpt{rq ´ Φpt{rqr ` Φpt{rqr ´ e´tX0 P0 , one gets the inequality }e´tXpt{rq ´ e´tX0 P0 } ď }e´tXpt{rq ´ Φpt{rqr } ` }Φpt{rqr ´ e´tX0 P0 }. Now using again (5.113) we obtain the estimate sup }e´tXpt{rq ´ e´tX0 P0 } ď tPra,bs

1 ` sup }Φpt{rqr ´ e´tX0 P0 } , r tPra,bs

for any compact interval ra, bs Ď R` . Therefore, the condition (b) of Lemma 5.38 is satisfied, so, applying this lemma we establish the uniform resolvent convergence of tXpsqusą0 to X0 when s Ñ `0. l

5.3.3 Key preliminary statement Now let us introduce the operator-valued families T ptq :“ f ptAq1{2 gptBqf ptAq1{2 , Rptq :“ p1 ´ T ptqq t 1{2

F ptq :“ gptBq

´1

,

f ptAqgptBq ´1

Sptq :“ p1 ´ F ptqq t

1{2

,

,

t ě 0,

(5.115)

t ą 0,

(5.116)

t ě 0,

(5.117)

t ą 0.

(5.118)

We also define the family of subspaces Hptq :“ ranpf ptAqq Ď H, for t ě 0. Let tQptqutě0 , denote the corresponding family of orthogonal projections from H onto the subspaces tHptqutě0 . By the continuity of Kato functions at `0 one gets s-lim Qptq “ 1. tÑ`0

(5.119)

Note that for each t ě 0 the projection Qptq commutes with A. We set f˜ptAq :“ f ptAq æ Hptq : Hptq Ñ Hptq, t ě 0. Then by definition one obtains that kerpf˜ptAqq “ t0u and that 0 ď f˜ptAq ď 1ptq,

t ě 0,

(5.120)

where restrictions 1ptq:“1æ Hptq. All this motivates the introduction of the family of operators tM ptqutą0 , defined as mappings M ptq : Hptq Ñ Hptq, by M ptq :“

1 ˜ 1 rf ptAq´1 ´ 1ptqs ` Qptqr1 ´ gptBqsQptq, t t

(5.121)


173

on the t-dependent domain dom M ptq “ ranpf˜ptAqq Ď Hptq.

(5.122)

Finally, we recall that 10 “ 1 æ H0 is the restriction of the unit operator on the subspace H0 “ dom A1{2 X dom B 1{2 . Note that if for the Kato function f satisfies f pxq ą 0, x P R, then one readily gets that Hptq “ H

and Qptq “ 1,

t ě 0.

Now we are in position to formulate and to prove the key preliminary statement necessary for the proof of Proposition 5.36. Proposition 5.40. Let A and B be non-negative self-adjoint operators in a Hilbert space H and let f p¨q and gp¨q be Kato functions of class K. Then the following conditions are equivalent: (i) limrÑ`8 suptPra,bs }T pt{rqr ´ P0 e´tH P0 } “ 0,

ra, bs Ď R` ,

(ii) limtÑ`0 }pλ1 ` Rptqq´1 ´ P0 pλ10 ` Hq´1 P0 } “ 0,

λ ą 0,

(iii) limtÑ`0 }Qptqpλ1ptq ` M ptqq´1 Qptq ´ P0 pλ10 ` Hq´1 P0 } “ 0, (iv) limtÑ`0 }pλ1 ` Sptqq

´1

´1

´ P0 pλ10 ` Hq

P0 } “ 0,

(v) limrÑ`8 suptPra,bs }F pt{rqr ´ P0 e´tH P0 } “ 0,

λ ą 0,

λ ą 0,

ra, bs Ď R` .

Proof. We prove that piq ñ piiq ñ piiiq ñ pivq ñ pvq ñ piq. piq ñ piiq To prove this implication one simply has to set Φptq :“ T ptq for t ě 0, Xptq :“ Rptq for t ą 0, and then apply Proposition 5.39. piiq ñ piiiq Since by definitions (5.115), (5.116) and (5.121) one gets Rptq “

1 p1 ´ Qptqq ` f ptAq1{2 M ptqf ptAq1{2 , t

we obtain the representation f ptAq1{2 p1 ` Rptqq´1 f ptAq1{2 “ Qptqtf˜ptAq´1 ` M ptqu´1 Qptq. Next, we note that f˜ptAq´1 ` M ptq “ p1 ` tqtM ptq `

1 QptqgptBqQptqu ě 1ptq, 1`t

which finally yields the representation f ptAq1{2 p1 ` Rptqq´1 f ptAq1{2 1 1 QptqtM ptq ` QptqgptBqQptqu´1 Qptq. “ 1`t 1`t

(5.123)

174

Chapter 5. Product formulae for Gibbs semigroups Now we can prove that lim }f ptAq1{2 p1 ` Rptqq´1 f ptAq1{2 ´ P0 p10 ` Hq´1 P0 } “ 0.

tÑ`0

(5.124)

To this end we use the representation f ptAq1{2 p1 ` Rptqq´1 f ptAq1{2 ´ P0 p10 ` Hq´1 P0 “ f ptAq1{2 tp1 ` Rptqq´1 ´ P0 p10 ` Hq´1 P0 uf ptAq1{2 ` rf ptAq1{2 ´ 1sP0 p10 ` Hq´1 P0 f ptAq1{2 ` P0 p10 ` Hq´1 P0 rf ptAq1{2 ´ 1s, which gives the estimate }f ptAq1{2 p1 ` Rptqq´1 f ptAq1{2 ´ P0 p10 ` Hq´1 P0 } ď }p1 ` Rptqq´1 ´ P0 p10 ` Hq´1 P0 } ` 2}r1 ´ f ptAq1{2 s ˆ P0 p10 ` Hq´1 P0 }.

(5.125) .

Since by the construction of the operator H “ A ` B in the sense of form-sum one has dom H 1{2 Ď dom A1{2 , we see that r1 ´ f ptAq1{2 sp10 ` Hq´1 h “ t1{2 r1 ´ f ptAq1{2 sptAq´1{2 A1{2 p10 ` Hq´1{2 p10 ` Hq1{2 ˆ p10 ` Hq´1 h, for h P H0 , which in turn yields the estimate }r1 ´ f ptAq1{2 sp10 ` Hq´1 h} ď t1{2 C1{2 }h}, h P H, where we used that the Kato function f p¨q has the property (5.16): C1{2 :“ supxą0 p1 ´ f pxq1{2 q{x1{2 ă `8. This implies: }r1 ´ f ptAq1{2 sp10 ` Hq´1 } ď C1{2 t1{2 .

(5.126)

Then by piiq, (5.125) and (5.126) we obtain (5.124). Define the operator Mptq on the domain Hptq “ dom M ptq “ dom pQptqgptBqQptqq by Mptq :“ M ptq ` QptqgptBqQptq{p1 ` tq. Then taking into account (5.123) and (5.124) one finds that › › lim ›QptqMptq´1 Qptq ´ P0 p10 ` Hq´1 P0 › “ 0.

(5.127)

Hence, to show that piiq ñ piiiq (for λ “ 1) we have to verify that › › lim ›Mptq´1 ´ t1ptq ` M ptqu´1 › “ 0 .

(5.128)

tÑ`0

tÑ`0


175

To this aim we first use the identity Mptq´1 ´ t1ptq ` M ptqu´1 " “ t1ptq ` M ptqu´1 1ptq ´

* 1 QptqgptBqQptq Mptq´1 , 1`t

to establish that Mptq´1 ´ t1ptq ` M ptqu´1 “ `

t t1ptq ` M ptqu´1 Mptq´1 1`t

1 t1ptq ` M ptqu´1 Qptqr1 ´ gptBqsQptqMptq´1 . 1`t

Therefore, the (5.128) holds if › › lim ›r1 ´ gptBqsQptqMptq´1 › “ 0.

(5.129)

tÑ`0

.

To prove (5.129) we note that, by the construction of the operator H “ A ` B, one has dom H 1{2 Ď dom B 1{2 . This allows to obtain the representation r1 ´ gptBqsQptqMptq´1 “ r1 ´ gptBqs QptqMptq 1{2

`t

r1 ` gptBq

ˆ p10 ` Hq

´1{2

1{2

(5.130) ´1

´1

Qptq ´ P0 p10 ` Hq

sr1 ´ gptBq ´1{2

p10 ` Hq

1{2

´1{2

sptBq

B

P0

(

1{2

P0 ,

and the estimate }r1 ´ gptBq1{2 sP0 p10 ` Hq´1 P0 } ď S1{2 t1{2 .

(5.131)

Here we used the property (5.18): S1{2 :“ supxą0 p1 ´ gpxq1{2 q{x1{2 ă `8 of the Kato function g. From (5.130) and (5.131) we obtain the estimate }r1 ´ gptBqsQptqMptq´1 } › › ď ›QptqMptq´1 Qptq ´ P0 p10 ` Hq´1 P0 › ` 2S1{2 t1{2 .

(5.132)

Applying (5.127) to (5.132) we obtain (5.129), which yields (5.128). Then from (5.127) and (5.128) we obtain piiiq for λ “ 1, and consequently for any λ ą 0. piiiq ñ pivq. The identity gptBq1{2 Qptqpλ1ptq ` M ptqq´1 QptqgptBq1{2 “ pλ1 ` Sptqq´1 F ptq ` λpλ1 ` Sptqq´1 gptBq1{2 Qptqr1 ´ f ptAqsQptqpλ1ptq ` M ptqq´1 ˆ QptqgptBq1{2

(5.133)

176


shows that pλ1 ` Sptqq´1 “ tSptqpλ1 ` Sptqq´1 ` gptBq1{2 Qptqpλ1ptq ` M ptqq´1 QptqgptBq1{2 ` λpλ1 ` Sptqq´1 gptBq1{2 rf ptAq ´ 1sQptqpλ1ptq ` M ptqq´1 QptqgptBq1{2 . This allows us to rewrite pivq as pλ1 ` Sptqq´1 ´ P0 pλ10 ` Hq´1 P0 “ tSptqpλ1 ` Sptqq´1 ( ` gptBq1{2 Qptqpλ1ptq ` M ptqq´1 Qptq ´ P0 pλ10 ` Hq´1 P0 gptBq1{2 ` rgptBq1{2 ´ 1sP0 pλ10 ` Hq´1 P0 gptBq1{2 ` P0 pλ10 ` Hq´1 P0 rgptBq1{2 ´ 1s ` λpλ1 ` Sptqq´1 gptBq1{2 rf ptAq ´ 1s ( ˆ Qptqpλ1ptq ` M ptqq´1 Qptq ´ P0 pλ10 ` Hq´1 P0 gptBq1{2 ` λpλ1 ` Sptqq´1 gptBq1{2 rf ptAq ´ 1sP0 pλ10 ` Hq´1 P0 gptBq1{2 . Then one gets the estimate }pλ1 ` Sptqq´1 ´ P0 pλ10 ` Hq´1 P0 } › › t ď ` 2 ›Qptqpλ1ptq ` M ptqq´1 Qptq ´ P0 pλ10 ` Hq´1 P0 › λ ` 2}r1 ´ gptBq1{2 sP0 pλ10 ` Hq´1 P0 } ` }r1 ´ f ptAqsP0 pλ10 ` Hq´1 P0 }. Combining piiiq, (5.126) and (5.131) we obtain pivq. pivq ñ pvq. To prove this part we set Φptq :“ F ptq for t ě 0, and Xptq :“ Sptq for t ą 0. Then one can apply Proposition 5.39. pvq ñ piq. Using (5.115), (5.117), we start with the identity F ptqgptBq1{2 f ptAq1{2 “ gptBq1{2 f ptAq1{2 T ptq, for t ě 0. Then F ptqr gptBq1{2 f ptAq1{2 “ gptBq1{2 f ptAq1{2 T ptqr , for t ě 0 and r ě 1, or f ptA{rq1{2 gptB{rq1{2 F pt{rqr gptB{rq1{2 f ptA{rq1{2 “ T pt{rqr`1 . It follows that T pt{rqr “ T pt{rqr p1 ´ T pt{rqq ` f ptA{rq1{2 gptB{rq1{2 F pt{rqr gptB{rq1{2 f ptA{rq1{2 ,


177

which leads to the representation T pt{rqr ´ P0 e´tH P0 “ T pt{rqr p1 ´ T pt{rqq ` f ptA{rq1{2 gptB{rq1{2 tF pt{rqr ´ P0 e´tH P0 ugptB{rq1{2 f ptA{rq1{2 ` f ptA{rq1{2 rgptB{rq1{2 ´ 1sP0 e´tH P0 gptB{rq1{2 f ptA{rq1{2 ` rf ptA{rq1{2 ´ 1sP0 e´tH P0 gptB{rq1{2 f ptA{rq1{2 ` P0 e´tH P0 rgptB{rq1{2 ´ 1sf ptA{rq1{2 ` P0 e´tH P0 rf ptA{rq1{2 ´ 1s. This readily implies the inequality }T pt{rqr ´ P0 e´tH P0 }

(5.134)

r

ď }T pt{rq p1 ´ T pt{rqq} ` }F pt{rqr ´ P0 e´tH P0 } ` 2}r1 ´ gptB{rq1{2 sP0 e´tH P0 } ` 2}r1 ´ f ptA{rq1{2 sP0 e´tH P0 }. By the spectral theorem the first term in the right-hand side of (5.134) is estimated as 1 }T pt{rqr p1 ´ T pt{rqq} ď , r ě 1. (5.135) 1`r Using for h P H0 the expression r1 ´ f ptA{rq1{2 se´tH h “ r´1{2 r1 ´ f ptA{rq1{2 sptA{rq´1{2 ptAq1{2 p10 ` tHq´1{2 ˆ p10 ` tHq1{2 e´tH h , and the fact that by condition the Kato function f satisfies C1{2 supxą0 p1 ´ f pxq1{2 q{x1{2 ă `8, we obtain

:“

}r1 ´ f ptA{rq1{2 se´tH h} ď r´1{2 C1{2 }p10 ` tHq1{2 e´tH } }h} ď r´1{2 C1{2 }h},

h P H0 .

Since 0 ď p1 ` xq1{2 e´x ď 1 for x ě 0, this yields }r1 ´ f ptA{rq1{2 sP0 e´tH P0 } ď r´1{2 C1{2 ,

(5.136)

for t ě 0 and r ě 1. Similarly we prove for t ě 0 and r ě 1 the corresponding estimate for the Kato function g: }r1 ´ gptB{rq1{2 sP0 e´tH P0 } ď r´1{2 S1{2 .

(5.137)

Therefore, the estimates (5.134), (5.135) and (5.136), (5.137) imply sup }T pt{rqr ´ P0 e´tH P0 } tPra,bs

ď sup }F pt{rqr ´ P0 e´tH P0 } ` tPra,bs

and so pvq yields piq.

2 1 pC ` S1{2 q, ` 1 ` r r1{2 1{2 l

178


Corollary 5.41. Under assumptions of Proposition 5.40 the conditions piq ´ pvq are equivalent to (vi) limnÑ8 suptPra,bs }pf ptA{nqgptB{nqqn ´ P0 e´tH P0 } “ 0, ra, bs Ď p0, 8q, (vii) limnÑ8 suptPra,bs }pgptB{nqf ptA{nqqn ´ P0 e´tH P0 } “ 0, ra, bs Ď p0, 8q.

5.3.4 Proof of Proposition 5.36 Proof. The above conditions on A, B imply that these operators are generators of C0 -semigroups. Then together with properties of the Kato functions f, g P K this yields for λ ą 0 in H that s-limpλ1 ` Sptqq´1 “ P0 pλ10 ` Hq´1 P0 , tÑ`0

dom H Ă H0 .

(5.138)

Now, from (5.133) we obtain, for t ą 0 and λ ą 0, the identity gptBq1{2 Qptqpλ1ptq ` M ptqq´1 QptqgptBq1{2 “ pλ1 ` Sptqq´1 F ptq ` λgptBq1{2 Qptqpλ1ptq ` M ptqq´1 Qptqr1 ´ f ptAqsgptBq1{2 pλ1 ` Sptqq´1 . Then using the relations s-lim gptBq1{2 “ s-lim f ptAq “ s-lim F ptq “ s-lim Qptq “ 1 tÑ`0

tÑ`0

tÑ`0

tÑ`0

and (5.138), we find for each λ ą 0 and for dom H Ă H0 the limit in H s-lim Qptqpλ1ptq ` M ptqq´1 Qptq “ P0 pλ10 ` Hq´1 P0 . tÑ`0

(5.139)

Note that by definitions (5.102) and (5.104) one has ϕ0 pxq ě ϕ0 pyq,

(5.140)

for 0 ă x ď y,

and 0 ď f pxq ď f0 pxq ď 1,

for x P R0` .

(5.141)

By virtue of (5.141) we get f ptAq ď f0 ptAq, t ě 0, which yields (see (5.119) and (5.120)) f˜ptAq ď f0 ptAqQptq, t ą 0. Then by the invertibility of the mappings f˜ptAq ą 0 and f0 ptAq ą 0, one deduces that f0 ptAq´1 Qptq ď f˜ptAq´1 . Consequently, in the subspaces tHptq “ QptqHutě0 we obtain the inequality 1 1 rf0 ptAq´1 ´ 1ptqsQptq ď rf˜ptAq´1 ´ 1ptqs, t t

t ą 0.

This inequality, the definition of M ptq (5.121), and the fact that p1 ´ gptBqq ě 0, yield the estimate rf0 ptAq´1 ´ 1ptqsQptq{t ď M ptq, t ą 0, or equivalently (see (5.104)) Aϕ0 ptAqQptq ď M ptq. Consequently, Qptqpλ1ptq ` M ptqq´1 Qptq ď pλ1 ` Aϕ0 ptAqq

´1

Qptq ď pλ1 ` Aϕ0 ptAqq

(5.142) ´1

,

t ą 0,


179

where we used that Qptq commutes with A. Recall that ϕ0 is a non-increasing function. Then (5.140) implies ϕ0 ptAqq ě ϕ0 pt0 Aqq for 0 ă t ď t0 and hence pλ1 ` Aϕ0 pt0 Aqq´1 ě pλ1 ` Aϕ0 ptAqq´1 . This inequality and (5.142) yield for 0 ă t ď t0 the estimate Qptqpλ1ptq ` M ptqqQptq ď pλ1 ` Aϕ0 pt0 Aqq´1 .

(5.143)

Note that since by definition (5.104) pλ1 ` Aϕ0 pt0 Aqq´1 “ t0 p1 ` pλt0 ´ 1qf0 pt0 Aqq´1 f0 pt0 Aq , condition (5.106) of the proposition implies pλ1 ` Aϕ0 pt0 Aqq´1 P C8 pHq,

for λt0 ą 1.

(5.144)

Now taking into account (5.143) and (5.144) we infer that convergence (5.139) gives the proof of condition piiiq in Proposition 5.40. Therefore, by Proposition 5.40 and Corollary 5.41, we obtain that: (a) The Trotter-Kato product formula (5.100) (vi) and all other versions (i),(v), (vii) converge in the operator-norm topology locally uniformly away from zero to the degenerate semigroup te´tH P0 utą0 . (b) For λ ą 0 the resolvent pλ10 ` Hq´1 P C8 pH0 q, where dom H Ă H0 Ď H, see (5.107). l We comment that condition (5.106) of Proposition 5.36 is quite implicitly related to the properties of the non-negative self-adjoint operator A. The following remark makes this more explicit. Remark 5.42. Recall that, by Definition 5.34 (5.102), 0 ď ϕ0 pxq ď 1, which implies 1 ` xϕ0 pxq ď 1 ` x, or by the definition of f0 (5.104), f0 pxq ě p1 ` xq´1 . Since by (5.106) f0 pt0 Aq P C8 pHq, this means that f0 pt0 Aq ě p1 ` t0 Aq´1 P C8 pHq,

t0 ą 0.

(5.145)

Thus, (5.106) implies that the resolvent of the operator A is compact. Although it is plausible, we caution the reader that the converse is certainly not true for generic Kato function, f P K. To provide a converse we consider a class of regular Kato functions Kr Ă K. Definition 5.43. Let f be a Kato function from K. We set bpxq :“ sup0ďsďx sf psq and rpxq :“ supsPrx,8q f psq, for x P R` . The Kato function f is called regular if limxÑ`8 bpxq{x “ 0 and 0 ď rpxq ă 1.

180


For instance, f pxq “ e´x and f pxq “ 1{p1 ` xq are regular Kato functions, whereas f pxq “ 1 ´ | sinpxq| is not, see Appendix C (Section C.3). Lemma 5.44. Let f P K and let A be a non-negative self-adjoint operator in a (infinite-dimensional ) Hilbert space H. Let p1 ` Aq´1 P C8 pHq, then f0 pAq P C8 pHq if and only if the Kato function f is regular. An obvious consequence of Proposition 5.36 and Lemma 5.44 is that the Trotter-Kato product formulae converge in the operator-norm topology locally uniformly away from zero for any regular f P Kr , any g P K, and any self-adjoint operator B ě 0 if the resolvent of the operator A is compact. It turns out that these assumptions are not only sufficient but also necessary. Proposition 5.45. Let .A ě 0 and B ě 0 be self-adjoint operators in a Hilbert space H such that H “ A ` B is self-adjoint in dom H Ă H0 Ď H. Then the TrotterKato product formulae converge in the operator-norm topology to the degenerate semigroup te´tH P0 utě0 locally uniformly away from zero for any regular Kato function f P Kr and for arbitrary Kato function g if and only if p1Àq´1 P C8 pHq. Proof. If p1 ` Aq´1 P C8 pHq, then by Lemma 5.44 one gets that f0 pAq P C8 pHq. Now applying Proposition 5.36 we verify the sufficient part of proposition. The converse part needs more preparation. The interested reader can find l the corresponding reference in Notes in Section 5.6 Comments about some other results when the operator compactness condition lifts the strong convergence of the Trotter-Kato product formulae to the convergence in the operator-norm topology one can also find in the Notes in Section 5.6. For completeness, we quote here a result when instead of regularity constrains on the class of admissible Kato functions K one imposes rather explicit conditions on the operators A and B in order to ensure the convergence of the Trotter-Kato product formulae in the operator-norm topology. Note that in the next statement the operators A, B and the pairs f , g of Kato functions are treated equally. Proposition 5.46. Let A ě 0 and B ě 0 be self-adjoint operators in a Hilbert space H and let f and g be Kato functions, from the class K1 , obeying the conditions C :“ sup xą0

xf pxq ă `8 1 ´ f pxq

and

S :“ sup xą0

xgpxq ă `8. 1 ´ gpxq

(5.146)

Then the Trotter-Kato product formulae converge to the degenerate semigroup te´tH P0 utě0 in the operator-norm topology locally uniformly away from zero and one also has p10 ` Hq´1 P C8 pH0 q, if p1 ` Aq´1 p1 ` Bq´1 P C8 pHq.

(5.147)

Here dom H Ă H0 Ď H, H0 “ P0 pHq and 10 “ 1æ H0 . For more details concerning the class K1 of functions obeying the conditions (5.146) we refer to Appendix C, Section C.3.


181

5.3.5 Rate of Convergence: non-self-adjoint semigroups We conclude Section 5.3 by two results that extend the operator-norm convergence of the exponential Trotter-Kato product formula to non-self-adjoint C0 semigroups. In contrast to Propositions 5.36, 5.45 and in particular to Proposition 5.46, these results are based on a definite hierarchy between generators A and B, where operator B is considered as a small perturbation. Proposition 5.47. Let te´tA utě0 be a holomorphic contraction semigroup on a Hilbert space H. Suppose that the m-sectorial generator A is invertible and B is an m-accretive operator in H such that dom Aα Ď dom B for some α P r0, 1q and dom A˚ Ď dom B ˚ . Then the Trotter product formula converges in the operatornorm topology to the C0 -semigroup te´tpA`Bq utě0 with rate estimated by › ›` › › ´tA{n ´tB{n ˘n e ´ e´tpA`Bq › (5.148) › e ď Oplnpnq{n1´α q pα ‰ 0q _ Opplnpnqq2 {nq pα “ 0q. For operators A and B as above, we recall that B is A-small with relative bound b ě 0 and a ě 0 if dom A Ď dom B

and }Bu} ď a}u} ` b}Au},

u P dom A.

(5.149)

For b ă 1 these conditions guarantee the existence of the perturbed semigroup te´tpA`Bq utě0 , see Section 1.7. Remark 5.48. The condition dom Aα Ď dom B in Proposition 5.47 restricts the smallness of the operator B even further. Indeed, since this condition implies }BA´α } ă 8, one gets that for any η ą 0 and α ă 1 }Bu} ď }BA´α }}Aα pA ` ηq´1 }p}Au} ` η}u}q,

u P dom A.

If A is the generator of a holomorphic semigroup, then for each α P r0, 1q there exists a constant Cα , such that }Aα pA ` ηq´1 } ď Cα {η 1´α for all η ą 0. Then we obtain the estimate }Bu} ď aη }u} ` bη }Au},

u P dom A,

where the relative bound bη :“ Cα }BA´α }{η 1´α of the operator B can be arbitrarily small for large η, cf. (5.149). This means that the unbounded perturbation B is infinitesimally small with respect to A. We denote this by B P P0` pAq, Definition 1.50. Other infinitesimally small perturbations from the class P0` are considered in Section 4.4 and Remark 5.31. To relax this severe smallness condition and improve the rate of convergence one can demand more restrictions on the C0 -semigroup generator A than it is done in Proposition 5.47. Then the following assertion gets true.

182


Proposition 5.49. Let A be a positive invertible self-adjoint operator in H. If B is an m-accretive operator such that dom A Ď dom B, dom A Ď dom B ˚ , and }Bu} ď b}Au}, ˚

}B u} ď b˚ }Au},

u P dom A, 0 ă b ă 1, u P dom A, 0 ă b˚ ă 1,

then the Trotter product formula converges in the operator-norm topology to the C0 -semigroup te´tpA`Bq utě0 with the error bound estimate for the rate given by › ›` ´ lnpnq ¯ ˘n › › . (5.150) › e´tA{n e´tB{n ´ e´tpA`Bq › ď O n Remark 5.50. Note that in Proposition 5.47 and in Proposition 5.49 the hierarchy of adjoint operators dom A˚ Ď dom B ˚ , is indispensable for ensuring the boundedness of the closure pA´1 Bq “ pB ˚ A˚´1 q˚ , cf. Remark 5.14. In Section 5.5 we use these results to prove the convergence of the exponential Trotter-Kato product formula for Gibbs semigroups and to obtain estimates of the rate of convergence in the trace-norm topology.

5.4

Trotter-Kato product formulae: trace-norm convergence, error bounds

In this section we start by an analysis of the exponential Trotter-Kato (or simply Trotter) product formula for Gibbs semigroups. The first result is that for these semigroups the convergence of the Trotter product formula can be strengthened from the operator-norm to the trace-norm convergence. The main tool for obtaining this improvement of convergence is the lifting lemma, see subsection 5.4.1, Lemma 5.51. For a certain class of Gibbs semigroups it allows to lift, for the Trotter product formula, the operator-norm convergence to convergence in the trace-norm topology due to a direct error bound estimates. In subsection 5.4.2, Trace-norm convergence without rate estimate, we consider the case of self-adjoint Gibbs semigroups and prove the trace-norm convergence of the Trotter product formula without error bound estimate. In this case the trace-norm limit may be a degenerate Gibbs semigroup, Proposition 5.53. Note that here the generators A and B in the Trotter approximants are not conditioned by any hierarchy. Next we use the operator-norm error bound from Proposition 5.8 and the lifting Lemma 5.51 to prove in Proposition 5.55 the trace-norm convergence of the Trotter product formula to the Gibbs semigroup with an error bound estimate. Here the self-adjoint generators A and B are subject to a smallness condition on B. In Proposition 5.56 we show that if the operator-norm error-bound is optimal, then the trace-norm error bound estimate inherits the same property. In subsection 5.4.3, Smallness conditions and optimal rate of convergence, we modify Proposition 5.8 by fractional-power relative smallness conditions for the

5.4. Trotter-Kato product formulae: trace-norm convergence

183

pair of self-adjoint operators A and B. Then they yield an optimal operator-norm error bound estimate for the rate of convergence of the Trotter-Kato product formulae. Note that these hierarchical fractional power conditions (cf. Remark 5.3) are less restrictive than the self-adjointness of the algebraic sum A ` B in Proposition . 5.19. Similarly to Proposition 5.36 and Proposition 5.45 they allow the sum A ` B in the wider sense of quadratic forms. Finally in subsection 5.4.4, Trotter-Kato product formulae: trace-norm convergence, we study sufficient conditions for the trace-norm convergence of the Trotter-Kato product formulae. This extension from the Trotter product formula requires some nontrivial conditions on the pair of generic Kato functions f, g P K (Definition 5.4). We mention here that the lifting Lemma 5.51 is well-suited for the Trotter formula, but not for the Trotter-Kato product formulae.

5.4.1

Lifting lemma

The following key lifting lemma allows us to lift the norm-convergence to the trace-norm convergence with the } ¨ }1 -error bound estimate. Lemma 5.51. Let A be an m-sectorial operator with vertex γ “ 0 and such that e´t <e A P C1 pHq for t ą 0. Let B be generator of a contraction C0 -semigroup such that the family te´tH utě0 is a Gibbs semigroup with }e´tH } ď 1. Let › › ˘m ›` › (5.151) εpm, tq :“ sup › e´ξA{m e´ξB{m ´ e´ξH › , p2m{p2m`1qqtďξďp2m{p2m´1qqt

for m “ 2, 3, . . ., and t ą 0. Then for each t0 ą 0 there are numbers L1 pt0 q and L2 pt0 q such that › › ›` ´tA{n ´tB{n ˘n › e ´ e´tH › › e 1

ď L1 pt0 qεprn{2s, t{2q ` L2 pt0 qε prpn ` 1q{2s, t{2q

(5.152)

for all n “ 2, 3, . . ., and for t ě t0 ą 0, that is, away from zero. Here rxs denotes the integer part of x ě 0. Proof. For n ą 1, we define two variables: kn :“ rn{2s ď n{2 and mn :“ rpn ` 1q{2s ě n{2. Then n “ kn ` mn , and we have the representation pe´tA{n e´tB{n qn ´ e´tH ı ” “ pe´tA{n e´tB{n qkn ´ e´kn tH{n pe´tA{n e´tB{n qmn ” ı ` e´kn tH{n pe´tA{n e´tB{n qmn ´ e´mn tH{n . By Lemma 4.43, we get the inequality: ›mn › ›` › › ˘mn ›› › › › › › › e´tA{n e´tB{n › ď ›e´tB{n › ›e´mn t <e A{4n › . 1

1

(5.153)

184


The self-adjoint operator <e A ě 0 (Remark 4.29) generates a contraction Gibbs semigroup te´t <e A utě0 and te´tB utě0 is also a contraction semigroup. Then since mn ě n{2, one obtains the uniform in n ą 1 estimate ›` › › ˘mn ›› › › › (5.154) › ď ›e´t <e A{8 › . › e´tA{n e´tB{n 1

1

On the other hand, since by }e´tH } ď 1 the Gibbs semigroup te´tH utě0 is a contraction semigroup and since pn{2 ´ n q ď kn ď n{2, where limnÑ8 n “ 0, we get (for example) the estimate › › › › › › › › (5.155) ›e´kn tH{n › ď ›e´tH{3 › . 1

1

Then (5.153)–(5.155) yield for n ą 1 the inequality › › › ›` ´tA{n ´tB{n ˘n ´ e´tH › e › e 1 ›› › ›` › ´tA{n ´tB{n ˘kn ´kn tH{n › › ´t <e A{8 › é e ď› e › › ›e 1 › ›` › › ˘m n › › › › ` ›e´tH{3 › › e´tA{n e´tB{n ´ e´mn tH{n › .

(5.156)

1

Let tn “ tkn {n and sn “ tmn {n, i.e., limnÑ8 tn “ limnÑ8 sn “ t{2, and let t0 ą 0. Then by definition (5.151) and by the estimate (5.156) one gets › › › ›` ´tA{n ´tB{n ˘n ´ e´tH › e › e 1 › ›` ›› › › ´tn A{kn ´tn B{kn ˘kn ›› ´ e´tn H q› ›e´t <e A{8 › e ď› e 1 › › › › › ´tH{3 › ›` ´sn A{mn ´sn B{mn ˘mn ´sn H › ` ›e e é › › › e ›1 › › › › › › › ď εpkn , t{2q ›e´t0 <e A{8 › ` ›e´t0 H{3 › εpmn , t{2q , (5.157) 1

1

for t ě t0 and n ą 1. Hence, we obtain the assertion (5.152), if we set L1 pt0 q :“ l }e´t0 <e A{8 }1 and L2 pt0 q :“ }e´t0 H{3 }1 , which are finite for each t0 ą 0. Corollary 5.52. Let te´t <e A utě0 and te´tH utě0 be eventually Gibbs semigroups away from t0 ą 0. Then (5.157) yields the estimate ›` › ˘n › › › e´tA{n e´tB{n ´ e´tH › 1

ď L1 ptqεprn{2s, t{2q ` L2 ptqε prpn ` 1q{2s, t{2q ,

(5.158)

for all n “ 2, 3, . . . and for t ą 8t0 , that is, away from 8t0 . We note that the threshold 8t0 in (5.158) have as its source estimates (5.154) and (5.155) which are originally not optimal.


185

5.4.2 Trace-norm convergence without rate estimate First we prove the convergence of the Trotter product formula away from zero in the trace-norm topology without estimate of the rate of convergence. Proposition 5.53. Let A ě 0 be the generator of the self-adjoint Gibbs semigroup tGt pAq “ e´tA utě0 . Then the exponential Trotter-Kato product formula converges in the trace-norm topology, away from zero, for Kato functions f pxq “ gpxq “ e´x and for any self-adjoint operator B ě 0, to a degenerate Gibbs semigroup: ` ˘n . (5.159) } ¨ }1 - lim e´tA{n e´tB{n “ e´tH P0 , H “ A ` B, nÑ8

where P0 is the orthogonal projection P0 : H Ñ dom H and t ą 0. Proof. Operator A is obviously m-sectorial and B. is the generator of a contraction semigroup. Then the form-sum operator H “ A ` B is self-adjoint in the subspace H0 :“ dom A1{2 X dom B 1{2 and generates a degenerate contraction semigroup }e´tH P0 } ď 1 on H. Moreover, it is known that the Trotter product formula converges strongly to e´tH P0 away from t “ 0 : ` ˘n s-lim e´tA{n e´tB{n “ e´tH P0 , nÑ8

where P0 is the orthogonal projection P0 : H Ñ H0 . Now we have to check (see Lemma 5.51) that e´tH P C1 pH0 q. To this aim, let tBk ě 0ukě1 Ă LpHq be a monotonically increasing to B sequence of positive bounded operators. Then the positivity of generator A implies .

e´tpA`Bq P0 ď e´tpA`Bk`1 q ď e´tpA`Bk q ď e´tA .

(5.160)

By monotonicity the weak operator limit on H yields w-lim e´tpA`Bk q “ e´tH P0 , kÑ8

.

H “ A ` B.

(5.161)

Note that by Proposition 4.45 the perturbed semigroup is positive and Gibbs: e´tpA`Bk q P C1,` pHq. Then by the lower weak semi-continuity of the trace (see Proposition 2.48(f) and Corollary 2.75) and by monotonicity (5.160) one gets 0 ď Tr e´tH P0 ď lim inf Tr e´tpA`Bk q ď Tr e´tA . kÑ8

(5.162)

Consequently, the positivity of the semigroups implies that }e´tH P0 }1 ď }e´tA }1 on H, or e´tH P C1 pH0 q. Note that by taking the Laplace transform of the Gibbs semigroup tGt pAqutě0 one gets that resolvent ż8 pλ1 ` Aq´1 “ dt e´tλ e´tA P C8 pHq, λ ą 0, (5.163) 0

186


is compact. Then Proposition 5.36 yields the operator-norm convergence of the Trotter product formula locally uniformly away from zero, and consequently › ›` › ´ξA{s ´ξB{s ˘s › e sup ´ e´ξH P0 › “ 0, lim εps, tq “ lim › e sÑ8

sÑ8 p2s{p2s`1qqtďξďp2s{p2s´1qqt

(5.164) for s “ 2, 3, . . ., and t ą 0. Therefore, by Lemma 5.51, (5.152) (for immediately Gibbs semigroups) and by the limit (5.164) we obtain › › ˘n ›` › lim › e´tA{n e´tB{n ´ e´tH P0 › “ 0, nÑ8

1

for t ě t0 and any t0 ą 0, that is away from zero.

l

Corollary 5.54. If tGt pAqutě0 is an eventually Gibbs semigroup away from t0 ą 0, then by Corollary 5.52 and (5.158) one gets that ›` › ˘n › › lim › e´tA{n e´tB{n ´ e´tH P0 › “ 0, nÑ8

1

away from 8t0 and that te´tH P0 utě0 is a degenerate eventually Gibbs semigroup away from 8t0 .

5.4.3

Smallness conditions and optimal rate of convergence

Since Lemma 5.51 is established for exponential Kato function, we shall prove the lifting of the operator-norm to the trace-norm convergence with the error-bound estimate only for f pxq “ gpxq “ e´x . To do that, we borrow the operator-norm error-bound from Proposition 5.8 for the self-adjoint A and B to prove under a smallness condition on B the following statement: Proposition 5.55. Suppose that the self-adjoint operators A and B satisfy conditions (5.10) and (5.11) (see (i), (ii) of Section 5.2) with b ă 1. If A is generator of the Gibbs semigroup tGt pAqutě0 , then the Trotter product formula converges in the trace-norm topology to the Gibbs semigroup tGt pA`Bqutě0 , with the error bound estimate for t ě t0 ą 0: › › lnpnq › › , ›pe´tA{n e´tB{n qn ´ e´tpA`Bq › ď c1 n 1

(5.165)

for some t0 -depended number c1 ą 0. Here n “ 6, 7, . . . . Proof. Since B P Pbă1 pAq, the semibounded by (5.10) operator H “ A ` B ě 2 1 with dom H “ dom A is self-adjoint, and it is the generator of holomorphic contraction semigroup. Since A is the generator of a Gibbs semigroup and B “ B ˚ ě 1, Proposition 4.45 shows that the perturbed semigroup is Gibbs, i.e., the operator H is the generator of the Gibbs semigroup tGt pHqutě0 with }e´tH } ď 1.


