Analysis of Toeplitz operators

Springer Monographs in Mathematics Albrecht Böttcher · Bernd Silbermann Analysis of Toeplitz Operators Second Edition...

Author: Albrecht Böttcher | Bernd Silbermann | Alexei Yurjevich Karlovich

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Springer Monographs in Mathematics

Albrecht Böttcher · Bernd Silbermann

Analysis of Toeplitz Operators Second Edition Prepared jointly with Alexei Karlovich

With 20 Figures

123

Albrecht Böttcher Bernd Silbermann Technische Universität Chemnitz Fakultät für Mathematik 09107 Chemnitz, Germany [email protected] [email protected]

Library of Congress Control Number: 2006922574

Mathematics Subject Classification (2000): 47-02, 15A15, 30D55, 42A50, 45E10, 46E25, 47B35, 47L15, 65F99 ISSN 1439-7382 ISBN-10 3-540-32434-8 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-32434-8 Springer-Verlag Berlin Heidelberg New York ISBN 3-540-52147-X 1st ed. Springer-Verlag Berlin Heidelberg New York (co-publication) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 © Akademie-Verlag Berlin 1990 First Edition Printed in Germany The use of general descriptive names, registered names, trademarks, 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. Typesetting by the authors using a Springer TEX macro package Production and data conversion: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

41/3100YL - 5 4 3 2 1 0

Preface to the Second Edition

Since the late 1980s, Toeplitz operators and matrices have remained a field of extensive research and the development during the last nearly twenty years is impressive. One encounters Toeplitz matrices in plenty of applications on the one hand, and Toeplitz operators confirmed their role as the basic elementary building blocks of more complicated operators on the other. Several monographs on Toeplitz and Hankel operators were written during the last decade. These include Peller’s grandiose book on Hankel operators and their applications and Nikolski’s beautiful easy reading on operators, functions, and systems, with emphasis on topics connected with the names of Hardy, Hankel, and Toeplitz. They also include books by the authors together with Hagen, Roch, Yu. Karlovich, Spitkovsky, Grudsky, and Rabinovich. Thus, results, techniques, and developments in the field of Toeplitz operators are now well presented in the monographic literature. Despite these competitive works, we felt that large parts of the first edition of the present monograph which is meanwhile out of stock - have not lost their fascination and relevance. Moreover, the first edition has received a warm reception by many colleagues and became a standard reference. This encouraged us to venture on thinking about a second edition, and we are grateful to the Springer Publishing House for showing an interest in this. The present book is a genuine second edition, which means that it differs from the original version but that it is not a completely new book. We left everything of the first edition, even in the cases where we felt that the material nowadays is not of the same brisance as two decades ago. However, we tried to incorporate as much as possible of the body of knowledge that has grown since about 1990. It is clearly impossible to take into account even only a moderate piece of all the developments in Toeplitz operators during the last ten or fifteen years. We therefore focussed our attention upon topics which are intimately related to those considered in the previous edition or on which we ourselves have worked some time. Many recent achievements were included together with a short comment into the list of references, some others, such as pseudospectra, phenomena caused by SAP symbols, or recent work on

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Preface to the Second Edition

Toeplitz and Wiener-Hopf determinants received own new sections in the text, partly with full proofs. We hope the present second edition will be welcomed by those who had no chance of purchasing a copy of the original edition and also by all those who are looking for a first or refreshed orientation about the present state of the art in the field.

Braga and Chemnitz, December 2005

Albrecht B¨ ottcher Alexei Karlovich Bernd Silbermann

Preface to the First Edition

This book was originally intended as an extended version of our book “Invertibility and Asymptotics of Toeplitz Matrices”, which appeared in 1983. We planned to discuss several topics in more detail, but our main concern was to incorporate a whole series of new results obtained during the last few years. However, we soon realized that the program we had in mind required new thoughts from both the methodological and substantial points of view, and so we decided to attempt writing a completely new book on the analysis of Toeplitz operators. There are at least two reasons for the continuous and increasing interest in Toeplitz operators. On the one hand, Toeplitz operators are of importance in connection with a variety of problems in physics, probability theory, information and control theory, and several other fields. Although we shall not embark on these problems, the selection of the material of the book is to a certain extent determined by such applications. On the other hand, besides the differential operators, Toeplitz operators constitute one of the most important classes of non-selfadjoint operators and they are a fascinating example of the fruitful interplay between such topics as operator theory, function theory, and the theory of Banach algebras. One main purpose of this book is to elucidate some of the ideas and methods illustrating just the latter aspect. The theory of Toeplitz operators is a very wide area and even a huge monograph can deal with only some selected topics. Our emphasis is on Toeplitz operators over the circle and over the torus (or, what is the same, discrete Wiener-Hopf operators over the half-axis and over the quarter-plane) viewed as concrete operators on concrete Banach spaces, and a central problem is to establish a relation between the functional-analytic properties of Toeplitz operators and the geometric properties of their symbols. The selection of the special topics has been determined by our own interests and competence. However, having chosen a topic, we try to present it in such a way that it may be taken as a systematic, exhaustive, and modern introduction to the well-known and by now classical results as well as a readable account of some recent developments. A glimpse of the table of contents provides an overall

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view of the material covered by the book. We merely want to add the following remarks. Chapter 1 contains a series of notations and definitions. The reader need not study this chapter very carefully; it suffices to glance through it, pick up some notations, and backtrack whenever the necessity arises. In Chapter 2 we begin by stating elementary properties of Toeplitz operators and finish by proving some of their rather deep-lying properties. This chapter mainly incorporates those results whose proof needs almost no “theory.” Moreover, it may be viewed as the trunk of a tree, the boughs and twigs of which are the concern of the forthcoming chapters. In Chapter 3 we start preparing the more delicate theory of Toeplitz operators. It is devoted to matrix functions which are locally sectorial in a very sensitive sense. This chapter also includes the study of the phenomenon of the asymptotic multiplicativity of approximate identities and Sarason’s theory of piecewise quasicontinuous functions. In this chapter we also devise and give a first application of some sort of machinery (or “philosophy”) that will be employed repeatedly in the remaining chapters: algebraization, essentialization, localization, determination of local spectra. Chapter 4 is concerned with the Hilbert space theory of block Toeplitz operators. There we prove Fredholmness and compute the index of Toeplitz operators whose symbol is locally sectorial over QC, describe Axler’s transfinite localization approach to maximal antisymmetric sets for C +H ∞ , present the theory of local Toeplitz operators due to Douglas, Clancey, Gosselin, study symbols with a specified local range (in particular, symbols with two or three essential cluster points), and develop a new approach to algebras generated by Toeplitz operators and related objects. Chapters 5 and 6 deal with block Toeplitz operators on weighted H p and p spaces, respectively. Ours is, to a great extent, a novel presentation of these topics. We provide new proofs of the classical results of the well-known monographs by Gohberg, Feldman and Gohberg, Krupnik, and we incorporate numerous results which are only known from mathematical journals, primarily Soviet ones. Chapter 7 is a self-contained and up to date theory of finite section method (reduction method) for Toeplitz operators. We prove that the finite section method is applicable to block Toeplitz operators on H 2 with symbols that are locally sectorial over QC, we develop a sufficiently simple theory for operators with piecewise continuous symbols on H p and p , we study symbols with singularities of Fisher-Hartwig type, and we conclude by proving the very recent and noteworthy result of Treil, according to which there are invertible Toeplitz operators on H 2 to which the finite section method is not applicable. Chapter 8 is a comprehensive treatment of Toeplitz operators over the quarter-plane and is, at least to a certain degree, a novelty in the monographic literature. We emphasize that we study both the Fredholm theory and the theory of finite section methods of quarter-plane operators with discontinuous symbols.


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Chapter 9 looks at Wiener-Hopf integral operators. There we point out the common features between Wiener-Hopf integral operators and their discrete analogues (the Toeplitz operators) but also dwell on the significant differences between these two classes of operators. In this chapter we also consider operators with almost periodic, semi-almost periodic, and other kinds of oscillating symbols. Chapter 10 is a systematic and self-contained theory of Toeplitz determinants. The material presented ranges from the classical Szeg˝o-Widom limit theorems to a proof of the conjecture of Fisher-Hartwig for some important special cases. We shall demonstrate that the very attractive field of Toeplitz determinants requires results from all foregoing chapters and may thus serve as a beautiful application of the functional analysis of Toeplitz operators. In particular, we shall show that some important problems on Toeplitz determinants can be solved by working with Toeplitz operators on the spaces H 2 () and pµ , so that passage from the Hilbert space theory to the Banach space theory does not turn out to be a purely academic matter. Let us also point out three peculiarities of the present monograph. First, Banach algebra techniques combined with local principles are our main tool for tackling Toeplitz operators. That such methods can be successfully applied to the study of the Fredholm theory of Toeplitz operators is wellknown from Douglas’ book. However, this approach has only recently proved to be a powerful technique of studying projection methods, harmonic approximation, or stable convergence (and thus index computation) for Toeplitz operators. Moreover, our consistent use of local Banach algebra technique will not only provide a unified technique of solving various problems related to Toeplitz operators, but will allow us to reformulate many classical results in pretty nice language. For instance, the well-known result that a Toeplitz operator with piecewise continuous symbol is Fredholm on H p or p if and only if the curve obtained from the essential range of its symbol by filling in certain circular arcs does not contain the origin reads in this language as follows: the local spectrum of the operator is either a point or a certain circular arc. Secondly, we shall consider Toeplitz operators on the (Hilbert) space H2 ∼ = 2 and on the weighted (Banach) spaces H p () and pµ . Each of these three situations has its peculiarities and requires its own techniques. While there are excellent and comprehensive discussions of the Hilbert space theory in the well-known monographs by Gohberg and Feldman, Douglas, and Nikolski, the same cannot be said about Banach space theory. We hope that the present book will fill this gap and, moreover, will also make a series of new contributions to the Hilbert space case. Thirdly, it should be explicitly noticed that our emphasis is on matrixvalued symbols. Many problems on Toeplitz operators have fairly fast solution in the scalar case, whereas substantial difficulties arise in the matrix case. Finally, to see what this book is all about, it should also be mentioned that the following topics are not touched upon: Toeplitz operators on domains, balls, or manifolds; pseudodifferential operators; Breuer-Fredholm results and

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generalized index theory; operator-valued symbols; invariant subspaces; linear algebra and computational mathematics of finite Toeplitz matrices. This list is naturally incomplete. Let it also be understood that this is a book on Toeplitz operators and not on Hankel or singular integral operators, although we pay due attention to these two classes of operators. We ventured on writing a book for both the beginner and the specialist. In the beginner’s interest we gave a rather full description of those topics which form the background to the theory of Toeplitz operators (e.g., some students of ours acknowledged the rather lengthy explanation of what a “fiber” is). We also provided the bulk of results with detailed proofs. We did this in order to teach the reader not only the results but also (or mainly) the techniques for proving them. This is, of course, not always in the beginner’s interest, but we hope the specialist will relish some details of these proofs. Some of the results and techniques are new and published here for the first time and are thus primarily addressed to the specialist. Many results are taken from the periodicals and are first cited with detailed proofs. We included a series of problems which we declared to be “open”. Some of them are well-known as open problems, some others are merely open in the sense that we have not found a solution within a few hours, days or weeks. In either case we followed the policy that we would better confess our own inability than hide something. We made the attempt of supplying all results with a source; however, the evolution of many theorems involves too many contributors, and so it may occur that our reference is not the right one. We hope that the reader will excuse our faulty referencing and we accept any criticism in this direction. Finally, we have labelled a lemma or theorem only when a name seems to have been attached to it by common usage. We wish to express our sincere appreciation to our colleagues Roland Hagen and Steffen Roch, who both read the bulk of the manuscript very carefully and eradicated not only a large number of mistakes and (sometimes serious) errors but helped with their criticism to essentially improve the book. We would also like to thank Mrs. Marianne Graupner and Mrs. Isolde Scholz for all the trouble they took in typing the entire manuscript. Finally, we are pleased to express our gratitude to the Akademie-Verlag Publishing House, especially to the Editor, Dr. Reinhard H¨ oppner, for inviting us to write this monograph and for the careful performance of the book. Special Acknowledgement. In May and June 1987, Naum Krupnik visited the Chemnitz University of Technology. During these two months he read the whole manuscript with great enthusiasm and made a large number of valuable remarks, a major part of which could still be incorporated into the text. We are extremely grateful to him for improving the book by his uncommon expertise.

Chemnitz, 1987

Albrecht B¨ ottcher, Bernd Silbermann

Contents

1

Auxiliary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Operator Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Operator Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Operator Matrices and Their Determinants . . . . . . . . . . . . . . . . 1.5 Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 C ∗ -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Local Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Lp and H p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 BM O and V M O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Smoothness Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 9 11 12 17 21 27 36 38 41

2

Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1 Multiplication Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2 Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3 Hankel Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.4 Invertibility of Toeplitz Operators on H 2 . . . . . . . . . . . . . . . . . . 57 2.5 Spectral Inclusion Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.6 The Connection Between Fredholmness and Invertibility . . . . . 71 2.7 Compactness of Hankel Operators and C + H ∞ Symbols . . . . 77 2.8 Local Methods for Scalar Toeplitz Operators . . . . . . . . . . . . . . . 85 2.9 Matrix Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.10 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3

Symbol Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.1 Local Sectoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.2 Asymptotic Multiplicativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.3 Piecewise Quasicontinuous Functions . . . . . . . . . . . . . . . . . . . . . . 128 3.4 Harmonic Approximation: Algebraization . . . . . . . . . . . . . . . . . . 140 3.5 Harmonic Approximation: Essentialization . . . . . . . . . . . . . . . . . 149

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Contents

3.6 3.7 3.8 3.9

Harmonic Approximation: Localization . . . . . . . . . . . . . . . . . . . . 152 Harmonic Approximation: Local Spectra . . . . . . . . . . . . . . . . . . . 157 Local Sectoriality Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

4

Toeplitz Operators on H 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.1 Fredholmness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.2 Stable Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.3 Index Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.4 Transfinite Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4.5 Local Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.6 Symbols with Specific Local Range . . . . . . . . . . . . . . . . . . . . . . . . 217 4.7 Toeplitz Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 4.8 The Role of the Harmonic Extension . . . . . . . . . . . . . . . . . . . . . . 240 4.9 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

5

Toeplitz Operators on H p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.1 General Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.2 Khvedelidze Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.3 Locally p, -Sectorial Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 5.4 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 5.5 P C Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 5.6 P2 C Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 5.7 Fisher-Hartwig Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 5.8 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

6

Toeplitz Operators on p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 6.1 Multipliers on Weighted p Spaces . . . . . . . . . . . . . . . . . . . . . . . . 287 6.2 Continuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.3 Piecewise Continuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 6.4 Analytic Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 6.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

7

Finite Section Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 7.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 7.2 C + H ∞ Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 7.3 Locally Sectorial Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 7.4 P C Symbols: p Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.5 P C Symbols: H p Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 7.6 Operators from algL(H 2 ) T (P C) . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 7.7 Fisher-Hartwig Symbols: H 2 () Theory . . . . . . . . . . . . . . . . . . . . 374 7.8 Fisher-Hartwig Symbols: pµ Theory . . . . . . . . . . . . . . . . . . . . . . . 378 7.9 Invertibility Versus Finite Section Method . . . . . . . . . . . . . . . . . 389 7.10 Pseudospectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 7.11 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

Contents

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8

Toeplitz Operators over the Quarter-Plane . . . . . . . . . . . . . . . . . 409 8.1 Function Classes on the Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 8.2 Elementary Properties of Quarter-Plane Operators . . . . . . . . . . 417 8.3 Continuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 8.4 The Invertibility Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 8.5 Bilocal Fredholm Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 8.6 P QC ⊗ P QC Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 8.7 Finite Section Method: Kozak’s Theory . . . . . . . . . . . . . . . . . . . . 454 8.8 Finite Section Method: Bilocal Theory . . . . . . . . . . . . . . . . . . . . . 461 8.9 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 8.10 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

9

Wiener-Hopf Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 9.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 9.2 Continuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 9.3 Piecewise Continuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 9.4 AP and SAP Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 9.5 Some Phenomena Caused by SAP Symbols . . . . . . . . . . . . . . . . 497 9.6 Other Oscillating Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 9.7 Finite Section Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 9.8 Operators over the Quarter Plane . . . . . . . . . . . . . . . . . . . . . . . . . 513 9.9 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

10 Toeplitz Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 10.1 The First Szeg˝o Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 10.2 Krein Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 10.3 Wiener-Hopf Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 10.4 The Strong Szeg˝o Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 537 10.5 Higher Order Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 10.6 Semirational Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 10.7 Nonvanishing Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 10.8 Selfadjoint Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 10.9 The Pure Fisher-Hartwig Singularity . . . . . . . . . . . . . . . . . . . . . . 570 10.10 Separation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 10.11 Fisher-Hartwig Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 10.12 More on Unbounded Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 10.13 Wiener-Hopf Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 10.14 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 10.15 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

1 Auxiliary Material

1.1 Operator Ideals 1.1. Bounded and compact operators. Let X and Y be Banach spaces. We denote by L(X, Y ) the linear space of all (bounded and linear) operators from X to Y . We let C∞ (X, Y ) ⊂ L(X, Y ) denote the collection of all compact operators from X into Y , and C0 (X, Y ) refers to the set of all finite-rank operators from X into Y , i.e., F is in C0 (X, Y ) if and only if F ∈ L(X, Y ) and dim F (X) < ∞. In the case X = Y we shall write L(X) = L(X, X), C∞ (X) = C∞ (X, X), C0 (X) = C0 (X, X). A sequence {An } of operators An ∈ L(X, Y ) is said to converge to an operator A ∈ L(X, Y ) (a) weakly, if f (An x) − f (Ax) → 0 for each x ∈ X and each functional f ∈ Y ∗; (b) strongly, if An x − AxY → 0 for each x ∈ X; (c) uniformly, if An − A → 0, where, for B ∈ L(X, Y ), B := sup Bx.

(1.1)

x≤1

Let {An } be a sequence of operators An ∈ L(X, Y ). Then one has the following. (d) If A ∈ L(X, Y ), if An x − AxY → 0 as n → ∞ for each x in a dense subset of X, and if sup An < ∞, then An converges strongly to A. n

(e) (Banach/Steinhaus). If {An x} is a convergent sequence in Y for each x ∈ X, then sup An < ∞, the operator A defined by Ax = lim An x n→∞

n

belongs to L(X, Y ), and A ≤ lim inf An . n→∞

2


(f) If A ∈ L(X, Y ), if An → A weakly, and K ∈ C∞ (Y, Z), then KAn → KA strongly. Equipped with the operator norm (1.1) the linear set L(X, Y ) becomes a Banach space and L(X) a Banach algebra. Then C∞ (X, Y ) is a closed subspace of L(X, Y ). In general, the subspace C0 (X, Y ) ⊂ L(X, Y ) is not closed, but its closure is contained in C∞ (X, Y ). Notice the following implications. A ∈ L(X, Y ), K ∈ C∞ (Y, Z), B ∈ L(Z, V ) =⇒ BKA ∈ C∞ (X, V ), A ∈ L(X, Y ), F ∈ C0 (Y, Z), B ∈ L(Z, V ) =⇒ BF A ∈ C0 (X, V ). In particular, C∞ (X) is a closed two-sided ideal of L(X) and C0 (X) is a twosided (but in general not closed) ideal of L(X). If X and Y are separable infinite-dimensional Hilbert spaces, we shall write X = H1 , Y = H2 , and in case X = Y simply X = Y = H. We then have the following. (g) The closure of C0 (H1 , H2 ) with respect to the operator norm (1.1) coincides with C∞ (H1 , H2 ). (h) L(H) is a C ∗ -algebra and C∞ (H) is a closed two-sided star-ideal of L(H). 1.2. The s-numbers. Given an operator A ∈ L(H1 , H2 ), we define for n in Z+ := {0, 1, 2, . . .} sn (A) := inf A − F : F ∈ C0 (H1 , H2 ), dim F (H1 ) ≤ n . The sequence {sn (A)}∞ n=0 is referred to as the sequence of the s-numbers of the operator A. The well known Horn lemma says that if K, L ∈ C∞ (H), n ∈ Z+ , and 1 ≤ p < ∞, then n n spj (KL) ≤ spj (K)spj (L). j=0

j=0

1.3. The Schatten-von Neumann classes. For 1 ≤ p < ∞, the collection of all operators K ∈ L(H1 , H2 ) satisfying 1/p Kp := spn (K) 1 is an integer. A simple computation shows that then ⎞⎤ ⎡ ⎛ p−1 j (−K) ⎠⎦ − I ∈ C1 (H). Rp (K) := ⎣(I + K) exp ⎝ j j=1 Thus, it is justified to define detp (I + K) := det(I + Rp (K)). One calls detp (I + K) the p-regularized determinant of I + K. (a) Let {λj (K)}j≥0 denote the sequence of the eigenvalues of K ∈ Cp (H), counted up to algebraic multiplicity. Then p−1 (−λj (K))l detp (I + K) = (1 + λj (K)) exp . l j≥0

l=1

(b) If K ∈ Cp (H), then tr (K p ) detp+1 (I + K) = detp (I + K) exp (−1)p . p In particular, for K ∈ C1 (H), ⎞ p−1 j tr (K ) ⎠ . detp (I + K) = det(I + K) exp ⎝ (−1)j j j=1 ⎛

(c) There exist constants Γp such that |detp (I + K)| ≤ exp(Γp Kpp ) for all K ∈ Cp (H). One may take Γ1 = 1, Γ2 = 1/2, and it is known that 1/p ≤ Γp ≤ e(2 + log p). The mapping Cp (H) → C, K → detp (I + K) is continuous. Moreover, for K, L ∈ Cp (H), |detp (I + K) − detp (I + L)| ≤ K − Lp exp Γp (Kp + Lp + 1)p . (d) If K, L ∈ C2 (H), then det2 (I + K)det2 (I + L) = det2 (I + K + L + KL) exp(tr KL).

1.2 Operator Determinants

7

(e) Let K ∈ Cp (H). Then detp (I + K) = 0 if and only if I + K is invertible in L(H). (f) (Plemelj-Smithies’ formula). Let K ∈ Cp (H), where p ≥ 1 is an integer. Then detp (I + zK) is an entire function (of z) whose power series expansion in the plain is given by ⎛ (p) ⎞ σ1 n − 1 0 . . . 0 ⎜ (p) (p) ⎟ ∞ ⎜ σ2 σ1 n − 2 . . . 0 ⎟ zn ⎜ det ⎜ . detp (I + zK) = 1 + .. .. .. ⎟ ⎟ n! ⎝ .. . . . ⎠ n=1

(p)

σn (p)

(p)

(p)

(p)

σn−1 σn−2 . . . σ1

(p)

where σj := 0 for j ≤ p − 1 and σj := tr (K j ) for j ≥ p. Note that this formula expresses det(I + K) in terms of tr (K j ) (j = 1, 2, . . .), and when tr (K), tr (K 2 ), . . . , tr (K p−1 ) are set equal to zero in this formula, we just get detp (I + K). 1.9. The Ehrhardt formula. If A, B ∈ L(H) and AB − BA ∈ C1 (H) then 1

det eA eB e−A−B = etr 2 (AB−BA) .

(1.3)

Proof. We first show that the function F (λ) := eλA eλB e−λ(A+B) −I is an entire C1 (H)-valued function. Put M = AB − BA. Clearly, it suffices to show that G(λ) := eλA eλB − eλ(A+B) is an entire C1 (H)-valued function. By definition, G(λ) =

λk+m λn Ak B m − (A + B)n . k! m! n! n≥0

k,m≥0

There exist exactly (k + m)!/(k! m!) possibilities to form a non-commutative product which contains exactly k factors A and exactly m factors B. Let Ik,m be an index set enumerating all these possibilities and denote the corresponding product by Pk,m (j) with j ∈ Ik,m . Then Pk,m (j) (A + B)n = k, m ≥ 0 k+m = n

j∈Ik,m

and hence G(λ) =

k,m≥0 j∈Ik,m

λk+m k m A B − Pk,m (j) . (k + m)!

The product Pk,m (j) can be transformed into Ak B m by at most km permutations that interchange only two neighboring elements. Consequently, Ak B m − Pk,m (j) = Xs(k,m,j) M Ys(k,m,j) s

8

1 Auxiliary Material (k,m,j)

where the sum is over at most km terms, and Xs propriate products of the A’s and B’s. In particular,

(k,m,j)

and Ys

are ap-

m−1 Xs(k,m,j) ∞ Ys(k,m,j) ∞ ≤ Ak−1 ∞ B∞ .

We claim that G(λ) =

k,m≥0 j∈Ik,m

λk+m (k,m,j) Xs M Ys(k,m,j) (k + m)! s

is an absolutely convergent series of operators in C1 (H) that converges uniformly with respect to λ on compact subsets of C. Indeed, this follows from the estimate |λk+m | Xs(k,m,j) M Ys(k,m,j) 1 (k + m)! s k,m≥0 j∈Ik,m

≤

|λ|k+m km m−1 Ak−1 ∞ B∞ M 1 (k + m)!

k,m≥0 j∈Ik,m

= M 1

Ak−1 Bm−1 ∞ ∞ |λ|k+m (k − 1)! (m − 1)!

k,m≥1

= |λ| M 1 eλa∞ eλB∞ . 2

Hence G(λ) takes values in C1 (H) and is an entire function. From what has just been proved we infer that f (λ) := det eλA eλB e−λ(A+B) is entire. Obviously, f (λ) = 0 for λ ∈ C. We take the logarithmic derivative of f and use the well known formula (det H(λ)) = tr H (λ)H −1 (λ) det H(λ) with H(λ) := eλA eλB e−λ(A+B) . What results is that f (λ) = tr A + eλA Be−λA − eλA eλB (A + B)e−λB e−λA f (λ) = tr A − eλA eλB Ae−λB e−λA = tr A − eλB Ae−λB . Differentiating again we obtain that f (λ) = tr eλB (−BA + AB)e−λB = tr (AB − BA). f (λ) Since f (0) = 0 and f (0)/f (0) = 0, we see that 2 λ f (λ) = exp tr (AB − BA) . 2

1.3 Fredholm Operators

9

1.10. The Pincus-Helton-Howe formula. If A, B ∈ L(H) and AB − BA is in C1 (H) then det eA eB e−A e−B = etr (AB−BA) . Proof. Clearly, (1.3) implies that 1

det eA+B e−A e−B = etr 2 (AB−BA)

(1.4)

and multiplication of (1.3) and (1.4) yields the desired formula.

1.3 Fredholm Operators 1.11. Definitions. Let X and Y be Banach spaces and let A ∈ L(X, Y ). We put Ker A = x ∈ X : Ax = 0 , Im A = A(X). The operator A is said to be normally solvable if Im A is closed in Y . A normally solvable operator A ∈ L(X, Y ) is called a Φ+ -operator (Φ− -operator ) if dim Ker A < ∞ (dim Coker A := dim(Y /Im A) < ∞). If A is both a Φ+ - and Φ− -operator, then it is called a Φ-operator (or Fredholm operator or Noetherian operator ) and the integer Ind A := dim Ker A − dim Coker A is referred to as the index of A. The collection of all Φ+ -operators from X to Y will be denoted by Φ+ (X, Y ), and Φ+ (X, X) will be abbreviated to Φ+ (X). A similar definition is made for Φ− (X, Y ), Φ− (X), Φ(X, Y ), Φ(X). 1.12. Basic properties of Fredholm operators. (a) For A ∈ L(X, Y ) the following are equivalent. (i) A ∈ Φ(X, Y ). (ii) There exist operators R, L ∈ L(X, Y ) such that AR − IY ∈ C∞ (Y ),

LA − IX ∈ C∞ (X).

(iii) There exists an operator B ∈ L(X, Y ) such that AB − IY ∈ C0 (Y ),

BA − IX ∈ C0 (X).

Here IX and IY are the identity operators on X and Y , respectively. Any operator B ∈ L(X, Y ) for which AB − IY ∈ C∞ (Y ), is called a regularizer of A.

BA − IX ∈ C∞ (X)

10


(b) (Fedosov’s formula). Let A ∈ Φ(X, Y ) and let B be any operator satisfying (iii) of (a). Then Ind A = tr (IX − BA) − tr (IY − AB). In particular, for X = Y we have Ind A = tr (AB − BA). (c) (Atkinson’s theorem). Let X, Y, Z be Banach spaces and let A ∈ Φ(X, Y ), B ∈ Φ(Y, Z). Then BA ∈ Φ(X, Z) and Ind BA = Ind B + Ind A. (d) Φ(X, Y ) is an open subset of L(X, Y ) and the mapping Ind : Φ(X, Y ) → Z is constant on the connected components of Φ(X, Y ). If A ∈ Φ(X, Y ) and K ∈ C∞ (X, Y ), then A + K ∈ Φ(X, Y ) and Ind (A + K) = Ind A. (e) Let X, Y, Z be Banach spaces and let A ∈ L(X, Y ), B ∈ L(Y, Z). Then the following implications are true. (i) A ∈ Φ± (X, Y ), B ∈ Φ± (Y, Z) =⇒ BA ∈ Φ± (X, Z). (ii) BA ∈ Φ+ (X, Z) =⇒ A ∈ Φ+ (X, Y ). (iii) BA ∈ Φ− (X, Z) =⇒ B ∈ Φ− (Y, Z). (iv) A ∈ Φ(X, Y ), BA ∈ Φ(X, Z) =⇒ B ∈ Φ(Y, Z). (v) B ∈ Φ(Y, Z), BA ∈ Φ(X, Z) =⇒ A ∈ Φ(X, Y ). (f) Property (d) remains valid if Φ is replaced by Φ+ or Φ− and the index is allowed to assume the values ±∞. Moreover, if A ∈ Φ± (X, Y ), then dim Ker (A + C) ≤ dim Ker A,

dim Coker (A + C) ≤ dim Coker A

whenever C has sufficiently small norm. (g) Let A ∈ Φ+ (X) and let K ∈ C0 (X) be any projection of X onto Ker A. Then there is a δ > 0 such that Ax + Kx ≥ δx ∀ x ∈ X. Conversely, if for an operator A ∈ L(X) there exist operators Kj ∈ C∞ (X) (j = 1, . . . , n) and a δ > 0 such that Ax +

n j=1

Kj x ≥ δx ∀ x ∈ X,

1.4 Operator Matrices and Their Determinants

11

then A ∈ Φ+ (X). (h) We have A ∈ Φ± (X, Y ) ⇐⇒ A∗ ∈ Φ∓ (Y ∗ , X ∗ ). Moreover, if A is in Φ(X, Y ) then dim Ker A∗ = dim Coker A,

dim Coker A∗ = dim Ker A,

whence Ind A∗ = −Ind A.

1.4 Operator Matrices and Their Determinants 1.13. Definitions. Given a linear space X, denote by XN the linear space of column-vectors of length N with components from X and let XN ×N denote the linear space of N × N matrices with entries from X. If X is a Banach space, XN can be made to a Banach space on defining a norm in XN by (x1 , . . . , xN ) XN := x1 X + . . . + xN X

(1.5)

or by choosing any norm in XN equivalent to that one. Every operator A ∈ L(XN ) may then be written as an operator matrix A = (Aij )N i,j=1 , where Aij ∈ L(X), that is, L(XN ) may be identified with (L(X))N ×N . It is easily seen that K ∈ Cp (XN ) if and only if all entries Kij of the matrix K = (Kij )N i,j=1 are in Cp (X). Let A = (Aij )N i,j=1 ∈ L(XN ). The determinant det A of A is the operator in L(X) which is defined by det A = (−1)p(σ) AN,σ(N ) . . . A1,σ(1) , (1.6) σ

where σ ranges over all permutations of {1, . . . , N } and p(σ) is the signature of the permutation σ. If the operators Aij do not commute pairwise, then the order in which the factors in each term of the sum (1.6) are arranged is of significance. The arrangement chosen in (1.6) corresponds to the expansion of determinants with respect to the last row, i.e., one has A11 A12 det = −A21 A12 + A22 A11 , A21 A22 ⎞ ⎛ A11 A12 A13 det ⎝ A21 A22 A23 ⎠ A31 A32 A33 A12 A13 A11 A13 A11 A12 = A31 det − A32 det + A33 det A22 A23 A21 A23 A21 A22 and so on. However, if we are only interested in whether det A is a Φ+ - or Φ− operator and if the entries of A pairwise commute up to a compact operator, then the answer will not depend on the arrangement of the factors in the terms of the sum (1.6). In the theorems below, [A, B] denotes the commutator AB − BA.

12


1.14. Theorem. Let X be a Banach space and let A = (Aij )N i,j=1 ∈ L(XN ). (a) If [Aij , Alm ] ∈ C∞ (X) for j = m, then det A ∈ Φ− (X) =⇒ A ∈ Φ− (XN ), A ∈ Φ+ (XN ) =⇒ det A ∈ Φ+ (X). (b) If [Aij , Alm ] ∈ C∞ (X) for i = l, then det A ∈ Φ+ (X) =⇒ A ∈ Φ+ (XN ), A ∈ Φ− (XN ) =⇒ det A ∈ Φ− (X). (c) If the entries of A pairwise commute modulo compact operators, then A ∈ Φ(XN ) (resp. Φ+ (XN ), Φ− (XN )) if and only if det A ∈ Φ(X) (resp. Φ+ (X), Φ− (X)). This theorem says nothing about the connection between the index of A and that of det A. Under the hypothesis of part (c) both equality and inequality of these indices are possible. The following theorem states sufficient conditions for equality. 1.15. Theorem (Markus/Feldman). (a) Let H be a Hilbert space, let A be in Φ(HN ), and suppose the entries of A pairwise commute modulo C1 (H). Then det A ∈ Φ(H) and Ind A = Ind det A. (b) Let X be a Banach space, let A ∈ Φ(XN ), and suppose the entries of A pairwise commute modulo C0 (X). Then det A ∈ Φ(X) and Ind A = Ind det A.

1.5 Banach Algebras 1.16. Invertibility and spectrum. Let A be a Banach algebra with identity e. Throughout the book we assume that the scalar field is C, that ab ≤ a b for all a, b ∈ A and that e = 1. An element a ∈ A is said to be left (right, resp. two-sided) invertible in A if there exists an element b ∈ A such that ba = e (ab = e, resp. ba = ab = e). Two-sided invertible elements will be simply called invertible. The collection of all invertible elements of A will be denoted by GA. Note that GA is open and

forms a group with respect to the multiplication in A. For a ∈ A, let exp a = n≥0 an /n!. (a) If A is commutative, then an invertible element a belongs to the connected component of GA containing the identity e if and only if a has a logarithm in A, i.e., if there exists an element b ∈ A such that a = exp b. The spectrum sp a (or sp(a)) of an element a ∈ A is the set of all λ ∈ C such that a − λe is not (two-sided) invertible in A. In order to emphasize that

1.5 Banach Algebras

13

invertibility in A is meant, we shall sometimes write spA a instead of sp a. For several applications the following result will often be useful. (b) Let B be a closed subalgebra of A and suppose e ∈ B. If a ∈ B, then spB a is the union of spA a and a (possible empty) collection of bounded connected components of the complement of spA a. 1.17. Ideals. Let A be a Banach algebra. A closed subalgebra J of A is called a closed left (resp. right) ideal of A if aj ∈ J (resp ja ∈ J) for all a ∈ A and j ∈ J. The closed two-sided ideals of A are the closed ideals which are both left and right. An ideal J of A is said to be proper if J = A. A proper closed left (right, resp. two-sided) ideal J of A is called a maximal left (right, resp. two-sided) ideal if it is not properly contained in any other proper left (right, resp. two-sided) ideal of A. (a) In a Banach algebra with identity every proper left (right, resp. twosided) ideal is contained in some maximal left (right, resp. two-sided) ideal. Maximal two-sided ideals will be simply referred to as maximal ideals. If J is a proper closed two-sided ideal of A, then the quotient algebra A/J is a Banach algebra under the norm a + J := inf a + j. j∈J

(b) If A is commutative and has an identity and if J is a maximal ideal of A then the quotient algebra A/J is a field, i.e., every nonzero element of A/J has an inverse. (c) The radical R(A) of A is the intersection of all maximal left ideals of A. If A has an identity, then R(A) is a closed two-sided ideal of A and R(A) coincides with the intersection of all maximal right ideals of A. A Banach algebra with identity whose radical consists only of the zero element will be called semisimple. 1.18. The maximal ideal space. Let A be a commutative Banach algebra with identity e. A multiplicative linear functional on A is a continuous linear mapping m : A → C which preserves multiplication (m(ab) = m(a)m(b) for all a, b ∈ A) and takes the value 1 at e (m(e) = 1). The kernel of m is the set of all a ∈ A for which m(a) = 0. There is a one-to-one correspondence between the multiplicative linear functionals and the maximal ideals of A: the kernel of every multiplicative linear functional is a maximal ideal and every maximal ideal is the kernel of some (uniquely determined) multiplicative linear functional. Therefore no distinction between multiplicative linear functionals and maximal ideals will be made. We denote the set of all multiplicative linear functionals on A by M (A). The formula a(m) = m(a) (m ∈ M (A)) assigns to each a ∈ A a function a : M (A) → C. This function is called the Gelfand be the set of all functions transform of a. Let A a, for a ∈ A. The Gelfand topology on M (A) is the coarsest (weakest) topology on M (A) continuous. It is the topology induced on M (A) that makes all functions a∈A thought of as a subset of the dual space A∗ , the latter space provided with the

14


weak-star topology. Thus, an open neighborhood base of a point m0 ∈ M (A) is formed by the sets Ua1 ,...,an ;ε (m0 ) = m ∈ M (A) : |ai (m) − ai (m0 )| < ε for i = 1, . . . , n = m ∈ M (A) : |m(ai ) − m0 (ai )| < ε for i = 1, . . . , n , where a1 , . . . , an ∈ A and ε > 0. The set M (A) equipped with the Gelfand topology is called the maximal ideal space of A. Note that M (A) is a com is a (not necessarily closed) subalgebra of pact Hausdorff space and that A C(M (A)). The mapping Γ : A → C(M (A)), a → a will be referred to as the Gelfand map. Notice that in general Γ is neither one-to-one nor onto. The kernel of Γ coincides with the radical of A. Thus, if A is semisimple, then the Gelfand map ⊂ C(M (A)). is one-to-one and hence an (algebraic) isomorphism of A onto A Therefore we shall then simply write a and A instead of a and A. Finally, recall that if A is a (not necessarily semisimple) commutative Banach algebra with identity and if a ∈ A, then the range of a coincides with a∞ ≤ a, where a∞ := max | a(m)|. the spectrum spA a and m∈M (A)

1.19. Singly generated algebras. A commutative Banach algebra A with identity e is said to be singly generated if there is an a ∈ A such that the linear hull of the set {e, a, a2 , . . .} is dense in A: A = clos lin{e, a, a2 , . . .}. In that case the maximal ideal space of A is naturally homeomorphic to spA a, where spA a has the topology induced from the inclusion spA a ⊂ C and C is regarded as furnished with the usual topology. 1.20. The Shilov boundary. Let A be a commutative Banach algebra with identity e. A closed subset F ⊂ M (A) is called a boundary if max | a(m)| = max | a(m)| ∀ a ∈ A.

m∈M (A)

m∈F

The intersection of all boundaries is also a boundary. It is called the Shilov boundary of A and denoted by ∂S M (A). (a) A point m0 ∈ M (A) is in ∂S M (A) if and only if for each open neigh such that a∈A borhood U ⊂ M (A) of m0 there exists an sup m∈M (A)\U

| a(m)| < sup | a(m)|. m∈U

(b) Let B be a closed subalgebra of A and let e ∈ B. Then each maximal ideal m ∈ ∂S M (B) is contained in some maximal ideal belonging to M (A). In other words, each multiplicative linear functional on B belonging to ∂S M (B) admits an extension to a multiplicative linear functional on A.

1.5 Banach Algebras

15

A subset N of A is said to consist of joint topological divisors of zero if inf

n

zai : z ∈ A, z = 1 = 0

i=1

for each finite subset {a1 , . . . , an } ⊂ N . (c) Every maximal ideal belonging to the Shilov boundary of A consists of joint topological divisors of zero. 1.21. Maximal antisymmetric sets. Let Y be a compact Hausdorff space and let B be a closed subalgebra of C(Y ) containing the constants. A subset S of Y is said to be an antisymmetric set for B in case every function in B which is real-valued on S is constant on S. It is clear that every subset of Y consisting of a single point is an antisymmetric set. It is easy to see that of antisymmetric sets having a nonempty intersection, if {Sα } is any family then the union α Sα is also an antisymmetric set, and that the closure of an antisymmetric set is again such a set. Thus the union of all antisymmetric sets containing a given point y ∈ Y is a nonempty closed antisymmetric set. Antisymmetric sets of this form are called maximal antisymmetric sets for B. Any two maximal antisymmetric sets are obviously disjoint and Y is the union of all maximal antisymmetric sets. It is also easily seen that the maximal antisymmetric sets are the equivalence classes of the equivalence relation on Y defined by y1 ∼ y2 if there exists an antisymmetric set containing both y1 and y2 . If A and B are closed subalgebras of C(Y ) containing the constants and if B ⊂ A, then obviously every maximal antisymmetric set for A is contained in some maximal antisymmetric set for B. For F a closed subset of Y and a ∈ C(Y ), we define dist(a, B) := inf max |a(y) − b(y)|, b∈B y∈Y

distF (a, B) := inf max |a(y) − b(y)|. b∈B y∈F

1.22. Theorem (Glicksberg). Let Y be a compact Hausdorff space and B be a closed subalgebra of C(Y ) containing the constants. Let S be the family of maximal antisymmetric sets for B. Then, for every a ∈ C(Y ), dist(a, B) = max distS (a, B) S∈S

and the maximum is attained at some S0 ∈ S. Proof. It suffices to prove that there is an S0 ∈ S such that dist(a, B) = distS0 (a, B). Let F denote the collection of all nonempty closed subsets F of Y for which dist(a, B) = distF (a, B). Obviously Y ∈ F. If C is a subfamily of F also belongs to F . Indeed, F totally ordered by inclusion, then E := F∈ C

if b ∈ B, then the set {y ∈ F : |a(y) − b(y)| ≥ dist(a, B)} is compact and nonempty (since distF (a, B) = dist(a, B)), and hence the intersection of the

16


chain of all such sets, namely {y ∈ E : |a(y) − b(y)| ≥ dist(a, B)}, is also nonempty; as this holds for each b ∈ B, it follows that distE (a, B) = dist(a, B), i.e., that E ∈ F. Zorn’s lemma therefore guarantees the existence of a minimal element S0 of F , and to complete the proof of the theorem it remains to show that S0 is an antisymmetric set for B. Suppose this is false. Then there is a b ∈ B such that b is both real-valued and non-constant on S0 . Replacing b by a suitable real-linear combination of b and 1, we may assume that min b(y) = 0 and max b(y) = 1. Put y∈S0

2 U = y ∈ S0 : 0 ≤ b(y) ≤ , 3

y∈S0

1 ≤ b(y) ≤ 1 . V = y ∈ S0 : 3

Since U and V are nonempty proper closed subsets of S0 , the minimality of S0 implies that there exist c, d ∈ B with max |a(y) − c(y)| < dist(a, B), y∈U

max |a(y) − d(y)| < dist(a, B). y∈V

(1.7)

n

For n ≥ 1, set bn = (1 − b)2 and gn = bn c + (1 − bn )d. Clearly bn , gn ∈ B and 0 ≤ bn ≤ 1 on S0 . Since at each point of S0 the value of gn is a convex combination of c and d, it follows from (1.7) that max |a(y) − gn (y)| < dist(a, B).

(1.8)

y∈U ∩V

Now on U \ V , where 0 ≤ b < 1/3, we have n

bn = (1 − bn )2 ≥ 1 − 2n bn ≥ 1 −

2 n 3

,

while on V \ U , where 2/3 < b ≤ 1, we have n

bn = (1 − bn )2 ≤

3 n 1 1 ≤ . n ≤ n 2 n n (1 + b ) 2 b 4

Thus gn converges uniformly to c on U \ V and uniformly to d on V \ U . Combining this with (1.7) and (1.8), we see that if n is taken large enough then max |a(y) − gn (y)| < dist(a, B), contradicting the fact that S0 belongs y∈S0

to the family F . 1.23. Corollary (Shilov/Bishop). Under the hypothesis of the preceding theorem, if a ∈ C(Y ) then a ∈ B ⇐⇒ a|S ∈ B|S

∀ S ∈ S.

Proof. Immediate from Theorem 1.22. Comment. Here a|S ∈ B|S means that there is a b ∈ B (depending on S) such that a(y) = b(y) for all y ∈ S.

1.6 C ∗ -Algebras

17

1.24. Fibers. Let A be a commutative Banach algebra with identity e and suppose B is a closed subalgebra of A containing e. Consider the mapping τ : M (A) → M (B),

α → α|B,

which assigns to each functional in M (A) its restriction to B. A little thought shows that τ is continuous. For β ∈ M (B), put Mβ (A) = α ∈ M (A) : α|B = β . The set Mβ (A) is referred to as the fiber of M (A) over β. Since τ is continuous and Mβ (A) = τ −1 ({β}), Mβ (A) is always a compact subset of M (A). Of course, it can happen that Mβ (A) = ∅. There are two situations of particular importance. (a) Suppose that for each β ∈ M (B) the fiber Mβ (A) is either empty or a singleton. Then τ is one-to-one and because M (A) is compact, it follows that τ is a homeomorphism of M (A) onto the compact subset τ (M (A)) of M (B). By identifying M (A) with τ (M (A)), we may therefore think of M (A) as a subset of M (B). (b) Suppose A = C(Y ), where Y is a compact Hausdorff space, and let B be a closed subalgebra of A which contains the constant functions and separates the points of Y . We then have M (A) = ∂S M (A) = Y . For β in M (B), the fiber Mβ (A) = y ∈ Y : b(y) = β(b) ∀ b ∈ B is either empty or a singleton, since B separates the point of Y . Therefore we may regard Y as a compact subset of M (B). Since b = max |b(y)| for y∈Y

all b ∈ B, the Shilov boundary of B is a closed subset of Y . Thus, we have ∂S M (B) ⊂ Y ⊂ M (B).

1.6 C ∗-Algebras 1.25. Definitions. A mapping a → a∗ of a Banach algebra A into itself is called an involution on A if a∗∗ = a,

(a + b)∗ = a∗ + b∗ ,

(ab)∗ = b∗ a∗ ,

(λa)∗ = λa∗

for all a, b ∈ A and λ ∈ C. A Banach algebra with an involution a → a∗ that satisfies aa∗ = a2 for all a ∈ A is called a C ∗ -algebra. If Y is a compact Hausdorff space and H is a Hilbert space, then C(Y ) and L(H) are C ∗ -algebras with their natural involutions.

18


1.26. Basic properties of C ∗ -algebras. (a) (Gelfand/Naimark). If A is a commutative C ∗ -algebra with identity, then the Gelfand map is an isometric star-isomorphism of A onto C(M (A)). (b) (Gelfand/Naimark). If A is any C ∗ -algebra, then there exists a Hilbert space H such that A is isometrically star-isomorphic to some C ∗ subalgebra of L(H). (c) If A is a commutative C ∗ -algebra with identity, then ∂S M (A) = M (A). (d) If A is a C ∗ -algebra with identity and B is a C ∗ -subalgebra of A containing the identity, then an element a ∈ B is left (right, resp. two-sided) invertible in A if and only if it is so in B. (e) Let A and B be C ∗ -algebras and let ϕ : A → B be an (algebraic) star-homomorphism. Then ϕ(a) ≤ a for all a ∈ A and the range of ϕ is closed in B. If, in addition, ϕ is one-to-one, then ϕ(a) = a for all a ∈ A. (f) Let A be a C ∗ -algebra and let J be a closed two-sided ideal of A. Then J is selfadjoint (that is, J ∗ = J) and A/J provided with the involution (a + J)∗ = a∗ + J and the usual quotient norm is a C ∗ -algebra. (g) Let A be a C ∗ -algebra, let B be a C ∗ -subalgebra of A, and let J be a closed two-sided ideal of A. Then B + J is a C ∗ -subalgebra of A and the C ∗ -algebras (B + J)/J and B/(B ∩ J) are isometrically star-isomorphic. 1.27. Fibers over C ∗ -algebras. Let A be a commutative Banach algebra with identity and B a closed subalgebra of A containing the identity. (a) By 1.20(b), for β in the Shilov boundary ∂S M (B) the fiber Mβ (A) is not empty. (b) In particular, if B is a C ∗ -subalgebra of A, then ∂S M (B) = M (B) and so Mβ (A) = ∅ for each β ∈ M (B). Via α1 ∼ α2 ⇐⇒ α1 |B = α2 |B

(α1 , α2 ∈ M (A))

an equivalence relation is given on M (A). The corresponding partition of M (A) into equivalence classes is nothing else than the partition of M (A) into fibers over M (B). This set of equivalence classes equipped with a natural topology is homeomorphic to M (B). (c) Now suppose Y is a compact Hausdorff space and B is a C ∗ -subalgebra of C(Y ) containing the constant functions. Note that M (C(Y )) = Y . For β ∈ M (B), denote by Yβ ⊂ Y the fiber Mβ (C(Y )). Then the maximal antisymmetric sets for B are the fibers Yβ , β ∈ M (B). This can be proved as follows. We first show that each antisymmetric set S for B is contained in some fiber Yβ . Assume the contrary, i.e., assume there are y1 , y2 ∈ S and two distinct points β1 , β2 ∈ M (B) such that y1 ∈ Yβ1 and y2 ∈ Yβ2 . Then, for every

1.6 C ∗ -Algebras

19

b ∈ B, b(y1 ) = b(β1 ) and b(y2 ) = b(β2 ). Since B is isometrically isomorphic to C(M (B)), there is a b ∈ B such that b(β1 ) = b(β2 ). Since B is a C ∗ algebra, both Re b and Im b belong to B, and we have (Re b)(β1 ) = (Re b)(β2 ) or (Im b)(β1 ) = (Im b)(β2 ). Thus, there is a real-valued function in B taking two distinct values on S, which is impossible if S is an antisymmetric set for B. So it remains to show that each fiber Yβ is an antisymmetric set for B. But this is obvious, since B|Yβ ∼ = C and so B|Yβ does not contain non-constant functions at all. In particular, Theorem 1.22 and Corollary 1.23 are valid if S is replaced by the collection of all fibers Yβ , β ∈ M (B). (d) Thus, if Y is a compact Hausdorff space and B and A are closed subalgebras of C(Y ) containing the constants C such that C ⊂ B ⊂ A ⊂ C(Y ), then each maximal antisymmetric set for A is contained in some maximal antisymmetric set for B. Finally, note that both C and C(Y ) are C ∗ -algebras whose maximal antisymmetric sets are the whole space Y and the singletons {y}, y ∈ Y , respectively. 1.28. Restriction algebras. Let A be a commutative C ∗ -algebra with identity. By the Gelfand-Naimark theorem 1.26(a), A is isometrically starisomorphic to C(Y ), where Y = M (A). If F is a closed subset of Y , then IF = {a ∈ A : a|F = 0} is obviously a closed (two-sided) ideal of A. Moreover, it can be shown that every closed ideal of A is of this form (see, e.g., Naimark [362, pp. 247–248]). Let A|F denote the algebra whose elements are the restrictions a|F (or, more precisely, a|F ), for a ∈ A. When endowed with the norm a|F = max |a(y)|, y∈F

A|F is a C ∗ -algebra which is isometrically star-isomorphic to C(F ) in a natural fashion. It is called the restriction of A to F . It is not difficult to show that the mapping ϕ : A/IF → A|F,

a + IF → a|F

is an isometric star-isomorphism of the C ∗ -algebra A/IF , equipped with the natural norm a + IF := inf a + g, g|F =0

onto the C ∗ -algebra A|F . Now let B be a closed subalgebra of A containing the identity. Via the Gelfand map, B is isometrically isomorphic to a closed subalgebra of C(Y )

20


and we shall identify B with that subalgebra. It is, in general, a delicate problem to decide whether or not B|F is a closed subalgebra of A|F ∼ = C(F ). A closed subset F of Y is called a peak set for B if there is a b ∈ B such that b(y) = 1 for all y ∈ F and |b(y)| < 1 for all y ∈ Y \F . It is easily seen that a finite or countable intersection of peak sets for B is again a peak set for B (if

∞ ∞ bn “peaks” on Fn then n=0 bn /2n “peaks” on n=0 Fn ). A closed subset of Y is called a weak peak set for B, if it is the (possibly uncountable) intersection of peak sets for B. One can show that every maximal antisymmetric set for B is a weak peak set for B. The role played by weak peak sets is illuminated by the following result: If F is a weak peak set for B, then B|F is a closed subalgebra of A|F ∼ = C(F ); moreover, B|F is isometrically isomorphic to the quotient algebra B/IFB , where IFB = {b ∈ B : b|F = 0}. 1.29. Matrix norms. Let A be a normed space. A norm in AN ×N , the linear space of all N × N matrices with entries from A, is said to be admissible if there are constants m, M > 0 such that m max aij ≤ aAN ×N ≤ M max aij i,j

i,j

for all a = (aij )N i,j=1 ∈ AN ×N . If A is a Banach space and if AN ×N is endowed with an admissible norm, then AN ×N is also a Banach space. If A is an algebra, then AN ×N is an algebra under the usual matrix operations. If A is a Banach algebra, then a norm in AN ×N will be called a Banach algebra norm if it is admissible and if ab ≤ a b for all a, b ∈ AN ×N . Now let A be a C ∗ -algebra. Then the mapping (aij ) → (a∗ji ) is an involution on AN ×N . A norm in AN ×N will be called a C ∗ -norm if it is a Banach algebra norm and if aa∗ = a2 for all a ∈ AN ×N . (a) If A is a C ∗ -algebra with identity, then there exists exactly one C ∗ norm in AN ×N . Proof. To see this, notice first that by / GAN ×N a2 := r(a∗ a) := max |λ| : λ ∈ C, a∗ a − λe ∈ a C ∗ -norm is given in AN ×N . Now let · 1 and · 2 be two C ∗ -norms in AN ×N and denote by A1N ×N and A2N ×N the algebra AN ×N endowed with the norm · 1 and · 2 , respectively. The identity mapping is clearly an algebraic star-isomorphism of A1N ×N onto A2N ×N . So 1.26(e) implies that it is an isometry, that is, a1 = a2 for all a ∈ AN ×N . Let the norm in CN be given by (zk )N k=1

:=

N

k=1

1/2 |zk |

2

.

(1.9)

1.7 Local Principles

21

The norm in (L(CN ))∗ , the dual space of L(CN ), is ϕ := sup |ϕ(z)| : z ∈ CN , z = 1 . We denote by Iij (i, j ∈ {1, . . . , N }) the operator in L(CN ) which is defined by Iij : (zn , . . . , zN ) → (0, . . . , 0, zj , 0, . . . , 0) where the zj occupies the i-th place. (b) Let A be a commutative C ∗ -algebra. Then there exists exactly one C ∗ norm in AN ×N . This norm can be given by aAN ×N

!" # N " " " ∗ = sup " aij ϕ(Iij )" : ϕ ∈ (L(CN )) , ϕ = 1 , i,j=1

A

where a = (aij )N i,j=1 . Proof. For a proof we refer to Sakai [446, 1.22.5]. Note that the linear space AN ×N can be naturally identified with the algebraic tensor product A ⊗ L(CN ). The above reference implies that there is precisely one C ∗ -norm in A ⊗ L(CN ), namely, the norm which generates the injective tensor product.

1.7 Local Principles 1.30. Definitions. Let A be a Banach algebra with identity e. A subset M ⊂ A is called a localizing class if (i) 0 ∈ / M, (ii) for any f1 , f2 ∈ M there exists a third element f ∈ M such that fj f = f fj = f (j = 1, 2). Two elements a, b ∈ A are said to be M -equivalent from the left (resp. from the right) if resp. inf f (a − b) = 0 . inf (a − b)f = 0 f ∈M

f ∈M

An element a ∈ A is called M -invertible from the left (resp. from the right) if there are a b ∈ A and an f ∈ M such that baf = f (resp. f ab = f ). A system {Mτ }τ ∈T of localizing classes is said to be covering if from each choice {fτ }τ ∈T (fτ ∈ Mτ ) there can be selected a finite number of elements fτ1 , . . . , fτm whose sum is invertible in A. Now suppose T is a topological space. Then a system {Mτ }τ ∈T of localizing classes will be said to be overlapping if

22


(iii) each Mτ is a bounded subset of A; (iv) f ∈ Mτ0 (τ0 ∈ T ) implies that f ∈ Mτ for all τ in some open neighborhood of τ0 ; (v) the elements of F := Mτ commute pairwise. τ ∈T

Let {Mτ }τ ∈T be an overlapping system of localizing classes. The commutant of F is the set Com F := a ∈ A : af = f a ∀ f ∈ F . It is clear that Com F is a closed subalgebra of A. For τ ∈ T , let Zτ denote the set of all elements in Com F which are Mτ -equivalent to zero both from the left and from the right. Note that Zτ is a closed (by virtue of (iii)) two-sided ideal of Com F which does not contain the identity (if e ∈ Zτ , then there are fn ∈ Mτ such that fn → 0 as n → ∞, and since there exist gn = 0 in Mτ such that fn gn = gn , it follows that fn ≥ 1, which is a contradiction). For each a ∈ Com F let aτ denote the coset a + Zτ of the quotient algebra Com F/Zτ . Finally, recall that a function f : Y → R given on a topological space Y is called upper semi-continuous at y0 ∈ Y if for each ε > 0 there is a neighborhood Uε ⊂ Y of y0 such that f (y) < f (y0 ) + ε whenever y ∈ Uε . The function f is said to be upper semi-continuous on Y if it is upper semicontinuous at each y ∈ Y . Equivalently, f is upper semi-continuous on Y if and only if {y ∈ Y : f (y) < α} is an open subset of Y for every α ∈ R. Notice that if Y is a compact Hausdorff space and f : Y → R is a bounded upper semicontinuous function on Y , then there is a y0 ∈ Y such that f (y0 ) = sup f (y). y∈Y

1.31. Lemma. (a) Let M be a localizing class, let a, a0 ∈ A, and suppose a and a0 are M -equivalent from the left (resp. from the right). Then a is M -invertible from the left (resp. from the right) if and only if so is a0 . (b) Let {Mτ }τ ∈T be a system of localizing classes having property 1.30(iii), let τ ∈ T and a ∈ Com F . Then a is Mτ -invertible in Com F from both the left and the right if and only if aτ ∈ G(Com F/Zτ ). Proof. (a) Let a be M -invertible from the left. Then there are b ∈ A and f ∈ M such that baf = f . Since a and a0 are M -equivalent from the left, there is a g ∈ M such that (a − a0 )g < 1/b. Choose h ∈ M so that f h = gh = h. Then ba0 h = bah − b(a − a0 )h = baf h − b(a − a0 )gh = h − uh = (e − u)h, where u := b(a−a0 )g, and because u < 1, we deduce that e−u ∈ GA. Thus, if we let v = (e − u)−1 b, then va0 h = h, which shows that a0 is M -invertible from the left.