187

Hence, the operators A, B and H satisfy the conditions of Lemma 5.51 and Proposition 5.8 for f pxq “ gpxq “ e´x , see (5.10). Then by virtue of (5.24) we get for (5.151) the estimate LF lnpsq εps, tq ď , (5.166) p1 ´ bq3 s where s “ 3, 4, . . ., uniformly for t ě 0. Since for n ě 6 we get from (5.166) that εprn{2s, t{2q ě εprpn ` 1q{2s, t{2q, inserting (5.166) into (5.152) we get assertion (5.165) uniformly away from t0 ą 0 (i.e. for t ě t0 ą 0) with t0 -dependent constant c1 :“

˘ 2LF LF ` ´t0 A{8 }e }1 ` }e´t0 H{3 }1 ď }e´t0 A{8 }1 , p1 ´ bq3 p1 ´ bq3

which is bounded for any t0 ą 0. Here we used that the perturbation B ě 1 yields H ě A and Corollary 4.24 to estimate the } ¨ }1 -norms. l The error bound estimate in Proposition 5.55 is evidently not optimal. Similar to analysis for operator-norm topology (Proposition 5.19) we can get an optimal trace-norm asymptotic rate for convergence of the Trotter product formula in the case of self-adjoint Gibbs semigroups. Proposition 5.56. Let A and B be non-negative self-adjoint operators in a Hilbert space H such that the operator sum C :“ A ` B on dom C “ dom A X dom B is also self-adjoint. Let A be the generator of the Gibbs semigroup tGt pAqutě0 . Then the operator C ě 0 is the generator of the Gibbs semigroup tGt pCq “ e´tC utě0 and : (a) the Trotter product formula › ´tA{n ´tB{n n › ›pe e q ´ e´tC ›1 “ Opn´1 q,

n Ñ 8,

(5.167)

converges in the trace-norm topology with the same rate Opn´1 q as its symmetrised version: › ´tB{2n ´tA{n ´tB{2n n › ›pe e e q ´ e´tC ›1 “ Opn´1 q, n Ñ 8, (5.168) locally uniformly away from t “ 0 (Definition 5.35); (b) the condition of self-adjointness and the error bound Opn´1 q are ultimate optimal. Proof. We remark that the condition e´tA P C1 pHq, t ą 0 is symmetric with respect to the choice A or B. To check that the operators A, B and H satisfy the conditions of Lemma 5.51 and Proposition 5.8 for f pxq “ gpxq “ e´x , one follows the same arguments as in the proof of Proposition 5.55. The extension to the symmetrised case uses exactly the same argument as the proof of Proposition 5.8. By virtue of (5.83) and (5.84), for the operator-norm rate of convergence we obtain for (5.151) the estimate εpn, tq “ Opn´1 q , t ą 0 ,

n Ñ 8,

188


uniformly away from zero. Then by the lifting Lemma 5.51 we obtain that (5.167) and (5.168) hold uniformly for t ě t0 ą 0. Finally, the optimality of (5.167) and (5.168) is a corollary of the optimality of the error bounds (5.83) and (5.84) of the convergence rate in operator norm. l Another result, when the trace-norm error bound for the Trotter product formula has an optimal estimate of the convergence rate, provides a certain clarification on Proposition 5.8 and Proposition 5.55. First we recall the fractional smallness conditions of subsection 5.2.5 (Proposition 5.27) that ensure the optimal asymptotic for the operator-norm rate convergence of the Trotter product formula. Proposition 5.57. Let the self-adjoint operators A ě 1 and B ě 0 in a Hilbert space H be such that for some α P p1{2, 1q and b P p0, 1q the following conditions are satisfied : (i) dom Aα Ă dom B α and }B α u} ď b}Aα u}, u P dom Aα ; (ii) dom H α Ă dom Aα . 9 Then semibounded from below operator H :“ A`B is the densely defined selfadjoint form-sum of A and B and for Kato functions f, g P Kα one gets the following operator-norm estimate: › › ›pf ptA{nqgptB{nqqn ´ e´tH › ď c2 , n ě 2 , (5.169) n2α´1 for some c2 ą 0, uniformly for t ě 0. Here the class of Kato functions Kα is specified in Definition 5.26, cf. Appendix C. The application of the operator-norm error bound (5.169) to estimate the trace-norm rate of convergence of the Trotter product formula is again based on our lifting strategy. It allows us to prove the following statement. Proposition 5.58. Suppose that in addition to the conditions of Propositions 5.57, (iii) the operator A is the generator of the immediate self-adjoint Gibbs semigroup te´tA utě0 . Then te´tH utě0 is also an immediate Gibbs semigroup and the Trotter product formula converges in the trace-norm with the rate error bound : }pe´tA{n e´tB{n qn ´ e´tH }1 ď

c3 2α´1 n

,

n ě 2,

(5.170)

for some c3 ą 0, uniformly away from zero. According to the lifting strategy, to prove (5.170) one has to check the conditions of Lemma 5.51. To continue, we note that the properties of operators A, B and H are evident by virtue of the conditions of Propositions 5.57. Moreover, the estimate of εpn, tq in (5.151) is ensured by Proposition 5.27 and Proposition 5.57 either due to (i), which gives for the Trotter-Kato product formulae an operator-norm rate of convergence of the order Opplnpnqq{n2α´1 q, or due to (i) and (ii), which yield the optimal rate of the order Op1{n2α´1 q.


189

Hence, it remains only to verify that the additional condition (iii) ensures that e´tH P C1 pHq for t ą 0. Note that, in contrast to the operator smallness and the operator-sum for H as in Proposition 5.55, now the self-adjoint H is defined as the form-sum in Propositions 5.57. Lemma 5.59. Condition (i) of Proposition 5.57 implies that there exists a bounded operator Y P LpHq obeying }Y } ă 1 such that for α P p1{2, 1q one has the representation H ´1 “ A´p1´αq p1 ` Y q´1 A´α “ A´α p1 ` Y ˚ q´1 A´p1´αq .

(5.171)

Proof. Since A ě 1 and α P p1{2, 1q, we set Y :“ pB α A´α q˚ pB 1´α A´p1´αq q. Then by (i) and by (5.92) with θ “ 1 ´ α, one gets }B p1´αq v} ď bp1´1{αq }Ap1´αq v}, v P dom Ap1´αq . This implies }B 1´α A´p1´αq } ď bp1´1{αq as well as }B α A´α } ď b, which is due to (i). Concequently, }Y } ď b2´1{α ă 1. Now, by monotonicity: dom Aα` Ď dom Aα´ for α´ ď α` , and by (5.93) we find that dom Aα Ď dom A1{2 “ dom H 1{2 Ď dom A1´α for 1{2 ă α ă 1. Therefore, pu, Hwq “ pA1{2 u, A1{2 wq ` pB 1{2 u, B 1{2 wq “ pAα u, Ap1´αq wq ` pB α A´α Aα u, B p1´αq A´p1´αq Ap1´αq wq “ pAα u, p1 ` Y qAp1´αq wq, for u P dom Aα and w P dom H. Hence, by the self-adjointness of Aα and if w P dom H, then p1 ` Y qAp1´αq w P dom Aα , and Hw “ Aα p1 ` Y qAp1´αq w . Consequently, together with }Y } ď b2´1{α ă 1 this yields the representation l (5.171), which in turn gives dom H Ă dom Aα for α P p1{2, 1q. Proof of Proposition 5.58. Since by conditions A ě 1 and (iii) the operator A´1 is compact, the same is true for A´p1´αq and for A´α . Therefore, by (5.171) the operator H ´1 is also compact. Consequently, generator H ě 1 has a discrete spectrum σpHq “ σd pHq with the only accumulation point of positive eigenvalues tλn pHquně1 at infinity, see Appendix A (Section A.6). 9 (see Corollary Note that since B ě 0, the minimax principle for H “ A`B ´tA ´tH for t ą 0. This proves ď Tr e 4.24) yields tλn pHq ě λn pAquně1 , or Tr e that e´tH P C1 pHq for t ą 0. Now we have all necessary ingredients of Lemma 5.51 to conclude that the trace-norm convergence of the Trotter product formula holds uniformly away from zero with the optimal rate (5.170), Remark 5.29. l

190


5.4.4 Trotter-Kato product formulae, trace-norm convergence The lifting Lemma 5.51 is well suited to prove (including the error bound estimates for the convergence rate) the trace-norm convergence of the Trotter product formula. Moreover, it is still efficient for trace-norm estimates in the case of non-self-adjoint generators A and B in the Trotter approximants, see Section 5.5 (subsection 5.5.3), but it is not very useful for the Trotter-Kato product formulae. To prove the operator-norm or the trace-norm convergence of the TrotterKato product formulae for a certain subclass of non-exponential Kato functions K away from zero, for example n

τ - lim pf ptA{nqgptB{nqq “ e´tH P0 , nÑ8

τ is } ¨ } _ } ¨ }1 ,

(5.172)

we need to impose additional conditions on the generic functions f, g P K (Definition 5.4 and Appendix C) involved in the Trotter-Kato product formulae approximants. Here τ stands for the topology of convergence, either operator-, or the trace-norm. Recall that to prove in Proposition 5.8 the operator-norm convergence of Trotter-Kato product formulae under assumption of smallness of B we put on the Kato functions K restrictions (iv),(5.15)-(5.19) and (v),(5.20). On the other hand, to prove the operator-norm convergence under conditions of fractional smallness (Proposition 5.27) we assumed that the Kato functions belong to the sub-class Kα , Definition 5.26 and Appendix C. In Proposition 5.45 without a smallness condition the operator-norm convergence holds if one of the Kato functions f, g is regular, Definition 5.43 and Appendix C. The aim of this subsection is to determine conditions on the pair f, g P K and on the pair A and B that ensure the convergence (5.172) in the trace-norm topology, (that is, τ is } ¨ }1 ), to a degenerate Gibbs semigroup te´tH P0 utě0 , where 9 H “ A`B. Note that Proposition 6.18 shows that to establish (5.172) in the }¨}1 -norm for any of the Trotter-Kato product formulae it is enough to prove convergence of the Trotter-Kato product formula only for one family of approximants, for example, for the family tpf ptA{nqgptB{nqqn uně1 . Remark 5.60. To proceed with selection from K admissible functions for the next proposition we assume that the generic Kato functions f, g P K verify the following additional condition (cf. (5.101) in Definition 5.34): (1)* f pxq is strictly positive and the auxiliary functions ˙ ˆ 1 ´1 ´1 and ψpxq :“ x´1 p1 ´ gpxqq , ϕpxq :“ s f pxq are monotone non-increasing for x P R` .

(5.173)

Then we say that f P K˚ , see Appendix C, Section C.2 (Comments 6 and 7).


191

densely defined self-adjoint operProposition 5.61. Let A and B be non-negative . ators in H and let the form-sum H “ A ` B be a self-adjoint operator defined in the subspace H0 Ď H. Suppose the Kato functions f, g P K˚ . If f ptAq P Cp pHq,

t ą 0,

1 ď p ă `8,

(5.174)

then the Trotter-Kato product formula (5.172) converge in the norm-topology τ “ } ¨ }2p , to the degenerate Gibbs semigroup te´tH P0 utě0 , locally uniformly away from t “ 0. In the Notes to Section 5.4 one finds a reference for the proof and some remarks. A few comments are in order. Condition f ptAq P Cp pHq directly relates the operator A and the Kato function f , but now the additional condition (5.173) excludes some natural choices for f . For example, f pxq “ e´x does not satisfy (5.173), whereas f pxq “ p1 ` x{κq´κ for 0 ă κ ď 1, does. This demonstrates how sensitive is the Trotter-Kato product formula (5.172) to the choice of admissible Kato functions. Besides the elimination of the exponential Kato function f the strongest topology of convergence is now the Hilbert-Schmidt norm } ¨ }2 under the rather severe requirement (5.174) that operator f ptAq must belong to the trace-class for t ą 0. To improve the convergence in Proposition 5.61 to the trace-norm topology τ “ } ¨ }1 , we formulate another type of conditions on the functions f, g P K and on the operators A and B. Proposition 5.62. Let A and B be non-negative densely defined self-adjoint oper. ators in H and let the form-sum H “ A ` B be a self-adjoint operator defined in the subspace H0 Ď H. Let f, g be arbitrary Kato functions. If f0 ptAq P Cp pHq,

t ą 0,

1 ď p ă `8,

(5.175)

then the Trotter-Kato product formulae (5.172) converge in the trace-norm topology τ “ } ¨ }1 to the degenerate Gibbs semigroup te´tH P0 utě0 , locally uniformly away from zero. Here the function f0 is given by (5.104) in Definition 5.34. We shall present the proofs of this assertion in Chapter 6 (Section 6.5). Therein we consider the problem of the convergence of the Trotter-Kato product formula in the more general framework of the symmetrically-normed ideals Cφ pHq. Then the trace-norm ideal C1 pHq “ Cφ1 pHq is simply a particular case corresponding to the special choice (6.20) of the symmetric norming function φ “ φ1 . In fact (5.175) imposes implicit conditions on the original f and A. However, since for the exponential f pxq “ e´x one gets f0 pxq “ p1 ` xq´1 , the condition (5.175) coincides with the definition of the p-generator f0 ptAq “ p1 ` tAq´1 P Cp pHq,

t ą 0,

1 ď p ă `8,

(5.176)

192


of a Gibbs semigroup, cf. Definition 4.26. In particular, (5.176) yields f ptAq “ e´tA P Cp pHq, t ą 0, but not converse. We conclude these observations by remarking that for the particular case of the exponential Kato function f pxq Proposition 5.62 can be improved. To this end one assumes instead of (5.175), that is, instead of (5.176), the weaker condition (5.174). Then using the Araki inequality (Appendix B, Section B.1) it is possible to refine the corresponding estimates up to the trace-norm convergence. For details see Section 6.5.

5.5 Product formulae: non-self-adjoint Gibbs semigroups In this section we extend the trace-norm convergence of the exponential TrotterKato product formula (i.e., the Trotter product formula) to the non-self-adjoint case. This is a subtle problem since self-adjointness served for important estimates. Our extension of the Trotter product formula to non-self-adjoint Gibbs semigroups covers essentially two cases. The first one concerns the Trotter product formula when the involved C0 semigroups have m-sectorial generators A and B. The strategy of the proof of the convergence in this case is based on the lifting the corresponding results for the self-adjoint generators. To this aim we propose the analytic extension method for holomorphic families of sectorial forms (generators) and of the corresponding semigroups. We develop this method in the preliminary subsection 5.5.1, Holomorphic families of generators and semigroups. In subsection 5.5.2 we use this method to establish the trace-norm compactness of the holomorphic families of Trotter approximants. Then in Proposition 5.78 we prove (without an estimate of the convergence rate) that the Trotter product formula converges in the trace-norm topology to a degenerate Gibbs semigroup. Note that in this statement the m-sectorial generators A and B are not subordinated. To control the rate of the trace-norm convergence we impose in subsection 5.5.3 more restrictive conditions of relative smallness on the pair of m-sectorial generators A and B. In Proposition 5.80 and Proposition 5.81 we establish the convergence of the Trotter product formula to Gibbs semigroups in the tracenorm topology, with estimates of the rate of convergence. Note that instead of the analytic extension method we use here the lifting Lemma 5.51. Following the idea of lifting and similar to Section 5.4 we first begin by establishing the convergence of non-self-adjoint exponential product formula in the operator-norm topology. Then we proceed with improving of this convergence to the trace-norm topology.

5.5. Product formulae: non-self-adjoint Gibbs semigroups

193

5.5.1 Holomorphic families of generators and semigroups We start with conditions imposed on the pair of generators A and B. Suppose A and B are densely defined m-sectorial operators in H with corresponding semiangles αA and αB belonging to r0, π{2q. We denote by a and b the densely defined closed sectorial sesquilinear forms associated to the operators A and B, respectively, see Definition 1.41 and Remark 4.29. Recall that if A is an m-sectorial operator with vertex γ “ 0 and semi-angle α P r0, π{2q, then we can associate with A a densely defined, closed, sectorial sesquilinear form aru, vs : dom a ˆ dom a Ñ C (with the same vertex γ “ 0 and semi-angle α P r0, π{2q), such that aru, vs “ pAu, vq,

u P dom Ap“ core aq, v P dom a.

Then arus :“ aru, us, for u P dom a, is the quadratic form associated with the sesquilinear form. Quadratic form determines the sesquilinear form uniquely by the polarisation identity aru, vs “

1 taru ` vs ´ aru ´ vs ` i aru ` i vs ´ i aru ´ i vsu. 4

Similarly to A, the adjoint operator A˚ is also m-sectorial with the same semiangle and the vertex. Then the associated densely defined, sectorial and closed sesquilinear form a˚ ru, vs “ pA˚ u, vq “ arv, us extended from dom A˚ to domain dom a˚ “ dom a, is the adjoint form to a. A form a is symmetric if a “ a˚ , i.e., aru, vs “ arv, us. By polarisation identity the sesquilinear form aru, vs is symmetric if and only if the quadratic form arus, u P dom a, is real-valued. For the sesquilinear form a we associate two symmetric forms <e a :“

1 pa ` a˚ q and 2

=m a :“

1 pa ´ a˚ q, 2i

such that one has the representation aru, vs “ <e aru, vs ` i =m aru, vs,

u, v P dom a.

(5.177)

Recall that these two forms are not real-valued and <e aru, vs ‰ <eparu, vsq, =m aru, vs ‰ =mparu, vsq, but they are real and imaginary parts for the quadratic form arus: <e arus “ <eparusq and

=m arus “ =mparusq,

u P dom a.

Note that the vertex γ and the semi-angle α are (not uniquely) determined by the sesquilinear form a via the inequalities <e arus ě γ,

| =m arus| ď tg αA <e arus,

u P dom a.

(5.178)

194


Proposition 5.63 (Representation Theorem). Let the sesquilinear form t : dom t ˆ dom t Ñ C be densely defined, closed, and sectorial on the domain dom t Ă H. Then there exists a unique m-sectorial operator T with dom T Ă dom t such that tru, vs “ pT u, vq for u P dom T , v P dom t, and dom T is a core of the form t. If, in addition, the form t is symmetric and non-negative: trus ě 0, u P dom t, then T is a non-negative self-adjoint operator such that dom T 1{2 “ dom t and tru, vs “ pT 1{2 u, T 1{2 vq, u, v P dom t. Moreover, a subset D Ă dom t is a core of t if and only if D is a core of T 1{2 . Corollary 5.64. The mapping t ÞÑ Tt p“ T q is a one-to-one correspondence between the set of all densely defined, closed sectorial forms and the set of all m-sectorial operators. The form t is bounded if and only if T is bounded, Tt˚ “ T ˚ , and t is symmetric it and only if operator T is self-adjoint. Since there are no maximal sectorial forms, there are advantages in using the Representation Theorem to exploit sesquilinear forms for constructing m-sectorial operators. The first is that a densely defined sectorial operator S naturally generates a closable sesquilinear form sru, vs “ pSu, vq for u, v P dom S. Let T “ Tt be the m-sectorial operator associated with the closed form t :“ sr. Then T is an extension of S such that dom S “ dom s is a core of t. Hence, even when the closure Sr is not m-sectorial, one can always associate with S a minimal m-sectorial extension T with dom T Ă dom t. The second advantage comes from the observation that in contrast to the case of the set of the closed operators CpHq, the set of the closed forms Cf pHq is a linear space. If a and b are densely defined, closed sectorial forms, then by the representation theorem there exists the m-sectorial operator C associated with operators A and B the closed form c “ a ` b. Since in turn there are m-sectorial . associated with the forms a and b, the operator C :“ A ` B is called the formsum of these operators. On the other hand, if a and b are densely defined, closed sectorial forms generated by densely defined sectorial operator A and B, then the . ` B ` Ă corresponding form-sum is an extension of the operator-sum: A B. A . Recall that the form-sum A ` B may exist and be well-defined even if dom A X dom B “ t0u. We have also seen another case, when for m-sectorial operators one has dom A X dom B Ă dom a X dom b “ H0 Ă H, that is, the operator sum A ` B. is not densely defined in H. Then the m-sectorial form-sum operator C “ A ` B is well-defined in the Hilbert space H0 “ P0 pHq, where P0 is the orthogonal projection P0 : H Ñ H0 “ dom C. Now we return to the form (5.177). Since a is closed and γ “ 0, the symmetric form <e aru, vs is also closed and <e aru, us ě 0. Then by Proposition 5.63 it defines a non-negative self-adjoint operator AR :“ <e A ě 0, which is called the real part of the operator A. Similar arguments for the closed non-negative symmetric form <e a˚ ru, vs yield also that AR “ <e A˚ . Note that for a bounded operator A P LpHq one readily obtains the identity AR “ pA ` A˚ q{2, although in general it is not true for unbounded operators. On the hand, by the definitions of a, a˚ and by the representation theorem we obtain


195

.

that the real part of A is the form-sum AR “ pA ` A˚ q{, which is a non-negative ? self-adjoint operator with dom AR “ dom a. We also recall that with help of the real part AR “ <e A the m-sectorial operator A with vertex γ “ 0 and semi-angle α P r0, π{2q has the following representation 1{2 1{2 A “ AR p1 ` i LqAR . (5.179) Here symmetric bounded operator L P LpHq is such that }L} ď tg α. Remark 5.65. By virtue of representation (5.179) the resolvent of the m-sectorial operator A is compact if and only if the resolvent of its real part AR is compact. Step 1. The first step in our analytic extension method is the complex extension of sectorial forms, cf. the representation (5.177). Let a and b be densely defined closed sesquilinear sectorial forms with vertex γ “ 0 and semi-angles αA , αB P r0, π{2q. We define two families of auxiliary closed sesquilinear forms for z P C by parametrisation of the imaginary parts az :“ <e a ` z =m a,

dom az “ dom a,

(5.180)

bz :“ <e b ` z =m b,

dom bz “ dom b.

(5.181)

Lemma 5.66. The forms z ÞÑ az and z ÞÑ bz are sectorial (with γ “ 0) and holomorphic functions in the strip DAB :“ tz P C : | <e z| ă mintctg αA , ctg αB uu.

(5.182)

Proof. Let x, y P R, z “ x ` iy, and u P dom a. Since the form a is sectorial, (5.178) yields |x =m arus| ď |x| tg αA <e arus. (5.183) Hence, for |x| tg αA ă 1 one gets <e az “ <e a`x =m a ą 0, i.e., γ “ 0, see (5.178). Moreover, for these values of x and u P dom a, we get <e az rus ě <e arus ´ |x =m arus| ě p1 ´ |x| tg αA q <e arus.

(5.184)

By (5.183) and (5.184), we obtain | =m az rus| “ |y =m arus| ď |y| tg αA <e arus ď

(5.185)

|y| tg αA <e az rus, 1 ´ |x| tg αA

and by (5.178) the form az is sectorial for z P tz P C : |x| ă ctg αA u. Similarly, the form bz is sectorial with γ “ 0 for |x| ă ctg αB . Therefore, the two forms az and bz are sectorial with γ “ 0 in the strip DAB defined by (5.182). Note that the families (5.180) and (5.181) are explicit functions of the complex variable z P C. Their derivatives with respect to this variable exist and have the form: Bz az rus “ =m arus,

u P dom a,

Bz bz rus “ =m brus,

u P dom b.

196


For each u P dom a, respectively u P dom b, they are linear holomorphic complex functions in the domain DAB . l Now we recall that a set of sesquilinear forms ttz uzPDĂC is called a holomorphic family of type (a) if it verifies two conditions: (1) Each tz is sectorial and closed with dense domain: dom tz “ Q, which is independent of z P D. (2) D Q z ÞÑ tz rus is holomorphic for each fixed u P Q. By the polarisation identity, this implies that the sesquilinear form tz ru, vs is holomorphic in z P D for each fixed pair u, v P Q. Recall that the concept of holomorphic family of operators is based on bounded-holomorphic properties of the corresponding resolvents. To this aim, consider a family of closed operators in a Banach space, tApzq P CpBquzPD , defined in a neighborhood of z0 P D. If ζ P C belongs to the resolvent set ρpAp0qq, then z ÞÑ Apzq is said to be holomorphic at z0 if there exists a disc Dε pz0 q :“ tz P C : |z ´ z0 | ă εu such that ζ P ρpApzqq and the resolvent Rζ pApzqq “ pApzq ´ ζ1q´1 is bounded-holomorphic for z P Dε pz0 q, ε ą 0. A special type of holomorphic family of operators that will be used in the text, is the family of type (A). Definition 5.67. A holomorphic family tApzquzPD of type (A) is defined by two conditions: (1) Each Apzq is closed with dense domain dom Apzq “ D, which is independent of z P D. (2) The mapping z ÞÑ Apzqu is a holomorphic vector-valued function for z P D and for every u P D. Conditions (1) and (2) imply that tApzquzPD is a holomorphic family in the bounded-holomorphic resolvent sense. The following criterion for type (A) is useful. Proposition 5.68. Let A be a closable operator with domain dom A “ D in a Banach space B. Let tApnq uně1 be operators with domains dom Apnq Ă D, and let a, b, c be non-negative constants such that }Apnq u} ď cn´1 pa}u} ` b}Au}q,

u P D, n P N.

Then for |z| ă 1{c and u P D the series Apzqu “

8 ÿ

z n Apnq u,

A :“ Ap0q ,

n“0

defines an operator Apzq with dom Apzq “ D. If |z| ă pb ` cq´1 , then operators ˜ Apzq are closable and the closures tApzqu tz:|z|ăpb`cq´1 u form a holomorphic family of type (A).


197

Corollary 5.69. (i) The forms z ÞÑ az (5.180) and z ÞÑ bz (5.181) are holomorphic families of type (a) in the strip DAB (5.182). Since az˚ “ az and b˚z “ bz , they are called self-adjoint holomorphic families. By the representation theorem (Proposition 5.63), there exist m-sectorial operators Apzq and Bpzq with γ “ 0, which are associated with the closed sectorial forms az and bz , z P DAB . (ii) These operators form resolvent bounded-holomorphic families, called holomorphic families of type (B). The operators tApzquzPDAB and tBpzquzPDAB are locally uniformly m-sectorial with vertex γ “ 0 and for the particular value z “ i yield Apiq “ A, Bpiq “ B. Therefore (Proposition 1.46) for each z P DAB the operators Apzq and Bpzq are generators of holomorphic contraction semigroups. (iii) Operator holomorphic families of type (B) inherit the properties of the form self-adjoint families. This yields A˚ pzq “ Apzq and B ˚ pzq “ Bpzq. Remark 5.70. Let the family tCpzquzPD be Ş holomorphic of type (A) and have non-empty resolvent sets such that ρD “ zPD ρpCpzqq ‰ H. Then Cpzq has a compact resolvent Rζ pCpzqq either for all z P D, or for no z. Let tCpzquzPD be a type (B) holomorphic family of (m-sectorial) operators. If the operator Cpz0 q has compact resolvent for z0 P D, then the resolvent of Cpzq is compact for any z P D. Step 2. The second step is the complex extension of the semigroup generators corresponding to sectorial forms. Proposition 5.71. Let tCpzquzPD be a holomorphic family of type (B) with vertex γ “ 0 and semi-angles αpzq ă π{2. Then for each z P D the operator Cpzq is the generator of a holomorphic contraction semigroup tUt pCpzqqutPSθpzq with semiangle θpzq P r0, π{2q. The function z ÞÑ Ut pCpzqq is holomorphic in z P D for any t in the open sector Sθ0 , with semi-angle θ0 :“ inf zPD pπ{2 ´ αpzqq. Proof. For each z P D the m-sectorial operator Cpzq P H pθpzq “ π{2 ´ αpzq, 0q is the generator of a holomorphic contraction semigroup, see Corollary 5.69(ii). Then by the Riesz-Dunford representation (1.67) for holomorphic semigroups we have ż 1 e´tζ dζ . (5.186) Ut pCpzqq “ 2πi Γ ζ1 ´ Cpzq RecallŞthat this integral is absolutely } ¨ }-convergent for t ą 0 if the contour Γ Ă zPD ρpCpzqq :“ ρD , running from infinity with arg ζ “ αmax ` ε, and then back to infinity with arg ζ “ ´pαmax ` εq, where αmax :“ supzPD αpzq for some 0 ă ε ă π{2 ´ αmax , see Section 1.5. Since tCpzquzPD is a holomorphic family, the resolvents tRζ pCpzqquzPD form a holomorphic family for any ζ P ρD . By Proposition 1.27 the generator Cpzq satisfies the condition (1.69), which yields

198


the estimate › › ż ˜ε › › › › M 1 1 ´2 1 ´1 › ›Bz pζ1 ´ Cpzqq´1 › “ › 1 dz pz ´ zq pζ1 ´ Cpz qq › 2πi › ď |ζ| r , γr

(5.187)

for any ζ P tζ P C : ´pαmax ` εq ă arg ζ ă αmax ` εu. Here γr is a small circle of radius r around z and › › ˜ ε :“ ›pζ1 ´ Cpz 1 qq´1 › . M sup ¯ ζPCzS π{2´θ0 `ε z 1 Pγr

The estimate (5.187) shows that the representation (5.186) is operator-norm differentiable under the integral since the integrand is operator-valued holomorphic complex function in D. Note that the arguments developed above are also valid for t P C such that | argpt ζq| ă π{2. Then we get the statement for any t in the open sector Sθ0 with semi-angle θ0 “ π{2 ´ αmax , which implies the assertion. l Corollary 5.72. Applying Proposition 5.71 to the particular case of generators tApzquzPDAB and tBpzquzPDAB in Corollary 5.69, we obtain that the contraction semigroups tUt pApzqqutPSθ0 and tUt pBpzqqutPSθ0 (5.188) 0 are holomorphic families for z P DAB , and for any t in the sector Sθ0 . Here, the domain of analyticity in z is defined as 0 DAB “ tz P C : | <e z| ă mintctg αA , ctg αB u ´ δ0 :“ ∆R ,

(5.189)

| =m z| ă ∆I u, for a small δ0 ą 0 and for some ∆I ą 1, cf. (5.182). Then by (5.185), the corresponding semi-angle θ0 ą 0 is defined by ´π ¯ ∆I maxttg αA , tg αB u tg ´ θ0 “ . (5.190) 2 1 ´ ∆R maxttg αA , tg αB u Step 3 (Vitali’s theorem). The last step in our analytic extension method is application of the Vitali theorem to holomorphic families of Trotter’s approximants. Remark 5.73. Recall that for holomorphic families of bounded operators in LpHq there is no distinction between uniform, strong, or weak operator analyticity. Moreover, these holomorphic operator-valued functions inherit some properties known from standard complex analysis. For example, let tΦn pzquně1 Ă LpHq be a sequence of operator-valued complex functions such that }Φn pzq} ă M, for all n ě 1 and z P D Ă C. If tΦn pzquně1 converges (in any of the three topologies) on a subset of D having a limit point in D, then by the Vitali theorem the limit limnÑ8 Φn pzq “ Φpzq exists for any z P D, the convergence is uniform on any compact K Ă D, and the limiting function Φpzq is holomorphic in D.


199

5.5.2 Norm holomorphic families of the Trotter approximants To continue our analysis of the convergence of the Trotter product formula in the trace-norm we recall the exponential case of Proposition 5.36. Proposition 5.74. Suppose A and B are non-negative densely defined self-adjoint . operators in H and the form-sum H “ A ` B is defined in the subspace H0 “ P0 pHq, where P0 is the orthogonal projection P0 : H Ñ dom A1{2 X dom B 1{2 . If the resolvent of A or of B is compact: p1 ` Aq´1 _ p1 ` Bq´1 P C8 pHq, then the Trotter product formula . ` ˘n } ¨ } - lim e´tA{n e´tB{n “ e´tpA`Bq P0 ,

nÑ8

(5.191)

converges in the operator-norm topology for t ą 0, locally uniformly away from zero. Now we can extend Proposition 5.74 to m-sectorial generators. Proposition 5.75. If A and B are m-sectorial operators (with vertex γ “ 0) in a Hilbert space H and if p1 ` Aq´1 _ p1 ` Bq´1 P C8 pHq, then the Trotter product formula converges in the operator-norm topology: . ` ˘n } ¨ } - lim e´tA{n e´tB{n “ e´tpA`Bq P0 ,

nÑ8

(5.192)

for any t P Sθ , where θ “ π{2 ´ maxtαA , αB u. The convergence is uniform on the . compact subsets of the sector Sθ . The generator H “ A ` B is the m-sectorial form-sum of A and B, and P0 is the orthogonal projection P0 : H Ñ H0 “ dom pHq. Proof. Following Corollary 5.69, we include the operators A and B into holomorphic type (B) self-adjoint families tApzquzPDAB and tBpzquzPDAB of m-sectorial operators with γ “ 0, which are generators of holomorphic contraction semigroups. Then using Corollary 5.72 we construct from the two families (5.188) a sequence of operator-valued uniformly bounded and holomorphic in the operator-norm topology Trotter product formula approximants ` ˘n z ÞÑ Φn pt, zq :“ e´tApzq{n e´tBpzq{n , }Φn pt, zq} ď 1, (5.193) 0 n P N, with domain of analyticity DAB , which is determined by (5.189) for any t in the sector Sθ0 with semi-angle defined by (5.190). Note that by the assumption of the proposition and by Remark 5.70 for D “ 0 DAB , the p1 ` Apzqq´1 and p1 ` Bpzqq´1 are holomorphic families of compact operators. In particular, the resolvents tp1 ` Apxqq´1 uxP∆R (or tp1 ` Bpxqq´1 uxP∆R ) are compact. Since the generators in (5.193) are self-adjoint holomorphic families of m-sectorial operators with γ “ 0, the operators Apxq and Bpxq are selfadjoint and non-negative for each x P ∆R . Then (5.180) and (5.181) yield that

200


dom a “ dom Apxq1{2 , dom b “ dom Bpxq1{2 , and that the self-adjoint non. negative form-sum Apxq ` Bpxq exists in domain D Ď H0 , which is dense in H0 “ pdom a X dom bq. Now we are in the position to apply Proposition 5.74 to the sequence of functions (5.193) for z “ x and t P Sθ . This yields .

} ¨ } - lim Φn pt, xq “ e´tpApxq`Bpxqq P0 , nÑ8

x P ∆R .

(5.194)

Therefore, the operator-norm convergence (5.194) of the operator-valued uniformly bounded and holomorphic family (5.193) on the interval ∆R , together with the 0 Vitali theorem imply the operator-norm convergence in (5.194) for any z P DAB . . 0 Since i P DAB , we obtain (5.192) as well as the equality H “ Apiq ` Bpiq. l The analytic extension method of Proposition 5.75 allows us to lift the operator-norm convergence of the Trotter-Kato product formulae for non-selfadjoint semigroups to the trace-norm convergence for non-self-adjoint Gibbs semigroups. The first step is the following assertion. Proposition 5.76. If A is an m-sectorial operator with vertex γ “ 0 such that e´t <e A P C1 pHq for t ą 0, then te´tApzq utě0 is a family of holomorphic contraction 0 Gibbs semigroups for z P DA , where 0 DA “ tz P C : | <e z| ă ∆R pAq “ ctg αA ´ δ0 ,

| =m z| ă ∆I pAqu,

(5.195)

for some δ0 ą 0 and ∆I pAq ą 1, in the sector Sθ0 pAq with the semi-angle θ0 pAq defined by the equation tg

`π 2

˘ ´ θ0 pAq “

∆I pAq tg αA . 1 ´ ∆R pAq tg αA

(5.196)

0 Moreover, the function z ÞÑ e´tApzq is } ¨ }1 -holomorphic in z P DA for any t P Sθ0 pAq .

Proof. Proposition 4.30 shows that e´tA P C1 pHq, for t ą 0. Then by representation (4.39) the operators <e A and A have compact resolvents. By Corollary 5.69 and Remark 5.70, tApzquzPDA0 is a holomorphic family of uniformly sectorial operators and Apz “ 0q “ <e A. Hence, Apzq has a compact resolvent and 0 Apzq P H pθ0 pAq, 0q for any z P DA , where θ0 pAq is defined by (5.196). The same is true for <e Apzq. Taking into account inequality (5.185) we obtain the estimate t1 ´ | <e z| tg αA u <e arus ď <e az rus ,

0 z P DA ,

u P dom a .

(5.197)

Then by the minimax principle for the self-adjoint operators <e Apzq and <e A, see Proposition 4.23 and Corollary 4.24, and by the condition e´t <e A P C1 pHq for


201

0 t ą 0, we conclude that e´t <e Apzq P C1 pHq for t ą 0 and z P DA . In addition, by Proposition 4.30 we have the estimate }e´tApzq }1 ď }e´t <e Apzq }1 . Finally, by (5.185), (5.195) and (5.196), the family tApzquzPDA0 consists of uniformly sectorial operators with numerical range NrpApzqq Ă Sπ{2´θ0 pAq . By Corollary 4.32 we get that tGt pApzqqutPSθ0 pAq is a holomorphic contraction Gibbs 0 semigroup for any z P DA . On the other hand, by Proposition 5.71, the family te´tApzq uzPDA0 is } ¨ }holomorphic (cf. Remark 5.73) for any t P Sθ0 pAq . Therefore, we have the representation ¿ 1 e´tApwq e´tApzq “ dw , (5.198) 2πi w´z γr

where the } ¨ }-convergent Cauchy integral is taken along a small circle γr “ tw P 0 C : |w ´ z| “ ru Ă DA . Since the trace-norm of the semigroup tGt pApzqqutPSθ0 pAq is bounded, using (5.197) one gets for w P γr the estimate › › › e´tApwq › }e´t <e Apwq }1 1 ›Bz › ď }e´βt <e A }1 2 , › w´z › ď 2 r r 1 where β :“ inf wPγr p1 ´ | <e w| tg αA q. Thus, the Cauchy integral (5.198) is } ¨ }1 0 differentiable, i.e., the function z ÞÑ e´tApzq is } ¨ }1 -holomorphic in DA for any t P Sθ0 pAq . l Now we can apply the analytic extension method to prove the trace-norm convergence of the Trotter-Kato product formulae for holomorphic Gibbs semigroups with non-self-adjoint m-sectorial generators A and B. This approach does not give an error estimate for the rate of convergence but treats the generators on the equal level and even for a trivial common domain: dom A X dom B “ t0u. To this aim we collect in Remark 5.77 the elements that we established for application of the Vitali theorem in the } ¨ }1 -topology. Remark 5.77. Let A and B be m-sectorial operators with vertex γ “ 0. Then according to definitions (5.180) and (5.181), and by virtue of Corollary 5.69, we constructed two holomorphic type (B) families of m-sectorial operators tApzquzPDA0 0 0 and tBpzquzPDB0 with DA and DB defined by (5.195), such that: (a) Apiq “ A and Bpiq “ B. (b) For x “ <e z P r´∆R , ∆R s, where ∆R “ mintctg αA , ctg αB u ´ δ0 , the nonnegative operators Apxq “ <e Apzq ě 0 and Bpxq “ <e Bpzq ě 0 are selfadjoint. (c) If e´t <e A and e´t <e B are trace-class operators for t ą 0, then by Proposition 5.76 the m-sectorial operators tApzquzPDA0 and tBpzquzPDB0 generate two 0 0 } ¨ }1 -holomorphic in DA and in DB families of holomorphic for t P Sθ0 pAq , respectively for t P Sθ0 pBq , contraction Gibbs semigroups. Here the sectors Sθ0 pAq , respectively Sθ0 pBq , are defined by condition (5.196).

202


Proposition 5.78. Let A and B be m-sectorial operators with vertex γ “ 0 in a Hilbert space H. If e´t <e A _ e´t <e B is trace-class operator for t ą 0, then the Trotter product formula for contraction Gibbs semigroups converges in the tracenorm topology: . ˘n ` } ¨ }1 - lim e´tA{n e´tB{n “ e´tpA`Bq P0 ,

nÑ8

(5.199)

for any t P Sθ , where θ “ π{2 ´ maxtαA , αB u. The convergence is uniform on . the compact subsets of the open sector Sθ . The generator H “ A ` B is the msectorial form-sum of A and B, and P0 is the orthogonal projection P0 : H Ñ H0 “ dom pHq. Proof. Essentially we argue as in the proof of Proposition 5.75. To this end we extend to the trace-norm topology the Vitali theorem about the sequence of the operator-valued functions identical with product formula approximants (5.193): ` ˘n 0 Φn pt, zq “ e´tApzq{n e´tBpzq{n , z P DAB , t P Sθ0 , n P N. (5.200) The assumption of the proposition, Corollary 5.72, Proposition 5.76 and the } ¨ }1 -continuity in z of the product e´tApzq e´tBpzq imply that the functions tz ÞÑ 0 e´tApzq e´tBpzq utPSθ0 are } ¨ }1 -holomorphic for z P DAB if either of exponentials (or both) belong to the ideal C1 pHq. Therefore, Φn pt, zq P C1 pHq and the function 0 z ÞÑ Φn pt, zq is } ¨ }1 -holomorphic in z P DAB for any t P Sθ0 . Since the conditions of the proposition are symmetric with respect to A and B, suppose that e´t <e A P C1 pHq. Then, by Lemma 4.43, the sequence 0 tΦn pt, zquně1 is uniformly bounded for z P DAB and t P Sθ0 in the trace-norm topology: }Φn pt, zq}1 ď }e´tBpzq{n }n }e´t <e Apzq{4 }1 ď }e´βt <e A{4 }1 .

(5.201)

0 Here we used that, by Corollary 5.72, for z P DAB the operator Bpzq is the gener´tBpzq ator of a contraction semigroup: }e } ď 1, and that by (5.197) and Corollary 4.24, the last inequality in (5.201) is valid for β “ inf zPDA0 p1 ´ | <e z| tg αA q. Note that for any real x P r´∆R , ∆R s, ∆R “ mintctg αA , ctg αB u ´ δ0 , the operators Apxq ě 0 and Bpxq ě 0 are self-adjoint, see Remark 5.77(b). Then by Proposition 5.53 the sequence tΦn pt, xquně1 converges in the } ¨ }1 -norm to e´tHpxq P0 for t P Sθ0 and for x P r´∆R , ∆R s. Here the m-sectorial operator 0 9 Hpzq “ Apzq`Bpzq, with z P DAB , is the form-sum corresponding to the sum of two closed densely defined sectorial forms (5.180) and (5.181). 0 The interval r´∆R , ∆R s is compact in DAB . Since the family tΦn pt, zquně1 0 is } ¨ }1 -holomorphic in z P DAB and } ¨ }1 -uniformly bounded in this domain by (5.201), the Vitali theorem (Remark 5.73) yields ` ˘n } ¨ }1 -lim e´tApzq{n e´tBpzq{n “ e´tHpzq P0 , t P Sθ0 , (5.202) nÑ8


203

0 0 for any z P DAB , one gets the assertion (5.199) in a (5.189). Since z “ i P DAB smaller sector θ0 ă θ, where θ0 is defined by (5.190). The convergence (5.199) for θ0 “ θ follows from (5.202) by shrinking domain 0 DAB , Remark 5.77. To this aim one takes in (5.190) the limits ∆R Ñ 0 and ∆I Ñ 1 that localise, in particular, the point z “ i. l

Corollary 5.79. Let A be an m-sectorial operator with semi-angle αA and vertex γ “ 0. If e´t <e A is a trace-class operator for t ą 0, then the Trotter product formula for contraction Gibbs semigroups converges in the trace-norm topology ˘n ` ˚ } ¨ }1 - lim e´tA {2n e´tA{2n “ e´t <e A , (5.203) nÑ8

for any t P Sθ , where θ “ π{2 ´ αA . The convergence is uniform on the compact subsets of the open sector Sθ .

5.5.3

Trace-norm convergence under smallness conditions

In this subsection we impose additional conditions on the generators A and B to prove the trace-norm convergence of the Trotter product formula with error bound estimates for the rate of convergence. Then we do not need a lifting via the analytic extension method, but the one that we used in Section 5.4, where the lifting Lemma 5.51 plays the key rôle. Proposition 5.80. Let A be an m-sectorial operator in a Hilbert space H. Suppose that <e A ą 0 and e´t <e A P C1 pHq for t ą 0. If B is an m-accretive operator in H such that dom Aα Ď dom B for some α P r0, 1q and dom A˚ Ď dom B ˚ , then the Trotter formula converges in the trace-norm with convergence rate estimates: › › ˘n › ›` (5.204) › e´tA{n e´tB{n ´ e´tpA`Bq › 1

ď Oplnpnq{n1´α q pα ‰ 0q _ Opplnpnqq2 {nq pα “ 0q, uniformly away from t “ 0. Proof. Since e´t <e A P C1 pHq for t ą 0, Proposition 4.30 shows that Gt pAq “ e´tA P C1 pHq is a holomorphic contraction Gibbs semigroup, which is generated by A. We express the fractional power 0 ă α ă 1 of A by the integral ż8 1 α A u“ dτ τ ´α´1 pGτ pAq ´ 1qu, u P dom A, Γp´αq 0 where τ α is chosen to be positive for τ ą 0. Notice that for any u P dom A this integral is convergent, thus dom A Ď dom Aα and consequently dom Aα Gt pAq P LpHq for t ą 0. It follows that ż ż8¯ 1 ´ t Aα Gt pAq “ ` dτ τ ´α´1 pGt`τ pAq ´ Gt pAqq. Γp´αq 0 t

204


Taking into account the estimate for the derivative of a holomorphic semigroup (Corollary 1.28 and Remark 1.31, for n “ 0) one gets that }Gt`τ pAq ´ Gt pAq} ď CA τ {t, and therefore Mα }Aα Gt pAq} ď α . (5.205) t Recall that a closed operator B belongs to the class of P0` -perturbations of the C0 -semigroup Ut pAq if dom B Ě

ď

ż1 Ut pAqH

and

dt}B Ut pAq} ă 8, 0

tą0

see Definition 1.50 and Definition 4.34. Since the condition dom Aα Ď dom B and (5.205) for the holomorphic semigroup tGt pAqutě0 yield the estimate ż1

ż1 dt }BA´α }}Aα e´tA }

dt }BGt pAq} ď 0

0

ż1 ď }BA

´α

dt

} 0

Mα ă 8, tα

the perturbation B P P0` Ă Pbă1 . Then by Proposition 4.44 the operator H :“ A ` B is the generator of the Gibbs semigroup tGt pA ` Bqutě0 . Since the assumptions of the proposition imply the operator-norm convergence (Proposition 5.47), with the convergence rate estimate, as well as the validity of the lifting Lemma 5.51, we use (5.148) to calculate εpm, tq in the inequality (5.151). Then the estimate (5.152) yields the rate of the trace-norm convergence (5.204). l Another case of lifting the operator-norm convergence of the Trotter product formula to the trace-norm topology is related to Proposition 5.49. Proposition 5.81. Let A be a non-negative self-adjoint operator in H such that pλ1 ` Aq´1 P Cp pHq for some λ P C and for a finite p ě 1. If B is an m-accretive operator such that dom A Ď dom B and dom A Ď dom B ˚ , with }Bu} ď b}Au}, ˚

}B u} ď b˚ }Au},

u P dom A, 0 ă b ă 1,

(5.206)

u P dom A, 0 ă b˚ ă 1,

(5.207)

then the Trotter product formula converges in the trace-norm topology with the error bound for the rate of convergence estimated by ˆ ˙ ›` › lnpnq › ´tA{n ´tB{n ˘n ´tpA`Bq › e é , (5.208) › e › ďO n 1 uniformly away from t “ 0.

5.6. Notes

205

Proof. By the hypotheses (5.206), (5.207) and by Definition 1.50 we have B P Pbă1 pAq. The m-accretive operator B generates the contraction semigroup tUt pBqutě0 . On the other hand, since pλ1Àq´1 P Cp pHq, Definition 4.26 and Proposition 4.27 show that the operator A is the self-adjoint p-generator for the holomorphic contraction Gibbs semigroup tGt pAqut with t P Sπ{2 . Then by Proposition 4.49 the operator A ` B P H pπ{2 ´ δ, 0q generates a holomorphic contraction Gibbs semigroup tGt pA ` Bqut for t P Sπ{2´δ . Therefore, the semigroups Gt pAq, Ut pBq and Gt pH “ A ` Bq, satisfy the corresponding requirements of the lifting Lemma 5.51. Since the conditions of the proposition imply the operator-norm convergence of the Trotter product formula (Proposition 5.49), with explicit estimate of the rate of convergence, we use (5.150) to calculate εpm, tq in inequality (5.151) of Lemma 5.51. Then the estimate (5.152) l yields the rate (5.208) of convergence in the trace-norm.

5.6 Notes Notes to Section 5.1. Proposition 5.1 is due to S. Lie (1875), who established it for matrices. The symmetrised version of the Trotter product formula in Proposition 5.1 was a source of many other generalisations for matrices and for the case of bounded generators, as well as for the so-called quantum Monte Carlo simulations, see e.g. [Suz96]. In this case the semigroups are norm-continuous, Section 1.4. The important step forward was done by H. Trotter [Tro59], who extended the Lie product formula to Banach spaces. A similar extension was formulated by Yu. L. Dalestskii [Dal60], [Dal61], in the context of solutions of operator-valued equations. The programme (a)-(e) outlined at the end of Section 5.1 for strongly continuous semigroups was formulated by V. A. Zagrebnov [Zag88], and then refined in [NZ90a]. Therein this programme was realised for the trace-norm convergence of the Trotter product formula first in the case of the Gibbs semigroups generated by Schrödinger operators. In [NZ90a], [NZ90b] H. Neidhardt and V. A. Zagrebnov extended this result to the abstract strongly continuous Gibbs semigroups and to convergence of the Trotter-Kato product formulae in the trace-norm topology. This and later the operator-norm convergence confirm the conventional wisdom [Zag03b] that a natural topology γ for the convergence of the product formulae inherits the topology in which the semigroup is continuous away from zero. (We discuss the concept of continuity away from zero in Section 4.2) Although for a long time, the convergence of the Lie-Trotter product formula for strongly continuous semigroups was known only in the strong operator topology γ “ s ([Tro59], [RS80]), as a trivial byproduct one obtains sufficient conditions for the operator-norm (γ “ } ¨ }) convergence for this formula in [NZ90a, NZ90b].

206


Moreover, these results were obtained in the more general context of the TrotterKato product formulae. A few years later, Dzh. L. Rogava in a short note [Rog93] announced the } ¨ }-convergence of the Trotter product formula for (non-Gibbs) strongly continuous self-adjoint contraction semigroups. This statement with improved rate of convergence and generalisation to the Trotter-Kato product formulae is due to H. Neidhardt and V. A. Zagrebnov [NZ98]. See also Chapter D.4 for further details concerning this episode. Notes to Section 5.2. Proposition 5.8 is due to H. Neidhardt and V. A. Zagrebnov [NZ98]. In the proof presented here we essentially follow this paper. For self-adjoint semigroups the proof of the operator-norm convergence (with estimate of the error bound for the rate) was improved in [ITTZ01]. This refinement allows to obtain the optimal estimate of the rate of convergence for the p β of Trotter product formula (Proposition 5.19) as well as for a special class K Kato functions for the Trotter-Kato product formulae (Proposition 5.25). Although these statements cover a majority of results for the self-adjoint case, they (in contrast to the method [NZ98]) are too restrictive to be applicable in the case of non-self-adjoint semigroups. The fractional power conditions and result (Proposition 5.27) are due to H. Neidhardt and V. A. Zagrebnov [NZ99a]. The optimality of the error bound estimate under the assumptions of Proposition 5.27 was proved by Hiroshi Tamura [Tam00], see Proposition 5.28. He also proved (Proposition 5.28) that the operatornorm convergence of the Trotter product formula cannot be extended (as conjectured in [NZ99a]) to the case α P p0, 1{2s in the abstract setting. It was shown that the case α “ 1{2 is an example where the Trotter product formula does not hold in the operator-norm topology, but still holds for convergence in the strong operator sense. The case of the fractional power α “ 1{2 was studied by H. Neidhardt and V. A. Zagrebnov in [NZ99b]. It is of special interest since one cannot expect to have operator-norm convergence without subordination of the generators A and B. This condition in Proposition 5.30 is the B 1{2 relative A1{2 -compactness. Proposition 5.27 was improved by T. Ichinose, H. Neidhardt and V. A. Zagrebnov in [INZ04], see Proposition 5.32. Their result extends Proposition 5.27 p β of Kato functions, see Definition 5.24. from the class Kα to a larger class K Moreover, it also relaxes the smallness conditions of Proposition 5.27. To this aim we modified accordingly the fractional conditions in Proposition 5.32. Note that Tamura’s optimal theorem (Proposition 5.28) still ensures that under the conditions of Proposition 5.32 the rate of convergence of Trotter-Kato product formulae is optimal for nonsymmetric case. For further discussion of optimality we refer to [IT09]. The explicit example (one-dimensional harmonic oscillator), in which the symmetric Lie-Trotter product formula converges as fast as for matrices, i.e., with the sharp, and thus optimal, rate Op1{n2 q, was proposed and studied in details by Y. Azuma and T. Ichinose

5.6. Notes

207

in [AzIch08]. Notes to Section 5.3. Proposition 5.36 is due to H. Neidhardt and V. A. Zagrebnov [NZ99b]. This paper establishes sufficient conditions that ensure the convergence for a class of the Trotter-Kato product formulae in the operator-norm topology without an estimate of the error bound for self-adjoint generators. In the proof presented here we essentially follow [NZ99b]. The result for the non-self-adjoint case with estimates presented in Proposition 5.47 is due to V. Cachia and V. A. Zagrebnov [CZ01a]. The line of reasoning is based on the method of [NZ98] and it heavily uses the analyticity and the infinitesimal smallness of m-accretive perturbations. Although the estimates are far from optimal, the method is strong enough to cover even the case of Banach spaces. Proposition 5.49 is due to V. Cachia, H. Neidhardt and V. A. Zagrebnov [CNZ01]. It drastically relaxes the smallness condition of [CZ01a]. The proof is again motivated by [NZ98]. Notes to Section 5.4. The key Lemma 5.51 for exponential Kato function is due to V. Cachia and V. A. Zagrebnov [CZ01c]. It allows us to prove the trace-norm convergence without error bound estimate for the self-adjoint Trotter product formula, see Proposition 5.53. This result is a generalisation of [NZ90a, NZ90b] and of [Hia95]. If one adds a smallness condition, then Lemma 5.51 lifts the operator-norm convergence of the self-adjoint Trotter product formula to the trace-norm convergence with an error bound estimate that coincides with that for the operator-norm convergence, see Proposition 5.55. Here we follow [CZ01c]. Propositions 5.56 and 5.58 show that Lemma 5.51 lifts the optimal operatornorm convergence error bound to the optimal trace-norm convergence error bound. The lifting Lemma 5.51 is well suited to prove the trace-norm convergence of the Trotter product formula, but it is not very useful for the Trotter-Kato product formulae. In [NZ90a] (Lemma 3.2) we established the first result in this direction under the conditions of Proposition 5.61. Note that Kato functions of the class K˚ were historically the first non-exponential functions proposed in [Kat74] for product formula approximants converging in the strong operator topology. To this aim he introduced an auxiliary functions which define the class K˚ . Under stronger (than (5.174)) condition (5.175) the class of Kato functions admissible for the trace-norm convergence is enlarged in Proposition 5.62 from K˚ to generic class K. This assertion was proven in [NZ90a] (Theorem 3.4). In [NZ99d] we developed further the auxiliary function control of K to produce more sufficient conditions ensuring the trace-norm convergence of Trotter-Kato product formulae, but now in the general framework of the symmetrically-normed ideals, see Chapter 6. Notes to Section 5.5. The analytic extension method is due to V. Cachia and V. A. Zagrebnov [CZ99], [CZ01c]. We proposed it for analysis of the Trotter product formula approximants in case of m-sectorial generators.

208


The Representation Theorem (Proposition 5.63) is proved in [Kat80], Ch.IX. In the rest of the preliminaries (Lemma 5.66, Corollary 5.69, Proposition 5.71 and Corollary 5.72) we develop the idea of analytic extension from [Kat78] and [CZ99]. Then an application of the Vitali theorem to the operator-norm holomorphic families (of approximants) allows us to prove Propositions 5.75. The first step in lifting the operator-norm convergence, established in Propositions 5.75, to convergence in the trace-norm is based on Propositions 5.76, [CZ01c]. Proposition 5.75 allows us also to prove (see Proposition 5.78) the nonself-adjoint Trotter product formula in the case of the trace-norm convergence without error bound estimate [CZ01c]. See an important in the context of the present book Corollary 5.79, in particular because of its application to inequality (3.45) in Section 3.3. We note that idea of the analytic extension method was formulated by B. Simon and published by T. Kato as Addendum in [Kat78]. It was realised in [CZ99] and in [CZ01c]. There we constructed extensions of non-self-adjoint holomorphic families of the Trotter product formula approximants (5.193) and then proved their convergence in operator-norm and in trace-norm topologies. In conclusion, we use the lifting Lemma 5.51 to prove the trace-norm convergence of the non-self-adjoint Trotter product formula with error bound estimates. In Proposition 5.80 we use the lifting lemma and the error bound estimate obtained in [CNZ01]. Another application of the lifting lemma for the Trotter product formula and non-self-adjoint Gibbs semigroups is Proposition 5.81. Both of these results are from [CZ01c]. See also a well-written review in [Ca10].

Chapter 6

Product formulae in symmetrically-normed ideals In this chapter we continuing realisation of the lifting error bound estimate programme outlined in Chapter 5. The central problem is still the proof and the estimate of the rate of convergence of the Trotter-Kato product formulae, but now in the general setting of symmetrically-normed ideals of compact operators, where a particular case important for the Gibbs semigroups is the trace-class. Section 6.1 serves for recalling a three-step strategy for lifting the convergence of operator-norm product formulae to convergence in the symmetric-norm topology. Since we consider the Trotter-Kato product formulae, the properties of various classes of Kato functions K are important for establishing the convergence. A class of self-dominated Kato functions has a particular importance for our arguments. Section 6.2 is a brief introduction to symmetrically-normed ideals, whereas Section 6.3 contains a recall about convergence in these ideals. The results presented in Section 6.4 involve the operator-norm convergence of the Trotter-Kato product formulae. Sufficient conditions for lifting them to the case of symmetric-norm topologies are formulated in terms of properties of auxiliary dominating Kato functions. In Section 6.5 the symmetric-norm convergence of the Trotter-Kato product formulae is proven for self-dominated Kato functions without reference to the operator-norm convergence. This statement (Proposition 6.34) is of special importance for the theory of Gibbs semigroups. To obtain the symmetric-norm error bound estimates we return in Section 6.6 to the lifting arguments, Proposition 6.38. This allows one to express the symmetric-norm error bound estimates via operator-norm error bound estimates. We find that the convergence rates in the two topologies coincide. Therefore, the optimality of the operator-norm rate remains valid in the case of the symmetric norm. © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1_6

209

210

Chapter 6. Product formulae in symmetrically-normed ideals

6.1 Preliminaries For non-negative self-adjoint operators A,B and Kato functions f, g P K we set F ptq :“ gptBq1{2 f ptAqgptBq1{2 ,

t ě 0,

(6.1)

T ptq :“ f ptAq1{2 gptBqf ptAq1{2 ,

t ě 0,

(6.2)

and for symmetrised self-adjoint families tF ptqutě0 and tT ptqutě0 . The question of convergence of the Trotter-Kato product formulae for Gibbs semigroups in C1 pHq, formulated in Section 5.4 (subsection 5.4.4), is part of the following general problem. Let φ be any symmetric norming function and Cφ pHq be the corresponding symmetrically-normed ideal of compact operators, with the norm denoted by } ¨ }φ , see Section 6.2. Then the main problem solved in the present chapter can be stated as follows: Find conditions that guarantee the converges of the Trotter-Kato product formulae locally uniformly away from t0 ą 0 (Definition 5.35) in the } ¨ }φ -topology on the ideal Cφ pHq, for example, for the Trotter-Kato approximants generated by tF ptqutě0 . Equivalently, this would mean that for any bounded interval rτ0 , τ s Ď pt0 , `8q there should exist r0 ě 1 (or n0 P N) such that e´tH P0 P Cφ pHq and F pt{rqr P Cφ pHq,

(6.3)

uniformly in t P rτ0 , τ s and in r ě r0 (or in n ě n0 ), and lim

sup }F pt{rqr ´ e´tH P0 }φ “ 0.

rÑ8 tPrτ ,τ s 0

.

(6.4)

Recall that the generator H “ A ` B is the form-sum of A and B, and P0 is the orthogonal projection P0 : H Ñ H0 “ dompHq. Similarly, one can extend this notion of convergence of the Trotter-Kato product formulae to the families tT ptqutě0 , tf ptAqgptBqutě0 and tgptBqf ptAqutě0 , though in the last two cases one has to use the discrete parameter n P N instead of the continuous parameter r ě 1. The continuous parameter r is suitable for self-adjoint approximants corresponding to the families (6.1) and (6.2). If the convergence holds for these families, then we say that the Trotter-Kato product formulae converge in the topology of Cφ pHq locally uniformly away from t0 for all families generated by f and g. It turns out (Proposition 6.18) that if the Trotter-Kato product formula converges locally uniformly away from t0 ą 0 in Cφ pHq for one of the families tF ptqutě0 , tT ptqutě0 , tf ptAqgptBqutě0 , tgptBqf ptAqutě0 , then the product formulae converge locally uniformly away from t0 in Cφ pHq for all approximants generated by these families. To solve the problem formulated in (6.3), (6.4) one has to provide answers to the following three sub-problems:

6.1. Preliminaries

211

First, find the conditions under which that the Trotter-Kato product formulae converge locally uniformly away from some t0 ą 0 in the operator-norm topology, i.e., for example (6.5) lim sup }F pt{rqr ´ e´tH P0 } “ 0, rÑ8 tPra,bs

for any bounded interval ra, bs Ă R` . Second, one has to guarantee that the sequence tF pt{rqr urě1 is locally uniformly bounded away from t0 ą 0 in the } ¨ }φ -topology. This means that there is r0 ě 1 such that for r ě r0 and t P rτ0 , τ s one has F pt{rqr P Cφ pHq and M prτ0 , τ sq :“ sup sup }F pt{rqr }φ ă `8,

(6.6)

rěr0 tPrτ0 ,τ s

for any bounded interval rτ0 , τ s Ď pt0 , `8q. Third, we have to verify that the limit e´tH belongs to the ideal Cφ pHq (or ´tH e P0 P Cφ pH0 q) for t ě t0 ą 0. In fact these steps are the ingredients of the lifting method for establishing the } ¨ }φ -convergence in (6.4). We have already used this approach in Section 5.4 for lifting the operator-norm convergence to the trace-norm convergence for the exponential Trotter-Kato product formula, see subsections 5.4.1–5.4.3. To solve the first sub-problem, we essentially use the results of Sections 5.2 and 5.3. The solution of the second sub-problem relies on Section 5.4 and especially on the remarks concerning the (non-exponential) Trotter-Kato approximants in subsection 5.4.4. To this aim we introduce the notion of dominated Kato functions KD , cf. Appendix C. Definition 6.1. We say that the generic Kato functions f and g are dominated by, ` ` ` D respectively, Borel measurable functions f D : R` 0 Ñ R0 and g : R0 Ñ R0 , if f pqxq1{q ď f D pxq and gpqxq1{q ď g D pxq, 0 ă q ď 1.

(6.7)

Therefore, if for some t0 ą 0 the condition F D pt0 q :“ g D pt0 Bq1{2 f D pt0 Aqg D pt0 Bq1{2 P Cφ pHq.

(6.8)

is satisfied, then one can prove that away from t0 the local uniform boundedness of tF pt{rqurě1 holds in Cφ pHq and, moreover, that the third sub-problem is also solvable: e´tH P0 P Cφ pHq for t ą t0 . A combination of these results with the operator-norm convergence of the Trotter-Kato product formulae established in Sections 5.2 and 5.3, yields a list of conditions that allow to establish the locally uniform } ¨ }φ -convergence of the Trotter-Kato product formulae away from t0 . We start this list by the notion of self-dominated Kato functions Ks-d Ă K, see Section C.3.

212


Definition 6.2. The Kato functions f, g P K obeying f pqxq1{q ď f pxq and gpqxq1{q ď gpxq,

0 ă q ď 1,

(6.9)

xf pxq xgpxq ă `8 and S :“ sup ă `8 , 1 ´ f pxq xą0 1 ´ gpxq

(6.10)

for x ě 0, and such that C :“ sup xą0

are called self-dominated. In this case we write f, g P Ks-d . Restricting now the class of admissible Kato functions to the self-dominated functions we obtain the main result. It states that there is a t0 ą 0 such that the Trotter-Kato product formulae converge in Cφ pHq locally uniformly away from t0 ą 0 if and only if for some positive s0 ď t0 and some integer p ě 1 we have F ps0 qp P Cφ pHq.

(6.11)

We recall that the trace-class C1 pHq is a particular symmetrically-normed ideal in the ring of compact operators C8 pHq. So this result yields also the proofs of assertions announced in subsection 5.4.4. Note that in addition we also get the following extension of results from Sections 5.2 and 5.3. For self-dominated functions Ks-d the Trotter-Kato product formulae converge uniformly away from zero in the operator-norm topology if for some t0 ą 0 the condition F pt0 q P C8 pHq is satisfied. In the next Section 6.2 we recall some basic facts about symmetric norming functions φ and the corresponding symmetrically-normed ideals Cφ . Section 6.3 contains certain indispensable information about convergence in the ideals Cφ pHq. Section 6.4 concerns the lifting approach and related results about the TrotterKato product formulae convergence in Cφ pHq. In Section 6.5 we indicate certain compactness conditions, which imply the } ¨ }φ -convergence of the Trotter-Kato product formulae. Section 6.6 is devoted to the lifting approach. We show that under suitable compactness assumptions the operator-norm error bounds can be lifted to error bounds in symmetrically-normed ideals. In particular, the operator-norm rate of convergence obtained in Sections 5.4 for the operator smallness conditions lifts to the same rate of convergence in symmetrically-normed ideals Cφ pHq. Section 6.7 is devoted to notes and references.

6.2

Symmetrically-normed ideals

8 Let c0 Ă l8 be the subspace of bounded sequences ξ “ tξj u8 of real j“1 P l numbers, that converge to zero. We denote by cf the subspace of c0 consisting of all sequences with a finite number of non-zero terms (finite sequences).

6.2. Symmetrically-normed ideals

213

Definition 6.3. A real-valued function φ : ξ ÞÑ φpξq defined on cf is called a norming function if it has the following properties: φpξq ą 0,

(6.12)

@ξ P cf , ξ ‰ 0,

φpαξq “ |α|φpξq,

@ξ P cf , @α P R,

(6.13)

@ξ, η P cf ,

(6.14)

φpξ ` ηq ď φpξq ` φpηq,

(6.15)

φp1, 0, . . .q “ 1.

A norming function φ is said to be symmetric if it has the additional property that φpξ1 , ξ2 , . . . , ξn , 0, 0, . . .q “ φp|ξj1 |, |ξj2 |, . . . , |ξjn |, 0, 0, . . .q (6.16) for any ξ P cf and any permutation j1 , j2 , . . . , jn of integers 1, 2, . . . , n. It turns out that for any symmetric norming function φ and for any elements ξ, η from the positive cone c` of non-negative and non-increasing sequences such that ξ, η P cf obey ξ1 ě ξ2 ě . . . ě 0, η1 ě η2 ě . . . ě 0, and n ÿ

ξj ď

j“1

n ÿ

ηj ,

n “ 1, 2, . . . ,

(6.17)

j“1

one has the Ky Fan inequality (6.18)

φpξq ď φpηq .

Therefore, inequality (6.18) together with the properties (6.12), (6.13) and (6.15) yield inequalities ξ1 ď φpξq ď

8 ÿ

ξj ,

` ξ P c` f :“ cf X c .

(6.19)

j“1

Note that the left- and right-hand sides of (6.19) are the simplest examples of symmetric norming functions on the domain c` f . We denote them by φ8 pξq :“ ξ1

and φ1 pξq :“

8 ÿ

ξj .

(6.20)

j“1

Then by Definition 6.3 the estimates (6.19) and (6.20) yield φ8 pξq :“ max |ξj |, jě1

φ1 pξq :“

φ8 pξq ď φpξq ď φ1 pξq,

8 ÿ

|ξj |, j“1

for all ξ P cf .

(6.21)

214

Chapter 6. Product formulae in symmetrically-normed ideals Next we denote by ξ ˚ :“ tξ1˚ , ξ2˚ , . . . u a decreasing rearrangement: ξ1˚ “ sup |ξj |, ξ1˚ ` ξ2˚ “ supt|ξi | ` |ξj |u, . . . , jě1

i‰j

of the sequence of absolute values t|ξn |uně1 , that is, ξ1˚ ě ξ2˚ ě . . . . Then ξ P cf implies ξ ˚ P cf and also φpξq “ φpξ ˚ q, ξ P cf , (6.22) see (6.16). Therefore, any symmetric norming function φ is uniquely determined by its values on the positive cone c` . Now, let ξ “ tξ1 , ξ2 , . . .u P c0 . We define ξ pnq :“ tξ1 , ξ2 , . . . , ξn , 0, 0, . . . u P cf .

(6.23)

If φ is a symmetric norming function, we set cφ :“ tξ P c0 : sup φpξ pnq q ă `8u,

(6.24)

ně1

which is a natural domain of φ. Therefore, one gets cf Ď cφ Ď c0 .

(6.25)

Note that by (6.16)-(6.18) and (6.24) one gets φpξ pnq q ď φpξ pn`1q q ď sup φpξ pnq q , for any ξ P cφ .

(6.26)

ně1

Therefore, by (6.26), the limit φpξq :“ lim φpξ pnq q, nÑ8

ξ P cφ ,

(6.27)

on the natural domain cφ exists and φpξq “ supně1 φpξ pnq q. This means that the symmetric norming function φ is a normal functional on the linear (over real numbers) set cφ Ă l8 . By virtue of (6.14) and (6.21), every symmetric norming function is continuous on cf : |φpξq ´ φpηq| ď φpξ ´ ηq ď φ1 pξ ´ ηq,

@ξ, η P cf .

(6.28)

Suppose that X is a compact operator on a Hilbert space H, i.e. X P C8 pHq. Then we denote by spXq :“ ts1 pXq, s2 pXq, . . .u , (6.29) the sequence of singular values of X, counting multiplicities. We always assume that s1 pXq ě s2 pXq ě . . . ě sn pXq ě . . . . (6.30) To define symmetrically-normed ideals of the ring of compact operators C8 pHq we introduce the notion of the symmetric norm.


215

Definition 6.4. Let I be a two-sided ideal of C8 pHq. A functional } ¨ }sym : I Ñ R` 0 is called a symmetric norm if besides the usual properties of the operator norm } ¨ }: }X}sym ą 0,

@X P I, X ‰ 0,

}αX}sym “ |α|}X}sym ,

(6.31)

@X P I, @α P C,

}X ` Y }sym ď }X}sym ` }Y }sym ,

@X, Y P I,

(6.32) (6.33)

it verifies also the following additional properties: X P I, A, B P LpHq,

(6.34)

for any rank-one operator X P I.

(6.35)

}AXB}sym ď }A}}X}sym }B}, }αX}sym “ |α|}X} “ |α| s1 pXq, If condition (6.34) is replaced by

}U X}sym “ }XU }sym “ }X}sym ,

X P I,

for any unitary operator U on H,

(6.36)

then, one arrives to the definition of the invariant symmetric norm } ¨ }inv . First, we note that the ordinary operator norm } ¨ } on any ideal I Ď C8 pHq is evidently a symmetric norm as well as an invariant norm. Second, we observe that every symmetric norm is automatically invariant. Indeed, for any unitary operators U and V one gets by (6.34) that }U XV }sym ď }X}sym ,

X P I.

(6.37)

Since X “ U ´1 U XV V ´1 , we also get }X}sym ď }U XV }sym , which together with (6.37) yields (6.36). Third, we claim that }X}sym “ }X ˚ }sym . For the proof, let X “ U |X| be the polar decomposition of the operator X P I. Since U ˚ X “ |X|, by (6.36) we obtain }X}sym “ }|X|}sym . The same argument applied to the adjoint operator X ˚ “ |X|U ˚ yields }X ˚ }sym “ }|X|}sym , which proves the claim. Now we can apply the concept of symmetric norming function to describe the symmetrically-normed ideals of the unital algebra of bounded operators LpHq, or in general, the symmetrically-normed ideals generated by symmetric norming functions. Recall that any proper two-sided ideal IpHq of the ring LpHq is contained in C8 pHq and contains the set KpHq of finite-rank operators: KpHq Ď IpHq Ď C8 pHq.

(6.38)

To clarify the relationships between symmetric norming functions and symmetrically-normed ideals we mention that on the cone c` there is an obvious one-to-one correspondence between functions φ (Definition 6.3) and the symmetric norms } ¨ }sym on IpHq. To carry on with a general setting we need the following relation.

216


Definition 6.5. Let cφ be the set of vectors (6.24) generated by a symmetric norming function φ. We associate with cφ the following subset of compact operators: Cφ pHq :“ tX P C8 pHq : spXq P cφ u,

(6.39)

which is a proper two-sided ideal of the algebra LpHq of all bounded operators on H. It is called a symmetrically-normed ideal generated by the symmetric norming function φ. Since the limit (6.27) yields φpξq “ supně1 φpξ pnq q, we define on the symmetrically-normed ideal Cφ pHq the functional X P Cφ pHq.

}X}φ :“ φpspXqq,

(6.40)

If one equips on the ideal IpHq “ Cφ pHq (Definition 6.4) the symmetric norm } ¨ }sym :“ } ¨ }φ , then this symmetrically-normed ideal becomes a Banach space. Hence, in accordance with (6.38) and (6.39) we obtain by (6.21) that KpHq Ď C1 pHq Ď Cφ pHq Ď C8 pHq.

(6.41)

Here the trace-class operators C1 pHq :“ Cφ1 pHq, where the symmetric norming function φ1 is defined in (6.20), (6.21), and consequently }X}φ ď }X}1 ,

X P C1 pHq.

(6.42)

Remark 6.6. By virtue of the Ky Fan inequality (6.18) and the definition of the symmetric norm (6.40), the so-called dominance property holds: if X P Cφ pHq, Y P C8 pHq and n n ÿ ÿ (6.43) sj pY q ď sj pXq, n “ 1, 2, . . . , j“1

j“1

then Y P Cφ pHq and }Y }φ ď }X}φ . Remark 6.7. To distinguish in (6.41) nontrivial ideals Cφ we need some criteria based on the properties of φ or of the norm } ¨ }φ . For example, any symmetric norming function (6.22) defined by φprq pξq :“

r ÿ

ξj˚ ,

ξ P cf ,

(6.44)

j“1

generates for fixed r P N a symmetrically-normed ideal, which is trivial in the sense that Cφprq pHq “ Cφ8 pHq “ C8 pHq. A criterion for the operator A to belong to a nontrivial ideal Cφ is M “ sup }Pm APm }φ ă 8,

(6.45)

mě1

where tPm umě1 is a monotonically increasing sequence of finite-dimensional orthogonal projectors on H that converges strongly to the identity operator. Note that for A P C8 condition (6.45) is trivial.


217

We consider now a couple of examples to explain the concept of the symmetrically-normed ideals Cφ pHq generated by symmetric norming functions φ and the rˆ ole of the trace functional, or simply trace Trp¨q, on these ideals. Example 6.8. The von Neumann-Schatten ideals Cp pHq (Chapter 2). These ideals of C8 pHq are generated by the symmetric norming functions φp pξq, where φp pξq :“

8 ´ÿ

|ξj |p

¯1{p ,

ξ P cf ,

(6.46)

j“1

coincides with the standard lp -norm for 1 ď p ă 8, or with the standard l8 -norm, φ8 pξq :“ sup |ξj |,

ξ P cf ,

(6.47)

1ďj

for p “ 8. Indeed, if we set tξj˚ :“ sj pXqujě1 for X P C8 pHq, then the symmetric norm }X}sym “ φp pspXqq (6.40) coincides with (2.37): 8 ´ÿ

}X}p “

sj pXqp

¯1{p .

(6.48)

j“1

The corresponding symmetrically-normed ideal Cφp pHq is identical to the von Neumann-Schatten class Cp pHq, see Definition 2.50. By Definition 2.41, for X P Cp pHq one can consider the functional |X| ÞÑ ř Tr |X| “ jě1 sj pXq ą 0. Then the trace-norm }X}1 :“ Tr |X| ě }X}, is finite on the trace-class operators C1 pHq and it is infinite for X P Cp ą 1 pHqzC1 pHq. We say that for p ą 1 the von Neumann-Schatten ideals admit no trace, whereas for p “ 1 the map: X ÞÑ Tr X, is continuous in the } ¨ }1 -topology. Note that since the trace is linear, the trace-norm: C1,` pHq Q X ÞÑ }X}1 is linear on the positive cone C1,` pHq of the trace-class operators. 0 Example 6.9. The symmetrically-normed ideals CΠ pHq and CΠ pHq. Let Π “ ` 8 tπj uj“1 P c be a non-increasing sequence of positive numbers with π1 “ 1. We associate with Π the function # + n ÿ 1 φΠ pξq “ sup řn ξj , ξ P cf . (6.49) ně1 j“1 πj j“1

Definition 6.3 shows that φΠ is a symmetric norming function, that is, φΠ pξq “ φΠ pξ ˚ q, where ξ ˚ is decreasing rearrangement of ξ P cf (6.22). Then the corresponding to (6.24) set cφΠ is defined by # + n ÿ 1 ˚ cφΠ :“ ξ P cf : sup řn ξj ă `8 . (6.50) ně1 j“1 πj j“1

218


Hence, the two-sided symmetrically-normed ideal CΠ pHq :“ CφΠ pHq of C8 pHq generated by symmetric norming function (6.49) consists of all compact operators X for which n ÿ 1 }X}φΠ :“ sup řn sj pXq ă `8. (6.51) ně1 j“1 πj j“1 By Definition 6.4 and (6.40), this formula defines on the ideal CΠ pHq of compact operators a symmetric norm }X}sym “ }X}φΠ . If the non-increasing sequence Π “ tπj ě 0u8 j“1 with π1 “ 1 satisfies 8 ÿ

πj “ `8 and

j“1

lim πj “ 0,

jÑ8

(6.52)

then the ideal CΠ pHq is nontrivial : CΠ pHq ‰ C8 pHq and CΠ pHq ‰ C1 pHq, see Remark 6.7. Moreover, by Lemma 3.5, (3.16), CΠ pHq contains as a proper subset 0 the (separable) symmetrically-normed ideal CΠ pHq given by 0 CΠ pHq :“ tX P CΠ pHq : lim řn nÑ8

n ÿ

1

j“1

πj

sj pXq “ 0u.

(6.53)

j“1

Thus, 0 C1 pHq Ă CΠ pHq Ă CΠ pHq Ă C8 pHq.

(6.54)

If in addition to (6.52) the sequence Π “ tπj u8 j“1 is regular, that is, n ÿ

πj “ Opnπn q,

n Ñ 8,

(6.55)

sn pXq “ Opπn q,

n Ñ 8,

(6.56)

j“1

then X P CΠ pHq if and only if

cf. condition (6.45). On the other hand, the asymptotics sn pXq “ opπn q,

n Ñ 8,

(6.57)

0 implies that X P CΠ pHq Ă CΠ pHq .

Remark 6.10. A natural choice of sequence tπj u8 j“1 that satisfies (6.52) is πj “ j ´α , 0 ă α ď 1. Note that if 0 ă α ă 1, then the sequence Π “ tπj u8 j“1 satisfies also (6.55). Consequently, the two-sided symmetrically-normed ideal CΠ pHq generated by the symmetric norming function (6.49) consists of the compact operators X whose singular values obey (6.56): sn pXq “ Opn´α q,

0 ă α ă 1, n Ñ 8.

(6.58)


219

Definition 6.11. Let α :“ 1{p , p ą 1. The symmetrically-normed ideal corresponding to (6.58), that is, Cp,8 pHq :“ tX P C8 pHq : sn pXq “ Opn´1{p q, p ą 1u,

(6.59)

is known as the weak -Cp ideal. By definition (6.51), the symmetric norm on the ideal Cp,8 pHq is equivalent for the regular case p ą 1 to }X}p,8 :“ sup ně1

1 n1´1{p

n ÿ

sj pXq.

(6.60)

j“1

Note that if p1 ą p, then definition (6.60) yields C1 pHq Ă Cp,8 pHq Ă Cp1 ,8 pHq Ă C8 pHq.

(6.61)

On the other hand, (6.48), (6.59) and (6.60), show that C1 pHq Ă Cp,8 pHq Ă Cq pHq,

for 1 ă p ă q .

(6.62)

Remark 6.12. The weak -Cp ideal defined for p “ 1 by C1,8 pHq :“ tX P C8 pHq :

n ÿ

sj pXq “ Oplnpnqq, n Ñ 8u,

(6.63)

j“1

is of a special interest. Note that since Π “ tj ´1 u8 j“1 does not satisfy (6.55), the characterisation sn pXq “ Opn´1 q is not true, see (6.56) and (6.58). In this case the equivalent symmetric norm can be defined on (6.63) as n ÿ 1 sj pXq. ně1 1 ` lnpnq j“1

}X}1,8 :“ sup

(6.64)

By (6.64), one readily gets that C1 pHq Q X ÞÑ }X}1 ě }X}1,8 . Therefore, similarly to (6.62), we obtain C1 pHq Ă C1,8 pHq. Example 6.13. With a non-increasing sequence of positive numbers Π “ tπj u8 j“1 , π1 “ 1, and with the decreasing rearrangement ξ ˚ of ξ P cf one can associate the symmetric norming function φπ given by φπ pξq :“

8 ÿ

πj ξj˚ ,

ξ P cf .

(6.65)

j“1

We denote the corresponding symmetrically-normed ideal by Cπ pHq :“ Cφπ pHq. If the sequence Π satisfies (6.52), then the ideal Cπ pHq is different from C8 pHq and from C1 pHq. If, in particular, πj “ j ´α , j “ 1, 2, . . . , for 0 ă α ď 1, then the

220


corresponding ideal is denoted by C8,p pHq, p “ 1{α. The norm on this ideal is given by 8 ÿ }X}8,p :“ (6.66) j ´1{p sj pXq, p ě 1. j“1

The symmetrically-normed ideal C8,1 pHq is called the Macaev ideal. It turns out that the dual of the Macaev ideal is C8,1 pHq˚ “ C1,8 pHq. We conclude this section by remarks concerning the problem of existence of a trace Trω p¨q on symmetrically normed ideals Cp,8 pHq, p ě 1. It should be a linear, positive, and unitarily invariant functional X ÞÑ Trω pXq, for any X P Cp,8 pHq. On the other hand, X does not admit the standard trace functional X ÞÑ TrpXq, if X P Cp,8 pHqzC1 pHq. In contrast to the linearity of the trace-norm }¨}1 on the positive cone C1,` pHq (Example 6.8), the map X ÞÑ }X}p,8 on the positive cone Cp,8,` pHq is not linear. Although this map is homogeneous: α X ÞÑ α}X}p,8 , α ě 0, it turns out that for X, Y P Cp,8,` pHq in general }X ` Y }p,8 ‰ }X}p,8 ` }Y }p,8 . However, on the space l8 there exists an appropriate state ω P Spl8 q such that for p “ 1 the mapping X ÞÑ Trω pXq :“ ωptp1 ` lnpnqq´1

n ÿ

sj pXqu8 n“1 q,

(6.67)

j“1

is linear and enjoys for X P C1,8 pHq, the properties of the trace. Note that ω is a linear positive normalised functional (state) on the Banach space l8 . Recall that the set of states Spl8 q belongs to pl8 q˚ , where pl8 q˚ is the dual of the Banach space l8 . For a particular choice of the state ω this gives the Dixmier trace Trω p¨q on the ideal C1,8 pHq, which in turn is known as the Dixmier ideal. Since the Dixmier trace is zero on the set of finite-rank operators KpHq (and hence on C1 pHq), it is a singular trace. We construct the trace Trω p¨q in Chapter 7. The Dixmier trace (6.67) is continuous in the topology defined by the norm (6.64). This property is basic for our discussion in Chapter 7 of the Trotter-Kato product formulae in the } ¨ }p,8 -topology, for p ě 1.