1.7 Local Principles

23

(b) Let aτ ∈ G(Com F/Zτ ). Then there is a b ∈ Com F such that ba − e is in Zτ . This implies that ba is Mτ -equivalent from the left to e, and from part (a) we deduce that ba and thus a is Mτ -invertible from the left. It can be shown similarly that a is Mτ -invertible from the right. Conversely, if there are b ∈ Com F and f ∈ Mτ such that baf = f , then (ba − e)f = 0, hence ba − e ∈ Zτ , and thus bτ aτ = eτ . 1.32. Theorem (Gohberg/Krupnik). Let A be a Banach algebra with identity, let {Mτ }τ ∈T be a covering system of localizing classes, and let a ∈ Com F . (a) Suppose that, for each τ ∈ T , a is Mτ -equivalent from the left (resp. right) to aτ ∈ A. Then a is left-invertible (resp. right-invertible) in A if and only if aτ is Mτ -invertible from the left (resp. right) for all τ ∈ T . (b) If the system {Mτ }τ ∈T has property 1.30(iii), then a ∈ GA if and only if aτ is invertible in Com F/Zτ for all τ ∈ T . (c) Let the system {Mτ }τ ∈T be overlapping. Then the mapping T → R+ ,

τ → aτ

is upper semi-continuous. If aτ0 ∈ G(Com F/Zτ0 ) for some τ0 ∈ T , then aτ ∈ G(Com F/Zτ ) for all τ in some open neighborhood of τ . Proof. (a) If a is left-invertible, then a is Mτ -invertible from the left for all τ ∈ T and hence, by Lemma 1.31(a), aτ is Mτ -invertible from the left for all τ ∈ T. Conversely, suppose aτ is Mτ -invertible from the left for all τ ∈ T . It follows again from Lemma 1.31(a) that a is Mτ -invertible from the left for all τ ∈ T . Thus, there are bτ ∈ A and fτ ∈ Mτ such that bτ afτ = fτ . Since m {Mτ }τ ∈T is covering, we can choose fτ1 , . . . fτm so that i=1 fτi is in GA. m Put s = i=1 bτi fτi . Then sa = bτi fτi a = bτi afτi = fτi i

i

i

−1 and it results that ( fτi ) is a left-inverse of a. (b) If aτ ∈ G(Com F/Zτ ) for all τ ∈ T , then a ∈ G(Com F ) and thus a ∈ GA by virtue of Lemma 1.31(b) and part (a) of the present theorem. On the other hand, if a ∈ GA, then clearly a−1 ∈ Com F , hence a ∈ G(Com F ) and thus aτ ∈ G(Com F/Zτ ) for all τ ∈ T . (c) Let τ0 ∈ T and ε > 0. Choose a z ∈ Zτ0 so that a + z < aτ0 + ε/2. Since z is Mτ0 -equivalent to zero from the left, there is an f ∈ Mτ0 such that zf < ε/2. From 1.30(iv) we deduce that f ∈ Mτ for all τ in some open neighborhood U (τ0 ) of τ0 . Put y = z − zf . If τ ∈ U (τ0 ), then there exists a g ∈ Mτ such that f g = g (1.30(ii)). Consequently, yg = zg−zf g = zg−zg = 0,

24


and since y ∈ Com F (1.30(v)), it follows that y ∈ Zτ for all τ ∈ U (τ0 ). Hence, aτ ≤ a + y for τ ∈ U (τ0 ). Thus, if τ ∈ U (τ0 ), then aτ − aτ0 < a + y − a + z +

ε ε ε ≤ y − z + = zf + < ε, 2 2 2

which proves the upper semi-continuity of τ → aτ at τ0 . Now suppose aτ0 ∈ G(Com F/Zτ0 ). Then there is a b ∈ Com F such that (ba − e)τ0 = ba − e + Zτ0 = 0,

(ab − e)τ0 = ab − e + Zτ0 = 0.

By what has just been proved, the mappings τ → (ba − e)τ and τ → (ab − e)τ are upper semi-continuous. Hence ba − e + Zτ = (ba − e)τ
0. Choose a1 , . . . , an ∈ A and z1 , . . . , zn ∈ N0 such that " " n " " ε " "a + (1.10) a z j j " < aN0 + . " 2 j=1

Define the open neighborhood Uε ⊂ M (B) of N0 as ! # n % zj (N )| < ε 2 aj + 1 for j = 1, . . . , n Uε = N ∈ M (B) : | j=1

and put yj = zj − zj (N )e. Then yj ∈ N and so " " n " " " a + aN ≤ " a y j j ". "

(1.11)

j=1

Now if N ∈ Uε , then (1.10) and (1.11) give " " " " n n " " " ε " " " " a + − a + a y a y aN − aN0 ≤ " j j" j j" + " " 2 j=1 j=1 " " " " n " ε " n " ε " " " " ≤" aj (yj − zj )" + = " zj (N )aj " " + 2 < ε. 2 j=1 j=1 This proves the upper semi-continuity of aN at N0 . The second part of the assertion can be proved as the corresponding assertion of Theorem 1.32(c). (c) Let L be any maximal left ideal of A. From

Lemma 1.34 we know that N := L ∩ B is a maximal ideal of B. If x = zk ak (zk ∈ N, ak ∈ A), then x= ak zk ∈ L (because L is a left ideal), and it follows that JN ⊂ L. Let πN denote the canonical homomorphism of A onto AN = A/JN . Clearly, πN (L) is a left ideal of AN . If πN (L) would not be proper, then there would exist a y ∈ L such that πN (y) = eN , whence y − e ∈ JN ⊂ L, and thus e ∈ L, which contradicts the maximality of L. Consequently, πN (L) is a proper left

1.8 Lp and H p

27

ideal of AN . By 1.17(a), πN (L) is contained in some maximal left ideal LN −1 −1 of AN . We claim that πN (LN ) = L. It is clear that L ⊂ πN (LN ) and that −1 −1 πN (LN ) is a left ideal of A. If πN (LN ) would contain the identity e, then eN = πN (e) would belong to LN , which is impossible, since LN is maximal. −1 (LN ) is a proper left ideal of A containing the maximal left ideal Thus, πN −1 (LN ). L, and this implies that L = πN JN and let L be any maximal left ideal of A. By what Now let x ∈ N ∈M (B)

has just been proved, there is an N ∈ M (B) and a maximal left ideal LN of −1 (LN ). Because πN (x) = 0, we have πN (x) ∈ LN and AN such that L = πN hence, x ∈ L. Thus, x belongs to each maximal left ideal of A and since A was supposed to be semisimple, it follows that x = 0. & (d) If A is a C ∗ -algebra, then so is the direct sum AN . The canonN ∈M (B)

ical homomorphisms πN : A → AN produce a star-homomorphism ' AN , a → (aN )N ∈M (B) . π:A→ N ∈M (B)

Since a C ∗ -algebra is always semisimple, the above proved part (c) implies that π is one-to-one, and so 1.26(e) shows that π is an isomorphism, i.e., a = sup aN for all a ∈ A. Finally, in view of the upper semi-continuity N ∈M (B)

of aN (part (b)), the sup may be replaced by max. Remark. If A and B are C ∗ -algebras, then JN = A for every N ∈ M (B). Indeed, if N0 ∈ M (B), then there is a b ∈ B such that b(N0 ) = 1 and 0 < b(N ) < 1 for N = N0 , and part (d) of the above theorem gives 1 = b =

max bN =

N ∈M (B)

max b(N )eN = eN0 ,

N ∈M (B)

hence eN0 = 0, and thus JN0 = A. If A = C (the continuous functions on the unit circle T) and B = CA (the disk algebra), then M (B) = clos D (the closed unit disk) and it is easily seen that, for z ∈ clos D, Jz = C if and only if z ∈ T. This is a special case of a more general result: if A is a commutative Banach algebra with identity element, then JN = A for all N belonging to the Shilov boundary of M (A).

1.8 Lp and H p 1.36. The spaces Lp . Let T be the complex unit circle. We denote by Lp (1 ≤ p < ∞) the Banach space of (classes of) complex-valued measurable functions on T summable (with respect to Lebesgue measure) in the p-th power. We let C (resp. L∞ ) denote the C ∗ -algebra of all continuous (resp.

28


[classes of] measurable and essentially bounded) functions on T. Clearly, C ⊂ L∞ ⊂ Lr ⊂ Ls ⊂ L1 for 1 ≤ s ≤ r ≤ ∞. The norm in Lp (1 ≤ p ≤ ∞) will be denoted by · p . Given f ∈ L1 we define its Fourier coefficients fn (n ∈ Z) by ( 2π 1 f (eiθ )e−inθ dθ. fn = 2π 0

Notice that f ∈ L2 if and only if n∈Z |fn |2 < ∞ and that f 22 = n∈Z |fn |2 . Moreover, L2 is a Hilbert space and if we define the functions χn (n ∈ Z) by χn (t) = tn (t ∈ T), then {χn }n∈Z is an orthogonal basis in L2 . 1.37. The harmonic extension. Let D be the open unit disk in the complex plane. The harmonic extension of a function f ∈ L1 is the function f defined in D by f(reiθ ) = fn r|n| einθ (0 ≤ r < 1, 0 ≤ θ < 2π). n∈Z

Note that f can also be defined via the Poisson integral, ( 2π 1 iθ f (re ) = kr (θ − t)f (eit ) dt, 2π 0 where

1 − r2 1 − 2r cos θ + r2 is the Poisson kernel. Sometimes it will be convenient to denote the harmonic extension f by hf . For fixed r ∈ [0, 1), the function f(reiθ ) may be viewed as a function given on the unit circle T = {eiθ : 0 ≤ θ < 2π}; we denote this function by hr f of fr , that is, kr (θ) =

fr (eiθ ) = (hr f )(eiθ ) = f(reiθ ) = (hf )(reiθ ). We shall occasionally think of functions in Lp as extended harmonically into D, that is, as given on the closed unit disk clos D. 1.38. Basic properties of the harmonic extension. (a) The harmonic extension f of a function f ∈ L1 is harmonic in D, i.e., f ∈ C ∞ (D) and ∆f = 0, where ∆ is the Laplace operator. (b) Let f ∈ Lp . If 1 ≤ p ≤ ∞, then sup hr f p ≤ f p r∈[0,1)

and if 1 ≤ p < ∞, then hr f − f p → 0 as

r → 1 − 0.

1.8 Lp and H p

If f ∈ C, then

hr f − f ∞ → 0 as

29

r → 1 − 0,

and if f ∈ L∞ , then hr f converges to f in the weak-star topology of L∞ = (L1 )∗ , that is, ( 2π ( 2π iθ iθ (hr f )(e )g(e ) dθ → f (eiθ )g(eiθ ) dθ (r → 1 − 0) 0

0

for every g ∈ L1 . (c) (Fatou’s theorem). Let F be a harmonic function in D, put Fr (eiθ ) = F (reiθ ), and suppose sup Fr p < ∞, where 1 ≤ p ≤ ∞. Then the non-tangential r∈[0,1)

limit f (eiθ ) := lim F (z) z→eiθ

exists and is finite almost everywhere on T, and the “boundary function” f belongs to Lp . If 1 < p ≤ ∞, then the harmonic extension f of f coincides with F , but if p = 1, then all one can say is that there is a complex measure dσ on T singular with respect to Lebesgue measure such that ( 2π 1 kr (θ − t) dσ(t), F (reiθ ) = f(reiθ ) + 2π 0 where kr is the Poisson kernel. 1.39. The spaces H p . For 1 ≤ p ≤ ∞ put H p = {f ∈ Lp : fn = 0 for all n < 0}, CA = {f ∈ C : fn = 0 for all n < 0}. Thus, H p (the Hardy spaces) and CA (the disk algebra) are closed subspaces of Lp and C, respectively. Clearly, CA ⊂ H ∞ ⊂ H r ⊂ H s ⊂ H 1

for

1 ≤ s ≤ r ≤ ∞.

Moreover, H ∞ and CA are Banach algebras. H 2 is a Hilbert space and we

2 have f ∈ H if and only if n∈Z+ |fn |2 < ∞; the set {χn }n∈Z+ forms an orthogonal basis in H 2 . 1.40. The analytic extension. If f ∈ H 1 , then the harmonic extension f is an analytic function in D and it is therefore also referred to as the analytic extension. Thus, for f ∈ H 1 one has fn z n (z ∈ D). f(z) = n≥0

30


Some properties of the analytic extension of H p functions immediately result from 1.38(b). Fatou’s theorem can be made more precise as follows. (a) Let F be an analytic function in D, put Fr (eiθ ) = F (reiθ ), and suppose that sup Fr p < ∞, where 1 ≤ p ≤ ∞. Then the non-tangential limit r∈[0,1)

f (eiθ ) := lim F (z) z→eiθ

exists and is finite almost everywhere on T, and the “boundary function” f is in H p . If 1 ≤ p ≤ ∞, then the analytic extension f of f coincides with F . That, unlike the L1 situation, the singular measure dσ does not appear in the H 1 case is, at long last, due to the following fact, which is of great importance in connection with many other problems, too. (b) (The F. and M. Riesz theorem). A function in H 1 vanishes either almost everywhere or almost nowhere on T. In what follows we shall frequently identify functions in H p with their analytic extension in D. 1.41. Inner-outer factorization. A function ϕ ∈ H ∞ is called an inner function if |ϕ(t)| = 1 for almost all t ∈ T. A function g ∈ H 1 is said to be an outer function if its analytic extension can be represented in the form ( 2π iθ 1 e +z iθ log ψ(e g(z) = c exp ) dθ , (1.12) 2π 0 eiθ − z where c ∈ T, ψ ∈ L1 , ψ ≥ 0 a.e. on T, log ψ ∈ L1 . The inner-outer factorization theorem says the following. (a) Every function f ∈ H p (1 ≤ p ≤ ∞) which is not identically zero has a factorization of the form f = ϕg where ϕ ∈ H ∞ is an inner function and g ∈ H p is an outer function. This factorization is unique up to a multiplicative constant. A remarkable property of H 1 functions is that log |f | ∈ L1 whenever f does not vanish identically. The outer function g occurring in the inner-outer factorization theorem can be obtained through (1.12) with ψ = |f |. Examples of inner functions are χn (t) = tn (n ≥ 0), t+1 α−t (α ∈ D). Sa (t) = exp a (a > 0), bα (t) = t−1 1 − αt Functions of the form bα1 . . . bαn (αj ∈ D) are also inner and are called finite be a sequence of complex numbers such that Blaschke products. Let {αn }∞ n=1

∞ 0 < |α1 | ≤ |α2 | ≤ . . . < 1 and n=1 (1 − |αn |) < ∞. Then the infinite product b(z) =

∞ |αn | αn − z αn 1 − αn z n=1

(1.13)

1.8 Lp and H p

31

converges uniformly in each disk |z| ≤ R < 1, one has |b(z)| < 1 for |z| < 1, and the non-tangential limits of b have modulus 1 a.e. on T. Thus b is an inner function. Each αn is a zero of b, with multiplicity equal to the number of times it occurs in the sequence, and b(z) has no other zeros in D. A function of the form |αn | αn − z b(z) = z m αn 1 − αn z n

is called a Blaschke product. Here m ∈ Z+ , αn ∈ D, and (1 − |αn |) < ∞. The set {αn } may be finite or even empty. In the latter case b(z) := z m . In this connection notice the following: if f ∈ H 1 , f ≡ 0, and if α1 , α2 , . . . are the zeros of f in D, repeated according to multiplicity, then (1 − |αn |) < ∞. n

A function of the form ( S(z) = exp −

π −π

eiθ + z dµ(θ) eiθ − z

(z ∈ D),

where µ is a positive measure on [−π, π] singular with respect to Lebesgue measure, can also be shown to be inner. Such functions are called singular inner functions. If µ is the unit mass at 0, we obtain z+1 S1 (z) = exp . z−1 A singular inner function has no zeros in D, but the radial limits lim S(z) are z→t

zero for each t belonging to the (closed) support of the measure µ. The canonical factorization theorem states the following. (b) Every inner function ϕ ≡ 0 has a unique factorization ϕ = cBS where c ∈ T, B is a Blaschke product, and S is a singular inner function. Here are some important properties of outer functions. (c) If g ∈ H 1 and h ∈ H 1 are outer and if |g| = |h| a.e. on T, then g = λh a.e. on T with some constant |λ| = 1. (d) If f ∈ Lp (1 ≤ p ≤ ∞), f ≥ 0 a.e. on T, and if log f ∈ L1 , then there exists an outer function g ∈ H p such that f = |g| a.e. on T. In particular, if f is a real-valued function in L∞ and ess inf f (t) > 0, then there is an outer t∈T

function g ∈ H ∞ such that f = |g| a.e. on T. (e) If f ∈ H 1 is outer then f(z) = 0 for z ∈ D. If S is a set, f S denotes the set {f s : s ∈ S}. Functions of the form

32

1 Auxiliary Material N

f n tn

(t ∈ T)

n=0

are referred to as analytic polynomials and the (linear) set of all analytic polynomials will be denoted by PA . (f) Let f ∈ H ∞ . Then the following are equivalent: (i) f is outer; (ii) f PA is dense in H 2 ; (iii) f H 2 is dense in H 2 . We finally mention some characterizations of the outer functions in H ∞ that are invertible in L∞ . (g) For f ∈ H ∞ the following are equivalent: (i) f ∈ GL∞ and f is outer; (ii) inf |f(z)| > 0; z∈D

(iii) f ∈ GH ∞ ; (iv) f H 2 = H 2 . 1.42. The Riesz projection. A function of the form N

f n tn

(t ∈ T)

n=−N

is called a Laurent polynomial and the (linear) set of all Laurent polynomials will be denoted by P. The Riesz projection is the operator P which is defined on P by N N P : fn tn → f n tn . n=0

n=−N

A famous theorem of M. Riesz states that cp :=

P f p f ∈P,f =0 f p sup

is finite if 1 < p < ∞ and infinite if p = 1 or p = ∞. Thus, if 1 < p < ∞ then P extends from the dense subset P of Lp to a bounded projection of the whole Lp onto H p . In 2000, Hollenbeck and Verbitsky [286] proved that cp = 1/ sin(π/ max{p, q}) where 1/p + 1/q = 1. ◦

p In the case p = 2, P is the orthogonal projection of L2 onto H 2 . Let H− p denote the kernel of P and put Q = I − P . Then L decomposes into the · ◦

p and direct sum H p +H−

H p = Im P = Ker Q,

◦

p H− = Im Q = Ker P.

1.8 Lp and H p

33

◦

◦

p p Furthermore, let H p := {f ∈ H p : f0 = 0} and H− :=H− +C. Clearly, ◦

·

p p Lp =H p + H− . It is obvious that H− = {h : h ∈ H p } (1 ≤ p ≤ ∞), where p h(t) := h(t) (t ∈ T). Therefore H− will occasionally also be denoted by H p . ◦

q (1/p + 1/q = 1) in the The dual space of H p (1 < p < ∞) is Lq / H− following sense: the general form of a functional G ∈ (H p )∗ is given by

G(f ) =

1 2π

(

2π

f (eiθ )g(eiθ ) dθ,

(1.14)

0

where g ∈ H q ; g is uniquely determined by G, and ◦ q . G(H p )∗ = inf g + hLp : h ∈H−

It can be easily checked that, for 1 < p < ∞, 1 gH q ≤ G(H p )∗ ≤ gH q , cq

(1.15)

where cq = P L(Lq ) . Note that c2 = 1, and consequently, for p = q = 2 we have gH 2 = G(H 2 )∗ . We also remark that every G ∈ (H 1 )∗ can be represented in the form (1.14) with some g ∈ L∞ . Finally, note that P is given in terms of Fourier coefficients by (P f )n = fn for n ≥ 0,

(P f )n = 0 for n < 0.

Also notice that the operator P − Q = 2P − I is nothing else than the Cauchy singular integral operator S over T, ( ϕ(τ ) 1 dτ (t ∈ T), (1.16) (Sf )(t) = πi T τ − t the integral understood in the Cauchy principal-value sense. 1.43. The conjugate function. If F is a real-valued harmonic function in D, then there is a real-valued harmonic function G in D such that F + iG is analytic in D. Any two such functions G differ only by a real constant and the function G for which G(0) = 0 is called the conjugate function of F and denoted by F) . If F is a complex-valued harmonic function in D, then F can be written as F = U + iV , where U and V are real-valued, and F) is defined ) + iV) . as U Now let f ∈ L1 . One can show that the conjugate function of the harmonic extension hf possesses non-tangential limits almost everywhere on T. This “boundary function” is then referred to as the conjugate function of f and denoted by f). Thus, f)(eiθ ) = lim (hf ))(z). z→eiθ

34


If f ∈ L1 and (hf )(reiθ ) =

fn r|n| einθ

(0 ≤ r < 1,

0 ≤ θ < 2π),

n∈Z

then (hf ))(reiθ ) = −i

(sign n)fn r|n| einθ ,

n∈Z

where sign 0 := 0. Also notice the formula ( 2π 1 qr (θ − ϕ)f (eiϕ ) dϕ, (hf ))(reiθ ) = 2π 0 where qr (θ) =

2r sin θ . 1 − 2r cos θ + r2

If f ∈ P, then f) is also in P. Using the Riesz projection one can write f) = −i(P f − Qf − f0 ) = −i(2P f − f − f0 ).

(1.17)

We therefore deduce from the M. Riesz theorem in 1.42 that the supremum of f)p /f p over f ∈ P \ {0} is finite for 1 < p < ∞ and infinite for p = 1 and p = ∞. Thus, the operator of conjugation, f → f), is bounded on Lp for 1 < p < ∞. In particular, (1.17) makes a correct sense for arbitrary f ∈ Lp (1 < p < ∞). The conjugation operator f → f) is frequently also referred to as the Hilbert transform. It can be given by the formula ( 2π θ−t 1 cot (1.18) f)(eiθ ) = f (eit ) dt, 2π 0 2 which is correct if the integral is interpreted in the Cauchy principal-value sense. Finally, we use (1.17) to define the Riesz projection on L1 , that is, for f ∈ L1 we define the measurable function P f as 12 (f + if)) + 12 f0 . 1.44. Lp and H p with weight. It will be convenient to denote the Lebesgue measure on T by dm. By a weight we understand a Lebesgue measurable function w : T → [0, ∞] such that the pre-image w−1 ({0, ∞}) has Lebesgue measure zero. Let w be a weight belonging to Lp (1 < p < ∞). The weighted Lp space Lp (w) is the Banach space of all functions f on T for which f w ∈ Lp , i.e., ( 1/p p p f p,w := |f | w dm < ∞. T

p

p

The weighted H space H (w) is defined as the closure in Lp (w) of the linear hull of the set {χ0 , χ1 , χ2 , . . .}. Sometimes H p (w) will also be denoted by

1.8 Lp and H p

35

◦

p Lp+ (w). H− (w) will refer to the Lp (w)-closure of the linear hull of the set ◦

◦

p (w) will occasionally be written as Lp− (w). {χ−1 , χ−2 , . . .} and H− It can be shown that P is dense in Lp (w) (1 < p < ∞). The following statements are equivalent:

(i) There is a constant cp,w depending only on p and w such that P f p,w ≤ cp,w f p,w

∀ f ∈ P.

(ii) The Riesz projection P generates a bounded operator on Lp (w). ◦

p (w) and H p (w). (iii) Lp (w) decomposes into the direct sum of H−

Of course, here the Riesz projection can be replaced by the singular integral operator (Hilbert transform). 1.45. The Helson-Szeg˝ o theorem. This theorem establishes an interesting necessary and sufficient conditions for the boundedness of the Riesz projection P on L2 (w). Suppose w is a weight in L2 . Then P is bounded on L2 (w) if and only if w can be represented in the form w = eu+)v , where u and v are real-valued functions in L∞ and v∞ < π/4. 1.46. The Hunt-Muckenhoupt-Wheeden condition. This is a necessary and sufficient condition for the boundedness of the Riesz projection P on Lp (w) whose nature is completely different from the condition occurring in the Helson-Szeg˝o theorem. Let 1 < p < ∞ and let w be a weight in Lp . Then for P to be bounded on L (w) it is necessary and sufficient that w−1 ∈ Lq (1/p + 1/q = 1) and that 1/p ( 1/q ( 1 1 p −q sup w dm w dm < ∞, (1.19) |I| I |I| I I p

where the supremum is over all subarcs of the circle T and |I| is the arc length of I. For concrete applications the following simple observation is useful: to show that the supremum in (1.19) is finite it suffices to show that the supremum over all subarcs I satisfying |I| ≤ δ, where δ is any positive number, is finite. If one asks for a condition on the weight w that ensures that P be bounded o type on both Lp (w) and Lq (w) (1/p + 1/q = 1), then again a Helson-Szeg˝ criterion can be given. The sufficiency portion of the following result was established by Simonenko [492], its necessary part is due to Krupnik [328]. Its proof is given in Gohberg, Krupnik [232, Chap. 12, Theorem 5.2]. Let 2 ≤ p < ∞ and 1/p + 1/q = 1. Suppose w is a weight in Lp . Then for P to be in L(Lp (w)) ∩ L(Lq (w)) it is necessary and sufficient that w be of the form w = eu+)v , where u and v are real-valued functions in L∞ and v∞ < π/(2p).

36


1.9 BM O and V M O 1.47. Definitions. The Lebesgue measure of a measurable subset E of T will * be denoted by |E|, that is |E| = E dm.* For f ∈ L1 and I a subarc of T, define the mean value fI by fI := (1/|I|) I f dm. For 0 < δ ≤ 2π let ( 1 Mδ (f ) := sup |f (ζ) − fI | dm(ζ), |I|≤δ |I| I where I ranges over subarcs of T only, and put M0 (f ) := lim Mδ (f ), δ→0

f ∗ := M2π (f ).

A function f ∈ L1 is said to have bounded mean oscillation, or to be in BM O, if f ∗ < ∞, and is said to have vanishing mean oscillation, or to belong to V M O, if f ∗ < ∞ and M0 (f ) = 0. 1.48. Basic properties of BM O and V M O. (a) If f ∈ L1 and if for each subarc I of T with |I| ≤ δ there is a constant αI such that ( 1 |f − αI | dm ≤ M, |I| I then Mδ (f ) ≤ 2M . (b) If f ∈ L1 and Mδ (f ) < ∞, where 0 < δ < 2π, then f ∈ BM O and M2π (f ) ≤ (2π/δ)6 Mδ (f ) (the estimate is very generous). (c) L∞ ⊂ BM O and f ∗ ≤ f ∞ for f ∈ L∞ . (d) If f ∈ L1 and f ∗ = 0, then f =const. (e) L∞ ⊂ BM O ⊂

Lp ,

C ⊂ V M O ⊂ BM O.

1≤p 0 such that for all a ∈ BM O, A1 M0 (a) ≤ distBM O (a, V M O) ≤ A2 M0 (a). Some much deeper results on BM O and V M O can be stated as follows. (g) (Charles Fefferman). BM O = {u + v) : u, v ∈ L∞ } and there is an absolute constant B such that every f ∈ BM O can be written as f = u + v) with u∞ ≤ Bf ∗ , v∞ ≤ Bf ∗ . (h) (Sarason). V M O = {u + v) : u, v ∈ C} and there is an absolute constant B such that every f ∈ V M O can be written as f = u + v) with u, v ∈ C and u∞ ≤ Bf ∗ , v∞ ≤ Bf ∗ .

1.9 BM O and V M O

37

(k) The conjugation operator (and thus also the Cauchy singular integral operator and the Riesz projection) is bounded on the following pairs of spaces: L∞ → BM O,

BM O → BM O,

C → V M O,

V M O → V M O.

(l) We have BM O = {u + P v : u, v ∈ L∞ }, and

V M O = {u + P v : u, v ∈ C}

f BM O = inf{u∞ + v∞ : u, v ∈ L∞ , u + P v = f }

is an equivalent norm in BM O. BM O(R) is defined as the set of all functions F ∈ L1loc (R) for which ( 1 F ∗ := sup |F (x) − FI | dx < ∞, I |I| I * where the sup is over all bounded intervals I ⊂ R, FI := (1/|I|) I F (x) dx, and |I| is the Lebesgue measure of I. For f ∈ L1 , define F ∈ L1loc (R) by i−x F (x) = f (x ∈ R). i+x We are now in a position to state two further important properties of BM O. (m) f ∈ BM O ⇐⇒ F ∈ BM O(R). (n) (John/Nirenberg). If F ∈ BM O(R), then, for every interval I ⊂ R and every λ > 0, + −cλ 1 ++ {x ∈ I : |F (x) − FI | > λ}+ ≤ C exp |I| F ∗ with absolute constants C and c. Finally, V M O(R) is the collection of all functions F ∈ BM O(R) for which ( 1 M0 (F ) := lim sup |F (x) − FI | dx = 0. δ→0 |I|≤δ |I| I If we let U C(R) denote the uniformly continuous functions on R, then (p) V M O(R) coincides with the set of all functions F ∈ BM O(R) for which inf F − G∗ : G ∈ U C(R) ∩ BM O(R) = 0. For F ∈ L1loc (R) and x ∈ R define Fx ∈ L1loc (R) by Fx (t) = F (t − x). (q) If F ∈ BM O(R), then F ∈ V M O(R) ⇐⇒ F − Fx ∗ → 0 as x → 0.

38


We conclude the present register with a technical result. (r) Let F ∈ BM O(R) and let I and J be finite intervals of R. (i) If I ⊂ J and |J| ≤ 2|I|, then |FI − FJ | ≤ 2F ∗ . (ii) If I ⊂ J and |J| > 2I, then |FI − FJ | ≤ c log

|J| |I|

F ∗

with some absolute constant c. (iii) If |I| = |J|, then dist(I, J) |FI − FJ | ≤ c log 2 + F ∗ |I| with some absolute constant c.

1.10 Smoothness Classes 1.49. Spaces of p type. Let α, β be any real numbers and let p, r be real numbers satisfying 1 ≤ p, r < ∞. We denote by r,p β,α the Banach space of all sequences ϕ = {ϕn }n∈Z of complex numbers for which ϕr,p;β,α :=

∞

1/r |ϕ−n | (n + 1) r

rβ

+

n=1

∞

1/p |ϕn | (n + 1) p

pα

< ∞.

n=0

p p p,p α,α will be also written as α (Z); an equivalent norm in α (Z) is

ϕp,α :=

1/p |ϕn |p (|n| + 1)pα

.

n∈Z p p p,p 0,0 = 0 (Z) will be abbreviated to (Z) and · p will always refer to the norm 1/p p ϕp := |ϕn | . n∈Z

The collection of all functions in L1 whose sequence of Fourier coefficients r,p belongs to r,p α,β (α ≥ 0, β ≥ 0) will be denoted by F α,β . The norm of a funcr,p r,p tion in F α,β is defined as the α,β -norm of its sequence of Fourier coefficients. p,p p p 1 F p,p α,α and F 0,0 will be abbreviated to F α and F , respectively. F and 1,1 α,β F α,β will be denoted by W and W , respectively (the W is for Wiener). Notice that F 2 is L2 .

1.10 Smoothness Classes

39

Let pα denote the closed subspace of pα (Z) defined by pα = ϕ ∈ pα (Z) : ϕn = 0 for n < 0 . p0 will be abbreviated to p . We shall frequently think of pα as a space of one-sided sequences, that is, we let pα

! # 1/p p pα = ϕ = {ϕn }n∈Z+ : ϕp,α := |ϕn | (n + 1) 0 and 1 ≤ p ≤ ∞ we define the Besov class Bpα as ! # ( π |t|−1−αp ∆nt f pLp dt < ∞ (p < ∞), Bpα = f ∈ Lp : −π , α ∞ −α n B∞ = f ∈ L : sup |t| ∆t f L∞ < ∞ (p = ∞), t =0

where n is any integer such that n > α and where ∆nt := ∆t ∆n−1 , t (∆t f )(eiθ ) := f (ei(θ+t) ) − f (eiθ ), for θ, t ∈ R. Note that this definition does not depend on the choice of n, n > α. α are nothing else than the H¨ older-Zygmund classes, that The classes B∞ is, for 0 < α < 1 we have α B∞ = f ∈ C : |f (t1 ) − f (t2 )| ≤ Mf |t1 − t2 |α ∀ t1 , t2 ∈ T , for α = 1 we have 1 B∞ = f ∈ C : |f (ei(θ+t) ) + f (ei(θ−t) ) − 2f (eiθ )| ≤ Mf |t|

∀ θ, t ∈ R ,

and on denoting by C n (n ∈ Z+ ) the class of n times continuously differentiable functions on T, we can finally write α α−n (n < α ≤ n + 1). B∞ = f ∈ C n : f (n) ∈ B∞ α For this reason, if α is not an integer, we shell henceforth denote B∞ by C α .

Later we shall need the following two facts. (a) If 1 1/p, then Bpα ⊂ C. (b) If 1 0, then P is dense in Bpα . α will be denoted by cα . The closure of P in C α = B∞ α If α ≤ 0, we think of Bp as a space of sequences of complex numbers. Namely, we define Bpα as the linear space of all sequences f = {fn }n∈Z such that {(|n| + 1)−s fn }n∈Z is the sequence of the Fourier coefficients of some s−|α| function belonging to Bp , s − |α| > 0. This definition does not depend on the choice of the number s > |α|. If we identify the functions in Bpα (α > 0) with their Fourier coefficients sequence, we have the following: the mapping Is which sends a sequence {ϕn }n∈Z of complex numbers into the sequence {(|n| + 1)−s ϕn }n∈Z maps Bpα one-to-one onto Bpα+s for every 1 ≤ p ≤ ∞, α ∈ R, s ∈ R. Thus, on defining a norm in Bpα for α ≤ 0 by f Bpα = Is f Bpα+s we make Bpα become a Banach space and this definition is correct, since it is independent of the choice s > |α|. If 1 ≤ p < ∞ and s ∈ R, then the dual space (Bpα )∗ of Bpα coincides with Bq−α (1/p + 1/q = 1) in the following sense: the general form of a functional f ∈ (Bpα )∗ is given by

1.11 Notes and Comments

f (ϕ) =

ϕn fn

41

(ϕ = {ϕn }n∈Z ∈ Bpα ),

n∈Z

where {fn }n∈Z ∈ Bq−α . The space B2α admits a very simple description: (|n| + 1)2α |fn |2 < ∞. f ∈ B2α ⇐⇒ n∈Z

Also notice that f ∈ Bpα ⇐⇒

∞

nαp−1 distLp (f, Pn ) < ∞,

n=1

where Pn denotes the set of all Laurent polynomials of degree at most n. Finally, let (Bpα )A := f ∈ Bpα : fn = 0 for n < 0 . Thus (Bpα )A is a closed subspace of Bpα . For α > 0 this space can be characterized as the class of all functions f ∈ Lp such that ( 1 (1 − r)(n−α)p−1 (hr f )(n) pLp dr < ∞ (p < ∞), 0

sup (1 − r)n−α (hr f )(n) L∞ < ∞ (p = ∞),

0 2, we have " " " M (a) − M a(m) ϕ" ≤ M (a − a(m) )ϕ2 ≤ a − a(m) ∞ ϕ2 = o(1) p as m → ∞ for every ϕ ∈ 0 (Z). Because of (2.4), M (a(m) )p ≤ sp V1 (a(m) ) + a(m) ∞ ≤ sp V1 (a) + a∞ . Thus, M (a) ∈ L(p (Z)) and (2.4) holds for general a. (g) M p is a Banach algebra with respect to the norm ap := M (a)L(p (Z)) . Proof. It remains to show that M p is complete. Let {a(m) }m∈Z+ be a Cauchy sequence in M p . By virtue of (d), {a(m) }m∈Z+ is a Cauchy sequence in L∞ and, consequently, there is an a ∈ L∞ such that a(m) −a∞ → 0 as m → ∞. Since L(p (Z)) is complete, there is an A ∈ L(p (Z)) such that M (a(m) ) − Ap → 0 as m → ∞. Because M (a)ϕ = Aϕ for ϕ ∈ 0 (Z), we finally conclude that A = M (a).

2.2 Toeplitz Operators 2.6. Definition. The operator T (a) defined for a ∈ L∞ and 1 < p < ∞ by T (a) : H p → H p ,

f → P (af )

(2.5)

is obviously bounded and T (a)L(H p ) ≤ cp a∞ , where cp is the norm of the Riesz projection on Lp . This operator is called the Toeplitz operator on H p generated by the function a. If a ∈ M p , then the operator T (a) given on p = p (Z+ ) (1 ≤ p < ∞) as T (a) : p → p ,

ϕ → P (a ∗ ϕ)

(2.6)

is clearly bounded and T (a)L(p ) ≤ M (a)L(p (Z)) . Here P denotes the discrete Riesz projection. The operator defined by (2.6) is called the Toeplitz operator on p generated by the function a.

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2 Basic Theory

The function a generating the Toeplitz operators (2.5) and (2.6) is usually referred to as the symbol of the corresponding operator. Toeplitz operators on p are sometimes also called discrete Wiener-Hopf operators. For p = 2, the Toeplitz operators defined on H 2 and 2 by (2.5) and (2.6), respectively, are unitarily equivalent through the isomorphism H 2 → 2 , ϕn χn → {ϕn }n∈Z+ . (2.7) n∈Z+

Therefore we shall frequently identify these operators without mentioning this explicitly. For f ∈ H p , g ∈ H q , ϕ ∈ p , ψ ∈ q (1/p + 1/q = 1) let ( 1 (f, g) := f g dm, (ϕ, ψ) := ϕn ψn . 2π T n∈Z+

It is clear that the operators (2.5) and (2.6) satisfy (T (a)χj , χk ) = ak−j ,

(T (a)ej , ek ) = ak−j

∀ j, k ∈ Z+ .

The following theorem states that every bounded operator on H p resp. p with this property is a Toeplitz operator and, moreover, relates the norm of a Toeplitz operator with the norm of the multiplication operator generated by the same function. 2.7. Theorem (Brown/Halmos). (a) Let A ∈ L(H p ) (1 < p < ∞) and suppose there is a sequence {an }n∈Z of complex numbers such that (Aχj , χk ) = ak−j for all k, j ∈ Z+ . Then there exists an a ∈ L∞ such that A = T (a) and {an } is the Fourier coefficient sequence of a. Moreover, a∞ ≤ T (a)L(H p ) ≤ cp a∞ ,

(2.8)

where cp is the norm of P on Lp . (b) Let a ∈ L(p ) (1 ≤ p < ∞) and suppose there is a sequence {an }n∈Z of complex numbers such that (Aej , ek ) = ak−j for all k, j ∈ Z+ . Then there exists an a ∈ M p such that A = T (a) and {an } is the Fourier coefficient sequence of a. Moreover, T (a)L(p (Z+ )) = M (a)L(p (Z)) .

(2.9)

Proof. (a) For n ≥ 0, define bn ∈ Lp as bn := χ−n (Aχn ). Then bn p ≤ A. Since Lp = (Lq )∗ , the Banach-Alaoglu theorem implies that there is a b ∈ Lp such that bp ≤ A and some subsequence {bnk } of {bn } converges to b in the weak topology on Lp . In particular, (bnk , χj ) → (b, χj ) for all j ∈ Z, and because (bnk , χj ) = (Aχnk , χnk +j ) = aj whenever nk + j ≥ 0, it follows that (b, χj ) = aj

∀ j ∈ Z.

(2.10)

2.2 Toeplitz Operators

51

Now define the mapping B by B : P → Lp ,

f → bf.

(2.11)

If f, g ∈ P, then, by virtue of (2.10), (Bf, g) is equal to (M (χ−n )AM (χn )f, g) whenever n is chosen large enough. Hence |(Bf, g)| ≤ lim sup |(M (χ−n )AM (χn )f, g)| ≤ A f p gq n→∞

and thus Bf p = sup |(Bf, g)| : g ∈ P, gq ≤ 1 ≤ A f p for all f ∈ P. This shows that the linear mapping (2.11) extends to an operator B ∈ L(Lp ) with B ≤ A. Again from (2.10) we deduce that (Bχj , χk ) = (b, χk−j ) = ak−j for all j, k ∈ Z. Now Proposition 2.2 gives the existence of an a ∈ L∞ such that B = M (a) and {an } is the Fourier coefficient sequence of a. Since both (T (a)χj , χk ) and (Aχj , χk ) equal ak−j , it follows that A = T (a). Finally, because M (a) = B ≤ A = T (a), the norm equality in Proposition 2.2 gives the first “≤” in (2.8). The second “≤” in (2.8) is trivial. (b) Since Aen ∈ p for all n ∈ Z+ , it is obvious that the sequence {an }n∈Z belong to p (Z). After defining B as ! # aj−k ϕk B : 0 (Z) → p (Z), {ϕj }j∈Z → j∈Z

k∈Z

the rest of the proof is completely analogous to (a).

2.8. Corollary. (a) If a ∈ L∞ , then T (a)L(H 2 ) = T (a)L(2 ) = a∞ . (b) If a ∈ M p , then T (a)L(p ) = aM p . (c) If 1 < p < ∞, 1/p + 1/q = 1, and if a ∈ M p , then T (a) ∈ L(q ) and T (a)L(q ) = T (a)L(p ) . The adjoint T ∗ (a) ∈ L(q ) is equal to T (a). (d) If 1 < p < ∞, 1/p + 1/q = 1, and if a ∈ M p , then T (a) ∈ L(r ) for all r ∈ [p, q] and γ a∞ ≤ T (a)L(r ) ≤ a1−γ ∞ aM p ≤ aW ,

where γ = p|r − 2|/(r|p − 2|). Proof. Combine 2.4, 2.5, and 2.7.

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2 Basic Theory

2.9. Shift operators. Recall that χn is given by χn (t) = tn (t ∈ T). The operators M (χ1 ) and T (χ1 ) are usually referred to as the bilateral and unilateral shift, respectively, because they act on p (Z) resp. p (Z+ ) (1 ≤ p < ∞) by the rules M (χ1 ) : {ϕi }i∈Z → {ϕi−1 }i∈Z , T (χ1 ) : {ϕ0 , ϕ1 , . . .} → {0, ϕ0 , ϕ1 , . . .}. Also as usual, we put U := M (χ1 ) and V := T (χ1 ), and, for n ∈ Z+ , we let U n = M (χn ),

U −n = M (χ−n ),

V n = T (χn ),

V (−n) = T (χ−n ).

Then obviously U ±n = (U ±1 )n , V n = (V 1 )n , V (−n) = (V (−1) )n , and U −n U n = U n U −n = I,

U ∗ = U −1 ,

V (−n) V n = I,

V ∗ = V (−1) .

Note that V n V (−n) is not the identity operator. It is clear that U ±n are isometries on Lp and p (Z) and that the operators V n are isometries on H p (1 < p < ∞) and p (Z+ ) (1 ≤ p < ∞). In particular, the range of V n is always closed. It is easy to see that V (−n) is onto, dim Ker V (−n) = n, dim Coker V n = n, V n is one-to-one. Thus, Ind T (χk ) = −k for all k ∈ Z.

2.3 Hankel Operators 2.10. Definitions. The definition of Hankel operators is slightly complicated by the circumstance that there is neither a definition in general use nor a unique notation for them in the literature and that there is in fact no compelling reason for adopting such a unique definition or notation. Besides the projections P and Q = I − P , we now need a third operator J, the so-called flip operator. This is the (obviously isometric) operator acting on Lp (1 < p < ∞) by the formula 1 1 (Jf )(t) = f fn t−n−1 (t ∈ T) = t t n∈Z

and, accordingly, acting on p (Z) (1 ≤ p < ∞) by the rule (Jϕ)n = ϕ−n−1

(n ∈ Z).

For a ∈ L∞ , we define the Hankel operator H(a) on H p (1 < p < ∞) by H(a) : H p → H p ,

f → P M (a)QJf.

We let H() a) denote the operator given by

2.3 Hankel Operators

H() a) : H p → H p ,

53

f → JQM (a)P f

and refer to H() a) also as a Hankel operator (see 2.15 below). It is clear that both H(a) and H() a) are bounded whenever a ∈ L∞ and 1 < p < ∞. p Given a ∈ M (1 ≤ p < ∞) with Fourier coefficients sequence {an }n∈Z we analogously define H(a) : p → p , H() a) : p → p ,

ϕ → P M (a)QJϕ, ϕ → JQM (a)P ϕ

and call H(a) and H() a) the Hankel operators on p generated by the function a. It is easily seen that H(a) and H() a) are bounded on p . Moreover, it is p readily verified that their action on can be given by the following formulas: ! # H(a) : p → p , {ϕj }j∈Z+ → aj+k+1 ϕk , j∈Z+

k∈Z+

H() a) : p → p ,

{ϕj }j∈Z+ →

!

# a−j−k−1 ϕk

k∈Z+

. j∈Z+

If p = 2, the Hankel operators defined on H 2 and 2 are again unitarily equivalent through the isomorphism (2.7). The operators resulting from Hankel operators by omitting the flip operator, that is, the four operators ◦

p → H p, P M (a)Q : H− ◦

p QM (a)P : H p → H− ,

P M (a)Q : Qp (Z) → P p (Z), QM (a)P : P p (Z) → Qp (Z)

will occasionally also be referred to as Hankel operators. If a ∈ L∞ , then (H(a)χj , χk ) = aj+k+1 for all j, k ∈ Z+ . The following theorem describes the bounded operators on H p with this property and provides an important norm estimate for Hankel operators on H p . 2.11. Theorem (Nehari). Let A ∈ L(H p ) (1 < p < ∞) and suppose there is a sequence {an }n∈N of complex numbers such that (Aχj , χk ) = aj+k+1 for all j, k ≥ 0. Then there is a function b ∈ L∞ such that A = H(b) and the n-th Fourier coefficient bn of b is equal to an for all n ≥ 1. Moreover, distL∞ (b, H ∞ ) ≤ H(b)L(H p ) ≤ cp distL∞ (b, H ∞ ), where cp = P L(Lp ) . In particular, for p = 2 we have H(b)L(H 2 ) = distL∞ (b, H ∞ ).

(2.12)

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2 Basic Theory

Proof. Note that AL(H p ) equals sup{|G(Af )| : f H p ≤ 1, G(H p )∗ ≤ 1}. So we deduce from (1.15) that 1 AL(H p ) ≤ Φ(A) ≤ AL(H p ) , cp

(2.13)

where Φ(A) := sup{|(Af, g)| : f H p ≤ 1, gH q ≤ 1} and 1/p + 1/q = 1. The hypothesis implies that (Af, g) = (Aχ0 , f c g)

∀ f, g ∈ PA ,

where f c (t) = f (1/t) (t ∈ T). It is obvious that f c H p = f H p . Thus, Φ(A) = sup |(Aχ0 , f g)| : f H p ≤ 1, gH q ≤ 1 . We claim that f g : f H p ≤ 1, gH q ≤ 1 = h ∈ H 1 : hH 1 ≤ 1 .

(2.14)

H¨ older’s inequality immediately gives the inclusion “⊂”. To get the reverse inclusion note first that, by 1.41(a), every h ∈ H 1 factors as h = ϕhe , where 1/p ϕ ∈ H ∞ is inner and he ∈ H 1 is outer. From 1.41(e) we see that he ∈ H p 1/q 1/p 1/q and he ∈ H q . Thus, if we let f = ϕhe and g = he , then h = f g and f H p = gH q = hH 1 . This completes the proof of (2.14). Taking into account (2.14) we obtain Φ(A) = sup{|(Aχ0 , h)| : hH 1 ≤ 1}. Hence, the mapping ( 1 1 (Aχ0 )h dm C : H → C, h → 2π T ∗ is a linear functional belonging to H 1 and we have C 1 ∗ = Φ(A). Again from 1.42 we conclude that there is a c ∈ L∞ such that ( ( (Aχ0 )h dm = ch dm ∀ h ∈ H 1 . T

H

T

Letting h = χn (n ≥ 0), we get (Aχ0 , χn ) = an+1 = cn , and thus the function b = χ1 c ∈ L∞ has the desired property: bn = an for n ≥ 1. We are left with the norm estimate (2.12). Any extension of C to a bounded ) on L1 is given by linear functional C ( ) : L1 → C, f → C ϕf dm T

with some ϕ ∈ L∞ and we have C

H1

∗ ≤ C ) (L1 )∗ = ϕ∞ . Due to

)0 ∈ (L1 )∗ of C such that the Hahn-Banach theorem there is an extension C )0 . Thus, C = C

2.3 Hankel Operators

Φ(A) = C

H1

55

∗ = inf C ) (L1 )∗ = inf ϕ∞

) is an extension of C (where, in fact, the inf can be replaced by min). But C if and only if ( (ϕ − Aχ0 )h dm = 0 ∀ h ∈ H1 T

⇐⇒

(ϕ − Aχ0 )n = 0

⇐⇒ ⇐⇒

(χ1 ϕ)n − an = 0 χ1 ϕ − b ∈ H ∞ ,

∀n≥0 ∀n≥1

therefore, Φ(A) = inf ϕ∞ : χ1 ϕ − b ∈ H ∞ = inf χ1 ϕ∞ : χ1 ϕ − b ∈ H ∞ = distL∞ (b, H ∞ ) and now (2.13) gives (2.12). Remark. Let the hypothesis of the preceding theorem be satisfied. Then Aχ0 ∈ H p and so there exists a function a ∈ H p whose n-th Fourier coefficient is an (n ∈ N). One can now formulate the following criterion for the boundedness of A: A ∈ L(H p ) if and only if a ∈ BM O. Indeed, if A ∈ L(H p ), then, by the above theorem, a = P b for some b ∈ L∞ and thus a ∈ BM O by 1.48(k); on the other hand, if a ∈ BM O, then, by virtue of 1.48(l), there are u, v ∈ L∞ such that a = u + P v, whence u = P u, so a = P b with b = u + v ∈ L∞ , and the above theorem implies the boundedness of A. 2.12. Open problem. Establish a Nehari theorem for p . The conjecture is that for this case in the preceding theorem “b ∈ L∞ ” must be replaced by “b ∈ M p ” and the norm H(b)L(p ) is equivalent (or, maybe, even equal) to distM p (b, H ∞ ∩ M p ). The following two sections are intended to give a first idea of the connection between multiplication, Toeplitz, and Hankel operators. Moreover, the formulas stated in 2.14, thought being very simple, will be of extreme importance for all what follows. 2.13. Decomposition of the multiplication operator. L2 decomposes ◦

into the orthogonal sum of H 2− and H 2 . Accordingly, every operator in L(L2 ) can be represented as a 2 × 2 operator matrix, ◦ ◦ 2 2 AB H − → H− . : 2 CD H H2 This applies, in particular, to the multiplication operator M (a) ∈ L(L2 ). In the corresponding matrix representation we meet operators closely related to Toeplitz and Hankel operators:

56

2 Basic Theory

M (a) =

QM (a)Q P M (a)Q

⎛ ◦ ⎞ ⎛ ◦ ⎞ 2 H H2 : ⎝ −⎠ → ⎝ −⎠. P M (a)P H2 H2

QM (a)P

(2.15)

Clearly, the P M (a)P in the right lower corner is nothing else than the Toeplitz operator T (a). The operators QM (a)P and P M (a)Q differ from H() a) and H(a), respectively, only by the flip operator. Finally, if we define T () a) on H 2 as T () a) = JQM (a)QJ, then the QM (a)Q in the left upper corner is equal to JT () a)J. In terms of Fourier coefficients we have ! # aj−i ϕj . T () a) : 2 → 2 , {ϕi }i∈Z+ → i∈Z+

j∈Z+

Thus, the operator matrix in (2.15) can be written as ⎛ ◦ ⎞ ⎛ ◦ ⎞ 2 JT () a)J JH() a) H H2 : ⎝ −⎠ → ⎝ −⎠. M (a) = H(a)J T (a) H2 H2 In more detail, if we express things via Fourier coefficients, the 6 × 6 matrix in the center of M (a) equals ⎛ ⎞ a0 a−1 a−2 a−3 a−4 a−5 ⎜ a1 a0 a−1 a−2 a−3 a−4 ⎟ ⎜ ⎟ ⎜ a2 a1 a0 a−1 a−2 a−3 ⎟ ⎜ ⎟ ⎜a a a a a a ⎟. 1 0 −1 −2 ⎟ ⎜ 3 2 ⎝ a4 a3 a2 a1 a0 a−1 ⎠ a5 a4 a3 a2 a1 a0 ◦

·

·

p Since Lp =H− + H p (1 < p < ∞) and p (Z) = Qp (Z) + P p (Z) (1 ≤ p < ∞), all what has been said above applies to the case p = 2 as well. In particular, a) defined by if a ∈ L∞ or a ∈ M p , then the operators T ()

T () a) : H p → H p ,

f → JQM (a)QJf

and T () a) : → , p

p

{ϕi }i∈Z+ →

! j∈Z+

# aj−i ϕj

(2.16) i∈Z+

will be bounded on H p and p , respectively. Finally, notice the obvious identity M (a) = P M (a)P + P M (a)Q + QM (a)P + QM (a)Q.

(2.17)

This is merely a translation of the representation (2.15) into another language. The four operators on the right of (2.17) are the building stones of the operators

2.4 Invertibility of Toeplitz Operators on H 2

M (a)P + M (b)Q,

57

P M (a) + QM (b),

which are usually called singular integral operators when considered as acting on Lp and paired convolution operators when considered as acting on p (Z). 2.14. Proposition. Let a, b ∈ L∞ resp. a, b ∈ M p . Then T (ab) = T (a)T (b) + H(a)H()b), H(ab) = T (a)H(b) + H(a)T ()b).

(2.18) (2.19)

In particular, if the positive Fourier coefficients of a = a− and the negative Fourier coefficients of b = b+ vanish, then for every c ∈ L∞ resp. c ∈ M p , T (a− cb+ ) = T (a− )T (c)T (b+ ).

(2.20)

Proof. We have T (ab) = P M (ab)P = P M (a)M (b)P = P M (a)P M (b)P + P M (a)QM (b)P = P M (a)P · P M (b)P + P M (a)QJ · JQM (b)P and this is (2.18). Similarly, H(ab) = P M (ab)QJ = P M (a)M (b)QJ = P M (a)P · P M (b)QJ + P M (a)QJ · JQM (b)QJ, which is (2.19). To complete the proof, note that the conditions imposed upon a− and b+ imply that H(a− ) = 0 and H(b. + ) = 0. 2.15. Important remark. The ) a used in 2.10, 2.13 and 2.14 has nothing to do with the conjugate function of a (in the sense of 1.43). Given a measurable function a on T we now define ) a by ) a(t) = a(1/t) (t ∈ T). Moreover, this point of view eliminates any confusion that might arise in connection with 2.10 and definitions in 2.13. For instance, H() a) may be thought of as both JQM (a)P and as P M () a)QJ, and T () a) may be interpreted as both JQM (a)QJ and P M () a)P . In either case, both is the same. The notation ) a is in general use for the conjugate function of a as well as for the function given by ) a(t) = a(1/t) (t ∈ T). As a rule, henceforth ) a will always refer to the function ) a(t) = a(1/t) unless it is explicitly indicated that ) a means the conjugate function.

2.4 Invertibility of Toeplitz Operators on H 2 We first show how the expressions for the norms T (a)L(H 2 ) and H(a)L(H 2 ) obtained in Theorems 2.7 and 2.11 can be used to derive results on the invertibility of Toeplitz operators on H 2 .

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2 Basic Theory

2.16. Definition. Let a ∈ L∞ . If there exist a real number ε > 0 and a complex number c of modulus 1 such that Re (ca) ≥ ε a.e. on T, then a is called sectorial. It is obvious that a sectorial function is necessarily invertible in L∞ . It is also easy to see that a is sectorial if and only if so is a/|a|. Thus, sectoriality is a matter of the argument. Also notice that a ∈ GL∞ is sectorial if and only if a/|a| can be written as ei(v+c) where c ∈ R and v ∈ L∞ is a real-valued function with v∞ < π/2. Finally, it is easily verified that a function a ∈ GL∞ is sectorial if and only if distL∞ (a/|a|, C) < 1, where distL∞ (g, C) := inf{g − c∞ : c ∈ C}. 2.17. Theorem (Brown/Halmos). If a ∈ L∞ is sectorial then T (a) is invertible on H 2 . Proof. It is readily seen that there are a sufficiently small real number δ > 0 and a complex number c of modulus 1 such that 1 − δca∞ < 1. From Theorem 2.7 (or, more explicitly, from its Corollary 2.8(a)) we get I − δcT (a)L(H 2 ) < 1. This implies the invertibility of δcT (a) and thus also that of T (a). In order to state the consequence of Nehari’s theorem we have promised above, we need two propositions which are interesting on their own and will often be applied in the following. 2.18. Proposition (Wintner). Let h ∈ H ∞ . Then T (h) ∈ GL(H 2 ) ⇐⇒ h ∈ GH ∞ . Proof. If h−1 ∈ H ∞ , then, by Proposition 2.14, T (h−1 ) ∈ L(H 2 ) is the inverse of T (h). Conversely, if T (h) ∈ GL(H 2 ), then the equation T (h)f = P (hf ) = hf = g must have a solution f ∈ H 2 for every g ∈ H 2 . Thus hH 2 = H 2 and 1.41(g) completes the proof. 2.19. Proposition. Let a ∈ GL∞ . Then T (a) ∈ GL(H 2 ) ⇐⇒ T (a/|a|) ∈ GL(H 2 ). Proof. Since a−1 ∈ L∞ , we deduce from 1.41(d) that there is an outer function h ∈ GH ∞ such that |a−1 |1/2 = |h|. So a/|a| = hah and, by Proposition 2.14, T (a/|a|) = T (h)T (a)T (h). Due to the preceding proposition T (h) and T (h) = T ∗ (h) are invertible on H 2 and this gives the assertion at once. Remark. This proposition reduces the invertibility problem for Toeplitz operators on H 2 to the case of unimodular symbols. In other words, the invertibility of a Toeplitz operator on H 2 is exclusively dictated by the behavior of the argument of its symbol.


59

2.20. Theorem (Widom/Devinatz). Let ϕ ∈ L∞ be a unimodular function, i.e., |ϕ| = 1 a.e. on T. Then (a) T (ϕ) is left-invertible on H 2 ⇐⇒ distL∞ (ϕ, H ∞ ) < 1, (b) T (ϕ) is right-invertible on H 2 ⇐⇒ distL∞ (ϕ, H ∞ ) < 1, (c) T (ϕ) is invertible on H 2 ⇐⇒ distL∞ (ϕ, GH ∞ ) < 1. Proof. (a) We have M (ϕ)P = P M (ϕ)P + QM (ϕ)P and since |ϕ| = 1, it follows that ) 2 f 2 = ϕf 2 = P (ϕf )2 + Q(ϕf )2 = T (ϕ)f 2 + H(ϕ)f for every f ∈ H 2 . The operator T (ϕ) is left-invertible on H 2 if and only if there exists an ε > 0 such that εf 2 ≤ T (ϕ)f 2 for all f ∈ H 2 (recall 1.12(g)). ) < 1, which, Consequently, T (ϕ) is left-invertible on H 2 if and only if H(ϕ) by Nehari’s theorem 2.11 for p = 2, is the same as " " ) − h"∞ : h ∈ H ∞ 1 > dist(ϕ, ) H ∞ ) = inf "ϕ " " )−) h"∞ : h ∈ H ∞ = inf "ϕ = inf ϕ − h∞ : h ∈ H ∞ = dist(ϕ, H ∞ ). (b) Since T ∗ (ϕ) = T (ϕ), this is immediate from (a). (c) Suppose T (ϕ) ∈ GL(H 2 ). Then, by (a), there is an h ∈ H ∞ such that ϕ − h∞ < 1 and it remains to show that h ∈ GH ∞ . We have (Corollary 2.8(a)) (2.21) I − T (ϕh) = 1 − ϕh∞ = ϕ − h∞ < 1, this implies the invertibility of T (ϕh) = T (ϕ)T (h) = T ∗ (ϕ)T (h),

(2.22)

and because T ∗ (ϕ) is invertible, so also is T (h). From Proposition 2.18 we deduce that h ∈ GH ∞ . Now suppose h ∈ GH ∞ and ϕ−h∞ < 1. Then (2.21) holds and therefore the operator (2.22) is invertible. By Proposition 2.18, T (h) ∈ GL(H 2 ), hence T ∗ (ϕ) ∈ GL(H 2 ) and thus T (ϕ) ∈ GL(H 2 ). 2.21. Lemma. Suppose B is a subset of L∞ with the property that cb ∈ B whenever c ∈ C \ {0} and b ∈ B. Let ϕ ∈ L∞ be a unimodular function. Then distL∞ (ϕ, B) < 1 if and only if there are a function b ∈ B and a sectorial function s ∈ GL∞ such that ϕ = bs. Proof. If distL∞ (ϕ, B) < 1, then 1 − ϕ−1 b∞ = ϕ − b∞ < 1 for some b ∈ B. Hence ϕ−1 b is equal to a function s whose (essential) range is contained in some disk with center 1 and radius less than 1. Thus, ϕ−1 b = s with sectorial s, so ϕ = bs−1 and it remains to observe s−1 is sectorial whenever s is so.

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2 Basic Theory

Let ϕ = bs, where b ∈ B and s is sectorial. There is a c ∈ C \ {0} such that the (essential) range of cs−1 is contained in some disk with center 1 and radius less than 1. This implies that ϕ − cb∞ = 1 − ϕ−1 cb∞ = 1 − cs−1 ∞ < 1, whence distL∞ (ϕ, B) < 1.

2.22. Corollary. Let a ∈ GL∞ . Then the operator T (a) is left (right, resp. two-sided ) invertible on H 2 if and only if a/|a| = hs, where h ∈ H ∞ ∩ GL∞ (h ∈ H ∞ ∩ GL∞ , resp. h ∈ GH ∞ ) and s ∈ GL∞ is sectorial. Proof. Combine Theorem 2.20, Proposition 2.19, and Lemma 2.21. For several applications the following restatement of Theorem 2.20(c) is useful. 2.23. Theorem (Widom/Devinatz). Let a ∈ GL∞ . Then T (a) is invertible on H 2 if and only if a/|a| = ei()u+v+c)

a.e. on

T,

(2.23)

∞

where c ∈ R, u and v are real-valued functions in L , and v∞ < π/2. Here u ) refers to the conjugate function of u. Proof. Put ϕ = a/|a|. By virtue of Proposition 2.19, T (a) is invertible if and only if T (ϕ) is invertible. Let T (ϕ) be invertible. Due to Corollary 2.22 there is an h ∈ GH ∞ such that ϕh is sectorial. Thus, ϕh = |h|e−iv with some real-valued function v ∈ L∞ for which v∞ < π/2. Hence ϕ = (|h|/h)eiv = (h/|h|)eiv .

(2.24)

Since h is outer, there is an analytic logarithm log h in D (see 1.41(g), (ii)). The real part of log h is u(z) := log |h(z)| (z ∈ D) and log h can be written as log h(z) = u(z) + i) u(z) + ic

(z ∈ D)

with some c ∈ R. Consequently, h(z) = eu(z) ei()u(z)+c)

(z ∈ D).

∞

Because h ∈ H , the non-tangential limit of eu(z) = |h(z)| exists a.e. on T and equals |h(t)|. Therefore the non-tangential limit of ei()u(z)+c) also exists a.e. on T and is equal to h(t)/|h(t)|. In other words, h/|h| = ei()u+c) , and (2.24) gives (2.23) with u = log |h|, which is clearly in L∞ . Now let ϕ = a/|a| be of the form (2.23). Put ψ = ei(v+c) ,

h = ei(u+i)u)/2 .

It is obvious that h ∈ GH ∞ and that ϕ = (1/h)ψh, whence T (ϕ) = T (1/h)T (ψ)T (h). But the operators T (1/h) and T (h) are invertible by Proposition 2.18, while T (ψ) is invertible due to Theorem 2.17. Thus, T (ϕ) is invertible, too.


61

2.24. Remark. The Widom-Devinatz theorems solve the invertibility problem in H 2 for Toeplitz operators (with symbols in GL∞ ) completely. However, given an a ∈ GL∞ it is, in general, by no means easy to decide whether there is an outer function h ∈ GL∞ such that a − h∞ < 1 or to check whether a can be represented in the form (2.23). This is the reason for a great part of all further investigations devoted to the invertibility of Toeplitz operators. The main goal of these investigations is to obtain invertibility criteria, or, equivalently, descriptions of the spectrum, in terms of geometric data of the symbol. The Widom-Devinatz theorems answer the question in an analytic language. Nevertheless, there are situations in which Theorem 2.23 can be almost directly applied to decide whether a given Toeplitz operator is invertible or not. It can also be used to produce interesting examples of invertible Toeplitz operators. We shall demonstrate this in Proposition 2.26 below. 2.25. The class C(T◦ ). Let T◦ denote the punctured circle T \ {−1} and let C(T◦ ) denote the collection of all functions on T which are continuous at every point t ∈ T◦ . We denote by CU (T◦ ) the unimodular and by CR(T◦ ) the real-valued functions in C(T◦ ). Every a ∈ CU (T◦ ) can be written as a = eib with b ∈ CR(T◦ ) and b is uniquely determined by a up to an additive constant of the form 2kπ, k ∈ Z. Given a ∈ CU (T◦ ) choose any b ∈ CR(T◦ ) for which a = eib and define the real-valued function a# ∈ C(R) as i−x a# (x) = b (x ∈ R). i+x The behavior of a# (x) as x → ±∞ provides a good picture of the argument of a near the possible discontinuity of a at −1. We write a# (+∞) = −∞ if lim a# (x) = −∞; the notations a# (+∞) = +∞, a# (−∞) = +∞, x→+∞

and a# (−∞) = −∞ are defined analogously. The function a# is said to be bounded from above (below) at −∞ if there exist both M ∈ R and x0 ∈ R such that a# (x) ≤ M

∀ x < x0

(a# (x) ≥ M

∀ x < x0 ).

It is clear that these definitions express properties of the function a that do not depend on the particular choice of the function b ∈ CR(T◦ ) which defines a# . 2.26. Proposition. (a) There are a ∈ CU (T◦ ) such that a# (+∞) = +∞, a# (−∞) = +∞ and T (a) is invertible on H 2 . (b) There exist functions a ∈ CU (T◦ ) such that a# (+∞) = +∞, a# is not bounded neither from above nor from below at −∞ and T (a) is invertible on H 2 . (c) If a ∈ CU (T◦ ) and if a# (+∞) = +∞ and a# is bounded from above at −∞, then T (a) is not invertible on H 2 .

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2 Basic Theory

(d) Let a ∈ CU (T◦ ) and suppose a# = ψ + δ, where δ ∈ L∞ (R), ψ is monotonous on (−∞, 0) and (0, ∞), and ψ(±∞) = +∞. Then if T (a) is invertible, we have a# (x) = O(log |x|)

as

|x| → +∞.

Proof. (a) Let w be a conformal mapping of D onto the region π π . Ω1 = z = x + iy ∈ C : y > | tan x|, − < x < 2 2 There is a point t0 ∈ T such that |w(z)| → ∞ as z → t0 , z ∈ D. Without loss of generality assume t0 = −1. Define (2.25) w(t) := lim w(z) t ∈ T◦ z→t,z∈D

and, for t ∈ T◦ , put a(t) := eiIm w(t) . Then a ∈ CU (T◦ ), a# (±∞) = +∞, and, of course, a ∈ GL∞ . Since a is of the form (2.23) with u = Re w ∈ L∞ (so u ) = Im w + const) and v = 0, we deduce that T (a) is invertible on H 2 . (b) Now let Ω2 be the region Ω2 = z = x + iy ∈ C : y > − cot x, 0 < x < 2π and let S be the countable union of vertical half-lines given by S=

∞ / 1 z = x + iy ∈ C : x = , y ≤ n . n n=1

Then Ω3 := Ω2 \ S is a simply connected region. Let w denote a conformal mapping of D onto Ω3 and without loss of generality suppose |w(z)| → ∞ as z → −1, z ∈ D. Define w on T◦ as in (2.25) and put a := eiIm w on T◦ . Then a has all the required properties and T (a) is invertible by Theorem 2.23. (c) Assume the contrary, that is, assume T (a) ∈ GL(H 2 ). Then a can be written in the form (2.23) and it follows from the Fefferman theorem 1.48(g) that the argument of a is in BM O. From 1.48(m) we deduce that a# is in BM O(R). But a function a# with the properties required in the hypothesis cannot be in BM O(R). This can be seen, for instance, as follows. Assume a# ∈ BM O(R). Then the function g(ξ) := (1/2) a# (ξ)−a# (−ξ) , ξ ∈ R, is also in BM O(R). Since g is odd, gI must be zero for every I of the form I = (−x, x), and therefore we have ( 1 x |g(ξ)| dξ =: N < ∞. (2.26) sup x>0 x 0 *x Put G(x) := 0 |g(ξ)| dξ. Obviously, G(0) = 0, G(x) ≥ 0 for x ≥ 0, and (2.26) says that


G(x) ≤ N x

for

x > 0.

63

(2.27)

It is precisely the conditions that a# (+∞) = +∞ and that a# be bounded from above at −∞ which imply that g(ξ) → +∞ as ξ → +∞. This in turn ensures the existence of a (sufficiently large) x0 > 0 such that ( 2x0 G(2x0 ) − G(x0 ) = |g(ξ)| dξ ≥ (2N + 1)x0 x0

(apply the mean-value theorem). Hence, G(2x0 ) G(x0 ) + (2N + 1)x0 (2N + 1)x0 1 = ≥ =N+ , 2x0 2x0 2x0 2 which contradicts (2.27) and completes the proof. (d) As in the proof of part (c) we deduce that a# ∈ BM O(R), whence ψ = a# − δ ∈ BM O(R). Put g(x) := ψ(1/x) (x = 0). It is not difficult to see from 1.48(m) that g is also in BM O(R). The assertion can now be derived from the John-Nirenberg theorem 1.48(n) as follows. There is an x0 > 0 such that g(x) > 0 for x ∈ (−x0 , x0 ). Define ( x0 1 g(x) dx. g0 = 2x0 −x0 We now conclude from 1.48(n) that, for λ > 0, + + + x ∈ (−x0 , x0 ) : |g(x) − g0 | > λ + ≤ Ce−c0 λ with some constants C and c0 independent of λ. Hence, if we define x1 (λ) ∈ (0, x0 ) and x2 (λ) ∈ (0, x0 ) by g(−x1 (λ)) = g(x2 (λ)) = g0 + λ (note that g is monotonous on (−x0 , 0) and (0, x0 )), then x1 (λ) + x2 (λ) ≤ Ce−c0 λ (again use monotonicity). So xi (λ) ≤ Ce−c0 λ , whence log xi (λ) ≤ log C − c0 λ, and therefore g(±xi (λ)) = g0 + λ ≤ g0 +

log C log xi (λ) − c0 c0

(i = 1, 2).

On replacing xi (λ) by x we get g(x) ≤ A log(1/|x|) for all x ∈ (−x3 , x3 ) with some x3 > 0 (once more take into account monotonicity). Thus, ψ(x) = O(log |x|) as |x| → ∞, and consequently, a# (x) = ψ(x) + δ(x) = O(log |x|)

as |x| → ∞.

Remark. This proposition, though being a simple consequence of the WidomDevinatz theorem 2.23 obtained by invoking some deep BM O results in a luxurious way, is already concerned with geometric data of the symbol. It says, roughly speaking, that if a ∈ CU (T◦ ) has a discontinuity of oscillating type at −1 then

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2 Basic Theory

(a), (b) T (a) may be invertible if a has the possibility of changing the orientation of the oscillation (=rotation ) into the opposite direction when passing through −1; (c) T (a) cannot be invertible if the orientation of the oscillation is preserved when passing through −1; (d) T (a) cannot be invertible if the oscillation is allowed to alter its orientation into the opposite direction when passing through −1 but is, in addition, required to be “monotonous” and “sufficiently fast”. Let us still dwell a bit on symbols a ∈ CU (T◦ ) for which a# is an even function. We saw that if a# (x) tends monotonically to infinity as x → ±∞, then T (a) is not invertible unless a# increases sufficiently slowly. We shall soon be in a position to decide whether T (a) is invertible if the limits a# (±∞) exist and are finite (this corresponds to the situation in which a ∈ CU (T◦ ) is continuous or has a jump discontinuity at −1). Much more difficulties arise for the “intermediate case,” e.g., for the cases where a# approaches +∞ sufficiently slowly or where the limits a# (±∞) do not exist at all. For instance, if a# (x) = cos x,

a# (x) = log log |x|,

or a# (x) = log log |x| + cos x

(x ∈ R, x large) we have situations of that kind. The spectral inclusion theorems we are now going to derive can be viewed as a first step towards the description of invertibility of Toeplitz operators in geometrical language.

2.5 Spectral Inclusion Theorems 2.27. Definitions. The essential range R(a) of a function a ∈ L∞ is the spectrum of a considered as an element of the C ∗ -algebra L∞ . Equivalently, R(a) is the set of all λ ∈ C such that {t ∈ T : |a(t) − λ| < ε} has positive (Lebesgue) measure for every ε > 0. Let X be a Banach space and let π denote the canonical homomorphism of L(X) onto the Calkin algebra L(X)/C∞ (X). For A ∈ L(X), the spectrum sp A of A is defined by sp A := spL(X) A = λ ∈ C : A − λI ∈ GL(X) and the essential spectrum spess A of A is defined as spess A := spL(X)/C∞ (X) (πA) = λ ∈ C : A − λI ∈ Φ(X) . The essential norm of A is given by Aess := πAL(X)/C∞ (X) = inf A + K : K ∈ C∞ (X) .

2.5 Spectral Inclusion Theorems

65

In order to avoid confusion, we shall sometimes write spΦ(X) A and AΦ(X) for spess A and Aess , respectively. Note that obviously spess A ⊂ sp A and Aess ≤ A for every A ∈ L(X). 2.28. Proposition. (a) If a ∈ L∞ and 1 < p < ∞, then M (a) ∈ GL(Lp ) if and only if a ∈ GL∞ . In other words, spL(Lp ) M (a) = R(a). (b) If a ∈ M p and 1 ≤ p < ∞, then M (a) ∈ GL(p (Z)) if and only if a ∈ GM p . Consequently, spL(p (Z)) M (a) = spM p a ⊃ R(a). Proof. (a) If a ∈ GL∞ and b ∈ L∞ is the inverse of a, then M (b) ∈ L(Lp ) is the inverse of M (a). Conversely, suppose M (a) ∈ GL(Lp ). Then the equation M (a)b = 1 has a solution b ∈ Lp and we have ab = 1. Let B ∈ L(Lp ) denote the inverse of M (a). So a·Bf = f for all f ∈ P, whence Bf = bf for f ∈ P, and this implies that (Bχj , χk ) equals the (k − j)-th Fourier coefficient of b. The assertion now follows from Proposition 2.2. (b) If a ∈ GM p and b ∈ M p is the inverse of a, then M (b) ∈ L(p (Z)) is the inverse of M (a). Now suppose M (a) ∈ GL(p (Z)). By virtue of 2.5(a) it suffices to consider the case 1 ≤ p ≤ 2. The invertibility of M (a) implies that the equation = {ϕn } ∈ p (Z). Since p (Z) ⊂ 2 (Z), we M (a)ϕ = e0 has a solution ϕ

conclude that the function b = n∈Z ϕn χn belongs to L2 and that ab = 1. Thus, for a sequence ψ = {ψi } ∈ 0 (Z) the inverse B of M (a) is given by bi−j ψj (i ∈ Z). (Bψ)i = j∈Z

Due to the boundedness of B we !"! # " " sup " bi−j ψj j∈Z

have " # " " : ψ ∈ 0 (Z), ψp ≤ 1 < ∞, "

i∈Z p

which implies that b ∈ M p by the definition of M p . 2.29. Proposition. (a) Suppose a ∈ L∞ and 1 < p < ∞. If M (a) ∈ Φ+ (Lp ) or M (a) ∈ Φ− (Lp ), then M (a) ∈ GL(Lp ). (b) Suppose a ∈ W . If M (a) ∈ Φ+ (1 (Z)) or M (a) ∈ Φ− (1 (Z)), then M (a) ∈ GL(1 (Z)). (c) If a ∈ M p , 2 ≤ p < ∞, and M (a) ∈ Φ+ (p (Z)), then M (a) belongs to GL(p (Z)). (d) If a ∈ M p , 1 ≤ p < ∞, and M (a) ∈ Φ(p (Z)), then M (a) ∈ GL(p (Z)). Open problem. We are embarrassed to report that we have not been able to prove (c) for 1 < p < 2, although there seems to be no reason that (c) be false in that case.

66

2 Basic Theory

Proof. (a) Suppose M (a) ∈ Φ+ (Lp ); otherwise pass to the adjoint operator and take into account 1.12(h). We first show that Ker M (a) = {0}. Let af = 0 for some f ∈ Lp , f = 0. Then f χn ∈ Ker M (a) for all n ∈ Z and it is easily seen that the system {f χn }n∈Z is linearly independent in Lp (if f p = 0 for some p ∈ P, then f = 0 a.e. on T). It would follow that dim Ker M (a) = ∞, which is a contradiction. Thus Ker M (a) = {0}. Consequently, Im M ∗ (a) = Im M (a) = Lq (1/p + 1/q = 1) and there is a b ∈ Lq such that ab = 1 a.e. on T, and it follows that Ker M (a) = {0}. Thus M ∗ (a) ∈ GL(Lq ), whence M (a) ∈ GL(Lp ). (b) Suppose M (a) ∈ Φ+ (1 (Z)). As in the proof of part (a), one can see that then necessarily Ker M (a) = {0}. Thus, by 1.12(g), there is a δ > 0 / GL(1 (Z)). such that abW ≥ δbW for all b ∈ W . Now assume M (a) ∈ −1 Then a ∈ / GW , since otherwise M (a ) would be an inverse of M (a). But the maximal ideal space, T, of the Banach algebra W coincides with its Shilov boundary. Therefore, a is a topological divisor of zero, that is, there exists a sequence {bn }∞ n=1 of functions bn ∈ W such that bn W = 1 and abn W → 0 as n → ∞. We arrived at a contradiction. Now let M (a) ∈ Φ− (1 (Z)). Since c0 (Z) := ϕ = {ϕn }n∈Z : ϕn → 0 as |n| → ∞ is a predual of 1 (Z), we conclude from 1.12(h) that M (a) ∈ Φ+ (c0 (Z)). Assume M (a)ψ = 0, ψ ∈ c0 (Z), ψ = 0. Then M (a)(ψ ∗ en ) = 0 for all n ∈ Z, where (ψ ∗ en )i := ψn−i (i ∈ Z). We claim that the system {ψ ∗ en }n∈Z is linearly independent in c0 (Z). To see this, let π ∈ 0 (Z) and assume ψ ∗ π = 0. Let p ∈ PA denote the polynomial whose Fourier coefficients sequence is π, assume p(t) = q(t)(t − α) (t ∈ T) with q ∈ PA and α ∈ C, and let ∈ 0 (Z) be the Fourier coefficients sequence of q. Then ξ := ψ ∗ ∈ c0 (Z) and we have ξ ∗ (e1 − αe0 ) = 0, i.e., ξn−1 = αξn (n ∈ Z). If α = 0, then ξ = 0, and in case α = 0 we have ξ−n = αn ξ0 and ξn = (1/α)n ξ0 (n ∈ Z+ ), which also implies that ξ = 0. On repeating this argument with in place of π etc., we finally see that πn = 0 for n = 0. This proves the linear independence of the system {ψ ∗ en }n∈Z . Thus, what results is that Ker M (a) = {0} in c0 (Z). Consequently, Im M (a) = 1 (Z), hence there is a b ∈ W such that ab = 1, whence M (a) ∈ GL(1 (Z)). (c) As in the proof of the Φ− -part of (b) we conclude that M (a) has a trivial kernel in p (Z) whenever M (a) ∈ Φ+ (p (Z)). Therefore M (a) is onto on q (Z) (1/p + 1/q = 1). In particular, there is a ψ ∈ q (Z) such that M (a)ψ = e0 . Since q (Z) ⊂ 2 (Z), the function whose Fourier coefficients sequence is ψ belongs to L2 and we have af = 1. It follows that a = 0 a.e. on T. Thus, if M (a)ϕ = 0 for some ϕ ∈ q (Z) ⊂ 2 (Z), then, again by passing into L2 , we have ϕ = 0. So Ker M (a) = {0} in q (Z), hence M (a) ∈ GL(q (Z)), and thus M (a) ∈ GL(p (Z)).


67

(d) For p = 1 and 2 ≤ p < ∞ this is immediate from (b) and (c), respectively. If 1 < p < 2 and M (a) ∈ Φ(p (Z)), then M (a) ∈ Φ− (p (Z)), whence M (a) ∈ Φ+ (q (Z)) (2 < q < ∞) and (c) applies again. 2.30. Theorem (Hartman/Wintner). (a) Let a ∈ L∞ and 1 < p < ∞. If T (a) ∈ Φ+ (H p ) or T (a) ∈ Φ− (H p ), then a ∈ GL∞ . Consequently, R(a) ⊂ spΦ(H p ) T (a). (b) If a ∈ M p , 1 ≤ p < ∞, and T (a) ∈ Φ(p ), then a ∈ GM p . Consequently, R(a) ⊂ spM p a ⊂ spΦ(p ) T (a). Proof. (a) Let T (a) ∈ Φ+ (H p ) and denote by K any (finite-rank) projection of H p onto Ker T (a). By 1.12(g), there is a δ > 0 such that T (a)f p + Kf p ≥ δf p

∀ f ∈ H p.

This implies that P M (a)P gp + P KP gp + δQgp ≥ δgp

∀ g ∈ Lp .

Hence, if we let U denote the bilateral shift, then P M (a)P U n gp + P KP U n gp + δQU n gp ≥ δU n gp

∀ g ∈ Lp ,

and since U ±n are isometries, we obtain for all g ∈ Lp , U −n P M (a)P U n gp + P KP U n p + δU −n QU n gp ≥ δgp .

(2.28)

The operators U −n P U n are uniformly bounded on Lp , and because, obviously, U −n P U n f converges in Lp to f for every f ∈ P, we deduce from 1.1(d) that U −n P U n converges strongly to the identity operator. Thus, U −n QU n → 0 strongly on Lp , U −n P M (a)P U n = U −n P U n M (a)U −n P U n → M (a) strongly on Lp . Because U n converges weakly to zero on Lp , we get, by 1.1(f), P KP U n → 0

strongly on Lp .

Thus, (2.28) gives that M (a)gp ≥ δgp for all g ∈ Lp . So M (a) ∈ Φ+ (Lp ) and Propositions 2.29(a) and 2.28(a) imply that a ∈ GL∞ . For T (a) ∈ Φ− (H p ) passage to the adjoint yields the desired result. (b) The proof is the same as that of part (a).

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Remark 1. Propositions 2.29 and 2.28 also imply that the following implications hold: a ∈ W, T (a) ∈ Φ+ (1 ) ∪ Φ− (1 ) =⇒ a ∈ GW ; a ∈ M p , 2 ≤ p < ∞, T (a) ∈ Φ+ (p ) =⇒ a ∈ GM p ; a ∈ M p , 1 ≤ p ≤ 2, T (a) ∈ Φ− (p ) =⇒ a ∈ GM p . To prove that T (a) ∈ Φ− (1 ) =⇒ a ∈ GW pass first to the predual c0 of 1 and notice that U −n QU n → 0 strongly on c0 . Remark 2. The Hartman-Wintner theorem shows that in Proposition 2.19 and Theorem 2.23 the hypothesis that a be invertible is redundant. Both results can be stated in the form “Let a ∈ L∞ . Then T (a) is invertible on H 2 if and only if a ∈ GL∞ and ....” Remark 3. Suppose a, b ∈ L∞ and 1 0 such that " −n " "U M (a)P + M (b)Q U n g "p + KU n gp ≥ δgp , " " n "U M (a)P + M (b)Q U −n g " + KU −n gp ≥ δgp p for all n ≥ 0 and g ∈ Lp , and because U ±n → 0 weakly on Lp as n → ∞ and U −n P U n → I,

U −n QU n → 0,

U n P U −n → 0,

U n QU −n → I

strongly on Lp as n → ∞, we have M (a)gp ≥ δgp ,

M (b)gp ≥ δgp

for all g ∈ Lp and the assertion follows as above.

Thus, when investigating Fredholmness or invertibility of singular integral operators (over the unit circle) we may a priori assume that the coefficients are in GL∞ . Moreover, we then have M (a)P + M (b)Q = M (b) M (b−1 a)P + Q = M (b) P M (b−1 a)P + Q QM (b−1 a)P + I (2.29) and since QM (b−1 a)P +I is always invertible (the inverse is I −QM (b−1 a)P ), we arrive at the following conclusion. Let a, b ∈ L∞ . Then M (a)P + M (b)Q is in Φ(Lp ) (resp. GL(Lp )) if and only if a, b ∈ GL∞ and T (b−1 a) is in Φ(H p ) (resp. GL(H p )). In the case of Fredholmness, Ind M (a)P + M (b)Q = Ind T (b−1 a).


69

This shows in what a sense the study of Fredholmness and invertibility for singular integral operators over the unit circle (and thus over smooth curves) is equivalent to the study of the corresponding problems for Toeplitz operators. We now extend Proposition 2.18 to the case p = 2. Note that passage to adjoints yields results for anti-analytic symbols, that is, for h ∈ H ∞ . 2.31. Proposition. (a) If 1 < p < ∞ and h ∈ H ∞ , then T (h) ∈ GL(H p ) ⇐⇒ h ∈ GH ∞ . (b) If 1 ≤ p < ∞ and h ∈ M p ∩ H ∞ , then T (h) ∈ GL(p ) ⇐⇒ h ∈ GM p and h is outer. Proof. The implications “⇐=” follow as in the proof of Proposition 2.18. So we are left with the reverse implications. Theorem 2.30 gives that h ∈ GL∞ resp. h ∈ GM p . Thus, by 1.41(g) it remains to show that h−1 ∈ H ∞ . The identity (2.19) implies that −1 −1 −1 H ) h +H ) h T (h). H ) h ) h =T ) h −1 −1 But H ) h ) T (h) = 0, and h = H(1) = 0 and H ) h) = 0, whence H ) h −1 ) = 0. Because ) h−1 = (h−1 )), since T (h) is invertible, it results that H h −1 ∞ we conclude that h ∈ H . 2.32. Proposition. Let a ∈ L∞ and 1 < p < ∞. Then T (a) is in GL(H p ) (Φ± (H p ) resp. Φ(H p )) if and only if a ∈ GL∞ and T (a/|a|) is in GL(H p ) (Φ± (H p ) resp. Φ(H p )). Moreover, if a ∈ GL∞ , then dim Ker T (a) = dim Ker T (a/|a|),

dim Coker T (a) = dim Coker T (a/|a|).

Proof. It follows from Theorem 2.30(a) that a may be assumed to belong to GL∞ . As in the proof of Proposition 2.19 we see that a/|a| = hah for some h ∈ GH ∞ . Since T (a/|a|) = T (h)T (a)T (h), the preceding proposition implies all assertions. 2.33. Theorem (Brown/Halmos). If a ∈ L∞ , then spL(H 2 ) T (a) ⊂ conv R(a),

(2.30)

where conv R(a) is the closed convex hull of R(a). Proof. Immediate from Theorem 2.17.

Remark. We shall see later that if E is any subarc of T and χE is the characteristic function of E, neither spL(H p ) T (χE ) nor spL(p ) T (χE ) is contained in conv R(χE ) = [0, 1] for 1 < p < ∞ and p = 2.

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2.34. Real-valued continuous functions. For p = 2, Theorems 2.30 and 2.33 together give that R(a) ⊂ spess T (a) ⊂ sp T (a) ⊂ conv R(a).

(2.31)

This is all what is needed to derive the following: if a is a real-valued continuous function, then T (a) ∈ GL(H 2 ) ⇐⇒ T (a) ∈ Φ(H 2 ) ⇐⇒ a(t) = 0 ∀ t ∈ T and

spΦ(H 2 ) T (a) = spL(H 2 ) T (a) = min a(t), max a(t) . t∈T

t∈T

Note that both the spectrum and the essential spectrum are completely described via geometric data of the symbol. 2.35. Connectedness of the spectrum. A powerful tool for obtaining information about the spectra of Toeplitz operators are the following results. (a) (Widom). If a ∈ L∞ then spL(H p ) T (a) is connected. (b) (Douglas). If a ∈ L∞ then spΦ(H 2 ) T (a) is connected. Corollary 2.40 below implies that the boundary of spL(H p ) T (a) is contained in spL(H p ) T (a). Using this it is easy to derive the connectedness of the spectrum of a Toeplitz operator from the connectedness of the essential spectrum. Open problems. Is spΦ(H p ) T (a) always connected? We conjecture that the answer is yes and that a check of the proof in Widom [563] and Douglas [162] will indicate the modification needed to obtain the desired result. The following problem seems to us to lie essentially deeper: what can be said about the connectedness of the spectra of a Toeplitz operator on p ? We do not know any symbol in M p generating a Toeplitz operator whose spectra are disconnected. 2.36. Real-valued symbols. If a ∈ L∞ is real-valued, then spΦ(H 2 ) T (a) = spL(H 2 ) = ess inf a(t), ess sup a(t) . t∈T

t∈T

This result is due to Hartman and Wintner, too. Proof. Combine (2.31) and 2.35(b). There is a simple direct proof, which goes as follows. Let λ ∈ R and put b = a − λ. We must show that sign b = const whenever T (b) ∈ Φ(H 2 ). If Ind T (b) = κ, then Ind T (b) = Ind T (b) = Ind T ∗ (b) = −κ, whence κ = 0. Coburn’s theorem, which will be proved below (Corollary 2.40 for p = 2), therefore shows that we may assume that T (b) is invertible. Then the equation

2.6 The Connection Between Fredholmness and Invertibility

71

◦

2 T (b)f = 1 has a solution f ∈ H 2 . So bf = 1 + g with g ∈H− , and we obtain, for n ≥ 1, ( ( ( b|f |2 χn dm = bf f χn dm = (1 + g)f χn dm = 0. T

T

*

T

Since b|f |2 is real-valued, it follows that T b|f |2 χn dm = 0 for all n ∈ Z \ {0}, so b|f |2 = const, that is, sign b = const. 2.37. The boundary of conv R(a). For a ∈ L∞ , denote by a the harmonic extension of a into D. The following result of Wolff is sometimes very useful to get further information about the spectrum of a Toeplitz operator. Let a ∈ L∞ and let λ belong to the boundary of conv R(a). Then λ ∈ spL(H 2 ) T (a) ⇐⇒ λ ∈ clos a(D). An application of this result will be given in 4.75 and 4.78.

2.6 The Connection Between Fredholmness and Invertibility 2.38. Theorem (Coburn). A nonzero bounded Toeplitz operator has a trivial kernel or a dense range. The precise statement is as follows. (a) If a ∈ L∞ and if a does not vanish identically, then the kernel of T (a) in H p (1 < p < ∞) or the kernel of T (a) in H q (1/p + 1/q = 1) is trivial. (b) If a ∈ M p and if a does not vanish identically, then the kernel of T (a) in p (1 ≤ p < ∞) or the kernel of T (a) in q (1/p + 1/q = 1) is trivial. Proof. (a) Assume there are f+ ∈ H p , g+ ∈ H q , f+ = 0, g+ = 0 such that T (a)f+ = 0, T (a)g+ = 0. The F. and M. Riesz theorem 1.40(b) implies that ◦

f+ = 0 and g+ = 0 a.e. on T. Put f− := af+ and g− := ag+ . Then f− ∈H p , ◦

◦

◦

1 g− ∈H q , and so g− f+ ∈H 1 , g+ f− ∈H− . But g− f+ = ag+ f− , whence g+ f− = g− f+ = 0. Since f+ = 0 a.e. on T, we conclude that g− = 0 a.e. on T, and since g− = ag+ and g+ = 0 a.e. on T, it follows that a = 0 a.e. on T, which contradicts the hypothesis of the theorem.

(b) Let first 1 < p < ∞. Since the assertion is symmetric in p and q (recall 2.5(a), (b)), we may assume that 1 < p ≤ 2. Let T (a)ϕ+ = 0, T (a)ψ+ = 0, where ϕ+ ∈ p , ψ ∈ q , ϕ+ = 0, ψ+ = 0. Put ϕ− := M (a)ϕ+ and ψ− := M (a)ψ+ . Then ϕ− ∈ p (Z), ψ− ∈ q (Z), (ϕ− )n = (ψ− )n = 0 for all n ≥ 0. s (Z) (1 ≤ r ≤ ∞, 1/r + 1/s = 1) the convolution ϕ ∗ ψ For ϕ ∈ r (Z) and ψ ∈

defined by (ϕ ∗ ψ)i := j∈Z ϕi−j ψj belongs to ∞ (Z). For ϕ ∈ r (Z) define ϕ ∈ r (Z) by (ϕ)n = ϕn . Thus, we have

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(ψ− ∗ ϕ+ )n = 0 ∀ n ≤ 0,

(ψ+ ∗ ϕ− )n = 0 ∀ n ≤ 0

and because ψ− ∗ ϕ+ = (M (a)ψ+ ) ∗ ϕ+ = ψ+ ∗ (M (a)ϕ+ ) = ψ+ ∗ ϕ− , it follows that (ψ− ∗ ϕ+ )n = (ψ+ ∗ ϕ− )n = 0 for n ∈ Z. Since ψ+ = 0, we have (ϕ− )n = 0 for all n ∈ Z. Thus, M (a)ϕ+ = 0, and since ϕ+ ∈ p (Z) ⊂ 2 (Z), we deduce that af+ = 0 a.e. on T, where f+ ∈ H 2 is the function whose Fourier coefficients sequence is ϕ+ . The function f+ has a non-vanishing Fourier coefficient and therefore, by the F. and M. Riesz theorem 1.40(b), f+ = 0 a.e on T. This gives a = 0 a.e. on T and we arrived at a contradiction. Since M 1 is the Wiener algebra W , minor modifications of the proof also yield the result for p = 1. Notice that Toeplitz operators with symbols in W are obviously bounded on ∞ . Recall that, for a ∈ L1 , the function a is defined by a(t) = a(t) (t ∈ T). 2.39. Lemma. Let a ∈ L∞ , 1 < p < ∞, 1/p + 1/q = 1. Then T (a) is Fredholm (invertible ) on H p if and only if T (a) is Fredholm (invertible ) on H q . In the case of Fredholmness one has dim Ker T (a) = dim Coker T (a),

dim Ker T (a) = dim Coker T (a). ◦

Remark. Some care is in order, since the dual of H p is Lq / H q− and not H q . Nevertheless, all is easy. Proof. The hypothesis that a be in L∞ ensures that all operators occurring ◦

·

◦

·

p q + H p and Lq =H− + H q , we have are bounded. Since Lp =H−

T (a) ∈ Φ(H p ) ⇐⇒ P M (a)P + Q ∈ Φ(Lp ), T (a) ∈ Φ(H q ) ⇐⇒ P M (a)P + Q ∈ Φ(Lq ) and this is true with Φ replaced by GL. But (Lp )∗ is Lq and (P M (a)P + Q)∗ is easily seen to be P M (a)P + Q. This implies all assertions of the lemma. 2.40. Corollary. A Toeplitz operator is invertible if and only if it is Fredholm and has index zero. More explicitly: if a ∈ L∞ and 1 < p < ∞, then T (a) ∈ GL(H p ) ⇐⇒ T (a) ∈ Φ(H p ) and Ind T (a) = 0; if a ∈ M p and 1 ≤ p < ∞, then T (a) ∈ GL(p ) ⇐⇒ T (a) ∈ Φ(p ) and Ind T (a) = 0.


73

Proof. The previous lemma and Theorem 2.38 imply that Ind T (a) = dim Ker T (a) − dim Ker T (a) = 0 if and only if dim Ker T (a) = dim Ker T (a) = 0. 2.41. The index of a continuous function. Let a ∈ C and suppose a has no zeros on T. Then there is a b ∈ CR(T◦ ) (recall 2.25) such that a = |a|e2πib . The increment of b as the result of a circuit around T counter-clockwise is an integer and depends only on a, i.e., it does not depend on the particular choice of b. This integer is referred to as the index (or winding number ) of a and is denoted by ind a. If a ∈ C has no zeros on T, then a/|a| is a continuous function belonging to CU (T◦ ). Therefore the limits lim (a/|a|)# (x) =: (a/|a|)# (±∞) exist, are x→±∞

finite, and its difference is an integral multiple of 2π. It is easily seen that ind a is nothing else than 1 1 0 (a/|a|)# (+∞) − (a/|a|)# (−∞) . 2π Note that ind χn = n, where χn (t) = tn (t ∈ T). Here are two important properties of the index. (a) If a, b ∈ C and a(t)b(t) = 0 for all t ∈ T, then ind (ab) = ind a + ind b. (b) If a, d ∈ C, a(t) = 0 for all t ∈ T, and d/a∞ < 1, then a(t)+d(t) = 0 for all t ∈ T and ind (a + d) = ind a. If a is continuously differentiable and does not vanish on T, then ( ( 2π iθ 1 a (t) a (e ) iθ 1 dt = e dθ. ind a = 2πi T a(t) 2π 0 a(eiθ ) Thus, if a ∈ C has no zeros on T, then by the above property (b) and by 1.38(b), ( 1 (hr a) (t) ind a = lim ind hr a = lim dt. r→1−0 r→1−0 2πi T (hr a)(t) Also notice the following. (c) If a is a rational function without poles and zeros on the unit circle T, then ind a = z − p, where z and p are the numbers of zeros and poles (counted up to multiplicity) of a in D, respectively. In the language of Banach algebras we have the following. (d) If a ∈ GC, then ind a = 0 if and only if a belongs to the connected component of GC containing the identity. Given a Banach algebra A of continuous functions on T that contains the constants we shall say that the maximal ideal space of A is T if

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(i) the general form of a multiplicative linear functional on A is given by ϕ : A → C, ϕ(a) = a(τ ), where τ ranges over T; (ii) the Gelfand topology on T coincides with the usual topology on T. The notion of the index allows us to specialize a result of Shilov as follows. (e) If A is a Banach algebra of continuous functions on T that contains the constants and whose maximal ideal is T, then every a ∈ GA of index zero has a logarithm log a ∈ A and, consequently, by 1.16(a), belongs to the connected component of GA containing the identity. We are now in a position to establish criteria for Fredholmness and invertibility of Toeplitz operators with continuous symbols on H p and p . 2.42. Theorem. Let a ∈ C and 1 < p < ∞. Then (a) H(a) ∈ C∞ (H p ); (b) T (a) ∈ Φ(H p ) if and only if a(t) = 0 for all t ∈ T; if T (a) is Fredholm on H p , then T (a−1 ) is a regularizer of T (a) and Ind T (a) = −ind a; (c) T (a) ∈ GL(H p ) if and only if a(t) = 0 for all t ∈ T and ind a = 0. Proof. (a) There are an ∈ P (for example, the Fejér means of a) such that a − an ∞ → 0 as n → ∞. Then H(an ) has finite rank and since H(a) − H(an )p = P M (a − an )QJp ≤ c2p a − an ∞ , H(a) is compact on H p . (b) The implication “=⇒” follows from Theorem 2.30. So suppose a = 0 on T. By Proposition 2.14, −1 T (a−1 )T (a) = I − H(a−1 )H() (2.32) a), T (a)T (a−1 ) = I − H(a)H ) a and since, by (a), all Hankel operators occurring are compact, it follows that T (a) ∈ Φ(H p ) and that T (a−1 ) is a regularizer of T (a). So we are left with the index formula. Let T (a) ∈ Φ(H p ) and ind a = n. Then, by 2.41(a), ind (χ−n a) = 0. Hence, by 2.41(d), χ−n a belongs to the connected component of GC containing the identity. As T (f )L(H p ) ≤ cp f ∞ , the mapping T : GC → Φ(H p ), f → T (f ) is continuous. Consequently, the operator T (χ−n a) must be in the connected component of Φ(H p ) containing I and 1.12(d) gives Ind T (χ−n a) = 0. Because T (χ−n a) equals T (χ−n )T (a) or T (a)T (χ−n ), we deduce from Atkinson’s theorem 1.12(c) and from 2.9 that 0 = Ind T (χ−n a) = Ind T (χ−n ) + Ind T (a) = n + Ind T (a). (c) Immediate from (b) and Corollary 2.40. Remark. Theorem 2.30 even implies that a(t) = 0 for all t ∈ T provided T (a) ∈ Φ+ (H p ) or T (a) ∈ Φ− (H p ).


75

2.43. The classes Cp and M p . For 1 ≤ p < ∞, let Cp denote the closure in M p of the Laurent polynomials: Cp = closM p P. Clearly, C1 = W and C2 = C. Note that Cp is a closed subalgebra of M p . For p ∈ (1, 2) ∪ (2, ∞), let M p denote the collection of all functions a ∈ L∞ which belong to M p) for all p) in some neighborhood of p, i.e., $ (M p+ε ∩ M p−ε ). M p := ε>0

Finally, let M 1 = M 1 = W and M 2 = M 2 = L∞ . The following Proposition 2.45 intends to give an alternative description of Cp and to provide a better understanding of which functions belong to Cp . However, neither the definition of M p nor that proposition are needed to prove Propositions 2.46 and Theorem 2.47. 2.44. Lemma. Let a ∈ M p and let σn a denote the n-th Fejér mean of a, n |j| (σn a)(t) = 1− a j tj n + 1 j=−n

(t ∈ T).

Then σn aM p ≤ aM p for all n ≥ 0. Proof. For θ ∈ (−π, π], let sin2 (n + 1)θ/2 1 Kn (θ) = 2π(n + 1) sin2 (θ/2) denote the n-th Fejér kernel and define the function ax by ax (eiθ ) := a(ei(θ−x) ). Thus, ( π

(σn a)(eiθ ) =

−π

ax (eiθ )Kn (x) dx.

(2.33)

It is easy to see that M (ax ) = D−x M (a)Dx , where Dx is the isometry Dx : p → p ,

{ϕj }j∈Z+ → {eijx ϕj }j∈Z+ .

Therefore, if ϕ ∈ 0 (Z), then the function (−π, π] → p ,

x → Kn (x)M (ax )ϕ

is continuous. This and (2.33) enable us to write M (σn a)ϕ as a Bochner integral: ( π M (ax )ϕKn (x) dx. M (σn a)ϕ = −π

Hence,

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( M (σn a)ϕp ≤

π −π

M (ax )ϕp Kn (x) dx

≤ D−x p M (a)p Dx p ϕp

(

π −π

Kn (x) dx

= M (a)p ϕp , for all ϕ ∈ P, and consequently, M (σn a)p ≤ M (a)p . 2.45. Proposition. If 1 ≤ p < ∞, then Cp = closM p (C ∩ M p ). Proof. There is nothing to prove for p = 1 or p = 2. Thus let p ∈ (1, 2)∪(2, ∞). We first show that C ∩ M p ⊂ Cp . By virtue of 2.5(b), we may without loss of generality assume that 2 0. Let σn a denote the n-th Fejér mean of a. From 2.5(e) we get γ M (a − σn a)p ≤ a − σn a1−γ ∞ M (a − σn a)p+ε ,

where 1/p = γ/(p + ε) + (1 − γ)/2. Since a ∈ C, we know that a − σn a∞ → 0 as n → ∞, and Lemma 2.44 applied to a ∈ M p+ε shows that M (a−σn a)p+ε remains bounded as n → ∞. Thus, the inclusion C ∩ M p ⊂ Cp is proved. Now it is easy to see that the asserted equality holds: Cp = closM p P ⊂ closM p (C ∩ M p ) ⊂ closM p Cp = Cp . 2.46. Proposition. Let 1 ≤ p < ∞. (a) The maximal ideal space M (Cp ) of Cp is T. Thus, if a ∈ Cp , then a ∈ GCp ⇐⇒ a(t) = 0 ∀ t ∈ T.

(2.34)

(b) The connected component of GCp containing the identity coincides with the functions in GCp of index zero. Proof. (a) By virtue of 2.5(d), we have |a(τ )| ≤ a∞ aM p for all τ ∈ T and hence m : Cp → C, a → a(τ ) defines a multiplicative linear functional on Cp . Conversely, let m : Cp → C be a multiplicative linear functional. Put τ := m(χ1 ). It is easy to see that spM p (χ1 ) = T, whence τ ∈ T. This implies that m(f ) = f (τ ) for every f ∈ P and thus m(a) = a(τ ) for every a ∈ Cp , the closure of P. That the Gelfand topology on T coincides with the usual one on T can be checked in a standard way. (b) This follows from Shilov’s theorem 2.41(e). A proof which does not invoke that theorem is as follows. If a ∈ GCp has index zero, then, by 2.41(b), c ∈ GCp and ind c = 0 whenever c ∈ Cp and a − cM p is sufficiently small. Since P is dense in Cp , among these c’s there is a c ∈ P. Using 2.41(a) and 2.41(c) it is not difficult to see that c factors into a product of functions of the form χ1 − α and 1 − βχ−1 , where |α| > 1 and |β| < 1. But such functions are in the connected component of GCp containing the identity, hence so is c and therefore a, too.

2.7 Compactness of Hankel Operators and C + H ∞ Symbols

77

2.47. Theorem. Let a ∈ Cp and 1 ≤ p < ∞. Then (a) H(a) ∈ C∞ (p ); (b) T (a) ∈ GL(p ) if and only if a(t) = 0 for all t ∈ T; if T (a) is Fredholm on p , then T (a−1 ) is a regularizer of T (a) and Ind T (a) = −ind a; (c) T (a) ∈ GL(p ) if and only if a(t) = 0 for all t ∈ T and ind a = 0. Proof. (a) By the definition of Cp , there are an ∈ P such that H(a) − H(an )p ≤ a − an M p = o(1)

as n → ∞,

which gives the compactness of H(a) on p . (b) The implication “=⇒” follows from Theorem 2.30. Taking into account (2.34), one can obtain the implication “⇐=” as in the proof of Theorem 2.42. Proposition 2.46(b) shows that the argument applied in the proof of Theorem 2.42 can be used to verify the index formula in the case at hand, too. (c) This is immediate from (b) and Corollary 2.40. Remark 1. The index formula can also be proved with the help of the argument that will be used in the proof of Theorem 2.66 below. Remark 2. One can show, e.g., as in the proof of 2.29(b) or by using a perturbation argument (see the proof of Theorem 2.74 below), that a ∈ GCp whenever a ∈ Cp and T (a) ∈ Φ+ (p ) or T (a) ∈ Φ− (p ).

2.7 Compactness of Hankel Operators and C + H ∞ Symbols 2.48. Definition. For 1 < p < ∞, let Ap := a ∈ L∞ : H(a) ∈ C∞ (H p ) ,

Bp := a ∈ M p : H(a) ∈ C∞ (p ) .

2.49. Lemma. Ap and Bp are closed subalgebras of L∞ and M p , respectively. Proof. It is clear that Ap and Bp are linear spaces. From Proposition 2.14 we have H(ab) = T (a)H(b) + H(a)T ()b), which shows that ab ∈ Ap resp. ab ∈ Bp whenever a, b ∈ Ap resp. a, b ∈ Bp . Thus Ap and Bp are algebras. If an ∈ Bp , b ∈ M p , an − bp → 0 as n → ∞, then H(b) − H(an )L(p ) ≤ b − an p = o(1) as n → ∞, whence H(b) ∈ C∞ (p ). This shows that Bp is closed. The closedness of Ap can be proved analogously. 2.50. Theorem. (a) If a ∈ Ap , then T (a) ∈ Φ(H p ) ⇐⇒ a ∈ GAp . (b) If a ∈ Bp , then T (a) ∈ Φ(p ) ⇐⇒ a ∈ GBp .

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Proof. (a) Let a ∈ Ap . Then a, a−1 ∈ Ap and, by (2.18), T (a−1 )T (a) − I = −H(a−1 )H() a) ∈ C∞ (H p ), T (a)T (a−1 ) − I = −H(a)H ) a−1 ) ∈ C∞ (H p ), whence T (a) ∈ Φ(H p ). Conversely, suppose T (a) ∈ Φ(H p ). From Theorem 2.30 we know that then a ∈ GL∞ and it remains to show that a−1 ∈ Ap . Let RT (a) = I + K, where R ∈ L(H p ) is a regularizer of T (a) and K ∈ C∞ (H p ). Formula (2.19) gives −1 0 = H(aa−1 ) = T (a)H(a−1 ) + H(a)T ) , a hence, by acting with R from the left, −1 0 = H(a−1 ) + KH(a−1 ) + RH(a)T ) , a and it follows that H(a−1 ) ∈ C∞ (H p ), as desired. (b) The proof is the same. 2.51. Definition. For 1 < p < ∞, define Cp as in 2.43 and let Hp∞ := H ∞ ∩M p . It is obvious that Hp∞ is a closed subalgebra of M p . Let alg (Cp , Hp∞ ) denote the smallest closed subalgebra of M p containing Cp and Hp∞ . It is clear that alg (Cp , Hp∞ ) coincides with alg (χ1 , Hp∞ ), the smallest closed subalgebra of M p containing the set Hp∞ and the function χ1 . The discontinuous function (1 − χ1 )iβ (β ∈ R \ {0}) can be shown to belong to Hp∞ for all 1 < p < ∞ (Theorem 6.45). This shows that alg (Cp , Hp∞ ) is strictly larger than Cp . Of course, alg (Cp , Hp∞ ) contains Cp + Hp∞ = f + g : f ∈ Cp , g ∈ Hp∞ . It is an absolutely unexpected fact, the discovery of which goes back to Sarason, that alg (Cp , Hp∞ ) is actually equal to Cp + Hp∞ . Our next goal is to prove this. The proof will be based on the following interesting lemma. 2.52. Lemma (Zalcman/Rudin). Let X be a Banach space and let E and F be closed subspaces of X. Suppose {Sn }n∈Z+ is a sequence of operators Sn ∈ L(X) with the following properties: (i) Sn L(X) ≤ M for all n ∈ Z+ ; (ii) Sn (X) ⊂ E for all n ∈ Z+ ; (iii) Sn (F ) ⊂ F for all n ∈ Z+ ; (iv) Sn u − u → 0 as n → ∞ for all u ∈ E. Then E + F is a closed subspace of X.


79

Proof. Let x ∈ closX (E + F ). Then

there are uk ∈ E, vk ∈ F such that ∞ uk + vk ≤ 1/2k for k ≥ 2 and x = k=1 (uk + vk ). For each k ≥ 2 choose k nk so that uk − Snk uk ≤ 1/2 . If we let xk := uk + vk , then obviously xk = (uk − Snk uk + Snk xk ) + (vk − Snk vk ). We have u )k := uk − Snk uk + Snk xk ∈ E and ) uk ≤

1 1+M + Snk xk ≤ k 2 2k

(note that xk ≤ 1/2k ). Furthermore, v)k := vk − Snk vk ∈ F and ) vk ≤ ) uk + xk ≤

2+M . 2k

∞

∞ )k and k=1 v)k are absolutely convergent. Let Consequently, the series k=1 u u ∈ E and v ∈ F denote their sums. What results is that x=

∞ k=1

xk =

∞

() uk + v)k ) = u + v ∈ E + F.

k=1

2.53. Theorem. Cp + Hp∞ is a closed subalgebra of M p and Cp + Hp∞ = alg (Cp , Hp∞ ) = alg (χ1 , Hp∞ ).

(2.35)

Proof. We apply the preceding lemma with X = M p , E = Cp , F = Hp∞ , and Sn ∈ L(M p ) given by Sn a = σn a, where σn a denotes the n-th Fejér mean of a. It is clear that (ii) and (iii) are satisfied. Lemma 2.44 shows that (i) is fulfilled. Finally, given a ∈ Cp and ε > 0 choose f ∈ P so that a − f M p < ε/3. Then σn a − aM p ≤ σn (a − f )M p + σn f − f M p + f − aM p ≤ a − f M p + σn f − f W + f − aM p 2ε + σn f − f W < 3 (Lemma 2.44 and 2.5(d)) and σn f − f W < ε/3 whenever n is large enough. Thus, the requirement (iv) is also met and it follows that Cp + Hp∞ is closed. Now let a, b ∈ Cp + Hp∞ . There are an , bn ∈ P + Hp∞ such that a − an M p → 0,

b − bn M p → 0 as

n → ∞.

It is obvious that an bn ∈ P +Hp∞ , and since an bn −abM p → 0 as n → ∞ and Cp + Hp∞ is closed, we conclude that ab ∈ Cp + Hp∞ . Consequently, Cp + Hp∞ is an algebra. Once we know that Cp + Hp∞ is a closed subalgebra of M p , the equalities (2.35) are obvious.

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2 Basic Theory

2.54. Theorem (Hartman/Adamyan/Arov/Krein). Let a ∈ L∞ and 1 < p < ∞. Then distL∞ (a, C + H ∞ ) ≤ H(a)Φ(H p ) ≤ cp distL∞ (a, C + H ∞ ), where cp = P L(Lp ) . In particular, Ap = C + H ∞ , that is, H(a) ∈ C∞ (H p ) ⇐⇒ a ∈ C + H ∞ . Proof. We have distL∞ (a, C + H ∞ ) = inf a − f − h∞ : f ∈ C, h ∈ H ∞ = inf

inf a − f − h∞ = inf distL∞ (a − f, H ∞ )

f ∈C h∈H ∞

c∈C

≥

1 inf H(a − f )L(H p ) cp c∈C

=

1 inf H(a) − H(f )L(H p ) cp f ∈C

=

1 H(a)Φ(H p ) cp

(Theorem 2.11)

(Theorem 2.42(a)).

Now let V = T (χ1 ). Since (V n )∗ = T (χ−n ) converges strongly to zero on H p as n → ∞, we conclude that KV n → 0 as n → ∞ for every K ∈ C∞ (H p ) (see 1.3(d)). Thus, if K ∈ C∞ (H p ) then H(a) − K ≥ (H(a) − K)V n ≥ H(a)V n − KV n = H(χ−n a) − KV n ≥ distL∞ (χ−n a, H ∞ ) − KV n (Theorem 2.11) = distL∞ (a, χn H ∞ ) − KV n = distL∞ (a, C + H ∞ ) − KV n , whence H(a)Φ(H p ) ≥ distL∞ (a, C + H ∞ ). Since C + H ∞ is closed, we have distL∞ (a, C + H ∞ ) = 0 if and only if a ∈ C + H ∞. Remark. The above theorem implies the following compactness criterion for Hankel operators: if a ∈ L∞ and 1 < p < ∞, then H(a) ∈ C∞ (H p ) if and only if P a ∈ V M O. Indeed, if a ∈ C + H ∞ , then P a ∈ P (C) ⊂ V M O by 1.48(k), and if P a ∈ V M O, then P a = u + P v with u, v ∈ C by virtue of 1.48(l), which shows that a = u + v + Q(a − v) is in C + H ∞ . 2.55. Corollary. If a ∈ C + H ∞ , then T (a) ∈ Φ(H p ) ⇐⇒ a ∈ G(C + H ∞ ). If T (a) ∈ Φ(H p ), then T (a−1 ) is a regularizer of T (a).


81

Proof. Immediate from Theorems 2.50 and 2.54 and formula (2.18). 2.56. Open problem. Establish the analogue of Theorem 2.54 for p . In this connection recall 2.12. It is clear that Cp + Hp∞ ⊂ Bp , but we have not been able to prove that Bp ⊂ Cp + Hp∞ . Nevertheless we shall show that the p version of Corollary 2.55 holds (see Theorem 2.60 below). 2.57. Definition. Put R = p/q : p ∈ PA , q ∈ PA , q(t) = 0 ∀t ∈ T . Note that R is the restriction to the unit circle T of the set of all rational functions defined on the whole plane C and having no poles on T. 2.58. Theorem (Kronecker). (a) Let 1 < p < ∞ and a ∈ L∞ . Then H(a) ∈ C0 (H p ) ⇐⇒ a ∈ R + H ∞ . (b) Let 1 < p < ∞ and a ∈ M p . Then H(a) ∈ C0 (p ) ⇐⇒ a ∈ R + Hp∞ . Proof. We first prove that the operator H(a) has finite rank for a ∈ R (and thus for a ∈ R+H ∞ resp. a ∈ R+Hp∞ ). This is obvious if a is a polynomial. If a = 1/(χ1 −λ) with some λ ∈ C, then H(a) = 0 for |λ| < 1 and rank H(a) = 1 for |λ| > 1. So application of the formula H(bc) = T (b)H(c) + H(b)T () c) shows that H(a) has finite rank if a is of the form polynomial /(χ1 − λ)n (n ∈ Z+ , λ ∈ C), and decomposition into partial fractions gives the assertion for all a ∈ R. Now suppose rank H(a) = r < ∞. This implies that the first r + 1 columns of the matrix (aj+k+1 )∞ j,k=0 are linearly dependent, where an denotes the n-th Fourier coefficient of a. Hence, if we let b :=

∞

ar+k+1 χk

(∈ H 2 ),

k=0

then there exist complex numbers λ0 , λ1 , . . . , λr such that at least one of them is non-zero and λ0 (a1 + a2 χ1 + . . . + ar χr−1 + χr b) +λ1 (a2 + a3 χ1 + . . . + ar χr−2 + χr−1 b) + . . . + λr b = 0. It follows that (λ0 χr + λ1 χr−1 + . . . + λr )b is a polynomial and therefore b must be a rational function. Since b ∈ H 2 , b cannot have poles on T, whence b ∈ R. If a ∈ L∞ resp. a ∈ M p , then χ−r−1 a − b belongs to H ∞ resp. M p ∩ H ∞ = Hp∞ , which shows that a ∈ R + H ∞ resp. a ∈ R + Hp∞ .