6.3

Convergence in Cφ pHq-ideals

Let X ě 0 be a compact operator. Let λpXq “ tλj pXqu8 j“1 be the sequence of its eigenvalues, counting multiplicities. In this case they coincide with the singular values λpXq “ spXq and we always assume that we leave the decreasing arrangement: (6.68) λ1 pXq ě λ2 pXq ě . . . ě 0. Definition 6.14. If X ě 0 and Y ě 0 are two compact operators, then we say that they are log-ordered, and write, Y ă log X,

(6.69)

6.3. Convergence in Cφ pHq-ideals if

n ź

λj pY q ă

j“1

221

n ź

λj pXq,

n “ 1, 2, . . . .

(6.70)

j“1

Lemma 6.15. Let X and Y be non-negative self-adjoint operators and let φ be a symmetric norming function. If X P Cφ pHq and Y ălog X, then Y P Cφ pHq and }Y }φ ă }X}φ .

(6.71)

Proof. By (6.69) and (6.70) we get that n ÿ

lnpλj pY qq ă

j“1

n ÿ

n “ 1, 2, . . . .

lnpλj pXqq,

(6.72)

j“1

Now let x` :“ maxpx, 0q for any x P R and denote µj p¨q :“ lnpλj p¨qq. Then one gets the representation ż µj p¨q du eu pµj p¨q ´ uq` . e (6.73) “ R

Note that n ÿ

pµj p¨q ´ uq`

j“1

#ř “

k j“1 pµj p¨q

´ uq,

0,

k “ the largest integer such that µk p¨q ě u , if µ1 p¨q ă u ,

which together with (6.72) implies n ÿ

pµj pY q ´ uq` “

j“1

k ÿ

µj pY q ´ k u

(6.74)

j“1 k ÿ

ď

µj pXq ´ k u ď

n ÿ

pµj pXq ´ uq` .

j“1

j“1

Therefore, (6.73) and (6.74) yield n ÿ

elnpλj pY qq ă

j“1

whence

n ÿ

elnpλj pXqq ,

n “ 1, 2, . . . ,

j“1 n ÿ

j“1

λj pY q ă

n ÿ

λj pXq,

n “ 1, 2, . . . .

(6.75)

j“1

Then by (6.75) and by the dominance property of Cφ pHq (Remark 6.6), we find that Y P Cφ pHq and }Y }φ ă }X}φ . l

222

Chapter 6. Product formulae in symmetrically-normed ideals Below we shall apply Lemma 6.15 in a slightly modified form.

Lemma 6.16. Let X, X0 , Y and Y0 be non-negative self-adjoint operators and let φ be a symmetric norming function. If for some r ě 1 one has X r ď X0 and 1{2 1{2 Y r ď Y0 and if Y0 X0 Y0 P Cφ pHq, then ˘r ` 1{2 1{2 1{2 (6.76) Y XY 1{2 ă log Y0 X0 Y0 , ` 1{2 ˘r Y XY 1{2 P Cφ pHq and ` ˘r 1{2 1{2 (6.77) } Y 1{2 XY 1{2 }φ ď }Y0 X0 Y0 }φ . Proof. By the Araki inequality (Section B.1), we have ` 1{2 ˘r Y XY 1{2 ălog Y r{2 X r Y r{2 , r ą 1.

(6.78)

Since X r ď X0 , we get Y r{2 X r Y r{2 ď Y r{2 X0 Y r{2 .

(6.79)

Moreover, since Y r ď Y0 , there exists a contraction Γ0 : ran Y0 Ñ H such that 1{2

Y r{2 “ Γ0 Y0

,

(6.80)

which leads to 1{2

1{2

Y r{2 X r Y r{2 ď Γ0 Y0 X0 Y0 Γ˚0 .

(6.81)

This in turn implies that ` ˘ ` ˘ ` 1{2 1{2 1{2 1{2 ˘ λj Y r{2 X r Y r{2 ď λj Γ0 Y0 X0 Y0 Γ˚0 ď λj Y0 X0 Y0 ,

(6.82)

for j “ 1, 2, . . . , and we finally obtain ` 1{2 ˘r 1{2 1{2 Y XY r{2 ălog Y r{2 X r Y r{2 ă log Y0 X0 Y0 .

(6.83)

Now applying Lemma 6.15 we complete the proof.

l

The next lemma provides a sufficient condition that allows to lift the strong operator convergence to convergence in the } ¨ }φ -topology on the symmetricallynormed ideal Cφ pHq. Lemma 6.17. Let X “ X ˚ P Cφ pHq, Y “ Y ˚ P Cp pHq and Z “ Z ˚ P LpHq. If tZptqutě0 , Zptq “ Zptq˚ is a family of bounded self-adjoint operators such that s-lim Zptq “ Z, tÑ`0

(6.84)

then lim sup }pZpt{rq ´ ZqY X}φ “ lim sup }XY pZpt{rq ´ Zq}φ “ 0,

rÑ8 tPr0,τ s

for any τ P p0, 8q.

rÑ8 tPr0,τ s

(6.85)

6.4. Lifting for Trotter-Kato product formulae

223

Proof. Note that by (6.84) one has s-limrÑ8 Zpt{rq “ Z uniformly in t P r0, τ s. Since Y P Cp pHq, this implies lim sup }pZpt{rq ´ ZqY } “ 0.

rÑ8 tPr0,τ s

(6.86)

By virtue of (6.34) one gets the estimate }pZpt{rq ´ ZqY X}φ ď }pZpt{rq ´ ZqY }}X}φ , which in conjunction with (6.86) yields (6.85).

l

Although straightforward, this line of reasoning for lifting to a stronger topology of convergence will be useful in the following.

6.4

Lifting for Trotter-Kato product formulae

This section contains some basic results about the lifting approach motivated by Lemma 6.17. Essentially they are statements about the Cφ pHq-norm convergence of the Trotter-Kato product formulae conditioned by the operator-norm convergence. Note that sufficient conditions for this lifting of the topology of convergence require that the admissible Kato functions K satisfy domination condition, see Definition 6.1. We start by showing that it is enough to establish the convergence of the Trotter-Kato product formula for only one representative of the families of operators tF ptqutě0 , tT ptqutě0 , tf ptAqgptBqutě0 , or tgptBqf ptAqutě0 . . Recall that the generator H “ A ` B is the form-sum of A and B, and P0 is the orthogonal projection P0 : H Ñ H0 “ dom H. Proposition 6.18. Let A and B be non-negative self-adjoint operators in a Hilbert space H and let f, g be Kato functions of class K. If the Trotter-Kato product formula for approximants corresponding to one representative of the families tF ptqutě0 , tT ptqutě0 , tf ptAqgptBqutě0 , or tgptBqf ptAqutě0 , converges locally uniformly away from t0 ą 0 in Cφ pHq, then the Trotter-Kato product formulae converge locally uniformly away from t0 in Cφ pHq for all families of approximants generated by f and g. Proof. We prove the chain of implications of convergence tF ptqutě0 tT ptqutě0 ùñ tf ptAqgptBqutě0 ùñ tgptBqf ptAqutě0 ùñ tF ptqutě0 . tF ptqutě0 ùñ tT ptqutě0 . To this aim we use the representation T pt{rqr “ T pt{rqp f ptA{rq1{2 gptB{rq1{2 F pt{rqn´1 gptB{rq1{2 f ptA{rq1{2 .

ùñ

(6.87)

for t ě 0 and r ě 1, where r :“ n ` p, n “ 1, 2 . . . , 0 ď p ă 1. Since F pt{rqn´1 “ F ptn {pn ´ 1qqn´1 ,

(6.88)

224


where tn :“ tpn ´ 1q{pn ` pq ď τ01 , we obtain from (6.87) that T pt{rqr P Cφ pHq if tn P rτ01 , τ s Ă pt0 , 8q. Since tn P rτ01 , τ s Ă pt0 , 8q is equivalent to t P rp1 ` p`1 p`1 1 1 n´1 qτ0 , p1 ` n´1 qτ s, we find that for rτ0 , τ s Ď pτ0 , τ s and for sufficiently large r p`1 p`1 1 one has rτ0 , τ s Ď rp1 ` n´1 qτ0 , p1 ` n´1 qτ s and T pt{rqr P Cφ pHq for t P rτ0 , τ s. To prove that the Trotter-Kato product formula for tT ptqutě0 converges locally uniformly away from t0 in Cφ pHq we note that this kind of convergence for tF ptqutě0 implies sup }F pt{pn ´ 1qqn´1 ´ e´tH P0 }φ “ 0.

lim

(6.89)

nÑ8 tPrτ 1 ,τ s 0

Replacing t by tn we get lim

sup

nÑ8 t Prτ 1 ,τ s n 0

}F ptn {pn ´ 1qqn´1 ´ e´tH P0 }φ “ 0,

(6.90)

and since tn P rτ01 , τ s for t P rτ0 , τ s, we obtain lim

sup }F pt{rqn´1 ´ e´tH P0 }φ “ 0.

(6.91)

nÑ8 tPrτ ,τ s 0

Taking into account the representation T pt{rqr ´ e´tH P0 p

(6.92) 1{2

“ T pt{rq f ptA{rq ˆ tF pt{rq

n´1

é

1{2

gptB{rq

P0 ugptB{rq1{2 f ptA{rq1{2

´tH

` T pt{rqp f ptA{rq1{2 gptB{rq1{2 e´tH P0 tgptB{rq1{2 ´ 1uf ptA{rq1{2 ` T pt{rqp f ptA{rq1{2 gptB{rq1{2 e´tH P0 tf ptA{rq1{2 ´ 1u ` T pt{rqp f ptA{rq1{2 tgptB{rq1{2 ´ 1ue´tH P0 ` T pt{rqp tf ptA{rq1{2 ´ 1ue´tH P0 ` T pt{rqp tf ptA{rq1{2 ´ 1ue´tH P0 ` tT pt{rqp ´ 1ue´tH P0 , we obtain the estimate }T pt{rqr ´ e´tH P0 }φ ď }F pt{rq

n´1

é

´tH

(6.93) P0 }φ ` 2}t1 ´ gptB{rq

1{2

` 2}t1 ´ f ptA{rq

1{2

ue

´tH

´tH

ue

p

P0 }φ ` }t1 ´ T pt{rq ue

P0 }φ

´tH

P0 }φ .

Since s-limtÑ`0 f ptAq1{2 “ 1, s-limtÑ`0 gptBq1{2 “ 1 and s-limtÑ`0 T ptq “ 1, Lemma 6.17 shows that lim

sup }t1 ´ gptB{rq1{2 ue´tH P0 }φ

rÑ8 tPrτ ,τ s 0

“ lim

sup }t1 ´ f ptA{rq1{2 ue´tH P0 }φ “ 0 ,

rÑ8 tPrτ ,τ s 0

(6.94)


225

and lim

sup }t1 ´ T pt{rqp ue´tH P0 }φ

rÑ`8 tPrτ ,τ s 0

ď lim

(6.95)

sup }t1 ´ T pt{rque´tH P0 }φ “ 0 .

rÑ`8 tPrτ ,τ s 0

Combining (6.91), (6.93), (6.94) and (6.95), one gets the proof of the locally uniform convergence away from t0 of the Trotter-Kato product formula for tT ptqutě0 . tT ptqutě0 ùñ tf ptAqgptBqutě0 . Using the representation pf ptA{nqgptB{nqqn “ f ptA{nq1{2 T pt{nqn´1 f ptA{nq1{2 gptB{nq,

(6.96)

and arguing as in the proof that tF ptqutě0 ùñ tT ptqutě0 , one verifies that the Trotter-Kato product formula converges locally uniformly away from t0 ą 0 in Cφ pHq. tf ptAqgptBqutě0 ùñ tgptBqf ptAqutě0 . The proof is based on the representation pgptB{nqf ptA{nqqn “ gptB{nqpf ptA{nqgptB{nqqn´1 f ptA{nq,

(6.97)

and the same argument. tgptBqf ptAqutě0 ùñ tF ptqutě0 . The assertion is a consequence of the representation F pt{rqr “ F pt{rqp gptA{rq1{2 f ptA{rqpgptB{rqf ptA{rqqn´1 gptB{rq1{2 , where r “ n ` p, n “ 1, 2, . . . , 0 ď p ă 1, arguing again as above.

(6.98) l

Proposition 6.18 shows that it is enough to consider and to prove convergence of the Trotter-Kato product formulae in the } ¨ }φ -norm only for one family, for example, for the family tF ptqutě0 . Proposition 6.19. Let A and B be non-negative self-adjoint operators in a Hilbert space H and let f, g be Kato functions in K. If (a) the Trotter-Kato product formula for tF ptqutě0 and for given f, g converges locally uniformly away from zero in the operator-norm; (b) the family of the approximants tF pt{rqur is bounded locally uniformly away from some t0 ą 0 in Cφ pHq; (c) the semigroup e´tH P Cφ pH0 q, for t ą t0 , then the Trotter-Kato product formulae converge locally uniformly away from t0 in Cφ pHq for all families generated by the Kato functions f and g.

226


Proof. If t P rτ0 , τ s Ă pt0 , 8q, then there is an α P p0, 1q such that τ01 :“ ατ0 ą t0 and αt P rτ01 , τ s Ă pt0 , 8q. Since F pt{rqαr “ F pαt{αrqαr ,

t ě 0,

(6.99)

there is a r0 ě 1 such that F pt{r0 qαr0 P Cφ pHq and }F pt{rqαr }φ ď M prτ01 , τ sq,

(6.100)

for t P rτ0 , τ s and r ě r0 , see (b). Since e´tH P Cφ pHq for t ą t0 (c), the operator e´tH is compact for t ą 0, which yields that the spectrum of H is discrete with the only accumulation point at infinity. Denoting by EH p¨q the spectral measure of H, we consider the decomposition e´tH “ e´tH EH pr0, N qq ` e´tH EH prN, 8qq,

(6.101)

N “ 1, 2, . . . , t ě 0 . Since the spectrum of H is discrete, the projection EH pr0, N qq is a finitedimensional operator for each N “ 1, 2, . . . . Hence for any N “ 1, 2, . . . one gets lim sup }EH pr0, N qqP0 pF pt{rqαr ´ e´αtH q}φ “ 0. (6.102) rÑ8 tPr0,τ s

Since HEH prN, 8qq ě N EH prN, 8qq, we find that }e´p1´αqtH EH prN, 8qq} ď e´p1´αqτ0 N ,

N “ 1, 2, . . . ,

(6.103)

for t P rτ0 , τ s. Taking into account the estimate }e´p1´αqtH EH pr0, N qqpF pt{rqαr ´ e´αtH P0 q}φ ď }e

´p1´αqtH

αr

´αtH

EH prN, 8qq} }F pt{rq }φ ` }e

(6.104) (

P0 }φ ,

as well as (6.100) and (6.103) we finally obtain }e´p1´αqtH EH prN, 8qqpF pt{rqαr ´ e´αtH P0 q}φ ( ď e´p1´αqτ0 N M prτ01 , τ sq ` }e´ατ0 H }φ ,

(6.105)

for N “ 1, 2, . . . , t P rτ0 , τ s Ď pτ01 , τ s and r ě r0 . Using the representation ¯ ´ F pt{rqr ´ e´tH P0 “ F pt{rqp1´αqr ´ e´p1´αqtH P0 F pt{rqαr (6.106) ˘ ` ` e´p1´αqtH EH pr0, N qqP0 F pt{rqαr ´ e´αtH P0 ` ˘ ` e´p1´αqtH EH prN, 8qqP0 F pt{rqαr ´ e´αtH P0 ,


227

we argue as in the proof of Lemma 6.17. From (6.34), (6.100), and (6.105) we obtain the estimate sup }F pt{rqr ´ e´tH P0 }φ

(6.107)

tPrτ0 ,τ s

ď M prτ01 , τ sq sup }F pt{rqp1´αqr ´ e´p1´αqtH P0 } tPrτ0 ,τ s

` sup }EH pr0, N qqP0 pF pt{rqαr ´ e´αtH P0 q}φ tPrτ0 ,τ s

( ` e´p1´αqτ0 N M prτ01 , τ sq ` }e´ατ0 H }φ , for N “ 1, 2, . . . , t P rτ0 , τ s and r ě r0 . For any ą 0 we find an integer N “ 1, 2, . . . such that ( (6.108) e´p1´αqτ0 N M prτ01 , τ sq ` }e´ατ0 H }φ ď . Then we fix this integer N . Since for given f, g the Trotter-Kato product formula converges locally uniformly away from zero in the operator norm (a), we obtain for sufficiently large r that M prτ01 , τ sq sup }F pt{rqp1´αqr ´ e´p1´αqtH P0 } ď ,

(6.109)

tPrτ0 ,τ s

and sup }EH pr0, N qqP0 pF pt{rqαr ´ e´αtH P0 q}φ ď .

(6.110)

tPrτ0 ,τ s

Hence, for any ą 0 we get sup }F pt{rqr ´ e´tH P0 }φ ď 3 ,

(6.111)

tPrτ0 ,τ s

for the integer N , which is fixed above, and sufficiently large r. Therefore, for tF ptqutě0 the Trotter-Kato product formula converges locally uniformly away from l t0 ą 0 in Cφ pHq. Applying Proposition 6.18 we complete the proof. To verify that the family of approximants tF pt{rqr urě1 is locally uniformly bounded away from t0 ą 0 in Cφ pHq we need the following lemma. Lemma 6.20. Let A and B be non-negative self-adjoint operators in a Hilbert space ` D H and let f D : R` : R0` Ñ R` 0 be bounded Borel measurable 0 Ñ R0 and g D p functions such that F pt0 q P Cφ pHq for t0 ą 0 and some integer p ě 1. If the Kato functions f and g are dominated by, respectively, f D and g D , then F pt{rqr P Cφ pHq,

(6.112)

}F pt{rqr }φ ď }F D pt0 qp }φ ,

(6.113)

and for pt0 ď t ď prt0 and r ě p.

228


Proof. Since f and g are dominated by f D and g D , the inequality (6.7) yields f ptA{rqr ď f ptA{rqrt0 {t ď f D pt0 Aq,

(6.114)

gptB{rqr ď gptB{rqrt0 {t ď g D pt0 Bq,

(6.115)

and for t0 ď t ď rt0 and r ě 1. Setting X0 :“ f D pt0 Aq and Y0 :“ g D pt0 Bq we obtain from Lemma 6.16 that F pt{rqr ă log F D pt0 q, (6.116) for t0 ď t ď rt0 and r ě 1. Thus, F pt{rqrp ă log F D pt0 qp ,

(6.117)

which yields F pt{rqrp P Cφ pHq and by Lemma 6.15 }F pt{rqrp }φ ď }F pt0 qp }φ ,

(6.118)

for t0 ď t ď rt0 and r ě 1. Since F pt{rqrp “ F ppt{rpqrp , we get (6.112) and l (6.113) for pt0 ď t ď prt0 and r ě p. From Lemma 6.20 we immediately see that for any interval rτ0 , τ s Ď ppt0 , `8q there exist an r0 ě 1 such that for r ě r0 and t P rτ0 , τ s the conditions F pt{rqr P Cφ pHq and (6.6) are satisfied. This means that the sequence tF pt{rqr urě1 is locally uniformly bounded away from pt0 in Cφ pHq. Moreover, one gets that M prτ0 , τ sq ď }F D pt0 qp }φ for sufficiently large r ě 1 and any interval rτ0 , τ s Ď ppt0 , `8q. Next, we show that under the assumptions of Lemma 6.20 one has e´tH P Cφ pH0 q for t ą pt0 . To this end we note that if the Kato function f is dominated by f D , then, by (6.7), (6.119) f px{rqr ď f D pxq, r ě 1, for x ě 0. Since limrÑ8 f px{rqr “ e´x for each x ě 0, we find that e´x ď f D pxq,

(6.120)

for x ě 0. In other words, if f D dominates the Kato function f , then it also dominates the exponential Kato function fˆpxq “ e´x , x ě 0, cf. Appendix C. Lemma 6.21. Let A and B be non-negative self-adjoint operators in a Hilbert space H, and let f D : R0` Ñ R0` and g D : R0` Ñ R0` be bounded Borel measurable functions such that F D pt0 qp P Cφ pHq for some t0 ą 0 and some integer p ě 1. If the Kato functions f and g are dominated by f D and g D , respectively, then e´tH P Cφ pH0 q,

(6.121)

}e´tH P0 }φ ď }F D pt0 qp }φ ,

(6.122)

and for t ě pt0 .


229

Proof. Since fˆpxq “ gˆpxq “ e´x , x ě 0, are dominated by f D and g D , respectively, we find by Lemma 6.20 that Fˆ pt{rqr P Cφ pHq

(6.123)

and }Fˆ pt{rqr }φ ď }F D pt0 qp }φ ,

r ě 1,

(6.124)

t ě 0.

(6.125)

for pt0 ď t ď prt0 and r ě p, where Fˆ ptq :“ e´tB{2 e´tA e´tB{2 ,

Using the Araki inequality (Appendix B) and formula (3.20) one obtains that e´tH P0 ă log Fˆ pt{nqn ,

n “ 1, 2, . . . ,

(6.126)

for t ą 0, where the left-hand side is the operator-norm limit of Proposition 6.19(a). Then by (6.123), (6.124), and Lemma 6.15 we complete the proof of (6.121) and (6.122). l Now, by virtue of Propositions 6.18, 6.19 and of Lemmata 6.20, 6.21 we obtain the following result: Proposition 6.22. Let A and B be non-negative self-adjoint operators in a Hilbert ` ` ` D space H and let f D : R` 0 Ñ R0 and g : R0 Ñ R0 be bounded Borel measurable D p functions such that F pt0 q P Cφ pHq for some t0 ą 0 and for some integer p ě 1. Suppose (a) the Kato functions f, g P K are dominated by f D ,g D , respectively; (b) the Trotter-Kato product formulae converge locally uniformly away from zero in the operator norm for the family tF ptqutě0 . Then the Trotter-Kato product formulae converge locally uniformly away from pt0 in Cφ pHq for all families of approximants generated by Kato functions f and g. Proof. By Lemma 6.21, we obtain that e´tH P0 P Cφ pHq for t ą pt0 . Lemma 6.20 implies that for each bounded interval rτ0 , τ s Ď ppt0 , 8q there is an r0 ě 1 such that the approximants F pt{rqr P Cφ pHq for r ě r0 and t P rτ0 , τ s. By (6.113) one has the upper bound suprěr0 suptPrτ0 ,τ s }F pt{rqr }φ ď }F D pt0 q}φ ă 8, which ensures that the family tF pt{rqr urě1 is locally uniformly bounded away from pt0 in Cφ pHq. Then applying Proposition 6.19 we obtain that the Trotter-Kato product formulae for tF ptqutě0 converge locally uniformly away from pt0 in Cφ pHq. Finally, using Proposition 6.18 we conclude that this holds for all families generated by f l and g, which are dominated by f D and g D .

230


6.5 Product formulae in Cφ pHq-ideals In this section we study conditions that ensure the Cφ pHq-norm convergence of the Trotter-Kato product formulae without reference to the operator-norm convergence. As we established in Sections 5.3, 5.4, and 6.4 the properties of the Kato functions are crucial for controlling the convergence. Thus, one has first to specify a set of admissible Kato functions f, g from the generic class K. In the next statement we impose on the pair f, g the regularity condition on f , keeping g P K arbitrary, see Appendix C, Section C.3. Proposition 6.23. Let A and B be non-negative self-adjoint operators in a Hilbert ` D space H and let f D : R` 0 Ñ R0 be a Borel measurable function such that f pt0 Aq P Cφ pHq for some t0 ą 0. If the Kato function f is regular and dominated by f D , and g P K is an arbitrary Kato function, then the Trotter-Kato product formulae converge locally uniformly away from t0 in Cφ pHq for all families of approximants generated by f, g. Proof. Since for any dominating function f D one has e´x ď f D pxq, x ě 0 (6.120), the condition f D pt0 Aq P Cφ pHq yields e´t0 A P Cφ pHq, t0 ą 0, which evidently implies p1 ` t0 Aq´1 P C8 pHq. By Proposition 5.45 (or Proposition 5.36), this last observation together with the regularity of f ensure that the Trotter-Kato product formulae converge locally uniformly away from zero in the operator-norm topology. Note that any Kato function g is evidently dominated by g D pxq “ 1. Then condition f D pt0 Aq P Cφ pHq yields F D pt0 qp P Cφ pHq for any p ě 1 and t0 ą 0. Consequently, by the lifting Proposition 6.22, the Trotter-Kato product formulae converge locally uniformly in Cφ pHq away from some t0 ą 0 for all families of the approximants generated by the Kato functions f, g from the statement of the proposition. l To continue with the next assertion we first give a useful characterisation of the self-dominated Kato functions, see Definition 6.2 and Section C.3. Lemma 6.24. A Borel function f P K is a self-dominated Kato function if and ` only if there is a non-increasing function h : R` 0 Ñ R0 , satisfying lim hpxq “ 1,

xÑ`0

(6.127)

such that f admits the representation f pxq “ e´xhpxq ,

x ě 0.

(6.128)

The function h is unique. Proof. Let f be a self-dominated Kato function and let t :“ qx, x ě 0 for 0 ă q ď 1, which yields 0 ď t ď x. Then by (6.9) we obtain that f ptq ď f pxqt{x ,

0 ď t ď x, x ą 0,

(6.129)

6.5. Product formulae in Cφ pHq-ideals

231

whence f ptq1{t ď f pxq1{x ,

0 ă t ď x.

(6.130)

If f pxq “ 0 for some x ą 0, then one gets f ptq “ 0 for 0 ă t ď x, which is impossible by (5.12) in Definition 5.4. Therefore, f pxq ‰ 0 and one can define 1 hpxq :“ ´ lnpf pxqq ě 0, x

x ą 0.

(6.131)

Taking into account (6.130) we get hptq ě hpxq, for 0 ă t ď x. Thus h is a non-negative and non-increasing function. Note that f pxq ‰ 1 in a small vicinity of x “ 0. Otherwise, there would exist a sequence txn ą 0u8 n“1 such that limnÑ8 xn “ 0 and f pxn q “ 1. This is impossible because of (5.12), since then one would get f 1 p`0q “ lim

nÑ8

f pxn q ´ 1 “ 0, xn

instead of f 1 p`0q “ ´1. Therefore, the representation hpxq “ ´

˘ 1 ´ f pxq ` ln f pxq1{p1´f pxqq , x

(6.132)

makes sense for sufficiently small x ą 0. Since lim y 1{p1ýq “ e´1 ,

yÑ1´0

by (6.132) and f 1 p`0q “ ´1 one gets (6.127). Now the representation (6.128) follows from the definition (6.131). Conversely, let f be given by (6.128), where h is a non-increasing function which satisfies (6.127). Since h is non-increasing, we get for x ě 0 and 0 ă q ď 1 that f pqxq “ e´qxhpqxq ď e´qxhpxq “ f pxqq , (6.133) which verifies (6.9). Furthermore, by (6.127) one gets f p0q “ 1 and 8 ÿ f pxq ´ 1 1 “ p´1qn xn´1 hpxqn , x n! n“1

which leads to the estimate ˇ ˇ 8 ÿ ˇ ˇ 1 ´ f pxq 1 ˇď ˇ ` hpxq xn´1 , ˇ ˇ x n! n“2

x ą 0,

x ą 0.

(6.134)

(6.135)

Then by (6.127) we immediately obtain f 1 p`0q “ ´1. Thus (6.128) defines a self-dominated Kato function. The uniqueness of h is obvious by construction. l

232


Corollary 6.25. If f is a self-dominated Kato function, then f0 pxq ď

1 , 1 ´ lnpf pxqq

(6.136)

for x ě 0, where f0 is defined by (5.102) and (5.104). Proof. By definition (5.102) and representation (6.128) one has ˆ ˙ ˘ 1 1 1 ` thptq ϕ0 pxq “ inf ´ 1 “ inf e ´1 , 0ătďx t 0ătďx t f ptq

(6.137)

that yields f0 pxq “

1 1 1 ď “ , 1 ` xϕ0 pxq 1 ` xhpxq 1 ´ lnpf pxqq

for x ě 0.

(6.138) l

Lemma 6.26. Let A be a non-negative self-adjoint operator in a Hilbert space H and let f be a self-dominated Kato function. If f pt0 Aq P Cφ pHq for some t0 ą 0, then f0 pt0 Aq P C8 pHq, (6.139) where f0 is defined by (5.102) and (5.104). Proof. Since f pt0 Aq P Cφ pHq implies f pt0 Aq P C8 pHq, where 0 ď f pt0 Aq ď 1, we see that the positive operator K :“

1 P C8 pHq, 1 ` lnp1{f pt0 Aqq

(6.140)

is also compact. By Corollary 6.25, the positive bounded operator f0 pt0 Aq is dominated by the compact operator (6.140): f0 pt0 Aq ď p1 ´ lnpf pt0 Aqqq´1 “ K. Hence, f0 pt0 Aq is in turn a compact operator (see Section 2.4), which yields (6.139). l Besides the representation established in Lemma 6.24, the following property of the Kato functions is indispensable. Lemma 6.27. Let the function f P K obey the inequality f pxq ď dpxq for a Borel function dpxq and x ě 0. If d is a self-dominated Kato function, then in turn the function f : (1) is dominated by d; (2) satisfies the condition C0 :“ sup xą0

xf pxq ă `8. 1 ´ f pxq

(6.141)

6.5. Product formulae in Cφ pHq-ideals

233

Proof. (1) Since f pxq ď dpxq, by (6.9) for the self-dominated Kato function d one gets f pqxq1{q ď dpqxq1{q ď dpxq, 0 ă q ď 1, x ě 0. (6.142) Hence, the function f is dominated by d in the sense of Definition 6.1. (2) Since f pxq ď dpxq and the self-dominated Kato function d satisfies (6.10) for some constant C0,d , we obtain the estimates: C0 “ sup xą0

x f pxq x dpxq ď sup “: C0,d ă 8. 1 ´ f pxq xą0 1 ´ dpxq

This proves (6.141) and the lemma.

(6.143) l

Note that similarly to Proposition 6.23, in the next assertion we consider the case of an arbitrary g P K. However, instead of regularity of f , we study Kato functions that are bounded from above by some self-dominated functions. Although these conditions sound very different from regularity, they lead to the same conclusion as Proposition 6.23. Proposition 6.28. Let A and B be non-negative self-adjoint operators in a Hilbert space H and let d be a self-dominated Kato function such that dpt0 Aq P Cφ pHq for some t0 ą 0. Then for any Kato function f obeying f pxq ď dpxq, x ě 0 and for arbitrary g P K the Trotter-Kato product formulae converge in Cφ pHq locally uniformly away from t0 for all approximants generated by f and g. Proof. By Lemma 6.27, the inequality f pxq ď dpxq implies that f is dominated by the self-dominated Kato function d: f pqxq1{q ď dpxq “ f D pxq, 0 ă q ď 1. On the other hand, by f pxq ď dpxq and by Definition 5.34, see (5.102), one gets ˆ ˙ ˆ ˙ 1 1 1 1 ´ 1 ě inf ´ 1 “ ϕ0,d pxq, ϕ0 pxq “ inf 0ătďx t 0ătďx t f ptq dptq which in turn by definition (5.104) implies that f0 pxq “

1 1 ď “ f0,d pxq. 1 ` xϕ0 pxq 1 ` xϕ0,d pxq

(6.144)

By Lemma 6.26, for a self-dominated Kato function d P Cφ pHq we obtain compactness: f0,d pt0 Aq P C8 pHq and then by (6.144), the inequalities 0 ď f0 pt0 Aq ď f0,d pt0 Aq P C8 pHq. Therefore, the self-adjoint operator f0 pt0 Aq is also compact. Since f0 pt0 Aq P C8 pHq, Proposition 5.36 ensures that the Trotter-Kato n product formula for the approximantes tpf ptA{nqgptB{nqq uně1 converges in the operator-norm topology locally uniformly away from zero (t0 “ 0) to the degenerate semigroup te´tH P0 utą0 .

234


Recall that any Kato function g is dominated by g D pxq “ 1, Appendix C. Then condition dpt0 Aq P Cφ pHq yields that for F D ptq “ g D ptBq1{2 f D ptAqg D ptBq1{2 “ dptAq and any p ě 1, t0 ą 0, one has F D pt0 qp P Cφ pHq. Therefore, by the lifting Proposition 6.22, the Trotter-Kato product formulae converge in the topology of Cφ pHq locally uniformly away from some t0 ą 0 for all families of the product approximants generated by the Kato functions f, g l verifying the conditions of the proposition. Until now no additional conditions were imposed on the Kato function g. But under certain conditions, there are new formulations implying the convergence of the Trotter-Kato product formulae in the ideal Cφ pHq. Proposition 6.29. Let A and B be non-negative self-adjoint operators in a Hilbert ` ` ` D space H and let f D : R` 0 Ñ R0 and g : R0 Ñ R0 be Borel measurable functions D such that F pt0 q P Cφ pHq for some t0 ą 0. If the Kato functions f and g are dominated by f D and g D , respectively, and if they satisfy conditions (6.10): C0 :“ sup xą0

xf pxq ă `8 1 ´ f pxq

and

S0 :“ sup xą0

xgpxq ă `8, 1 ´ gpxq

then the Trotter-Kato product formulae converge in Cφ pHq locally uniformly away from t0 , for all approximants generated by f and g. Proof. First we check that under hypothesis of the proposition, p1 ` Aq´1 p1 ` Bq´1 P C8 pHq.

(6.145)

Arguing as in the proof of Lemma 6.21 we find (in accordance with (6.123)) that e´t0 B{2 e´t0 A e´t0 B{2 P C8 pHq.

(6.146)

Setting C :“ e´t0 A{2 e´t0 B{2 we get C ˚ C P C8 pHq, which yields e´t0 A{2 e´t0 B{2 P C8 pHq and e´t0 B{2 e´t0 A{2 e´t0 B{2 P C8 pHq. Iterating, we obtain

m

e´t0 A{2 e´t0 B{2 P C8 pHq,

(6.147)

for m “ 1, 2, . . . . This evidently gives e´tA e´t0 B{2 P C8 pHq,

(6.148)

for any t ą 0. Similarly, we prove that k

e´tA e´t0 B{2 P C8 pHq,

(6.149)

for t ą 0 and k “ 0, 1, 2, . . . . Therefore, e´tA e´sB P C8 pHq,

(6.150)

6.5. Product formulae in Cφ pHq-ideals for t ą 0 and s ą 0. Taking into account the representation ż8ż8 p1 ` Aq´1 p1 ` Bq´1 “ dt ds e´pt`sq e´tA e´sB , 0

235

(6.151)

0

we obtain from (6.150) and (6.151) that (6.145) holds. By Proposition 5.46, the conditions (6.10) and the property (6.145) are indispensable for the proof that the Trotter-Kato product formulae converge to the degenerate semigroup te´tH P0 utě0 in the operator-norm topology, locally uniformly away from zero. Moreover, by Proposition 5.46 we also obtain that p10 ` Hq´1 P C8 pH0 q. Then Proposition 6.22 completes the proof of the assertion. l Instead of (6.10) and the compactness (6.145) of the product of resolvent, in the next statement the main condition on admissible functions of class K that ensures the convergence of the Trotter-Kato product formulae is formulated in terms of self-dominantness. Proposition 6.30. Let A and B be non-negative self-adjoint operators in a Hilbert ` D : R0` Ñ R` space H and let f D : R` 0 Ñ R0 and g 0 be self-dominated Kato D functions such that condition F pt0 q P Cφ pHq is satisfied for some t0 ą 0. If the Kato functions f and g obey f pxq ď f D pxq and gpxq ď g D pxq for x ě 0, then the Trotter-Kato product formulae converge locally uniformly away from t0 in Cφ pHq for all families of approximants generated by f and g. Proof. This statement is essentially a corollary of Proposition 6.29. First, we note that the upper bounds f pxq ď f D pxq and gpxq ď g D pxq for x ě 0, imply that the functions f and g are dominated by f D and g D , respectively, see Lemma 6.27, (6.142). Second, since f D and g D are self-dominated Kato functions, Lemma 6.27, (6.141), proves the conditions (6.10). l Corollary 6.31. Let A and B be non-negative self-adjoint operators in a Hilbert space H such that p1 ` Bq´1{2 p1 ` Aq´1 p1 ` Bq´1{2 P Cφ pHq.

(6.152)

If f and g are Kato functions such that f pxq ď 1{p1 ` xq and gpxq ď 1{p1 ` xq for x ě 0, then the Trotter-Kato product formulae converge locally uniformly away from zero (sic! ) in Cφ pHq for all families of approximants generated by f and g. Proof. First of all we note that condition (6.152) implies p1 ` t0 Bq´1{2 p1 ` t0 Aq´1 p1 ` t0 Bq´1{2 P Cφ pHq,

(6.153)

for any t0 ą 0. Further, we set f D pxq :“ p1 ` xq´1 and g D pxq :“ p1 ` xq´1 for x ě 0. The Kato functions f D and g D are obviously self-dominated, i.e., they obey conditions (6.9) and (6.10). Therefore, by Lemma 6.27, the Kato functions f and

236


g satisfy the condition (6.10) too. Now, applying Proposition 6.30 one finishes the proof. l Corollary 6.32. Let A and B be non-negative self-adjoint operators in a Hilbert space H such that e´t0 B{2 e´t0 A e´t0 B{2 P Cφ pHq,

t0 ą 0.

(6.154)

If f and g are Kato functions such that f pxq ď e´x and gpxq ď e´x for x ě 0, then the Trotter-Kato product formula converges locally uniformly away from t0 in Cφ pHq for all families generated by f and g. Proof. We set f D pxq “ g D pxq “ e´x , x ě 0. The exponential Kato functions f D and g D are obviously self-dominated, that is, satisfy conditions (6.9) and (6.10), and by Lemma 6.27 the Kato functions f and g satisfy the condition (6.10) too. l Then Proposition 6.30 completes the proof. Note that Proposition 6.30 allows us to state, in some sense, a converse. Proposition 6.33. Let A and B be non-negative self-adjoint operators in a Hilbert space H and let f and g be self-dominated Kato functions. Then the Trotter-Kato product formulae converge in Cφ pHq locally uniformly away from some t0 ą 0 for all families of the product approximants generated by f and g if and only if there exist s0 ą 0 and an integer p ě 1 such that F ps0 qp P Cφ pHq. Proof. If the Trotter-Kato product formulae converge locally uniformly away from t0 in Cφ pHq for all families of approximants generated by f and g, then they converge in particular for the family tF ptqutě0 . Hence, there exists t ą t0 such that for sufficiently large r one has F pt{rqr P Cφ pHq. Setting p “ r and s0 :“ t{r ą 0, we find that F ps0 qp P Cφ pHq. Thus, the necessity is proven. To prove the converse, we note that if F ps0 qp P Cφ pHq for some integer p ě 1, then F ps0 q P C8 pHq, cf. (6.146). Now, following the proof of Proposition 6.29 we obtain (6.145). By Proposition 5.46, the self-dominants of the Kato functions f , g and the property (6.145) ensure that the Trotter-Kato product formulae converge in the operator-norm topology locally uniformly away from zero. Finally, setting t0 “ ps0 and taking into account Proposition 6.22, one completes the proof of the l sufficiency. Proposition 6.33 allows to establish the following result, important for the Gibbs semigroups theory. Proposition 6.34. Let A and B be non-negative self-adjoint operators in a Hilbert space H and let f and g be self-dominated Kato functions. Then there is a t0 ě 0 such that the Trotter-Kato product formulae converge locally uniformly away from t0 in C1 pHq for all families of the product approximants generated by f, g if and only if there exists an s0 ą 0 and an integer p such that F ps0 q P Cp pHq.