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2.59. Corollary. (a) R + Hp∞ is an algebra. (b) If a ∈ R + H ∞ , then T (a) ∈ Φ(H p ) ⇐⇒ a ∈ GL∞ and a−1 ∈ R + H ∞ . (c) If a ∈ R + Hp∞ , then T (a) ∈ Φ(p ) ⇐⇒ a ∈ GL∞ and a−1 ∈ R + Hp∞ . Proof. (a) This follows from Theorem 2.58 together with the formula (2.19). (b), (c) Combine the reasoning in the proof of Theorem 2.50 with Theorem 2.58 (also take into account 1.12(a), (iii)). 2.60. Theorem. Let 1 < p < ∞ and a ∈ Cp + Hp∞ . Then T (a) ∈ Φ(p ) ⇐⇒ a ∈ G(Cp + Hp∞ ). If T (a) ∈ Φ(p ), then T (a−1 ) is a regularizer of T (a). Proof. If a and a−1 are in Cp + Hp∞ , then H(a) and H(a−1 ) are compact, and so the argument used in the proof of Theorem 2.50 gives the implication “⇐=” and shows that T (a−1 ) is a regularizer of T (a). Now suppose T (a) ∈ Φ(p ). Then, by Theorem 2.30(b), a ∈ GM p and it remains to prove that a−1 ∈ Cp + Hp∞ . By the definition of Cp , there are functions bn ∈ R+Hp∞ such that a−bn M p → 0 as n → ∞. If n is sufficiently large, then T (bn ) ∈ Φ(p ) (by 1.12(d)) and so Corollary 2.59 implies that p p ∞ b−1 n ∈ R+Hp . But if bn converges to an element a ∈ GM in the norm of M , −1 −1 p −1 then bn converges to a in the norm of M . Hence a ∈ closM p (R + Hp∞ ) and since Cp + Hp∞ is closed, it follows that a−1 ∈ Cp + Hp∞ . Our next concern is an invertibility criterion for C + H ∞ . We first need a property of the harmonic extension. Recall that, for f ∈ L∞ and 0 < r < 1, the function fr ∈ C is defined by fr (eiθ ) := f(reiθ ). 2.61. Lemma. If c ∈ C and a ∈ L∞ , then (ca)r − cr ar ∞ → 0 as r → 1 − 0. Proof. Since cr − c∞ → 0 it suffices to show that (ca)r − car ∞ → 0. Let kr denote the Poisson kernel (see 1.37). Then ( 2π 1 (ca)r (eiθ ) − c(eiθ )ar (eiθ ) = [c(eit ) − c(eiθ )]a(eit )kr (θ − t) dt. 2π 0 Given ε > 0 there is a δ > 0 such that |c(eit ) − c(eiθ )| < ε whenever |t − θ| < δ and so + +( ( 2π + + + + it iθ it ≤ ε [c(e ) − c(e )]a(e )k (θ − t) dt |a(eit )|kr (θ − t) dt + + r + + |t−θ|δ + |τ |>δ ≤ 2c∞ a∞ ε. This completes the proof.

2.62. Theorem (Douglas). (a) If a, b ∈ C + H ∞ then (ab)r − ar br ∞ → 0 as

r → 1 − 0.

a is bounded (b) Let a ∈ C + H ∞ . Then a ∈ G(C + H ∞ ) if and only if away from zero in some annulus near T, i.e., if and only if there exist δ > 0 and ε > 0 such that | a(z)| > ε for 1 − δ < |z| < 1. Proof. (a) Let a = c + h, b = d + g, where c, d ∈ C and h, g ∈ H ∞ . It is clear that (hg)r = hr gr . From the preceding lemma we deduce that lim (cd)r −cr dr ∞ = lim (cg)r −cr gr ∞ = lim (hd)r −hr dr ∞ = 0,

r→1−0

r→1−0

r→1−0

and this gives the assertion at once. (b) Let a ∈ G(C + H ∞ ) and b = a−1 . Then, by (a), ar br − 1∞ → 0 as r → 1 − 0, and since br ∞ is bounded from above (by b∞ ), it follows that |ar | must be bounded away from zero if r is close enough to 1. a(z)| > ε for 1 − δ < |z| < 1. Then Now let a ∈ C + H ∞ and assume | |a| ≥ ε a.e. on T and so a ∈ GL∞ . Because a ∈ C + H ∞ , there are hn ∈ H ∞ such that χ−n hn → a as n → ∞ in the norm of L∞ . Part (a) and the fact 2n is bounded that a is bounded away from zero near T imply that each h away from zero in some annulus 1 − δn < |z| < 1. So 1/hn is bounded and 2n can there be written as analytic in 1 − δn < |z| < 1. Consequently, 1/h the sum of a function which extends to be bounded and analytic in D and a function which extends to be bounded and analytic in |z| > 1 − δn . This decomposition yields a representation of 1/hn as the sum of a function in H ∞ ∞ ∞ and a function in C. Thus h−1 and therefore χn h−1 n ∈ C +H n ∈ C +H . ∞ ∞ −1 −1 But if χ−n hn → a ∈ GL in the norm of L , then χn hn → a in the L∞ -norm. Since C + H ∞ is closed, it follows that a−1 ∈ C + H ∞ . We finally compute the index of Toeplitz operators with C + H ∞ and Cp + Hp∞ symbols. By virtue of Corollary 2.40 this solves the invertibility problem for these operators. 2.63. Definition. Let a ∈ L∞ and suppose a is bounded away from zero in some annulus near T. Then there are ε > 0 and δ > 0 such that |ar (eiθ )| ≥ ε for all r ∈ (1 − δ, 1) and θ ∈ [0, 2π). By 2.41(b), the mapping (1 − δ, 1) → Z, r → ind ar is continuous, and because (1 − δ, 1) is connected, ind ar must be constant for r ∈ (1 − δ, 1). That constant value of ind ar will be denoted by ind {ar }.

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2 Basic Theory

2.64. Theorem. Let h ∈ H ∞ and 1 < p < ∞. Then T (h) ∈ Φ(H p ) if and only if h = bg where b is a finite Blaschke product and g ∈ GH ∞ . If T (h) is Fredholm, then h is bounded away from zero in some annulus near T and Ind T (h) = −ind {hr }. Proof. If h = bg with b ∈ GC and g ∈ GH ∞ , then T (h) = T (b)T (g) ∈ Φ(H p ) by Theorem 2.42 and Proposition 2.31. Conversely, suppose T (h) ∈ Φ(H p ). Then h ∈ G(C + H ∞ ) due to Corollary 2.55. In view of 1.41(a), 1.41(b), we have h = bSg, where b is a Blaschke product, S is a singular inner function, and g ∈ GH ∞ . Thus bS ∈ G(C + H ∞ ) and by virtue of Theorem 2.62(b), b(z)S(z) must be bounded away from zero in some annulus near T. Since the radial limit of S(z) vanishes at the points in the support of the singular measure defining S, it follows that S = 1, and b(z) is bounded away from zero in an annulus near T only if b is a finite Blaschke product. The index formula can be derived as follows: Ind T (h) = Ind T (b) + Ind T (g) (Atkinson) = Ind T (b) (Proposition 2.31) = −ind b (Theorem 2.42) = −ind {br } (1.38(b) and 2.41(b)) = −ind {br } − ind {gr } (1.41(g)) = −ind {(bg)r } (Lemma 2.61 and 2.41(b)).

2.65. Theorem (Douglas). Let a ∈ C + H ∞ and 1 < p < ∞. Then T (a) is a is bounded away from zero in some annulus near T, in Φ(H p ) if and only if and in that case Ind T (a) = −ind {ar }. Proof. The Fredholm criterion follows by combining Corollary 2.55 and Theorem 2.62(b). So it remains to prove the index formula. There is a number ε > 0 with the following property: if b − a∞ < ε, then T (b) ∈ Φ(H p ), Ind T (b) = Ind T (a), b is bounded away from zero in some annulus near T, and ind {br } = ind {ar } (recall 1.12(d), 1.38(b), 2.41(b)). Among these b’s we can find a b ∈ C + H ∞ of the form b = χ−n h, n ∈ Z+ , h ∈ H ∞ . Because T (b) = T (χ−n )T (h) ∈ Φ(H p ), it follows that T (h) ∈ Φ(H p ) and the preceding theorem gives (2.36) Ind T (h) = −ind {hr }. The desired index formula can now be verified as follows: Ind T (a) = Ind T (χ−n h) = Ind T (χ−n ) + Ind T (h) (Atkinson) = n + Ind T (h) (2.9) = n − ind {hr } (equality (2.36)) = −ind {(χ−n )r } − ind {hr }

2.8 Local Methods for Scalar Toeplitz Operators

85

= −ind {(χ−n )r hr } (2.41(a)) = −ind {(χ−n h)r } (Lemma 2.61 and 2.41(b)) = −ind {ar }. a is 2.66. Theorem. Let a ∈ Cp + Hp∞ and 1 < p < ∞. If T (a) ∈ Φ(p ), then bounded away from zero in some annulus near T and Ind T (a) = −ind {ar }. Proof. Suppose 1 < p < 2 and let 1/p + 1/q = 1. From Theorem 2.60 we know that a ∈ G(Cp + Hp∞ ). Hence, due to 2.5(c), (d), a ∈ G(Cs + Hs∞ ) for all s ∈ [p, q] and thus, again by Theorem 2.60, T (a) ∈ Φ(s ) for all s ∈ [p, q]. Given an operator A ∈ Φ(s ) denote by αs (A) the dimension of the kernel of A in s and by Inds A the index of A considered as operator on s . Since p ⊂ 2 ⊂ q , we have αp (T (a)) ≤ α2 (T (a)),

α2 (T (a)) ≤ αq (T (a)),

hence Indp T (a) = αp (T (a)) − αq (T (a)) ≤ α2 (T (a)) − α2 (T (a)) = Ind2 T (a) = −ind {ar }. On repeating this argument with a−1 in place of a we arrive at the inequality Indp T (a−1 ) ≤ −ind {(a−1 )r } = ind {ar } (recall Theorem 2.62(a)). But T (a−1 ) is a regularizer of T (a). So Indp T (a) = −Indp T (a−1 ) ≥ −ind {ar }, and we finally get Indp T (a) = −ind {ar }. If 2 < p < ∞, then 1 < q < 2 and from the equality Indp T (a) = −Indq T (a) = ind {ar } = −ind {ar } we get the desired formula.

2.8 Local Methods for Scalar Toeplitz Operators 2.67. The local distance at a point. Given a ∈ L∞ and an open subarc U of T denote by a|U the restriction of a to U regarded as an element of L∞ (U ). For τ ∈ T, let Uτ denote the family of all open subarcs of T containing the point τ . The local distance of a, b ∈ L∞ at τ ∈ T is defined as distτ (a, b) = inf a|U − b|U L∞ (U ) . U ⊂Uτ

Clearly, if a|U = b|U for some neighborhood U of τ , then distτ (a, b) = 0. If the finite limits a(τ ±0) exist, then distτ (a, b) = 0 if and only if a(τ −0) = b(τ −0)

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2 Basic Theory

and a(τ + 0) = b(τ + 0). In particular, if a and b are continuous at τ , then distτ (a, b) = 0 if and only if a(τ ) = b(τ ). Let Rτ denote the collection of all functions f ∈ C such that 0 ≤ f ≤ 1 on T and f is identically 1 in some neighborhood of τ (depending on f ). It is not difficult to verify that distτ (a, b) = inf (a − b)c∞ . c∈Rτ

(2.37)

2.68. Theorem. Let 1 < p < ∞ and a ∈ L∞ . Assume that for each τ ∈ T there is an aτ ∈ L∞ such that distτ (a, aτ ) = 0 and T (aτ ) ∈ Φ(H p ). Then T (a) ∈ Φ(H p ). Proof. This theorem provides a good occasion of giving an application of Theorem 1.32. Put A = L(H p )/C∞ (H p ) and, for a ∈ L∞ , let T π (a) denote the coset in A containing T (a). If f ∈ C and b ∈ L∞ , then, by Proposition 2.14, T (f )T (b) = T (f b) − H(f )H()b) = T (b)T (f ) + H(b)H(f)) − H(f )H()b) and since H(f ) and H(f)) are compact on H p , we get T π (f )T π (b) = T π (f b) = T π (b)T π (f ).

(2.38)

For τ ∈ T, define Rτ as in 2.67 and put Mπτ := T π (f ) ∈ A : f ∈ Rτ . Using (2.38) it is easy to see that Mπτ is a localizing class in A. Given a of T we can choose a family {T π (fτ )}τ ∈T (fτ ∈ Rτ ), due to the compactness

finite subfamily {T π (fτj )}nj=1 such that g := j fτj ≥ 1 and Theorem 2.42(b) shows that T π (g) is invertible in A. Hence, {Mπτ }τ ∈T is a covering system of localizing classes in A. Also from (2.38) we deduce that T π (a) commutes with every T π (f ) in the union of all Mπτ . We finally have, again making use of (2.38), " " " " inf " T π (a) − T π (aτ ) T π (f )"A = inf "T π (a − aτ )f "A f ∈Rτ f ∈Rτ " " ≤ inf "T (a − aτ )f "L(H p ) ≤ cp inf (a − aτ )f ∞ = 0 f ∈Rτ

and analogously

f ∈Rτ

" " inf "T π (f ) T π (a) − T π (aτ ) "A = 0.

f ∈Rτ

In other words, T π (a) and T π (aτ ) are Mπτ -equivalent from the left and from the right at each τ ∈ T. Since T π (aτ ) is invertible in A, it is of course Mπτ invertible from the left and from the right. Thus, we have collected together all the things allowing us to apply Theorem 1.32. The conclusion is that T π (a) is invertible in A and this yields the assertion.


87

2.69. Theorem. Let 1 ≤ p < ∞ and a ∈ M p . Suppose for each τ ∈ T there exists an aτ ∈ M p such that distτ (a, aτ ) = 0 and T (aτ ) ∈ Φ(p ). Then T (a) ∈ Φ(p ). Proof. Since M 1 = M 1 = W , the case p = 1 is covered by Theorem 2.47. The case p = 2 is contained in the preceding theorem. Thus let 1 < p < 2; for 2 < p < ∞ the assertion can be proved analogously or can be obtained immediately from the case 1 < p < 2 by taking adjoints. Put A = L(p )/C∞ (p ) and for a ∈ M p let T π (a) := T (a) + C∞ (p ). It is clear that (2.38) holds for every b ∈ M p and every f ∈ Cp . Now, for τ = eiθ0 ∈ T, let Rτ := f ∈ C ∞ : 0 ≤ f ≤ 1, there is an ε > 0 (depending on f ) such that f (eiθ ) = 1 for |θ − θ0 | < ε, f (eiθ ) = 0 for |θ − θ0 | > 2ε, f is monotonically increasing for θ0 − 2ε < θ < θ0 − ε, f is monotonically decreasing for θ0 + ε < θ < θ0 + 2ε , Mπτ := T π (f ) ∈ A : f ∈ Rτ . As in the preceding proof it is readily seen that {Mπτ }τ ∈T is a covering system of localizing classes in A (note that C ∞ ∈ M p ) and that T π (a)T π (f ) = T π (f )T π (a) for every f ∈ Rτ . Since 1 < p < 2 and a, aτ ∈ M p , there is an r = rτ ∈ (1, p) such that a, aτ ∈ M r . Hence, " " " " inf " T π (a) − T π (aτ ) T π (f )" = inf "T π (a − aτ )f " A

f ∈Rτ

≤ inf (a − aτ )f M p ≤ inf (a − aτ )f 1−γ ∞ (a − f ∈Rτ

f ∈Rτ

A

f ∈Rτ

aτ )f γM r ,

where γ = r|p−2|/(p|r−2|) and the last estimate results from 2.5(e). To prove that T π (a) is Mπτ -equivalent from the left to T π (aτ ) it therefore remains to show that (a − aτ )f M r is bounded by a constant, K, as f varies over Rτ . But from 2.5(f) we obtain (a − aτ )f M r ≤ a − aτ M r f M r ≤ aM r + aτ M r sp f ∞ + V1 (f ) ≤ aM r + aτ M r sp · 3 =: K, since f ∞ = 1 and V1 (f ) = 2 for every f ∈ Rτ . Theorem 1.32 now completes the proof. 2.70. Remark. Let X be a Banach space and A ∈ L(X). Then A is said to be left-Fredholm (resp right-Fredholm) if A + C∞ (X) is left-invertible (resp. right-invertible) in L(X)/C∞ (X). A look at Theorem 1.32 shows that in the preceding two theorems the requirement that T (aτ ) be Fredholm for each τ ∈ T can be replaced by the requirement that T (aτ ) be left-Fredholm (resp. right-Fredholm) in order to deduce that T (a) be left-Fredholm (resp. right-Fredholm).

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The relation between left- and right-Fredholm operators and Φ± -operators is clarified by the following result of Yood [584]. (a) A ∈ L(X) is left-Fredholm if and only if A ∈ Φ+ (X) and if Im A is a complemented subspace of X. (b) A ∈ L(X) is right-Fredholm if and only if A ∈ Φ− (X) and if Ker A is a complemented subspace of X. Thus, in general, a Φ+ (Φ− )-operator need not be left-Fredholm (rightFredholm). However, for a bounded Hilbert space operator to be a Φ+ (Φ− )operator is equivalent to being left-Fredholm (right-Fredholm). 2.71. Definition. A function a ∈ L∞ is called sectorial on a subarc U of T if there is an ε > 0 and a c ∈ C of modulus 1 such that Re (ca) ≥ ε a.e. on U . A function a ∈ L∞ is said to be locally arcwise sectorial if for each τ ∈ T there is a subarc Uτ ∈ Uτ such that a is sectorial on Uτ . Since T is compact, a function a ∈ L∞ is locally arcwise sectorial if and only if T can be covered by a finite number of open subarcs Ui such that a is sectorial on each Ui . 2.72. Theorem. If a ∈ L∞ is locally arcwise sectorial then T (a) is Fredholm on H 2 . Proof. Immediate from Theorems 2.68 and 2.17. Remark. The index of a Toeplitz operator generated by a locally arcwise sectorial symbol is, loosely speaking, minus the winding number of the “sectorial cloud” associated with the symbol. We do not make precise what this means, but shall later provide another way of computing the index, namely, via the harmonic extension of the symbol. 2.73. The algebra P C. Let P C0 denote the collection of all piecewise continuous functions on T which have at most finitely many jumps. The closure of P C0 in L∞ is denoted by P C. A function a ∈ P C possesses finite limits a(t ± 0) everywhere on T and there are at most countably many t ∈ T such that a(t − 0) = a(t + 0). Note that P C is a C ∗ -algebra of L∞ . Given a ∈ P C0 define a function a2 : T × [0, 1] → C by the formula a2 (t, µ) = (1 − µ)a(t − 0) + µa(t + 0)

(t ∈ T,

µ ∈ [0, 1]).

(2.39)

The range of a2 is a continuous closed curve with a natural orientation; it is obtained from the (essential) range of a by filling in the straight line segment [a(t − 0), a(t + 0)] for each t ∈ T at which a has a jump. If this curve does not pass through the origin, we let ind a2 denote its winding number with respect to the origin. A more precise definition of ind a2 is as follows. With each finite subset S of T we associate a function ωS : T → T × [0, 1]. To construct this function let S = {eiθ1 , . . . , eiθR }, 0 ≤ θ1 < . . . < θR < 2π, put θR+1 := θR + 2π, and for j = 1, . . . , R let


89

θj + θj+1 θj + ϕj ϕj + θj+1 , ϕj = , ϕj = . 2 2 2 Then define, for j = 1, . . . , R and 0 ≤ λ ≤ 1, 1 i(θj +λ(ϕj −θj )) iθj 1 )= e , + λ , ωS (e 2 2 ϕj =

ωS (ei(ϕj +λ(ϕj −ϕj )) ) = (ei(θj +λ(θj+1 −θj )) , 1 − λ), i(ϕ +λ(θj+1 −ϕ )) iθj+1 1 j j ωS (e )= e , λ . 2 Given a ∈ P C0 denote by S(a) the finite subset of T formed by the points at which a has a jump. It is easily seen that a2 (t, µ) = 0 for all (t, µ) ∈ T×[0, 1] if and only if the continuous function a2 ◦ωS(a) : T → C does not vanish on T. In that case ind a2 is defined as ind (a2 ◦ ωS(a) ), where the latter “ind ” refers to the index as it was defined in 2.41. For a ∈ P C define a2 : T × [0, 1] → C again by (2.39). It is not difficult to see that the origin belongs to the range of a2 if and only if there is a sequence of functions an ∈ P C0 such that a−an ∞ → 0 and dist(0, range(an )2 ) → 0 as n → ∞. If a2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1], we choose any sequence of functions an ∈ P C0 with a − an ∞ → 0 as n → ∞ and define ind a2 as lim ind (an )2 . It can be easily seen that this limit always n→∞ exists and that it does not depend on the particular choice of the sequence {an }. 2.74. Theorem. Let a ∈ P C. Then T (a) ∈ Φ(H 2 ) ⇐⇒ a2 (t, µ) = 0

∀ (t, µ) ∈ T × [0, 1].

If T (a) is Fredholm, then Ind T (a) = −ind a2 . Proof. If a2 (t, µ) = 0 for all (t, µ) ∈ T×[0, 1], then a is locally arcwise sectorial and therefore T (a) ∈ Φ(H 2 ) due to Theorem 2.72. Our next objective is to prove the index formula. Thus, let a ∈ P C and a2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1]. If b ∈ P C0 is sufficiently close to a in the L∞ norm, then Ind T (a) = Ind T (b), b2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1] and ind a2 = ind b2 (the latter fact per definitionem!). So it remains to show that Ind T (b) = −ind b2 . Let t1 , . . . , tn denote the points on T at which b has jumps. Choose sufficiently small neighborhoods U1 , . . . , Un ⊂ T of the points t1 , . . . , tn and put U = U1 ∪ . . . ∪ Un . Then define c ∈ C as follows: let c = b on T \ U and on Ui let c be any continuous function such that c(Ui ) = b(Ui ) ∪ [b(ti − 0), b(ti + 0)]. The function d = b/c equals 1 on T \ U and it is easy to see that d is sectorial on U if only the neighborhoods U1 , . . . , Un have been chosen sufficiently small. Thus, by (2.18), ) T (b) = T (cd) = T (c)T (d) + H(c)H(d) with H(c) compact (Theorem 2.42(a)) and T (d) invertible (Theorem 2.17). It follows that Ind T (b) = Ind T (c), and since ind c = ind b2 , Theorem 2.42(b) completes the proof of the index formula.

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We now prove the implication “=⇒”. Let T (a) ∈ Φ(H 2 ) but assume there is a (t0 , µ0 ) ∈ T × [0, 1] such that a2 (t0 , µ0 ) = 0. Then T (b) ∈ Φ(H 2 ) whenever a − b∞ is sufficiently small and among these b’s there is a b ∈ P C0 such that b2 (t, µ) = 0. If b − c∞ and b − d∞ are small enough, then T (c) and T (d) are Fredholm and Ind T (b) = Ind T (c) = Ind T (d),

(2.40)

but it is easily seen that one can find such functions c and d in P C0 which satisfy c2 (t, µ) = 0, d2 (t, µ) = 0 ∀ (t, µ) ∈ T × [0, 1] and ind c2 − ind d2 = 1. From the index formula proved above we get Ind T (d) = Ind T (c) + 1 which contradicts (2.40). Remark. The perturbation argument used in this proof also applies to show that a2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1] if T (a) is a Φ+ - or Φ− -operator on the space H 2 . The following theorem is the “essentialization” of Theorem 2.20 and forms the basis for another local approach. 2.75. Theorem (Douglas/Sarason). Let ϕ ∈ L∞ be a unimodular function. Then (a) T (ϕ) ∈ Φ+ (H 2 ) ⇐⇒ distL∞ (ϕ, C + H ∞ ) < 1; (b) T (ϕ) ∈ Φ− (H 2 ) ⇐⇒ distL∞ (ϕ, C + H ∞ ) < 1; (c) T (ϕ) ∈ Φ(H 2 ) ⇐⇒ distL∞ (ϕ, G(C + H ∞ )) < 1. Proof. We abbreviate distL∞ to dist. (a) Let T (ϕ) ∈ Φ+ (H 2 ). If dim Ker T (ϕ) = 0, then T (ϕ) is left-invertible and so Theorem 2.20(a) gives dist(ϕ, H ∞ ) < 1. If dim Ker T (ϕ) = n > 0, then dim Ker T ∗ (ϕ) = 0 by Theorem 2.38. Thus, T (ϕ) is Fredholm of index n > 0. It follows that T (ϕχn ) is invertible, whence dist(ϕχn , H ∞ ) < 1 by Theorem 2.20(a), and thus dist(ϕ, χ−n H ∞ ) < 1. The proof of the implication “=⇒” is complete. Now suppose dist(ϕ, C + H ∞ ) < 1. Then dist(ϕχn , H ∞ ) < 1 for some n ≥ 0, and Theorem 2.20(a) yields the left-invertibility of T (ϕ)T (χn ). Since T (χn ) is Fredholm, it results that T (ϕ) is left-Fredholm. This proves the implication “⇐=”. (b) Take adjoints and apply (a). (c) If T (ϕ) is Fredholm, then T (χn ϕ) is invertible for some n ∈ Z (Corollary 2.40), hence dist(χn ϕ, GL∞ ) < 1 (Theorem 2.20(c)), and thus dist(ϕ, G(C + H ∞ )) < 1. On the other hand, if dist(ϕ, G(C +H ∞ )) < 1, then there are an n ≥ 0 and an h ∈ H ∞ ∩ G(C + H ∞ ) such that ϕ − χ−n h∞ < 1. Since T (h) ∈ Φ(H 2 )


91

(Corollary 2.55), we have h = bg, where b is a finite Blaschke product and g is in GH ∞ (Theorem 2.64). Consequently, dist(b−1 ϕχn , GH ∞ ) < 1, and so Theorem 2.20(c) implies that T (b−1 )T (ϕ)T (χn ) is invertible. Because T (b−1 ) and T (χn ) are Fredholm, it follows that T (ϕ) must also be Fredholm. 2.76. Corollary. Let a ∈ GL∞ . Then T (a) is in Φ+ (H 2 ) (Φ− (H 2 ) resp. Φ(H 2 )) if and only if a = bs, where b ∈ C + H ∞ (b ∈ C + H ∞ resp. b ∈ G(C + H ∞ )) and s ∈ GL∞ is sectorial. Proof. Combine Theorem 2.75, Proposition 2.32, and Lemma 2.21. Theorem 2.75 and its Corollary 2.76 do not answer the question on the Fredholmness of Toeplitz operators in terms of the geometric data of the symbol. The purpose of what follows in the next sections is to combine Theorem 2.75 with Glicksberg’s theorem 1.22 in order to make the things little bit more geometrical. Before doing this we need a few pieces of information about the maximal ideal space of L∞ and its decompositions. 2.77. M (L∞ ). The maximal ideal space M (L∞ ) of the Banach algebra L∞ will be denoted by X. Since L∞ is a C ∗ -algebra with respect to the involution a → a, where a(t) = a(t) (t ∈ T), L∞ is star-isometrically isomorphic to C(X). The Gelfand transform of a function a ∈ L∞ will also be denoted by a. Thus, if a ∈ L∞ and x ∈ X, then a(x) = x(a). The topological space X is totally disconnected in the following sense: the closure of every open set is again open. 2.78. L∞ fibers over M (C). The maximal ideal space of C is T: the general form of a functional in M (C) is given by vτ : C → C, where τ ∈ T. Let

f → f (τ ),

Xτ := Mτ (L∞ ) := x ∈ X : x|C = vτ .

It is easy to see that the fibers Xτ are homeomorphic to each other. Because Xτ , it follows that Xτ = ∅. This is also a consequence of 1.20(b) X = τ ∈T

(recall also 1.27(b)). Given a ∈ L∞ and an open subarc U of T denote by R(a|U ) the spectrum of the restriction of a to U regarded as an element of L∞ (U ). Equivalently, R(a|U ) is the set of all µ ∈ C such that {t ∈ U : |a(t) − µ| < ε} has positive (Lebesgue) measure for each ε > 0. Finally, recall that according to 2.67 distτ (a, b) = inf a|U − b|U L∞ (U ) U ∈Uτ

while in accordance with 1.21 distXτ (a, b) = max |a(x) − b(x)|. x∈Xτ

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2 Basic Theory

2.79. Proposition. (a) If a ∈ L∞ and τ ∈ T, then $ R(a|U ). a(Xτ ) = U ∈Uτ ∞

(b) If a, b ∈ L

and τ ∈ T, then distτ (a, b) = distXτ (a, b). In particular, distτ (a, b) = 0 ⇐⇒ a|Xτ = b|Xτ .

Proof. (a) A little thought shows that µ ∈ /

U ∈Uτ

R(a|U ) if and only if

∃ b, c ∈ L∞ : (a − µ)b + (χ1 − τ )c = 1.

(2.41)

If (2.41) holds, then (a(x) − µ)b(x) = 1 for all x ∈ Xτ , whence µ ∈ / a(Xτ ). On the other hand, if µ ∈ / a(Xτ ) then there is no x ∈ X such that a(x) = µ and χ1 (x) = τ . Thus, the closed ideal (a − µ)b + (χ1 − τ )c : b, c ∈ L∞ is not contained in any maximal ideal of L∞ , which gives (2.41). (b) Since distτ (a, b) = distτ (a − b, 0), it suffices to prove that max |f (x)| = distτ (f, 0)

x∈Xτ

for every f ∈ L∞ . By virtue of part (a), c(Xτ ) = {1} for every c ∈ Rτ (see 2.67). So max |f (x)| = max |f (x)c(x)| x∈Xτ

x∈Xτ

for every c ∈ Rτ , whence, by (2.37), max |f (x)| ≤ inf max |f (x)c(x)| = distτ (f, 0).

x∈Xτ

c∈Rτ x∈X

To establish the reverse inequality we need the following well known fact: if K1 ⊃ K2 ⊃ K3 ⊃ ... are compact nonempty subsets of a Hausdorff space, if ∞

n=1

Kn ⊂ Ω, and if Ω is open, then there is an n0 such that Kn0 ⊂ Ω.

Now put M = max |f (x)|. Given any ε > 0 we have, due to part (a), x∈Xτ ∞ $

R(f |Un ) ⊂ z ∈ C : |z| < M + ε ,

n=1

where Un = {t ∈ T : |t − τ | < 1/n}, and since each set R(f |Un ) is compact and nonempty (as the spectrum of f |Un ∈ L∞ (Un )), it follows that there is an U0 ∈ Uτ such that R(f |U0 ) ⊂ {z ∈ C : |z| < M + ε}. Now it is clear that there exists a c0 ∈ Rτ such that f c0 ∞ < M + ε, and therefore distτ (f, 0) = inf max |f (x)c0 (x)| = max |f (x)c0 (x)| = f c0 ∞ < M + ε. c∈Rτ x∈X

x∈X

Since ε > 0 can be chosen arbitrarily, we get distτ (f, 0) ≤ M .


93

Remark. Thus, a function a ∈ L∞ is continuous at a point τ ∈ T if and only if a(Xτ ) is a singleton. If a has a jump discontinuity at τ , then a(Xτ ) is a doubleton, but if a(Xτ ) is known to be a doubleton, then all one can say is that a has two essential cluster points at τ , which does, in general, not imply that a has a jump at τ . 2.80. QC. The largest C ∗ -subalgebra of C + H ∞ is denoted by QC and is referred to as the algebra of quasicontinuous functions. Thus QC = (C + H ∞ ) ∩ (C + H ∞ ). Although H ∞ ∩ H ∞ is the set of constant functions, QC is strictly larger than C. Indeed, let 1 Ω = z = x + iy ∈ C : 0 < x < 1, −2 < y < sin x and let ω map D conformally onto Ω. Then ω ∈ H ∞ and Re ω ∈ C, whence Im ω = iRe ω − iω ∈ C + H ∞ . Since Im ω is a real-valued function, Im ω ∈ C + H ∞ . But Im ω is obviously discontinuous and therefore Im ω ∈ QC \ C. Since QC is a C ∗ -algebra, we have, for c ∈ QC, c ∈ GQC ⇐⇒ c ∈ GL∞ . 2.81. L∞ fibers over M (QC). If ξ ∈ M (QC), then by virtue of 1.20(b) (or 1.27(b)) the fiber Xξ = Mξ (L∞ ) is not empty. To every ξ ∈ M (QC) there corresponds a τ ∈ M (C) = T such that ξ ∈ Mτ (QC), and it is clear that Mξ (L∞ ) ⊂ Mτ (L∞ ). We have / Mτ (L∞ ) = Mξ (L∞ ). ξ∈Mτ (QC)

Since QC = C, the partition

/

M (L∞ ) =

Mξ (L∞ )

ξ∈M (QC)

is a proper refinement of the partition / Mτ (L∞ ). M (L∞ ) = τ ∈T

Because the restriction of a function in C to a fiber Xτ (τ ∈ T) is constant, we have C + H ∞ |Xτ = H ∞ |Xτ

and

C + H ∞ |Xξ = H ∞ |Xξ .

(2.42)

We know from 1.27(c), (d) (in the setting Y = X, A = C +H ∞ , B = QC) that each maximal antisymmetric set for C + H ∞ is contained in some fiber Xξ , where ξ ∈ M (QC). Consequently, Corollary 1.23 implies that, for a ∈ L∞ ,

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2 Basic Theory

a ∈ C + H ∞ ⇐⇒ a|Xξ ∈ H ∞ |Xξ

∀ ξ ∈ M (QC).

This in turn gives that, for a ∈ L∞ , a ∈ QC ⇐⇒ a|Xξ = const ∀ ξ ∈ M (QC).

(2.43)

Note that the implications “=⇒” are trivial. Let B be a C ∗ -subalgebra of L∞ containing the constant functions. Then for each β0 ∈ M (B) the fiber Xβ0 = Mβ0 (L∞ ) is a peak set for B (recall 1.28). Indeed, because B is isometrically isomorphic to C(M (B)), there is an f ∈ B with f (β0 ) = 1 and 0 < f (β) < 1 for β = β0 , whence f |Xβ0 = 1 and 0 < f < 1 on X \ Xβ0 . In particular, for τ ∈ T, Xτ is a peak set for C and therefore for C + H ∞ , and for ξ ∈ M (QC), Xξ is a peak set for QC and thus also for C + H ∞ . So we deduce from 1.28 that C + H ∞ |Xτ and C + H ∞ |Xξ are closed subalgebras of L∞ |Xτ and L∞ |Xξ , respectively, for every τ ∈ T and ξ ∈ M (QC). Taking into account (2.42) we arrive at the conclusion that the algebras H ∞ |Xτ and H ∞ |Xξ are closed. It also follows that the algebras QC|Xτ and QC|Xξ are closed. Clearly, QC|Xξ is the complex field C, while 1.27(b) shows that M (QC|Xτ ) can be identified with Mτ (QC). We finally mention that both M (H ∞ |Xτ ) (which can be identified with the fiber Mτ (H ∞ ) of H ∞ over τ ∈ T = M (CA ) \ D) and M (H ∞ |Xξ ) are connected (τ ∈ T, ξ ∈ M (QC)). The connectedness of the first space is shown in Hoffman’s book [284] and that the second space is connected was proved by Gorkin [239, Corollary 2.9]. 2.82. Definition. Let a ∈ L∞ and let F be a closed subset of X = M (L∞ ). The Toeplitz operator T (a) will be said to be F -restricted invertible (left resp. right-invertible) if there is a b ∈ L∞ such that a|F = b|F and T (b) is invertible (left resp. right-invertible) on H 2 . If F is contained in some fiber Xτ (τ ∈ T), then T (a) is F -restricted invertible (left resp. right-invertible) if and only if there is a b ∈ L∞ such that a|F = b|F and T (b) is Fredholm (left resp. right-Fredholm) on H 2 (recall Remark 2.70). This follows from Corollary 2.40 together with the fact that continuous functions restricted to Xτ are constants. Proposition 2.79 shows that Theorem 2.68 for p = 2 may also be stated as follows: if T (a) is Xτ -restricted invertible for each τ ∈ T, then T (a) is Fredholm on H 2 . In this form the theorem was established by Douglas and Sarason [164] using a method which actually applies to prove the following much “more local” result. 2.83. Theorem (Axler). Let a ∈ L∞ and let B ⊂ C + H ∞ be a closed subalgebra of L∞ containing the constants. If T (a) is S-restricted invertible (left resp. right-invertible) for each maximal antisymmetric set S for B, then T (a) is Fredholm (left resp. right-Fredholm) on H 2 . Proof. Let T (a) be S-restricted left-invertible for some S ⊂ X. Then there is a b ∈ L∞ such that a|S = b|S and T (b) is left-invertible. By Theorem 2.30(a),


95

b ∈ GL∞ and hence a(x) = 0 for all x ∈ S. Proposition 2.32 implies that T (b/|b|) is left-invertible and so Theorem 2.75(a) gives distX (b/|b|, C + H ∞ ) = distL∞ (b/|b|, C + H ∞ ) < 1. Because b/|b| equals a/|a| on S, we have distS (a/|a|, C + H ∞ ) < 1.

(2.44)

If (2.44) holds for each maximal antisymmetric set S for B, then it also holds for each maximal antisymmetric set S for C + H ∞ , since the latter ones are contained in the former ones (see 1.27(d)). Thus, Theorem 1.22 gives that distL∞ (a/|a|, C + H ∞ ) = distX (a/|a|, C + H ∞ ) < 1, from Theorem 2.75(a) we deduce that T (a/|a) ∈ Φ+ (H 2 ) and once more applying Proposition 2.32 we see that T (a) ∈ Φ+ (H 2 ). The proof for the right-Fredholmness is analogous. Finally, if T (a) is Srestricted invertible for each maximal antisymmetric set S for B, then, by what has already been proved, T (a) is in both Φ+ (H 2 ) and Φ− (H 2 ), and hence in Φ(H 2 ). 2.84. Definitions. Let F be a closed subset of X = M (L∞ ). A function a ∈ L∞ is said to be sectorial on F if there are an ε > 0 and a c ∈ C of modulus 1 such that Re (ca(x)) ≥ ε for all x ∈ F . If a is sectorial on F , then a|F is obviously invertible in L∞ |F , and it is easy to see that a is sectorial on F if and only if so is a/|a|. Moreover, for a ∈ L∞ to be sectorial on F it is necessary and sufficient that a(x) = 0 for all x ∈ F and distF (a/|a|, C) < 1. Now let B be a closed subalgebra of C + H ∞ containing the constants. A function a ∈ L∞ will be called locally sectorial over B if it is sectorial on each maximal antisymmetric set for B. The most important special cases are B = C + H ∞ , B = QC, B = C, and B = C. So, by virtue of 1.27(c), a ∈ L∞ is locally sectorial over QC (resp. C) if and only if it is sectorial on each fiber Xξ , ξ ∈ M (QC) (resp. Xτ , τ ∈ T). The functions that are sectorial in the sense of Definition 2.16 are just the functions which are sectorial on X = M (L∞ ) or, equivalently, locally sectorial over C. Finally, from 1.27(d) (with Y = X) we deduce that if B ⊂ A, then a locally sectorial over B =⇒ a locally sectorial over A. 2.85. Theorem. If a ∈ L∞ is locally sectorial over a closed subalgebra B of C + H ∞ containing the constants then T (a) is Fredholm on H 2 . Proof. The hypothesis implies that a ∈ GL∞ and that a is locally sectorial over C + H ∞ . Hence

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distS (a/|a|, C + H ∞ ) ≤ distS (a/|a|, C) < 1 for each maximal antisymmetric set S for C + H ∞ . Since the maximal antisymmetric sets for C + H ∞ are the same as those for C + H ∞ , Theorem 1.22 and Theorem 2.75(a), (b) can be combined to obtain that T (a/|a|) ∈ Φ+ (H 2 ) ∩ Φ− (H 2 ) = Φ(H 2 ) and Proposition 2.32 completes the proof.

The following proposition provides an idea of what the different notions of local sectoriality involve. 2.86. Proposition. (a) If a ∈ L∞ is locally sectorial over C + H ∞ then a can be written as f s with f ∈ G(C + H ∞ ) and s ∈ GL∞ sectorial (on T). (b) Let B be a C ∗ -subalgebra of L∞ between C and QC. Then a is locally sectorial over B if and only if a can be represented as a = bs with b ∈ GB and s ∈ GL∞ sectorial (on T). (c) For a ∈ L∞ the following are equivalent: (i) a is locally sectorial over C; (ii) a = cs with c ∈ GC and s is sectorial; (iii) a is locally arcwise sectorial. Proof. (a) Theorem 1.22 gives distL∞ (a/|a|, C + H ∞ ) < 1,

distL∞ (a/|a|, C + H ∞ ) < 1,

so Theorem 2.75 shows that distL∞ (a/|a|, G(C + H ∞ )) < 1 and Lemma 2.21 ends the proof. (b) One half of the assertion can be proved as in (a). On the other hand, if a = bs with b ∈ GB and s sectorial, then a|S = (b|S)(s|S) for each maximal antisymmetric set S for B, and since these sets are just the fibers Xβ , β ∈ M (B) (1.27(c)), b|S is a nonzero constant and hence a is sectorial on S. (c) The implication (i) =⇒ (ii) follows from part (b) and the implication (iii) =⇒ (i) results from Proposition 2.79(a). Finally, if (ii) holds, then Re (γs(t)) ≥ ε for some ε > 0, some γ ∈ C, and for almost all t ∈ T. Hence, if τ ∈ T, Re γ/c(τ ) c(τ )s(t) ≥ ε > 0 for almost all t ∈ T and since c is continuous, Re

γ/c(τ ) c(t)s(t) ≥ ε > 0

for almost all t in some neighborhood of τ , which gives (iii).


97

Remark. Thus, Theorem 2.85 can also be proved as follows: if a ∈ L∞ is locally sectorial over C+H ∞ , then a = f s with f ∈ G(C+H ∞ ) and s sectorial, so T (a) = T (s)T (f )+ compact operator ((2.18) and Theorem 2.54), and since T (s) is invertible (Theorem 2.17) and T (f ) is Fredholm (Corollary 2.55), we conclude that T (a) is Fredholm. Our next concern is the index computation (and thus the solution of the invertibility problem) for Toeplitz operators whose symbol is locally sectorial over QC (or over any closed subalgebra of QC containing the constants). The key observations are Propositions 2.86(b) and the following generalization of Lemma 2.61. 2.87. Lemma (Sarason). If b ∈ QC and a ∈ L∞ , then (ba)r − br ar ∞ → 0 as

r → 1 − 0.

Proof. Let kr denote the Poisson kernel. Then + + ( 2π + 1 + [b(eit ) − br (eiθ )]a(eit )kr (θ − t) dt++ |(ba)r (eiθ ) − br (eiθ )ar (eiθ )| = ++ 2π 0 ( a∞ 2π ≤ |b(eit ) − br (eiθ )|kr (θ − t) dt 2π 0 1/2 ( 2π a∞ it iθ 2 ≤ |b(e ) − br (e )| kr (θ − t) dt 2π 0 1/2 a∞ (bb)r (eiθ ) − br (eiθ )br (eiθ ) = . 2π But if b ∈ QC, then b ∈ C + H ∞ and b ∈ C + H ∞ , whence, by virtue of Theorem 2.62(a), (bb)r − br br ∞ → 0 as r → 1 − 0. 2.88. Theorem. If a ∈ L∞ is locally sectorial over QC, then T (a) ∈ Φ(H 2 ), the harmonic extension a is bounded away from zero in some annulus near T, and Ind T (a) = −ind {ar }. Proof. Due to Proposition 2.86(b) we have a = bs, where b ∈ GQC and s ∈ GL∞ is sectorial (on T). So T (a) = T (b)T (s)+ compact operator ((2.18) and Theorem 2.54), and since T (b−1 ) is a regularizer of T (b) ((2.18) and Corollary 2.55) and T (s) is invertible (Theorem 2.17), it follows that T (a) ∈ Φ(H 2 ). Of course, the same conclusion might be also drawn from Theorem 2.85. Lemma 2.87 shows that ar − br sr ∞ → 0 as r → 1 − 0. If Re s ≥ ε > 0 a.e. on T, then Re s ≥ ε > 0 in D, because the Poisson kernel is positive. Hence, if s is sectorial then s is bounded away from zero in D and ind {sr } = 0.

(2.45)

If b ∈ GQC, then b is bounded away from zero in some annulus near T by Theorem 2.62. Thus, under our hypothesis, a is bounded away from zero in some annulus near T.

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The index formula can now be verified as follows: Ind T (a) = Ind T (b) + Ind T (s) = Ind T (b) = −ind {br } (Theorem 2.65) = −ind {br } − ind {sr } (by (2.45)) = −ind {(bs)r } (2.41(a), Lemma 2.87, 2.41(b)).

We finally show that for a relatively large class of symbols local sectoriality is not only sufficient but also necessary for the Fredholmness of the corresponding Toeplitz operator. 2.89. Definition. Let B be a C ∗ -subalgebra of QC containing the constants. We denote by P2 B the collection of all functions a ∈ L∞ which take at most two values on each fiber Xξ , ξ ∈ M (B). For instance, P2 C contains P C, and if E any measurable subset of T, then the functions aχE + b (χE being the characteristic function of E, a and b being in C) belong to P2 C. Further, P2 QC contains all functions of the form a=

n

pi qi ,

pi ∈ P2 C,

qi ∈ QC.

i=1

We shall see later that P QC, the closed subalgebra of L∞ generated by P C and QC, is also a subset of P2 QC (see Remark 1 of 3.36). 2.90. Lemma. Let B = C or B = QC and let ξ ∈ M (B). If ϕ ∈ L∞ is unimodular and ϕ(Xξ ) is a pair of antipodal points, then distXξ (ϕ, H ∞ ) = 1. Proof. Without loss of generality suppose ϕ(Xξ ) is the doubleton {−1, 1}. Assume there is an h ∈ H ∞ such that max |ϕ(x) − h(x)| ≤ 1 − δ < 1. Put E± := {x ∈ Xξ : ϕ(x) = ±1}. So |1 − h(x)| ≤ 1 − δ

∀ x ∈ E+ ,

x∈Xξ

|1 + h(x)| ≤ 1 − δ

∀ x ∈ E− .

(2.46)

Let B denote the restriction algebra L∞ |Xξ , which is closed by 1.28. From (2.46) we see that the spectrum of h|Xξ in B is contained in the union of two disks with center at +1 and −1 and radius 1 − δ, and, moreover, that each of these two disks contains a point of that spectrum, i.e., that there are z1 , z2 ∈ C such that Re z1 < 0, Re z2 > 0, z1 ∈ spB (h|Xξ ), z2 ∈ spB (h|Xξ ). Now put A := H ∞ |Xξ . From 2.81 we know that A is closed and M (A) is connected (Hoffman for B = C and Gorkin for B = QC). Consequently, spA (h|Xξ ) = h(M (A)) is a connected subset of C. By virtue of 1.16(b), spA (h|Xξ ) is the union of spB (h|Xξ ) and a (possibly empty) collection of bounded connected

2.9 Matrix Symbols

99

components of spB (h|Xξ ). However, the set {z ∈ C : |Re z| < δ/2} is contained in the unbounded complement of spB (h|Xξ ), hence {z ∈ C : |Re z| < δ/2} ∩ spA (h|Xξ ) = ∅. But this is a contradiction, since together with z1 and z2 some points of the stripe {|Re z| < δ/2} must belong to the (connected!) set spA (h|Xξ ). 2.91. Theorem. Let B = C or B = QC and let a ∈ P2 B. Then T (a) ∈ Φ(H 2 ) ⇐⇒ a is locally sectorial over B. Proof. The implication “⇐=” is immediate from Theorem 2.85 (or can be established as in the proof of Theorem 2.88). So we are left with the reverse implication. Let T (a) ∈ Φ(H 2 ). Then T (a/|a|) ∈ Φ(H 2 ) by Proposition 2.32, and hence distL∞ (a/|a|, C + H ∞ ) < 1 by Theorem 2.75(a). It follows that distXξ (a/|a|, H ∞ ) < 1 for each ξ ∈ M (B) (recall (2.42)). The preceding lemma shows that the singleton or doubleton (a/|a|)(Xξ ) cannot be a doubleton consisting of two antipodal points and this is equivalent to saying that a/|a| (and thus a itself) is sectorial on Xξ .

2.9 Matrix Symbols We conclude this chapter by stating some facts on Toeplitz operators with matrix symbols. We here confine ourselves to settling a few problems the solution of which merely requires minor modifications of the methods developed above for scalar Toeplitz operators. The more delicate questions on block Toeplitz operators will be deferred to the forthcoming chapters. ∞ 2.92. Definitions. Given a matrix function a = (ajk )N j,k=1 ∈ LN ×N the mulp tiplication operator M (a) is defined on LN (1 < p < ∞) by

M (a) :

LpN

→

LpN ,

(fk )N k=1

→

N

N M (akj )fj

j=1

k=1

p (recall the notations introduced in 1.13). For a ∈ MN ×N the operator M (a) p p is defined on N (Z) := ( (Z))N analogously. Here LpN and pN (Z) can be regarded as being equipped with the norms

f LpN :=

N

fj Lp ,

f = (fj )N j=1 ,

j=1

ϕpN (Z) :=

N

ϕj p (Z) ,

j=1

or with any norms equivalent to those ones.

ϕ = (ϕj )N j=1 ,

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2 Basic Theory

Similarly, if a ∈ L∞ N ×N , the Toeplitz operator T (a) and the Hankel operap (1 < p < ∞) by tor H(a) are given on HN p p T (a) : HN → HN ,

(fk )N k=1 →

N

N T (akj )fj

p p H(a) : HN → HN ,

(fk )N k=1 →

N

, k=1

j=1

N H(akj )fj

, k=1

j=1

p p and for a ∈ MN ×N an analogous definition is made for T (a) and H(a) on N (1 ≤ p < ∞). p and pN can be viewed as subspaces of LpN and pN (Z), The spaces HN respectively. Thus, whenever a norm in the latter two spaces is specified it will always be clear what the norm in the first two spaces is. Both the Riesz projection P : Lp → H p and the canonical projection P : p (Z) → p extend in a natural way to LpN and pN (Z). We denote these projections again by P . Thus, P = diag (P, . . . , P ). If the norm on pN (Z) is given by 1/p N N p ϕj p (Z) or ϕj p (Z) , j=1

j=1

then obviously P L(pN (Z)) = 1. In the same fashion the projection Q and the flip operator J are defined on LpN and pN (Z). We then have T (a) = P M (a)P |Im P,

H(a) = P M (a)QJ|Im P,

etc. In particular, formulas (2.18)–(2.20) remain true for the matrix case without any changes. The matrix function a will always be referred to as the symbol of the corresponding operator. With every ϕ ∈ pN we may associate a CN -valued sequence ψ ∈ p (Z+ , CN ) as follows: p j ∞ p if ϕ = (ϕk )N k=1 ∈ N where ϕk = {ϕk }j=0 ∈ , j N p then ψ = {ψj }∞ j=0 ∈ (Z+ , CN ) where ψj = (ϕk )k=1 ∈ CN .

So the Toeplitz operator on pN can also be thought of as acting on p (Z+ , CN ) by the rule ! #∞ ∞ → a ψ , T (a) : {ψj }∞ j−k k j=0 k=0

j=0

where an (n ∈ Z) denotes the N × N matrix ((ajk )n )N j,k=1 formed by the p N Fourier coefficients of a = (ajk )j,k=1 ∈ MN ×N . Therefore Toeplitz operators on pN are sometimes also called block Toeplitz operators. It is clear that M (a)

2.9 Matrix Symbols

101

and H(a) can be viewed as acting on p (Z, CN ) and p (Z+ , CN ), respectively, in a similar manner. p We define norms on L∞ N ×N and MN ×N by := M (a)L(L2N ) , aL∞ N ×N

aMNp ×N := M (a)L(pN ) .

p Provided with these norms L∞ N ×N and MN ×N (1 ≤ p < ∞) are (noncommutative) Banach algebras with identity I. Clearly, a ∈ GL∞ N ×N resp. p p ∞ a ∈ GMN ×N if and only if there is a b ∈ LN ×N resp. b ∈ MN ×N such that ∞ and ab = ba = I. It is also obvious that a ∈ GL∞ N ×N ⇐⇒ det a ∈ GL p p a ∈ GMN ×N ⇐⇒ det a ∈ GM . By virtue of 1.29(a),

= ess sup a(t)L(CN ) = max a(x)L(CN ) . aL∞ N ×N t∈T

x∈X

2.93. Theorem. (a) If 1 < p < ∞ and a ∈ L∞ N ×N , then M (a) ∈ Φ± (LpN ) ⇐⇒ M (a) ∈ GL(LpN ) ⇐⇒ a ∈ GL∞ N ×N , p T (a) ∈ Φ(HN ) =⇒ a ∈ GL∞ . N ×N p (b) If 1 ≤ p < ∞ and a ∈ MN ×N , then p M (a) ∈ Φ(pN (Z)) ⇐⇒ M (a) ∈ GL(pN (Z)) ⇐⇒ a ∈ GMN ×N , p p ∞ T (a) ∈ Φ(N ) =⇒ a ∈ GMN ×N =⇒ a ∈ GLN ×N .

Proof. The assertions about the multiplication operators can be proved by the same argument as in the scalar case (2.28, 2.29). After defining the bilateral shift U on LpN or pN (Z) as U = M (χ1 I) = diag (U, . . . , U ), the proof of Theorem 2.30 also works in the matrix case. 1 Note that the implication T (a) ∈ Φ(1N ) =⇒ a ∈ GMN ×N can also be verified by invoking Theorem 1.14(c). Indeed, we have, by (2.18), T (f )T (g) − T (g)T (f ) = −H(f )H() g ) + H(g)H(f))

(2.47)

for all f, g ∈ M 1 , and since M 1 = W , Theorem 2.47(a) shows that the occurring Hankel operators are compact. Important remark. A decisive distinction between the scalar case (N = 1) and the matrix case (N > 1) is that a Fredholm block Toeplitz operator of index zero is not necessarily invertible (compare Corollary 2.40). For instance, if a = diag (χ1 , χ−1 ), then obviously T (a) ∈ Φ(H22 ) although both dim Ker T (a) end dim Coker T (a) equal 1. 2.94. Theorem. (a) Let a ∈ (C + H ∞ )N ×N and 1 < p < ∞. Then H(a) is p ) and in C∞ (HN p ) ⇐⇒ det a ∈ G(C + H ∞ ); T (a) ∈ Φ(HN

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2 Basic Theory

p if T (a) is Fredholm on HN , then T (a−1 ) is a regularizer of T (a) and

Ind T (a) = Ind T (det a) = −ind {(det a)r }. (b) Let a ∈ (Cp + Hp∞ )N ×N and 1 < p < ∞. Then H(a) ∈ C∞ (pN ) and T (a) ∈ Φ(pN ) ⇐⇒ det a ∈ G(Cp + Hp∞ ); if T (a) is Fredholm on pN , then T (a−1 ) is a regularizer of T (a) and Ind T (a) = Ind T (det a) = −ind {(det a)r }. Proof. The compactness of the Hankel operators can be shown as in the scalar case. This and identity (2.47) allow us the application of Theorem 1.14(c). p resp. pN if and only if T (det a) What results is that T (a) is Fredholm on HN is so on H p resp. p , which, by Corollary 2.55 and Theorem 2.60, is equivalent to the invertibility of det a in C + H ∞ resp. Cp + Hp∞ . That T (a−1 ) is a regularizer of T (a) follows from (2.18). In view of Theorems 2.65 and 2.66 it remains to show that Ind T (a) = Ind T (det a). To this end approximate a sufficiently close in the norm of L∞ N ×N p ∞) ∞) resp. MN by b ∈ (R+H resp. b ∈ (R+H . Then, by 1.12(d), N ×N N ×N p ×N Ind T (a) = Ind T (b),

Ind T (det a) = Ind T (det b).

Taking into account identity (2.47) and Theorem 2.58 we see that the entries of T (b) commute modulo finite-rank operators. So Theorem 1.15(b) gives Ind T (b) = Ind T (det b). Remark. Obvious modifications of the proof show that part (b) is true for p = 1 if only Cp + Hp∞ is replaced by W = M 1 . p

2.95. Theorem. Let a ∈ MN ×N and 1 ≤ p < ∞. Assume for each τ ∈ T p

there exists an aτ ∈ MN ×N such that aτ |Xτ = a|Xτ and T (aτ ) ∈ Φ(pN ). Then T (a) ∈ Φ(pN ). Proof. This follows from applying Theorem 1.32. Put A = L(pN )/C∞ (pN ) and, p for a ∈ MN ×N , denote the coset of A containing T (a) by T π (a). For τ ∈ T, define Rτ as in the proof of Theorem 2.69 and put Fτ = {diag (f, . . . , f ) : f ∈ Rτ },

Mπτ = {T π (ϕ) : ϕ ∈ Fτ }.

It is readily seen that {Mπτ }τ ∈T is a covering system of localizing classes in A and that (2.48) T π (a)T π (ϕ) = T π (aϕ) = T π (ϕa) = T π (ϕ)T π (a) for every ϕ ∈ Fτ . As in the proof of Theorem 2.69 one can show that T π (a) τ

and T π (aτ ) are Mπτ -equivalent from the left and from the right for each τ ∈ T. Theorem 1.32 then gives the assertion.


103

∗ 2.96. Theorem. Let a ∈ L∞ N ×N and 1 < p < ∞, and let B be a C -subalgebra of QC containing the constants. Suppose for each ξ ∈ M (B) there is an aξ in p p L∞ N ×N such that aξ |Xξ = a|Xξ and T (aξ ) ∈ Φ(HN ). Then T (a) ∈ Φ(HN ). p p Proof. We shall derive this from Theorem 1.32. Put A = L(HN )/C∞ (HN ) p ∞ π and, for a ∈ LN ×N , let T (a) := T (a) + C∞ (HN ). For ξ ∈ M (B), define Rξ as the collection of all f ∈ B such that 0 ≤ f ≤ 1 and f is identically 1 in some neighborhood Uξ ⊂ M (B) of ξ. Then let

Fξ = {diag (f, . . . , f ) : f ∈ Rξ }, Mπξ = {T π (ϕ) : ϕ ∈ Fξ }. It is clear that (2.48) holds for ϕ ∈ Fξ and it is easy to see that {Mπξ }ξ∈M (B) ξ

is a covering system of localizing classes in A. We now show that T π (a) and T π (aξ ) are Mπξ -equivalent from the left. Choose ε > 0, set b = a − aξ , and let U = η ∈ M (B) : b(x)L(CN ) < ε ∀ x ∈ Xη . Assume U is not an open subset of M (B). Then there is an η ∈ U and a net ηi in M (B) such that ηi → η and such that for each i, there exists xi ∈ Xηi with b(xi ) ≥ ε. Taking a subnet, we can suppose that there is an x ∈ X such that xi → x. Since the mapping (y ∈ X) → (y|B ∈ M (B)) is continuous, it follows that x ∈ Xη . But b(x) ≥ ε, which is impossible for x ∈ Xη and η ∈ U . This contradiction shows that U is open. It is clear that ξ ∈ U , and hence there is a closed subset S of M (B) such that ξ ∈ S ⊂ U . So there exists an f ∈ B satisfying 0 ≤ f ≤ 1, f |S = 1, f |(M (B) \ U ) = 0. If we let ϕ = diag (f, . . . , f ), then ϕ ∈ Fξ and " " π " T (a) − T π (aξ ) T π (ϕ)" = T π (bϕ) ≤ cp bϕL∞ , N ×N where cp = P L(LpN ) . Since b(x)ϕ(x) < ε for all x ∈ X, and as ε > 0 can be chosen arbitrarily, it results that T π (a) and T π (aξ ) are Mπξ -equivalent from the left, as desired. It can be shown analogously that they are Mπξ -equivalent from the right. Now Theorem 1.32 completes the proof.

2.10 Notes and Comments 2.2–2.4. These facts are well known. The proof of Theorem 2.2 is patterned after Halmos [265, Problems 50 and 193]. 2.5. These results form only a little part of what is known about multipliers on p and similar spaces, and significant contributions to this topic have been made by many people. For more about this see, e.g., H¨ ormander [288], Hirschman [280], Zygmund [591], Nikolski [365], Gohberg, Krupnik [232], Duduchava [169], [173], Verbitsky [538]. Inequality 2.5(f) goes back to S. B. Stechkin.

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2 Basic Theory

2.6. Commemorative articles on life and work of Otto Toeplitz may be found in Gohberg et al. [235]. 2.7. See Brown, Halmos [125] or Halmos [265] for H 2 and Duduchava [169] for p . Let a ∈ C. Then neither the spectral radius of T (a) ∈ L(H p ) nor the spectral radius of T (a) + C∞ (H p ) ∈ L(H p )/C∞ (H p ) depend on p (see Theorem 2.42). However, one can show that T (a)L(H p ) depends on p. In particular, T (χ−1 )L(H p ) > 1 for all p = 2. Does the essential norm T (a)Φ(H p ) depend on p (see 2.27)? This problem is equivalent to the following question: Is T (χ−1 )Φ(H p ) equal 1 for all p ∈ (1, ∞)? For more about these things see Böttcher, Krupnik, Silbermann [98]. 2.9. We recorded these trivialities mainly to fix some notation and to have a reference. It is well known that shift operators have many remarkable and nontrivial properties. We only mention the following. Given any Hilbert space E let 2 (E) refer to the Hilbert space of all E-valued sequences {xn }n∈Z+ such

(−1) that xn 2E < ∞ and define VE as (−1)

VE

: 2 (E) → 2 (E),

{x0 , x1 , x2 , . . .} → {x1 , x2 , x3 , . . .}.

Then if A is any bounded linear operator on a Hilbert space H such that A ≤ 1 and An → 0 strongly as n → ∞, there exists a Hilbert space E (−1) with dim E = dim(I − A∗ A)H, an invariant subspace K ⊂ 2 (E) of VE , (−1) and a unitary operator W : H → K such that A = W −1 (VE |K)W . Thus, shifts turn out to be “universal operators.” If I − A∗ A has rank one, we may take E = C and identify 2 (E) with H 2 . The above result is then completed by Beurling’s theorem, which states that every (closed) invariant subspace K ⊂ H 2 of V (−1) , other than H 2 , is of the form K = H 2 θH 2 with some inner function θ. The beginner should consult Rudin [443] and Halmos [265] for these things; excellent presentations of this topic and of related questions are Nikolski [368] and Rosenblum, Rovnyak [442]. 2.10. Good discussions of the main facts about Hankel operators are Nikolski [369], Partington [375], Peetre [384] (this reference also contains an outline of the life of Hermann Hankel), Peller [389], Peller, Khrushchev [390], Power [403], [404]. 2.11. For p = 2, this theorem was established by Nehari [363]. The proof given in the text is due to Sarason [457]. It is this proof which makes the extension of Nehari’s theorem to the spaces H p to a relatively simple matter (this has also been observed by Peetre [384]; also see Peller [388] and Tolokonnikov [519]). Sarason’s proof was adopted in A. Karlovich [299] to extend the Nehari theorem to the case of Hardy type subspaces or rearrangement-invariant spaces. The proof of an analogue of (2.14) in A. Karlovich [299] is based on the inner-outer factorization 1.41 and the so-called Lozanovsky factorization of L1 functions.


105

There are at least three other proofs of Theorem 2.11: Nehari’s original one (which is quite complicated), the proof using the commutant lifting theorem (see Page [374], Clancey, Gohberg [138, Chap. VIII, Theorem 5.2], and Peller [389, Appendix 1.5]), and Parrott’s proof [376]. The latter two proofs also work in the matrix case; Parrott’s proof will be given in 4.32 and 4.33. Also see Bonsall [55]. Finally notice the following characterization of bounded positive Hankel operators due to Widom [564]: if µ is a positive Borel measure on [−1, 1] and if we let ( 1 xn dµ(x) µn = −1

and H[µ] = (µi+j )∞ i,j=0 , then H[µ] ∈ L(2 ) ⇐⇒ µn = O(1/n) (n → ∞) ⇐⇒ µ((−1, −x) ∪ (x, 1)) = O(1 − x) (x → 1). 2.14. Although similar (end equivalent) formulas had been used for a long time, identities (2.18), (2.19) appeared in Widom [569] for the first time. In connection with (2.20) we mention the following result of Brown, Halmos [125]: T (a)T (b) is a Toeplitz operator if and only if a or b is analytic. 2.17. Brown, Halmos [125] and Devinatz [151]. Corollary 4.2 generalizes this result to the matrix case. 2.18. Wintner [579]. Extensions to H p and p are in 2.31. It is clear from the proof in 2.31 that these results extend to the matrix case. 2.19–2.23. Widom [557], Devinatz [151]. There are difficulties in the matrix case, but see 4.35–4.38 and Devinatz, Shinbrot [155] or Speck [499]. For extensions to the spaces H p see 2.32, 5.3, 5.20, 5.22. We do not know a WidomDevinatz criterion for the spaces p . It should be noted here that the general invertibility problem for Toeplitz operators can be reduced to the special case that the symbol is of the form ω1 ω2 , where ω1 and ω2 are inner functions. Using certain factorization theorems of S. Axler, Th. H. Wolff, and D. Sarason one can show that for every function a ∈ GL∞ there exist inner functions ω1 and ω2 , an outer function h ∈ H ∞ , and a continuous function c such that a = ω1 ω2 hc and that T (a) ∈ Φ(H 2 ) (resp. GL(H 2 )) if and only if T (ω1 ω2 ) ∈ Φ(H 2 ) (resp. GL(H 2 )). However, it is by no means easy to decide whether T (ω1 ω2 ) is invertible or Fredholm. For a nice discussion of this topic see Nikolski [366], [367]. Let us also mention the following result of Lee and Sarason [335]. For an inner function ω let supp ω := {τ ∈ T : 0 ∈ ClH (ω, τ )} (see 3.72). If ω1 and ω2 are inner and supp ω1 = supp ω2 , then sp T (ω1 ω2 ) = clos D. 2.25–2.26. This material is taken from Böttcher [68]. See also 4.72, 4.73, 9.18. Parts (a) and (b) of Proposition 2.26 are well known. It may be that parts (c) and (d) of this proposition are known to specialists, but we have not found any reference. It should also be noted that Kats [310] considered the Riemann

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2 Basic Theory

boundary value problem f + = af − +g on T with a coefficient a ∈ C(T◦ ) in the class of all functions holomorphic in |z| < 1 resp. |z| > 1, bounded in |z| ≤ 1 resp. |z| ≥ 1, and continuous in {|z| ≤ 1, z = −1} resp. {|z| ≥ 1, z = −1}. He obtained necessary and sufficient conditions for the homogeneous problem to have a finite number of linearly independent solutions, computed this number, and studied the solvability of the inhomogeneous problem. 2.28–2.29. These results are well known, but apart from the case p = 2 (Halmos [265, Problem 52], Douglas [162, 4.24]) we do not know any reference. 2.30. Hartman and Wintner [267] showed that R(a) is contained in the spectrum spL(H 2 ) T (a), and Simonenko [496] proved that a ∈ GL∞ if T (a) is a Φ+ - or Φ− -operator on H p . The proof given here is based on arguments of Widom [556]. Also see 2.93. Finally, note that the normal solvability of T (a) on H p (1 < p < ∞) implies that either a ∈ GL∞ or a ≡ 0. This was proved by Leiterer [336] for p = 2 and by Heunemann [275] for general p. 2.31–2.33. See the notes to 2.17–2.23. 2.35. In 1963, Halmos posed the following as a test question for any theory of invertibility of Toeplitz operators: Is the spectrum of a Toeplitz operator necessarily connected? Widom showed that the answer is yes (in [562] for p = 2 using the Helson-Szeg˝o theorem 1.45 and in [563] for general p without using this theorem). The connectedness of spΦ(H 2 ) T (a) was first proved by Douglas [162, 7.45]. See also 4.68. 2.36. Hartman, Wintner [267]. It has been open for a long time whether an analogous result holds for quarter-plane Toeplitz operators (see Chapter 8); in Böttcher [71], we observed that an argument used by McDonald and Sundberg [354] in the context of Toeplitz operators on the disk also applies to half-plane Toeplitz operators and so, by 8.13 and 8.14, proves the connectedness of both the spectrum and the essential spectrum of quarter-plane Toeplitz operators with real-valued symbols. Open problems: Is the (essential) spectrum of T 2 (a) on H 2 (T2 ) connected for all a ∈ L∞ (T2 )? As we know that this is so for real-valued symbols, are there interesting applications of Theorem 4.100 to quarter-plane Toeplitz operators? Selfadjoint operators live in another world than the generically nonselfadjoint Toeplitz operators. However, if a Toeplitz operator is selfadjoint, one is faced with the same specific questions as for general selfadjoint operators (such as diagonalization or spectral projections). These questions are treated in the pioneering papers by Rosenblum [439], [440], [441] and Ismagilov [292] and in the book Rosenblum, Rovnyak [442]. A recent remarkable contribution to the matter was made by Vreugdenhil [552]. He showed, for example, that if a(t) = |t − 1|2 (t ∈ T), then the positive square root of T (a) is 3 4∞ 4 1 1 − . π 4(j + k + 2)2 + 1 4(j − k)2 + 1 j,k=0 See also Chapter 1 of B¨ ottcher, Grudsky [86].


107

2.37. Wolff [581]. See also 4.76. 2.38–2.40. Coburn [141] for p = 2, Simonenko [496] for H p , Duduchava [169] for p . For generalizations of Theorem 2.38 and for still another proof of 2.38(b) see Volberg, Tolokonnikov [551]. Vukotić [553] proved a more explicit form of 2.38(a): If T (a) ∈ L(H 2 ) is a nonzero Toeplitz operator which is not one-to-one, then T (a)(lin{P Ker T (a)}) = P, where lin S denotes the linear span of the set S and P Ker T (a) is defined as {pf : p ∈ P, f ∈ Ker T (a)}. 2.41. For (e) see Gelfand, Raikov, Shilov [212]. 2.42. This theorem is the culmination of several authors including Noether [370], Mikhlin [360], Gohberg [217], Simonenko [491], Krein [322], Calder´ on, Spitzer, Widom [129], Devinatz [151]. Note that Simonenko’s 1960 work [491] actually contains Theorems 2.68 and 2.72! Stegenga [507] proved that a Toeplitz operator T (a) is bounded on H 1 if and only if a ∈ L∞ ∩ BM O1/| log t| . Recently Virtanen [546] obtained an analogue of Theorem 2.42(b) for Toeplitz operators T (a) on H 1 provided a ∈ C ∩ V M O1/| log t| . Here BM O1/| log t| and V M O1/| log t| are certain generalizations of BM O and V M O. 2.43–2.47. Theorem 2.47 was established by Krein [322] (for symbols in W ) and by Gohberg, Feldman [220] (for symbols in Cp ). Lemma 2.44 (and its proof given here) as well as Proposition 2.46 are due to Nikolski [365]. 2.50. This is an extract of arguments due to Krein [323] and B¨ ottcher, Silbermann [106, Chap. IV]. 2.51–2.53. Sarason [451] was the first to observe that C + H ∞ is closed and thus an algebra. This (at the first glance unpretentious and rather curious) discovery is certainly one of the most significant achievements in mathematical analysis during the last decades, and it has stimulated and determined subsequent developments in various fields (in particular in the theory of Toeplitz operators) essentially. See also Sarason [452], [457], Douglas [162], Koosis [316], Garnett [211]. For the Zalcman-Rudin lemma see Koosis [316, Chap. VII, 3◦ ]. The observation that Cp + Hp∞ is a closed subalgebra of M p (p = 2) is due to the authors (B¨ ottcher, Silbermann [113]). 2.54. Hartman [266] showed that the Hankel operator H(a) is compact on H 2 if and only if a ∈ C + H ∞ , and Adamyan, Arov, Krein [2] established the equality H(a)Φ(H 2 ) = distL∞ (a, C + H ∞ ). The proof given here follows Axler, Berg, Jewell, Shields [14]. It makes use of the fact that C + H ∞ is closed. There are proofs of Hartman’s result (e.g. Hartman’s original one) which do not use the closedness of C + H ∞ . Thus the fact that C + H ∞ is closed is also a consequence of Hartman’s result (see Sarason [457, p. 102]). For more about compactness of Hankel operators (s-numbers, trace class criteria, vector-valued versions of Hartman’s theorem etc.) see Page [374], Peller [385] – [388], Peller, Khrushchev [390], Nikolski [366] – [369], Rochberg

108

2 Basic Theory

[434], Power [403], [404], Peetre [384], Treil [524], Havin, Khrushchev, Nikolski [269]. Widom [564] showed that for positive Hankel operators the following are equivalent (recall the notes to 2.11): (i) H[µ] ∈ C∞ (2 ), (ii) µn = o(1/n) (n → ∞), (iii) µ((−1, −x) ∪ (x, 1)) = o(1 − x) (x → 1). If a ∈ P C, then H(a)Φ(H 2 ) = distL∞ (a, C + H ∞ ) = distL∞ (a, C), which in particular implies that P C + H ∞ is a closed subset of L∞ . This was shown by Bonsall and Gillespie [56]. They also pointed out that P C + H ∞ is not an algebra. 2.55. This result (for p = 2) was obtained by Douglas [159], [162]. The proof presented here is essentially simpler than that of Douglas. 2.58. The theorem is Kronecker’s [325] and the proof in the text is after Gantmacher [210, Chap. XV, Section 10]. An alternative proof of this result (due to Axler) is contained in Sarason [457] and Peller [389, Chap. 1, Section 3]. See also the references listed in the notes to 2.54. Notice that the rank of H(a) is equal to the number of poles of the rational function P a (counted up to multiplicity). 2.59–2.60. These results are due to the authors. Corollary 2.59 appeared first in B¨ ottcher, Silbermann [106, 4.6] and Theorem 2.60 was first published in Böttcher, Silbermann [113]. 2.61–2.66. The results of 2.61–2.65 were established by Douglas [159], [162, 7.36] (for p = 2). Theorem 2.66 is new and due to the authors. Our presentation also relies on Sarason [452]. 2.67–2.69. Simonenko [491], [496] and Gohberg, Krupnik [232]. In the case p = 2 these results were also obtained by Douglas and Sarason [164], who used Glicksberg’s theorem 1.22 (see 2.83). 2.71–2.72. The notion of local sectoriality was introduced by Simonenko [496] and Douglas, Widom [166]. Theorem 2.72 is in Simonenko [491], Devinatz [151], Douglas, Widom [166]. 2.74. Widom [556], Simonenko [491], Devinatz [151], Gohberg [218]. 2.75. Douglas, Sarason [164]. 2.77–2.81. See Hoffman [284], Gamelin [209], Garnett [211], Gelfand, Raikov, Shilov [212]. The algebra QC was introduced by Douglas. 2.82. This definition is from Clancey, Gosselin [139]. 2.83. This theorem was established by Axler [12] using transfinite localization (Axler’s method will be described in Chapter 4). For B = C, the result goes over into Theorem 2.67 (p = 2). The proof presented here first appeared in Böttcher [69] and is new in the following sense: on the one hand it is not terribly new, since it mimics the argument used by Douglas and Sarason [164] to prove this theorem for B = C, and on the other hand it is strange that


109

Axler (who wrote his dissertation under the supervision of Sarason) did not at the very least mention this possibility of proving the theorem in [12]. 2.84–2.85. The term “locally sectorial over QC” had already been used by Douglas [161], a systematic study of symbols which are locally sectorial over C ∗ -algebras between C and QC has begun in Silbermann [483]. There Theorem 2.85 was established for B = QC. For B = C + H ∞ , Theorem 2.85 is due to Böttcher [69]. Note that Theorem 2.85 is neither an immediate consequence of Theorem 2.83 nor of Theorems 4.63 and 4.64. 2.86. Part (c) goes back to Simonenko [491] and Douglas, Widom [166], parts (a) and (b) were probably first proved in B¨ ottcher [69]. The fact that every function which is locally sectorial over QC can be written as a product of a function in GQC and a sectorial function is nontrivial and is a key result, which simplifies the theory of Toeplitz operators with symbols that are locally sectorial over QC substantially. We remark that it was the search for a proof of this result which led us to Glicksberg’s theorem 1.22 and, subsequently, to the proofs of Theorems 2.83 and 2.85 given here. For the matrix case see 3.7, 3.8, 4.31, and the remark after 4.49. 2.87. Sarason [456]. 2.88. The result was established in Silbermann [483], the proof presented here is from B¨ ottcher [69]. 2.90. For B = C, both the result and the proof are taken from Clancey [135] (see also Clancey, Morrel [140]). Once Gorkin [239] had shown that M (H ∞ |Xξ ) (ξ ∈ M (QC)) is connected, Silbermann [483] stated this lemma for B = QC. 2.91. See Silbermann [483] for B = QC. The present proof is taken from Böttcher [69]. 2.93. Simonenko [492]. Also see Devinatz, Shinbrot [155]. 2.94. Douglas [160] for p = 2 and Spitkovsky [502] for H p . 2.95–2.96. Both theorems are well known. Theorem 2.96 for B = C goes back to Simonenko [492]. It is clear that 1.32 is the appropriate tool to prove 2.95. That 2.96 (in the case B = QC) can be proved with the help of 1.32 is less obvious and requires an argument which was also used by Axler [12, pp. 39–40].

3 Symbol Analysis

3.1 Local Sectoriality 3.1. Definitions. Let F be a closed subset of X = M (L∞ ) and let a be in L∞ N ×N . The matrix function a is called analytically sectorial on F if there exist a real number ε > 0 and two invertible matrices b, c ∈ CN ×N such that Re (ba(x)c) ≥ ε for all x ∈ F , that is, Re (ba(x)cz, z) ≥ εz2

∀x∈F

∀ z ∈ CN ,

and is said to be geometrically sectorial on F if conv a(F ) ⊂ GCN ×N , that is, if each matrix in the closed convex hull of a(F ) is invertible. It is easy to see that a scalar-valued function (N = 1) is analytically sectorial on F if and only if it is geometrically sectorial on F . In this case the function is simply called sectorial on F , which is in accordance with 2.84. Functions which are (analytically or geometrically) sectorial on the whole maximal ideal space X will be called (analytically or geometrically) sectorial. In the scalar case this agrees with 2.16. Let B be a closed subalgebra of C + H ∞ containing the constants. A function a ∈ L∞ N ×N will be called (analytically or geometrically) locally sectorial over B if it is (analytically or geometrically) sectorial on each maximal antisymmetric set for B. In case B is a C ∗ -subalgebra (of QC), the fibers Xβ , β ∈ M (B), occupy the place of the maximal antisymmetric sets (see 1.27(c)). 3.2. Proposition. If a ∈ L∞ N ×N is analytically sectorial on a closed subset F of X, then a is geometrically sectorial on F .

)c) ≥ ε, then Re b λ a(x )c ≥ ε whenever λi ≥ 0 and Proof. If Re (ba(x i i i i

λ = 1. But if Re d ≥ ε > 0 for a matrix d ∈ C N ×N , then d ∈ GCN ×N . So, i i

since b, c ∈ GCN ×N , we conclude that i λi a(xi ) is invertible for all λi ≥ 0 such that i λi = 1.

112

3 Symbol Analysis

Remark. Azoff and Clancey [16] constructed an example of a matrix function a ∈ L∞ 2×2 which is geometrically sectorial but not analytically sectorial. The following lemma is needed to prove Theorem 3.4, which represents an important special case in which geometrical sectoriality implies analytic sectoriality. 3.3. Lemma. (a) Let Jλ be the m × m ⎛ λ 1 ⎜ 0 λ ⎜ Jλ = ⎜ ⎜ 0 0 ⎝... ... 0 0

Jordan cell ⎞ 0 ... 0 1 ... 0 ⎟ ⎟ λ ... 0 ⎟ ⎟ ... ... ...⎠ 0 ... λ

and suppose the origin does not belong to the line segment [λ, 1]. Then there are B, C ∈ GCm×m and δ > 0 such that Re (BC) ≥ δ > 0,

Re (BJλ C) ≥ δ > 0.

(b) Let E, F ∈ Cm×m and suppose det µE + (1 − µ)F = 0

∀µ ∈ [0, 1].

Then there are B, C ∈ GCm×m and δ > 0 such that Re (BEC) ≥ δ > 0, Proof. (a) Put ⎛ 0 ⎜ 0 ⎜ N =⎜ ⎜... ⎝ 0 0

1 0 ... 0 0

0 1 ... 0 0

... ... ... ... ...

⎞ 0 0 ⎟ ⎟ ...⎟ ⎟, 1 ⎠ 0

Re (BF C) ≥ δ > 0. ⎛

β m−1 β m−2 ⎜ 0 β m−2 ⎜ 0 Vβ = ⎜ ⎜ 0 ⎝ ... ... 0 0

β m−3 β m−3 β m−3 ... 0

... ... ... ... ...

β β β ... 0

⎞ 1 1 ⎟ ⎟ 1 ⎟ ⎟. ...⎠ 1

/ [λ, 1], there If β = 0, then Vβ N = βN Vβ , whence Vβ N Vβ−1 = βN . Since 0 ∈ are ν ∈ T and α > 0 such that Re ν ≥ α > 0,

Re (νλ) ≥ α > 0.

Consequently, if we let U = diag (ν, . . . , ν), then Re (U I) ≥ α > 0,

Re (U λI) ≥ α > 0.

Thus, Re (U Vβ IVβ−1 ) = Re (U I) ≥ α > 0, Re (U Vβ Jλ Vβ−1 ) = Re (U Vβ (λI + N )Vβ−1 ) = Re (U λI) + Re (βU N ) ≥

α >0 2

3.1 Local Sectoriality

113

if only β is sufficiently small. It follows that then B = U Vβ , C = Vβ−1 , δ = α/2 have the desired properties. (b) The hypothesis implies that E and F are invertible. There is a D in GCm×m such that D−1 E −1 F D = J is in the Jordan canonical form. We have det(µI + (1 − µ)J) = det(µD−1 E −1 ED + (1 − µ)D−1 E −1 F D) = det(D−1 E −1 ) det(µE + (1 − µ)F ) det D = 0 for all µ ∈ [0, 1]. The matrix J is block diagonal, J = block diag (Jλk ) with each Jλk of the form as in part (a). Because det(µI + (1 − µ)Jλk ) ∀ µ ∈ [0, 1], 0 = det(µI − (1 − µ)J) = k

we conclude that 0 ∈ / [λk , 1] for each k. So part (a) ensures the existence of matrices Bk , Ck and of numbers δk > 0 such that Re (Bk Ck ) ≥ δk > 0,

Re (Bk Jλk Ck ) ≥ δk > 0.

If we let B = block diag (Bk ), C = block diag (Ck ), then Re (B C ) ≥ δ > 0,

Re (B JC ) ≥ δ > 0

with some δ > 0. Now it is easily seen that the matrices B = B D−1 E −1 and C = DC have the desired properties. 3.4. Theorem (Clancey). Let F be a closed subset of X and let a ∈ L∞ N ×N be geometrically sectorial on F . If conv a(F ) is a line segment (i.e., a set of the form [z, w] = (1 − λ)z + λw : λ ∈ [0, 1] , where z, w ∈ CN ×N ), then a is analytically sectorial on F . Proof. Since F is compact and a is continuous on X, there are x1 , x2 ∈ F such that conv a(F ) = [a(x1 ), a(x2 )]. Because a is geometrically sectorial on F , the line segment [a(x1 ), a(x2 )] consists of invertible matrices only. So Lemma 3.3(b) can be applied to see that there are b, c ∈ GCN ×N and ε > 0 such that Re (ba(x1 )c) ≥ ε > 0, Re (ba(x2 )c) ≥ ε > 0. This implies that Re b λa(x1 ) + (1 − λ)a(x2 ) c ≥ ε > 0 ∀ λ ∈ [0, 1], whence Re (ba(x)c) ≥ ε > 0 for all x ∈ F .

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3 Symbol Analysis

Important convention. In what follows we shall mainly deal with matrix functions that are analytically sectorial on closed subsets of X. Therefore a matrix function which is analytically sectorial on a set F or over an algebra B will henceforth be simply called sectorial on F or over B, i.e., in the following “sectorial” always means “analytically sectorial.” The representations stated in Proposition 2.86 for scalar-valued locally sectorial functions played a crucial role in the local theory of scalar Toeplitz operators and it is therefore desirable to have analogous representations for locally sectorial matrix functions. We first show how the arguments used to prove Propositions 2.86(a), (b) can be extended to the matrix case. 3.5. Theorem (Machado/Szyma´ nski). Let Y be a compact Hausdorff space and B a closed subalgebra of C(Y ) containing the constants. Let S denote the family of maximal antisymmetric sets for B. Then, for every a ∈ [C(Y )]N ×N , dist(a, BN ×N ) = max distS (a, BN ×N ), S∈S

where for a closed subset F of Y , distF (a, BN ×N ) :=

inf

max a(y) − b(y)L(CN ) ,

b∈BN ×N y∈F

dist(a, BN ×N ) refers to distY (a, BN ×N ), and the norm on CN is the one given by (1.9). Proof. For a proof see Machado [342], Szyma´ nski [512], Burckel [127], Ransford [415], or Pérez Carreras and Bonet [391]. 3.6. Lemma. Let X = M (L∞ ) and let F be a closed subset of X. (a) A matrix function a ∈ L∞ N ×N is sectorial on F if and only if there are a d ∈ GCN ×N and an ε > 0 such that Re (a(x)d) ≥ ε for all x ∈ F . (b) A matrix function a ∈ L∞ N ×N is sectorial on F if and only if there is a d ∈ GCN ×N such that I − a(x)dL(CN ) < 1 for all x ∈ F . −1 (c) Let u ∈ GL∞ (x) = u∗ (x) for all N ×N be unitary-valued on F , i.e., u x ∈ F . Then u is sectorial on F if and only if distF (u, CN ×N ) < 1. ∞ (d) Let B be any subset of L∞ N ×N , let u ∈ GLN ×N be unitary-valued on −1 ∗ X (u (x) = u (x) for all x ∈ X), and suppose distX (u, B) < 1. Then there are a b ∈ B and a sectorial matrix function s ∈ GL∞ N ×N such that u = sb.

Proof. (a) If Re (ba(x)cz, z) ≥ δz2 for all x ∈ F with some b, c ∈ GCN ×N and δ > 0, then Re (a(x)c(b∗ )−1 b∗ z, b∗ z) = Re (a(x)cz, b∗ z) = Re (ba(x)cz, z) ≥ δz2 ≥ δb∗ −2 b∗ z2 ∀ z ∈ CN


115

and so d = c(b∗ )−1 and ε = δb∗ −2 have the desired property. (b) The “if” portion follows from the observation that Re (a(x)d) = I − Re (I − a(x)d) ≥ I − I − a(x)d. On the other hand, if a is sectorial on F , then, by part (a), there are c ∈ GCN ×N and δ > 0 such that Re (a(x)c) ≥ δ for all x ∈ F . Put α := max a(x)c (> 0) and ε := δ/α2 . Then, for zCN = 1, x∈F

(I − a(x)c)z2 = 1 − 2εRe (a(x)cz, z) + ε2 a(x)cz2 δ2 ≤ 1 − 2εδ + ε2 α2 = 1 − 2 < 1, α which gives the assertion with d = εc. (c) From part (b) we deduce that there is a d ∈ CN ×N such that I − u(x)d ≤ 1 for all x ∈ F . Consequently, u(x) − d∗ = u∗ (x) − d = u(x)(u∗ (x) − d) = I − u(x)d < 1 for all x ∈ F , i.e., distF (u, CN ×N ) < 1. Conversely, if there is a c ∈ CN ×N with u(x) − c < 1 − δ < 1 for all x ∈ F , then I − u(x)c∗ = u(x)(u∗ (x) − c∗ ) = u∗ (x) − c∗ < 1 − δ for x ∈ F , which implies that, for zCN = 1, 2Re (u(x)c∗ z, z) = 1 + u(x)c∗ z2 − (I − u(x)c∗ )z2 ≥ 1 − I − u(x)c∗ 2 ≥ δ > 0 ∀ x ∈ F. (d) If there is a b ∈ B with u − b < 1 − δ < 1, then I − bu−1 = (u − b)u∗ < 1 − δ, hence, for zCN = 1, 2Re (bu−1 z, z) = 1 + bu−1 z2 − (I − bu−1 )z2 ≥ 1 − I − bu−1 2 > δ > 0 and thus bu−1 = s with s satisfying Re s(x) ≥ δ/2 for all x ∈ X. It follows that u = s−1 b, and because Re (s−1 z, z) = Re (s−1 sy, sy) = Re (y, sy) ≥ =

δ −1 2 δ s z ≥ s−2 z2 2 2

s−1 must be sectorial (on X).

δ y2 2

∀ z ∈ CN ,

116

3 Symbol Analysis

3.7. Proposition. Let B be a C ∗ -subalgebra of L∞ containing the constants −1 (x) = u∗ (x) for all x ∈ X. and let u ∈ GL∞ N ×N be unitary-valued, that is, u Then for u to be locally sectorial over B it is necessary and sufficient that u be of the form u = sb with b ∈ GBN ×N and s ∈ GL∞ N ×N being sectorial. Proof. If u is locally sectorial over B, then, by Lemma 3.6(c), distXξ (u, BN ×N ) = distXξ (u, CN ×N ) < 1 ∀ ξ ∈ M (B), and Theorem 3.5 in conjunction with 1.27(c) implies that dist(u, BN ×N ) < 1. Now Lemma 3.6(d) gives that there are a b ∈ BN ×N and a sectorial ∞ s ∈ GL∞ N ×N such that u = sb. Since obviously b ∈ GLN ×N , we deduce from 1.26(d) that actually b ∈ GBN ×N . This proves the necessity portion. The sufficiency part is trivial. We now remove the restriction to unitary-valued matrix-functions by using other (even more elementary) techniques. 3.8. Theorem. Let B be a C ∗ -algebra between C and L∞ and let a ∈ L∞ N ×N . Then a is locally sectorial over B if and only if a is of the form a = sb where s ∈ GL∞ N ×N is sectorial and b ∈ GBN ×N . Proof. It suffices to prove the “only if” portion. So suppose a is locally sectorial over B. For ξ ∈ M (B), define Dξ := d ∈ CN ×N : I − a(x)dL(CN ) < 1 ∀ x ∈ Xξ . Lemma 3.6(b) implies that Dξ is nonempty. It is clear that Dξ is an open convex subset of CN ×N . If d ∈ Dξ , then d ∈ Dη for all η in some open neighborhood U (ξ) ⊂ M (B) of ξ; this follows from the upper semi-continuity of the mapping M (B) → R+ ,

ξ → max I − a(x)d, x∈Xξ

which, in turn, can be derived from Theorem 1.35(b) in the setting A = L∞ N ×N and B = {ϕIN ×N : ϕ ∈ B}. Associate with each ξ ∈ M (B) a matrix dξ ∈ Dξ and a neighborhood U (ξ) of ξ such that dξ ∈ Dη for all η ∈ U (ξ). Because M (B) is a compact Hausdorff space, it is a normal space and hence there are open neighborhoods U (ξ) such that ξ ∈ U (ξ) ⊂ clos U (ξ) ⊂ U (ξ). By the compactness of M (B), there are ξ1 , . . . , ξn in M (B) such that n U (ξi ). Consider the constant functions M (B) = i=1

fi : clos U (ξi ) → CN ×N ,

ξ → dξi .


117

Accept for a moment the validity of the following claim: If U and V are open subsets of M (B) such that U \ clos V = ∅, if g is a continuous function on clos U with g(ξ) ∈ Dξ for all ξ ∈ U , and if d is some matrix belonging to Dξ for all ξ in some open neighborhood W (clos V ) of clos V , then there is a continuous function h on clos (U ∪ V ) such that h(ξ) ∈ Dξ for all ξ ∈ U ∪ V . Hence, letting U = U (ξ1 ), V = U (ξ2 ), g = f1 , d = dξ2 , we get a continuous function h1 on clos [U (ξ1 ) ∪ U (ξ2 )] such that h1 (ξ) ∈ Dξ for all ξ ∈ U (ξ1 ) ∪ U (ξ2 ). Then the claim for U = U (ξ1 ) ∪ U (ξ2 ), V = U (ξ3 ), g = h1 , d = dξ3 gives a continuous function h2 on clos [U (ξ1 ) ∪ U (ξ2 ) ∪ U (ξ3 )] with h2 (ξ) ∈ Dξ for all ξ ∈ U (ξ1 ) ∪ U (ξ2 ) ∪ U (ξ3 ). Continuing, we finally arrive at a continuous function h on M (B) with h(ξ) ∈ Dξ for all ξ ∈ M (B), that is, we have an h ∈ BN ×N with I − a(x)h(x)L(CN ) < 1 ∀ x ∈ X.

(3.1)

From (3.1) we see that h ∈ GL∞ N ×N , whence h ∈ GBN ×N . Also by (3.1), the matrix function s = ah is sectorial. Hence, if we let b = h−1 , then a = sb is the desired factorization. It remains to prove the above claim. Since M (B) is a normal space, there is an open neighborhood W = W (clos V ) of clos V such that V ⊂ clos V ⊂ W (clos V ) ⊂ W (clos V ),

U := clos U \ W (clos V ) = ∅.

Put V = clos (V \ U ) and notice that U ∪ V ⊂ U ∪ V ∪ (W ∩ U ). The sets U and V are closed and U ∩ V = ∅. Hence, by Uryson’s extension theorem (see, e.g., Naimark [362, p. 43]), there exists a continuous function ϕ on clos (U ∪ V ) such that 0 ≤ ϕ ≤ 1, ϕ|U = 1, ϕ|V = 0. Extend g arbitrarily to a function on clos (U ∪ V ) and put h = gϕ + d(1 − ϕ). Because gϕ is continuous on clos U and vanishes identically on V = clos (V \ U ), it follows that h is continuous on clos (U ∪ V ). If ξ ∈ U , then h(ξ) = g(ξ) ∈ Dξ and if ξ ∈ V , then h(ξ) = d ∈ Dξ . Finally, if ξ ∈ W ∩ U , then h(ξ) is a convex linear combination of g(ξ) ∈ Dξ and d ∈ Dξ , which, by the convexity of Dξ , implies that h(ξ) ∈ Dξ . This completes the proof of our claim. We finally state a matrix analogue of 2.86(c). 3.9. Theorem. Let a ∈ L∞ N ×N . Then the following are equivalent: (i) a is locally sectorial over C; (ii) for each τ ∈ T there are open neighborhoods Uτ ⊂ T of τ , matrices bτ , cτ ∈ GCN ×N , and an ετ > 0 such that Re (bτ a(t)cτ ) ≥ ετ

for almost all

t ∈ Uτ ;

(iii) there are a finite number of open subarcs U1 , . . . , Un of T, matrices c1 , . . . , cn ∈ GCN ×N and an ε > 0 such that Re (a(t)ck ) ≥ ε

for almost all

t ∈ Uk ;

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3 Symbol Analysis

(iv) a = sψ with ψ ∈ GCN ×N and s ∈ GL∞ N ×N sectorial ; (v) a = ϕsψ with ϕ, ψ ∈ GCN ×N and s ∈ GL∞ N ×N sectorial. Proof. The implications (iv) =⇒ (v) =⇒ (i) are trivial and the implication (i) =⇒ (iv) results from the preceding theorem. The implication (iii) =⇒ (ii) is also trivial, while the implication (ii) =⇒ (iii) is a consequence of the compactness of T and of Lemma 3.6(a). Finally, the implication (iv) =⇒ (ii) can be verified as in the proof of Proposition 2.86(c) and the implication (ii) =⇒ (i) follows from Proposition 2.79(a). 3.10. Pn C. Let a ∈ L∞ N ×N and τ ∈ T. Each point of CN ×N which belongs to the (compact) set a(Xτ ) is called an essential cluster point of a at τ . (Pn C)N,N is defined as the set of all functions a ∈ L∞ N ×N which have at most n essential cluster points at each point of T. The set (Pn C)1,1 will be abbreviated to Pn C. Example. Let τ ∈ T and let E1 , . . . , En (n ≥ 2) be pairwise disjoint measurn Ek and U ∩ Ek has positive measure for able subsets of T such that T = k=1

each k ∈ {1, . . . , n} and each neighborhood U ⊂ T of τ . Then

n let α1 , . . . , αn be any pairwise distinct complex numbers and put a = k=1 αk χEk . Due to Proposition 2.79(a) we have a(Xτ ) = {α1 , . . . , αn } and a belongs to Pn C \ Pn−1 C. Let Ak denote the set of all x ∈ Xτ for which a(x) = ατ . It is easy to see that χEk (x) = 1 for x ∈ Ak and χEk (x) = 0 for x ∈ Xτ \ Ak . Note that, for N > 1 and n > 1, (Pn C)N,N is properly contained in (Pn C)N ×N , the collection of all N × N matrix functions whose entries are in Pn C. Indeed, if we let E1 , E2 , E3 , E4 be as in the above example and if α, β are distinct complex numbers, then the entries of the diagonal matrix function diag (αχE1 + αχE2 + βχE3 + βχE4 , αχE1 + βχE2 + αχE3 + βχE4 ) belong to P2 C, while the matrix function itself takes the four values diag (α, α),

diag (α, β),

diag (β, α),

diag (β, β)

on A1 , A2 , A3 , A4 , respectively. It is clear that P CN ×N (see 2.73) is contained in (P2 C)N ×N and that there are functions in (P2 C)N,N that do not belong to P CN ×N . Finally, it is obvious that (P1 C)N ×N = CN ×N . The following proposition characterizes the matrix functions in (P2 C)N,N that are locally sectorial over C. 3.11. Proposition. Let a ∈ (P2 C)N,N , and for τ ∈ T denote by a1τ and a2τ the essential cluster points of a at τ (it may be that a1τ = a2τ ). Then the following are equivalent: (i) a is locally sectorial over C;

3.2 Asymptotic Multiplicativity

119

(ii) for each τ ∈ T, the (possibly degenerate) line segment [a1τ , a2τ ] consists of invertible matrices only; (iii) det[(1 − µ)a1τ + µa2τ ] = 0

∀ (τ, µ) ∈ T × [0, 1].

Proof. The equivalence (ii) ⇐⇒ (iii) is trivial. To establish the equivalence (i) ⇐⇒ (ii), notice first that conv a(Xτ ) = [a1τ , a2τ ]. Therefore, by Proposition 3.2, (i) implies (ii). On the other hand, Theorem 3.4 (or Lemma 3.3(b)) shows that (i) is a consequence of (ii).

3.2 Asymptotic Multiplicativity 3.12. The Poisson kernels. For ϕ ∈ L1 , we defined hr ϕ (0 < r < 1), the Abel-Poisson means of the Fourier series (= harmonic extension), as r|l| ϕl eilx , x ∈ [0, 2π). (hr ϕ)(eix ) = l∈Z

Using the Poisson kernel, we have ( 2π kr (x − t)ϕ(eit ) dt, (hr ϕ)(eix ) =

x ∈ [0, 2π),

0

where kr (x) =

1 − r2 1 , 2π 1 − 2r cos x + r2

x ∈ R.

The slight change in notation (recall how kr was defined in 1.37) should not cause confusion. If we extend ϕ to a function Φ ∈ L1loc (R) periodically, i.e., if we set Φ(x) = ϕ(eix ), x ∈ R, then hr ϕ can be written as ( ∞ (hr ϕ)(eix ) = λK(λ(x − t))Φ(t) dt, x ∈ R, −∞

where K(x) = 1/(π(1 + x2 )) and λ = −1/ log r ∈ (0, ∞) (see Ahiezer [3, Section 62]). 3.13. The Fej´ er kernels. The Fejér (or Fejér-Cesaro) means σn ϕ, n ∈ N, of a function ϕ ∈ L1 are defined in terms of Fourier coefficients by n (σn ϕ)(e ) = 1− ix

l=−n

|l| n+1

ϕl eilx ,

x ∈ [0, 2π).

We also have (

2π

kn (x − t)ϕ(eit ) dt,

(σn ϕ)(eix ) = 0

x ∈ [0, 2π),

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3 Symbol Analysis

where the (Fejér) kernel kn is given by kn (x) =

sin2 ((n + 1)x/2) 1 , 2π(n + 1) sin2 (x/2)

x ∈ R.

Let Φ ∈ L1loc (R) denote the periodic extension of ϕ, Φ(x) = ϕ(eix ) for x ∈ R. Then ( ∞ (n + 1)K (n + 1)(x − t) Φ(t) dt, x ∈ R, (σn ϕ)(eix ) = −∞

with

2 sin2 (x/2) , π x2 (again see Ahiezer [3, Section 62]). K(x) =

x∈R

3.14. Approximate identities. The Poisson and Fejér kernels are typical examples of what is usually called an approximate identity. Let K be a function in L1 (R) which has the following properties: ( ∞ K(x) ≥ 0, K(x) = K(−x), K(x) dx = 1, (3.2) −∞

0 < ess inf K(x) ≤ ess sup K(x) < ∞, x∈(−π,π)

(3.3)

x∈(−π,π)

there is a constant M > 0 such that K(x) ≤

M for |x| ≥ 1. x2

(3.4)

The generalized sequence {Kλ }λ∈Λ , where Λ = {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N)

or

Λ = (r0 , ∞)

(r0 ∈ R+ ),

and Kλ (x) = λK(λx),

x ∈ R,

will be called the approximate identity generated by K. Given ϕ ∈ L1 let Φ ∈ L1loc (R) denote the periodic extension of ϕ. Then define ( ∞ (kλ Φ)(x) :=

−∞

λK(λ(x − t))Φ(t) dt,

x ∈ R,

and put (kλ ϕ)(eix ) := (kλ Φ)(x),

x ∈ [0, 2π).

Thus, for λ ∈ Λ, kλ Φ is a 2π-periodic function in L1loc (R) and kλ ϕ is a function in L1 = L1 (T). Finally we shall sometimes write kλ,t ϕ := (kλ ϕ)(t) (t ∈ T),

kλ,x Φ := (kλ Φ)(x) (x ∈ R).

The following facts are well known and can be verified without substantial difficulty.


121

(a) If a ∈ L∞ , then kλ a ∈ C for all λ ∈ Λ. (b) If a ∈ L∞ , then sup kλ a∞ ≤ a∞ . λ∈Λ

(c) If a ∈ L , then kλ a − aL2 → 0 as λ → ∞.

Let ϕ(x) = l ϕl eilx be a Laurent polynomial. Then ( ∞ ix λK(λt) ϕl eil(x−t) dt (kλ ϕ)(e ) = 2

−∞

=

ϕl eilx

l

=

l

where K(y) =

(

∞

ϕl K

l

(

∞

λK(λt)e−ilt dt

−∞

l eilx λ

K(x)e−ixy dx =

−∞

(

(x ∈ R),

(3.5)

∞

K(x) cos(yx) dx.

(3.6)

−∞

∈ C(R), K(±∞) So K = 0, K(0) = 1 and K(y) < 1 for y = 0. This can be used to derive the following fact, which will be needed later. (d) If (kλn χ1 )(eiθn ) → χ1 (τ ) = τ as n → ∞ (τ ∈ T), then eiθn → τ and λn → ∞ as n → ∞. If K is the Poisson kernel, then (d) says that eiθn → τ and rn = e−1/λn → 1 whenever rn eiθn → τ ; thus, in that case (d) is trivial. Finally, note that K(y) = e−|y| if K is the Poisson kernel, ! 1 − |y| (|y| ≤ 1) K(y) = if K is the Fejér kernel. 0 (|y| ≥ 1) 3.15. Asymptotic multiplicativity. Let A and B be subsets of L∞ and let {Kλ }λ∈Λ be an approximate identity. We say that {Kλ }λ∈Λ is asymptotically multiplicative on the pair (A, B) if kλ (ab) − (kλ a)(kλ b)∞ → 0 as

λ → ∞ ∀ a ∈ A,

∀ b ∈ B.

Thus, Lemma 2.61 says that the Poisson kernels are asymptotically multiplicative on the pair (C, L∞ ), Lemma 2.87 states that the Poisson kernels are even asymptotically multiplicative on the pair (QC, L∞ ), and the statement of Theorem 2.62(c) is the asymptotic multiplicativity of the Poisson kernels on the pair (C + H ∞ , C + H ∞ ). Minor and obvious modifications of the proof of Lemma 2.61 imply that every approximate identity is asymptotically multiplicative on the pair (C, L∞ ). The purpose of what follows is to show that this remains true for the pair (QC, L∞ ).

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3 Symbol Analysis

If an approximate identity is asymptotically multiplicative on the pair (H ∞ , H ∞ ), then it is so on the pair (C + H ∞ , C + H ∞ ). Thus, in that case the argument of the proof of Lemma 2.87 gives its asymptotic multiplicativity on the pair (QC, L∞ ). In this connection and as motivation for our further investigations notice the following. 3.16. Remark. In 1988, Wolf and Havin [580] showed that shifts and contractions of the Poisson kernel, i.e. kernels of the form K(x) =

ε 1 , 2 π ε + (h − x)2

are the only approximate identities which are asymptotically multiplicative on the pair (H ∞ , H ∞ ). Our starting point is the following characterization of QC. 3.17. Theorem (Sarason). QC = L∞ ∩ V M O. Proof. Let a ∈ L∞ ∩V M O. By 1.48(h), there are u, v ∈ C such that a = u+ v). v ∈ H ∞ . Therefore, Hence v) = a − u ∈ L∞ , and it follows that v + i) a = −i(v + i) v ) + (u + iv) ∈ H ∞ + C. If a ∈ L∞ ∩ V M O, then obviously a ∈ L∞ ∩ V M O, and the same reasoning gives a ∈ H ∞ + C. Thus a ∈ QC. Conversely, let a ∈ QC. Then a = b + ic, where b and c are real-valued u) + (v + iw) with functions in QC. Since b ∈ H ∞ + C, we have b = (u + i) ) = −w, whence u + i) u ∈ H ∞ and v + iw ∈ C. Because b is real-valued, u ) u = −u ) = w, ) and it results that b = u + v = w ) + v, where w ∈ C and v ∈ C. Again by 1.48(h), b ∈ V M O. The same argument gives that c ∈ V M O. Thus a ∈ V M O ∩ L∞ . 3.18. Lemma. Let ϕ ∈ L1 and Φ(x) := ϕ(eix ), x ∈ R. Then (a) ϕ ∈ BM O ⇐⇒ Φ ∈ BM O(R); (b) cΦ∗ ≤ ϕ∗ ≤ Φ∗ with some absolute constant c; (c) ϕ ∈ V M O ⇐⇒ Φ ∈ V M O(R). Proof. (a), (b) It is clear that ϕ ∈ BM O if Φ ∈ BM O(R) and that then ϕ∗ ≤ Φ∗ . So let ϕ ∈ BM O and let J be a finite interval of R. Without loss of generality assume J = [−δ, 2πn], where 0 < δ < 2π and n ∈ N. Put I = [−δ, 0]. Then ( ( ( 2π 1 1 |Φ − Φ[0,2π] | dt = |Φ − Φ[0,2π] | dt + n |Φ − Φ[0,2π] | dt |J| J δ + 2πn I 0 and we have


n δ + 2πn

(

2π

|Φ − Φ[0,2π] | dt ≤ 0

1 2π

(

123

2π

|Φ − Φ[0,2π] | dt ≤ ϕ∗ . 0

Further, 1 δ + 2πn

( |Φ − Φ[0,2π] | dt ≤ I

1 δ

( |Φ − ΦI | dt + I

1 δ + 2πn

( |Φ[0,2π] − ΦI | dt, I

and the first term is clearly not greater than ϕ∗ . The second term equals 2π δ δ δ ≤ |Φ[−2π,0] − ΦI | = const · log ϕ∗ |Φ[−2π,0] − ΦI | δ + 2πn 2π δ 2π (1.48(r), (i), (ii)), which is not greater than const·ϕ∗ . Now 1.48(a) completes the proof. (c) This is immediate from (a), (b) together with 1.48(f), (p). 3.19. The moving average. This is the approximate identity generated by K(x) =

1 2π

for

x ∈ (−π, π),

K(x) = 0 for x ∈ / (−π, π).

Note that ( kλ,x Φ = (kλ Φ)(x) = =

∞ −∞

λ 2π

(

Φ(t)λK λ(x − t) dt x+π/λ

Φ(t) dt = Φ[x−π/λ,x+π/λ] . x−π/λ

Moreover, the “norm” Φ∗ on BM O(R) is nothing else than sup sup kλ,x (|Φ − kλ,x Φ|) λ>0 x∈R

and if Φ ∈ BM O(R), then, by definition, Φ ∈ V M O(R) ⇐⇒ lim sup kλ,x (|Φ − kλ,x Φ|) = 0. λ→∞ x∈R

(3.7)

The proof of the following proposition shows how (3.7) can be used to study asymptotic multiplicativity. 3.20. Proposition. The moving average is asymptotically multiplicative on the pair (QC, L∞ ). Proof. Let ϕ ∈ QC and a ∈ L∞ . Let Φ ∈ L∞ (R) and A ∈ L∞ (R) denote the periodic extensions of ϕ and a, respectively. Then with K as in 3.19,

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3 Symbol Analysis

|(kλ ΦA)(x) − (kλ Φ)(x)(kλ A)(x)| + +( ∞ + + + [Φ(t) − (kλ Φ)(x)]A(t)Kλ (x − t) dt++ =+ −∞ ( ∞ ≤ A∞ |Φ(t) − (kλ Φ)(x)|Kλ (x − t) dt −∞

= A∞ kλ,x (|Φ − kλ,x Φ|). Theorem 3.17 and Lemma 3.18 imply that Φ ∈ L∞ (R) ∩ V M O(R) and so the assertion follows from (3.7). 3.21. Theorem. Let {Kλ }λ∈Λ be an approximate identity with generating kernel K. For ϕ ∈ L1 , define ϕK := sup sup kλ,t (|ϕ − kλ,t ϕ|). λ∈Λ t∈T

Then, if ϕ ∈ L1 ,

ϕ ∈ BM O ⇐⇒ ϕK < ∞.

Moreover, there are constants c1 and c2 depending only on K (and Λ) such that c1 ϕK ≤ ϕ∗ ≤ c2 ϕK . Proof. Put Φ(x) = ϕ(eix ), x ∈ R, and let ΦK := sup sup kλ,x (|Φ − kλ,x Φ|). λ∈Λ x∈R

Since in this definition sup may be replaced by x∈R

sup , we have ΦK = ϕK . x∈[0,2π)

Recall that Λ is either {λ0 , λ0 + 1, λ0 + 2, . . .}, where λ0 ≥ 1 is an integer, or the interval (λ0 , ∞), where λ0 > 0. First suppose ΦK < ∞. Let I be any interval of R whose length is less than δ0 := min{π, π/λ0 }: I = (x − δ, x + δ), 0 < δ < δ0 /2. Due to (3.3) there is a constant m > 0 such that K(τ ) ≥ m for (almost all) τ ∈ (−π, π). Choose a λ1 ∈ Λ so that π/(4δ) < λ1 < π/δ (this is always possible). Then, if t ∈ I ⊂ (x − π/λ1 , x + π/λ1 ), we have Kλ1 (x − t) = λ1 K(λ1 (x − t)) ≥ λ1 m > hence Kλ1 (x − t) ≥

πm 1 χ(−δ,δ) (x − t) 2 |I|

πm , 4δ

∀ t ∈ R,

and it follows that ( ( ∞ 1 1 |Φ(t) − kλ1 ,x Φ| dt ≤ |Φ(t) − kλ1 ,x Φ|χ(−δ,δ) (x − t) dt |I| I |I| −∞ ( ∞ 2 2 ΦK . ≤ |Φ(t) − kλ1 ,x Φ|Kλ1 (x − t) dt ≤ πm −∞ πm


125

Thus, 1.48(a) implies that Mδ0 (ϕ) = const · ΦK and 1.48(b) shows that ϕ ∈ BM O and ϕ∗ ≤ c2 ΦK . Now suppose ϕ ∈ BM O. By virtue of Lemma 3.18, Φ ∈ BM O(R). Let x ∈ R, λ ∈ Λ, and put Ij = [x − 2j /λ, x + 2j /λ] for j ≥ 0. Note that |Ij | = 2j+1 /λ. We have ( ∞ ( |Φ(t) − ΦI0 |Kλ (x − t) dt ≤ |Φ(t) − ΦI0 |Kλ (x − t) dt −∞

+

+

I0

∞ (

|Φ(t) − ΦIj |Kλ (x − t) dt

j=1 Ij \Ij−1 ∞ ( j=1

|ΦIj − ΦI0 |Kλ (x − t) dt.

Ij \Ij−1

(3.8)

For the first term in (3.8) we get ( ( |Φ(t) − ΦI0 | Kλ (x − t) dt ≤ K∞ λ |Φ(t) − ΦI0 | dt I0 I0 ( 2 |Φ(t) − ΦI0 | dt ≤ const · Φ∗ . = K∞ |I0 | I0 In view of (3.4) we have max

2j−1 ≤λ|x−t|≤2j

K λ(x − t) ≤

M (2j−1 )2

=

4M . 22j

(3.9)

Therefore, the second term in (3.8) is not greater than ( ∞ 4M λ j=1

22j

|Φ(t) − ΦIj | dt = Ij

( ∞ 4M λ 2j+1 1 |Φ(t) − ΦIj | dt 22j λ |Ij | Ij j=1

≤ 8M Φ∗

∞ 1 = 8M Φ∗ . 2j j=1

Finally, again using (3.9) we see that the third term in (3.8) is not greater than ( ∞ ∞ 4M λ 8M |Φ − Φ | dt = |ΦIj − ΦI0 | I I j 0 2j 2 2j Ij j=1 j=1 ≤

∞ 8M j=1

2j

const · jΦ∗

= const · Φ∗ . Thus, what we have shown is that

(by 1.48(r), (ii))

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3 Symbol Analysis

(

∞

sup sup λ∈Λ x∈R

−∞

|Φ(t) − ΦI0 |Kλ (x − t) dt ≤ const · Φ∗ ,

where I0 = [x − 1/λ, x + 1/λ]. But ( ∞ ( |Φ(t) − kλ,x Φ|Kλ (x − t) dt ≤ −∞

∞

(3.10)

|Φ(t) − ΦI0 |Kλ (x − t) dt

−∞ ( ∞

+ −∞

|ΦI0 − kλ,x Φ|Kλ (x − t) dt

and the first integral herein admits estimate (3.10), while +( ∞ + + + + |ΦI0 − kλ,x Φ| = + ΦI0 − Φ(t) Kλ (x − t) dt++ −∞ ( ∞ ≤ |ΦI0 − Φ(t)|Kλ (x − t) dt ≤ const · Φ∗ , −∞

the last “≤” again by (3.10), whence ( ∞ ( |ΦI0 − kλ,x Φ|Kλ (x − t) dt ≤ const · Φ∗ −∞

Thus,

(

∞

sup sup λ∈Λ x∈R

−∞

∞ −∞

Kλ (x − t) dt = const · Φ∗ .

|Φ(t) − kλ,x Φ|Kλ (x − t) dt ≤ cΦ∗

with some constant c depending only on K and Λ. It follows that ϕK = ΦK < ∞ and Lemma 3.18(b) gives that c1 ϕK ≤ ϕ∗ . 3.22. Lemma. Let {Kλ }λ∈Λ be an approximate identity and let ϕ ∈ V M O. Then kλ ϕ ∈ V M O for all λ ∈ Λ and ϕ − kλ ϕ∗ → 0 as λ → ∞. Proof. Put Φ(x) := ϕ(eix ) for x ∈ R. Then Φ ∈ V M O(R) by Lemma 3.18. For y ∈ R, let Φy (x) := Φ(x − y). Since ( ∞ ( ∞ (kλ Φ)(x) = Φ(t)Kλ (x − t) dt = Φ(x − y)Kλ (y) dy −∞

−∞

and since, by 1.48(q), the mapping R → V M O(R), y → Φy is continuous, kλ Φ can be written as a Bochner integral: ( ∞ kλ Φ = Φy Kλ (y) dy. −∞

This implies that kλ Φ ∈ V M O(R), whence, by Lemma 3.18, kλ ϕ ∈ V M O. Furthermore, we also have ( ∞ Φ − kλ Φ = (Φ − Φy )Kλ (y) dy −∞


and thus

(

Φ − kλ Φ∗ ≤ (

∞ −∞

127

Φ − Φy ∗ Kλ (y) dy

= |y|δ

Φ − Φy ∗ Kλ (y) dy.

Given any ε > 0 there is, by virtue of 1.48(q), a δ > 0 such that ( ( ε ∞ ε Φ − Φy ∗ Kλ (y) dy ≤ Kλ (y) dy = , 2 2 |y| 0 such that ( ( ε Φ − Φy ∗ Kλ (y) dy ≤ 2Φ∗ Kλ (y) dy ≤ 2 |y|>δ |y|>δ whenever λ > λ . Application of Lemma 3.18 completes the proof.

3.23. Theorem. Every approximate identity is asymptotically multiplicative on the pair (QC, L∞ ). Proof. Let ϕ ∈ QC and a ∈ L∞ , denote the periodic extensions by Φ and A, and note that Φ ∈ L∞ (R) ∩ V M O(R) (Theorem 3.17 and Lemma 3.18) and A ∈ L∞ (R). As in the proof of Proposition 3.20 we see that sup |(kλ ΦA)(x) − (kλ Φ)(x)(kλ A)(x)| ≤ A∞ sup kλ,x (|Φ − kλ,x Φ|). (3.11) x∈R

x∈R

We have sup kλ,x |Φ − kλ,x Φ| ≤ sup kλ,x |Φ − kµ Φ − kλ,x (Φ − kµ Φ)| x∈R x∈R + sup kλ,x |kµ Φ − kµ,x Φ| x∈R + sup kλ,x |kµ,x Φ − kλ,x (kµ Φ)| .

(3.12)

x∈R

The first term in (3.12) is not greater than Φ − kµ ΦK ≤ (1/c1 )Φ − kµ Φ∗ (Theorem 3.21) and consequently, by Lemma 3.22, there is a µ0 ∈ Λ such that this term is smaller than ε/3 for µ = µ0 . For µ = µ0 , the second term in (3.12) is not greater than ( sup |(kµ0 Φ)(t) − (kµ0 Φ)(x)|Kλ (x − t) dt x∈R |x−t|δ

Because kµ0 Φ ∈ L∞ (R) is uniformly continuous (by 3.14(a)), there is a δ > 0 such that the first term in (3.13) is smaller than ε/6, and having chosen this

128

3 Symbol Analysis

δ, we can find a λ1 ∈ Λ such that the second term in (3.13) becomes smaller than ε/6 for all λ > λ1 . Finally, for µ = µ0 , the third term in (3.12) is smaller than ε/3 for all λ > λ2 , since kλ f − f ∞ → 0 as λ → ∞ in case f ∈ C. Thus, the right-hand side of (3.11) is smaller than A∞ times an arbitrarily given ε > 0 whenever λ > max{λ1 , λ2 }. But this is the assertion.

3.3 Piecewise Quasicontinuous Functions We first state some results on the C ∗ -algebra P C of all piecewise continuous functions on T (recall 2.73). 3.24. Proposition. The maximal ideal space of P C is T × {0, 1} and the Gelfand map Γ : P C → C(T × {0, 1}) is given by (Γ f )(τ, 0) = f (τ − 0),

(Γ f )(τ, 1) = f (τ + 0).