6.6. Product formulae: error bound estimates

237

Proof. From Proposition 6.33 one gets that the condition F ps0 qp P C1 pHq has to be satisfied for some s0 ą 0 and some integer p ě 1. However, this is equivalent l to F ps0 q P Cp pHq. In particular, let f pxq “ gpxq “ 1{p1 ` xq or f pxq “ gpxq “ e´x , x ě 0. Then Proposition 6.34 shows that the Trotter-Kato product formulae converge in the trace-class norm locally uniformly away from some t0 for all families of approximants generated by f, g if and only if for some s0 ą 0 one has p1 ` s0 Bq´1{2 p1 ` s0 Aq´1 p1 ` s0 Bq´1{2 P Cp pHq,

(6.155)

e´s0 B{2 e´s0 A e´s0 B{2 P Cp pHq,

(6.156)

or for some p ě 1, respectively. In the case f pxq “ gpxq “ 1{p1 ` xq one gets in addition that the convergence in Proposition 6.34 holds away from zero, see Corollary 6.31. It is related to the fact that in this case (6.155) implies that either te´tA utě0 or te´tB utě0 is an immediately Gibbs semigroup, see Definition 4.1. Moreover, the notion of self-dominated Kato functions allows one to generalise our results in Proposition 5.36 and Proposition 5.46 about the operator-norm convergence of the Trotter-Kato product formulae locally uniformly away from zero, cf. Section 5.3. Proposition 6.35. Let A and B be non-negative self-adjoint operators in a Hilbert space H and let f and g be self-dominated Kato functions. If there is a t0 ą 0 such that F pt0 q P C8 pHq, then the Trotter-Kato product formulae converge locally uniformly away from zero in the operator-norm for all families of the product approximants generated by f, g. Proof. Following the proof of Proposition 6.29 one obtains (6.145). Then Proposition 5.46 shows that the Trotter-Kato product formulae converge in the operatorl norm locally uniformly away from zero, as needed.

6.6

Product formulae: error bound estimates

` or on Let tη : R` _ N Ñ R` 0 u be a non-negative real function defined on R N “ t1, 2, . . .u, such that lim ηpxq “ 0. (6.157) xÑ8

Definition 6.36. The function η is called the operator-norm error bound of the Trotter-Kato product formulae away from t0 ą 0 for the family tF ptqutě0 , if for any interval ra, bs Ď pt0 , `8q there exist an x0 ě 1, such that }F pt{xqx ´ e´tH P0 } ď ηpxq,

(6.158)

238


for t P ra, bs and x ě x0 , where x “ r _ n, is either the continuous parameter n P N, n ě n0 ě 1. r ě r0 ě 1, or the discrete parameter . Recall that the generator H “ A ` B is a form-sum of A and B, and P0 is the orthogonal projection P0 : H Ñ H0 “ dompHq. We note that in contrast to the real number x0 in (6.158), the error bound η does not depend on the interval ra, bs, that is, η is locally uniform away from t0 . Similarly, the notion of an operator-norm error bound can be extended to the families tT ptqutě0 , tf ptAqgptBqutě0 and tgptBqf ptAqutě0 , yet in the last two cases only the discrete parameter n P N makes sense for the approximants. Definition 6.37. The function ηφ is called a locally uniform error bound in Cφ pHq away from t0 ą 0 for the Trotter-Kato product formulae for the family tF ptqutě0 if for any interval rτ0 , τ s Ď pt0 , `8q there exist an xφ ě 1, such that F pt{xqx ´ e´tH P0 P Cφ pHq,

(6.159)

}F pt{xqx ´ e´tH P0 }φ ď ηφ pxq,

(6.160)

and for t P rτ0 , τ s and x ě xφ , where x “ r _ n. As above, this notion has an extension to approximants generated by the families tT ptqutě0 , tf ptAqgptBqutě0 and tgptBqf ptAqutě0 . The following lifting problem (cf. Section 5.4, Lemma 5.51) arises naturally in this setup: Assume that we know the operator-norm locally uniform error bound η (6.158) away from t0 ą 0. Can we find a locally uniform error bound ηφ away from t0 in a symmetrically-normed ideal Cφ pHq ? For simplicity, below we omit systematically the term locally uniform as an evidence. Next we prove the main lifting statement for the Trotter-Kato product formulae with locally uniform error bounds in the symmetrically-normed ideals Cφ pHq. Recall the trace-class ideal C1 pHq “ Cφ1 pHq, see Definition 6.5 and (6.20),(6.21) Proposition 6.38. Let A and B be non-negative self-adjoint operators in a Hilbert ` ` ` D space H, and let f D : R` 0 Ñ R0 and g : R0 Ñ R0 be Borel measurable functions D such that F pt0 q P Cφ pHq for some t0 ą 0. Assume that the Kato functions f and g are dominated by f D and g D , respectively. Then: If ηprq, r ě r0 “ 1 (continuous case), is an operator-norm error bound away from t0 ą 0 for the Trotter-Kato product formula for the selfadjoint family tF ptqutě0 (respectively for tT ptqutě0 ), then the function ηφ prq “ 2 }F D pt0 q}φ ηpr{2q, r ě 2, is an error bound in Cφ pHq away from 2t0 for the Trotter-Kato product formula for the family tF ptqutě0 (respectively, for the family tT ptqutě0 ). If ηpnq, n ě n0 (discrete case), is an operator-norm error bound away from t0 ą 0 for the Trotter-Kato product formula for tf ptAqgptBqutě0 , respectively for


239

tgptBqf ptAqutě0 , tF ptqutě0 or tT ptqutě0 , then the function ηφ pnq “ }F D pt0 q}φ tηprn{2sq ` ηprpn ` 1q{2squ,

n ě n0 ,

is error bound in Cφ pHq away 2t0 for the Trotter-Kato product formula for the family tf ptAqgptBqutě0 , respectively, for tgptBqf ptAqutě0 , tF ptqutě0 or tT ptqutě0 . Here rαs denotes the integer part of α ě 0. Proof. Continuous case. We use the representation (6.161) F pt{rqr ´ e´tH P0 ˘ ` ˘ ` ´tH{2 r{2 ´tH{2 ´tH{2 r{2 r{2 P0 F pt{rq ´ e “ F pt{rq ´ e P0 F pt{rq ` e P0 . Then Lemma 6.20 yields the estimate }F pt{rqr{2 }φ ď }F D pt0 q}φ ,

(6.162)

for t0 ď t{2 ď rt{2 and r{2 ě 1, while Lemma 6.21 yields }e´tH{2 P0 }φ ď }F D pt0 q}φ ,

(6.163)

for t ě 2t0 . Hence, (6.162) and (6.163) yield for (6.161) the estimate }F pt{rqr ´ e´tH P0 }φ ď 2}F D pt0 q}φ }F pt{rqr{2 ´ e´tH{2 P0 },

(6.164)

for 2t0 ď t ď rt0 and r ě 2. Since η is an operator-norm error bound (6.158) away from t0 , the estimate }F pt{rqr{2 ´ e´tH{2 P0 } ď ηpr{2q,

(6.165)

is valid for any interval rτ0 , τ s Ď p2t0 , 8q and r ě 2r0 . By condition, r0 “ 1. Then inserting (6.165) in (6.164) we find that ηφ prq :“ 2 }F D pt0 q}φ ηpr{2q, r P r2, `8q is the error bound in Cφ pHq away from 2t0 for approximants tF pt{rqr utą0,rě2 . In order to verify the statement for the family tT ptqutě0 , we have to show only that F D pt0 q P Cφ pHq implies T D pt0 q P Cφ pHq, where T D ptq :“ f D ptAq1{2 g D ptBqf D ptAq1{2 ,

t ě 0,

(6.166)

by a direct estimate. Let GD ptq :“ f D ptAq1{2 g D ptBq1{2 ,

t ě 0.

(6.167)

Using the polar decomposition GD ptq “ V ptq|GD ptq| “ V ptqF D ptq1{2 ,

t ě 0,

(6.168)

where V ptq is a partial isometry acting from ran GD ptq˚ onto ran GD ptq, we find that T D ptq “ GD ptqGD ptq˚ “ V ptqF D ptqV ptq˚ P Cφ pHq, (6.169)

240


for t ě 0. This yields }T D pt0 q}φ “ }F D pt0 q}φ . Discrete case. To prove the assertion for the family tf ptAqgptBqutě0 we use the decompositions n “ k ` m, k P N :“ t1, 2, . . . u, m “ 2, 3, . . . , n ě 3, and pf ptA{nqgptB{nqqn ´ e´tH P0 ` ˘ “ pf ptA{nqgptB{nqqk ´ e´ktH{n P0 pf ptA{nqgptB{nqqm ˘ ` ` e´ktH{n P0 pf ptA{nqgptB{nqqm ´ e´mtH{n P0 .

(6.170)

Since pf ptA{nqgptB{nqqm “ f ptA{nqgptB{nq1{2 F pt{nqm´1 gptBq1{2 ,

(6.171)

Lemma 6.20 shows that pf ptA{nqgptB{nqqm P Cφ pHq and }pf ptA{nqgptB{nqqm }φ ď }F D pt0 q}φ ,

(6.172)

for t0 ď pm ´ 1qt{n ď pm ´ 1qt0 and m ´ 1 ě 1. Note that Lemma 6.21 implies e´ktH{n P Cφ pHq and }e´ktH{n P0 }φ ď }F D pt0 q}φ , (6.173) for kt{n ě t0 . Hence, (6.170) yields the estimate }pf ptA{nqgptB{nqqn ´ e´tH P0 }φ D

(6.174) k

´ktH{n

ď }F pt0 q}φ }pf ptA{nqgptB{nqq ´ e

P0 }

` }F D pt0 q}φ }pf ptA{nqgptB{nqqm ´ e´mtH{k P0 }, for p1 ` pk ` 1q{pm ´ 1qqt0 ď t ď nt0 , m ě 2 and t ě p1 ` m{kqt0 . Since η is an operator-norm error bound away from t0 , by definition (6.158) for any interval ra, bs Ď pt0 , `8q there exist an n0 ě 1, such that }pf ptA{nqgptB{nqqk ´ e´ktH{n P0 } ď Cηpkq,

(6.175)

for kt{n P ra, bs ô t P rp1 ` m{kqa, p1 ` m{kqbs, and }pf ptA{nqgptB{nqqm ´ e´mtH{n P0 } ď ηpmq,

(6.176)

for mt{n P ra, bs ô t P rp1 ` k{mqa, p1 ` k{mqbs. Setting m :“ rpn ` 1q{2s and k “ rn{2s, n ě 3, we satisfy the conditions n “ k ` m and m ě 2 as well as limnÑ8 pk ` 1q{pm ´ 1q “ 1, limnÑ8 m{k “ 1 and limnÑ8 k{m “ 1. Hence, for any interval rτ0 , τ s Ď p2t0 , `8q we find that rτ0 , τ s Ď rp1`pk ` 1q{pm ´ 1qqt0 , nt0 s for sufficiently large n. Moreover, choosing rτ0 {2, τ {2s Ď pa, bq Ď pt0 , `8q we ensure that rτ0 , τ s Ď rp1`m{kqa, p1`m{kqbs and rτ0 , τ s Ď rp1`k{mqa, p1`k{mqbs for sufficiently large n too. Thus for any interval rτ0 , τ s Ď p2t0 , `8q there exist


241

an n0 ě 1, such that (6.174), (6.175) and (6.176) hold for t P rτ0 , τ s and n ě n0 . Therefore, from (6.174) we obtain the estimate }pf ptA{nqgptB{nqqn ´ e´tH P0 }φ

(6.177)

D

ď }F pt0 q}φ tηprn{2sq ` ηprpn ` 1q{2squ, for t P rτ0 , τ s Ď p2t0 , `8q and n ě n0 . Hence, ηφ pnq :“ }F D pt0 q}φ tηprn{2sq ` ηprpn ` 1q{2squ, is the error bound in Cφ pHq for the approximants tpf ptA{nqgptB{nqqn utą0,ně1 , away from 2t0 and for n ě n0 . Similarly one proves the result for the families tgptBqf ptAqutě0 , tF ptqutě0 and tT ptqutě0 . l Corollary 6.39. Note that if the rate of convergence in the operator-norm estimate η (6.158) is optimal, then so are the rates of convergence in the }¨}φ -norm estimates: ηφ prq and ηφ pnq. We next consider the application of Proposition 6.38 for the Kato functions of class Kα , see Definition 5.26 and Appendix C. They are suitable for pairs of C0 -semigroup generators A and B related by the fractional smallness conditions, subsection 5.2.5. Combining Proposition 6.38 with operator-norm estimates in Proposition 5.27, we obtain the following result. Proposition 6.40. Let the self-adjoint operators A ě 1 and B ě 0 in a Hilbert space H be such that for some α P p1{2, 1q and b P p0, 1q, dom Aα Ă dom B α and 9 }B α u} ď b}Aα u}, u P dom Aα , H :“ A`B. Let f D : R0` Ñ R0` and g D : R0` Ñ R0` be Kato functions such that F D pt0 q “ g D pt0 Bq1{2 f D pt0 Aqg D pt0 Bq1{2 P Cφ pHq for some t0 ą 0. Assume that the Kato functions f and g are dominated by f D and g D , respectively. (i) If for α P p1{2, 1q the functions f, g P Kα and b1{α C1{2α S1 ă 1,

(6.178)

then for some Γ1 ą 0 the inequality ηφ pnq ď Γ1 lnpnq{n2α´1 ,

n ě 2,

(6.179)

gives estimate for the error bound in Cφ pHq for the Trotter-Kato product formula for the family tf ptAqgptBqutě0 away from 2t0 . (ii) If in addition, for α P p1{2, 1q and (6.178), dom H α Ď dom Aα , then for some Γ2 ą 0 the inequality ηφ pnq ď Γ2 {n2α´1 ,

n ě 2,

(6.180)

242

Chapter 6. Product formulae in symmetrically-normed ideals gives estimate for the error bound in Cφ pHq for the Trotter-Kato product formula for the family tf ptAqgptBqutě0 away from 2t0 .

Proof. (i) Taking into account Proposition 5.27, with condition (i), and (5.94) we find that ηpnq “ c1 lnpnq{n2α´1 , n ě 2, (6.181) is an operator-norm error bound of the Trotter-Kato product formula for tf ptAqgptBqutě0 away from t0 ą 0. Applying now Proposition 6.38 we obtain that ηφ pnq :“ }F D pt0 q}φ tηprn{2sq ` ηprpn ` 1q{2squ, n P N, is the error bound in Cφ pHq for the family tf ptAqgptBqutě0 away from 2t0 . By a straightforward computation one finds a constant Γ1 :“ 22α c1 }F D pt0 q}φ such that ηφ pnq ď Γ1 lnpnq{n2α´1 , for n ě 2, as in (6.179). (ii) Similarly, if in addition to (i), condition dompH α q Ď dompAα q are satisfied, then by Proposition 5.27 ηpnq “ c{n2α´1 ,

n ě 2,

(6.182)

is an operator-norm error bound for the Trotter-Kato product formula for tf ptAqgptBqutě0 away from t0 ą 0. Therefore, applying Proposition 6.38 and proceeding as before, we prove the second estimate (6.180) of the proposition for Γ2 :“ 22α c }F D pt0 q}φ . Similarly, one proves the result for the families tgptBqf ptAqutě0 , tF ptqutě0 l and tT ptqutě0 . Note that the rates of convergence lnpnq{n2α´1 and 1{n2α´1 , in Proposition 6.40 coincide with the rates of convergence in the operator norm, cf. Corollary 6.39. Now the following remarks are in order (see also notes in Section 5.6 and in Section 6.7): Firstly, we remind that originally the operator-norm estimates have been proven in Proposition 5.8 under the assumption A ě 1 and B ě 1. A later revision of this proof shows that B ě 0 is sufficient. Therefore, we use this observation in Proposition 6.40. Secondly, we note that the fractional-power condition dom H α Ď dom Aα for some α P p1{2, 1q is satisfied if dom Aβ Ď dom B α for some β P p0, αq, in particular, if dom A1{2 Ď dom B α is valid.

6.7

Notes

Notes to Section 6.1. Here we follow the Introduction from [NZ99d]. The notions of dominated and self-dominated Kato functions (Definitions 6.1 and 6.2, or Appendix C) are also introduced in [NZ99d]. Notes to Section 6.2. Material concerning symmetrically-normed ideals can be found in [GK69] (Chapter III) and [Sim05] (Chapter 1). For the Ky Fan inequality, see [Fan51], or [GK69] (Chapter II).

6.7. Notes

243

Remarks 6.10, 6.12 and Example 6.13 serve as a preparation for the definition of the Dixmier trace, [Dix66]. We present the necessary details in Section 7.2. Notes to Section 6.3. The material presented in this section is partially motivated by results and observations from the paper [Ara90] and from the book [Sim05], Chapter 1. Notes to Section 6.4. The results of this section are essentially due to [NZ99d]. It turns out that assertions concerning the Gibbs semigroups (Chapter 5), [NZ90a], [NZ90b] (see also [Hia95]), and in general concerning the von Neumann-Schatten ideals, admit some extension to arbitrary symmetrically-normed ideals. For convergence of the Trotter-Kato product formulae in the topology of symmetrically-normed ideals away from t0 ą 0 it is enough that the self-adjoint operator F ptq (the transfer matrix ) belongs to one of these ideals for t “ t0 , provided the Kato functions f and g behave well. In particular, this is valid for the von Neumann-Schatten ideals and for dominated or self-dominated Kato functions. These observations improve the results in [NZ90a], [NZ90b] or [Hia95], where stronger conditions were assumed. Namely, either an auxiliary operator f0 ptAq for t ą 0 or, respectively, e´t0 A for some t0 ą 0, should belong to the von NeumannSchatten ideal. Of course, these conditions imply that the operator F ptq belongs to that ideal away from t0 ą 0. Notes to Section 6.5. Proposition 6.34 is a generalisation of a result in the paper [NZ90a]. Proposition 6.35 is a generalisation of the main result [NZ99b]. Concerning the class of admissible Kato functions f and g, the present results are more restrictive than in [NZ90a], [NZ90b] since it is assumed that the Kato functions are regular, dominated or self-dominated and admit a certain decaying behaviour at infinity, see Appendix C for details. Only Propositions 6.23 and 6.28 allow arbitrary Kato functions g. As a by-product, we also obtain the norm-convergence of the Trotter-Kato product formulae for the case when the operator F ptq is compact away from t0 ą 0. This completes the results of Section 5.3, see also H. Neidhardt and V. A. Zagrebnov [NZ99b]. Notes to Section 6.6. Here we follow Section 5 of [NZ99d]. The general idea is the same as the lifting approach proposed in Chapter 5 (Section 5.4) that we considered above. The results of this section easily yield the trace-norm convergence of the Trotter-Kato product formulae. For example, taking into account the operatornorm error bounds found in [DIT98], [IT98b] and applying Proposition 6.38, one immediately gets corresponding error bounds in the trace norm, as in Proposition 6.40.

Chapter 7

Product formulae in the Dixmier ideal In this chapter we discuss product formulae in the weak-C1 ideal C1,8 pHq that we introduced in Section 6.2. Since this ideal is a natural domain for the Dixmier trace, it is also called the Dixmier ideal.

7.1

Ideals and singular traces

The Dixmier trace and ideal arose from the question of whether the algebra LpHq of all bounded linear operators on a Hilbert space H admits a unique nontrivial trace. This problem was solved in the negative. On LpHq there exist singular traces that vanish on the finite-rank operators, and consequently, on the ideal of trace-class operators. Proposition 7.1. The space C1,8 pHq endowed with the norm } ¨ }1,8 (6.64) is a Banach space. The proof is quite standard, although tedious and long. For relevant references, see Section 7.4 (Notes to Section 7.1). Proposition 7.2. The space C1,8 pHq endowed with the norm } ¨ }1,8 is a Banach ideal in the algebra of bounded operators LpHq. Proof. It is sufficient to prove that if A and C are bounded operators, then B P C1,8 pHq implies ABC P C1,8 pHq. Recall that the singular values of the operator © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1_7

245

246

Chapter 7. Product formulae in the Dixmier ideal

ABC satisfy the estimate sj pABCq ď }A}}C}sj pBq. By (6.64), this yields }ABC}1,8 “ sup nPN

n ÿ 1 sj pABCq 1 ` lnpnq j“1

ď }A}}C} sup nPN

(7.1)

n ÿ 1 sj pBq “ }A}}C}}B}1,8 , 1 ` lnpnq j“1

which proves the proposition.

l

Recall that for any A P LpHq and all B P C1 pHq one can define a linear functional on C1 pHq given by trace TrH pABq, Definition 2.41. The set of these functionals tTrH pA ¨quAPLpHq is just the dual space C1 pHq˚ :“ LpC1 pHq, Cq of C1 pHq, equipped with the operator-norm topology. In other words, C1 pHq˚ “ LpHq, in the sense that the map A ÞÑ TrH pA ¨q is an isometric isomorphism of LpHq onto C1 pHq˚ . Remark 7.3. Using the duality pairing xA|By :“ TrH pABq,

(7.2)

one can also describe the space C1 pHq˚ , which is the predual of C1 pHq, that is, its dual LpC1 pHq˚ , Cq “ C1 pHq. To this aim, for each fixed B P C1 pHq we consider the functionals A ÞÑ TrH pABq on LpHq. It is known that they are not all continuous linear functional on the bounded operators LpHq, i.e., C1 pHq Ă LpHq˚ , but they yield the entire dual of the compact operators: C1 pHq “ C8 pHq˚ . Hence, C1 pHq˚ “ C8 pHq Ă LpHq. Now we note that under the duality relation (7.2) the Dixmier ideal C1,8 pHq is the dual of the Macaev ideal C1,8 pHq “ C8,1 pHq˚ , where C8,1 pHq “ tA P C8 pHq :

ÿ 1 sn pAq ă 8u, n ně1

(7.3)

see Example 6.13. By the same duality relation and by similar calculations one also p0q concludes that the predual of C8,1 pHq is the ideal C8,1 pHq˚ “ C1,8 pHq, defined by n ÿ p0q C1,8 pHq :“ tA P C8 pHq : sj pAq “ oplnpnqq, n Ñ 8u. (7.4) jě1

By virtue of (6.63) (Remark 6.12), the ideal (7.4) is not self-dual, since p0q

p0q

C1,8 pHq˚˚ “ C1,8 pHq Ą C1,8 pHq. The problem that motivated the construction of the Dixmier trace was related to the desire of introducing general definition of the trace, i.e., a linear, positive,

7.2. Dixmier trace

247

and unitarily invariant functional on a proper Banach ideal IpHq of the unital algebra of bounded operators LpHq, cf. Proposition 2.48. Since any proper twosided ideal IpHq of the ring LpHq is contained in the compact operators C8 pHq and contains the set KpHq of finite-rank operators (6.38), the domain of definition of the trace has to coincide with some ideal IpHq. Remark 7.4. The canonical trace TrH p¨q is nontrivial only on the trace-class ideal C1 pHq, see Example 6.8. We recall that it is characterised by the property of normality: TrH psupα Bα q “ supα TrH pBα q, for every directed increasing bounded family tBα uαP∆ of positive operators from the cone C1,` pHq of positive operators. Since every nontrivial normal trace on LpHq is proportional to the canonical trace TrH p¨q, the Dixmier trace (6.67), C1,8 Q X ÞÑ Trω pXq, is not normal. Definition 7.5. A trace on the proper Banach ideal IpHq Ă LpHq is called singular if it vanishes on the set KpHq. Since by Proposition 2.59(b) a singular trace is defined up to trace-class operators C1 pHq, then by Remark 7.4 it obviously is not normal.

7.2 Dixmier trace Recall that only the ideal of trace-class operators has the property that on its positive cone C1,` pHq :“ tA P C1 pHq : A ě 0u the trace-norm is linear, since }A ` B}1 “ Tr pA ` Bq “ Tr pAq ` Tr pBq “ }A}1 ` }B}1 for A, B P C1,` pHq, see Example 6.8. Then the uniqueness of the trace-norm allows one to extend the trace to the whole linear space C1 pHq. This however fails for other symmetricallynormed ideals, either because of lack of linearity, or owing to lack of boundedness. This problem motivates the Dixmier trace construction as a certain limiting procedure involving the } ¨ }1,8 -norm. Let C1,8,` pHq be the positive cone of the Dixmier ideal. One can try to construct on C1,8,` pHq a linear, positive, and unitarily invariant functional (called trace T ) via extension of the limit, called the Lim, of the sequence of the properly normalised finite sums of eigenvalues of the operator X: T pXq :“ LimnÑ8

n ÿ 1 λj pXq, 1 ` lnpnq j“1

X P C1,8,` pHq.

(7.5)

First, keeping in mind that Lim is essentially the limit, we note that for any unitary operator U : H Ñ H, the eigenvalues of X P C8 pHq are invariant: λj pXq “ λj pU XU ˚ q. Hence, this is also true for the sequence σn pXq :“

n ÿ j“1

λj pXq,

n ě 1,

(7.6)

248


and the Lim (7.5) (if it exists) inherits this property: T pU XU ˚ q “ T pXq. Since for α P R one has λj pαXq “ αλj pXq, the trace (7.5) is also homogeneous: T pαXq “ αT pXq. Similarly, we comment on positivity and on boundedness. Note that X ě 0 implies the positivity of eigenvalues tλj pXqujě1 . Therefore, σn pXq ě 0 and the Lim in (7.5) is a positive mapping: X ÞÑ T pXq ě 0 for X P C1,8,` pHq, which remains finite by virtue of the estimate T pXq ď }X}1,8 , (6.64). The next issue with the formula (7.5) for T pXq is its linearity. To approach it, we need to understand better the meaning of Lim. To proceed, we recall that if P : H Ñ P pHq is an orthogonal projection on a finite-dimensional subspace with dim P pHq “ n, then for any compact operator X ě 0 formula (7.6) gives σn pXq “ sup tTrH pXP q : dim P pHq “ nu .

(7.7)

P

Using (7.7) and the inequality (3.17), Lemma 3.5, we obtain σn pX ` Y q ď σn pXq ` σn pY q,

n P N,

(7.8)

which is valid for any pair of positive compact operators X, Y P C8,` pHq. To estimate (7.8) from above we consider for the pair X, Y P C8,` pHq and for a given ε ą 0 two orthogonal projections P1 and P2 satisfying the conditions dim P1 pHq “ dim P2 pHq “ n, Tr H pXP1 q ą σn pXq ´ ε

^

Tr H pY P2 q ą σn pY q ´ ε.

Then by taking the linear span P pHq :“ P1 pHq _ P2 pHq, that is, the orthogonal projection onto P1 pHq ` P2 pHq, we get Tr H ppX ` Y qP q “ Tr H pXP q ` Tr H pY P q

(7.9)

ě Tr H pXP1 q ` Tr H pY P2 q ě σn pXq ` σn pY q ´ 2 ε. Therefore, by virtue of dim P pHq ď 2n and of the arbitrariness of ε ą 0, (7.7) and (7.9) yield (7.10) σn pXq ` σn pY q ď σ2n pX ` Y q, n P N, which is a kind of anti-triangle inequality. Now to proceed with decoding the properties of (7.5) that would ensure the linearity of T p¨q, we introduce ξn pXq :“

1 σn pXq, 1 ` lnpnq

X P C1,8,` pHq,

(7.11)

and a non-negative functional ω lim : l8 Ñ R` 0 , such that ω lim ptξn pXquně1 q :“ LimnÑ8 ξn pXq,

X P C1,8,` pHq.

(7.12)

7.2. Dixmier trace

249

This functional is defined on the space of non-negative sequences tξn p¨quně1 P l8 generated by operators from the positive cone of the Dixmier ideal. Note that by (7.11) the inequalities (7.8) and (7.10) yield for n P N (7.13)

ξn pXq ` ξn pY q ě ξn pX ` Y q, ξn pXq ` ξn pY q ď

1 ` lnp2nq ξ2n pX ` Y q. 1 ` lnpnq

(7.14)

Since by definition (7.12) the functional ω lim p¨q incorporates the limit n Ñ 8, the inequalities (7.13), (7.14) would give a desired linearity of the trace T : T pX ` Y q “ T pXq ` T pY q,

(7.15)

if one can prove that for X, Y , as well as for X ` Y , the LimnÑ8 in (7.13), (7.14) exist and the limits of the right-hand sides coincide. To this aim we have to clarify further the properties (a), (b) and (c), of the functional ω lim p¨q. Remark 7.6. Firstly, summarising our remarks above, we see that the functional ω lim : l8 Ñ R is non-negative and homogeneous: (a) ω lim pα ηq “ α ω lim pηq ě 0,

for α ě 0, @η “ tηn ě 0unPN P l8 .

Secondly, examining (7.5) and p7.12q we dediuce that the functional ω lim pηp¨qq “ LimnÑ8 ηn p¨q is completely determined by the ”tail” behaviour of the sequences tηn ě 0uně1 P l8 . For example, ω lim pηq “ 0 for all η P c0 . Here c0 Ă l8 is the subspace of bounded sequences that converge to zero (Section 6.2). Therefore, c0 “ ker ω lim and any ξ P dom ω lim is defined modulo η P c0 , that is, ω lim pξ ` ηq “ ω lim pξq. Consequently, (b) ω lim pηq “ limnÑ8 ηn ,

if tηn ě 0unPN is convergent.

By virtue of (a) and (b), the definitions (7.5) and (7.11) imply that for X, Y P C1,8,` pHq we have T pXq “ ω lim ptξn pXqunPN q “ lim ξn pXq,

(7.16)

T pY q “ ω lim ptξn pY qunPN q “ lim ξn pY q,

(7.17)

nÑ8

nÑ8

T pX ` Y q “ ω lim ptξn pX ` Y qunPN q “ lim ξn pX ` Y q, nÑ8

(7.18)

if the limits in the right-hand sides of (7.16)–(7.18) exist. Now, to ensure that (7.15) holds one has to select the state ω lim in such a way that it allows to restore the equality of the right-hand sides of (7.13), (7.14), when n Ñ 8. To this aim we require that the state ω lim be dilation D2 -invariant. Definition 7.7. Define the dilation mapping D2 : l8 Ñ l8 , η ÞÑ D2 pηq by the rule D2 : pη1 , η2 , . . . ηk , . . .q Ñ pη1 , η1 , η2 , η2 , . . . ηk , ηk , . . .q, We say that ω lim is dilation D2 -invariant if for any η P l8

@η P l8 .

(7.19)

250


(c) ω lim pηq “ ω lim pD2 pηqq. We shall discuss the question of existence of the dilation D2 -invariant states (also called the invariant means) on the Banach space l8 in Remark 7.9. Let X, Y P C1,8,` pHq. Then applying the property (c) to the sequence η “ pξ2 , ξ4 , ξ6 , . . .q, where tξ2n :“ ξ2n pX ` Y qu8 n“1 , we obtain ω lim pηq “ ω lim pD2 pηqq “ ω lim pξ2 , ξ2 , ξ4 , ξ4 , ξ6 , ξ6 , . . .q .

(7.20)

8 the difference Note that for ξ “ tξn :“ ξn pX ` Y qun“1

D2 pηq ´ ξ “ pξ2 , ξ2 , ξ4 , ξ4 , ξ6 , ξ6 , . . .q ´ pξ1 , ξ2 , ξ3 , ξ4 , ξ5 , ξ6 , . . .q,

(7.21)

is the sequence that converges to zero if for n Ñ 8 one has ξ2n ´ ξ2n´1 Ñ 0. Then D2 pηq ´ ξ P c0 . Since by Remark 7.6(b) the state ω lim is determined modulo sequences that belong to c0 , the relations (7.18), (7.20) and (7.21) would imply ω lim ptξ2n pX ` Y quně1 q “ ω lim pD2 ptξ2n pX ` Y quně1 qq “ ω lim ptξn pX ` Y quně1 q. By (7.18), this would mean that lim ξ2n pX ` Y q “ lim ξn pX ` Y q,

nÑ8

nÑ8

which by estimates (7.13), (7.14) and limnÑ8 p1 ` lnp2nqq{p1 ` lnpnqq “ 1, would yield lim ξn pX ` Y q “ lim ξn pXq ` lim ξn pY q. (7.22) nÑ8

nÑ8

nÑ8

Combining (7.16), (7.17), (7.18) and (7.22), we obtain the linearity (7.15) of the functional T on the positive cone C1,8,` pHq provided that it is defined by the corresponding D2 -invariant state ω lim , or by some equivalent dilation-invariant mean. Therefore, to complete the proof of the linearity, it remains only to check that limnÑ8 pξ2n ´ ξ2n´1 q “ 0. To this end we note that, by definitions (7.6) and (7.11), one gets „  1 1 ξ2n ´ ξ2n´1 “ ´ σ2n´1 pX ` Y q 1 ` lnp2nq 1 ` lnp2n ´ 1q 1 ` λ2n pX ` Y q. (7.23) 1 ` lnp2nq Since X, Y P C1,8,` pHq, we have limnÑ8 λ2n pX`Y q “ 0, and also σ2n´1 pX`Y q ď Op1`lnp2n´1qq. Then taking into account that 1{p1`lnp2nqq´1{p1`lnp2n´1qq “ op1{p1 ` lnp2n ´ 1qq we conclude that the right-hand side of (7.23) converges to zero, as n Ñ 8.

7.2. Dixmier trace

251

Concluding this construction of the trace T p¨q (7.5) and following definition (7.12), we denote it by Trω pXq :“ ω lim pξpXqq,

X P C1,8,` pHq.

It is worth pointing out that by linearity (7.15) one can uniquely extend this functional from the positive cone C1,8,` pHq to the real subspace of the Banach space C1,8 pHq, and finally to the entire ideal C1,8 pHq. Definition 7.8. The Dixmier trace Trω pXq of the operator X P C1,8,` pHq is the value of the linear functional (7.5): Trω pXq “ LimnÑ8

n ÿ 1 λj pXq , 1 ` lnpnq j“1

(7.24)

where by (7.6) and (7.11) LimnÑ8 ξn p¨q is defined as a positive dilation-invariant functional ω lim pξp¨qq on l8 (7.12) that satisfies the properties (a), (b), and (c). Since any self-adjoint operator X P C1,8 pHq has the representation X “ X` ´X´ , where X˘ P C1,8,` pHq, one gets Trω pXq “ Trω pX` q´Trω pX´ q. Then for arbitrary Z P C1,8 pHq the Dixmier trace is Trω pZq “ Trω pRe Zq ` iTrω pIm Zq. Note that if X P C1,8,` pHq, then definition (7.24) of Trω p¨q together with the definition of the norm } ¨ }1,8 in (6.64), readily imply the estimate Trω pXq ď }X}1,8 , which in turn yields for arbitrary Z from the Dixmier ideal C1,8 pHq the inequality |Trω pZq| ď }Z}1,8 . (7.25) Remark 7.9. A decisive step in the construction of the Dixmier trace Trω p¨q (7.24) was the tacit assumption of the existence of the invariant mean (state) ω lim . To make this existence evident the following remarks are in order: (1) Although in our construction of ω lim we did not use this fact explicitly, we recall that the invariant functional can be included in the set Spl8 q of positive linear functionals (states) on the Banach space of bounded sequences l8 . Note that Spl8 q Ă pl8 q˚ , where the space pl8 q˚ (of all linear functionals on l8 ) is the dual of the Banach space of bounded sequences. Recall that positivity and linearity make the state continuous (equivalently, bounded) on l8 . So, one can consider the set of states ω P Spl8 q taking the positive linear functionals with unit norms. (2) Now let pl8 q˚ be equipped with the weak*-topology. Then by the BanachAlaoglu theorem, the convex set of states Spl8 q is compact in pl8 q˚ in the weak*topology. For any state ω P Spl8 q the relation ωpD2 pξqq “: pD˚2 ωqpξq defines dual D˚2 -dilation on the set of states. By definition (7.19) this map is such that D˚2 : Spl8 q Ñ Spl8 q, as well as continuous and affine (in fact linear). Then by the Markov-Kakutani theorem the dilation D˚2 has a fix point ω lim “ D˚2 ω lim ,

ω lim P Spl8 q.

This abstract observation justifies the existence of the invariant mean, or functional with the property (c) for D2 -dilation.

252


Note that Remark 7.9 has a straightforward extension to any Dk -dilation for k ą 2, which is defined similarly to (7.19). Since dilations for different k ě 2 commute, the extension of the Markov-Kakutani theorem shows that the commutative p , which coincides p “ D˚k ω family F “ tD˚k ukě2 has in Spl8 q a common fix point ω with ω lim . Therefore, Definition 7.8 of the Dixmier trace does not depend on the degree k ě 2 of the dilation Dk . For more details about different constructions of invariant functionals and the corresponding Dixmier trace on C1,8 pHq, see references in Section 7.4 (Notes to Section 7.2). Proposition 7.10. The Dixmier trace has the following properties: (a) For any bounded operator B P LpHq and any Z P C1,8 pHq one has Trω pZBq “ Trω pBZq. (b) The Dixmier trace is singular: Trω pCq “ 0 for any operator C P C1 pHq from the trace-class ideal, which is the closure of finite-rank operators KpHq in the } ¨ }1 -norm. (c) The Dixmier trace Trω : C1,8 pHq Ñ C is continuous in the } ¨ }1,8 -norm. Proof. (a) Since every operator B P LpHq is a linear combination of four unitary operators, it is sufficient to prove the equality Trω pZU q “ Trω pU Zq for a unitary operator U and moreover only for Z P C1,8,` pHq. Then the corresponding equality follows from the unitary invariance: sj pZq “ sj pZU q “ sj pU Zq “ sj pU ZU ˚ q, of singular values of the positive operator Z for all j ě 1. (b) Since C P C1 pHq yields }C}1 ă 8, definition (7.6) for tλj p|C|qujě1 implies σn p|C|q ď }C}1 for any n ě 1. Then by Definition 7.8 one gets Trω pCq “ 0. The proof of the last part of the statement is standard. (c) Since the ideal C1,8 pHq is a Banach space and Trω : C1,8 pHq Ñ C a linear functional, it is sufficient to consider the continuity at X “ 0. Hence, suppose the sequence tXk ukě1 Ă C1,8 pHq converges to X “ 0 in } ¨ }1,8 -topology, that is, by (6.64) 1 lim }Xk }1,8 “ lim sup σn pXk q “ 0. (7.26) kÑ8 kÑ8 nPN 1 ` lnpnq Since (7.25) implies |Trω pXk q| ď }Xk }1,8 , the assertion follows from (7.26). l Therefore, by Proposition 7.10(b) the Dixmier construction gives an example of a singular trace in the sense of Definition 7.5.

7.3

Product formulae

Consider first the trace-class ideal Cφ1 pHq “ C1 pHq, where the symmetric norming function φ1 is defined by (6.20), (6.21). Then under the assumptions of the general Proposition 6.22, the Trotter-Kato product formula convergence in the trace-norm: } ¨ }1 - lim F pt{rqr “ e´tH , rÑ`8

(7.27)

7.3. Product formulae

253

locally uniformly away from zero for Kato functions of class K (Appendix C, Section C.1). Since the trace Tr is a continuous functional on the ideal C1 pHq, the limit (7.27) implies Trpe´tH q “ lim TrpF pt{rqr q “ lim TrpT pt{rqr q rÑ`8

rÑ`8

(7.28)

“ lim Trppf ptA{nqgptB{nqqn q “ lim TrppgptB{nqf ptA{nqqn q, nÑ8

nÑ`8

where the limits of approximants for the families tT ptqutě0 , tf ptqgptqutě0 , and tgptqf ptqutě0 follow from the results of Section 6.4, Proposition 6.18. Now we consider the Dixmier ideal: Cφ pHq|φ“p1,8q “ C1,8 pHq, (6.63). Proposition 7.11. Let A and B be non-negative self-adjoint operators on a Hilbert ` ` ` D space H and let f D : R` 0 Ñ R0 and g : R0 Ñ R0 be bounded Borel measurable D p functions such that F pt0 q P Cφ pHq for some t0 ą 0 and some integer p ě 1. Suppose the Kato functions f and g are dominated by f D and g D , respectively, and the Trotter-Kato product formula converges in the operator-norm locally uniformly away from zero for self-adjoint approximants tF pt{rqr urě1 . Then for the Dixmier trace, similarly to (7.28), it holds that Tr ω pe´tH q “ lim Tr ω pF pt{rqr q “ lim Tr ω pT pt{rqr q rÑ`8

rÑ`8

(7.29)

“ lim Tr ω ppf ptA{nqgptB{nqqn q “ lim Tr ω ppgptB{nqf ptA{nqqn q. nÑ`8

nÑ8

The limits are locally uniform away from pt0 for all families of approximants generated by f and g. Proof. The argument is close to that used for Proposition 6.22. Recall that, by Lemma 6.21 e´tH P Cφ pHq for t ą pt0 . Lemma 6.20 implies that for each bounded interval rτ0 , τ s Ď ppt0 , 8q there is an r0 ě 1 such that F pt{rqr P Cφ pHq for r ě r0 and t P rτ0 , τ s. By (6.113), we obtain suprěr0 suptPrτ0 ,τ s }F pt{rqr }φ ď }F D pt0 q}φ ă 8, which shows that the sequence tF pt{rqr urě1 is bounded in Cφ pHq locally uniformly away from pt0 . Now applying Proposition 6.19 we get that the Trotter-Kato product formula (7.30) } ¨ }1,8 - lim F pt{rqr “ e´tH , rÑ`8

converges locally uniformly away from pt0 . Since by Proposition 7.10(c) the Dixmier trace Trω : C1,8 pHq Ñ C is continuous in the } ¨ }1,8 -norm topology, the limit (7.30) implies Tr ω pe´tH q “ lim Tr ω pF pt{rqr q. rÑ`8

(7.31)

The rest of (7.29) follows from (7.31) and Proposition 6.18 of Section 6.4. There it is shown that (7.30) yields the same limit in the symmetrically normed ideals Cφ pHq for the families tT ptqutě0 , tf ptqgptqutě0 , and tgptqf ptqutě0 and for the corresponding choice of r _ n in the product formulae. l

254


Different types of sufficient conditions that ensure assumptions of Proposition 7.11 are formulated in Proposition 6.28, Proposition 6.29, Proposition 6.30, Corollary 6.31, and Corollary 6.32. Corollary 7.12. The } ¨ }φ -norm estimates ηφ prq and ηφ pnq of Proposition 6.38 determine the rate of convergence | Tr ω pe´tH q ´ Tr ω pF pt{xqx q| ď Cω ηω pxq,

(7.32)

of the Dixmier traces in (7.31) for x “ r _ n, where r P R` or n P N. In fact, the rate is the same (modulo Cω and r _ n) for all Trotter-Kato approximants generated by tF ptqutě0 , tT ptqutě0 , tf ptqgptqutě0 , and tgptqf ptqutě0 . Consider for concreteness the approximants tpF pt{xqqx ux“r_n . Recall that by inequalities (6.160) and (7.25) we get the estimate | Tr ω pe´tH q ´ Tr ω pF pt{xqx q| ď }e´tH ´ F pt{xqx }1,8 ď C η1,8 pxq,

(7.33)

for t P rτ0 , τ s and x ě x0 , where either x “ r P R` , or x “ n P N. Then the rate ηω p¨q “ η1,8 p¨q. Corollary 7.13. Under assumptions of Proposition 6.38 the estimate of the rates of } ¨ }1,8 -convergence in (7.33) are entirely determined by the operator-norm error bound η. Hence, the lifting of operator-norm estimates to the error bound estimates in the C1,8 pHq-topology for t ą pt0 coincide (up to factors) with ηφ prq or ηφ pnq for φ “ p1, 8q. Therefore, by Corollary 7.12, ηω prq :“ 2 }F D pt0 q}1,8 ηpr{2q, r P R` ,

or

D

ηω pnq :“ }F pt0 q}1,8 tηprn{2sq ` ηprpn ` 1q{2squ,

(7.34) n P N,

where rss denotes the entire part of s ą 0. If the operator-norm estimate η is optimal, then, similarly to ηφ , the rate ηω is also optimal. Note that Proposition 6.40 gives the explicit expressions for the rates of convergence (7.34) and the Trotter-Kato formulae for all families tF ptqutě0 , tT ptqutě0 , tf ptqgptqutě0 , and tgptqf ptqutě0 . Proposition 7.14. Assume that the Kato functions f and g are dominated, respectively, by f D and g D and that the corresponding function F D ptq is such that F D pt0 q P Cφ pHq for some t0 ą 0. Let for some α P p1{2, 1s the Kato functions f and g belong to the class Kα . Suppose that b1{α C1{2α S1 ă 1. Then the function (7.33) η1,8 pnq ď Γ1 lnpnq{n2α´1 ,

7.4. Notes

255

for n ą 1, is the error bound in C1,8 pHq away from 2t0 for the Trotter-Kato product formulae for all families tF ptqutě0 , tT ptqutě0 , tf ptqgptqutě0 , and tgptqf ptqutě0 . Here the rate of convergence is lnpnq{n2α´1 . If in addition, besides f, g P Kα and (6.178), for some α P p1{2, 1q the condition dom H α Ď dom Aα is satisfied, then the function (7.33) η1,8 pnq ď Γ2 {n2α´1 , for n ą 1, is the error bound in C1,8 pHq away from 2t0 for the Trotter-Kato product formulae for all families tF ptqutě0 , tT ptqutě0 , tf ptqgptqutě0 , and tgptqf ptqutě0 . Here the rate of convergence is improved to 1{n2α´1 . More examples of explicit estimates for the rate of convergence in the Dixmier ideal can be found by lifting the operator-norm error bounds for the Trotter-Kato product formulae from Section 5.2.

7.4 Notes Notes to Section 7.1. The contents of this section are standard. The section starts as a continuation of Section 6.2 by a discussion of two key definitions: the normal trace and the singular trace. Here we follow in part [Zag19]. For more details, including the proof of Proposition 7.1, the reader is referred to the book [LSZ12], which is a rather complete source on this topic. Notes to Section 7.2. Recall that the Dixmier trace and Dixmier ideal arose from the question of whether the algebra LpHq of all bounded linear operators on a Hilbert space H admits a unique nontrivial trace. J. Dixmier solved this problem in the negative in a short note [Dix66]. He proved the existence on LpHq of (singular) traces that vanish on the ideal of trace-class operators. The key steps of his construction are presented in Section 7.2. It obviously yields a singular trace in the sense of Definition 7.5. The example of construction is taken from the lecture notes [Su08]. We consider the simplest illustration based on the proof of the existence of the dilation D2 -invariant functional (mean) ω lim . The reader can find in [Su08] examples of other constructions of invariant functionals based, e.g., on the Banach limit, or on the Ces` aro mean. Note that in our presentation we did not assume a priori that the functional ω lim is a state, i.e., that it is a positive linear functional with unit norm on the Banach space l8 , see Remark 7.9. In fact, instead of continuity, the argument for proving of linearity needs only Remark 7.6(b) that the state ω lim is determined modulo sequences that belong to the set c0 , see [Zag19]. For more details and more examples we refer to the review article [CaSu06] and to the lecture notes [Su08]. Notes to Section 7.3. The Trotter-Kato product formulae in the Dixmier ideal C1,8 pHq were considered for the first time by H. Neidhardt and V. A. Zagrebnov

256


in [NZ99d]. There under some sufficient conditions the convergence of Dixmier traces for the Trotter-Kato product formulae were announced without proof, as well as without hypothesis about estimates of the rate of convergence. Here we developed indispensable arguments and made them explicit by proving in Proposition 7.11 that the Trotter-Kato product formulae convergence in the } ¨ }1,8 -topology. To establish these results with an estimate of the rate of convergence one needs more arguments. In Proposition 7.14 the Trotter-Kato product formulae in Dixmier ideal C1,8 pHq with an estimate of the rate of convergence in the } ¨ }1,8 -topology is proved following a standard scheme of lifting arguments developed in Chapter 5 and Chapter 6. To this end we use the lifting of the estimates of the rate of convergence due to Proposition 6.38.

Appendix A. Spectra of closed operators Although in this chapter we essentially focus on linear operators in a Hilbert space H, the information below also concerns bounded and unbounded operators in a Banach space B. We shall make clear if certain facts concern only a specific type of operators or a particular space. By LpH1 , H2 q we denote the Banach space of bounded linear operators from a Hilbert space H1 to a Hilbert space H2 , and we set LpHq :“ LpH, Hq. Similar notations will be used for the general case of Banach spaces B1 , B2 . Notice that we do not assume (a priory) that the domains of involved operators are dense, but do it whenever needed.

A.1

Resolvents and spectra

(a) Let A with domain dom A Ď H be a closed linear operator in a Hilbert (or Banach) space H (or B) and let Aζ :“ A ´ ζ1 for ζ P C. Then the map Aζ : dom A Ñ ran Aζ is said to be continuous in dom A if for any sequence tun uně1 Ă dom A such that limnÑ8 un “ 0, we have limnÑ8 Aζ un “ 0. The ran Aζ of this closed map is in general not a subspace, but a non-closed linear subset (manifold) of H (or B). Since in H the orthogonal complement pran Aζ qK is closed (i.e., it is a subspace of H), we can define the deficiency (or defect number) of the closed operator Aζ by defAζ :“ dimpran Aζ qK . (b) Below we would need the following version of the closed graph theorem: Let the operator A P CpHq with domain dom A be closed in H. Then, saying that dom A being a subspace of H is equivalent to saying that A is continuous on dom A. The latter is the same as the boundedness of A on dom A: there exists a ą 0 such that }Au} ď a}u}, u P dom A. We denote this by A P Lpdom A, Hq. Therefore, a closed unbounded operator A P CpHq must have an unclosed domain. (c) Note that the range ran A is a subspace of H if the operator A is closed and injective, that is, satisfies }Au} ě c}u} , for some c ą 0 and all u P dom A. In © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1

257

Appendix A. Spectra of closed operators

258

turn, this property of the range implies the existence of the closed inverse operator A´1 : ran A Ñ dom A, which is continuous: A´1 P Lpran A, dom Aq, on this range. We recall that if in addition ran A “ H, then A is surjective (onto map). Together with injectivity, this makes this map bijective. Note that the general observations (b) and (c) are valid in Banach spaces. (d) For a closed linear operator A in a Banach space B we call ρpAq “ tζ P C : Aζ : dom A Ñ B is bijectiveu ô tAζ : dom A Ñ B is injectiveu ^ tAζ : dom A Ñ B is surjectiveu, the resolvent set. Then, by definition, the spectrum of A is the set σpAq :“ CzρpAq. By the closed graph theorem, for ζ P ρpAq the resolvent of A RA pζq “ pAζ q´1 : ρpAq Ñ LpBq, is a bounded operator-valued function, although with independent of ζ and (in general) unclosed range: ran RA pζq “ dom A, see (c). (e) Note that ζ0 P ρpAq if and only if there exists δ0 ą 0 such that }Aζ0 u} ě δ0 }u} for any u P dom A. Then }RA pζ0 qw} ď δ0´1 }w}, w P B, and for other points ζ P ρpAq one obtains the resolvent identity RA pζq ´ RA pζ0 q “ pζ ´ ζ0 q RA pζq RA pζ0 q. For a fixed ζ0 P ρpAq this identity can be regarded as an equation for the unknown operator-valued function ζ ÞÑ RA pζq. It is solvable in the open disc Dζ0 pδ0 q :“ tζ P C : |ζ ´ ζ0 | ă δ0 u. This solution is a uniformly operator-norm convergent in the disc Dζ0 pδ0 q power series RA pζq “

8 ÿ

pζ ´ ζ0 qn RA pζ0 qn`1 .

n“0

Therefore, the resolvent is the piecewise holomorphic (in the operator-norm topology) function RA : ζ ÞÑ RA pζq defined in any connected component of the open resolvent set ρpAq. (f) Note that the power series representation for the resolvent implies for ζ P ρpAq the estimate from below }RA pζq} ě 1{distpζ, σpAqq, i.e., the spectrum σpAq is a closed subset of C and its topological boundary BσpAq Ă σpAq. (g) If the operator A is bounded with norm }A}, then the fact that the series RA pζq “ ´

8 1 ÿ pA{ζqn , ζ n“0

converges for tζ P C : |ζ| ą }A}u Ď ρpAq and the Liouville theorem for the operator-valued holomorphic functions in C show that the spectrum σpAq ‰ H. Note that it is compact and lies in the closed disc Dr of radius r “ }A}.

ppAq of σpAq A.2. Core σ

259

(h) Recall that rpAq :“ supt|ζ| : ζ P σpAqu is the spectral radius of A, and one has rpAq ď }A}. We note that rpAq “ limnÑ8 }An }1{n . Hence, rpAq “ }A} if A P LpHq is self-adjoint. In the opposite case it is possible that rpAq “ 0 (e.g., for (quasi )-nilpotent operators), whereas }A} ‰ 0. See below the example of the Volterra operator. (i) Note that the assumption of closeness of the linear operator A makes the notion of spectrum of A nontrivial. Indeed, suppose A is not closed. To check whether the resolvent set of A is non-empty, suppose that for some ζ P C the operator Aζ´1 is defined on B and that Aζ´1 P LpBq. Then Aζ´1 is closed, which implies that Aζ is also closed, and so is A. This contradiction shows that ρpAq “ H, that is, the spectral problem for nonclosed A is trivial : the solution is always σpAq “ C. However, there are examples of closed operators with spectrum σpAq “ C and by (g) they are not bounded operators. ˆ which is a bounded (j) It is often useful to consider a pseudo-resolvent Rpζq, operator-valued function on some domain D Ă C. For motivation, note that the strong operator limit of the resolvents tRk pζqukě1 ˆ Rpζq “ s-lim Rk pζq, kÑ8

ζ P Ds Ă C,

does not need to be the resolvent of an operator. Here Ds denotes the domain of strong convergence in the regular set for all tRk pζqukě1 . However, this strong limit satisfies the resolvent equation for ζ, ζ0 P Ds , and it is called pseudo-resolvent. ˆ We recall that the pseudo-resolvent Rpζq is the resolvent of a closed operator ˆ A if and only if ker Rpζq “ t0u for any ζ P Ds . ˆ is Let Ds be non-empty. Then there is an alternative: either the limit Rpζq ˆ not invertible for any ζ P Ds , or Rpζq “ RA pζq is the resolvent of a unique closed operator A and one also gets that Ds “ ρpAq X Db . Here Db is the domain of norm-boundedness of the sequence tRk pζqukě1 .

A.2

ppAq of σpAq Core σ

(a) By virtue of A.1(a), the deficiency of Aζ is equal to defAζ :“ dimpran Aζ qK . If dom A is dense in H, then the adjoint to Aζ , Aζ ˚ “ A˚ ´ ζ1, is well defined for any ζ P C, and one has the orthogonal decomposition H “ ran Aζ ‘ ker Aζ ˚ . Therefore, the deficiency of Aζ can be expressed using the closed kernel ker Aζ ˚ by the formula: defAζ “ dimpker Aζ ˚ q. Note that if A is closed, then the adjoint operator A˚ is in turn densely defined in H and then one has that Aζ ˚˚ “ Aζ . (b) Let A be a closed linear operator in H and let ran Aζ Ă H be closed for ζ P C. Then one can define on the ran Aζ a continuous inverse A´1 ζ P Lpran Aζ , Hq. These points ζ constitute the set of quasi-regular points ρppAq of the operator A.

260


Note that the set ρppAq is open and that the integer-valued function ζ ÞÑ defAζ is constant on each of the connected components of ρppAq. (c) Let ρpAq :“ tζ P ρppAq : defAζ “ 0u denote the set of the regular points of the operator A. Then for ζ P ρpAq the map Aζ : dom A Ñ H is bijective with ran Aζ “ H and by the closed graph theorem the inverse operator (resolvent) is bounded: A´1 P BpHq. Therefore, by A.1(d) the set of regular points coincides ζ with the resolvent set and ρpAq Ď ρppAq. (d) The set σpAq :“ CzρpAq is known as the spectrum of the closed operator ppAq :“ Czp A, whereas the complement of the set of the quasi-regular points, σ ρpAq, ppAq Ď σpAq and that is called the core of the spectrum σpAq. It is clear that σ both of these sets are closed.

A.3 Subsets of the spectrum σpAq (a) By A.1(d) the mapping Aζ : dom A Ñ B, for ζ P σpAq, is not bijective with ran Aζ “ B. This is equivalent to the statement: tAζ : dom A Ñ B is injectiveu _ tAζ : dom A Ñ B is surjectiveu, that is, either the mapping Aζ is not one-to-one, or it is not onto. Now note that the following three statements are equivalent: tAζ : dom A Ñ B is surjectiveu ô tran Aζ “ Bu ô tran Aζ is dense in Bu ^ tran Aζ is closed in Bu. Therefore, the statement tζ P σpAqu is equivalent to the following three-part assertion: tζ P σpAqu ô tAζ : dom A Ñ B is not injectiveu _ tran Aζ is not dense u _ tran Aζ is not closed u . Either of these three parts implies the statement tζ P σpAqu. (b) Since the first part: tAζ : dom A Ñ B is not injectiveu, implies a nontrivial ker Aζ , the point spectrum σp pAq of A is the subset of σpAq defined as σp pAq :“ tζ P C : ker Aζ ‰ t0uu . This subset consists of eigenvalues of A, and ker Aζ contains the associated eigenvectors. In σp pAq Ď σpAq we distinguish the subset of eigenvalues of infinite multiplicity: σp8 pAq :“ tζ P C : dimpker Aζ q “ 8u. (c) The second part: tran Aζ is not denseu ô tran Aζ ‰ Bu, implies that for the injective map Aζ with closed range we have pran Aζ qK ‰ H. Hence, def Aζ ‰ 0,

A.3. Subsets of the spectrum σpAq

261

and the operator Aζ is boundedly invertible on ran Aζ . Then by the definitions of the quasi-regular set ρppAq and the resolvent set ρpAq (see A.2) the corresponding set of ζ is σ pAq. σres pAq :“ ρppAqzρpAq “ σpAqzp The subset σres pAq Ă σpAq is called the residual spectrum of A. Since for ζ P σres pAq the operator Aζ remains injective, the point and the residual spectra are disjoint: σp pAq X σres pAq “ H. (d) The last remark has an important corollary: if a densely defined in a Hilbert space H operator A is self-adjoint, then σres pAq “ H. To show this we note that if ζ P σres pAq, that is tran Aζ ‰ Hu, then by the orthogonal decomposition and by formula in A.2(a) one gets ker Aζ ˚ ‰ H, which together with A.3(b) yields ζ P σp pA˚ q. Since A “ A˚ , the spectrum σpAq Ď R. Therefore, ζ P σp pAq and consequently σp pAq X σres pAq ‰ H leads to a contradiction with the last remark in A.3(c). (e) The third part: tran Aζ is not closedu, motivates the definition of the continuous spectrum σcont pAq of A as the subset σcont pAq :“ tζ P C : ker Aζ “ t0u ^ ran Aζ ‰ ran Aζ “ Bu. First we note that the intersection σp pAq X σcont pAq can be non-empty, ppAq and see examples below (Section A.4(f)). Next, we remark that σp pAq Ă σ σcont pAq Ă σ ppAq, that is, in fact one has σ ppAq “ σp pAq Y σcont pAq. Indeed, the inclusion σp pAq Y σcont pAq Ď σ ppAq is obvious. Now suppose that ζ R σp pAqYσcont pAq. Then by the definitions of the spectra σp pAq and σcont pAq, the map Aζ is injective and ran Aζ is a subspace. Therefore, A´1 ζ P Lpran Aζ , Bq. Then by A.2(b)-(d) the point ζ is quasi-regular: ζ P ρppAq, that yields σp pAqYσcont pAq Ě σ ppAq. ppAq found in (e) and the definition (c) of (f) The representation of the core σ ppAq “ H. Thus we infer that the residual spectrum σres pAq yield σres pAq X σ σpAq “ σp pAq Y σcont pAq Y σres pAq, and that σres pAq “ σpAq X ρppAq. Therefore, the residual spectrum is an open set, which consists of components of ρppAq with nontrivial deficiencies def Aζ ‰ 0. Equivalently, ζ P σres pAq implies that the closed ran Aζ ‰ B and A´1 P ζ Lpran Aζ , Bqq. r Ą A (i.e., dom A Ă dom A) r is a closed extension of a (g) Suppose that A r closed operator A. Then σ ppAq Ă σ ppAq. Indeed, by the representation of the core σ ppAq found in (e) one gets the following alternative: ´1 tζ P σ ppAqu ñ tD A´1 ζ u _ tAζ R LpBqu.

262


rζ , the extension cannot reduce the core of the spectrum σ ppAq “ Since Aζ Ă A r σp pAq Y σcont pAq. Moreover, it is clear that σp pAq Ď σp pAq. On the other hand, r may be larger or smaller than σcont pAq. We the continuous spectrum σcont pAq note that the latter case is possible only if r r ñ tζ P σ 8 pAqu. tζ P σcont pAqzσcont pAqu p r Ą A such Now we recall that if ζ P σres pAq, then there exists an extension A r that ζ P ρpAq. On the other hand, if ζ P ρpAq, then for this ζ and any nontrivial r Indeed, since ran Aζ “ B, then for any r Ą A one gets ζ P σp pAq. extension A r rζ u. Hence, u P dom Az dom A there exists vector w P dom A such that Aζ w “ A r for u ‰ w one gets the non-zero element u ´ w P ker Aζ .

A.4 Approximate and essential spectra (a) Let A : dompAq Ñ B, be a closed operator in a Banach space B. Recall that if for ζ P C there exists a sequence tun uně1 Ă dom A such that }un } “ 1 and limnÑ8 }Aζ un } “ 0, then ζ belongs to the set σapp pAq, which is called the approximate spectrum (or approximate point spectrum) of A. (b) We note that according to this definition the approximate spectrum coincides with the core of the spectrum: σapp pAq “ σ ppAq. On the other hand, the definition is constructive and it helps to establish many results, especially in a Hilbert space. We start, however, with one general result valid in a Banach space. If A : dom A Ñ B is a closed operator, then the (topological) boundary of the ppAq. Recall that operator spectrum belongs to the core of the spectrum: BσpAq Ď σ the non-empty operator spectrum σpAq is a closed set. Let ζ0 P BσpAq Ď σpAq. Then there exists a sequence tζn uně1 Ă ρpAq such that limnÑ8 ζn “ ζ0 . Using the estimate in A.1(f) one can find a vector x P B such that limnÑ8 }RA pζn qx} “ 8. If we define xn :“ RA pζn qx{}RA pζn qx}, then the sequence txn uně1 Ă dom A. Moreover, it is such that }xn } “ 1 and }Aζ0 xn } “ }Aζn xn ´ pζn ´ ζ0 qxn } Ñ 0 for n Ñ 8. Hence, we get ζ0 P σapp pAq. (c) Now let A : dompAq Ñ H, be a self-adjoint operator in the Hilbert space H. Recall that σres pAq “ H, see A.2(d). Since by A.2(e),(f) one has ppAq Y σres pAq, the spectrum of a self-adjoint operator A coincides with σpAq “ σ the core: σpAq “ σ ppAq “ σp pAq Y σcont pAq. Hence, it is completely defined by the approximate spectrum as in (a). A refinement of definition (a) in a Hilbert space (Weyl’s criterion) allows to distinguish in the spectrum of a self-adjoint operator an important subset σess pAq Ď σpAq, called the essential spectrum of A. (d) To this aim we define first the discrete spectrum σd pAq Ď σp pAq. It is the simplest subset of the point spectrum, which consists of isolated eigenvalues

A.4. Approximate and essential spectra

263

of finite multiplicity. Then the essential spectrum of a self-adjoint operator A is the complementary set of σd pAq in σpAq: σess pAq “ σpAqzσd pAq “ pσp pAqzσd pAqq Y σcont pAq. The Weyl criterion. Let A : dom A Ñ H, define a self-adjoint operator in H. If for a real number λ P R there exist an orthonormal sequence of vectors ten uně1 Ă dompAq, t}en } “ 1 : en Ken`1 uně1 , such that limnÑ8 }Aλ en } “ 0, then λ belongs to σess pAq. We note that by the Weyl criterion this subset of the spectrum σpAq is closed: σess pAq “ σess pAq. If A is self-adjoint, then λ P σpAq belongs to the essential spectrum σess pAq unless it lies in σd pAq. (e) Let A be a self-adjoint operator. Recall that a symmetric operator K is A-compact if there exists ζ P ρpAq such that KRA pζq P C8 pHq. In this case the operator A ` K with dompA ` Kq “ dom A is self-adjoint and σess pA ` Kq “ σess pAq. This stability of the essential spectrum is known as the Weyl theorem. Moreover, if A and A1 are two self-adjoint operators such that the difference of their resolvents, pA ´ ζ1q´1 ´ pA1 ´ ζ1q´1 , is compact for some ζ P ρpAq X ρpA1 q, then σess pAq “ σess pA1 q. (f) To understand better the approximate and essential spectra we consider a few examples. We denote by σp˚ pAq :“ σp pAqzσd pAq the complement of a discrete spectrum in the point spectrum. (1f) Let A be the self-adjoint operator given by Aen “ λn en , λn “ n´1 , on the orthonormal basis ten uně1 in a Hilbert space H. Then by (d) one gets that σpAqzt0u “ σd pAq with multiplicity one and σess pAq “ t0u. To elucidate the nature of the unique non-isolated point λ “ 0 of σpAq one can either formally deduce from (d) and from σp˚ pAq “ H that 0 P σcont pAq, or infer the same by referring to the approximate spectrum, see (a)–(c), and to the fact that non-isolated point λ “ 0 belongs to the boundary BσpAq Ď σp pAq Y σcont pAq, but not to the point spectrum σp pAq. (2f) Let operator A0 acts as A0 e1 “ 0, A0 en “ µn en , µn “ pn ´ 1q´1 , n ě 2, on the orthonormal basis ten uně1 in a Hilbert space H. Then by the definitions of the point and discrete spectra one obtains that σpA0 q “ σp pA0 q “ t0u Y σd pA0 q. Therefore, again by (d) the non-isolated point t0u “ σess pA0 q, but now 0 P σp pA0 q and σcont pA0 q “ H. (3f) Let the bounded self-adjoint operator A1 acts as A1 en “ rn en , n ě 1, on the orthonormal basis ten uně1 in a Hilbert space H, where trn uně1 Ă p0, 1q is the set of all rational numbers in the open interval p0, 1q. Then by the definitions of the point and discrete spectra we have σp pA1 q ‰ H, with non-isolated eigenvalues of multiplicity one and σd pA1 q “ H. Therefore, σpA1 q “ σess pA1 q “ σp pA1 q Y σcont pA1 q . Note that for any λ P p0, 1qzσp pA1 q there exists a sequence trnk ukě1 Ă σp pA1 q such that limkÑ8 rnk “ λ. Then by the definition of the essential

264


spectrum, }pA1 ´ λ1qenk } ď }pA1 ´ rnk 1qenk } ` }prnk ´ λqenk }. This yields that λ P σess pA1 qzσp pA1 q “ σcont pA1 q and in accordance with (c) BσpA1 q Ď σp pA1 q Y σcont pA1 q, that is, we have σpA1 q “ r0, 1s.

A.5 Fredholm operators and essential spectrum (a) Another approach to the essential spectrum is via Fredholm operators. Recall that an operator A P LpBq is called Fredholm if ker A and the quotient space coker A :“ B{ ran A, also known as the cokernel of A, are finite-dimensional. It is worth mentioning that cokerA ”measures” the deficiency of surjection and in this way takes into account an eventual residual spectrum of A. Indeed, below we give an example of the Volterra operator V with non-empty σres pV q. It is known that if A is a Fredholm operator, then ran A is a subspace of B and that if K is compact, then operator 1 ` K is Fredholm. (b) Let A be a closed unbounded operator with dense domain dom A Ă B in a Banach space B. Then A is a Fredholm operator if ker A is finite-dimensional, ran A is closed, and the cokernel coker A is also finite-dimensional. One defines the essential spectrum σess pAq of a closed operator A to be the set of ζ P C for which the operator Aζ “ A ´ ζ1 is not Fredholm. (c) This definition covers the one in A.4(c),(d) for the case of self-adjoint operators and allows a non-empty residual spectrum. Note that stability A.4(e) of the essential spectrum is valid for perturbations K, which are A-compact, i.e., KRA pζq P C8 pBq, in Banach spaces. The nature of the essential spectrum of self-adjoint operator A in a Hilbert space is easy to describe: any λ P σpAq lies in σess pAq unless it is a point of σd pAq.

A.6

Spectrum of compact operators

(a) Compact operators constitute an instructive example to explain basic notions concerning spectra of closed operators. Recall that a bounded operator on a Banach space B is compact, K P C8 pBq, if it maps bounded sets of B into precompact sets, cf. with the completely continuous operators, see Notes to Section 2.1. By A.1(h) the spectrum σpKq belongs to the disc in C with the (spectral) radius rpAq ď }A}. Another general remark is that t0u Ď σpKq. Otherwise, K would be boundedly invertible: K ´1 K “ 1, with a compact operator in the lefthand side, which is obviously impossible. (b) Recall that by the Fredholm alternative: if the operator K P C8 pBq and ζ P Czt0u, then ζ P ρpKq Y σd pKq. Moreover, the spectrum σd pKq has no accumulation point different from ζ “ 0.

A.7. Example: the Volterra operator

265

(c) As we learned from examples of Section A.4(f) the classification of the spectral point ζ “ 0 may be a subtle matter. We consider it first in a Hilbert space H. A general result for self-adjoint compact operators K “ K ˚ follows from A.4 (d),(e). Let N “ λ 1, for λ “ 0, be the zero-operator. The spectrum of this bounded self-adjoint operator contains only one point, σpN q “ t0u, which is an eigenvalue of infinite multiplicity: σp8 pN q “ t0u. Hence, by A.4(d), σpN q “ σess pN q. Since the stability of essential spectrum A.4(e) yields σess pN q “ σess pN `Kq, one obtains the accumulation point t0u “ σess pKq. Note that this observation also agrees with the argument A.4(b),(c) since the accumulation point t0u belongs to the topological boundary BσpKq. (d) We stress the point that the arguments in (c) do not imply that σess pKq coincides with σp8 pN q. Examples A.4(1f) and A.4(2f) of two self-adjoint compact operators show that the accumulation point t0u does always belong to essential spectrum, but to its different parts. For the non-self-adjoint case, or for compact operators in Banach spaces, the classification of the spectral point t0u gets even more complicated, since in general one has t0u Ď σess pKq Y σres pKq. To this aim we consider in subsection A.7 the example of the Volterra operator. (e) We conclude by a remark concerning unbounded operators in a Banach space B. Let A be a closed operator in B such that the resolvent RA pζq exists and is compact for some ζ P C. Then RA pζq P C8 pBq for every ζ P ρpAq, the spectrum σpAq “ σd pAq, and the corresponding eigenvalues are such that limkÑ8 |ζk | “ 8. Since this implies that the spectral radius rpAq “ 8, the operator A is unbounded.

A.7

Example: the Volterra operator

(a) The Volterra operator V : H Ñ H on the Hilbert space H “ L2 r0, 1s is the integral operator żt ds upsq, u P dompV q “ L2 r0, 1s. pV uqptq :“ 0

To estimate the norm }V } we note that żt

ż1

2

a ˇ2 upsq ˇ cospπs{2q a cospπs{2q 0 0 * ż 1 "ż t żt 2 dt ds1 cospπs1 {2q ds2 |ups2 q| {cospπs2 {2q ď ˇ dtˇ

}V u} “

0

2 “ π

ds

0

ż1

0

ż1 dt sinpπt{2q|upsq|2 {cospπs{2q “

ds 0

s

ˆ ˙2 2 }u}2 . π

The norm }V } “ 2{π, corresponds to the vector u˚ psq :“ cospπs{2q.


266

(b) Recall that the linear operator A P LpHq belongs to the Hilbert-Schmidtclass if there exists an orthonormal basis ten unPN such that }A}22 :“

8 ÿ

}Aen }2 ă 8 .

n“1

Here }A}2 is the Hilbert-Schmidt norm of A, which (if finite) does not depend on the choice of the basis. Note that the set of Hilbert-Schmidt operators C2 pHq is a ˚ideal in LpHq and that it is also a subset of the compact operators: C2 pHq Ă C8 pHq. Moreover, if the operators A1 , A2 belong to C2 pHq, then the product A1 A2 belongs to the trace-class ˚-ideal C1 pHq Ă C2 pHq Ă C8 pHq. definition to the Volterra operator on L2 r0, 1s for the orthonorApplying this ? mal basis ten pxq “ 2 sinpπnxqunPZ , one finds that }V en }2 “ 3{pn2 π 2 q. Therefore, the Volterra operator, together with its adjoint V ˚ , are Hilbert-Schmidt operators, and as such they are compact. Note that V is not normal and that by explicit calculations, the self-adjoint operator V ˚ V P C1 pL2 r0, 1sq, with σd pV ˚ V q “ tλk “ 4{rp2k `1q2 π 2 sukě0 and σpV ˚ V q “ σd pV ˚ V qYt0u. Since }V ˚ V } “ }V }2 and since for self-adjoint operators the spectral radius rpV ˚ V q “ }V ˚ V } “ maxkě0 λk , we obtain }V } “ 2{π, as in (a). (c) The spectrum of the compact V has the canonical structure σpV q “ σd pV q Y t0u. To study the discrete part σd pV q one notes that żt |V n u|ptq ď

ds |upsq| 0

pt ´ sqn´1 1 ď }u} , pn ´ 1q! pn ´ 1q!

which implies that }V n } ď 1{pn ´ 1q! . Then the spectral radius rpV q “ limnÑ8 p1{pn ´ 1q!q1{n “ 0. Hence, V is a quasi-nilpotent compact operator with σd pV q “ H. Recall that if rpAq “ 0 for a self-adjoint operator, then A “ 0, cf. with (b) for A “ V ˚ V , but this is not the case for the Volterra operator. Although V is quasi-nilpotent and its spectrum contains only zero point, σpV q “ 0, one has }V } ‰ 0. Consequently, Proposition 2.45 (Lidskiˇı’s spectral trace theorem) yields Tr V “ ΛpV q “ 0, for a nontrivial set of matrix elements tpek , V ek qukě1 and for any choice of the orthonormal basis tek ukě1 in L2 r0, 1s. (d) To study the zero spectral point of non-self-adjoint V we recall that t0u Ď σp˚ pV q Y σcont pV q Y σres pV q, see A.4(f), A.6(d), and note that according to the definition in (a) the range ran V “ AC0 r0, 1s, is the set of absolutely continuous functions, which are zero at t “ 0. Now let ζ “ 0 P σp˚ pKq with the eigenvector u0 , i.e., one has V u0 “ 0. Since V u0 P AC0 r0, 1s, this implies that u0 “ 0 in L2 r0, 1s. Hence, σp˚ pKq “ H. Next we note that the closure of ran V in L2 r0, 1s gives AC0 r0, 1s “ L2 r0, 1s. So, the range of the Volterra operator Vζ“0 is dense in the Hilbert space H. Therefore, σres pV q “ H and thus σcont pV q “ t0u.

A.8. Example: unbounded operators

267

(e) The Volterra operator V : B Ñ B , on the Banach space B “ Cr0, 1s with the sup-norm } ¨ }8 , is defined by the same integral operator as in (a). Then for any f P Cr0, 1s and for the identity function r0, 1s Q t ÞÑ 1ptq “ 1, we obtain the estimates 1 1 |pV n f qptq| ď }f }8 and }V n } ě }V n 1}8 “ . n! n! Hence, the operator norm }V n } “ 1{n! and the spectral radius rpV q “ 0. Therefore, the spectrum of this bounded operator with }V } “ 1 is σpV q “ t0u. Note that the range ran V of this operator is the set AC0 r0, 1s of absolutely continuous functions such that pV f qpt “ 0q “ 0. Since |pV f qptq| ď }f }8 t1{2 and |pV f qptq ´ pV f qpsq| ď }f }8 |t ´ s|1{2 , then by the Arzela-Ascoli theorem the image V pB1 q of the unit ball B1 :“ tf P B : }f }8 ď 1u is a compact set. So, V is a compact operator on Cr0, 1s and σpV q “ t0u implies that σd pV q “ H. Now, to decide between the three disjoint possibilities: t0u “ σp pV q Y σcont pV q Y σres pV q, we claim first that t0u ‰ σp pV q follows along the same line of reasoning as in (d). Second, note that ranpVζ“0 q “ AC0 r0, 1s is not dense in Cr0, 1s in the } ¨ }8 -norm. Therefore, ran Vζ“0 ‰ B, and consequently t0u “ σres pV q.

A.8

Example: unbounded operators

By the resolvent arguments in A.1(g) spectra of bounded operators are always non-empty. This is not the case for unbounded operators as it is shown below by two extreme examples of operators with σ “ H and with σ “ C. (a) Let the unbounded operator A0 be defined in the Hilbert space H “ L2 r0, 1s on dom A0 “ W01,2 p0, 1q by pA0 uqpxq :“ í Bx upxq,

u P dom A0 .

W01,2 p0, 1q

The Sobolev space “ tu P ACr0, 1s : Bx u P L2 p0, 1q ^ up0q “ 0u. We note that dom A0 is dense in L2 r0, 1s and that the operator A0 is closed. By inspection one sees that for any ζ P C the equation pA0 ´ζ1qu “ 0 for u P dom A0 , has only trivial solution, that is, kerpA0 ´ ζ1q “ t0u. Therefore, σp pA0 q “ H and pA0 ´ ζ1q : dom A0 Ñ H is injective, see A.3(b). Since for any v P L2 r0, 1s there exists the vector uv P W01,2 p0, 1q (defined by the Volterra operator V : v ÞÑ uv ), the map A0 ´ζ1 is bijective: ran Aζ “ ran Aζ “ H. By A.3(c)-(e), this yields σcont pA0 q “ H and σres pA0 q “ H. Therefore, the spectrum σpA0 q is empty. (b) Let the unbounded operator A1 be defined in the Hilbert space H “ L2 r0, 1s by pA1 uqpxq “ íBx upxq, u P dom A1 “ W 1,2 p0, 1q. Here the domain is the Sobolev space W 1,2 p0, 1q “ tu P ACr0, 1s : Bx u P L2 p0, 1qu. Note that the operator A1 is closed. Since for any ζ P C we have pA1 ´ ζ1qeζ “ 0 for eζ :“ eiζx P dom A1 ,

268


kerpA1 ´ ζ1q ‰ t0u and by A.3(b) one gets that σpA1 q “ σp pA1 q “ C.

A.9 Spectral mapping theorem for semigroups (a) Let A be a closed unbounded operator in the Banach space B, with dense domain dom A Ă B. Recall that if ζ R σpAq, then the resolvent spectral mapping theorem holds: σpRA pζqq “ t0u Y tλ ´ ζq´1 : λ P σpAqu. (b) Let A be the generator of the strongly continuous semigroup tUt pAqutě0 on B. Recall that the relationship between the spectra σpAq and σpUt pAqq is not as simple as in the resolvent spectral mapping theorem. In general, one can prove for the C0 -semigroups only the following form of this theorem: σpUt pAqq Ě te´t z : z P σpAqu. One can prove a converse statement under some additional continuity assumptions on the C0 -semigroups. (c) To this aim we establish the following version of the spectral mapping theorem. Let the C0 -semigroup tUt pAqutě0 generated by the unbounded operator pt,α pAq :“ A be quasi-bounded of the type ω0 : }Ut pAq} ď M eω0 t , M ě 1. Let U Ut pAqRA pαq, where α R σpAq. Then for t ą 0 and <e α ă ´ω0 , we obtain pt,α pAqq “ t0u Y te´t z pz ´ αq´1 : z P σpAqu. σpU (d) The next result is due to the improved continuity condition. Suppose the C0 -semigroup tUt pAqutě0 is eventually norm-continuous, i.e., there exists t0 ą 0 such that t ÞÑ Ut pAq is a norm-continuous function for t ą t0 . Then σpUt pAqq “ te´t z : z P σpAqu. (e) Let us assume more: the C0 -semigroup tUt pAqutě0 is eventually compact, that is, there exists t0 ą 0 such that t ÞÑ Ut pAq P C8 pBq for t ą t0 . Then there exists a direct sum decomposition B “ B0 ‘ B1 such that: (1) The subspaces B0 and B1 are invariant under the semigroup tUt pAqutě0 . (2) dimpB0 q ă 8 and the restriction St :“ Ut pAq æ B1 satisfies the asymptotics }St } “ ope´βt q as t Ñ 8, for all β ą 0. (3) The spectrum σpAq “ σd pAq, and if dimpBq “ 8, then σpUt pAqq “ t0u Y te´t σpAq u. (4) The eigenvalues tzk P σd pAqukě1 are such that limkÑ8 <e zk “ 8.

A.10. Notes

269

For the proof one observes that the eventually compact C0 -semigroup tUt utě0 is in fact eventually norm-continuous. Indeed, let Ut0 P C8 pBq and let B1 :“ Ut0 pB1 q denote the closure of the compact image of the unit ball B1 Ă B. Since pt, f q ÞÑ Ut f is jointly continuous, for any given ε ą 0 there exists δ ą 0 such that }Ut f ´ f } ă ε for all f P B1 and 0 ď t ă δ. Let t0 ď t1 ď t ă t1 ` δ and }f } ă 1. Then }Ut f ´ Ut1 f } “ }Ut1 ´t0 pUt´t1 ´ 1qUt0 f } ď }Ut1 ´t0 } ε . Therefore, }Ut ´ Ut1 } ď }Ut1 ´t0 }ε and the C0 -semigroup tUt utě0 is normcontinuous for t0 ă t ă 8. (f) Counterexamples show that the opposite inclusion to the inclusion in (b) does not hold in general. However, there is a partial result in this direction. Let tUt pAqutě0 be a quasi-bounded C0 -semigroup on a Banach space B generated by the unbounded operator A. Then the converse to inclusion (b) concerns only the point and the residual spectra and reads as σp pUt pAqq “ t0u Y te´t z : z P σp pAqu, σres pUt pAqq “ t0u Y te´t z : z P σres pAqu. (g) More versions of the spectral mapping theorem one finds in Section 4.1: Lemma 4.9 and Propositions 4.11, 4.12, and also in Section 4.2: Propositions 4.21, 4.22.