An open neighborhood base of (τ, 0) is formed by the sets −iε (τ e , τ ] × {0} ∪ (τ e−iε , τ ) × {1} , 0 < ε < π, and an open neighborhood base of (τ, 1) is formed by the sets [τ, τ eiε ) × {1} ∪ (τ, τ eiε ) × {0} , 0 < ε < π, where (a, b) denotes the open subarc of T whose endpoints are a and b and whose length is less than π, [a, b) := {a} ∪ (a, b), (a, b] := (a, b) ∪ {b}. Proof. It is clear that (τ, 0) and (τ, 1) are in M (P C). Conversely, let v be in M (P C). Then v belongs to some fiber Mτ (P C) of M (P C) over τ ∈ M (C) = T. Every function f ∈ P C can be written as f = cχτ +g, where c ∈ C, χτ is the characteristic function of the arc (τ, τ eiπ/2 ), and g is a function in P C which is continuous at τ . The spectrum spP C (χτ ) is obviously the doubleton {0, 1}. Therefore v(χτ ) must be equal either 0 or 1. If v(χτ ) = 0, then v(f ) = g(τ ) = f (τ − 0) for all f ∈ P C, and if v(χτ ) = 1, then v(f ) = c + g(τ ) = f (τ + 0) for all f ∈ P C. Thus, Mτ (P C) is the doubleton {(τ, 0), (τ, 1)}. The assertion concerning the Gelfand topology of M (P C) can be checked straightforwardly. 3.25. Proposition. Let τ ∈ T and let F be a closed subset of Xτ = Mτ (L∞ ). Then the restriction algebra P C|F is either isometrically isomorphic to the complex field C or is a singly generated C ∗ -algebra whose maximal ideal space is the doubleton {0, 1} with the discrete topology. Proof. Every f ∈ P C can be written as f = cχτ + g, where c, χτ , g are as in the proof of the preceding proposition. Let χτF denote the restriction χτ |F . Then f |F = cχτF + g(τ ), and therefore P C|F coincides with the algebra of all functions of the form cχτF + d, where c, d ∈ C. If χτF = const on F , then

3.3 Piecewise Quasicontinuous Functions

129

P C|F ∼ = C. If χτF is not constant on F , then the range of χτF is {0, 1}, and it is easily seen that P C|F = {cχτF + d : c, d ∈ C} is closed (hence, a C ∗ -algebra), that it is generated by χτF , and that the spectrum of χτF in P C|F is {0, 1}. It remains to recall 1.19. 3.26. Lemma. Let χU be the characteristic function of the upper half-circle {eiθ : 0 < θ < π} and put H(x) := χU (eix ) (x ∈ R). Let K be an approximate identity. Then for every µ ∈ (0, 1) there is a ν ∈ R such that (kλ H)(ν/λ) → µ as λ → ∞. Proof. For definiteness, assume µ ∈ [1/2, 1). Then there is a ν ≥ 0 such that *ν K(x) dx = µ, and we have, as λ → ∞, −∞ ν ν ( (2n+1)π ( ν/λ−2nπ − x dx = = Kλ Kλ (x) dx (kλ H) λ λ n∈Z 2nπ n∈Z ν/λ−(2n+1)π ( ν ( ν/λ Kλ (x) dx + o(1) = K(x) dx + o(1) = (

ν/λ−π ν

=

ν−λπ

K(x) dx + o(1) = µ + o(1). −∞

Our next concern is to provide some information about the C ∗ -algebra QC of all quasicontinuous functions on T (see 2.80). 3.27. Lemma. Let {Kλ }λ∈Λ be an approximate identity and let ν ∈ R. Then, for ϕ ∈ QC, ( π/λ ν λ Φ(x) dx → 0 (kλ Φ) − λ 2π −π/λ

as

λ → ∞,

where Φ(x) = ϕ(eix ) (x ∈ R). Proof. Let ε > 0 be given arbitrarily. By virtue of Theorem 3.23 there is a λ0 ∈ Λ such that ( ∞+ ν +2 ν ν + ν +2 + + + + − x dx = (kλ |Φ|2 ) − +(kλ Φ) +Φ(x) − (kλ Φ) + Kλ + λ λ λ λ −∞ ≤ kλ (ϕϕ) − (kλ ϕ)(kλ ϕ)∞ < ε2 for all λ ∈ Λ, λ > λ0 . Hence, by the Cauchy-Schwarz inequality, ( ∞+ ν + ν + + − x dx < ε (3.14) +Φ(x) − (kλ Φ) +Kλ λ λ −∞ for λ > λ0 . Let Iλ denote the interval λν − πλ , λν + πλ . Then (recall the proof of Theorem 3.21)

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3 Symbol Analysis

Kλ

ν

ν πm 1 −x ≥ χ(−π/λ,π/λ) −x ∀ x ∈ R. λ 2 |Iλ | λ

Thus, we obtain from (3.14) that ( + ν + 1 2 + + ε + dx < +Φ(x) − (kλ Φ) |Iλ | Iλ λ πm Because

∀ λ > λ0 .

( + + ν + ν + 1 + + + + +Φ(x) − (kλ Φ) +ΦIλ − (kλ Φ) +≤ + dx, λ |Iλ | Iλ λ

ν (kλ Φ) − ΦIλ → 0 as λ → ∞. (3.15) λ Taking into account that Φ ∈ V M O(R) (Theorem 3.17 and Lemma 3.18) and using 1.48(r), (iii), it is not difficult to see that we deduce that

Φ(−π/λ,π/λ) − ΦIλ → 0 as

λ → ∞.

(3.16)

On combining (3.15) and (3.16) we get the assertion. 3.28. M (QC). Let A be a C ∗ -subalgebra of L∞ (note that QC is a C ∗ subalgebra of L∞ ). Clearly, M (A) can be regarded as a subset of A∗ , the dual of A. We shall always think of A∗ as being equipped with its weak-star topology. Let {Kλ }λ∈Λ be an approximate identity. Each µ = (λ, t) ∈ Λ × T induces a functional δµ ∈ A∗ given by δµ : A → C,

a → (kλ a)(t),

and therefore Λ × T may be viewed as a subset of A∗ . By virtue of 3.14(b), Λ × T is contained in the unit ball of A∗ . If Λ = (r0 , ∞), then Λ × T can be identified with a circular annulus or a punctured disk in a natural way, and if Λ = {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N), then Λ × T is the countable disjoint union of copies of the circle T. The weak-star closure of a set S ⊂ A∗ will be denoted by closA∗ S. 3.29. Proposition. Let A be a C ∗ -algebra of L∞ and let {Kλ }λ∈Λ be an approximate identity. Then (a) M (A) ⊂ closA∗ (Λ × T),

(b) if C ⊂ A ⊂ QC, one has M (A) = closA∗ (Λ × T) \ (Λ × T). Proof. (a) Let ξ0 ∈ M (A). Any A∗ -neighborhood of ξ0 is of the form U = Uε;a1 ,...,an (ξ0 ) = ξ ∈ A∗ : |ξ(ai ) − ξ0 (ai )| < ε ∀ i = 1, . . . , n ,


131

where ε > 0 and a1 , . . . , an ∈ A. We must show that there is a µ ∈ Λ × T such that δµ ∈ U . Put a = |a1 − ξ0 (a1 )| + . . . + |an − ξ0 (an )|. Since A is a / GA, C ∗ -algebra, a belongs to A. By construction, ξ0 (a) = 0. Therefore a ∈ hence a ∈ / GL∞ , so ess inf |a(t)| = 0. It follows that there is a sequence t∈T

{µn } = {(λn , tn )} ⊂ Λ × T such that λn → ∞ and δµn a → 0 as n → ∞ (this is an immediate consequence of Corollary 3.57 below; this claim also results from the fact that, for a ∈ L∞ , (kλ a)(t) → a(t) a.e. on T as λ → ∞, for whose proof see, e.g., Ahiezer [4, pp. 133–137]). Since |ai − ξ0 (ai )| ≤ a on T for each i, we have |σµ ai − ξ0 (ai )| ≤ δµ a for all µ ∈ Λ × T and each i. The conclusion is that there exists a µ0 ∈ Λ × T such that |δµ0 ai − ξ0 (ai )| ≤ δµ0 a < ε for each i. But this is the assertion. (b) It is clear that δµ ∈ / M (A) for µ ∈ Λ × T, since, by (3.5), 1 −1 K [kλ (χ−1 χ1 )](eix ) − (kλ χ−1 )(eix )(kλ χ1 )(eix ) = 1 − K > 0. λ λ Now let ξ0 ∈ closA∗ (Λ × T) \ (Λ × T), let a ∈ A, b ∈ A, and ε > 0. By Theorem 3.23, there is a λ0 ∈ Λ such that sup sup |δλ,t (ab) − δλ,t (a)δλ,t (b)| < ε. λ>λ0 t∈T

Then choose a µ1 = (λ1 , t1 ) ∈ Λ × T so that λ1 > λ0 and δµ1 ∈ Uε;a,b,ab (ξ0 ) (the proof of part (a) shows that this is always possible). We have |ξ0 (ab) − ξ0 (a)ξ0 (b)| ≤ |ξ0 (ab) − δµ1 (ab)| + |δµ1 (ab) − δµ1 (a)δµ1 (b)| +|δµ1 (a) − ξ0 (a)| |δµ1 (b)| + |ξ0 (a)| |δµ1 (b) − ξ0 (b)| ≤ ε + ε + ε|δµ1 (b)| + |ξ0 (a)|ε and since ε > 0 can be chosen arbitrarily, we get ξ0 (ab) = ξ0 (a)ξ0 (b), that is, ξ0 ∈ M (A). Remark 1. In particular, taking the approximate identity generated by the Poisson kernel, we see that the open unit disk D can be naturally identified with a subset of A∗ and that, for C ⊂ A ⊂ QC, M (A) is the weak-star closure of D minus D. Remark 2. Notice how simple the things are for C ∗ -algebras. The unit disk D can be identified with a subset M (H ∞ ) via the harmonic extension (which is multiplicative on H ∞ ). That M (H ∞ ) is contained in the weak-star closure of D (i.e., that M (H ∞ ) ⊂ clos(H ∞ )∗ D, whence, obviously, M (H ∞ ) = clos(H ∞ )∗ D) is Carleson’s corona theorem! 3.30. QC fibers over M (C). For τ ∈ T = M (C), let Mτ+ (QC) (resp. Mτ− (QC)) denote the set of all ξ ∈ Mτ (QC) such that ϕ(ξ) = 0 whenever ϕ ∈ QC and lim sup |ϕ(t)| = 0 (resp. lim sup |ϕ(t)| = 0). For a function a ∈ L1 t→τ +0

t→τ −0

and numbers τ = eiθ ∈ T, λ ∈ (1, ∞) define

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3 Symbol Analysis

(mλ a)(τ ) =

λ 2π

(

θ+π/λ

a(eix ) dx.

(3.17)

θ−π/λ

Thus, {mλ a}λ∈(1,∞) arises from a by applying the moving average. In accordance with 3.28, we may identify (1, ∞) × {τ } with a subset of QC ∗ : with (λ, τ ) ∈ (1, ∞) × {τ } we associate the functional δ(λ,τ ) : QC → C,

a → (mλ a)(τ ).

Let Mτ0 (QC) denote the points in M (QC) that lie in the weak-star closure of (1, ∞) × {τ }, i.e., Mτ0 (QC) = M (QC) ∩ closQC ∗ (1, ∞) × {τ } . It is clear that Mτ0 (QC) is a compact subset of the fiber Mτ (QC). Now let {Kλ }λ∈(1,∞) be any approximate identity. For fixed ν ∈ R, the set Kν ⊂ (1, ∞) × R consisting, by definition, of all ordered pairs of the form (λ, ν/λ) (λ ∈ (1, ∞)) may be viewed as a subset of QC ∗ by identifying (λ, ν/λ) with the functional ν , δ(λ,ν/λ) : QC → C, a → (kλ A) λ where A(x) := a(eix ) (x ∈ R). Note that if {Kλ }λ∈(1,∞) is the moving average and ν = 0, then Kν is just the set (1, ∞) × {1} considered in the preceding paragraph. The following lemma shows that the points in M (QC) which lie in the weak-star closure of Kν are just the points in M10 (QC). 3.31. Lemma. M (QC) ∩ closQC ∗ Kν = M10 (QC). Proof. If ξ ∈ M10 (QC), then for each QC ∗ -neighborhood U = Uε;ϕ1 ,...,ϕn (ξ) = η ∈ QC ∗ : |η(ϕj ) − ϕj (ξ)| < ε ∀ j = 1, . . . , n there is a λ1 ∈ (1, ∞) such that δ(λ1 ,1) ∈ U , i.e., + + ( + + λ1 π/λ1 + + Φj (x) dx+ < ε ∀ j = 1, . . . , n, +ϕj (ξ) − + + 2π −π/λ1

(3.18)

(3.19)

where Φj (x) = ϕj (eix ). So Lemma 3.27 implies that there is a λ2 ∈ (1, ∞) such that + ν + + + (3.20) +ϕj (ξ) − (kλ2 Φj ) + < 2ε ∀ j = 1, . . . , n. λ2 This shows that ξ is in the weak-star closure of Kν . Conversely, let ξ ∈ M (QC) be in the weak-star closure of Kν . Then obviously ξ ∈ M1 (QC). Given any neighborhood U of ξ of the form (3.18), there is a λ2 ∈ (1, ∞) satisfying (3.20) with ε in place of 2ε. Again by Lemma 3.27, there exists a λ1 ∈ (1, ∞) such that (3.19) holds with 2ε in place of ε. This implies that ξ ∈ M10 (QC).


133

3.32. Definition. For ϕ ∈ L1 and τ = eiθ ∈ T, the integral gap γτ (ϕ) of ϕ at τ is defined by + ( + ( θ + 1 θ+δ + 1 + + γτ (ϕ) := lim sup + ϕ(eix ) dx − ϕ(eix ) dx+ . + + δ δ δ→0+0 θ θ−δ If J is a subarc of T, we define V M O(J) in the natural manner (see 1.47 for J = T). As before, V M O := V M O(T). 3.33. Lemma. (a) If ϕ ∈ V M O, then γτ (ϕ) = 0 for each τ ∈ T. (b) If ϕ ∈ V M O(a, τ ) ∩ V M O(τ, b) and γτ (ϕ) = 0, then ϕ ∈ V M O(a, b). (c) If ϕ ∈ QC, ϕ|Mτ0 (QC) = 0, and if p ∈ P C, then γτ (pϕ) = 0. Proof. (a) Let Φ(x) = ϕ(eix ) (x ∈ R). Without loss of generality assume τ = 1. Then, by Lemma 3.18 and 1.48(r), (iii), + ( + ( +1 δ + 1 0 + + Φ(x) dx − Φ(x) dx+ ≤ const · M2δ (Φ) = o(1) + +δ 0 + δ −δ as δ → 0, which is the assertion. (b) Let τ = eiθ . Fix ε > 0 and choose c > 0 so that ( 1 |ϕ − ϕI | dm < ε |I| I whenever I is a subarc of (a, τ ) or of (τ, b) satisfying |I| < c, and so that + ( + ( + 1 θ+δ + 1 θ + + Φ(x) dx − Φ(x) dx+ < ε + +δ θ + δ θ−δ whenever δ < c. We show that if I is any subarc (a, b) such that |I| < c, then ( 1 |ϕ − ϕI | dm < 6ε. |I| I It suffices to consider the case where τ ∈ I. First let τ be the center of I. Put I− = I ∩ (a, τ ) and I+ = I ∩ (τ, b). Then |ϕI+ − ϕI− | < ε and ϕI = (1/2)(ϕI+ + ϕI− ), so that |ϕI± − ϕI | < ε/2. Consequently, ( ( ( 1 1 1 |ϕ − ϕI | dm = |ϕ − ϕI | dm + |ϕ − ϕI | dm |I| I 2|I+ | I+ 2|I− | I− ( ( 1 1 ε ε |ϕ − ϕI+ | dm + |ϕ − ϕI− | dm, < + + 4 4 2|I+ | I+ 2|I− | I− which is less than 3ε/2. In fact this is also true for the case where |I| < 2c.

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3 Symbol Analysis

Now suppose I is any subarc of (a, b) containing τ , and let I0 be the smallest subarc of T that contains I and whose center is τ . By choosing c > 0 sufficiently small we can guarantee that I0 ⊂ (a, b). So the preceding estimate applies to I0 , and therefore ( ( 1 1 |ϕ − ϕI0 | dm ≤ |ϕ − ϕI0 | dm < 3ε. |I| I 2|I0 | I0 Thus, + ( + ( ( + 1 + 1 1 + + ϕ dm − ϕI dm+ ≤ |ϕ − ϕI0 | dm < 3ε. |ϕI − ϕI0 | = + |I| I |I| I 0 |I| I * The two preceding inequalities give (1/|I|) I |ϕ − ϕI | dm < 6ε, as desired. (c) Without loss of generality assume τ = 1. Every p ∈ P C can be written as p = cχ + g, where c ∈ C, χ is the characteristic function of the upper half-circle, and g ∈ P C is continuous at τ = 1. By Theorem 3.17 and (a), γ1 (gϕ) = 0. It remains to show that γ1 (χϕ) = 0. We have + ( + +2 δ + + + Φ(x) dx+ 2γ1 (χϕ) = lim sup + + δ→0 + δ 0 + ( + ( +1 δ + 1 0 + + ≤ lim sup + Φ(x) dx − Φ(x) dx+ + + δ δ δ→0 0 −δ + ( + +1 + δ + + +2 lim sup + Φ(x) dx+ . (3.21) + δ→0 + 2δ −δ The first term equals γ1 (ϕ), which is zero by (a). Let + ( + + + +1 + 1 ( δn + + δ + + + + Φ(x) dx+ = lim + Φ(x) dx+ . lim sup + n→∞ + + + + 2δ 2δ n −δn δ→0 −δ By the compactness of the unit ball of QC ∗ in the weak-star topology, there are a ξ ∈ M10 (QC) and a subsequence {δnk } of {δn } such that + + + 1 ( δnk + + + lim + Φ(x) dx+ = lim (mπ/δnk ϕ)(1) = ϕ(ξ) = 0, + k→∞ k→∞ + 2δnk −δ n k

i.e., the second term of (3.21) is zero, too.

3.34. Proposition (Sarason). If τ ∈ T, then Mτ+ (QC) ∩ Mτ− (QC) = Mτ0 (QC),

Mτ+ (QC) ∪ Mτ− (QC) = Mτ (QC).


135

Proof. Without loss of generality assume τ = 1. Let ξ ∈ M10 (QC). If ϕ ∈ QC and lim sup |ϕ(t)| = 0, then t→1+0

+ ( + +1 δ + + + lim sup + Φ(x) dx+ = 0, + δ→0 + δ 0

whence, by Lemma 3.33(a) and Theorem 3.17, + ( 0 + +1 + + lim sup + Φ(x) dx++ = 0. δ −δ δ→0 + + ( + +1 δ + + lim sup + Φ(x) dx+ = 0, + + 2δ δ→0 −δ

So

and therefore

+ ( + +1 + δ + + |ϕ(ξ)| ≤ lim sup + Φ(x) dx+ = 0. + δ→0 + 2δ −δ

− It follows that ξ ∈ M1+ (QC). It can be shown similarly that ξ ∈ M 1 (QC). / M1− (QC) ∩ Now suppose ξ ∈ M1 (QC) \ M10 (QC). We show that then ξ ∈ M1+ (QC) ; this will give the first equality in our proposition. There is a ϕ in QC such that ϕ(ξ) = 0 and ϕ|M10 (QC) = 0. Let p ∈ P C be continuous except for a jump at 1, with p(1 + 0) = 1 and p(1 − 0) = 0. Then pϕ ∈ V M O(T \ {1}) and Lemma 3.33(c) shows that γ1 (pϕ) = 0. Therefore, by Lemma 3.33(b) and Theorem 3.17, pϕ is in QC, and hence so also is (1 − p)ϕ. The function pϕ vanishes on M1− (QC) while (1 − p)ϕ vanishes on M1+ (QC). But since ϕ = pϕ + (1 − p)ϕ and ϕ(ξ) = 0, it is impossible for ξ to belong to both M1− (QC) and M1+ (QC). To establish the second equality, suppose ξ ∈ M1 (QC) \ M1+ (QC). Then there is a ϕ ∈ QC such that 0 ≤ ϕ ≤ 1, ϕ(ξ) = 0, ϕ|M1+ (QC) = 0. We claim that lim sup ϕ(t) = 0. To see this, let x := lim sup ϕ(t) and notice first that t→1+0

t→1+0

x = lim sup ϕ(η) : η ∈ Mt (QC), t ∈ (1, eiθ ) . θ→0

It follows that there is a sequence {ηn } ∈ Mtn (QC) such that tn → 1 and ϕ(ηn ) → x. Due to the compactness of M (QC), there are a subsequence of {ηn }, again denoted by {ηn }, and an η ∈ M1 (QC) such that ηn → η in the weak-star topology. If ϕ ∈ QC and lim sup |f (t)| = 0, then f (η) = t→1+0

lim f (ηn ) = 0. Thus, η ∈ M1+ (QC). This shows that ϕ(η) = 0, and since

n→∞

both x and ϕ(η) are limits of ϕ(ηn ), we conclude that x = 0, as desired. Now let ψ be any function in QC with lim sup |ψ(t)| = 0. Then ϕψ is continuous at t→1−0

1 and takes the value 0 there. It follows that ϕψ|M1 (QC) = 0, in particular, ϕ(ξ)ψ(ξ) = 0. Because ϕ(ξ) = 0, we get ψ(ξ) = 0. Thus, ξ ∈ M1− (QC), and the proof is complete.

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3 Symbol Analysis

3.35. P QC. This is the smallest closed subalgebra of L∞ containing P C and QC, i.e., P QC = alg (P C, QC). Note that P QC is a C ∗ -subalgebra of L∞ . The functions in PQC are referred to as piecewise quasicontinuous functions.

Let P QC0 denote the collection of all finite sums of the form i pi qi , where pi ∈ P C0 and qi ∈ QC. If ξ ∈ M (QC), then, by 1.27(b), the fiber Mξ (P QC) is not empty. Given y ∈ Mξ (P QC) put τ = y|C and v = y|P C (clearly, τ = ξ|C = v|C). Since v belongs to Mτ (P C) and since Mτ (P C) is the doubleton {(τ, 0), (τ, 1)} (Proposition 3.24), we have either g(y) = pi (τ − 0)qi (ξ) ∀ g = pi qi ∈ P QC0 (3.22) i

or g(y) =

i

pi (τ + 0)qi (ξ)

∀g=

i

pi qi ∈ P QC0 .

(3.23)

i

Hence, if Mξ (P QC) would contain three distinct functionals, then two of them would coincide on P QC0 , and since P QC0 is dense in P QC, those two functionals would also coincide on P QC. The conclusion is that Mξ (P QC) contains at most two points. There is a natural mapping w : M (P QC) → M (QC) × {0, 1}, which is given as follows: for y ∈ M (P QC) let ξ = y|QC, τ = y|C, and v = y|P C; if v = (τ, 0) (resp. v = (τ, 1)), i.e., if y satisfies (3.22) (resp. (3.23)), define w(y) = (ξ, 0) (resp. w(y) = (ξ, 1)). This mapping is clearly one-to-one and therefore M (P QC) may be identified with a subset of the set M (QC)×{0, 1}. 3.36. Theorem (Sarason). Let ξ ∈ M (QC). Then (a) a(Mξ (L∞ )) = a(Mξ (P QC)) for all a ∈ P QC; (b) P QC|Mξ (L∞ ) = P C|Mξ (L∞ ); (c) Mξ (P QC) = {(ξ, 0)} for ξ ∈ Mτ− (QC) \ Mτ0 (QC), Mξ (P QC) = {(ξ, 1)} for ξ ∈ Mτ+ (QC) \ Mτ0 (QC); (d) Mξ (P QC) = {(ξ, 0), (ξ, 1)} for ξ ∈ M 0 (QC), and if {λn } ⊂ (1, ∞) is any sequence such that (λn , τ ) → ξ in the weak-star topology on QC ∗ , then for every a ∈ P QC the limits λn n→∞ π

(

θ0

lim

θ0 −π/λn

a(eix ) dx,

λn n→∞ π

(

θ0 +π/λn

lim

a(eix ) dx,

(3.24)

θ0

where τ = eiθ0 , exist and are equal to a(ξ, 0) and a(ξ, 1), respectively. Proof. (a) For x ∈ Mξ (L∞ ), put y = x|P QC. It is clear that y ∈ Mξ (P QC). Therefore, if a ∈ P QC, then a(x) = x(a) = y(a) = a(y) and so a(Mξ (L∞ )) ⊂ a(Mξ (P QC)). On the other hand, by 1.20(b) and 1.26(c), each functional y ∈ Mξ (P QC) extends to a functional x ∈ Mξ (L∞ ). So the same argument as above shows that a(Mξ (P QC)) ⊂ a(Mξ (L∞ )).


137

(b) If g = i pi qi ∈ P QC0 and x ∈ Mξ (L∞ ), then g(x) = i pi (x)qi (ξ), hence qi (ξ)(pi |Mξ (L∞ )) ∈ P C|Mξ (L∞ ). g|Mξ (L∞ ) = i

If a ∈ P QC, then there are gn ∈ P QC0 such that a − gn ∞ → 0 as n → ∞, and since P C|Mξ (L∞ ) is closed (Proposition 3.25), it follows that a|Mξ (L∞ ) is in P C|Mξ (L∞ ). (c) Let y ∈ Mξ (P QC) and suppose (3.22) holds. We show that then ξ is in Mτ− (QC). Let ϕ ∈ QC and assume lim sup |ϕ(t)| = 0. If p ∈ P C is continuous t→τ −0

except for a jump at τ with p(τ − 0) = 1 and p(τ + 0) = 0, then obviously, γτ (pϕ) = 0, and so Lemma 3.33(b) and Theorem 3.17 imply that pϕ ∈ QC. Since pϕ is continuous at τ and takes the value 0 there, we have (pϕ)(ξ) = 0. Because of (3.22), (pϕ)(ξ) = (pϕ)(y) = p(τ − 0)ϕ(ξ) = ϕ(ξ), and it follows that ϕ(ξ) = 0. Thus ξ ∈ Mτ− (QC). / Mξ (P QC). This Consequently, if ξ ∈ Mτ (QC) \ Mτ− (QC), then (ξ, 0) ∈ and Proposition 3.34 prove the second equality of (c). The first equality can be proved analogously. (d) Without loss of generality assume τ = 1. If a ∈ QC, then λn n→∞ 2π

(

π/λn

a(ξ) = lim hence

a(eix ) dx,

−π/λn

+ + + λ ( π/λn + + n + ix lim sup + a(e ) dx − a(ξ)+ + n→∞ + π 0 + + ( ( π/λn +λ + π/λn λ + n + n ≤ lim sup + a(eix ) dx − a(eix ) dx+ + + π 2π n→∞ 0 −π/λn + + + λ ( π/λn + + n + + lim sup + a(eix ) dx − a(ξ)+ + n→∞ + 2π −π/λn =

1 γ1 (a) + 0 = 0 (by 3.33(a)), 2

(3.25)

which implies that the second limit in (3.24) exists and equals a(ξ). If a ∈ P C0 , then obviously ( λn π/λn lim a(eix ) dx = a(τ + 0). (3.26) n→∞ π 0

To see that the second limit in (3.24) exists for any a = i pi qi ∈ P QC0 1 (pi ∈ P C0 , qi ∈ QC), note first that, for any ϕ ∈ L ,

138

3 Symbol Analysis

λn π

(

π/λn

ϕ(eix ) dx = (m2λn Φ)(δn ),

0

where δn := π/(2λn ), Φ denotes the periodic extension of ϕ, and we make use of notation (3.17). So + + + + + + pi (τ + 0)qi (ξ)+ lim sup +(m2λn A)(δn ) − + + n→∞ i ≤ lim sup |(m2λn Pi Qi )(δn ) − (m2λn Pi )(δn )(m2λn Qi )(δn )| i

+

i

n→∞

lim sup |(m2λn Pi )(δn )(m2λn Qi )(δn ) − pi (τ + 0)qi (ξ)| n→∞

and the first upper limit on the right is zero by virtue of the asymptotic multiplicativity of the moving average on the pair (L∞ , QC) (Proposition 3.20), while the second upper limit equals zero due to (3.25) and (3.26). It follows that the limit ( λn π/λn a(eix ) dx (3.27) y(a) := lim n→∞ π 0 exists for every a ∈ P QC0 and that (3.23) holds. This shows that y is linear and multiplicative on P QC0 . Since obviously |y(a)| ≤ a∞ for all a ∈ P QC0 , y extends to a linear and multiplicative (bounded) functional y) on P QC, and (3.23) implies that y) = (ξ, 1). Thus, (ξ, 1) ∈ Mξ (P QC). Finally, if a ∈ P QC and b ∈ P QC0 , then + + ( + + λn π/λn + + a(eix ) dx+ lim sup +y)(a) − + + π n→∞ 0 + + ( + + λn π/λn + + ix ≤ |) y (a) − y)(b)| + lim sup +y)(b) − b(e ) dx+ + π 0 n→∞ + + + ( +λ + π/λn + n + + lim sup + [b(eix ) − a(eix )] dx+ + n→∞ + π 0 ≤ a − b∞ + 0 + b − a∞ = 2a − b∞ and a − b∞ can be made as small as desired. The conclusion is that the limit (3.27) exists for every a ∈ P QC and that it is equal to y)(a) = a(ξ, 1). This completes the proof for (ξ, 1). The proof for (ξ, 0) is analogous. Remark 1. We observed in 3.35 that Mξ (P QC) contains at most two points. This and part (a) of the above theorem imply that a function in P QC takes at most two values on each fiber Mξ (L∞ ). The same is true for functions in (P QC)N ×N .


139

Remark 2. One can show that Mτ0 (QC) is a proper subset of Mτ (QC) (Sarason [456, p. 824]). This shows that the mapping w introduced in 3.35 is not onto. Remark 3. We saw in 3.35 that M (P QC) can be identified with a subset of M (QC)×{0, 1}. The preceding theorem shows which points of M (QC)×{0, 1} belong to M (P QC). The Gelfand topology on M (P QC) can now be described as follows. For ξ ∈ M (QC) let V(ξ) denote the family of open neighborhoods of ξ. For ξ ∈ Mτ (QC) and V ∈ V(ξ) let Vτ = V ∩ Mτ (QC) and let Vτ+ and Vτ− denote the sets of points in V that lie above the semicircles {eiθ : arg τ < θ < arg τ + π} and {eiθ : arg τ − π < θ < arg τ }, respectively. Then, if ξ ∈ Mτ+ (QC), the sets 0 1 (Vτ × {1}) ∪ (Vτ+ × {0, 1}) ∩ M (P QC), V ∈ V(ξ), form an open neighborhood base for (ξ, 1). If ξ ∈ Mτ− (QC), the sets 0 1 (Vτ × {0}) ∪ (Vτ− × {0, 1}) ∩ M (P QC), V ∈ V(ξ) form an open neighborhood base for (ξ, 0). 3.37. Pn QC. The collection of all matrix functions a ∈ L∞ N ×N which have the property that a(Mξ (L∞ )) contains at most n points for each ξ ∈ M (QC) will be denoted by (Pn QC)N,N . In the case N = 1 we shall write Pn QC in place of (Pn QC)1,1 . From (2.43) we know that (P1 QC)N,N = QCN ×N . It is obvious that P2 C ⊂ P2 QC and from Remark 1 in the preceding section it follows that P QC ⊂ P2 QC. Let a ∈ P QC and let m be any positive real number. We claim that the set {τ ∈ T : γτ (a) > m} is finite. Indeed, there is a function b ∈ P QC0 such that

a − b∞ < m/2, so γτ (b) > m/2 whenever γτ (a) > m. But b is a finite sum i pi qi , where pi ∈ P C0 and qi ∈ QC. This implies that γτ (b) = 0 for all τ at which all pi ’s are continuous and this proves our claim. Now let χE be the characteristic function of the set E=

∞ / n=1

En ,

En = eiθ ∈ T :

1 22n+1

< |θ|
0 such that " " " " n m m " " n " " "i " " ajk " ≤ γ " i(ajk )" (3.28) " " j=1 k=1

B

j=1 k=1

B

for every finite collection of elements ajk ∈ A. Any submultiplicative mapping i : A → B which satisfies (3.28) will also be called γ-submultiplicative. If i is a γ-submultiplicative quasi-embedding then ie is obviously a γ-submultiplicative embedding. 3.39. Definitions. Let Λ be either of the sets {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N) or (r0 , ∞) (r0 ∈ R+ ). We let A∞ N,N denote the collection of all (generalized) sequences {aλ }λ∈Λ of continuous matrix functions aλ ∈ CN ×N such that < ∞: sup aλ L∞ N ×N

λ∈Λ

A∞ N,N

! # ∞ := {aλ }λ∈Λ : aλ ∈ CN ×N , sup aλ LN ×N < ∞ . λ∈Λ

On defining α{aλ } := {αaλ }, {aλ } + {bλ } := {aλ + bλ }, {aλ }{bλ } := {aλ bλ }, {aλ }∗ := {a∗λ },

3.4 Harmonic Approximation: Algebraization

{aλ } := sup aλ L∞ := sup M (aλ )L(L2N ) N ×N λ∈Λ

141

(3.29)

λ∈Λ

∗ we make A∞ N,N become a C -algebra. Put

AN,N :=

∞ {aλ } ∈ A∞ N,N : there exists an a ∈ LN ×N such that

M (aλ ) → M (a), M ∗ (aλ ) → M ∗ (a) strongly on L2N as λ → ∞ .

Here the asterisk refers to the adjoint operator. It can be checked straightfor∞ ∞ wardly that AN,N is a C ∗ -subalgebra of A∞ N,N . Denote A1,1 and A1,1 by A ∞ and A, respectively. Because (3.29) is an admissible norm in AN,N and AN,N ∞ (recall 1.29), we have A∞ N,N = AN ×N and AN,N = AN ×N . Therefore we shall ∞ henceforth AN,N and AN,N denote by A∞ N ×N and AN ×N , respectively. If Λ = (r0 , ∞), then A∞ N ×N and AN ×N can be thought of as algebras of matrix functions given on an annulus or a punctured disk. However, for certain reasons it will be more advantageous to work with algebras of generalized sequences, although for a moment this seems to be an unnecessary complication. Finally, given an approximate identity {Kλ }λ∈Λ define kλ a for a = ∞ N (ajk )N j,k=1 ∈ LN ×N as (kλ ajk )j,k=1 . 3.40. Proposition. Let {Kλ }λ∈Λ be an approximate identity. (a) If a ∈ L∞ N ×N , then {kλ a}λ∈Λ ∈ AN ×N . (b) The mapping K : L∞ N ×N → AN ×N , a → {kλ a}λ∈Λ is a 1-submultiplicative isometry. Proof. (a) First let N = 1. From 3.14(a), (b) we deduce that {kλ a} ∈ A∞ and sup M (kλ a)L(L2 ) ≤ M (a)L(L2 )

(3.30)

λ

for every a ∈ L∞ . Since, by 3.14(c), ( M (kλ a)χn −

M (a)χn 2L2

= T

|kλ a − a|2 dm → 0 (λ → ∞)

for all n ∈ Z (χn (t) := tn ), we have M (kλ a)ϕ − M (a)ϕL2

as

λ → ∞ ∀ ϕ ∈ P.

(3.31)

But (3.30) and (3.31) in conjunction with 1.1(d) imply that M (kλ a) → M (a) strongly on L2 . The argument applies to a in place of a, which completes the proof for N = 1. The assertion for general N follows from the fact that the norm in A∞ N ×N is admissible. (b) If N = 1, then, by part (a), 1.1(e), 3.14(b),

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3 Symbol Analysis

a∞ = M (a) ≤ lim inf M (kλ a) ≤ sup M (kλ a) = sup kλ a ≤ a∞ , λ→∞

λ

which shows that {kλ a}A = a∞

λ

∀ a ∈ L∞ .

Now let N > 1. Then the norms in both AN ×N and in 1.29(b). Hence, for a = (ajk ) ∈ L∞ N ×N , " " " " " {k a }ϕ(I ) {kλ a}AN ×N = sup " λ jk jk " " ϕ

(3.32)

L∞ N ×N

can be written as

(by 1.29(b))

A

j,k

"! #" " " " k a ϕ(I ) = sup " λ jk jk " " ϕ A j,k " " " " = sup " ajk ϕ(Ijk )" (by (3.32)) " " ϕ

L∞

j,k

= aL∞ N ×N

(again by 1.29(b)).

(3.33)

Thus, K is an isometry. If aij is a finite collection of matrix functions in L∞ N ×N , then "! #" " " " " " " " " kλ " a = aij " (by (3.33)) ij " " ∞ " " AN ×N LN ×N i j i j " " " " " " " " " " " = "M aij " = " M (aij )" " i

j

i

j

i

j

" " " " " ≤ lim inf " M (kλ aij )" " (by part (a) and 1.1(e)) λ→∞ i j " " " " " " " " " " " ≤ sup " M (kλ aij )" = sup "M kλ aij " " λ

" ! #" " " " =" k a λ ij " " i

j

λ

i

j

. AN ×N

This proves that K is 1-submultiplicative.

3.41. Definitions. Let A and B be Banach algebras and let i : A → B be a mapping of A into B. We denote by i(A) the image (range) of i; alg i(A) the closed subalgebra generated by i(A), that of B is, 5m n alg i(A) := closB j=1 k=1 i(ajk ) : ajk ∈ A ; Qi (A) the quasicommutator ideal of alg i(A), that is, the smallest closed two-sided ideal of alg i(A) containing all elements of the form i(ab) − i(a)i(b), a ∈ A, b ∈ A. To avoid confusion, alg i(A) will be sometimes denoted by algB i(A).


143

3.42. Theorem. Let A and B be Banach algebras and suppose i : A → B is a submultiplicative quasi-embedding. Then alg i(A) decomposes into the direct sum of i(A) and Qi (A): ·

alg i(A) = i(A) + Qi (A). Proof. The set F :=

! n m

i(ajk ) : n, m ∈ Z+ , ajk

# ∈A

j=1 k=1

is dense in alg i(A). Define the linear mapping S : F → F by S: i(ajk ) → i ajk . j

j

k

k

The hypothesis that i be implies that S is well defined, i.e., 5

submultiplicative

5 if f = j k i(bjk ) = l m i(clm ) then i bjk = i clm , j

k

l

m

and that S is bounded on F . Therefore S extends to a bounded linear mapping on the whole algebra alg i(A) into itself. Since S(i(a)) = i(a) for every a ∈ A, it follows that S is a projection on alg i(A) and that i(A) ⊂ S(alg i(A)). The hypothesis that i(A) be closed in B implies that actually i(A) = S(alg i(A)). We claim that the closed set Ker S is a two-sided ideal in alg i(A). Let b ∈ Ker S and c ∈ alg i(A). Then there are sequences {bn }, {cn } ⊂ F such that bn → b, cn → c, S(bn ) → 0 and, hence, bn cn → bc. Again from the submultiplicativity of i we deduce that S(bn cn ) ≤ γS(bn )S(cn ) ≤ γS(bn ) S(cn ), and therefore S(bc) = 0. It can be shown in the same way that S(cb) = 0. Thus, Ker S is a closed two-sided ideal in alg i(A). Our next objective is to prove that Ker S = Qi (A). Since Ker S has been shown to be an ideal, we have Qi (A) ⊂ Ker S. To get the reverse inclusion, let b ∈ Ker S and choose a sequence {bn } ⊂ F such that bn → b and S(bn ) → 0. A little thought shows that bn − S(bn ) ∈ Qi (A) and passage to the limit n → ∞ yields that b ∈ Qi (A). Thus Ker S ⊂ Qi (A). ·

Putting the things together, we have alg i(A) = Ker S + Im S (because S is a bounded projection on alg i(A)) and at the same time Ker S = Qi (A) and Im S = i(A), which is the assertion.

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3 Symbol Analysis

3.43. Definitions. Let the hypothesis of Theorem 3.42 be fulfilled. We denote by the (necessarily continuous) projection of alg i(A) onto i(A) parallel to Qi (A); from the proof of the preceding theorem it is seen that Si ≤ γ in case i is γ-submultiplicative; i(−e) the linear homeomorphism given (correctly) by i(−e) : i(A) → A/Ker i, i(a) → a + Ker i; Smbi the continuous and linear mapping Smbi : alg i(A) → A/Ker i, B → i(−e) Si (B); Qi (A) and that it is clear Ker Smb i =

that 5 5 Smbi j k i(ajk ) = j k ajk + Ker i (the sum and the products finite); the linear homeomorphism given (correctly) by σi σi : alg i(A)/Qi (A) → A/Ker i, B + Qi (A) → Smbi (B).

Si

Thus, we have the following commutative diagram:

In the case where i is an embedding (Ker i = {0}) we regard Smbi as the mapping (3.34) Smbi : alg i(A) → A, A → i(−1) Si (A), where i(−1) is the inverse of i viewed as acting from A onto i(A). 3.44. Corollary. Suppose the hypotheses of Theorem 3.42 are satisfied. Then σi is a homeomorphic isomorphism of alg i(A)/Qi (A) onto A/Ker i. Proof. It remains to show that σi is multiplicative. Let A, B ∈ alg i(A). Due to Theorem 3.42 we have A = i(a) + K and B = i(b) + L with a, b ∈ A and K, L ∈ Qi (A). Hence, AB = i(a)i(b) + N = i(ab) + M with certain N, M ∈ Qi (A) and thus, Si (A) = i(a),

Si (B) = i(b),

Si (AB) = i(ab).

Consequently, σi ((A + Qi (A))(B + Qi (A))) = σi (AB + Qi (A)) = Smbi (AB) = i(−e) Si (AB) = i(−e) i(ab) = ab + Ker i = (a + Ker i)(b + Ker i) = i(−e) i(a) · i(−e) i(b) = i(−e) Si (A) · i(−e) Si (B) = Smbi (A)Smbi (B) = σi (A + Qi (A))σi (B + Qi (A)).


145

Before applying 3.42 and 3.44 to the concrete situation given by 3.40 we need two further results of technical nature. 3.45. Definitions. Let A be a Banach algebra and let S be a subset of A. We denote by algA S the smallest closed subalgebra of A containing S; closidA S the smallest closed two-sided ideal of A containing S. It is clear that algA S = closA

! n m

# sjk : sjk ∈ S ,

j=1 k=1 ! n

# aj sj bj : sj ∈ S, aj ∈ A, bj ∈ A

closidA S = closA

j=1

For a ∈ A, we let a ⊗ Ijk denote the element in AN ×N whose jk entry is a and all other entries of which are zero. 3.46. Lemma. Let A be a Banach algebra and suppose an (admissible) Banach algebra norm is given in AN ×N . (a) If C is a closed two-sided ideal in A, then CN ×N is a closed two-sided ideal in AN ×N . (b) If S is a subset of A, then (algA S)N ×N = algAN ×N SN ×N . (c) If S is a subset of A, then (closidA S)N ×N = closidAN ×N SN ×N . Proof. (a) Obvious. (b) It is clear that the right-hand side is contained in the left-hand side. In order to establish the reverse inclusion, we must show that s1 . . . sm ⊗ Ijk belongs to algAN ×N SN ×N for all s1 , . . . , sm ∈ S. This is obvious for m = 1 or for j = k. the general case follows from the identity s1 . . . sm ⊗ Ijk = (s1 ⊗ Ijk )(s2 . . . sm ⊗ Ikk ). (c) That the right-hand side is a subset of the left-hand side is trivial. The reverse inclusion results from the identity asb ⊗ Ijk = (a ⊗ Ijk )(s ⊗ Ikk )(b ⊗ Ikk ).

3.47. Lemma. Let A and B be Banach algebras, let i : A → B be a linear mapping, suppose AN ×N and BN ×N are endowed with (admissible) Banach algebra norms, and define iN ×N : AN ×N → BN ×N ,

(ajk ) → (i(ajk )).

Then (algB i(A))N ×N = algBN ×N iN ×N (AN ×N ), (Qi (A))N ×N = QiN ×N (AN ×N ).

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3 Symbol Analysis

Proof. The first equality follows immediately from Lemma 3.46(b). The second one will follow from Lemma 3.46(c) as soon as we have shown that (S1,1 )N ×N equals SN,N , where Sm,m is the collection of all quasicommutators im×m (ab) − im×m (a)im×m (b),

a, b ∈ Am×m .

The inclusion (S1,1 )N ×N ⊂ SN,N is a consequence of the identity i(ab) − i(a)i(b) ⊗ Ijk = iN ×N (a ⊗ Ijk )(b ⊗ Ikk ) − iN ×N (a ⊗ Ijk )iN ×N (b ⊗ Ikk ) and the reverse inclusion results from the observation that [iN ×N (ab) − iN ×N (a)iN ×N (b)]jk =i ajl blk − i(ajl )i(blk ) = [i(ajl blk ) − i(ajl )i(blk )]. l

l

l

3.48. Definitions. Let A be a closed subalgebra of L∞ containing the constants. Then AN ×N is a closed subalgebra of L∞ N ×N . Let {Kλ }λ∈Λ be an approximate identity and define the mapping K as in Proposition 3.40. Put alg K(AN ×N ) := algAN ×N K(AN ×N ), and let QK (AN ×N ) be the smallest closed two-sided ideal of alg K(AN ×N ) containing all elements of the form {kλ (ab) − (kλ a)(kλ b)}λ∈Λ ,

a, b ∈ AN ×N

(these definitions are in accordance with 3.41 and 3.45). Lemma 3.47 tells us that alg K(AN ×N ) = (alg K(A))N ×N , QK (AN ×N ) = (QK (A))N ×N

(3.35) (3.36)

(note that the K in Proposition 3.40 is actually the KN ×N ). Proposition 3.40 and Theorem 3.42 give that ·

alg K(AN ×N ) = K(AN ×N ) + QK (AN ×N ),

(3.37)

and since K is 1-submultiplicative, we have SK = 1 (recall 3.43). If {aλ } is in AN ×N , then there is an a ∈ L∞ N ×N such that M (aλ ) → M (a) strongly on L2N ; the (obviously linear and bounded) mapping AN ×N → L∞ N ×N which assigns that a to the sequence {aλ } will be denoted by LA for the meanwhile. 3.49. Proposition. (a) SK = K ◦ LA |alg K(AN ×N ). (b) SmbK = LA |alg K(AN ×N ). (c) Let {aλ } ∈ alg K(AN ×N ). Then {aλ } ∈ QK (AN ×N ) ⇐⇒ LA ({aλ }) = 0.


Proof. (a) If {aλ } =

j

kλ ajk

147

∈ alg K(AN ×N ),

k

the sum and the products are finite, then SK ({aλ }) = kλ = (K ◦ LA )({aλ }), ajk j

k

because M (aλ ) → M

j

ajk

k

strongly (see the proof of Proposition 3.40(a)). The continuity of SK and K ◦ LA give the assertion for general {aλ } ∈ alg K(AN ×N ). (b), (c) Immediate from (a).

3.50. Definition. Put NN,N := {aλ } ∈ A∞ → 0 as λ → ∞ N ×N : aλ L∞ N ×N and given a closed subalgebra A of L∞ let A NN,N = NN,N ∩ alg K(AN ×N ).

It is easy to see that NN,N is a closed two-sided ideal of both A∞ N ×N and A is a closed two-sided ideal of alg K(AN ×N ). It is clear AN ×N and that NN,N A = that NN,N = NN ×N , where N := N1,1 . This and (3.35) imply that NN,N A A A A NN ×N := (N )N ×N , where N := N1,1 . Therefore we shall henceforth write A , respectively. NN ×N and NNA×N in place of NN,N and NN,N 3.51. Proposition. (a) Let B be a closed subalgebra of QC containing the constants. Then QK (BN ×N ) = NNB×N . (b) Let B be a C ∗ -subalgebra of QC. Then the mapping σK : alg K(BN ×N )/NNB×N → BN ×N ,

{aλ } + NNB×N → LA ({aλ })

is an isometric star-isomorphism. (c) Let B be a C ∗ -subalgebra of L∞ . If QK (BN ×N ) ⊂ NN ×N , then B is a subset of QC. Proof. (a) From Proposition 3.49(a) we deduce that NNB×N ⊂ Ker SK = QK (BN ×N ). On the other hand, the asymptotic multiplicativity of {Kλ } on the pair (QC, QC) (Theorem 3.23) implies that QK (BN ×N ) ⊂ NNB×N .

148

3 Symbol Analysis

(b) Part (a), Corollary 3.44, and Proposition 3.49(b) give that σK is a homeomorphic star-isomorphism. Finally, 1.26(e) shows that it is even an isometry. (c) By (3.36), it suffices to consider the case N = 1. If a ∈ B, then a ∈ B and so {kλ a}{kλ a} − {kλ (aa)} ∈ N , whence |kλ a|2 − kλ (|a|2 )∞ → 0 (λ → ∞) ∀ a ∈ B.

(3.38)

Let A(x) := a(eix ) (x ∈ R). Then, for x ∈ R, ( ∞ kλ,x (|A − kλ,x A|) = |A(t) − (kλ A)(x)|Kλ (x − t) dt −∞

( ≤ (

∞ −∞ ∞

= −∞

1/2 |A(t) − (kλ A)(x)|2 Kλ (x − t) dt 1/2 |A(t)|2 Kλ (x − t) dt − |(kλ A)(x)|2

1/2 = kλ,x (|A|2 ) − |kλ,x A|2 , hence, for t ∈ T, 1/2 kλ,t (|a − kλ,t a|) ≤ kλ,t (|a|2 ) − |kλ,t a|2 . Consequently, taking into account (3.38) we see that for each ε > 0 there is a λ0 ∈ Λ, λ0 > 1, such that aλ0 :=

sup

sup kλ,t (|a − kλ,t a|) < ε.

λ>λ0 ,λ∈Λ t∈T

The first part of the proof of Theorem 3.21 shows that M2π/λ0 (a) ≤ caλ0 with some constant c independent of a and λ0 (recall 1.47). Therefore lim M2π/λ (a) = 0,

λ→∞

it follows that a ∈ V M O, and since a ∈ L∞ , we have a ∈ QC by Theorem 3.17. Remark. In case {Kλ }λ∈Λ is the approximate identity generated by the Poisson kernel, we shall write H, alg H(AN ×N ), etc. in place of K, alg K(AN ×N ), etc. Theorem 2.62(a) shows that the equality QK (BN ×N ) = NNB×N also holds for B = C + H ∞ .

3.5 Harmonic Approximation: Essentialization

149

3.5 Harmonic Approximation: Essentialization 3.52. Theorem. Suppose (a) A, B are Banach algebras and i : A → B is a γ-submultiplicative quasi-embedding; (b) J is a closed two-sided ideal of alg i(A); (c) Si (J) ⊂ J. Define the mapping iπ by iπ : A → alg i(A)/J,

a → i(a) + J.

Then (d) iπ is a γ-submultiplicative quasi-embedding and Ker iπ = {a ∈ A : i(a) ∈ J}; (e) alg iπ (A) = alg i(A)/J; (f) σiπ is a homeomorphic isomorphism of the algebra alg iπ (A)/Qiπ (A) onto the algebra A/Ker iπ . Proof. (d) It is clear that iπ is linear and continuous. Let π denote the canonical projection of alg i(A) onto alg i(A)/J. Since i(A) + J = Ker π(I − Si ) is closed, π|i(A) is normally solvable (see Gohberg, Krupnik [232, Chap. 4, Theorem 2.1]) and so iπ (A) is closed. We have a ∈ Ker iπ if and only if i(a) ∈ J. Hence, if a ∈ Ker iπ and b ∈ A, then i(ab) = Si (i(a)i(b)) ∈ J because of (c), and it follows that Ker iπ is a (necessarily closed) two-sided ideal of A. It remains to show that iπ is γ-submultiplicative. Since " " " " "π " " " " "i " ajk " = "i ajk + J " " " j j k k " " " " " = "Si i(ajk ) + J " " j

k

" " " " " S = inf " i(a ) + G jk " i " G∈J

i

k " " " " " ≤ inf "Si i(ajk ) + Si (H)" " H∈J j k " " " " " ≤ Si inf " i(a ) + H jk " " H∈J j k " " " " " = Si " i(a ) + J jk " " j k " " " " π " = Si " i (a ) jk " " i

k

(due to (c))

(3.39)

150

3 Symbol Analysis

and because Si ≤ γ, the γ-submultiplicativity follows. (e) The assertion is that the finite product-sums iπ (ajk ) = i(ajk ) + J, ajk ∈ A, j

j

k

k

are dense in alg i(A)/J, which can be checked straightforwardly. (f) Immediate from Corollary 3.44. 3.53. Corollary. (a) If, in addition to the hypotheses 3.52(a)–(c), i is 1submultiplicative, then, for all a ∈ A, " " iπ (a) = "Siπ iπ (a) " = inf iπ (a) + G. G∈Qiπ (A)

(b) If, in addition to the hypotheses 3.52(a)–(c), i is 1-submultiplicative and both A and alg i(A) are C ∗ -algebras, then, for all a ∈ A, " " iπ (a) = "Smbiπ iπ (a) " = inf π a + g. g∈Ker i

Proof. (a) Since iπ (a) = Siπ (iπ (a) + G) for every G ∈ Qiπ (A), we have " " π "π " "Siπ i (a) + G " ≤ "i (a) + G" ≤ iπ (a). inf iπ (a) = inf G∈Qiπ (A)

G∈Qiπ (A)

(b) Theorem 3.52(f) and 1.26(e) imply that inf G∈Qiπ (A)

iπ (a) + G =

inf

g∈Ker iπ

a + g

and it remains to apply part (a). 3.54. Proposition. Let AπN ×N := AN ×N /NN ×N . Then π Kπ : L∞ N ×N → AN ×N ,

a → {kλ a}π := {kλ a} + NN ×N

is a 1-submultiplicative isometry. Proof. Theorem 3.52(d) applied with A = L∞ N ×N , B = AN ×N , i = K, J = NN ×N (whose hypotheses are satisfied due to 3.40 and 3.49(a)) shows that Kπ is 1-submultiplicative. If a ∈ L∞ N ×N and {cλ } ∈ NN ×N , then M (kλ a + cλ ) → M (a) strongly (see the proof of Proposition 3.40), so that ! # a ≤ inf lim inf kλ a + cλ : {cλ } ∈ NN ×N λ→∞ ! # ≤ inf sup kλ a + cλ : {cλ } ∈ NN ×N λ

= {kλ a}π ≤ {kλ a} = a, which implies that Kπ is an isometry.

3.5 Harmonic Approximation: Essentialization

151

Remark. Let A be a closed subalgebra of L∞ . In view of Theorem 3.52, alg Kπ (AN ×N ) equals alg K(AN ×N )/NNA×N . We also have ·

alg Kπ (AN ×N ) = Kπ (AN ×N ) + QKπ (AN ×N ), SKπ = 1, and alg Kπ (AN ×N )/QKπ (AN ×N ) is homeomorphically (isometrically in case A is a C ∗ -algebra) isomorphic to AN ×N , because Ker Kπ = {0} by the preceding proposition. 3.55. Definition. Let {aλ }λ∈Λ ∈ A∞ N ×N . The (generalized) sequence {aλ } is said to be bounded away from zero (abbreviated as bafz ) if there exists a λ0 ∈ Λ such that (a) aλ ∈ GL∞ N ×N for all λ ∈ Λ, λ > λ0 ,

(b)

sup λ>λ0 ,λ∈Λ

∞ a−1 λ LN ×N < ∞.

Since aλ ∈ CN ×N , (a) is equivalent to the requirement that det aλ (t) = 0 for all t ∈ T, λ ∈ Λ, λ > λ0 . It is easily seen that the following equivalences are true: {aλ } bafz ⇐⇒ {det(aλ )} bafz ⇐⇒ {aλ }π ∈ G(A∞ N ×N /NN ×N ). Of particular importance is the case where aλ = kλ a for some a ∈ L∞ N ×N . For instance, the statement of Theorem 2.62(b) is that if a ∈ (C + H ∞ )N ×N , then a ∈ G(C + H ∞ )N ×N if and only if {kλ (det a)} is bounded away from zero, where {Kλ } is the approximate identity generated by the Poisson kernel. 3.56. Theorem. Let {aλ } ∈ alg K(AN ×N ), where A is a C ∗ -subalgebra of L∞ containing the constants. Then {aλ } bafz ⇐⇒ {aλ }π ∈ G(alg Kπ (AN ×N )). Proof. We suppress the subscript N × N . The implication “⇐=” is obvious. So we are left with the implication “=⇒”. Thus, let {aλ }π = {aλ } + N ∈ G(A∞ /N ). Because alg K(A) is a C ∗ -subalgebra of A∞ , it follows, that alg K(A) + N is a C ∗ -subalgebra of A∞ (1.26(g)), hence (alg K(A) + N )/N is a C ∗ -subalgebra of A∞ /N (1.26(f)). Therefore {aλ } + N belongs to G((alg K(A) + N )/N ) (1.26(d)). Taking into account 1.26(g) once more, we obtain that {aλ } + N A is in G(alg K(A)/N A ) = G(alg Kπ (A)). 3.57. Corollary. Let {aλ } ∈ alg K(L∞ N ×N ) be bounded away from zero. Then SmbK ({aλ }) ∈ GL∞ N ×N . Proof. The preceding theorem implies that {aλ }π is in G(alg Kπ (L∞ N ×N )). ) such that b a = I + c with some Hence, there is a {bλ } ∈ alg Kπ (L∞ λ λ λ N ×N {cλ } ∈ NN∞×N . From (3.37) we deduce that bλ = kλ b + dλ and aλ = kλ a + fλ , ∞ where {dλ }, {fλ } ∈ QK (L∞ N ×N ) and b, a ∈ LN ×N . By the definition of SmbK , we have a = SmbK ({aλ }) (see also Proposition 3.49). Thus

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3 Symbol Analysis

kλ b · kλ a + kλ b · fλ + dλ · kλ a + dλ · fλ = I + cλ . Now let SK act on both sides of this equality and take into account that SK |QK (L∞ N ×N ) = 0 (Proposition 3.49). What results is that {kλ (ba)} = {I}. Now 3.14(c) yields ba = I, i.e., that a is left-invertible in L∞ N ×N . It can be shown in the same way that a is right-invertible.

3.6 Harmonic Approximation: Localization 3.58. Definitions. Suppose (a) A is a C ∗ -subalgebra of L∞ containing the constants and F is a closed subset of M (A); let A = AN ×N ; (b) B is a C ∗ -algebra with identity and i : A → B is a 1-submultiplicative isometry. If a = (ajk )N j,k=1 ∈ A, then a|F = 0 means that ajk |F = 0 for all j, k. Define If = {a ∈ A : a|F = 0},

JF = closidalg i(A) i(IF ),

πF : alg i(A) → alg i(A)/JF ,

G → G + JF ,

iF : A → alg i(A)/JF ,

a → i(a) + JF .

3.59. Lemma. Let A and B be Banach algebras, let i : A → B be a submultiplicative quasi-embedding, and let I be a closed two-sided ideal of A. Put J = closidalg i(A) i(I). Then (a) Si (J) ⊂ J,

(b)

i(a) ∈ J ⇐⇒ a ∈ I.