A.10

Notes

Notes to Section A.1–Section A.4 and to Section A.6. Although standard, this Appendix A recalls some useful key notations, definitions and facts from the spectral theory. This will facilitate understanding the text of the book. In our presentation of Sections A.1–A.4 and partially of Section A.6 we followed essentially [BS87] (Chapters 3 and 9), as well as [HN01] (Chapter 9) and [Kat80] (Chapter 8, §1). Notes to Section A.5, Section A.6 and to Section A.9. In the presentation of Sections A.5, A.6, and A.9 we were motivated by Chapters 4 and 8 from [Dav07] and by [EN00] (Chapter IV).

Appendix B. More inequalities B.1

The Araki inequality

(a) Recall that if A P LpHq is a positive bounded self-adjoint operator on H, then by the spectral representation of A and the elementary formula ż sinpπθq 8 tθ´1 s sθ “ dt , s ą 0, 0 ă θ ă 1, π s`t 0 the fractional power of A admits the representation ż sinpπθq 8 tθ´1 A Aθ “ dt , 0 ă θ ă 1, π A ` t1 0

(B.1)

where equality (B.1) is in the sense of sesquilinear forms and we have pu, Aθ uq “ }Aθ{2 u}2 , for u P H. Lemma B.1 (L¨ owner-Heinz inequality). Let A, B P LpHq be positive operators such that 0 ă B ă A. Then B θ ă Aθ

for any 0 ă θ ă 1.

(B.2)

Proof. The condition 0 ă B ă A implies that for t ą 0 the operators: A ` t1 ą 0 and B ` t1 ą 0, and hence, pA ` t1q1{2 and pB ` t1q1{2 , are invertible. Moreover, for any u P H one also gets that pu, pB ` t1quq ă pu, pA ` t1quq. Hence, for u “ pA ` t1q´1{2 w and any w P H, pw, pA ` t1q´1{2 pB ` t1qpA ` t1q´1{2 wq ă pw, wq. Therefore, the operator Cptq :“ pA ` t1q´1{2 pB ` t1quqpA ` t1q´1{2 ă 1. is invertible. Then Cptq´1 ą 1 and consequently pA ` t1q´1 ă pB ` t1q´1 . © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1

(B.3) 271

Appendix B. More inequalities

272 By (B.3) it follows that

ApA ` t1q´1 “ 1 ´ tpA ` t1q´1 ą 1 ´ tpB ` t1q´1 “ BpB ` t1q´1 .

(B.4)

Therefore, representation (B.1) and inequality (B.4) give the L¨ owner-Heinz inequality (B.2) for positive bounded operators. l Corollary B.2. The inequality (B.2) is equivalent to }B θ{2 u}2 ă }Aθ{2 u}2 , u P H. This yields that }B θ{2 u} ă }Aθ{2 u}, for 0 ă θ ă 1. Therefore, if the positive bounded operators A, B satisfy }Bu} ă }Au}, for all u P H, then }Bu}2 ă }Au}2 , i.e. B 2 ă A2 . Now applying the inequality (B.2) one gets }B θ u} ă }Aθ u}, which is often also called the L¨ owner-Heinz inequality. 8 (b) Let X P C8 pHq be a compact operator and let σpXq “ tλj pXquj“1 be the corresponding sequence of eigenvalues, counting multiplicities. For X ě 0 we assume that λ1 pXq ě λ2 pXq ě . . . ě 0. In this case the singular values spXq “ tsj pXqu8 j“1 coincide with σpXq. Recall that two positive compact operators X, Y are said to be log-ordered , and one writes Y ă log X, if n ź j“1

λj pY q ă

n ź

λj pXq,

n “ 1, 2, . . . ,

j“1

see Definition 6.14. Proposition B.3 (Araki log-order inequality). For every two positive compact operators A, B P C8 pHq pB 1{2 AB 1{2 qr ă log B r{2 Ar B r{2 ,

(B.5)

r ą 1.

Proof. First, we define two invertible for ε ą 0 operators, Aε “ A ` ε1 and Bε “ B ` ε1. The spectra σpAε q and σpBε q consist of isolated eigenvalues tλj pAε q ą 8 8 εuj“1 and tλj pBε q ą εuj“1 of finite multiplicity and (eventually) the point ε “ λ8 pAε q “ λ8 pBε q, see Section A.6. Recall that ε ÞÑ λj pAε q and ε ÞÑ λj pBε q are continuous functions, including at ε “ `0. Second, we define for r ą 1 two positive bounded operators with discrete spectra: ´ ¯ Xε :“ Bεr{2 Aεr Bεr{2

and Yε :“ Bε1{2 Aε Bε1{2

r

.

(B.6)

8 Note that the spectra tλj pXε qu8 j“1 and tλj pYε quj“1 are also continuous functions of ε ě 0. For each n P N and bounded operator Z P LpHq with a discrete spectrum 8 we introduce the n-fold tensor product space Hbn “ H b H b ¨ ¨ ¨ b H tλj pZquj“1 and denote by Z bn “ Z b Z b ¨ ¨ ¨ b Z the corresponding tensor product of operators. Let E n : Hbn Ñ E n Hbn , be the projection onto the subspace of totally antisymmetric vectors. Then Z ba n :“ Z bn E n is the restriction of the operator

B.1. The Araki inequality

273

Z bn on this subspace. By this construction the largest eigenvalue ΛpZ ba n q of the operator Z ba n is given by ΛpZ ba n q “

n ź

(B.7)

λj pZq.

j“1

Therefore, the log-order (B.5) is equivalent to ΛpY ba n q ď ΛpX ba n q,

(B.8)

or, by the continuity of the spectra σpXεba n q and σpYεba n q at ε “ `0, to inequality ΛpYεba n q ď ΛpXεba n q.

(B.9)

To prove (B.9) we introduce on Hbn for ε ą 0 two axillary invertible and positive operators αε :“ Aε b Aε b ¨ ¨ ¨ b Aε

and βε :“ Bε b Bε b ¨ ¨ ¨ b Bε .

(B.10)

Since the tensor and operator products commute, restricting the operators (B.10) to the antisymmetric subspace E n Hbn one obtains the following two identities: βεr{2 αεr βεr{2 “ Xεba n

and pβε1{2 αε βε1{2 qr “ Yεba n .

(B.11)

Consequently, (B.11) yields for the corresponding largest eigenvalues, Λpβεr{2 αεr βεr{2 q “ ΛpXεba n q and

Λppβε1{2 αε βε1{2 qr q “ ΛpYεba n q. r{2

r{2

To proceed, we rescale the operator βε αεr βε

(B.12)

in such a way that

Λpβεr{2 αεr βεr{2 q “ 1.

(B.13)

We also note that the operators (B.10) are invertible and that (B.13) implies r{2 r{2 βε αεr βε ď 1. Then one gets αεr ď βε´r . Since 1{r ă 1, the Löwner-Heinz inequality (B.2) for θ “ 1{r yields αε ď βε´1 , whence (B.14) βε1{2 αε βε1{2 ď βε1{2 βε´1 βε1{2 “ 1. Since r ą 1, definition (B.11) and inequality (B.14) yield pβε1{2 αε βε1{2 qr “ Yεba n ď 1.

(B.15)

Therefore, (B.12) together with (B.13) and (B.15) imply (B.9). Then by the continuity of the spectra σpXεba n q and σpYεba n q at ε “ `0 we deduce (B.8), which proves the Araki log-order (inequality) (B.5) for compact operators A, B P C8 pHq. l Corollary B.4. Note that inequality (B.5) is equivalent to the log-order in the form pB q{2 Aq B q{2 q1{q ă log pB r{2 Ar B r{2 q1{r ,

0 ă q ď r.

(B.16)

Appendix B. More inequalities

274

B.2 The Araki-Lieb-Thirring inequality in symmetrically-normed ideals Proposition B.5. For every two positive compact operators A, B P C8 pHq one has the inequality: (B.17) }B 1{2 AB 1{2 qr }φ ď }B r{2 Ar B r{2 }φ , r ą 1, in the symmerically-normed ideals Cφ pHq. Proof. The result (B.17) follows from the Araki inequality (B.5) and Lemma 6.15. l Proposition B.6. Let ϕ : R0` Ñ R` 0 be a monotone increasing function such that ϕp0q “ 0 and ξ ÞÑ ϕpeξ q is convex in ξ P R. Then the inequality (B.17) can be generalised as follows: }ϕppB 1{2 AB 1{2 qr q}φ ď }ϕpB r{2 Ar B r{2 q}φ ,

r ą 1.

(B.18)

Now setting ϕptq “ tq for q ą 0 one obtains the Araki-Lieb-Thirring inequality }pB 1{2 AB 1{2 qrq }φ ď }pB r{2 Ar B r{2 qq }φ ,

r ą 1,

(B.19)

in the symmetrically-normed ideals Cφ pHq.

B.3 Notes Notes to Section B.1. The log-order inequality (B.5) was first proved by E. H. Lieb and W. Thirring in [LiTh76] for matrices as an inequality between traces. Using Araki’s arguments ([Ara90], Theorem 1) one can generalise the log-order inequality (B.5) to positive compact operators ([Hia95], Proposition 2.1), see Proposition B.3 and Corollary B.4. owner-Heinz inequality Similarly to [Ara90], our proof is based on the L¨ [BS87], Ch.10, and the arguments are close to the one developed in [LiSe10], Sec.4.5. Notes to Section B.2 Another generalisation due to Araki ([Ara90], Theorem 2) is the trace inequality Tr ϕppB 1{2 AB 1{2 qr q ď Tr ϕpB r{2 Ar B r{2 q,

r ą 1.

(B.20)

Recall that here the operator-valued function X ÞÑ ϕpXq P C1 pHq (in general for unbounded X ě 0) is such that ϕp0q “ 0 and ξ ÞÑ ϕpeξ q is convex in ξ P R0` . In Propositions B.5 and B.6 we extend the Araki-Lieb-Thirring inequality (B.20) from the trace norm to symmetrically-normed ideals Cφ pHq.

Appendix C. Kato functions The functions named above were introduced by T. Kato to extend the Trotter product formula to non-exponential approximants tpf ptA{nqgptB{nqqn uně1 . Here the Borel measurable functions f, g : r0, 8q Ñ r0, 1s are assumed to imitate the exponential function t ÞÑ e´t for t ě 0 in the vicinity of zero and monotonically decrease to zero at infinity. Soon, it became clear that to prove even the strong operator convergence of non-exponential Trotter product formulae one has to impose additional conditions on this natural and simple class. For example, to ensure the convergence for generators A and B subordinated by a relative smallness condition one has to tune correspondingly the properties of the generic Kato functions f, g P K. For the reader’s convenience Section C.1 collects a list of the key Kato functions, which are mentioned in the book, with comments and proofs that were omitted in the text. We note that the conditions for the convergence of Trotter-Kato product formulae are often formulated in the text not directly for the Kato functions f, g, but after mapping them into auxiliary functions f0 , g0 . The constructions and properties of the latter are presented in Section C.2. There are two kinds of properties to distinguish in the description of the Kato and auxiliary functions: the global behaviour, including at infinity, and the local behaviour in the vicinity of t “ `0. To this aim we introduce in Section C.3 the concepts of regularity and domination. There we define also the corresponding classes of Kato functions. In the Notes in Section C.4, one finds more details about the origin of particular classes of the Kato functions, as well as further references.

pβ C.1 Classes: K, Kα , K Since monomials in the product formulae are defined by at least two Kato functions, in definition below we consider a pair of them. Definition C.1. Two Borel measurable functions f, g, defined on R0` “ r0, 8q and © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1

275

Appendix C. Kato functions

276 which satisfy the conditions 0 ď f pxq ď 1, 0 ď gpxq ď 1,

f p0q “ 1, gp0q “ 1,

f 1 p`0q “ ´1, 1

g p`0q “ ´1,

(C.1) (C.2)

are called generic Kato functions. We denote the ring of monomials generated by f, g, including their fractional powers, by K. Elementary examples of the individual Kato functions and the monomials that, in turn, satisfy conditions (C.1) (or (C.2)) are: f pxq “ e´x , f pxq “ p1 ` qxq´1{q , gpxq “ p1 ´ | sin x|q and f pxqgpxq “ p1 ` qxq´1{q p1 ´ | sin x|q, f pxqgpxq “ e´x p1 ` qxq´1{q , f pxq1{2 gpxqf pxq1{2 “ e´x{2 p1 ` qxq´1{q e´x{2 , etc., for q ą 0. We introduced this class of Kato functions K in Section 5.2 preparing its specification Kα“1 for the case of pairs A and B subordinated by the Kato smallness condition of the operator B with respect to the operator A, with the relative bound b ă 1. Therefore, we recall first the definition of the class Kα for α P p1{2, 1s. Definition C.2. For each α P p1{2, 1s we denote by Kα the class of Kato functions f, g P K defined by the following conditions: xf pxq1{2α ă `8, xą0 1 ´ f pxq 1 :“ sup p1 ´ f pxqq ă `8, x xą0 ˇˆ ˙ ˇ ˇ 1 1 ˇˇ ˇ :“ sup ˇ f pxq ´ ă `8, 2ˇ 1 ` x x xą0 1 :“ sup p1 ´ gpxqq ă `8, x xą0 ˇˆ ˙ ˇ ˇ 1 1 ˇˇ :“ sup ˇˇ gpxq ´ ă `8. 1 ` x x2 ˇ xą0

C1{2α :“ sup C1 C2 S1 S2

(C.3) (C.4) (C.5) (C.6) (C.7)

To link the relative Kato smallness of B (b ă 1) with properties of the Kato functions f, g P Kα we consider also the additional condition b1{α C1{2α S1 ă 1,

(C.8)

which is indispensable for Proposition 5.8 (α “ 1) and for Proposition 5.27 (α P p1{2, 1q). Comments: 1. If f P Kα , then one evidently has C1{2α ě 1 and S1 ě 1. Then the left-hand side of (C.8) 0 ď b ă pC1{2α S1 q´1 ď 1. For large C1{2α ě 1 and S1 ě 1 the condition (C.8) is nontrivial for b. On the other hand, for f pxq “ gpxq “ e´x one gets C1{2α S1 “ 1.

C.2. Auxiliary functions and K˚

277

2. The proofs of the statements in Proposition 5.8 and Proposition 5.27 are valid for all Kato functions from Kα“1 and respectively from K1{2ăαă1 if the relative bound for the generator B is b ă 1. 3. The operator-norm convergence in the case α “ 1{2 requires the relative A-compactness of the operator B and Kato functions from the class K1 , see Proposition 5.30 and Notes to Section C.4. 4. Note that (C.3) and, hence the tuning condition (C.8), exclude from the class Kα such elementary function as f pxq “ p1 ` xq´1 , because in this case C1{2α “ 8. This is a consequence of the subordination of smallness between the generators A and B. Now let us consider the opposite case, when there is no subordination between A and B. Suppose that A, B are non-negative self-adjoint operators in H such that the operator sum C :“ A ` B is self-adjoint on the domain dom C :“ dom A X dom B. Recall (see Proposition 5.25) that the Trotter-Kato product formulae conp β“2 . verge in the operator norm for the Kato functions from the class K p β if: Definition C.3. We say that h P K (i) h : r0, 8q Ñ r0, 1s is a Borel-measurable function such that hp0q “ 1, and h1 p`0q “ ´1; (ii) there exist ε ą 0 and δpεq ă 1, such that hpsq ď 1 ´ δpεq for s ě ε, and rhsβ :“ sup są0

|hpsq ´ 1 ` s| ă 8, sβ

for 1 ă β ď 2.

p β are hpxq “ e´x and hpxq “ p1 ` a´1 xqá , The standard examples of h P K p β is larger than Kα . for a ą 0, where h denotes either f or g. The class K Comments: pβ . 5. In contrast to Kα , one has f pxq “ p1 ` xq´1 P K 6. For α “ 1 Definition C.2 yields as a local condition for f (and the same for g) the estimate |f pxq ´ 1 ` x| ď C˜2 x2 . This estimate coincides with that for rhsβ“2 .

C.2

Auxiliary functions and K˚

For many reasons, see Section 5.3, it is convenient (and in fact necessary) to use certain auxiliary functions f0 , g0 , associated with Kato functions f, g P K. To this aim and to study the properties of f0 , g0 we recall first their construction, see Definition 5.34.


278

Definition C.4. First we associate with the Kato functions f, g P K the pair of functions ˆ ˙ 1 ϕpxq :“ x´1 (C.9) ´1 and ψpxq :“ x´1 p1 ´ gpxqq . f pxq Here we assume that f pxq ą 0. Using (C.9), we define a pair of auxiliary functions by ˆ ˙ 1 0 ď ϕ0 pxq :“ inf ϕpsq “ inf s´1 ´ 1 ď 1, (C.10) 0ăsďx 0ăsďx f psq 0 ď ψ0 pxq :“ inf ψpsq “ inf s´1 p1 ´ gpsqq ď 1. 0ăsďx

0ăsďx

(C.11)

Here we make the convention that 0´1 :“ `8. Finally, we set for x P R` 0: " 1, if x “ 0, f0 pxq :“ (C.12) p1 ` xϕ0 pxqq´1 , if x ą 0, and

" g0 pxq :“

1, if x “ 0, p1 ´ xψ0 pxqq, if x ą 0.

(C.13)

Comments: 1. Since f P K, definitions (C.10) and (C.12) yield ˆ ˙ ˆ ˙ 1 1 ´ 1 ď t´1 ´ 1 ď 1, 0 ď ϕ0 pxq “ t´1 f0 ptq f ptq which implies 0 ď f pxq ď f0 pxq ď 1, x P R` 0 . Similarly (C.11) and (C.13) yield 0 ď gpxq ď g0 pxq ď 1, x P R` 0. Therefore, since for f P K it holds that limxÑ`0 f pxq “ 1, we get also that limxÑ`0 f0 pxq “ 1 and similarly limxÑ`0 g0 pxq “ 1. 2. Note that (C.10) implies for δ ą 0 that ϕ0 pxq :“ inf ϕpsq ě 0ăsďx

inf

0ăsďx`δ

ϕpsq “ ϕ0 px ` δq,

i.e., the function ϕ0 pxq is monotone non-increasing on R` 0 . Moreover, lim ϕ0 pxq “ lim

xÑ`0

inf ϕpsq “ lim ϕpxq “ 1,

xÑ`0 0ăsďx

xÑ`0

(C.14)

where limxÑ`0 ϕpxq “ limxÑ`0 p1 ´ f pxqq{x “ 1 by (C.9) and because of f P K. By Comment 1, (C.14), and definition (C.10), we obtain for the derivative ˆ ˙ 1 1 ´1 f0 p`0q “ lim x ´1 xÑ`0 f0 pxq ϕ0 pxq “ ´ lim ϕ0 pxq f0 pxq “ ´1. “ ´ lim xÑ`0 xÑ`0 1 ` tϕ0 pxq

C.3. Regularity, domination: Kr , KD , Ks-d , K1

279

3. Arguing in much the same way for the function g, we deduce that the function 1 ψ0 pxq is monotone non-increasing on R` 0 and the right derivative g0 p`0q “ ´1. 4. Summarising Comments 1.–3., we conclude that the auxiliary functions f0 , g0 associated with f, g P K are themself Kato functions. Moreover, the corresponding functions ϕ0 , ψ0 are monotone non-increasing, whereas the functions ϕ, ψ (C.9) a priori are not. x Consider for example f pxq “ e´x , x P R` 0 . Then ϕpxq “ pe ´1q{x, ϕ0 pxq “ 1, ´1 and f0 pxq “ p1 ` xq . 5. Besides the inequalities in Comment 1 that involve the functions f and f0 , we use in the text many other properties of them. For example: – 0 ď ϕ0 pxq ď 1 (C.10) implies that 1 ` xϕ0 pxq ď 1 ` x, for x P R` 0 , or by definition of f0 , that always f0 pxq ě p1 ` xq´1 . – For any f P K one has limnÑ8 f px{nqn “ e´x . 6. In order to prove Proposition 5.61 we restricted the set of Kato functions K by a condition, that is a modification of (C.9). This defines a news class of Kato functions K˚ , as below. Definition C.5. We say that f, g P K belong to the class K˚ , if the the pair of functions ϕ, ψ associated with f, g P K are such that ˆ ˙ 1 ´1 ϕpxq :“ s ´1 and ψpxq :“ x´1 p1 ´ gpxqq f pxq are monotone non-increasing for x P R` .

(C.15)

7. As noted in Comment 4. by adding the condition (C.15) one eliminates, for example, the exponential Kato function f pxq “ e´x , whereas f pxq “ p1 ` x{κq´κ P K˚ for 0 ă κ ď 1.

C.3

Regularity, domination: Kr , KD , Ks-d , K1

As we have observed (for example in Chapter 5) the important part of conditions ensuring the convergence of Trotter-Kato product formulae in norm topologies are formulated using auxiliary functions f0 , g0 . These conditions have an impact (restriction) on the choice of the set of original Kato functions f, g P K admissible for these assertions. An example dealing with this issue is dealt within Lemma 5.44 of Section 5.4. Let f P K and A be a non-negative self-adjoint operator in a (infinite dimensional) Hilbert space H. Let p1 ` Aq´1 be compact. Is the operator f0 pAq also compact ?


280

Note that since f0 pxq ě p1 ` xq´1 (Section C.2, Comment 5) and, hence, f0 pAq ě p1 ` Aq´1 , the converse statement follows easily. It turns out the affirmative answer to the question is possible for the class of regular Kato functions Kr Ă K, see Lemma 5.44. Definition C.6. Let f be a Kato function from K. We set bpxq :“ sup0ďsďx sf psq and rpxq :“ supsPrx,8q f psq, for x P R` . The Kato function f is called regular if limxÑ`8 bpxq{x “ 0 and 0 ď rpxq ă 1. Comments: 1. Note that, for instance, f pxq “ e´x and f pxq “ 1{p1 ` xq are regular Kato functions, whereas f pxq “ 1 ´ | sinpxq| is obviously not regular. Therefore, regularity controls the global behaviour of the Kato functions, including at infinity: xϕ0 pxq ě p1 ´ rpδqqx{bpxq Ñ `8, x Ñ 8, and δ ą 0. 2. It is instructive to compare this control at infinity with the ”tail” control of f P Kα (Comment 4. Section C.1) and of f P K˚ (Comment 7. Section C.2) One of the key ingredients of the lifting method (see Sections 6.1 and 6.4) is a control of the local uniform boundedness (6.6) of the sequence of TrotterKato product approximants in an appropriate norm-topology. To develop this observation, we introduce the notions of dominated and self-dominated functions. This concern f, g as well as the auxiliary functions f0 , g0 . Definition C.7. We say that two generic Kato functions f, g P K are dominated by ` D the Borel measurable functions f D : R0` Ñ R` : R` 0 and g 0 Ñ R0 , respectively, if f pqxq1{q ď f D pxq and gpqxq1{q ď g D pxq, 0 ă q ď 1. (C.16) We denote the sub-class of dominated Kato functions by KD . Comments: 3. It is trivial, but useful to note that any Kato function f P K is dominated by f D pxq “ 1 for x ě 0. A concept related to domination is self-domination. Definition C.8. The Kato functions f and g with f pqxq1{q ď f pxq and gpqxq1{q ď gpxq,

0 ă q ď 1,

(C.17)

xgpxq xf pxq ă `8 and S :“ sup ă `8. 1 ´ f pxq xą0 1 ´ gpxq

(C.18)

for x ě 0, are called self-dominated if C :“ sup xą0

We denote this class of Kato functions by Ks-d .

C.3. Regularity, domination: Kr , KD , Ks-d , K1

281

Comments: 4. Note that C ď C1{2α (C.3). Therefore, in contrast to Comment 4 in Section C.1, one has f pxq “ p1 ` xq´1 P Ks-d . Hence, for self-dominated Kato functions the ”tail” control is weaker than in Kα , see Comment 2. 5. By Comment 5 in Section C.1, and by (C.16) one obtains that e´x “ limqÑ0 f pq xq1{q ď f D pxq. Since e´x “ pe´qx q1{q , this means that if f D dominates the Kato function f , then it also dominates the exponential Kato function fˆpxq “ e´x , x ě 0. 6. Similarly, if f P Ks-d , then by (C.17) it dominates fˆpxq. 7. By Lemma 6.24, any function f P Ks-d has a unique representation as f pxq “ e´xhpxq , x ě 0, where h : R0` Ñ R0` is non-increasing and limxÑ`0 hpxq “ 1. 8. By this representation and by virtue of definition (C.10) ˆ ˙ ¯ 1 1 1 ´ thptq ϕ0 pxq “ inf ´ 1 “ inf e ´1 , 0ătďx t 0ătďx t f ptq which yields for x ě 0 f0 pxq “

1 1 1 ď “ ď 1. 1 ` xϕ0 pxq 1 ` xhpxq 1 ´ lnpf pxqq

Therefore, combining these estimates with Comment 1 in Section C.2, we obtain for any f P Ks-d the following chain of inequalities: 0 ď f pxq ď f0 pxq ď t1 ´ lnpf pxqqu´1 ď 1.

(C.19)

9. Recall an important property of self-dominated Kato functions, see Lemma 6.26. Let A be a non-negative self-adjoint operator in a Hilbert space H and let f P Ks-d . Then f pt0 Aq P Cφ pHq for some t0 ą 0, implies f0 pt0 Aq P C8 pHq. 10. Let s P Ks-d be self-dominated. Then f pxq ď spxq implies f pqxq1{q ď pspxqq1{q ď spxq, 0 ă q ď 1, that is, the function f is dominated by s. Definition C.9. The class of Kato functions f and g obeying for x ě 0 the conditions C :“ sup xą0

xf pxq xgpxq ă `8 and S :“ sup ă `8, 1 ´ f pxq 1 xą0 ´ gpxq

will be denoted by K1 .

(C.20)


282 Comments:

11. Functions of the class K1 appeared in the formulation of sufficient conditions for the operator-norm convergence of Trotter-Kato product formulae in Proposition 5.30 and Proposition 5.46. 12. Since these statements concern only the operator-norm topology, one does not need the first part of the assumptions of the dominance in Definition C.8.

C.4

Notes

Notes to Section C.1. Definition C.1 of the class K is due to T. Kato. It appeared for at the first time in [Kat74] and then in his seminal paper [Kat78] about what is now called the strong operator convergent Trotter-Kato product formulae for an arbitrary pair of non-negative self-adjoint generators. To lift this convergence to the trace-norm topology we introduced in [NZ90a], [NZ90b] Definition C.2 for α “ 1. In the framework of the class Kα“1 the operator-norm convergent TrotterKato product formulae with error bound estimate for the rate of convergence were proved in [NZ98]. The operator-norm convergence of the Trotter-Kato product formulae and the rate estimate in the class K1{2ăαă1 were established in [NZ99a]. It was subsequently shown that for the Trotter product approximants this estimate is optimal [Tam00]. The optimal rate of convergence for α “ 1 was established in [ITTZ01] for p β (Definition C.3), proposed by Takashi Ichinose and Hideo Tamura in the class K [IT01]. For α “ 1{2 the conditions for the operator-norm convergence of the TrotterKato product formulae were formulated in [NZ99b], see also Proposition 5.30. The class of admissible Kato functions considered there coincides with K1 , Section C.3, but there are no results about the error bound estimate. Counterexamples show that for α ă 1{2 the operator-norm convergence, in general, does not hold. Notes to Section C.2. The auxiliary functions ϕ and ψ (C.9) appeared already in the first paper [Kat74] by Kato on this subject. There he proved the strong operator convergence only for the class K˚ , Definition C.5, which unfortunately excludes the exponential function. In a subsequent paper [Kat78], Kato proposed a completely different proof that improves his result for the strong topology by extending the class K˚ to the class K. Under the conditions of Proposition 5.61 we need to restrict to the class K˚ for the proof of the trace-norm convergence, see [NZ90a], Lemma 3.2. In [NZ90b] and then in [NZ99d] we developed further the auxiliary functions control of K to produce other sufficient conditions: Kr , KD , Ks-d , K1 , permitting the exponential p β proposed in function. Another elegant way to do this is to consider the class K [IT01]. This way is closer to Kato’s original approach [Kat78] to the problem.

C.4. Notes

283

Notes to Section C.3. The concepts of regularity (Definition C.6), domination (Definition C.7), and self-domination (Definition C.8) originated in the analysis of sufficient conditions for the convergence of the Trotter-Kato product formulae in symmetrically-normed ideals by means of auxiliary functions in [NZ99d]. The leading idea is to optimise the choice of the admissible/appropriate Kato functions by taking into account the conditions on the pairs of generators A and B involved into the Trotter-Kato approximants. The ultimate aim is to ensure in this way the convergence in the corresponding norm topology. For example, in the definition of the class K1 (Definition C.9) one does not need the conditions corresponding to self-dominated functions (Definition C.8) since the convergence in Proposition 5.30 and Proposition 5.46 concerns only the operator-norm topology.

Appendix D. Lie-Trotter-Kato product formulae: comments on the bibliography D.1

From the strong to the norm convergence

We recall that since the work of Sophus Lie (1875) one knows that for any pair of finite square matrices A, B P MpRd q on the d-dimensional Euclidean space Rd , one has for n Ñ 8 the estimates › › ˘n › ›` (D.1) › e´tA{n e´tB{n ´ e´tpA`Bq › ď Op1{nq, ›` › › ´tA{2n ´tB{n ´tA{2n ˘n › ´ e´tpA`Bq › ď Op1{n2 q. e (D.2) e › e } ¨ } is ˘any norm on the matrix space MpRd q. Note that the power `Here ´tA{n ´tB{n n e e is called the Lie product (or the product approximant), and the ˘n ` power e´tA{2n e´tB{n e´tA{2n is called the symmetric (or symmetrised ) Lie product (or approximant). For the proof see Section 5.1 to understand the reason of the difference in the convergence rate between (D.1) and (D.2). Generalisations of the Lie-type product formula to infinite-dimensional spaces appeared in several papers by Yu. L. Daletskiˇi (see, e.g., [Dal60], [Dal61]) in connection with a path-integral representation (known as the Feynman-Kac formula) for solutions of operator differential evolution equations. But the first abstract result was due to H. Trotter [Tro59], who extended the Lie product formula to strongly continuous contraction semigroups on a Banach space B. The convergence of the Lie-Trotter product formula was proved in the strong operator topology. Let A and B be generators of contraction semigroups in a Banach space B; if the closure C of the operator sum A ` B is the generator of a contraction © Springer Nature Switzerland AG 2019 V. A. Zagrebnov, Gibbs Semigroups, Operator Theory: Advances and Applications 273, https://doi.org/10.1007/978-3-030-18877-1

285

Appendix D. Lie-Trotter-Kato product formulae

286 semigroup, then

˘n ` s-lim e´tA{n e´tB{n “ e´tC , nÑ8

t ě 0.

(D.3)

In [Nel64] E. Nelson pointed out the importance of the Trotter product formula for semigroups generated by Schrödinger operators, and for the proof of the Feynman-Kac formula. To him also belongs a simple proof of the Trotter theorem, in the case where A and B are two non-negative self-adjoint operators in a separable Hilbert space H such that the operator sum A ` B is also self-adjoint [Nel64]. It is striking that much later it was proved in [ITTZ01] that Nelson’s conditions ensure convergence of the product formula in the operator-norm topology, with an error bound estimate for the rate of convergence that is optimal, since it coincides with the rate in (D.1). A beautiful and elegant way to treat the Lie-Trotter product formula in the framework of the general theory of strongly continuous contractions is due to P. Chernoff [Che68, Che74]. In particular, he proposed a new way to consider perturbations (sums) of linear operators in a Hilbert space. Since the right-hand side of the Trotter product formula defines a semigroup, the corresponding generator can be associated with a ”sum” of the generators in the left-hand side, see [Che74], and [Far67a, Far67b], where perturbations of linear propagators were also considered. Further progress was achieved by T. Kato [Kat78]. In 1978 he obtained the following important result: Let A ě 0 and B ě 0 be non-negative self-adjoint operators in a separable Hilbert space H. Denote by H0 the subspace H0 :“ dom A1{2 X dom B 1{2 . .

It may happen that dom A X dom B “ t0u, but the form-sum H “ A ` B is well defined on the subspace H0 . Under these conditions the Trotter product formula converges strongly to the degenerate semigroup te´tH P0 utą0 , locally uniformly away from zero. That is, one has ` ˘n s-lim e´tA{n e´tB{n “ e´tH P0 , t ą 0, (D.4) nÑ8

uniformly in t P ra, bs, for any ra, bs Ă p0, `8q. Here P0 denotes the orthogonal projection from H onto H0 . Further, T. Kato [Kat74, Kat78] discovered that the Trotter product formula is valid not only for the exponential function e´x , x ě 0, but also for a whole class of Borel measurable functions f and g that are defined on r0, 8q and satisfy the conditions 0 ď f pxq ď 1, 0 ď gpxq ď 1,

f p0q “ 1, gp0q “ 1,

f 1 p`0q “ ´1, 1

(D.5)

g p`0q “ ´1.

(D.6)

t ą 0,

(D.7)

Kato proved that in this case one always has n

s-lim pf ptA{nqgptB{nqq “ e´tH P0 , nÑ8

D.1. From the strong to the norm convergence

287

uniformly in t P ra, bs, for any ra, bs Ă p0, `8q. These product formulae for pairs f, g are known as the Trotter-Kato product formulae for generic Kato functions f, g P K, [NZ90a, NZ90b]. More details about classes of Kato functions are provided in Appendix C. The next important step concerning the Trotter product formula is related to the following result by Dzh. L. Rogava in 1993 [Rog93]: Let A and B be two bounded from below self-adjoint operators in a separable Hilbert space H. If dom A Ă dom B and the operator C “ A ` B is self-adjoint, then › c lnpnq ›` ´tA{n ´tB{n ˘n › e e ´ e´tC › ď ? , n ą 1, (D.8) n locally uniformly in t P r0, T s , 0 ă T ă 8. This operator-norm convergence of the Trotter product formula with error bound estimate (D.8) is uniform in t ě 0 if the operators A and B are positive. Since Rogava discovered that the Trotter product formula may converge in the operator norm topology, the question of the error bound estimate and its dependence on the pair A, B becomes an interesting problem. Trying to understand the Rogava statement, the following result was established in [Zag95]: let A and B be two positive self-adjoint operators such that B is bounded, B P LpHq, then for self-adjoint C “ A ` B ›` ´tA{n ´tB{n ˘n › c plnpnqq2 › e , e ´ e´tC › ď n

n ą 1,

(D.9)

uniformly in t ě 0. It was conjectured that the improved error bound estimate (D.9) is owing to the boundedness of B. In 1997, T. Ichinose and Hideo Tamura [IT97] found that if A and B are two non-negative self-adjoint unbounded operators in a separable Hilbert space H such that dom Aα Ă dom B, 0 ď α ă 1, (D.10) then for self-adjoint C “ A ` B the error bound for the rate of convergence for the Trotter product formula is of the order Opn´2{p3`αq q, when n Ñ 8. Then in 1998, H. Neidhardt and V. A. Zagrebnov [NZ98] relaxed the IchinoseTamura operator smallness condition (D.10) and generalised the Trotter product formula to the Trotter-Kato product formulae for unbounded operator B with a better error bound estimate than (D.9). Namely, suppose A ě 1 and B ě 1 be self-adjoint operators such that unbounded operator B, with dom A Ă dom B, is Kato-small with respect to A for the relative A-bound b ă 1, that is, }Bu} ď a }u} ` b }Au} ,

u P dom A, a, b ě 0.

(D.11)

Then for self-adjoint C “ A ` B we proved that ›` › ˘ › f ptA{nqgptB{nq n ´ e´tC › ď c lnpnq , n

n ą 1,

(D.12)

288


uniformly in t ě 0, where the Borel measurable functions f, g P Kα satisfy, besides the generic Kato conditions (D.5) and (D.6), some additional requirements (5.15)– (5.20). For details about the sub-class Kα of the generic Kato functions K, see Appendix C. Note that the Ichinose-Tamura condition (D.10) on A and B implies that the relative bound b ą 0 is infinitesimally small, that is, B P P0` , see Section 4.4. In the subsequent paper [IT98a], T. Ichinose and H. Tamura improved their estimate to pc lnpnqq{n, and extended it to the case where the generator B “ Bptq is timedependent. Their condition (D.10) has to be uniform in t P r0, T s , 0 ă T ă 8, and in addition they require that › › ›A´α pBptq ´ BpsqqA´α › ď d |t ´ s| , d ą 0, (D.13) for positive operators A and Bp¨q, when t, s P r0, T s. This was the first Trottertype result about the operator-norm convergent product formula for propagators. The paper [IT98a] generalises a previous result in [Far67b] about the strong operator topology convergent product formula for propagators to the operator-norm topology under a time-dependent condition (D.10): dom Aα Ă dom Bptq,

0 ď α ă 1 for t P r0, T s .

D.2 Norm convergence: optimal rate In 1999, H. Neidhardt and V. A. Zagrebnov [NZ99a] considered, instead of (D.11), the optimal fractional power conditions on the pair of self-adjoint non-negative generators A and B for the Trotter-Kato product formula (D.12). In particular, they proved the following assertion: Let A ě 1 and B ě 0 be two self-adjoint operators such that B α is Katosmall with respect to Aα , with the relative bound b ă 1: }B α u} ď b }Aα u} ,

u P dom Aα , b ě 0,

(D.14)

and dom H α Ă dom Aα ,

(D.15)

.

for some α P p1{2, 1q, where H “ A ` B. Then ›` › ˘ › f ptA{nqgptB{nq n ´ e´tH › ď

c , n2α´1

n ą 1,

(D.16)

uniformly in t ě 0. Hiroshi Tamura [Tam00] proved that this estimate is optimal. More precisely, there exist operators A and B satisfying the statement above, such that one also gets ›` ´tA{n ´tB{n ˘n › ct › e e ´ e´tH › ě 2α´1 , for α P p1{2, 1q, (D.17) n

D.2. Norm convergence: optimal rate for large n, whereas

›` ´tA{n ´tB{n ˘n › › e ´ e´tH › ě dt , e

289

(D.18)

if α P p0, 1{2s. Here ct and dt are positive continuous functions for t ą 0. Note that the case α “ 1{2 is an example where the Trotter product formula does not hold in the operator norm, while it still converges strongly by Kato’s theorem [Kat78]. On the other hand, inequality (D.18) proves a conjecture in [NZ99a] that the operator norm convergence of (D.16), and of all other TrotterKato formulae, cannot be extended to the case α P p0, 1{2s without additional conditions. Some of these (sufficient) conditions were found by H. Neidhardt and V. A. Zagrebnov in [NZ99b, NZ99c]. Their first result concerns a generalisation of the main proposition by Chernoff (Theorem 1.1 in [Che74]) to the operator norm topology. This result gives a criterion for the convergence of the Trotter-Kato product formulae in the operator norm, which implies the following assertion: Let A ě 0 and B ě 0 be two self-adjoint operators in a separable Hilbert space H. The Trotter-Kato product formulae converge in the operator norm topology if one of the following conditions is satisfied [NZ99b]: (i) p1 ` Aq´1 P C8 pHq; (ii) A ě 1 and B 1{2 A´1{2 P C8 pHq; (iii) p1 ` Aq´1 p1 ` Bq´1 P C8 pHq. In particular, condition (ii) ensures the operator norm convergence of the Trotter-Kato formula (D.16) for α “ 1{2. odinger semigroups on The abstract condition (iii) covers the case of Schr¨ H “ L2 pRd q with potentials increasing at infinity considered in the papers [Hel95, DS97, ITak97, DIT98, Tak97, ITak98, ITak00], giving a rate not better than order Op1{nq, see review in [IT09]. Note that in these references neither p1 ` Aq´1 , nor p1 ` Bq´1 are compact, but their product is. Note also that the operator norm convergence was proven in [NZ99b] without error bound estimate. The optimality of the convergence rate was elucidated by T. Ichinose, Hideo Tamura, Hiroshi Tamura, and V. A. Zagrebnov in [ITTZ01]. This was the next after [NZ99a] result about the optimal convergence rate for the Trotter-Kato product formulae. In [ITTZ01] we proved the following statement: Let A and B be non-negative self-adjoint operators in a Hilbert space H such that C “ A ` B is also self-adjoint. Suppose, in addition to (D.5) and (D.6), that the Borel functions f and g are decreasing and satisfy the conditions |f pxq ´ 1 ` x| ă 8, xβ xą0 |gpxq ´ 1 ` x| sup ă 8, xβ xą0

sup

(D.19) (D.20)

p β“2 . For details about the Ichinose-Tamura class K pβ for β “ 2, that is, f, g P K of the Kato-functions, see Appendix C. Then the Trotter-Kato product formulae

290


(D.12) converges in the operator norm topology uniformly in t P r0, T s, for 0 ă T ă 8, with the upper error bound estimate Op1{nq, instead of Oplnpnq{nq. The optimality of the rate Op1{nq is shown in [ITTZ01] via simple noncommutativity of generators A and B, which gives Op1{nq for the estimate of the Trotter-Kato product formulae (D.12) from below, this means that the rate Op1{nq is sharp. Similarly to [NZ99a], the same conclusion is also valid (sic!) for symmetric Trotter-Kato approximants with the same rate of convergence Op1{nq . Based on an example from Tamura’s optimal theorem [Tam00], it was shown in [ITTZ01] that symmetrisation itself does not improve the rate of convergence because there p 2 satisfying the conditions are operators A, B and (exponential) functions f, g P K of [ITTZ01] such that for t ą 0 and n ą 1 one has › › ›pgptB{2nqf ptA{nqgptB{2nqqn ´ e´tC › ě Lptq{n , Lptq ą 0. (D.21) This is an unexpected result if one compares it with the Lie product formula (D.2), which improves the rate of convergence after symmetrisation. Therefore, the rate of convergence of the Trotter-Kato product formulae under the [ITTZ01] conditions is optimal. This abstract result covers almost all cases of the Trotter-Kato product formulae for self-adjoint semigroups mentioned above. The only exception is the one where the Neidhardt-Zagrebnov fractional power conditions are involved [NZ99a]. We note also that condition [ITTZ01]: the operator C :“ A ` B is selfadjoint if A and B are self-ajoint, is quite subtle and cannot be relaxed. In the paper [Tam00], Proposition 3, Hiroshi Tamura constructed example in which the operator-norm convergence of the Trotter product formula does not hold (cf.(D.18)): › › ›pgptB{2nqf ptA{nqgptB{2nqqn ´ e´tC › ě Dt , Dt ą 0, (D.22) even though the operator sum C “ A ` B, is essentially self-adjoint and the self-adjoint operator B is A-form bounded with a relative bound less than one. Of course, in this case the product formula (D.22) still converges strongly according to Kato’s theorem [Kat78], or by Trotter’s theorem [Tro59] for exponential functions f, g. Another result by T. Ichinose, H. Neidhardt, and V. A. Zagrebnov [INZ03, INZ04] (INZ) is also proved with the optimal error bound. on a Hilbert space Let A and B be non-negative self-adjoint operators . such that their densely defined form-sum H “ A ` B obeys the condition dom H α Ď dom Aα X dom B α , for some α P p1{2, 1q. If in addition A and B satisfy dom A1{2 Ď dom B 1{2 , then the nonsymmetric and symmetric Trotter product formulae converge in the operator norm: ›` › ˘ › e´tB{2n e´tA{n e´tB{2n n ´ e´tH › “ Opn´p2α´1q q, (D.23) ›` ´tA{n ´tB{n ˘n › ´tH › ´p2α´1q › e q, (D.24) e “ Opn é

D.2. Norm convergence: optimal rate

291

uniformly in t P r0, T s, 0 ă T ă 8 as n Ñ 8, both with the same optimal error bound. The same holds true if one replaces the exponential function in the ˆ β , see Appendix C. products (D.23), (D.24) by functions from the Kato class K The INZ result improves the previous one in [NZ99a] by relaxing the assumption of smallness of B α with respect to Aα to the milder assumption dom A1{2 Ď dom B 1{2 and by extending the admissible class of the Kato functions ˆβ. from Kα to K If instead of dom A1{2 Ď dom B 1{2 one has a stronger condition: B 1{2 is relatively compact with respect to A1{2 (condition (ii) of [NZ99b], see above), then we have operator norm convergence of the Trotter-Kato product formulae, but without any error bound estimate. In [NZ99b] we conjectured that instead of the relative compactness the relative form boundedness with the relative bound b “ 0` , is sufficient for this convergence. The optimality problem was treated by Ichinose and Tamura in [IT04] for odinger C0 -semigroups on H “ L2 pRd q. This was an important step in the Schr¨ resolution of the [ITTZ01] no-go problem (D.21) concerning the improvement of the convergence rate by symmetrisation of the Trotter product formula. In [IT04] it is proved that for generators A “ ´∆ and B “ V , where V is a non-negative function Rd Q x ÞÑ V pxq growing polynomially at infinity and hence verifying the conditions of Proposition 5.19, the symmetric Trotter product formula converges in the operator norm with the rate Opn´2 q. This result advises against a prejudice that for any pair of noncommuting unbounded operators A and B verifying the conditions of Proposition 5.19 the rate of convergence Opn´1 q is necessarily the best possible for the symmetric Trotter product formula. In fact, if one imposes on A, B additional conditions (here p1 ` Aq´1 p1 ` Bq´1 P C8 pL2 pRd qq) compared with the [ITTZ01] Proposition 5.19 (subsection 5.2.4), then the symmetrisation could improve the rate of convergence up to the asymptotic Opn´2 q. Moreover, in [AzIch08], taking a simple example of the one-dimensional harmonic oscillator: H “ A ` B, with A “ ´Bx2 , B “ V pxq “ x2 in H “ L2 pRq, it is proved by explicit calculations that the estimate of the symmetric Trotter formula from below is also of the order Opn´2 q:

›` › ˘n c C ď › e´tB{2n e´tA{n e´tB{2n ´ e´tH › ď 2 . 2 n n

Here 0 ă c ă C, locally uniformly for t ě 0. Hence the rate Opn´2 q is sharp, and so optimal. A natural hypothesis is that the same rate Opn´2 q is optimal for any non-negative growing polynomially at infinity potential V pxq. For more details see review in [IT09].

292


D.3 Norm convergence: non-self-adjoint semigroups and Banach spaces On the other hand, we are actually rather far from the optimal error bound estimates for the rate of convergence of product formulae in the case of non-self-adjoint semigroups, or of semigroups on a Banach space. In paper [CZ01a] V. Cachia and V. A. Zagrebnov proved the following result: Let A generate a holomorphic contraction semigroup on a Banach space B and let B be the generator of a contraction semigroup such that for some α P r0, 1q, dom Aα Ď dom B

and

dom A˚ Ď dom B ˚ .

(D.25)

Then operator C “ A ` B is generator of the holomorphic semigroup and the Trotter product formula converges in the operator norm with the error bound estimates ›` › ˘ › e´tA{n e´tB{n n ´ e´tC › ď Mαą0 lnpnq , n ą 1, (D.26) n1´α ›` ´tA{n ´tB{n ˘n › plnpnqq2 › e e ´ e´tC › ď Mα“0 , n ą 1, (D.27) n locally uniformly in t ě 0. We note that in the case α “ 0 the operator B in (D.25) is by definition bounded : B P LpBq. The asymptotics Opplnpnqq2 {nq in (D.27) was established by the same argument as in [Zag95] for a Hilbert space. This argument was not based on the self-adjointness as the proof in [NZ98], but on analytic extension of the semigroup te´tA utě0 . So, it was not unexpected that the rates of convergence in (D.9) and in (D.27) coincide. If α ą 0, then (D.25) (similar to the Ichinose-Tamura condition (D.10)) implies that the operator B is infinitesimally A-small, i.e., B P P0` . But in contrast to the improved self-adjoint estimate pc lnpnqq{n, [IT98a], the rate of convergence (D.26) in the non-self-adjoint setting rests less sharp and it is α-dependent. In [CZ99] the same authors showed that some results of [NZ99a], [NZ99b], and [NZ99c] can be extended to m-sectorial generators A and B, but without error bound estimates. In [CZ01b] they introduced the notion of quasi-sectorial contractions and have shown that for them the Chernoff approximation theorem [Che68] is valid in the operator norm topology. The Trotter product formulae for a class of non-self-adjoint contraction semigroups with error bound estimate was proved by V. Cachia, H. Neidhardt, and V. A. Zagrebnov in [CNZ01, CNZ02]. In [CNZ01], they proved the following generalisation of [NZ98]: Let A be a positive self-adjoint operator, and let B be an m-accretive (i.e., closed and maximal accretive) operator such that B and B ˚ are Kato-small with respect to A, with relative bounds less than 1. Then the Trotter product formulae converge in the operator norm topology with error bound estimate Oplnpnq{nq, uniformly in t ě 0.

D.4. Trace-norm convergence

293

In [CNZ02] it is proved that if one replaces the above conditions on the m-accretive operator B by the fractional conditions: dom pH ˚ qα Ď dom Aα X dom pB ˚ qα ‰ t0u,

(D.28)

for some α P p0, 1s, then the Trotter product formula converges in the operator norm with error bound estimate Opn´α lnpnqq uniformly in t ě 0. A similar estimate is also true for H ˚ “ pA ` Bq˚ , that is, for adjoint semigroups generated by A “ A˚ , B ˚ , and H ˚ . Note that in the latter case we do not require the smallness of B ˚ with respect to A as in [CNZ01], but the weaker condition (D.25). We must stress the point that just the A-smallness of B (even with relative bound less than 1) is not sufficient for the existence of a nontrivial operator A`B ˚ . The paper [CNZ02] starts with an example in which for these conditions one gets dom A X dom B ˚ “ t0u. So, in this case H ˚ ‰ A ` B ˚ . For more details see a nice short book [Ca10]. Recently we revised the operator-norm convergence (D.26), (D.27) of the Trotter product formula on a Banach space [NSZ18a]. The operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is an infinitesimally small generator of a contraction semigroup (D.26), in particular, if B is a bounded operator (D.27). Inspired by our studies of the evolution semigroups in Banach spaces, we show that in general the operatornorm convergence of the Trotter product formula fails, even for bounded operators B, if A is not generator of a holomorphic semigroup. We proposed examples that in this case by varying B the rate of the operator-norm convergence of the Trotter product formula can be made arbitrary slow. For further reading we suggest a short review [NSZ18b].

D.4

Trace-norm convergence

Since the present book is about the Gibbs semigroups, which are trace-norm continuous away from t0 ě 0, the bibliographic comments about this topology are dispersed throughout the Notes at the end of Chapters 4–6 and Chapter B.3. So, I would like to add here only few more comments on the question of convergence of semigroups and product formulae in the trace-norm topology. As far as I know, it was D. Uhlenbrock, [Uhl71], who introduced for the first time the notion the Gibbs semigroups (Definition 4.1). He was motivated by the one-parameter Gibbs exponential for the density-matrix operator in Quantum Statistical Mechanics. Then this concept was supported in papers [ANB75] and [Zag80]. The problem of the possible convergence of the Trotter product formula for the Gibbs semigroups in the trace-norm topology was suggested to the author by Robert A. Minlos. A positive answer in the quantum statistical mechanics setup for the Schr¨ odinger (Gibbs) semigroups was obtained in [Zag88]. The first abstract


294

result was due to H. Neidhardt and V. A. Zagrebnov [NZ90a, NZ90b]. We proved the following assertion: Let A ě 0 and B ě 0 be two non-negative self-adjoint operators in a separable Hilbert space H. Let H0 denote the subspace H0 :“ dom A1{2 X dom B 1{2 . Further, let f, g be the Borel functions defined on r0, 8q satisfying conditions: 0 ď f pxq ď 1,

f p0q “ 1,

f 1 p`0q “ ´1,

0 ď gpxq ď 1,

gp0q “ 1,

g 1 p`0q “ ´1,

that is, they are generic Kato functions: f, g P K, Appendix C. Then the TrotterKato product formula n

} ¨ }1 - lim pf ptA{nqgptB{nqq “ e´tH P0 , nÑ8

t ą 0,

(D.29)

converges locally uniformly away from zero in the trace-norm topology on C1 pHq if f0 ptAq P Cp pHq, for t ą 0, and 1 ď p ă 8. Here P0 is the orthogonal projection from H onto H0 and the function f0 : R` Ñ R` is completely determined by f in (C.12), Appendix C. For example, if f pxq “ e´x , then f0 pxq “ p1 ` xq´1 . Therefore, the operator A is a p-generator since p1 ` tAq´1 P Cp pHq for t ą 0, Definition 4.26. We remaind that, incidentally, (D.29) implies the operator-norm convergence. Thus, the operator-norm convergence of the product formulae for Gibbs semigroups was establishes before [Rog93]. The trace-norm Trotter product formula with error bound estimates of the convergence rate appeared first for the case of the Schrödinger (Gibbs) semigroups in [IT98b]. Then estimates of the rate of trace-norm convergence for the TrotterKato product formulae and abstract Gibbs semigroups were obtained in [NZ99d] and in [CZ01c]. They were established by lifting the corresponding operator-norm error bound estimates. We consider this lifting method in Chapter 5. The study of the convergence of the Trotter-Kato product formulae in general framework of symmetrically-normed ideals was initiated by the paper [NZ99d]. Therein a programme for treating the case of the Dixmier ideal was mentioned in concluding remarks. Chapter 7 is a partial realisation of this programme, see also [Zag19].

D.5

Unitary product formulae

This book is about the Gibbs semigroups, which are trace-norm continuous away from t0 ě 0. Therefore we do not discuss product formulae that involve, or have

D.5. Unitary product formulae

295

as their limits, the unitary groups of operators. Thus, we conclude this appendix by only few remarks to indicate some known results about the unitary product formulae. The first one is due to H. Trotter [Tro59]. Proposition D.1 (Unitary Trotter product formula). Let A, B and C be self-adjoint generators of unitary groups on H. Suppose that algebraic sum Cu “ Au ` Bu,

(D.30)

is valid for all u P D, where D Ă pdom A X dom Bq is a core of the operator C, i.e., the operator A ` B is essentially self-adjoint. Then the unitary group tUC ptq “ eítC utPR can be approximated in the strong operator topology on H by the Trotter product formula eítC u “ lim peítA{n eítB{n qn u, nÑ8

u P H,

(D.31)

for all t P R. Here generator C :“ pA ` Bq æ D is the closure of the algebraic sum (D.30). Note that similar to the case of semigroup Trotter product formula (D.3), the unitary product formula (D.31) generally does not hold in the operator norm. There are examples showing that conditions of Proposition D.1 are neither sufficient, nor necessary for convergence of the unitary product formula in the operatornorm topology, see [Ich03], [Che74] §5. Since the unitary groups involved in (D.31) do not belong to the trace-class C1 pHq, this note is also valid for convergence in the trace-norm topology. Recall that a systematic treatment of convergence of the unitary Trotter product formula in the operator-norm topology can be based on the commutator conditions [IT04c]. Let A and B be self-adjoint operators in a Hilbert space H with domains dom A and dom B such that the operator sum A ` B is essentially self-adjoint on dom AXdom B. Both A and B may not be semi-bounded, that is, neither bounded form below, nor from above. We denote the unique self-adjoint extension, which is the closure of A ` B, by C :“ A ` B. Here the case when the A ` B itself is already self-adjoint is included. Assume that there exists a dense set D Ă dom A X dom B such that (d1) eítA , eítB : D Ñ D for t P R. (d2) There exists a symmetric operator E on D such that ` ˘ i pBu, Avq ´ pAu, Bvq “ pEu, vq, for every u, v P D. Remark D.2. The operator E is formally represented in the form of the commutator E “ irA, Bs “ ipAB ´ BAq.


296

Proposition D.3. Let A and B be self-adjoint operators with domains dom A and dom B, respectively. Suppose that operator A ` B is essentially self-adjoint on dom A X dom B and denote by the C its unique self-adjoint extension. If conditions (d1) and (d2) are satisfied and operator E in (d2) admits a bounded extension on H, then ›` ítA{n ítB{n ˘n › › e (D.32) e ´ eítC › “ Opn´1 q, ›` ítB{2n ítA{n ítB{2n ˘n › › e e ´ eítC › “ Opn´1 q, (D.33) e as n Ñ 8, locally uniformly in t P R. Since the unitary groups involved in (D.32), (D.33) do not belong to trace-class C1 pHq, the Proposition D.3 does not allow us to use the lifting method of Section 5.2 for improving the operator-norm to the trace-norm convergence. To proceed further with analysis of unitary versus semigroup product formulae we recall the Kato result (D.4) for H0 “ H and H “ C, ˘n ` (D.34) e´tC “ s-lim e´tA{n e´tB{n , t P R` . nÑ8

In this case intersection dom A1{2 Xdom B 1{2 is dense in H and self-adjoint operator . C :“ A ` B is the form-sum of A and B. Now one can poses a question, whether the Trotter product formula (D.34) remains valid for the imaginary parameter it, t P R, see [Che74, Remarks pp. 90-91] and [JL00, Problem 11.3.9]. Note that if A and B are non-negative selfadjoint operators in H and the limit in the right-hand side of (D.31) exists for all t P R, then dom A1{2 X dom B 1{2 is dense in H , see [JL00, Proposition 11.7.3]. Moreover, applying Proposition D.1 we find that formula (D.31) is valid if the operator C “ A ` B, i.e. the closure is self-adjoint. However, if A ` B is not essentially self-adjoint, then all attempts to verify the Trotter product formula (D.34) for t ě 0 substituted by it, t P R, have failed so far. One of the ways out was proposed by T. Ichinose in [Ich80]. His main idea was to find for product formulae some appropriate non-exponential approximants a la Trotter-Kato. In [ENZ11] we generalised this scheme and proved that for non` negative self-adjoint operators A and B, such that the intersection dompA1{2 q X dompB 1{2 q is dense in H one gets ` ˘n eítC “ s-lim φptA{nqψptB{nq , (D.35) nÑ8

locally uniformly in t P R, where φ and ψ are some admissible functions such that <epφpxqq ě 0, =mpφpxqq ď 0 and =mpψpxqq ď 0 for x P R` . The admissible functions are defined by the properties |ϕpxq| ď 1,

x P r0, 8q,

ϕp0q “ 1,

and ϕ1 p`0q “ í,

cf. Kato functions K in Appendix C (Section C.1). These functions were introduced in [EINZ07] in the context of the Zeno product formula, which is tightly related

D.5. Unitary product formulae

297

with the general unitary product formulae. The typical examples are: ϕpxq “ eíx and ϕpxq “ p1 ` ixq´1 , for x P R. Another way to legitimate the unitary product formulae is to consider their convergence in topology, which is different than the operator strong, or norm, topology. In [ENZ11] we show that the unitary Trotter product formula makes sense in the L2 -topology, that is, it holds żT › ›2 ˘n › ›` lim › eítA{n eítB{n h ´ eítC h› dt “ 0

nÑ8 ´T

(D.36)

for h P H and any T ą 0, and for the standard conditions: A, B are non-negative . self-adjoint operators in a Hilbert space H such that the form-sum A ` B is a densely defined operator and self-adjoint operator C is its closure. Using the concept of the holomorphic Kato functions [ENZ11] we also proved the Trotter-Kato product formula in the L2 -topology. For example, under the same conditions on the operators, we obtain żT › ›2 ˘n › ›` g pitB{nq h ´ eítC h› dt “ 0 lim › frpitA{nqr

nÑ8 ´T

(D.37)

for h P H and any T ą 0. Here f, g are holomorphic Kato functions and fr, gr are Borel measurable extensions of f and g on the imaginary axis, see [ENZ11] for details. Similarly one gets the Trotter-Kato product formulae for other approximants.

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Index C0 -semigroup, 6, 101 ˚-ideal, 43, 45 p-generator, 115, 191 analytic extension method, 192, 207 analyticity operator-norm, 200 sectorial forms, 195 trace-norm, 133, 201 weak, strong, uniform, 198 approximants non-exponential, 275 product formulae, 147 symmetrised, 164, 165 Trotter, 165, 192, 199 Trotter-Kato, 164, 199 approximate spectrum, 262 approximats nonsymmetrised, 146 symmetrised, 146 Banach space, 1, 59, 101, 292 Cp , 64, 65 dual, 9, 19 of bounded operators, 64 weak convergence, 19, 78 weak topology, 19 Bochner integral trace-norm convergent, 129 Borel function, 151, 286, 288 canonical form of compact operator, 50, 55, 65, 86 Chernoff approximation theorem, 167, 171, 292

closure of operator sum, 285 compact operator, 42 canonical form, 50, 55, 65, 86 eigenvalues, 47 self-adjoint, 49, 51 singular values, 51 spectrum, 47, 49, 110 trace class, 52 compact resolvent, 111, 117, 200 compact semigroup, 105, 140 continuity in operator norm, 146 of absolute value, 74 of multiplication, 70, 74, 99 strong, 2, 5, 38 uniform, 15 weak˚ , 21 continuous spectrum, 30, 46 contraction semigroup, 6, 35 generator, 17, 32, 38, 117, 148 strongly continuous, 286 convergence away from t0 , 184 eventually Gibbs semigroup, 100, 186 away from zero, 22, 89, 183 uniformly, 189, 205 in Cφ -ideal, 222 in operator norm, 67, 68, 70 in weak operator topology, 66, 67 locally uniformly, 75, 210 away from t0 , 169 away from zero, 170, 191 of Dixmier traces, 253


309

310 strong operator, 67, 106, 222 trace-norm, 207 trace-norm: without rate estimate, 185 vector norm, 42 weak in Hilbert space, 42 convergence rate optimal in Cφ pHq-norm, 241 optimal in operator norm, 160, 162 optimal in trace norm, 182, 186, 207 optimal ITTZ, 206, 290 optimal of Dixmier traces, 254 optimal, fractional conditions, 163, 166, 207, 288, 291 core for operator, 14 of spectrum, 260 Darboux-Riemann sum, 107 deficiency, 28 density matrix, 140 differentiability in operator norm, 1, 17 of vector-valued functions, 4 dilation D2 , 249 D2 -invariance, 249 Dixmier ideal, 245 trace, 245 dominance property, 216 dominated convergence theorem in Cp pHq, 66, 78 in the lp -space, 69 Lebesgue, 77 error bound estimate, 160, 292 in the Cφ pHq-topology, 238 in the operator-norm, 203, 287 in the trace-norm topology, 186

Index operator-norm estimate, 237, 287, 289 optimal estimate, 150, 161, 182, 292 essential spectrum, 31 Euler formula, 2, 39 exponential operator-valued, 186 strongly continuous, 2 family holomorphic of bounded operators, 23, 198 holomorphic of type (A), 36, 96, 134, 137, 196 holomorphic of type (a), 196 holomorphic of type (B), 197 } ¨ }1 -uniformly bounded, 202 of m-sectorial operators, 202 Feynman-Kac formula, 285 form closed, 194 closed sectorial, 193, 197 holomorphic, 195, 197 non-negative, 117, 194 representation theorem, 194 sectorial, 195, 197 sesquilinear, 271 closed sectorial, 117, 193, 194 symmetric, 117, 194 form-sum, 117, 194, 286 fractional power conditions, 288, 290, 293 function (symmetric) norming, 213 Borel, 148, 151, 286 convex, 83 norm holomorphic, 1 operator-valued exponential, 1, 5, 149, 286 holomorphic, 16, 198 norm differentiable, 15, 16 symmetric norming, 210 continuous, 214

Index

311 normal, 214

Gel’fand transform, 103 generator, 11, 103 m-accretive, 32 m-sectorial, 141, 192, 207 p-generator, 115, 141 core, 14, 121 numerical range, 39 of class H pθ, ω0 q, 27 of class Qp1, 0q, 35 of class QpM, ω0 q, 12–14 of compact semigroup, 114 self-adjoint, 113, 114 of contraction semigroup, 17, 32, 38, 117, 148 of eventually Gibbs semigroup, 116 of Gibbs semigroup, 101, 115 of holomorphic semigroup, 23, 32, 117, 148, 197 of quasi-bounded semigroup, 27, 32, 127 of strongly continuous semigroup, 11 resolvent of, 17 self-adjoint, 36 spectrum, 23, 101 Gibbs semigroup, 20, 28, 66, 99 p-generator, 115, 116 self-adjoint, 115 degenerate, 89, 116, 182, 185 eventually, 100 generator, 101, 115 holomorphic, 99, 132, 200 immediately, 100 non-self-adjoint, 99 product formulae, 182 quasi-bounded, 101 self-adjoint, 130 self-adjoint, 99, 141 trace-norm continuity, 100 trace-norm holomorphic, 115 Ginibre-Gruber inequality, 141

Gram-Schmidt orthogonalisation, 47, 48 graph norm, 14 Hölder inequality in Cp pHq, 63, 90 Hausdorff theorem, 28 Hilbert space, 1, 20, 28, 41 inner product, 20 norm, 41 separable, 1, 287 Hilbert-Schmidt theorem, 57 Hille-Yosida theorem, 39 Hille-Yosida-Phillips theorem, 39 holomorphic function norm, 1 trace norm, 128 holomorphic semigroup, 39, 111 bounded, 26 bounded of semi-angle θ, 22 contraction, 32, 115, 119, 292 generator, 23, 27, 38, 117, 197 Gibbs, 99, 115, 132, 200 non-self-adjoint, 201 of semi-angle θ, 27 quasi-bounded, 27 strongly, 25 Ichinose-Tamura conditions, 288, 292 ideal Cp,8 pHq, 219 Cp pHq, 217 Dixmier, 220, 245 Macaev, 220 maximal space, 104 nontrivial, 216, 218 of trace-class operators, 59 predual, 246 symmetrically-normed, 81, 210, 214 von Neumann-Schatten, 59, 64 weak-Cp , 219 ideals

Index

312 von Neumann-Schatten, 41, 52, 78, 98, 140 immediately compact, 106 inequalities Weyl-Horn, 82, 86, 97 inequality Araki, 222, 272, 273 Araki-Lieb-Thirring, 274 Bernstein, 98 Bogoliubov convexity, 93 older, 63, 90 Cp -H¨ Ginibre-Gruber, 141 Golden-Thompson, 97 H¨ older, 92 Klein, 96 Klein, canonical, 97 Ky Fan, 97, 98, 213, 242 L¨ owner-Heinz, 272, 273 } ¨ }p -triangle, 61, 63 Peierls-Bogoliubov, 92 Peierls-Bogoliubov, general, 95 triangle, 69 integral Bochner, 8, 120, 124, 139 Cauchy, 24, 201 Riesz-Dunford, 22 Stieltjes, 83 strong Riemann, 7 integration of vector-valued functions, 6 isometry, 42 partial, 42, 51 ITTZ self-adjointness, 290 no-go problem, 291 Kato condition, 288 Kato function dominated, 211, 228 generic, 148 regular, 179 self-dominated, 212, 237, 280 Kato functions, 147, 162, 277 K, 282

Kα , 163, 206, 241 K˚ , 190, 279, 282 p β , 162, 206, 282 K KD , 211, 280 Kr , 179, 280 Ks-d , 212, 280 additional conditions, 279 dominated, 211 Kato-Rellich theorem, 130 Kato-small, 33, 130, 147, 292 Laplace transform, 8, 13, 28, 127 Lidskiˇı trace theorem, 57 Lie product formula, 144 lifting to Cφ -topology, 238 analytic continuation method, 203 approach, 223, 243 error bound estimates, x lemma, 183, 192, 208, 238 method, 255 to C1,8 -topology, 254, 256 to norm convergence, 68 to trace-norm, viii, 132, 294 Lim, 247 maximal Abelian subalgebra, 103 maximal ideal space, 103 method of analytic continuation, 144, 195 of Approximating Hamiltonian, 93 of lifting, 143, 211 minimax principle, 53, 130, 200 for unbounded operators, 113 multiplication } ¨ }p -continuity, 74, 99 strong continuity, 70, 74 trace-norm continuity, 116, 131 multiplicity finite, 47, 49, 102 infinite, 49, 50

Index Neumann series, 33, 37 norm } ¨ }p -triangle inequality, 61 } ¨ }1,8 , 219, 247 } ¨ }1 , 59, 217 } ¨ }8,p , 220 } ¨ }φ , 216 } ¨ }p,8 , 219 on Hilbert space, 41 symmetric, 215, 216 numerical range, 28, 30, 39 operator psq-limit, 69 psq-order, 69 m-accretive, 31, 292 m-sectorial, 31, 117, 119, 127, 183, 193, 197, 202 absolute value, 42, 69 } ¨ }p -continuity, 74 norm continuity, 74 s-continuity, 74 accretive, 31 algebraic sum, 124 closable, 139, 152 closed, 2, 194, 257 compact, see compact operator completely continuous, 42, 78, 264 continuous spectrum, 30 core, 14, 121 deficiency, 28 essential spectrum, 31 essentially self-adjoint, 290 exponential, 88, 90 finite-rank, 215 form-bounded, 290 form-sum, 163, 164, 166, 168, 188, 189, 202 fractional power, 271 invariant norm, 215 isometry, 42 maximally accretive, 31 non-negative, 41

313 norm convergence, 70 normal, 119 numerical range, 28, 30, 39 of finite rank, 44 of rank r, 51 orthogonal projection, 42, 75 finite-rank, 66 partial isometry, 42, 51 point spectrum, 30 polar decomposition, 42, 55 positive, 287 quasi-m-accretive, 31 quasi-m-sectorial, 31, 115 quasi-accretive, 31 quasi-sectorial contraction, 292 rank-one, 52, 60 real part of, 117, 194 relatively compact, 166, 291 residual spectrum, 30 Schr¨ odinger, 205 sectorial, 31, 200 self-adjoint, 41, 151 non-negative, 117, 194 singular value, see singular value spectrum, see spectrum strong convergence, 67, 106 symmetric norm, 215 trace class, 52 trace-class, 217 unbounded, 17, 117, 194 closed, 30 self-adjoint, 113 Volterra, 49, 57, 127, 259, 265 weak convergence, 66, 67 operator-norm convergence, 146, 287, 289 estimates, 241 operators form-sum, 195 holomorphic family, 196 log-ordered, 221, 272 orthonormal basis, 44, 49, 54 orthonormal set, 57

Index

314 parabolic differential equation, 39 partition function, 140 perturbation P-perturbation, 34, 120, 141 P0 -perturbation, 34 P0` -perturbation, 122 Pbă1 -perturbation, 141 bounded, 33, 141 Kato-small, 33, 287, 288, 292 relatively bounded, 33 resolvent-based method, 123, 132 semigroup-based method, 123, 136 unbounded infinitesimally small, 33, 122, 166, 181 of class PpAq, 122 of class P0` pAq, 34, 122, 181 of class Pb , 135, 141 of class Pb pAq, 34 point spectrum, 30, 46 polar decomposition, 42, 55 polarisation identity, 41, 193 positive cone C1,` , 72, 217, 247 C1,8,` , 247 sequences c` , 213 product approximation, 144 symmetrised, 144 product formula, 160 error bound estimate, 150 Lie-Trotter, 89, 146 nonsymmetric, 160, 291 symmetric, 160, 291 Trotter, 182, 186, 204, 287 product formulae error bound estimate, 186 in the C1,8 -topology, 256 in the trace norm, 182 optimal, 288, 291 for Gibbs semigroup, 182 for unitary groups, 295 in Banach space, 207, 292

in the C1,8 -topology, 254, 256 in the Dixmier ideal, 254 rate of convergence, 143, 181, 186 Trotter-Kato, 209, 210, 223, 230, 243, 286, 289 projection of finite rank, 66 orthogonal, 42, 45, 75 propagator, 288 quantum Gibbs state, 140 quantum statistical mechanics, 140, 293 rank of operator, 51 rate of convergence optimal, 147, 160 relative bound, 33, 122 relative form boundedness, 291 representation Riesz-Dunford, 23, 38, 96, 197 spectral, 37, 150 representation theorem first, 196 for non-negative forms, 117, 194 residual spectrum, 30, 46 resolvent, 2, 46 compact, 111, 114, 200 of generator, 17 resolvent identity, 103 resolvent set, 2, 46, 101 Riesz-Dunford calculus, 22, 96 Riesz-Dunford integral, 22 Riesz-Dunford representation, 23, 38, 197 self-adjoint families, 197 semi-angle, 117, 193 semigroup bounded, 12 compact, 105, 111, 140 contraction, 6, 35, 286 non-self-adjoint, 292

Index strongly continuous, 17 degenerate, 21, 22, 99, 147, 180, 235 space of strong continuity, 22 eventually compact, 105 eventually Gibbs, 99, 100, 140, 184 eventually norm-continuous, 39 exponentially bounded, 12 generator, see generator Gibbs, 81 holomorphic, see holomorphic semigroup, 39 image, 105 immediately compact, 105, 106 immediately continuous, 39 immediately Gibbs, 99, 100, 140 non-self-adjoint, 292 norm continuous, 14, 18, 105 quasi-bounded of type ω0 , 12–14 Schr¨ odinger, 162, 205, 291 self-adjoint, 111 strongly continuous, 5, 11, 39 trace-norm differentiable, 131 sequence finite cf , 212 norm convergent, 43 regular, 218 weakly convergent, 43 series Dyson-Phillips, 129 set bounded, 42 orthonormal, 49, 57 precompact, 42, 44, 106 weakly compact, 43 singular values, 41, 50 of compact operator, 51, 214 space c0 , 212 cφ , 214 nonseparable, 6 spectral measure, 41

315 point of continuous nonconstancy, 113 spectral radius, 50 spectral representation, 37, 119, 150 spectral theorem, 50 spectrum, 22, 30, 46, 49, 78 and resolvent, 258 approximate, 262 continuous, 30, 46, 261 core, 260 essential, 31, 264 Fredholm operator, 264 multiplicity of eigenvalue, 47, 49, 102 of closed operator, 257 of compact operator, 47, 110 of generator, 101 point, 30, 46 residual, 30, 46, 261 Volterra operator, 266 square-root lemma, 42, 81, 97 state D2˚ -dilation invariant, 251 ω lim , 249 Stieltjes integral, 83 strong closure of KpHq, 45 symmetrically-normed ideals Araki-Lieb-Thirring inequality, 274 definition, 214 theorem Cauchy, 24 closed graph, 120 dominated convergence, 66, 78 in the lp -space, 69 Fubini, 28 Hausdorff, 28 Hilbert-Schmidt, 57 Hille-Yosida, 39 Hille-Yosida-Phillips, 39 Kato-Rellich, 130 Markov-Kakutani, 251 spectral mapping, 112, 268

Index

316 Tamura, 206, 288, 290 Vitali, 198, 202 topology C1,8 , 220 C1 , 217 Cp , 60 operator norm, 43, 287 trace norm, 186, 205, 217 weak, 43 weak*, 251 trace, 56 T , 247, 251 basic properties, 58 Dixmier, 220, 243, 255 Dixmier Trω , 252 matrix, 57, 88 singular, 247, 255 spectral, 57, 88 weakly semi-continuous, 89 trace class, 52 trace norm, 59 trace-class operators positive cone, 58 trace-norm topology, 186, 205 triangle inequality, 69 Trotter product formula, 118, 186 operator-norm convergence, 182 trace-norm convergence, 204 unitary, 295 Trotter-Kato product formula exponential, 182 Trotter-Kato product formulae, 148, 286, 289 operator-norm convergence, 146 strong convergence, 146 trace-norm convergence, 182, 186 uniform boundedness principle, 6, 9, 18, 70 unitary product formula in the L2 -topology, 297 non-exponetial, 296 operator-norm convergence, 295

strong convergence, 295 vertex, 117, 183, 193 Volterra operator, 49 von Neumann-Schatten classes, 52, 65 von Neumann-Schatten ideals, 52, 98, 140 weak convergence in Hilbert space, 42 Weyl-Horn inequalities, 82, 86, 97 Yosida approximant, 39 Zorn’s lemma, 103

List of Symbols

pAq-type paq-type pBq-type pu, vq aru, vs az |A| Apzq 9 A`B AR :“ <e A Aptq Aζ psq

AăB B B˚ C C` C´ C1 pHq Cp pHq (1 ď p ă 8) C8 pHq

holomorphic family holomorphic family holomorphic family inner product in H : pαu, βvq “ αβpu, vq sesquilinear form in H type (a) holomorphic family absolute value of the operator A type (B) holomorphic family form-sum of A and B real part of A Aptq :“ t´1 p1 ´ Ut q for t ą 0 Aζ :“ A ´ ζ1 for ζ P C

36, 196 196 197 20, 28 117, 193 195 42, 81 197 117 117 7 12, 29

psq-operator order Banach space dual space of B complex numbers right half-plane of C left half-plane of C trace-class operators on H von Neumann-Schatten classes compact operators on H

69 1 19 1 2 32 52 60 42


317

List of Symbols

318 C1,8 pHq Cφ pHq core A dom A dom An dom A def T EK pλq tGt utě0 H H “ M ‘ MK HnK H pθ, ω0 q 1 ImpU q =m a K KD Kr Ks-d KpHq Kr pHq ker U LpB, Bq LpB, Cq LpHq L8 pBq lp L2 pRd q limtÓ0 limnÑ8 limnÑ8 } ¨ }- lim s-lim psq- lim w-lim NrA

Dixmier ideal symmetrically-normed ideal core of the operator A operator domain domain of the operator An closure of dom A deficiency (defect) of the operator T spectral measure of the operator K Gibbs semigroup Hilbert space orthogonal direct sum orthogonal complement of the subspace Hn generators of holomorphic semigroups unit operator image of the map U : X Ñ Y imaginary part of the sesquilinear form a generic Kato function dominated Kato function regular Kato function self-dominated Kato function finite-rank operators on H operators with dimension of range r kernel of the operator U bounded linear operators on B bounded linear functionals on B bounded linear operators on H completely continuous operators on B space space limtÑ`0 ô {limit t Ñ 0 ^ t ą 0} lim sup lim inf operator-norm limit strong operator limit psq-operator limit weak limit in H numerical range of the operator A

279, 311 273 14, 155 2 3, 23 7 28 41 99 28 42, 50 45 27, 29 2 105 117 148, 276 211, 280 179, 280 212, 280 44 44 42 1 19 30, 41 78 78, 217 265, 289 3 10, 102 58, 89 43 3 69 42 28

List of Symbols P0` Pb Pn QpM, ω0 q q˚p ran A <e a R R´ R` Y t0u Rζ pAq :“ pA ´ ζ1q´1 Sθ tsn pAquně1 tÓ0 Tr : C1 pHq Ñ C Trω : C1,8 pHq Ñ C Ut pAq tUt utě0 ˆz U w

un Ñ u ˜ X rxs N N0 “ N Y t0u ρpAq σpAq σcont pAq σp pAq σres pAq Bt a ‘ 9 ` } ¨ }p }¨} } ¨ }- Bt }¨} }¨} ^ _ ô AñB æ

319 operators with relative bound b “ 0` operators with relative bound b ą 0 orthogonal projection on subspace Hn ě 0 generators of quasi-bounded semigroups convolution range of the operator A real part of the sesquilinear form a real numbers negative real semi-axis p´8, 0q non-negative real semi-axis r0, 8q “ R0` resolvent of A at ζ open sector of angle θ in C` singular values of operator A denotes t Ñ `0 trace on C1 pHq Dixmier trace semigroup generated by A one-parameter semigroup Laplace transform of semigroup Ut weak vector convergence Gelfand transformation of X integer part of x ě 0 natural numbers t1, 2, . . .u non-negative integers resolvent set of A spectrum of operator A continuous spectrum of A point spectrum of A residual spectrum of A derivative with respect to t orthogonal difference orthogonal direct sum form-sum norm on Cp pHq operator norm operator-norm derivative vector norm on B vector norm on H and or equivalent, if and only if A yields (implies) B restriction

33, 122 33, 122 45 12, 29 129 29, 42, 258 117 2 2 2 2, 46 22 51, 82 3 56 220, 251 7 5 8 19, 43 103 11 12 12 2, 46, 258 22, 258 30, 261 30, 261 30, 261 2 29 42 89, 117 60 1 2 3 41 37, 248 116, 253 21, 240, 258 21, 173 14, 42

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Smarandache Semigroups

Semigroups of Matrices

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Rings and Semigroups

Semigroups Theory and Applications

Semigroups and their applications

Harmonic Analysis on Semigroups

Co-Semigroups and Applications

Co-semigroups and applications

Words, Semigroups, Transductions

R-diagonal dilation semigroups

Harmonic analysis on semigroups

Words, semigroups, transductions

One-Parameter Semigroups

Topics in operator semigroups

Semigroups in Geometrical Function Theory

Lectures in Rings and Semigroups

Bisimple type a -semigroups-I

Quantum Dynamical Semigroups and Applications

Nonlinear Evolution Operators and Semigroups

Analytical methods for Markov semigroups

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