Proof. (a) J is the closure of the linear hull of all elements of the form i(a1 ) . . . i(am )i(c)i(b1 ) . . . i(bm ) where aj , bj ∈ A and c ∈ I. But Si i(a1 ) . . . i(am )i(c)i(b1 ) . . . i(bm ) = i(a1 . . . am cb1 . . . bm ), which is in i(I) and therefore in J. (b) If i(a) ∈ J, then i(a) = Si (i(a)) must be in the closure of the linear hull of all elements of the form i(a1 . . . am cb1 . . . bm ), aj , bj ∈ A, c ∈ I. Thus i(a) ∈ i(I), i.e., a ∈ I. 3.60. Definitions. Suppose 3.58(a), (b) are fulfilled. Theorem 3.52 and the preceding lemma show that iF is a 1-submultiplicative quasi-embedding and that Ker iF = IF . Thus, ·

alg iF (A) = alg i(A)/JF = iF (A) + QiF (A)

(3.40)

3.6 Harmonic Approximation: Localization

153

and σiF is an isometric star-isomorphism of alg iF (A)/QiF (A) onto A/IF (Corollary 3.53(b)). The algebra alg iF (A) will be referred to as the local algebra (associated with F ⊂ M (A)) and if a ∈ A, then iF (a) will be called a local object. The local spectrum sp(iF (a)) is the spectrum of iF (a) as an element of alg iF (A). 3.61. Theorem. Let 3.58(a), (b) be satisfied. Then, for a ∈ A, iF (a) = a|F , where a|F := max a(x)L(CN ) and CN is endowed with the norm (1.9). x∈F

Proof. Due to Corollary 3.53(b) we have . iF (a) = inf a + gL∞ N ×N g∈IF

Let IF1 := {a ∈ A : a|F = 0}. Then, by Lemma 3.46(b), inf a + gL∞ = N ×N

g∈IF

a + gL∞ . N ×N

inf

1) g∈(IF N ×N

(3.41)

We mentioned in 1.28 that the mapping ϕ : A/IF1 → A|F,

a + IF1 → a|F

is an isometric star-isomorphism. Therefore, ϕN ×N : (A/IF1 )N ×N → (A|F )N ×N ,

(ajk + IF1 ) → (ajk |F )

is a star-isomorphism. A C ∗ -norm in (A/IF1 )N ×N is given by a + (IF1 )N ×N 1 :=

inf

1) h∈(IF N ×N

a + hL∞ N ×N

and a C ∗ -norm in (A|F )N ×N is a|F 2 := max a(x)L(CN ) . x∈F

Thus, if a ∈ A = AN ×N , then by virtue of 1.26(e), " " a|F 2 = "ϕN ×N a + (IF1 )N ×N "2 = a + (IF1 )N ×N 1 . Recall (3.41) to see that the proof is complete.

3.62. Corollary. Suppose 3.58(a), (b) are fulfilled. If a ∈ A is sectorial on F , then iF (a) is in G(alg iF (A)). Proof. Lemma 3.6(b) shows that there is a matrix d ∈ GCN ×N such that I − a(x)dL(CN ) < 1 for all x ∈ F . So Theorem 3.61 gives that iF (I) − iF (ad)alg iF (A) < 1. Since iF (I) is the identity in alg iF (A), it follows that iF (ad) = iF (a)d is in G(alg iF (A)), which implies the invertibility of iF (a) at once.

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3 Symbol Analysis

3.63. Corollary. Suppose 3.58(a), (b) hold. If iF (a) is left or right or twosided invertible in alg iF (A), then a|F is invertible in the restriction algebra A|F . Proof. There is a BF ∈ alg iF (A) such that BF iF (a) = iF (I). From (3.40) we deduce that BF = iF (b) + CF with certain b ∈ A and CF ∈ QiF (A). Hence, iF (b)iF (a) − iF (I) = −CF iF (a) ∈ QiF (A) and it follows that iF (ba − I) = SiF (iF (b)iF (a) − iF (I)) = 0. So Theorem 3.61 implies that (ba − I)|F = 0, i.e., that det b(x) det a(x) = 1 for all x ∈ F . This gives the invertibility of a|F in A|F immediately. 3.64. Corollary. Let 3.58(a), (b) be satisfied and let N = 1. Then, if a ∈ A, a(F ) ⊂ sp (iF (a)) ⊂ conv a(F ). Proof. Immediate from the two preceding corollaries.

3.65. Lemma. Let A be a commutative C ∗ -algebra with identity and let B be a C ∗ -subalgebra of A containing the identity. Then, for ξ ∈ M (B), closidA c ∈ B : c(ξ) = 0 = a ∈ A : a|Mξ (A) = 0 . Proof. We know from 1.28 that there is a closed subset Fξ of M (A) such that (3.42) closidA c ∈ B : c(ξ) = 0 = IFξ . We must show that Fξ = Mξ (A). Fξ ⊂ Mξ (A): Let α ∈ Fξ . If c ∈ B, then c − c(ξ) ∈ B and (c − c(ξ))(ξ) = 0. Hence c − c(ξ) ∈ IFξ , and so (c − c(ξ))(α) = 0, i.e., c(α) = c(ξ). Consequently, α ∈ Mξ (A). Mξ (A) ⊂ Fξ : Let α ∈ Mξ (A). Because of (3.42), , n IF = closA ak ck : ak ∈ A, ck ∈ B, ck (ξ) = 0, n ∈ Z+ . k=1

Since

a k ck

k

(α) =

ak (α)ck (ξ) = 0

k

for all finite sums k ak ck ∈ IFξ , we have g(α) = 0 for all g ∈ IFξ . If α would not be in F , then there would exist a g ∈ IFξ such that g(α) = 0. This contradiction shows that α ∈ Fξ . 3.66. Definitions. Suppose (a) A is a commutative C ∗ -algebra with identity I and B is a C ∗ -subalgebra of A containing I; put A = AN ×N and let C = {cIN ×N ∈ A : c ∈ B}, where IN ×N is the N × N identity matrix in A;

3.6 Harmonic Approximation: Localization

155

(b) B is a C ∗ -algebra with identity and i : A → B is a 1-submultiplicative isometry. For ξ ∈ M (B), define Iξ = closidA {cIN ×N ∈ C : c(ξ) = 0},

Jξ = closidalg i(A) i(Iξ ),

πξ : alg i(A) → alg i(A)/Jξ ,

G → G + Jξ ,

iξ : A → alg i(A)/Jξ ,

a → i(a) + Jξ .

By Theorem 3.52 and Lemma 3.59, iξ is a 1-submultiplicative quasi-embedding whose kernel is Iξ and alg iξ (A) coincides with alg i(A)/Jξ . Lemma 3.65 and Lemma 3.46 show that Iξ = IMξ (A) , Jξ = JMξ (A) , πξ = πMξ (A) , iξ = iMξ (A)

(3.43) (3.44)

whenever A is a C ∗ -subalgebra of L∞ (recall 3.58). 3.67. Theorem. Suppose that, in addition to 3.66(a), (b), the following holds: (c) i(caIN ×N ) = i(cIN ×N )i(aIN ×N ) ∀ c ∈ B,

∀ a ∈ A.

Then if Y ∈ alg i(A), Y ∈ G(alg i(A)) ⇐⇒ πξ Y ∈ G(alg iξ (A))

∀ ξ ∈ M (B).

Proof. The hypotheses imply that i(C) is a closed subalgebra of the center of alg i(A) which is isometrically isomorphic to B. Hence, to each N ∈ M (i(C)) there corresponds a ξ ∈ M (B) such that N = {i(cIN ×N ) : c ∈ B, c(ξ) = 0}. Thus JN := closidalg i(A) N = closidalg i(A) i(Iξ ) =: Jξ and the assertion follows from (the C ∗ -version of) Theorem 1.35(a). 3.68. Definitions. Let a ∈ L∞ N ×N , let {Kλ }λ∈Λ be an approximate identity, and let F be a closed subset of M (L∞ ). The sequence {kλ a} is said to be F restricted bounded away from zero if there is a b ∈ L∞ N ×N such that a|F = b|F and {kλ b} is bounded away from zero. We are now in a position to establish one of the main results of the present chapter. 3.69. Corollary. Let B be a C ∗ -algebra of QC containing the constants, let {Kλ }λ∈Λ be an approximate identity, and let a ∈ L∞ N ×N . (a) If {kλ a} is Mξ (L∞ )-restricted bounded away from zero for each ξ in M (B), then {kλ a} is bounded away from zero. (b) If a is locally sectorial over B, then {kλ a} is bounded away from zero.

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3 Symbol Analysis

Proof. We apply Theorem 3.67 with A = L∞ , B = AπN ×N , i = Kπ . The hypothesis 3.66(a) is clearly satisfied and 3.66(b) is fulfilled due to Proposition 3.54. The hypothesis 3.67(c) requires that Kπ (ca) = Kπ (c)Kπ (a) ∀ c ∈ b,

∀ a ∈ L∞

(here we abbreviate cIN ×N and aIN ×N to c and a, respectively). But this is nothing else than the requirement that kλ (ca) − (kλ c)(kλ a)∞ → 0

(λ → ∞) ∀ c ∈ B,

∀ a ∈ L∞ ,

which is equivalent to the asymptotic multiplicativity of {Kλ } on the pair (B, L∞ ). Theorem 3.23 therefore shows that 3.67(c) is also satisfied. Thus, by virtue of Theorem 3.67 and Theorem 3.56, it suffices to show that Kξπ (a) (Kξπ := (Kπ )ξ ) is invertible in alg Kξπ (L∞ N ×N ) for each ξ ∈ M (B). Under the hypothesis (a), we deduce from (3.43), (3.44) that Kξπ (a) = π π ∞ Kξπ (b) with some b ∈ L∞ N ×N such that Kξ (b) ∈ G(alg K (LN ×N )) (Theoπ π rem 3.56), whence Kξ (a) ∈ G(alg Kξ (LN ×N )) (Theorem 3.67). Under the hypothesis (b), Corollary 3.62 applied with F = Mξ (L∞ ) in conjunction with (3.43), (3.44) gives the invertibility of Kξπ (a) in alg Kξπ (L∞ N ×N ). Remark. For (b) see also 4.31. 3.70. Theorem. Let B be a C ∗ -algebra with identity and let i : L∞ N ×N → B be a 1-submultiplicative isometry such that i(ϕa) = i(ϕ)i(a) for all ϕ ∈ QCN ×N and a ∈ L∞ N ×N . For τ ∈ M (C) = T and ξ ∈ M (QC), put i(cIN ×N ) : c ∈ C, c(τ ) = 0 , Jτ = closidalg i(L∞ N ×N ) Jξ = closidalg i(L∞ i(ϕIN ×N ) : ϕ ∈ QC, ϕ(ξ) = 0 , N ×N ) and define πτ , πξ , iτ , iξ as in 3.66. Then if Y ∈ alg i(L∞ N ×N ) and τ ∈ T, ∞ πτ Y ∈ G(alg iτ (L∞ N ×N )) ⇐⇒ πξ Y ∈ G(alg iξ (LN ×N )) ∀ ξ ∈ Mτ (QC).

Proof. Theorem 3.52 and Lemma 3.59 imply that ∞ iτ : L∞ N ×N → alg i(LN ×N )/Jτ

is a 1-submultiplicative quasi-embedding whose kernel is Iτ := a ∈ L∞ N ×N : a = cIN ×N , c ∈ C, c(τ ) = 0 . By virtue of Lemma 3.65 and 1.28 we have L∞ N ×N /Iτ = AN ×N , where A = L∞ |Mτ (L∞ ). Therefore, the mapping ieτ defined (correctly) by ieτ : AN ×N → alg i(L∞ N ×N )/Jτ ,

a|Mτ (L∞ ) → iτ (a)

3.7 Harmonic Approximation: Local Spectra

157

is a 1-submultiplicative embedding (recall 3.38). From 1.26(e) we deduce that ieτ is even an isometry. Put B = QC|Mτ (L∞ ) and recall that by 2.81 the maximal ideal space of B is M (B) = Mτ (QC). For ξ ∈ Mτ (QC), let Jξτ := closidalg ieτ (AN ×N ) ieτ (bIN ×N ) : b ∈ B, b(ξ) = 0 . e ∞ It is clear that ieτ (AN ×N ) = iτ (L∞ N ×N ), alg iτ (AN ×N ) = alg iτ (LN ×N ), and Jξτ = closidalg iτ (L∞ iτ (ϕIN ×N ) : ϕ ∈ QC, ϕ(ξ) = 0 . (3.45) N ×N )

Now we apply Theorem 3.67 with i = ieτ and A, B as above. What results is that if Y ∈ alg iτ (L∞ N ×N ), then τ ∞ τ Y ∈ G(alg iτ (L∞ N ×N )) ⇐⇒ Y + Jξ ∈ G(alg iτ (LN ×N )/Jξ ) ∀ ξ ∈ Mτ (QC).

But Jτ is obviously a closed two-sided ideal in Jξ and a little thought shows that Jξ /Jτ coincides with Jξτ (take into account (3.45)). Therefore τ ∞ alg iτ (L∞ N ×N )/Jξ = (alg i(LN ×N )/Jτ )/(Jξ /Jτ )

is naturally isomorphic to ∞ alg i(L∞ N ×N )/Jξ = alg iξ (LN ×N ),

which completes the proof.

3.71. Corollary. Let the hypotheses of Theorem 3.70 be satisfied. If Y is in alg i(L∞ N ×N ), then sp (Y ) =

/

/

sp (πτ Y ) =

τ ∈T

and, for τ ∈ T, sp (πτ Y ) =

sp (πξ Y )

(3.46)

ξ∈M (QC)

/

sp (πξ Y ).

(3.47)

ξ∈Mτ (QC)

Proof. Equalities (3.46) follow from Theorem 3.67 and equality (3.47) results from Theorem 3.70.

3.7 Harmonic Approximation: Local Spectra 3.72. Cluster sets. Let {Kλ }λ∈Λ be an approximate identity and let {aλ } be in alg K(L∞ ). Recall that Λ × T can be viewed as a subset of QC ∗ and that M (QC) = (closQC ∗ (Λ × T)) \ (Λ × T) (3.28 and 3.29). Let τ ∈ T and ξ ∈ M (QC). We define

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3 Symbol Analysis

ClK ({aλ }, Λ × T) as the set of all z ∈ C such that there is a sequence {(λn , tn )} ⊂ Λ × T with aλn (tn ) → z as n → ∞;

δ

ClK ({aλ }, T)

as the set of all z ∈ C such that there is a sequence {(λn , tn )} ⊂ Λ × T with λn → ∞ and aλn (tn ) → z as n → ∞;

ClK ({aλ }, τ )

as the set of all z ∈ C such that there is a sequence {(λn , tn )} ⊂ Λ × T with λn → ∞, tn → τ , and aλn (tn ) → z as n → ∞;

ClK ({aλ }, ξ)

as the set of all z ∈ C with the following property: for each ε > 0 and for each QC ∗ -neighborhood U of ξ there is a (λ, t) ∈ (Λ × T) ∩ U such that |z − aλ (t)| < ε.

In these definitions “Cl ” is for “cluster.” Given {aλ } ∈ alg K(L∞ ) define by δ {aλ } : Λ × T → C, (µ, t) → aµ (t),

{aλ }

and for ν ∈ Λ, τ = eiθ0 ∈ T, ε > 0 put Λν := λ ∈ Λ : λ > ν , (τ − ε, τ + ε) := t = eiθ ∈ T : |θ − θ0 | < ε . It is easily seen that ClK ({aλ }, Λ × T) = clos δ {aλ } (Λ × T), $ ClK ({aλ }, T) = clos δ {aλ } (Λν × T), ν>0

ClK ({aλ }, τ ) =

$ $

clos δ {aλ } Λ × (τ − ε, τ + ε) .

(3.48) (3.49) (3.50)

ν>0 ε>0

It is also clear that ClK ({aλ }, ξ) =

$

clos δ {aλ } (Λ × T) ∩ Uε;q1 ,...,qn (ξ) ,

ε;q1 ,...,qn

the intersection over all ε > 0 and q1 , . . . , qn ∈ QC; here, of course, Uε;q1 ,...,qn (ξ) := η ∈ QC ∗ : |η(qi ) − ξ(qi )| < ε ∀ i = 1, . . . , n . Each neighborhood of this form contains some neighborhood of the form U1;q (ξ), where q is a non-negative function in QC: for instance take q=

1 |q1 − ξ(q1 )| + . . . + |qn − ξ(qn )| . ε

Thus, ClK ({a}, ξ) =

$ q∈QC

clos δ {aλ } (Λ × T) ∩ U1;q (ξ) .


159

For a ∈ L∞ , the sets ClK ({kλ a}, . . .) will be simply denoted by ClK (a, . . .) and will be referred to as the cluster sets of a on Λ × T, on T, at τ ∈ T, or at ξ ∈ M (QC) (associated with the approximate identity {Kλ }λ∈Λ ). If {Kλ }λ∈(0,∞) is generated by the Poisson kernel, then ClK (a, . . .) will be written as ClH (a, . . .). In that case we have the familiar cluster sets of the harmonic extension a(ζ) (ζ ∈ D) of a: a(ζn ) → z as n → ∞ , ClH (a, T) = z ∈ C : ∃ {ζn } ⊂ D such that |ζn | → 1, a(ζn ) → z as n → ∞ , ClH (a, τ ) = z ∈ C : ∃ {ζn } ⊂ D such that ζn → τ, ClH (a, ξ) = z ∈ C : ∀ ε > 0 ∀ QC ∗ -neighborhood U of ξ ∃ ζ ∈ D ∩ U such that | a(ζ) − z| < ε In this situation, Λ × T can be identified with a circular annulus, and if we let Λ = [0, ∞) (λ = 0 corresponds to ζ = 0), then Λ × T can be identified with D. For the harmonic extension it is obvious from (3.48)–(3.50) that ClH (a, D), ClH (a, T), and ClH (a, τ ) (τ ∈ T) are connected, compact, nonempty subsets of C. 3.73. Proposition. Let {aλ } ∈ alg K(L∞ ). (a) δ {aλ } : Λ × T → C is continuous on Λ × T equipped with the product topology of Λ ⊂ R and T. (b) If Λ×T is connected, then ClK ({aλ }, Λ×T), ClK ({aλ }, T), ClK ({aλ }, τ ) (τ ∈ T) are connected, compact, and nonempty. / ClK ({aλ }, τ ), (c) ClK ({aλ }, T) = τ ∈T

ClK ({aλ }, τ ) =

/

ClK ({aλ }, ξ)

(τ ∈ T).

ξ∈Mτ (QC)

Proof. (a) It suffices to prove that for each a ∈ L∞ the function δ {kλ a} is continuous on Λ × T. But this results from 3.14(a) and the fact that ( ∞ |Kλ (x) − Kµ (x)| dx → 0 as λ → µ, −∞

which is readily checked for continuous kernels K and extends to general kernels K by the density of the continuous functions with compact support in L1 (R). (b) If Λ × T is connected, then so are Λν × T and Λν × (τ − ε, τ + ε). By virtue of part (a), clos δ {aλ } (Λ × T),

clos δ {aλ } (Λν × T),

clos δ {aλ } (Λν × (τ − ε, τ + ε))

are therefore connected, compact, nonempty sets. So the assertion is a consequence of the following well known fact, which can be found in many textbooks: if K1 ⊃ K2 ⊃ K3 ⊃ . . . are connected, compact, and nonempty subsets of a Hausdorff space, then K =

∞

n=1

Kn is connected, compact, and nonempty.

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3 Symbol Analysis

(c) The first equality is obvious from the definition. Let us show the second equality. Suppose z ∈ ClK ({aλ }, τ ), (λn , tn ) ∈ Λ × T, λn → ∞, tn → τ , and aλn (tn ) → z. Since the sequence {(λn , tn )} is contained in a ball of QC ∗ centered at zero (3.14(b)), by the Banach-Alaoglu theorem there is a ξ ∈ QC ∗ and a subsequence of {(λn , tn )}, which will again be denoted by {(λn , tn )}, such that (λn , tn ) → ξ in the weak-star topology. If ϕ, ψ ∈ QC, then by Theorem 3.23, ξ(ϕψ) − ξ(ϕ)ξ(ψ) equals lim (kλn ϕψ)(tn ) − (kλn ϕ)(tn ) · (kλn ψ)(tn ) = 0, n→∞

whence ξ ∈ M (QC), and if c ∈ C, then ξ(c) = lim (kλn c)(tn ) = c(τ ), n→∞

whence ξ ∈ Mτ (QC). It is clear that z ∈ ClK ({aλ }, ξ). Now let z ∈ ClK ({aλ }, ξ) for some ξ ∈ Mτ (QC). Then for each ε > 0 and each n ∈ N there is a (λn , tn ) ∈ (Λ×T)∩U1/n;χ1 (ξ) such that |aλn (tn )−z| < ε. Since (λn , tn ) ∈ U1/n;χ1 (ξ) and ξ(χ1 ) = τ , we have |(kλn χ1 (tn )−τ | < 1/n, and 3.14(d) implies that λn → ∞ and tn → τ . Consequently, z ∈ ClK ({aλ }, τ ). 3.74. Open problems. Is ClK ({aλ }, ξ) (ξ ∈ M (QC)) connected whenever the set Λ × T is so? It would be sufficiently interesting to know the answer for the case that K is the Poisson kernel and aλ = kλ a (a ∈ L∞ ). Under what conditions the conclusion that ClK ({aλ }, T) or ClK ({aλ }, τ ) (τ ∈ T) is connected remains valid when the hypothesis that Λ × T be connected is dropped? For instance, are ClK ({kλ a}, T) and ClK ({kλ a}, τ ) connected for every a ∈ L∞ in case {Kλ }λ∈Λ is generated by the Fejér kernel and Λ = {1, 2, 3, . . .}? 3.75. Notation. Let {Kλ }λ∈Λ be an approximate identity. As in 3.66, for τ ∈ M (C) = T and ξ ∈ M (QC), let Jτ = closidalg K(L∞ ) {kλ c}λ∈Λ : c ∈ C, c(τ ) = 0 , Jξ = closidalg K(L∞ ) {kλ ϕ}λ∈Λ : ϕ ∈ QC, ϕ(ξ) = 0 . For {aλ } ∈ alg K(L∞ ), let {aλ }π , {aλ }πτ , {aλ }πξ denote the coset in ∞

alg Kπ (L∞ ) = alg K(L∞ )/N L , alg Kτπ (L∞ ) = alg Kπ (L∞ )/Jτ , alg Kξπ (L∞ ) = alg Kπ (L∞ )/Jξ , respectively, containing {aλ }.


161

3.76. Theorem. Let {aλ }λ∈Λ ∈ alg K(L∞ ). Then (a) sp ({aλ }) = ClK ({aλ }, Λ × T); (b) sp ({aλ }π ) = ClK ({aλ }, T); (c) sp ({aλ }πτ ) = ClK ({aλ }, τ )

(τ ∈ T);

(d) sp ({aλ }πξ ) = ClK ({aλ }, ξ)

(ξ ∈ M (QC)).

Proof. (a) We have z∈ / sp ({aλ }) ⇐⇒ {aλ − z} ∈ G(alg K(L∞ )) ⇐⇒ {aλ − z} ∈ G(A∞ ) (3.39 and 1.26(d)) ⇐⇒ aλ (t) − z = 0 such that ⇐⇒

inf (λ,t)∈Λ×T

∀ (λ, t) ∈ Λ × T and

|(aλ (t) − z)−1 | ≤ M

∃M >0

∀ (λ, t) ∈ Λ × T

|aλ (t) − z| > 0

⇐⇒ z ∈ / ClK ({aλ }, Λ × T). (b) The assertion follows from part (c) combined with Corollary 3.71 and Proposition 3.73(c). However, there is a simple straightforward proof: z ∈ sp ({aλ }π ) ⇐⇒ {aλ − z} not bafz (Theorem 3.56) ⇐⇒ ∃ (λn , tn ) ∈ Λ × T : λn → ∞, aλn (tn ) → z as n → ∞ ⇐⇒ z ∈ ClK ({aλ }, T). (c) Taking into account Corollary 3.71 and Proposition 3.73(c) this is seen to be an immediate consequence of part (d). The proof we shall give for part (d) also works in the case at hand (where it is even a bit simpler); this provides a possibility of proving the assertion in a more direct way. (d) To establish the inclusion “⊃” it suffices to show that 0 ∈ / ClK ({aλ }, ξ) whenever {aλ }πξ is invertible. Thus, let {aλ }πξ be invertible. Then there are {bλ } ∈ alg K(L∞ ),

{cjλ } ∈ alg K(L∞ ),

ϕj ∈ QC

(j = 1, . . . , n)

such that ϕj (ξ) = 0 and " " 1 j " " {cλ }π {kλ ϕj }π " < . "{bλ }π {aλ }π − 1π − 8 j ∞

It follows that there is a {dλ } ∈ N L such that " 2 " j " " {cλ }{kλ ϕj } − {dλ }" < . "{bλ }{aλ } − 1 − 8 j

162

3 Symbol Analysis

Since there is a λ0 ∈ Λ such that dλ < 1/8 for all λ > λ0 , we have " " j 3 " " ∀λ > λ0 , cλ · kλ ϕj " < "bλ aλ − 1 − 8 ∞ j hence

3 + {cjλ } · |(kλ ϕj )(t)| 8 j=1 for all λ > λ0 and t ∈ T. Put ε1 = 1/ 8n max {cjλ } . If n

|bλ (t)aλ (t) − 1|
0 is sufficiently small, then, by 3.14(d), (Λ × T) ∩ Uε2 ;χ1 (ξ) ⊂ Λλ0 × T.

(3.51)

Consequently, if ε := min{ε1 , ε2 } and U := Uε;χ1 ,ϕ1 ,...,ϕn (ξ), then there is no (λ, t) ∈ (Λ × T) ∩ U such that |aλ (t)| < δ. But this implies that 0 is not in ClK ({aλ }, ξ), as desired. We now prove the inclusion “⊂”. To do this, assume 0 ∈ sp({aλ }πξ ) but 0 ∈ / ClK ({aλ }, ξ). So there are a QC ∗ -neighborhood U1 of ξ and an m > 0 such that (3.52) |aλ (t)| ≥ m ∀ (λ, t) ∈ (Λ × T) ∩ U1 . Now choose ε1 > 0 so that ε1 < m/10. Since {aλ }πξ is not invertible in the (commutative) C ∗ -algebra alg Kξπ (L∞ ), we deduce from 1.20(c) that there exists a {uλ } ∈ alg K(L∞ ) such that {uλ }πξ = 1 and {uλ }πξ {aλ }πξ < ε1 .

(3.53)

Hence there are {cjλ } ∈ alg K(L∞ ) and ϕj ∈ QC (j = 1, . . . , n) such that ϕj (ξ) = 0 and " " j " " {cλ }π {kλ ϕj }π " < 2ε1 , "{uλ }π {aλ }π − j ∞

and there is a {dλ } ∈ N L

such that


" " j " " sup "uλ aλ − cλ kλ ϕj − dλ "

λ∈Λ

j

∞

163

< 3ε1 .

There exists a λ0 ∈ Λ such that dλ ∞ < ε1 for all λ > λ0 . If ε2 > 0 is sufficiently small, then (3.51) holds, and so |dλ (t)| < ε1 for all (λ, t) in (Λ × T) ∩ U2 , where U2 := Uε2 ;χ1 (ξ). Thus, + + j + + cλ (t)(kλ ϕj )(t)+ < 4ε1 ∀ (λ, t) ∈ (Λ × T) ∩ U2 . +uλ (t)aλ (t) − j

Further, there is a QC ∗ -neighborhood U3 of ξ such that {cjλ } |(kλ ϕj )(t)| < ε1 ∀ (λ, t) ∈ (Λ × T) ∩ U3 , j

(recall that ϕj (ξ) = 0), whence |uλ (t)aλ (t)| < 5ε1

∀ (λ, t) ∈ (Λ × T) ∩ U2 ∩ U3 ,

and finally, by (3.52), |uλ (t)|
0 (recall 3.72 to see that this is possible). Thus, we have proved that |uλ (t)|
λ0 . In the language of 3.30, this means that |δ(λ,ν/λ) H − µ| < ε for all (λ, ν/λ) ∈ Kν with λ > λ0 . If δ1 < δ, then the neighborhood Uδ1 ;ϕ,χ1 (ξ) = η ∈ QC ∗ : |η(ϕ) − ϕ(ξ)| < δ1 , |η(χ1 ) − 1| < δ1 is contained in Uδ;ϕ (ξ). By Lemma 3.31, there is a (λ, ν/λ) ∈ Kν ∩ Uδ1 ;ϕ,χ1 (ξ), and by choosing δ1 sufficiently small we can guarantee that λ > λ0 (see 3.14(d)). Thus, there exists a (λ, t) ∈ (Λ × T) ∩ Uδ;ϕ (ξ) with |(kλ χ)(t) − µ| < ε, which completes the proof.

166

3 Symbol Analysis

3.8 Local Sectoriality Continued 3.80. Theorem. Let {Kλ }λ∈Λ be any approximate identity and suppose Λ is connected. (a) If a ∈ L∞ N ×N , τ ∈ T, and conv a(Xτ ) is a line segment, then a is sectorial on Xτ ⇐⇒ {kλ a}πτ ∈ G(alg Kτπ (L∞ N ×N )) ⇐⇒ 0 ∈ / ClK {det(kλ a)}, τ . (b) If a ∈ (P QC)N ×N and ξ ∈ M (QC), then a is sectorial on Xξ ⇐⇒ {kλ a}πξ ∈ G(alg Kξπ (L∞ N ×N )) ⇐⇒ 0 ∈ / ClK {det(kλ a)}, ξ . Proof. In both cases the second equivalence “⇐⇒” and the first implication “=⇒” follow from Theorem 3.76 and Corollary 3.62, respectively. So we are left with the first implication “⇐=”. Let β be τ (∈ T) or ξ(∈ M (QC)) and suppose conv a(Xβ ) = [E, F ]. By Corollary 3.63, E and F are invertible matrices. Due to Theorem 3.4, the sectoriality of a on Xβ will follow as soon as we have shown that det(µE + (1 − µ)F ) = 0 for all µ ∈ [0, 1]. There is no loss of generality in assuming that E is the identity matrix I and that F is the Jordan canonical form, F = J (see the proof of Lemma 3.3(b)). Let ajj ∈ L∞ (j = 1, . . . , N ) denote the diagonal entries of a. Since a(x) is an upper-triangular matrix for each x ∈ Xβ , we have {det(kλ a)}πβ =

N

{kλ ajj }πβ .

j=1

Hence, if {kλ a}πβ is invertible, so also is {kλ ajj }πβ for each j. It is clear that conv ajj (Xβ ) = [1, θj ], where θj is an eigenvalue of J. Thus, by Corollary 3.78(a) for β = τ and by Theorem 3.79 for β = ξ, the line segments [1, θj ] do not contain the origin. This gives det(µI + (1 − µ)J) =

N

(µ + (1 − µ)θj ) = 0

∀ µ ∈ [0, 1].

j=1

3.81. Open problems. Does the first implication “⇐=” of part (b) in Theorem 3.80 hold under the hypothesis that a ∈ L∞ N ×N and conv a(Xξ ) is a line segment? Does that implication hold for every a ∈ (P2 QC)N ×N ? Equivalently: is Theorem 3.79 true with P QC replaced by P2 QC? Sufficiently interesting special case: is Theorem 3.79 valid for a = χE , the characteristic function of a measurable subset E of T (well, say of the “simple” kind as in 3.37)? Note that, by Corollary 3.62 and Theorem 3.76, the first implications “=⇒” as well as the second equivalences “⇐⇒” of Theorem 3.80 hold for every a ∈ L∞ N ×N .

3.8 Local Sectoriality Continued

167

3.82. Corollary. Let {Kλ }λ∈Λ be any approximate identity and suppose Λ is connected. (a) If a ∈ L∞ N ×N and conv a(Xτ ) is a line segment for each τ ∈ T, then a is locally sectorial over C ⇐⇒ {kλ a} bafz ⇐⇒ 0 ∈ / ClK {det(kλ a)}, T . (b) If a ∈ (P QC)N ×N , then a is locally sectorial over QC ⇐⇒ {kλ a} bafz ⇐⇒ 0 ∈ / ClK {det(kλ a)}, T . Proof. Theorem 3.80, Theorem 3.67 (or Corollary 3.71), and Proposition 3.73(c). Remark. By Corollary 3.69(b) and Theorem 3.76 the first implications “=⇒” and the second equivalences “⇐⇒” hold for every a ∈ L∞ N ×N . 3.83. Definition. Let OCN ×N denote the (closed) set of non-invertible matrices, OCN ×N = CN ×N \ GCN ×N . Note that OC1×1 is nothing else but the origin in C. For a ∈ (P QC)N ×N and τ = eiθ ∈ T, let βτ (a, δ) (δ > 0) denote the distance between OCN ×N and the line segment ( ( 1 δ 1 θ+δ ix ix a(e ) dx, a(e ) dx δ θ−δ δ θ and put βτ (a) := lim inf βτ (a, δ). δ→0

3.84. Proposition. Let a ∈ (P QC)N ×N , let τ ∈ T, and let {Kλ }λ∈(1,∞) be an approximate identity. (a) βτ (a) > 0 ⇐⇒ a is sectorial on Xξ for all ξ ∈ Mτ0 (QC). (b) If a ∈ GL∞ N ×N , then βτ (a) > 0 ⇐⇒ a is sectorial on Xξ for all ξ ∈ Mτ (QC) ⇐⇒ 0 ∈ / ClK {det(kλ a)}, τ . (c) If N = 1, then βτ (a) > 0 ⇐⇒ a is sectorial on Xξ for all ξ ∈ Mτ (QC) ⇐⇒ 0 ∈ / ClK (a, τ ). Proof. The second equivalences in (b) and (c) follow from Theorem 3.80 and Proposition 3.73(c). Without loss of generality assume τ = 1.

168

3 Symbol Analysis

(a) Let β1 (a) = 0. Then, by a compactness argument, there are a sequence {δn } of positive numbers tending to zero and a matrix b ∈ OCN ×N such that the limits ( 0 ( δn 1 1 ix a0 = lim a(e ) dx, a1 = lim a(eix ) dx (3.57) n→∞ δn −δ n→∞ δn 0 n exist and b ∈ [a0 , a1 ]. Put λn = π/δn . Due to the compactness of the unit ball in QC ∗ and Proposition 3.29(b) there are a subsequence {λnk } of {λn } and a ξ ∈ M10 (QC) such that (λnk , 1) → ξ (recall 3.30). From Theorem 3.36(d) we deduce that a0 = a(ξ, 0) and a1 = a(ξ, 1). Thus [a(ξ, 0), a(ξ, 1)] contains b ∈ OCN ×N , and so Proposition 3.2 in conjunction with Theorem 3.36(a), (d) implies that a is not sectorial on Xξ . Now suppose there is a ξ ∈ M10 (QC) such that a is not sectorial on Xξ . Then, again by Proposition 3.2 and Theorem 3.36(a), (d), there is a b in OCN ×N belonging to [a(ξ, 0), a(ξ, 1)]. If {λn } ⊂ (1, ∞) is any sequence such that (λn , 1) → ξ and if we set δn = π/λn , then the limits (3.57) exist and are equal to a(ξ, 0) and a(ξ, 1), respectively (Theorem 3.36(d)). Thus, β1 (a) = 0. (b) The (first) implication “⇐=” is immediate from (a). To get the reverse implication notice that a(Xξ ) is a singleton for ξ ∈ M1 (QC)\M10 (QC) (Theorem 3.36(a), (c)) and that therefore the invertibility of a yields the sectoriality of a on Xξ . (c) The (first) implication “⇐=” is again a consequence of (a). So suppose β1 (a) > 0. Then a is sectorial on Xξ for each ξ ∈ M10 (QC) by virtue of (a). Assume there is a ξ0 ∈ M1 (QC) \ M10 (QC) such that a is not sectorial on Xξ0 . Since a(Xξ0 ) is a singleton (Theorem 3.36(a), (c)), we have a(x) = 0 for all x ∈ Xξ0 . Proposition 3.29(a) for A = L∞ shows that there are (λn , tn ) in (1, ∞) × T such that |(mλn a)(tn )| = |(mλn a)(tn ) − a(x0 )|
0 ∀ τ ∈ T. (b) If a ∈ P QC, then a is locally sectorial over QC ⇐⇒ βτ (a) > 0

∀ τ ∈ T.


169

Proof. (a) Notice that locally sectorial matrix functions are necessarily in GL∞ N ×N . So the assertion is a straightforward consequence of Proposition 3.84(b). (b) Immediate from Proposition 3.84(c). Remark. We must confess that we have not been able to remove the assumption “a ∈ GL∞ N ×N ” in (a).

3.9 Notes and Comments 3.1. Matrix functions which are analytically sectorial over C were first studied by Simonenko [492]. Matrix functions which are geometrically sectorial over C were introduced by Douglas and Widom [166], who also raised the question of whether geometric sectoriality implies analytic sectoriality. Azoff and Clancey [16] then showed that the answer is no in general. 3.3–3.4. Clancey [134]. 3.6–3.10. Theorem 3.9 is known. Theorem 3.8 is due to the authors but its proof makes essential use of an argument by Roch [421]. 3.11. Toeplitz operators with P2 C symbols were first considered by Clancey [134] and Clancey, Morrel [140]. Douglas [160], [161] studied operators whose symbol a has the property that a(Xξ ) is contained in some straight line segment for each ξ ∈ M (QC). This class of symbols contains P2 QC. See also Silbermann [483]. 3.12–3.14. All the facts stated here can be found in Ahiezer [3], for example. 3.15. The asymptotic multiplicativity of the Poisson kernel on the pair (C + H ∞ , C + H ∞ ) was discovered by Douglas [159] and its asymptotic multiplicativity on the pair (QC, L∞ ) by Sarason [456]. 3.17. Sarason [454]. 3.21–3.22. These results are certainly known to specialists. The proof of Theorem 3.21 is patterned after Garnett [211, Theorem VI.1.2]. The proof of this theorem motivates why it is more convenient to define kλ ϕ as a convolution over the real line rather than over the circle. 3.23. Böttcher, Silbermann [112]. 3.24–3.25. Well-known. 3.26–3.36. All these results as well as the basic ideas for their proofs are due to Sarason [456]. Our minor contribution is the extension of Sarason’s results to the case of arbitrary approximate identities and the simplification of some of his arguments.

170

3 Symbol Analysis

3.38–3.71. This approach was developed by Silbermann [481], [483]. Theorem 3.42 and Corollary 3.44 had been earlier established in B¨ ottcher, Silbermann [106], and similar results and ideas are also in Clancey [136]. In connection with Definition 3.60 we remark that local Toeplitz operators were introduced by Douglas [161]. The results of 3.52, 3.61, 3.62–3.64, 3.67 can be found in Silbermann [481], [483]. There Corollary 3.69 is proved for the Poisson kernel; its extension to arbitrary approximate identities was obtained in B¨ ottcher, Silbermann [112]. 3.72–3.79. Böttcher, Silbermann [112]. 3.80–3.84. These results were established in Silbermann [481], [482], [483] and B¨ ottcher, Silbermann [112]. The parts (b) of 3.80 and 3.82 as well as 3.84 were proved by Sarason [456] for N = 1 and the Poisson kernel.

4 Toeplitz Operators on H 2

4.1 Fredholmness We begin by applying Theorem 3.42 to the Fredholm theory of Toeplitz operators. Although this chapter is concerned with Toeplitz operators on H 2 ∼ = 2 , p p we first state some results for Toeplitz operators on H and , because there do not arise any substantial difficulties when passing from the case p = 2 to the case p = 2. 4.1. Proposition. (a) If 1 < p < ∞, then p T : L∞ N ×N → L(HN ),

a → T (a)

is a submultiplicative embedding. (b) If 1 ≤ p < ∞ and if the norm in the space pN (Z) is chosen so that P L(pN (Z)) = 1, then p p T : MN ×N → L(N ),

a → T (a)

is a 1-submultiplicative isometry. Proof. (a) Let ajk be a finite collection of functions in L∞ N ×N . Then " " " " " T (ajk )" " " j

p L(HN )

k

" !" # " " p " " = sup " P M (ajk )P f " : f ∈ HN , f HNp ≤ 1 Lp j k N " !" # " " p " " = sup " P M (ajk )P g " : g ∈ LN , P gLpN ≤ 1 j

k

Lp N

" " " 1 " " " ≥ P M (a )P jk " " p , cp L(L ) j k

N

(4.1)

172


where cp = P L(LpN ) . Let U ±n = M (χ±n I) denote the bilateral shifts on LpN . Then " " " " P M (ajk )P " " j

k

" " " " = "U −n P M (ajk )P U n " j

k

" " " " =" (U −n P U n )(U −n M (ajk )U n )(U −n P U n )" j

k

j

k

" " " " =" (U −n P U n )M (ajk )(U −n P U n )" " " " " ≥" M (ajk )" j

(U −n P U n → I strongly + 1.1(e))

k

" " " " ajk ". = "M j

(4.2)

k

N For 1 ≤ n ≤ N , define En ∈ L(LpN ) by En : (fk )N k=1 → (δnk fk )k=1 . Given ∞ ∈ L , we have f = (fmn )N m,n=1 N ×N " " " " 2 ) = " = M (f ) E M (f )E f L∞ " 2 m n L(L N ×N N

≤ const

L(LN )

m,n

M (fmn )L(L2 )

m,n

= const

M (fmn )L(Lp )

(Proposition 2.2)

m,n

≤ const

Em M (f )En L(LpN )

m,n

≤ const M (f )L(LpN ) . Hence, (4.1), (4.2), (4.3) give " " " " T (ajk )" " j

k

p L(HN )

(4.3)

" " " " ≥ const " ajk " j

k

L∞ N ×N

.

(4.4)

In particular, T (a)L(HNp ) ≥ const aL∞ N ×N

∀ a ∈ L∞ N ×N .

(4.5)

∀ a ∈ L∞ N ×N .

(4.6)

On the other hand, it is readily seen that T (a)L(HNp ) ≤ const aL∞ N ×N

The mapping T is clearly linear, (4.5) shows that T is one-to-one, (4.6) implies that T is continuous, and (4.5), (4.6) together imply that T has a closed image.

4.1 Fredholmness

173

Thus, T is an embedding. Finally, by combining (4.4) and (4.6) we see that T is submultiplicative. (b) The proof is essentially the same as that of part (a). We have " " " " " " " " T (ajk )" p ≥ " P M (ajk )P " p " j

L(N )

k

" " " " ≥ "M ajk " j

k

j

L(p N (Z))

L(N (Z))

k

" " " " ≥ "T ajk " j

k

L(p N)

,

which gives the 1-submultiplicativity of T . Since, in particular, T (a)L(pN ) ≥ M (a)L(pN (Z)) ≥ T (a)L(pN ) and, by definition, M (a)L(pN (Z)) = aMNp ×N , we finally conclude that T is an isometry. 2 4.2. Corollary. If a ∈ L∞ N ×N is sectorial then T (a) is invertible on HN .

Proof. This follows from the preceding proposition and Corollary 3.62, applied 2 ), i = T . with A = L∞ , F = X, B = L(HN Remark. Recall that “sectorial” means “analytically sectorial.” Azoff and Clancey [16] showed that there exist geometrically sectorial matrix functions 2 a ∈ L∞ 2×2 such that T (a) is not even semi-Fredholm on H2 . 4.3. Corollary. (a) If 1 < p < ∞ and if A is a closed subalgebra of L∞ N ×N , then · algL(HNp ) T (A) = T (A) + QT (A) and algL(HNp ) T (A)/QT (A) is homeomorphically isomorphic to A. p (b) If 1 ≤ p < ∞ and if A is a closed subalgebra of MN ×N , then ·

algL(pN ) T (A) = T (A) + QT (A) and algL(pN ) T (A)/QT (A) is homeomorphically isomorphic to A. Proof. Proposition 4.1, Theorem 3.42, Corollary 3.44. The next proposition provides a description of the quasicommutator ideal QT (L∞ N ×N ). In accordance with 3.43, we let ST denote the projection of ∞ ∞ alg T (L∞ N ×N ) onto T (LN ×N ) parallel to QT (LN ×N ). Analogously ST is unp derstood on alg T (MN ×N ). p and pN as 4.4. Proposition. For n ≥ 0, define V n and V (−n) on HN

V n = T (χn I),

V (−n) = T (χ−n I).


174

(a) If 1 < p < ∞ and A ∈ algL(HNp ) T (L∞ N ×N ), then the strong limit s- lim V (−n) AV n n→∞

exists and equals ST (A). p (b) If 1 ≤ p < ∞ and A ∈ algL(pN ) T (MN ×N ), then the strong limit

s- lim V (−n) AV n n→∞

exists and equals ST (A). (c) Under the hypotheses of (a) or (b), p (−n) A ∈ QT (L∞ AV n = 0. N ×N ) (resp. QT (MN ×N )) ⇐⇒ s- lim V n→∞

(d) The only compact Toeplitz operator on p ∞ Moreover, if a ∈ MN ×N and b ∈ LN ×N , then T (a)L(pN ) = T (a)Φ(pN ) ,

p HN

or pN is the zero operator.

T (b)L(HNp ) ≤ cp T (b)Φ(HNp ) ,

where cp := P L(Lp ) . p , then Proof. (a) If f ∈ HN T (ajk ) V n f = P U −n P M (ajk )P U n P f, V (−n) j

j

k

k

and the arguments used in the proof of Proposition 4.1(a) show that this p to converges in the norm of HN ajk P f = T ajk f = ST T (ajk ) f. PM j

k

j

k

j

k

This proves the assertion for the case where A is a finite product-sum of Toeplitz operators. The general case now follows from 1.1(d). (b) The proof is that of part (a). (c) Immediate from (a) resp. (b). (d) If K is a compact operator, then V (−n) KV n → 0 uniformly, because p → 0 strongly. It is clear that V (−n) T (c)V n = T (c) for every c ∈ MN V ×N ∞ resp. c ∈ LN ×N . Thus, by 1.1(e), " " T (c) ≤ lim inf "V (−n) (T (c) + K)V n " n→∞ ≤ sup P V (−n) T (c) + K V n = P T (c) + K (−n)

n

for every compact operator K.

4.1 Fredholmness

175

4.5. Proposition. (a) Let 1 < p < ∞ and let B be a closed subalgebra of C + H ∞ containing C. Then p QT (BN ×N ) = C∞ (HN ).

(b) Let 1 < p < ∞ and let B be a closed subalgebra of Cp + Hp∞ containing Cp . Then QT (BN ×N ) = C∞ (pN ). Proof. (a) Formula (2.18) and Theorem 2.42(a) imply that QT (BN ×N ) ⊂ p p ). So it remains to show that C∞ (HN ) ⊂ QT (CN ×N ). By virtue of C∞ (HN Lemma 3.47 we may assume that N = 1. Let K ∈ C∞ (H p ) and notice first that (I − T (χn )T (χ−n ))K converges uniformly to K as n → ∞, because T (χn )T (χ−n ) → 0 strongly. Since I −T (χn )T (χ−n ) has finite rank, it remains to show that C0 (H p ) ⊂ QT (C). This on its hand will follow as soon as we have shown that every operator L ∈ L(H p*) of the form Lf = (f, χk )χn (k, n ≥ 0) is in QT (C), where (f, χk ) := 1/(2π) T f χ−k dm. But this is immediate from the identity L = T (χn ) T (χ1 χ−1 ) − T (χ1 )T (χ−1 ) T (χ−k ). (b) The proof is the same. Remark. Let Lf := (f, g)e0 , where g = (1, 1, . . .) ∈ ∞ . Then L ∈ C0 (1 ), but it can be shown that L is not in algL(1 ) T (W ). Using Proposition 4.4(c) one can easily prove that QT (W ) = C∞ (1 ) ∩ algL(1 ) T (W ). We are now in a position to prove the following important spectral inclusion theorem (recall Theorems 2.30 and 2.93). 2 4.6. Corollary. If A ∈ algL(HN2 ) T (L∞ N ×N ) is Fredholm on HN , then

SmbT (A) ∈ GL∞ N ×N . 2 ∞ In particular, if a ∈ L∞ N ×N and T (a) ∈ Φ(HN ), then a ∈ GLN ×N .

Proof. We suppress the subscript N . If A ∈ Φ(H 2 ), then A + C∞ (H 2 ) belongs to G(L(H 2 )/C∞ (H 2 )). Proposition 4.5 implies that C∞ (H 2 ) ⊂ alg T (H 2 ) and so 1.26(d) shows that A + C∞ (H 2 ) must be invertible in alg T (L∞ )/C∞ (H 2 ). Moreover, Proposition 4.5 even states that C∞ (H 2 ) ⊂ QT (L∞ ), therefore A + QT (L∞ ) must belong to G(alg T (L∞ )/QT (L∞ )). Corollary 4.3 applied with A = L∞ N ×N completes the proof. The next two corollaries settle the Fredholm theory for operators belonging to the closed algebra generated by Toeplitz operators with C + H ∞ symbols. 4.7. Corollary. (a) Let 1 < p < ∞ and let B be a closed subalgebra of C +H ∞ containing C. Then

176

4 Toeplitz Operators on H 2 ·

p algL(HNp ) T (BN ×N ) = T (BN ×N ) + C∞ (HN ) p and algL(HNp ) T (BN ×N )/C∞ (HN ) is homeomorphically isomorphic to BN ×N .

(b) Let 1 < p < ∞ and let B be a closed subalgebra of Cp + Hp∞ containing Cp . Then ·

algL(pN ) T (BN ×N ) = T (BN ×N ) + C∞ (pN ) and algL(pN ) T (BN ×N )/C∞ (pN ) is isometrically isomorphic to BN ×N (as a p subalgebra of MN ×N ). Proof. Corollary 4.3 and Proposition 4.5 show that the corresponding algebras are homeomorphically isomorphic. On combining Proposition 4.1(b) and Proposition 4.4(d) we see that in the case (b) the isomorphism is even isometric. 4.8. Corollary. (a) Let 1 < p < ∞ and A ∈ algL(HNp ) T ((C + H ∞ )N ×N ). Then p ) ⇐⇒ det SmbT (A) ∈ G(C + H ∞ ). A ∈ Φ(HN (b) Let 1 < p < ∞ and A ∈ algL(pN ) T ((Cp + Hp∞ )N ×N ). Then A ∈ Φ(pN ) ⇐⇒ det SmbT (A) ∈ G(Cp + Hp∞ ). (c) Under the hypotheses (a) or (b), if A is Fredholm then Ind A = Ind T (det SmbT (A)). Proof. (a), (b) The implications “⇐=” follow from Corollary 4.7. To prove the reverse implications, note first that, again by Corollary 4.7, A = T (SmbT (A)) + K with some compact operator K. Thus, if A is Fredholm, then so is T (SmbT (A)) and Theorem 2.94 completes the proof. (c) Note that

Ind A = Ind T (SmbT (A)) + K = Ind T SmbT (A)

and apply Theorem 2.94. 4.9. Remark. If A ∈ algL(1N ) T (WN ×N ), then A ∈ Φ(1N ) ⇐⇒ det SmbT (A) ∈ GWN ×N . The implication “⇐=” results from Corollary 4.3(b) and the remark in 4.5. That Ind A = Ind T (SmbT (A)) = −ind det SmbT (A) can be shown as above, and so an index perturbation argument yields the implication “=⇒”. We now show how the machinery developed in Chapter 3 to study harmonic approximation can be applied to the Fredholm theory of (block) 2 . Toeplitz operators on HN

4.1 Fredholmness

177

4.10. Essentialization. We know from Proposition 4.1 that the mapping 2 T : L∞ N ×N → L(HN ),

a → T (a)

2 is a 1-submultiplicative isometry. Proposition 4.5 tells us that C∞ (HN ) is ∞ contained in alg T (LN ×N ) and is therefore a closed two-sided ideal of that algebra. Finally, Proposition 4.4 shows that 2 ST (K) = s- lim V (−n) KV n = 0 ∀ K ∈ C∞ (HN ), n→∞

because V n → 0 weakly (recall 1.1(f)). So Theorem 3.52 can be used to see that the mapping π ∞ T π : L∞ N ×N → alg T (LN ×N ),

2 a → T π (a) := T (a) + C∞ (HN )

is a 1-submultiplicative quasi-embedding. From Proposition 4.1(b) and Proposition 4.4(d) we deduce that T π is even an isometry. Since alg T π (L∞ N ×N ) is 2 2 )/C∞ (HN ), we conclude from a C ∗ -subalgebra of the Calkin algebra L(HN 1.26(d) that, for a ∈ L∞ N ×N , 2 ) ⇐⇒ T π (a) ∈ G(alg T π (L∞ T (a) ∈ Φ(HN N ×N )).

(4.7)

4.11. Localization. Let F be a closed subset of X = M (L∞ ). In accordance with 3.58 we define IF = a ∈ L∞ JFπ = closidalg T π (L∞ T π (IF ). N ×N : a|F = 0 , N ×N ) Lemma 3.59(a) implies that ST π (JFπ ) ⊂ JFπ . Consequently, again by Theorem 3.52, the mapping π ∞ TFπ : L∞ N ×N → alg TF (LN ×N ),

a → TFπ (a) := T π (a) + JFπ

is a 1-submultiplicative quasi-embedding with Ker TFπ = IF (Lemma 3.59(b)). Keeping in mind 3.60, we call TFπ (a) (a ∈ L∞ N ×N ) a local Toeplitz operator. Theorem 3.61 specializes to give that TFπ (a) = a|F ∀ a ∈ L∞ N ×N .

(4.8)

In case F is a fiber Xξ , where ξ ∈ M (B) and B is a C ∗ -subalgebra of L∞ containing the constants, we also have from (3.43) IXξ = closidL∞ cIN ×N : c ∈ B, c(ξ) = 0 , N ×N where IN ×N denotes the N × N identity matrix. In that case we abbreviate π π , TX (a) to Iξ , Jξπ , Tξπ (a), respectively. IXξ , JX ξ ξ Local Toeplitz operators will be studied in some more detail later. In the meanwhile we confine ourselves to stating the following.

178


4.12. Theorem. Let B be a C ∗ -subalgebra of QC containing the constants. Then if a ∈ L∞ N ×N , 2 ) ⇐⇒ Tξπ (a) ∈ G(alg Tξπ (L∞ T (a) ∈ Φ(HN N ×N )) ∀ ξ ∈ M (B).

Proof. This is a consequence of (4.7) and Theorem 3.67, applied with A = L∞ , 2 2 )/C∞ (HN ), i = T π . Note that T π (ca) = T π (c)T π (a) for all c in B = L(HN ∞ B ⊂ C + H and a ∈ L∞ by virtue of formula (2.18) and Theorem 2.42(a). 4.13. Corollary. Let B be a C ∗ -subalgebra of QC containing the constants and let a ∈ L∞ N ×N . (a) If for each fiber Xξ , ξ ∈ M (B), there exists a bξ ∈ L∞ N ×N such that 2 2 ), then T (a) ∈ Φ(HN ). a|Xξ = bξ |Xξ and T (bξ ) ∈ Φ(HN 2 (b) If a is locally sectorial over B, then T (a) ∈ Φ(HN ).

Proof. (a) If T (bξ ) is Fredholm, then Tξπ (bξ ) is invertible in alg Tξπ (L∞ N ×N ) by the preceding theorem. From (4.8) we deduce that Tξπ (a) = Tξπ (bξ ), and again applying the preceding theorem we get the Fredholmness of T (a). (b) We have Tξπ (a) ∈ G(alg Tξπ (L∞ N ×N )) for all ξ ∈ M (B) due to Corollary 3.62. Note that part (a) of this corollary is identical with Theorem 2.96. For (b) see also 4.31.

4.2 Stable Convergence 4.14. Definition. Let Λ be either of the index sets Λ = {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N) or Λ = (r0 , ∞) (r0 ∈ R+ ). Let X be a Banach space, let A ∈ L(X), and let {Aλ }λ∈Λ be a (generalized) sequence of operators Aλ ∈ L(X). The sequence {Aλ }λ∈Λ is said to converge stably to A if (a) Aλ → A strongly on X as λ → ∞, (b) there is a λ0 ∈ Λ such that Aλ ∈ GL(X) for all λ > λ0 , (c) sup A−1 λ L(X) < ∞. λ>λ0

For instance, if {Kλ }λ∈Λ is an approximate identity and a ∈ L∞ N ×N , then M (kλ a) converges stably to M (a) on L2N if and only if {kλ a} is bounded away from zero (see 3.55 for (b), (c) and Proposition 3.40(a) for (a)). Here our concern is the study of questions connected with the stable convergence 2 . On this basis we shall then propose an approach to of T (kλ a) to T (a) on HN establishing index formulas for Toeplitz operators with locally sectorial matrix symbol.

4.2 Stable Convergence

179

∞ 4.15. Algebraization. Let BN,N denote the collection of all (generalized) 2 sequences {Aλ }λ∈Λ of operators Aλ ∈ L(HN ) such that sup Aλ L(HN2 ) < ∞. λ∈Λ

On defining α{An } := {αAn }, {Aλ } + {Bλ } := {Aλ + Bλ }, {Aλ }{Bλ } := {Aλ Bλ }, {An }∗ := {A∗n }, and {Aλ } := sup Aλ L(HN2 )

(4.9)

λ∈Λ

∞ we make BN,N become a C ∗ -algebra. Let

∞ 2 : there exists an A ∈ L(HN ) such that BN,N = {Aλ } ∈ BN,N ∗ ∗ 2 Aλ → A and Aλ → A strongly on HN as λ → ∞ . ∞ It is not difficult to see that BN,N is a C ∗ -subalgebra of BN,N . If {Aλ } ∈ BN,N converges stably to its strong limit A, then A is in2 vertible. Indeed, we have f ≤ A−1 λ Aλ f for all f ∈ HN , hence −1 Aλ f ≥ (1/M )f with M := sup Aλ , and passage to the limit λ → ∞ λ∈Λ

2 ; the stable convergence of Aλ to A gives Af ≥ (1/M )f for all f ∈ HN ∗ implies the stable convergence of Aλ to A∗ , and so the same argument applied 2 ; the conclusion to A∗λ yields the inequality A∗ f ≥ (1/M )f for all f ∈ HN is that A must be invertible.

4.16. Essentialization. Now put MN,N = {Aλ } ∈ BN,N : Aλ L(HN2 ) → 0 as λ → ∞ , 2 JN,N = {Aλ } ∈ BN,N : Aλ = K + Cλ , K ∈ C∞ (HN ), {Cλ } ∈ MN,N . It can be checked straightforwardly that MN,N is a closed two-sided ideal of ∞ and BN,N and that JN,N is a closed two-sided ideal of BN,N . both BN,N ∞ ∞ ∞ := B1,1 ), we Since (4.9) defines an admissible norm on BN ×N (with B ∞ ∞ have BN,N = BN ×N , BN,N = BN ×N , MN,N = MN ×N , JN,N = JN ×N , where B := B1,1 , M := M1,1 , J := J1,1 . For {Aλ } ∈ BN ×N , let {Aλ }πM and {Aλ }πJ denote the cosets {Aλ } + MN ×N and {Aλ } + JN ×N , respectively. A little thought shows that, for {Aλ } ∈ BN ×N , ∞ Aλ converges strongly to its limit ⇐⇒ {Aλ }πM ∈ G(BN ×N /MN ×N ).

4.17. Proposition. Let {Aλ } ∈ BN ×N and let A denote the strong limit of Aλ as λ → ∞. Then the following are equivalent: (i) Aλ converges stably to A. (ii) {Aλ }πM ∈ G(BN ×N /MN ×N ). 2 (iii) A ∈ GL(HN ) and {Aλ }πJ ∈ G(BN ×N /JN ×N ).

180


Proof. We drop the subscript N × N . (i) ⇐⇒ (ii): Since B/M is a C ∗ -subalgebra of B∞ /M, {Aλ }πM is invertible in B/M if it is invertible in B∞ /M (1.26(d)). (i)+(ii) =⇒ (iii): The invertibility of A was shown in 4.15 and the invertibility of {Aλ }πJ is a consequence of (ii). (iii) =⇒ (ii): There is a {Bλ } ∈ B such that K ∈ C∞ (H 2 ),

Bλ Aλ = I + K + Cλ ,

{Cλ } ∈ M.

Passage to the strong limit λ → ∞ gives BA = I + K. Hence (Bλ − KA−1 )Aλ = I + K + Cλ − KA−1 Aλ = I + K + Cλ − KA−1 A + Cλ = I + Cλ + Cλ with some {Cλ } ∈ M, because KA−1 ∈ C∞ (H 2 ) and A∗λ → A∗ strongly (recall 1.3(d)). It follows that {Aλ }πM is left-invertible in B/M. The rightinvertibility can be shown analogously. 4.18. alg T K(AN ×N ). Let A be a closed subalgebra of L∞ containing the constants, put A = AN ×N , and let {Kλ } be an approximate identity. (a) If a ∈ L∞ N ×N , then {T (kλ a)} ∈ BN ×N . This follows from the fact that {kλ a} ∈ AN ×N (Proposition 3.40(a)). (b) The mapping defined by T K : L∞ N ×N → BN ×N ,

a → {T (kλ a)}

is a 1-submultiplicative isometry. That T K is an isometry follows from the equalities = sup kλ aL∞ = sup T (kλ a)L(HN2 ) , aL∞ N ×N N ×N λ

(4.10)

λ

the first of which holds by virtue of Proposition 3.40(b), while the second is true because of Proposition 4.1(b) with p = 2. Since " " " " " " " " aij " = "T aij " (by (4.10)) " T kλ i

j

i j " " " " T (aij )" ≤ " i

(Proposition 4.1(b))

j

" " " " ≤ lim inf " T (kλ aij )" λ→∞

i

j

" " " " ≤ sup " T (kλ aij )" λ

" i j " " " T (kλ aij ) ", =: " i

j

it follows that T K is 1-submultiplicative.

(by 1.1(e))


181

(c) So Theorem 3.42 gives that ·

alg T K(A) = T K(A) + QT K (A).

(4.11)

Because T K is 1-submultiplicative, the projection ST K has norm 1. Let LB denote the (linear and continuous) mapping which is defined by 2 ), LB : BN ×N → L(HN

{Bλ } → s−lim Bλ . λ→∞

The mappings ST K and SmbT K , which are given at finite product-sums (correctly) by {T (kλ aij )} → T kλ aij , ST K : i

j

SmbT K :

i

i

{T (kλ aij )} →

j

i

j

aij ,

j

can then be represented in the form ST K = T K ◦ SmbT ◦ LB |alg T K(A), SmbT K = SmbT ◦ LB |alg T K(A). (d) The restriction of LB to alg T K(A) will be denoted by Φ. It is clear that Φ is a continuous algebraic homomorphism whose range is dense in alg T (A). Thus, if A is a C ∗ -algebra, then Φ is a star-homomorphism of alg T K(A) onto alg T (A) (see 1.26(e)). Notice that Φ(QT K (A)) is contained in QT (A). (e) Define the mapping Ψ at finite product-sums by {T (kλ aij )} → {kλ aij }. Ψ: i

j

i

j

Because " " " " " " " " {kλ aij }" := sup " kλ aij " " i

j

λ

i

j

" " " " kλ aij " (Proposition 4.1(b)) = sup "T λ

i

j

" " " " T (kλ aij )" ≤ sup " λ

i

(Proposition 4.1(b))

j

" " " " {T (kλ aij )}", =: " i

j

it follows that the mapping Ψ extends to a continuous algebraic homomorphism of alg T K(A) into alg K(A). Moreover, if A is a C ∗ -algebra, then

182


Ψ is a star-homomorphism of alg T K(A) onto alg K(A). We always have Ψ (QT K (A)) ⊂ QK (A). Finally, given µ ∈ Λ define the continuous algebraic homomorphism Fixµ by Fixµ : alg T K(A) → alg T (A), {Aλ } → Aµ . It is easy to see that we can now write (Ψ {Aλ })µ = (SmbT ◦ Fixµ ){Aλ }. (f) Thus, we have the following picture:

π 4.19. alg T Kπ M (AN ×N ) and alg T KJ (AN ×N ). (a) For a closed subalge∞ bra of L containing C, put A = AN ×N and

MA N ×N = MN ×N ∩ alg T K(A),

JNA×N = JN ×N ∩ alg T K(A)

A (also recall Lemma 3.47). Thus, both MA N ×N and JN ×N are closed two-sided ideals of alg T K(A).

(b) We have ST K (JNA×N ) = {0}. Indeed, if {Bλ } ∈ JNA×N , then Bλ = 2 ) ⊂ QT (A) (Proposition 4.5) and {Cλ } ∈ MN ×N , K + Cλ with K ∈ C∞ (HN whence ST K {Bλ } = (T K ◦ SmbT ◦ LB ){Bλ } = (T K ◦ SmbT )(K) = T K(0) = 0. (c) So Theorem 3.52 gives the following: the mappings π T KM : A → alg T K(A)/MA N ×N ,

a → {T (kλ a)} + MA N ×N ,

π T KJ : A → alg T K(A)/JNA×N ,

a → {T (kλ a)} + JNA×N ,

are 1-submultiplicative quasi-embeddings and π alg T KM (A) = alg T K(A)/MA N ×N ,

π alg T KJ (A) = alg T K(A)/JNA×N .


183

π π The mappings T KM and T KJ are actually isometries: if {T (kλ a)} ∈ JNA×N , 2 ) and this can only happen if a = 0 then T (kλ a) converges to T (a) ∈ C∞ (HN (Proposition 4.4(d)). For {Bλ } ∈ alg T K(A), we denote the cosets {Bλ } + A π π MA N ×N and {Bλ } + JN ×N by {Bλ }M and {Bλ }J , respectively.

(d) Let θ denote the (continuous) algebraic homomorphism defined by π π θ : alg TM K(A) → alg T KJ (A),

{Bλ }πM → {Bλ }πJ .

(e) The mapping Φ defined in 4.18(d) maps the ideal JNA×N into the ideal 2 ). We can therefore define the quotient mapping C∞ (HN Φπ : alg TJπ K(A) → alg T π (A), which is the extension to the whole algebra of the mapping given at finite product-sums (correctly) by Φπ :

i

j

{T (kλ aij )}πJ →

i

T π (aij ).

j

Clearly, Φπ (QT KπJ (A)) ⊂ QT π (A). Finally, Φπ is a surjective algebraic starhomomorphism whenever A is a C ∗ -algebra. (f) The mapping Ψ introduced in 4.18(c) has the property that Ψ (JNA×N ) ⊂ Indeed, if {K + Cλ } ∈ JNA×N , then

NNA×N .

(Ψ {K + Cλ })µ = (SmbT ◦ Fixµ ){K + Cλ } = SmbT (K + Cµ ) = SmbT (Cµ ) and {SmbT (Cλ )}λ∈Λ is obviously in NNA×N whenever {Cλ }λ∈Λ ∈ MA N ×N . Thus, the quotient mapping π Ψ π : alg T KJ (A) → alg Kπ (A)

can be defined. It is the extension to the whole algebra of the mapping given on finite product-sums (correctly) by Ψπ :

{T (kλ aij )}πJ → {kλ aij }π . i

j

i

j

Again we have Ψ π (QT KπJ (A)) ⊂ QKπ (A), and if A is a C ∗ -algebra, then Ψ π is a surjective algebraic star-homomorphism. (g) Thus, the “quotient picture” or the “essentialization” of 4.18(f) looks as in the following diagram:

184


4.20. Theorem. Let A be a C ∗ -subalgebra of L∞ containing C, let {Bλ } be in alg T K(AN ×N ), and let B ∈ alg T (AN ×N ) denote the strong limit of Bλ as λ → ∞. Then the following are equivalent: (i) Bλ converges stably to B. π (AN ×N )). (ii) {Bλ }πM ∈ G(alg T KM 2 π ) and {Bλ }πJ ∈ G(alg T KJ (AN ×N )). (iii) B ∈ GL(HN

Proof. The proof is a combination of that of Proposition 4.17 and some standard C ∗ -arguments as they have already been used in the proof of Theorem 3.56. The following proposition provides two important situations in which the quasicommutator ideal of alg T K(AN ×N ) can be identified. 4.21. Proposition. (a) Let B be a closed subalgebra of QC containing C and let {Kλ }λ∈Λ be any approximate identity. Then QT K (BN ×N ) = JNB×N . (b) Let B be a closed subalgebra of C + H ∞ containing C and let {Kλ }λ∈Λ be the approximate identity generated by the Poisson kernel. Then QT H (BN ×N ) = JNB×N . Proof. By virtue of Lemma 3.47 it suffices to consider the case N = 1. If {Aλ } ∈ J B , then ST K {Aλ } = 0 by 4.19(b). Consequently, J B ⊂ QT K (B). Now let ϕ, ψ ∈ B. Then, by formula (2.18), T (kλ ϕ)T (kλ ψ) − T (kλ (ϕψ)) ) = T [(kλ ϕ)(kλ ψ) − kλ (ϕψ)] − H(kλ ϕ)H(kλ ψ)

(4.12)


185

) where f)(t) := f (1/t) for t ∈ T). The first term in (note that (kλ ψ)) = kλ ψ, (4.12) converges uniformly to zero as λ → ∞, since {Kλ } is asymptotically multiplicative on the pair (B, B) (Theorem 3.23 resp. Theorem 2.62(a)). Further, we have ψ) = c + h with c ∈ C and h ∈ H ∞ . For fixed λ, the operator a → kλ a is bounded on L2 , and because it maps PA into PA (by (3.5)), it maps H 2 into H 2 . Hence, kλ h ∈ H ∞ and therefore ) = H(kλ ϕ)H(kλ c) H(kλ ϕ)H(kλ ϕ) converges uniformly to H(ϕ)H(c) ∈ C∞ (H 2 ), since H(kλ ϕ) converges strongly to H(ϕ) and H(kλ c) converges uniformly to H(c). Thus, we have shown that every quasicommutator in alg T K(B) belongs to J , which implies that QT K (B) ⊂ J B . 4.22. Corollary. (a) Let {Aλ } ∈ alg T K(QCN ×N ), where {Kλ }λ∈Λ is any approximate identity. Then Aλ converges stably to its strong limit A as λ → ∞ 2 ). if and only if A ∈ GL(HN (b) Let {Aλ } ∈ alg T H((C + H ∞ )N ×N ). Then Aλ converges stably to its 2 ). strong limit A as λ → ∞ if and only if A ∈ GL(HN Proof. (a) We showed in 4.15 that A is invertible if Aλ converges stably to A. To get the reverse implication it suffices by virtue of Theorem 4.20 to show that {Aλ }πJ ∈ G(alg T K(QCN ×N )/JNQC ×N ), which, in view of Proposition 4.21, Corollary 3.44, and the fact that Ker T K = {0}, is equivalent to the requirement that SmbT K {Aλ } be in GQCN ×N . But we know from 4.18(c) that SmbT K {Aλ } = SmbT A, and the invertibility of SmbT A results from Corollary 4.6 in conjunction with 1.26(d). 2 ), then SmbT A is invertible in (C + H ∞ )N ×N by Corol(b) If A ∈ GL(HN lary 4.8. As in the proof of part (a), this implies that {Aλ }πJ is invertible in π ((C + H ∞ )N ×N ), which, in turn, gives the invertibility of {Aλ }πJ in alg T HJ π alg T HJ (L∞ N ×N ). So Theorem 4.20 applies again.

4.23. Localization. Let F be a closed subset of X = M (L∞ ), let A be a closed subalgebra of L∞ containing C, and put A = AN ×N . (a) In accordance with 3.58 we define π (IF ). IF = a ∈ A : a|F = 0 , JFπ = closidalg T KπJ (A) T KJ From Lemma 3.59 and Theorem 3.52 we deduce that T KFπ : A → alg T KFπ (A) = alg T Kπ (A)/JFπ , a → {T (kλ a)}πF := {T (kλ a)}πJ + JFπ is a 1-submultiplicative quasi-embedding whose kernel is IF . If F is a fiber Xξ , where ξ ∈ M (B) and B is a C ∗ -algebra of L∞ containing the constants, π and {Aλ }πXξ to T Kξπ and {Aλ }πξ , respectively. we abbreviate T KX ξ

186


(b) It can be checked straightforwardly that Φπ and Ψ π (defined in 4.19(e) and 4.19(f)) map JFπ into the corresponding ideal JFπ of alg T π (A) and alg Kπ (A). Therefore we can define the quotient mappings ΦπF and ΨFπ in a natural way. These mappings are continuous algebraic homomorphisms, which are surjective whenever A is a C ∗ -algebra. (c) So we arrive at the following “localization” of the diagram 4.18(f):

4.24. Theorem. Let {Kλ }λ∈Λ be any approximate identity, let F be a closed subset of X, and let a ∈ L∞ N ×N . Then the following spectral inclusions hold: sp TFπ (a) ⊂ sp {T (kλ a)}πF ⊃ sp {kλ a}πF ∪ ∪ ∪ sp (a|F ) = sp (a|F ) = sp (a|F ). In the case N = 1 we have sp (a|F ) = a(F ) and, in addition, sp {T (kλ a)}πF ⊂ conv a(F ). Proof. The inclusions in the first row follow from the fact that ΦπF and ΨFπ are algebraic homomorphisms. The vertical inclusions result from Corollary 3.63. Finally, the last inclusion is a consequence of Corollary 3.62 (or Corollary 3.64). 4.25. Lemma. Suppose ϕ ∈ QC and let {Kλ }λ∈Λ be any approximate identity. Then T (kλ (ϕa)) − T (kλ ϕ)T (kλ a) ∈ J . Proof. Due to Theorem 3.23, T (kλ (ϕa)) − T (kλ ϕ · kλ a)L(H 2 ) → 0. By formula (2.18), T (kλ ϕ · kλ a) − T (kλ ϕ)T (kλ a) = H(kλ ϕ)H(kλ ) a). We have ϕ = c + h with c ∈ C and h ∈ H ∞ , and since kλ h ∈ H ∞ (see the proof of Proposition 4.21), we conclude that a) = H(kλ c − c)H(kλ ) a) + H(c)H(kλ ) a). H(kλ ϕ)H(kλ )

(4.13)

4.3 Index Computation

187

Because H ∗ (kλ ) a) converges strongly to H ∗ () a) (see the proof of Proposition 3.40(a)) and kλ c converges uniformly to c, it follows that the first term in (4.13) converges uniformly to zero while the second converges uniformly to H(c)H() a) ∈ C∞ (H 2 ). 4.26. Theorem. Let {Kλ }λ∈Λ be any approximate identity and let a be in ∗ L∞ N ×N . Suppose B is a C -subalgebra of QC containing the constants. (a) If for each fiber Xξ , ξ ∈ M (B), there is a bξ ∈ L∞ N ×N such that 2 ), a|Xξ = bξ |Xξ and T (kλ bξ ) converges stably to T (bξ ) and if T (a) ∈ GL(HN then T (kλ a) converges stably to T (a). (b) If a is locally sectorial over B, then for T (kλ a) to converge stably to 2 . T (a) it is necessary and sufficient that T (a) be invertible on HN Proof. Without loss of generality suppose C ⊂ B. (a) Theorem 4.20(a) (with A = L∞ ) and Theorem 3.67 (with A = L∞ , π π (L∞ B = alg T KJ N ×N ), i = T KJ ), whose hypothesis (c) is fulfilled by the preceding lemma, imply that {T (kλ bξ )}πξ is invertible in alg T Kξπ (L∞ N ×N ) for each ξ ∈ M (B). Theorem 3.61 shows that {T (kλ a)}πξ = {T (kλ bξ )}πξ and so, π (L∞ again by Theorem 3.67, {T (kλ a)}πJ ∈ G(alg T KJ N ×N )). Now the assertion follows from Theorem 4.20. (b) We have {T (kλ a)}πξ ∈ G(alg T Kξπ (L∞ N ×N )) as a consequence of Corollary 3.62. So it remains to apply Theorem 3.67 and Theorem 4.20.

4.3 Index Computation We now establish an index formula for operators A belonging to a relatively large subclass of alg T (L∞ N ×N ). However, note that unless A is a scalar Toeplitz operator, A = T (a) with a ∈ L∞ , the knowledge of an index formula does, in general, not solve the invertibility problem. In particular, although it is an easy matter to derive the index formulas for A ∈ alg T (C) or A ∈ T (CN ×N ) (recall Corollary 4.8), it is a very delicate question to decide whether such an operator is invertible. 4.27. Definition. Let {Kλ }λ∈Λ be an approximate identity whose index set is Λ = {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N) or Λ = (r0 , ∞) (r0 ∈ R+ ). In either case, conv Λ, the convex hull of Λ, is a connected subset of R+ . Now let {aλ }λ∈Λ ∈ alg K(L∞ N ×N ) and assume aλ is well defined for all λ ∈ conv Λ. For example, if aλ = kλ b, where b ∈ L∞ N ×N , then aλ is defined for all λ ∈ R+ in a natural manner. If Λ is connected, then aλ is of course also well defined for all λ ∈ conv Λ, since Λ = conv Λ. We shall say that {aλ }λ∈Λ is bounded away from zero on conv Λ if there is a λ0 ∈ conv Λ such that {aλ }λ∈(λ0 ,∞) is bounded away from zero. In that case det aλ (t) = 0 for all (λ, t) ∈ (λ0 , ∞) × T. By Proposition 3.73(a) and 2.41(b), ind det aλ depends

188


on λ ∈ (λ0 , ∞) continuously, and since (λ0 , ∞) is connected, it follows that lim ind det aλ is a ind det aλ is constant for λ > λ0 . This implies that λ→∞,λ∈Λ

well-defined integer. This integer will simply be denoted by ind {det aλ }. Note that Corollary 3.69(b) actually states that {kλ a} is bounded away ∗ from zero on conv Λ whenever a ∈ L∞ N ×N is locally sectorial over a C subalgebra of QC. A similar remark can be made for Theorem 4.26(b). 4.28. Theorem. Let {Aλ } ∈ alg T K(L∞ N ×N ), put A := Φ{Aλ } := s−lim Aλ ∈ alg T (L∞ N ×N ) , λ→∞ {aλ } := Ψ {Aλ } ∈ alg K(L∞ N ×N ) and suppose {Aλ }πJ ∈ G(alg T KJ (L∞ N ×N )) and Λ is connected. Then 2 (a) A ∈ Φ(HN ),

(b) {aλ } is bounded away from zero, (c) Ind A = −ind {det aλ }. Proof. (a), (b) Since Φπ and Ψ π are algebraic homomorphisms, the invertibility of {Aλ }πJ implies that of both Aπ = Φπ {Aλ }πJ and {aλ }π = Ψ π {Aλ }πJ . This in turn gives (a) and (b) at once. (c) Let Ind A = κ. Define χ(t) = diag (tκ , 1, . . . , 1), t ∈ T. Then T (χ) 2 2 ) and Ind T (χ) = −κ. Consequently, AT (χ) ∈ Φ(HN ) and belongs to Φ(HN 2 Ind AT (χ) = 0. It follows that there is an R0 ∈ C∞ (HN ) such that AT (χ)+R0 2 is invertible. Because C∞ (HN ) equals QT (CN ×N ) (Proposition 4.5), there is even a finite product-sum 2 T (ϕij ) ∈ C∞ (HN ), ϕij ∈ CN ×N , R= i

j

such that 2 ). (4.14) AT (χ) + R ∈ GL(HN Put {Rλ } := i j T (kλ ϕij ) . Since ϕij ∈ CN ×N , it follows that Rλ converges uniformly to the compact operator R as λ → ∞ and hence

5

∞

{Rλ } ∈ JNC×N ⊂ JNL×N .

(4.15)

It is clear that Aλ T (kλ χ) + Rλ → AT (χ) + R

strongly.

Because of (4.15), {Aλ T (kλ χ) + Rλ }πJ = {Aλ T (kλ χ)}πJ = {Aλ }πJ {T (kλ χ)}πJ

(4.16)

4.3 Index Computation

189

and since {Aλ }πJ is invertible by our hypothesis and {T (kλ χ)}πJ has the inverse {T (kλ χ−1 )}πJ (Lemma 4.25 for ϕ ∈ C and a ∈ C), we conclude that π {Aλ T (kλ χ) + Rλ }πJ ∈ G(alg T KJ (L∞ N ×N )).

(4.17)

Now (4.14), (4.16), (4.17), and Theorem 4.20 give that π {Aλ T (kλ χ) + R}πM = {Aλ T (kλ χ) + Rλ }πM ∈ G(alg T KM (L∞ N ×N )).

In particular, Aλ T (kλ χ) + R must be invertible for λ > λ0 , whence 0 = Ind (Aλ T (kλ χ) + R) = Ind Aλ − κ

(4.18)

for λ > λ0 . Since {aλ } = Ψ {Aλ }, we have, for λ > λ0 , aλ = (SmbT ◦ Fixλ ){Aµ } = SmbT Aλ , which implies that ST (Aλ ) = T (Aλ ). Thus, Aλ = T (aλ ) + Kλ with 2 ), so Ind Aλ = Ind T (aλ ) = −ind det aλ some Kλ ∈ QT (CN ×N ) = C∞ (HN (Theorem 2.94 for the case of a continuous symbol), and therefore (4.18) finally gives ind det aλ = −κ for λ > λ0 . This ends the proof. 4.29. Remark. Let {Kλ }λ∈(1,∞) be the approximate identity generated by the Poisson kernel. Taking into account Theorem 2.62(a) it is easily seen that if a ∈ G(C + H ∞ ), then {T (kλ a−1 )}πJ is the inverse of {T (kλ a)}πJ in π (L∞ ). Thus, the preceding theorem immediately implies that alg T KJ Ind T (a) = −ind {kλ a} ∀ a ∈ G(C + H ∞ ),

(4.19)

a fact which had already been established in the proof of Theorems 2.64 and 2.65. Further, once (4.19) has been proved, Corollary 4.8(c) is all what then is needed to deduce that Ind A = −ind {kλ det SmbT A} whenever A ∈ algL(HN2 ) T (C + H ∞ )N ×N is Fredholm. 4.30. Corollary. Let B be a C ∗ -subalgebra of QC containing the constants and let {Kλ }λ∈Λ be any approximate identity. If a ∈ L∞ N ×N is locally sectorial over B, then 2 ), (a) T (a) ∈ Φ(HN

(b) {kλ a} is bounded away from zero on conv Λ, (c) Ind T (a) = −ind {det(kλ a)}. Proof. (a) and (b) follow from Corollary 4.13(b) and Corollary 3.69(b), respectively, (c) will follow from Theorem 4.28 as soon as we have proved π (L∞ that {T (kλ a)}πJ is invertible in alg T KJ N ×N ). But this follows from Corolπ lary 3.62 (which shows that {T (kλ a)}ξ is in G(alg T Kξπ (L∞ N ×N )) for each ξ ∈ M (B)) in conjunction with Theorem 3.67 and Lemma 4.25.

190


4.31. Remark. Theorem 3.8 offers another possibility of proving the previous corollary. We have a = sb where s ∈ GL∞ N ×N is sectorial and b ∈ GBN ×N . There are c, d ∈ GCN ×N and an ε > 0 such that, for z ∈ CN and t ∈ T, Re c(kλ s)(t)dz, z = kλ [Re (csdz, z)] (t) ≥ (kλ [εz2 ])(t) = εz2 , which implies that kλ s ∈ GCN ×N is sectorial for all λ ∈ Λ and that {kλ s} is bafz on conv Λ. From Theorem 3.23 we deduce that (kλ b)(kλ b−1 ) − IL∞ → 0 as N ×N

λ → ∞,

and this shows that {kλ b} is bafz on conv Λ. Since, again by Theorem 3.23, kλ (sb) − (kλ s)(kλ b)L∞ → 0 as N ×N

λ → 0,

(4.20)

we see that {kλ a} is bafz on conv Λ. Because T (a) = T (s)T (b)+ compact operator, since T (s) is invertible (Corollary 4.2) and T (b) is Fredholm (a regularizer is T (b−1 )), it follows that 2 ) and that Ind T (a) = Ind T (b). Thus, it remains to prove that T (a) ∈ Φ(HN Ind T (b) = −ind {det(kλ a)}. Taking into account (4.20) it is easily seen that ind {det(kλ a)} = ind {det(kλ s)(kλ b)} = ind {det(kλ s)} + ind {det(kλ b)} = ind {det(kλ b)} (recall that kλ s is sectorial). Because, by Theorem 3.23, ind {det(kλ b)} = ind {kλ (det b)}, we are left with the equality Ind T (b) = −ind {kλ (det b)}. Theorem 1.14(c) shows that T (det b) ∈ Φ(H 2 ). Choose matrix functions → 0 as n → ∞. Then, obviously, bn ∈ (PA +H ∞ )N ×N such that b−bn L∞ N ×N det b − det bn L∞ → 0 as n → ∞, and we have Ind T (b) = Ind T (bn ),

Ind T (det b) = Ind T (det bn )

(4.21)

for all sufficiently large n. Since the operator entries of T (bn ) commute modulo finite-rank operators (Proposition 2.14), we deduce from Theorem 1.15(a) that Ind T (bn ) = Ind T (det bn ). So (4.21) implies that Ind T (b) = Ind T (det b). Hence, it remains to show that Ind T (ϕ) = −ind {kλ ϕ} for every ϕ ∈ GB. If B = C, then T (kλ ϕ) converges uniformly to T (ϕ), so that the desired index equality is an immediate consequence of Theorem 2.42. Thus let B = QC. Let Ind T (ϕ) = m, then T (ϕχm ) is invertible (Corollary 2.40). Put ψ = ϕχm and notice that ψ ∈ GQC. Since T (ψ −1 ) is a regularizer of T (ψ), it follows that T (ψ −1 ) also has index zero and is therefore invertible. Consequently, T (ψ)T (ψ −1 ) = I − H(ψ)H(ψ)−1 )

4.4 Transfinite Localization

191

is also invertible. We have T (kλ ψ)T (kλ ψ −1 ) = I − H(kλ ψ)H(kλ ψ)−1 ), and since H(kλ ψ) and H(kλ ψ)−1 ) converge uniformly to H(ψ) and H(ψ)−1 ), respectively (note that ψ and ψ)−1 are in C +H ∞ ), we see that T (kλ ψ)T (kλ ψ −1 ) must be invertible for all λ large enough. Hence ind kλ ψ = −Ind T (kλ ψ) = 0 for all sufficiently large λ (Theorem 2.42) and since, by Theorem 3.23, ind {kλ ψ} = ind {kλ ϕ} + ind {kλ χm } = ind {kλ ϕ} + m, we finally obtain that ind {kλ ϕ} = −m, as desired.

4.4 Transfinite Localization We now present the transfinite induction approach to maximal antisymmetric sets for C + H ∞ and shall give two applications of this approach: the first consists in proving Axler’s theorem 2.83 for the matrix case and the second is the determination of the norm of a “local Hankel operator.” We begin by extending Theorems 2.11 and 2.54 to block Hankel operators on Hilbert spaces. The bulk of the work that is necessary to prove Nehari’s theorem for the matrix case is done by the following theorem. 4.32. Theorem (Parrott). Let H and K be Hilbert spaces with the orthogonal decompositions H = H1 ⊕ H2 and K = K1 ⊕ K2 , and let MX ∈ L(H, K) have the operator matrix K1 XC H1 → : H2 K2 AB with respect to these decompositions. Then "# " " !" " 0 0 " " 0C " " . " " " , inf MX = max " AB " " 0B " X∈L(H1 ,K1 ) Proof. For a proof we refer to Parrott [376] or Power [399]. 4.33. Theorem. If a ∈ L∞ N ×N , then ∞ (a, HN H(a)L(HN2 ) = distL∞ ×N ). N ×N

Proof. We first show that there is a sequence a0 , a−1 a−2 , . . . of N × N matrices such that H(a) = Sn , where Sn ∈ L(2N (Z)) is defined as the operator whose (block) matrix representation with respect to the decomposition 2N (Z) = 2N (Z− ) ⊕ 2N (Z+ ) is given by

192


⎞ 0 0 0 ⎜ . . . a−n+1 a−n . . . a−2n+1 ⎟ ⎟ ⎜ ⎟ ⎜ .. .. .. 0 ⎟ ⎜ . . . ⎟ ⎜ ⎠ ⎝ . . . a0 a−1 . . . a−n a0 . . . a−n+1 0 H(a)J ⎛

(recall 2.13). This, on its hand, will follow once we have shown that the following inductive process is valid: define Rn ∈ L(2N (Z)) for n = 0, 1, 2, . . . by ⎞ ⎛ a−n+1 a−n+2 a−n+3 . . . ⎟ ⎜a ⎟ ⎜ −n+2 a−n+3 . . . Rn = ⎜ ⎟; ⎠ ⎝ a−n+3 . . . ... then, given Rn there is an a−n ∈ CN ×N such that Rn+1 = Rn . But this is an immediate consequence of Parrott’s theorem 4.32, because in the case at hand " " " " " 0 0 " " 0C " " "=" " " A B " " 0 B " = Rn . Thus, the existence of the sequence a0 , a−1 , a−2 , . . . with desired property is proved. It is easily seen that {Sn ϕ} is convergent for every ϕ ∈ 2N (Z) with finite support, and an ε/3 argument then gives the convergence of {Sn ϕ} for every ϕ ∈ 2N (Z). So 1.1(e) implies that the operator S defined by Sϕ = lim Sn ϕ n→∞

is bounded on 2N (Z) and that S ≤ H(a). Now Proposition 2.2 can be applied to deduce that there is a b ∈ L∞ N ×N such that the n-th matrix Fourier coefficient of b equals an (n ∈ Z) and that S = M (b). Therefore, bL∞ := M (b) = S ≤ H(a) N ×N ∞ ∞ and since b − a ∈ HN ×N , it follows that dist(a, HN ×N ) ≤ H(a). As the reverse inequality is obvious, we are done.

4.34. Theorem. If a ∈ L∞ N ×N , then ∞ (a, CN ×N + HN H(a)Φ(HN2 ) = distL∞ ×N ). N ×N

Proof. Once the matrix analogue of Nehari’s theorem has been established, the proof is almost literally the one given for Theorem 2.54 (with p = 2 and cp = 1). Our next objective is to state some results on Toeplitz operators with unitary-valued symbols, which almost immediately follow from the preceding two theorems.


193

4.35. Lemma. Let A be a C ∗ -algebra with identity element e. Suppose u ∈ GA satisfies u−1 = u∗ and p, q ∈ A satisfy p2 = p = p∗ , q 2 = q = q ∗ , p + q = e. Then pup + q is left-invertible in A ⇐⇒ qup < 1; pup + q is right-invertible in A ⇐⇒ puq < 1. Proof. By the Gelfand-Naimark theorem 1.26(b), we may assume that A is a C ∗ -subalgebra of L(H), where H is some Hilbert space, that u is a unitary operator and p, q are orthogonal and complementary projections on H. From 1.26(d) we know that an operator belonging to a C ∗ -subalgebra of L(H) is left (resp. right) invertible in that C ∗ -algebra if and only if it is left (resp. right) invertible in L(H). The image Im p of p is equal to Ker q and is therefore a closed subspace of H. It is easy to see that pup + q is left invertible if and only if pup|Im p is left-invertible on Im p. For f ∈ Im p, we have f 2 = uf 2 = p(uf )2 + q(uf )2 = (pup)f 2 + (qup)f 2 .

(4.22)

But pup|Im p is left-invertible on Im p if and only if there is an ε > 0 such that (pup)f 2 ≥ εf 2 for all f ∈ Im p, and due to (4.22) this is valid if and only if qup < 1. The assertion on the right-invertibility follows by taking adjoints. 4.36. Corollary. Let u ∈ GL∞ N ×N be unitary-valued. Then the following are equivalent: 2 (i) T (u) is left (resp. right) invertible on HN ; ∞ ∞ (ii) dist(u, HN ×N ) < 1 (resp. dist(u, HN ×N ) < 1); ∞ (iii) u = sh (resp. u = hs), where s ∈ GL∞ N ×N is sectorial and h ∈ HN ×N ∞ (resp. h ∈ HN ×N ).

Moreover, one has 2 ∞ T (u) ∈ GL(HN ) ⇐⇒ dist(u, GHN ×N ) < 1 ∞ ∞ ⇐⇒ u = sh, where s ∈ GLN ×N is sectorial and h ∈ GHN ×N .

Proof. If T (u) is left-invertible, then so is P M (u)P + Q, the preceding lemma ∞ gives QM (u)P L(L2N ) = H() u)L(HN2 ) < 1, and hence dist(u, HN ×N ) < 1 by Theorem 4.33. The implication (ii) =⇒ (iii) results from Lemma 3.6(d). Finally, if (iii) holds then h−1 ∈ L∞ N ×N and so formula (2.20) and Corollary 4.2 show that T (h−1 )T −1 (s) (resp. T −1 (s)T (h−1 )) is a left (resp. right) inverse of T (u). Now suppose T (u) is invertible. By what has just been proved, there is an ∞ h ∈ HN ×N such that u − h < 1. So, due to Proposition 4.1(b), I − T (u∗ h) = I − u∗ h = u − h < 1,

194


which implies the invertibility of T (u∗ h) = T ∗ (u)T (h). Hence T (h) is in 2 ), consequently the equation T (h)f = hf = IN ×N has a solution GL(HN 2 ∞ f ∈ HN ×N and from Theorem 2.93 we deduce that f ∈ LN ×N . Thus, ∞ ∞ f ∈ HN ×N , i.e., h ∈ GHN ×N . ∞ ∞ If dist(u, GHN ×N ) < 1, then u = sh with s ∈ GLN ×N sectorial and ∞ h ∈ GHN ×N by virtue of Lemma 3.6(d). Finally, if u has such a representation, then T (u) is obviously invertible (formula (2.20) and Corollary 4.2). 4.37. Corollary. Let u ∈ GL∞ N ×N be unitary-valued. Then 2 ∞ (a) T (u) ∈ Φ+ (HN ) ⇐⇒ dist(u, CN ×N + HN ×N ) < 1 ⇐⇒ u = sb with ∞ ∞ s ∈ GLN ×N sectorial and b ∈ CN ×N + HN ×N ; 2 ∞ (b) T (u) ∈ Φ− (HN ) ⇐⇒ dist(u, CN ×N + HN ×N ) < 1 ⇐⇒ u = bs with ∞ ∞ s ∈ GLN ×N sectorial and b ∈ CN ×N + HN ×N ; 2 ∞ (c) T (u) ∈ Φ(HN ) ⇐⇒ dist(u, G(CN ×N + HN ×N )) < 1 ⇐⇒ u = sb with ∞ ∞ s ∈ GLN ×N sectorial and b ∈ G(CN ×N + HN ×N ). 2 ). Then P M (u)P +Q+C∞ (L2N ) is left-invertible Proof. (a) Let T (u) ∈ Φ+ (HN 2 2 in L(LN )/C∞ (LN ) (recall Remark 2.70), Lemma 4.35 gives the inequality

u)Φ(HN2 ) < 1, QuP + C∞ (L2N )Φ(L2N ) = H() ∞ and Theorem 4.34 implies that dist(u, CN ×N + HN ×N ) < 1. If this distance estimate holds, then, by virtue of Lemma 3.6(d), u = sb with s, b as desired. Finally, if u has such a representation, then T (b−1 )T −1 (s) is a left regularizer of T (u).

(b) Analogous (or take adjoints). 2 (c) Suppose T (u) ∈ Φ(HN ). We know from (a) that there is a matrix ∞ function b ∈ CN ×N + HN ×N with u − b < 1. In view of 4.10,

= u − bL∞ < 1, I − T (u∗ b)Φ(HN2 ) = I − u∗ bL∞ N ×N N ×N 2 so T ∗ (u)T (b) = T (u∗ b) + compact operator is in Φ(HN ), and hence T (b) is in 2 ∞ Φ(HN ). Theorem 2.94 now gives the invertibility of b in CN ×N + HN ×N , and the desired distance estimate follows. That the distance estimate yields the required factorization is a consequence of Lemma 3.6(d), and if u possesses that factorization u = sb, then T (b−1 )T −1 (s) is a regularizer of T (u).

4.38. Remark. Pousson [398] and Rabindranathan [409] showed that ev∞ ery a ∈ GL∞ N ×N can be factored in the form a = uh with h ∈ GHN ×N and a unitary-valued function u ∈ GL∞ . Using this result one can see N ×N 2 if and only that a Toeplitz operator is Fredholm (resp. invertible) on HN if its symbol is of the form sb where s ∈ GL∞ N ×N is sectorial and b is in ∞ ∞ ) (resp. in GH ). Indeed, if a = sb with s sectorial and b G(CN ×N + HN ×N N ×N ∞ ), then T (a) = T (s)T (b)+ compact operator, which gives in G(CN ×N + HN ×N


195

the Fredholmness of T (a). On the other hand, if T (a) = T (uh) = T (u)T (h) is Fredholm, then T (u) must also be Fredholm, and from Corollary 4.37(c) ∞ we deduce that u = sb where s is sectorial and b ∈ G(CN ×N + HN ×N ). It is ∞ clear that bh is also in G(CN ×N + HN ×N ), so that a = s(bh) is the desired factorization. The argument is analogous for the case of invertibility. 4.39. Corollary. Let u ∈ GL∞ N ×N be unitary-valued. (a) If for each maximal antisymmetric set S for C + H ∞ there exists a 2 unitary-valued uS ∈ GL∞ N ×N such that u|S = uS |S and T (uS ) ∈ Φ+ (HN ) 2 2 2 (resp. Φ− (HN )), then T (u) ∈ Φ+ (HN ) (resp. Φ− (HN )). 2 (b) If u is locally sectorial over C + H ∞ , then T (u) ∈ Φ(HN ). 2 Proof. (a) Let T (uS ) ∈ Φ+ (HN ) for each S in question. Then Corollary 4.38(a) gives the estimate ∞ (4.23) distS (u, CN ×N + HN ×N ) < 1, 2 ). and Theorem 3.5 with Corollary 4.37(a) implies that T (u) ∈ Φ+ (HN

(b) In this case (4.23) results from Lemma 3.6(b), and since the maximal antisymmetric sets for C +H ∞ are the same as those for C +H ∞ , the assertion follows from Theorem 3.5 and Corollary 4.37(a), (b). We now turn to Axler’s method of transfinite localization. In particular, using this method we shall remove the twice occurring “unitary-valued” in the previous corollary. 4.40. Transfinite induction. Let W be a set and let < be a relation on W . The set W is said to be ordered by the relation < if this relation is irreflexive (i.e., there is no u ∈ W such that u < u), connected (i.e., for any u, v ∈ W either u = v, or u < v, or v < u holds), and transitive (i.e., u, v, w ∈ W , u < v, v < w always implies that u < w). A nonempty subset U of W is said to have a first element if there is a u ∈ U such that u < v for all v ∈ U with u = v. A set W is said to be well ordered by a relation < if W is ordered by the relation < and each nonempty subset of W has a first element. It follows from the axiom of choice that every set can be well ordered. Let W be a well-ordered set. As usual, the first element of W will be denoted by 1 and w + 1 will denote the first element of the set {v ∈ W : w < v}. We say that w ∈ W has a predecessor if there is a v ∈ W such that w = v + 1. The principle of transfinite induction consists in the following: to show that a statement (*) holds for all elements of a well-ordered set W it suffices to show that (i) (*) holds for w = 1, (ii) if w ∈ W and (*) holds for all v < w, then (*) holds for w.

196


We shall usually break (ii) into two cases: (a) w has a predecessor and (b) w does not have a predecessor. 4.41. Transfinite decomposition of M (L∞ ). Let W be a set whose carM (L∞ ) and let < be a relation on W which makes W into a dinality is 22 well-ordered set. We use transfinite induction to define for each w ∈ W a partition ∆w of X = M (L∞ ). Thus if w ∈ W , then ∆w will be a collection of disjoint subsets of X whose union is X. The partitions ∆w are defined as follows. (i) ∆1 is the partition of X whose only element is X. (ii) Suppose that w has a predecessor v and that ∆v has been defined. Define an equivalence relation on X by saying that x is equivalent to y if (a) there exists an S ∈ ∆v such that x ∈ S and y ∈ S and (b) f (x) = f (y) for every f ∈ C + H ∞ such that f |S is real-valued. The elements of ∆w are now defined to be the equivalence classes of X under this equivalence relation. (iii) Suppose that w has no predecessor and that ∆v is defined for all v < w. Define an equivalence relation on X by saying that x is equivalent to y if for each v < w there exists an Sv ∈ ∆v such that x ∈ Sv and y ∈ Sv , and then define the elements of ∆w as the equivalence classes of X under this equivalence relation. It is clear that if v < w then ∆w is a refinement of ∆v in the sense that for each Sw ∈ ∆w there exists an Sv ∈ ∆v such that Sw ⊂ Sv . Also note that each S ∈ ∆w is a closed subset of X. The following proposition identifies ∆2 := ∆1+1 and shows that the above construction terminates with the partition of X into maximal antisymmetric sets for C + H ∞ . 4.42. Proposition. (a) ∆2 is the partition of X into fibers over M (QC). (b) There exists a w ∈ W such that each S ∈ ∆w is a maximal antisymmetric set for C + H ∞ . Proof. (a) x is equivalent to y ⇐⇒ f (x) = f (y) ∀ f ∈ C + H ∞ real-valued ⇐⇒ f (x) = f (y) ∀ f ∈ QC real-valued ⇐⇒ f (x) = f (y) ∀ f ∈ QC. (b) For x ∈ X, let Sx be the maximal antisymmetric set for C +H ∞ which contains x. Using transfinite induction we first show that for each w ∈ W there is an S ∈ ∆w such that Sx ⊂ S. (i) This is obvious for w = 1. (ii) Let w = v + 1 and suppose Sx ⊂ Sv ∈ ∆v . If y is in Sx , then y is equivalent to x: indeed, if f ∈ C + H ∞ and f |Sv is real-valued, then f |Sx is real-valued, so f |Sx must be constant, whence f (y) = f (x). This implies that there is an S ∈ ∆w which contains all y ∈ Sx .


197

(iii) Suppose w has no predecessor and that for each v < w there is an Sv ∈ ∆v such that Sx ⊂ Sv . If y ∈ Sx , then y ∈ Sv for all v < w, so y is equivalent to x, and hence there exists an S ∈ ∆w such that y ∈ S and x ∈ S, which gives the inclusion Sx ⊂ S. Thus if w ∈ W and S ∈ ∆w and S is an antisymmetric set for C + H ∞ , then S is a maximal antisymmetric set for C + H ∞ . If w ∈ W is such that ∆w = ∆w+1 , then the definition of ∆w+1 implies that each S ∈ ∆w is an antisymmetric set for C + H ∞ . Consequently, to prove the proposition it suffices to show that there is a w ∈ W such that ∆w = ∆w+1 . Assume this is false. Then for each w ∈ W there is a set Sw ∈ ∆w+1 \ ∆w . ∞ Clearly, if w = v, then Sw = Sv . Hence the mapping W → 2M (L ) , w → Sw is one-to-one. However, the cardinality of W is too large for there to exist ∞ any injective mappings from W into 2M (L ) . This contradiction completes the proof. We now state some lemmas in order to prepare the proof of Theorem 4.48. Recall the terminology introduced in 4.10 and 4.11. For A ∈ alg T (L∞ N ×N ) and S a closed subset of X, let Aπ and AπS denote the cosets in alg T π (L∞ N ×N ) ), respectively, which contain A. and alg TSπ (L∞ N ×N 4.43. Lemma. Suppose w ∈ W has no predecessor and let {Sv }v<w be a Sv . collection of sets Sv ∈ ∆v such that Su ⊂ Sv for v < u. Put Sw := v<w π π Then Sw ∈ ∆w and JSw = clos JSv . v<w

Proof.That Sw belongs to ∆w is easily verified. It is clear that both JSπw and JSπv are closed two-sided ideals of alg T π (L∞ ). It is also clear that clos v<w π JSπw ⊂ clos JSv . To see the opposite inclusion let N = 1 (which in view of v<w

Lemma 3.46 is no loss of generality) and let a ∈ L∞ be such that a|Sw = 0. Let ε > 0 and define U = {x ∈ X : |a(x)| < ε}. Since U is an open set and Sw ⊂ U is the intersection of the compact sets Sv (v < w), which satisfy Su ⊂ Sv for v < u, there exists a v < w such that Sv ⊂ U . Choose b ∈ L∞ so that b|Sv = 0, π π b|(X \ U ) = 1, 0 ≤ b ≤ 1. Then a(1 − b) < ε and thusT (a) − T (ab) < ε. But T π (ab) ∈ JSπv , and so dist T π (a), clos JSπv < ε. It follows that v<w π π JSv and thus JSπw ⊂ clos JSv . T π (a) ∈ clos v<w

v<w

4.44. Lemma. If w ∈ W and S ∈ ∆w , then S is a weak peak set for C + H ∞ . Proof. We prove this by transfinite induction. (i) The case w = 1 is trivial. (ii) Suppose that w ∈ W does not have a predecessor and that the lemma holds for all v < w. Let S ∈ ∆w . Then for each v < w there exists an Sv ∈ ∆v

198


such that S ⊂ Sv . The definition of ∆w shows that S =

Sv . Since each v<w

Sv is a weak peak set, S is also a weak peak set. (iii) Now suppose that w has a predecessor v and that the lemma holds for v. Let S ∈ ∆w and let Sv ∈ ∆v be such that S ⊂ Sv . Thus, Sv is a weak peak set for C + H ∞ and therefore (C + H ∞ )|Sv is a closed algebra. By the definition of ∆w , there is a λ ∈ R such that $ x ∈ Sv : f (x) = λ , S= f

the intersection over all f ∈ C + H ∞ whose restriction to Sv is real-valued. But x ∈ Sv : f (x) = λ = x ∈ Sv : 1 − ε(f (x) − λ)2 = 1 (ε > 0 sufficiently small) is a peak set for (C + H ∞ )|Sv , and so S must be a weak peak set for (C + H ∞ )|Sv . A result from the theory of function algebras (see, e.g., Gamelin [209, Chap. II, Corollary 12.9]) now implies that S is a weak peak set for C + H ∞ . 4.45. Definition. Let v ∈ W and Sv ∈ ∆v . Then, by 1.28 and the preceding lemma, both (C + H ∞ )|Sv and (C + H ∞ )|Sv are closed subalgebras of L∞ |Sv . Let QSv denote the C ∗ -subalgebra of L∞ |Sv defined by QSv := (C + H ∞ )|Sv ∩ (C + H ∞ )|Sv . Note that if v > 1, then QSv is not equal to (C + H ∞ ) ∩ (C + H ∞ ) |Sv = QC|Sv ∼ = C. Also notice that S1 = X and thus QS1 = (C + H ∞ )|X ∩ (C + H ∞ )|X is nothing else than QC. 4.46. Lemma. Let v ∈ W and Sv ∈ ∆v . Then DSv := TSπv (ϕ) : ϕ ∈ L∞ , ϕ|Sv ∈ QSv is a C ∗ -subalgebra of the center of alg TSπv (L∞ ). Proof. Since QSv is a closed subalgebra of L∞ |Sv , it follows from Theorem 3.61 (with A = L∞ , B = L(H 2 )/C∞ (H 2 ), i = T π , F = Sv , N = 1) that DSv is a closed subspace of alg TSπv (L∞ ). To see that DSv is contained in the center of alg TSπv (L∞ ), let ϕ ∈ L∞ be such that ϕ|Sv ∈ QSv and let a ∈ L∞ . Let ϕ|Sv = h1 |Sv = h2 |Sv , where h1 and h2 are functions in C + H ∞ . Then TSπv (ϕ)TSπv (a) − TSπv (a)TSπv (ϕ) = TSπv (h2 )TSπv (a) − TSπv (a)TSπv (h1 ) = TSπv (h2 a) − TSπv (ah1 ) = TSπv (ϕa) − TSπv (aϕ) = 0, and it follows that DSv ⊂ Cen (alg TSπv (L∞ )). If ϕ|Sv ∈ QSv and ψ|Sv ∈ QSv , then TSπv (ϕ)TSπv (ψ) = TSπv (ϕψ) and thus DSv is an algebra.


199

4.47. Lemma. Suppose w ∈ W has a predecessor v and let Sv ∈ ∆v . (a) The maximal ideal space M (QSv ) of QSv can be identified with the set {S ∈ ∆w : S ⊂ Sv } and the Gelfand map is given by Γ : QSv → C({S ∈ ∆w : S ⊂ Sv }),

f → f |S.

(b) For S ∈ ∆w and S ⊂ Sv , define ES := closidalg TSπ (L∞ ) TSπv (ϕ) : ϕ ∈ L∞ , ϕ|Sv ∈ QSv , ϕ|S = 0 , v ES := closidalg TSπ (L∞ ) TSπv (a) : a ∈ L∞ , a|S = 0 . v

Then ES = ES . (c) If Sw ∈ ∆w and Sw ⊂ Sv , then ESw is isometrically isomorphic to JSπw /JSπv and alg TSπv (L∞ )/ESw is isometrically isomorphic to alg TSπw (L∞ ). Remark. In the proof of Theorem 4.48, when we shall be applying the local principle 1.35, the ideal ES will appear and there will be a point at which we must show that alg TSπv (L∞ )/ESw ∼ = alg TSπw (L∞ ). However, the latter isomorphism is not obvious. What is “obvious” is the isomorphism alg TSπv (L∞ )/ES w ∼ = alg TSπw (L∞ ) (see the proof of part (c)). This is the justification of part (b) of the present lemma. Proof. (a) This follows from 1.27(b). (b) What we must prove is that ES ⊂ ES . Let a ∈ L∞ and suppose a|S = 0. Let ε > 0 be given arbitrarily. For σ ∈ M (QSv ), let Mσ (L∞ |Sv ) denote the fiber of L∞ |Sv over σ and put U = σ ∈ M (QSv ) : |a(x)| < ε ∀ x ∈ Mσ (L∞ |Sv ) . Assume that U is not an open subset of M (QSv ). Then there is a σ ∈ U and a net σi in M (QSv ) such that σi → σ and such that for each i there exists an xi ∈ Mσi (L∞ |Sv ) with |a(xi )| ≥ ε. Taking a subnet, we can assume that there is an x ∈ M (L∞ |Sv ) such that xi → x and it follows that x is even in Mσ (L∞ |Sv ) (recall that the mapping τ : M (L∞ |Sv ) → M (QSv ) which sends a functional to its restriction functional is continuous). However, |a(x)| ≥ ε, which contradicts our assumption that σ ∈ U . The conclusion is that U is an open subset of M (QSv ). Since S ∈ ∆w and S ⊂ Sv , there is exactly one σ ∈ M (QSv ) such that S ⊂ Mσ (L∞ |Sv ) (recall part (a)). Because a|S = 0, it is clear that σ ∈ U . Thus, there is an f ∈ QSv such that f (σ) = 0, f |(M (QSv )\U ) = 1, 0 ≤ f ≤ 1. Let h be a function in C + H ∞ for which h|Sv = f . Note that the choice of f implies that a(1 − h)|Sv < 1. Thus, ε > TSπv (a) − TSπv (ah) = TSπv (a) − TSπv (a)TSπv (h).

200


But TSπv (h) ∈ ES , so TSπv (a)TSπv (h) ∈ ES , whence dist(TSπv (a), ES ) ≤ TSπv (a) − TSπv (a)TSπv (h) < ε. Letting ε go to zero, we conclude that TSπv (a) ∈ ES . (c) We have JSπv = closidalg T π (L∞ ) T π (a) : a ∈ L∞ , a|Sv = 0 , JSπw = closidalg T π (L∞ ) T π (a) : a ∈ L∞ , a|Sw = 0 , ES v = closidalg TSπ (L∞ ) TSπv (a) : a ∈ L∞ , a|Sw = 0 . v

A little thought therefore shows that JSπw /JSπv ∼ = ES w and so part (b) gives π π ∼ that JSw /JSv = ESw (note that all algebras occurring are C ∗ -algebras and take into account 1.26(e)). The second assertion now results as follows: alg TSπv (L∞ )/ESw ∼ = (alg T π (L∞ )/JSπv )/(JSπw /JSπv ) ∼ alg T π (L∞ )/J π = alg T π (L∞ ). = Sw Sw π 4.48. Theorem (Axler). Let A ∈ alg T (L∞ N ×N ) and suppose A is not left π ∞ (right, resp. two-sided) invertible in alg T (LN ×N ). Then there exists a collection {Sw }w∈W of subsets of X such that

(a) Sw ∈ ∆w for each w ∈ W ; (b) Sw ⊂ Sv if v < w; (c) if w ∈ W , then AπSw is not left (right, resp. two-sided) invertible in alg TSπw (L∞ N ×N ). Proof. For the sake of definiteness, let us prove the theorem for the case of left-invertibility. The collection {Sw }w∈W of subsets of X that satisfies conditions (a)–(c) will be defined by transfinite induction. (i) For w = 1, let S1 = X. Then JSπ1 = {0} and so conditions (a)–(c) are obviously satisfied. (ii) Suppose that w has no predecessor, that Sv has been defined for v < w, Sv . It is and that conditions (a)–(c) are satisfied for v < w. Put Sw := v<w

obvious that (b) holds for w and (we mentioned this in Lemma 4.43) it can be easily verified that (a) is also true for w. Now assume (c) does not hold. Thus, AπSw is left-invertible in the algebra ∞ π π π π alg TSπw (L∞ N ×N ). So there is a B ∈ alg T (LN ×N ) for which B A − I ∈ JSw . ∞ Lemma 4.43 shows that, for D ∈ alg T (LN ×N ),


201

/ DSπw := dist(Dπ , JSπw ) = dist Dπ , clos JSπv , v<w

such that BSπv AπSv − alg TSπv (L∞ N ×N ), which

ISπv < 1. Conseand hence there is a v < w π π implies that AπSv is quently, BSv ASv is invertible in left-invertible in that algebra. This, however, contradicts the induction hypothesis. Thus, we have proved that (c) holds for w. (iii) Now suppose w ∈ W has a predecessor v, that Sv has been defined and satisfies (a)–(c). Thus, AπSv is not left-invertible in alg TSπv (L∞ N ×N ). We apply Theorem 1.35(a) (in the setting A = alg TSπv (L∞ N ×N ), B = DSv := {TSπv (ϕIN ×N ) : ϕ ∈ L∞ , ϕ|Sv ∈ QSv }, a = AπSv ). Lemma 4.46 tells us that DSv is a C ∗ -subalgebra of Cen (alg TSπv (L∞ N ×N )). By Theorem 3.61 (in the setting A = L∞ , B = L(H 2 )/C∞ (H 2 ), i = T π , F = Sv , N = 1), the mapping QSv → DSv , ϕ|Sv → TSπv (ϕIN ×N ) is an isometric isomorphism. It follows that M (DSv ) can be identified with M (QSv ) = {S ∈ ∆w : S ⊂ Sv } (Lemma 4.47(a)). So the ideal of the algebra alg TSπv (L∞ N ×N ) generated by Sw ∈ M (QSv ) (i.e., the JSw in the terminology of 1.33) coincides with closid TSπv (ϕIN ×N ) ∈ DSv : ϕ|Sw = 0 = closid TSπv (ϕIN ×N ) : ϕ ∈ L∞ , ϕ|Sv ∈ QSv , ϕ|Sw = 0 , that is, with (ESw )N ×N , where ESw is defined as in Lemma 4.47(b). Thus, what results is that there exists an Sw ∈ ∆w (property (a)) such that Sw ⊂ Sv (property (b)) and AπSv + (ESw )N ×N is not left-invertible in alg TSπv (L∞ N ×N )/(ESw )N ×N . Lemma 4.47(c) in conjunction with Lemmas 3.46 and 3.47 finally implies that AπSw is not left-invertible in alg TSπw (L∞ N ×N ) (property (c)). 4.49. Corollary. (a) Let A ∈ alg T (L∞ N ×N ) and w ∈ W . Then A is left (right, 2 if and only if AπS is left (right, resp. tworesp. two-sided) Fredholm on HN sided) invertible in alg TSπ (L∞ N ×N ) for all S ∈ ∆w . ∞ (b) Let a ∈ L∞ N ×N . If for each maximal antisymmetric set S for C + H ∞ 2 there exists an aS ∈ LN ×N such that a|S = aS |S and T (aS ) ∈ Φ+ (HN ) (resp. 2 2 2 )), then T (a) ∈ Φ+ (HN ) (resp. Φ− (HN )). Φ− (HN ∞ 2 then T (a) ∈ Φ(HN ). (c) If a ∈ L∞ N ×N is locally sectorial over C + H

Proof. (a) The “if” part is immediate from the preceding theorem, the “only if” portion results from that theorem in conjunction with 1.26(d). (b) Note that TSπ (a) = TSπ (aS ), take into account Proposition 4.42(b), and apply part (a). (c) Corollary 3.62 gives the invertibility of TSπ (a) in alg TSπ (L∞ N ×N ) for each maximal antisymmetric set S for C + H ∞ , so that Proposition 4.42(b) and part (a) apply once more.

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Remark. Combining 4.49(c) and Remark 4.38 we see that every matrix function which is locally sectorial over C +H ∞ can be written in the form sb where ∞ s ∈ GL∞ N ×N is sectorial and b is in G(CN ×N + HN ×N ). Of course, it would be desirable to have a more direct proof of this result (see the proof of 2.86(a)). 4.50. Compactness of quasicommutators. (a) Let a, b ∈ L∞ and let B be a C ∗ -subalgebra of QC containing the constants. Suppose for each ξ ∈ M (B) either a|Xξ ∈ H ∞ |Xξ or b|Xξ ∈ H ∞ |Xξ . Then T (ab) − T (a)T (b) = H(a)H()b) ∈ C∞ (H 2 ).

(4.24)

Indeed, if a|Xξ = h|Xξ where h ∈ H ∞ , then Tξπ (ab) − Tξπ (a)Tξπ (b) = Tξπ (hb) − Tξπ (h)Tξπ (b) = 0, the situation is analogous for b|Xξ ∈ H ∞ |Xξ , so $ T π (ab) − T π (a)T π (b) ∈

Jξπ ,

ξ∈M (B)

and Theorem 1.35(c) gives the assertion (note that alg T π (L∞ ) as a C ∗ -algebra is semisimple). (b) Axler [12] also established the following theorem. Let A ∈ alg T (L∞ ). Then there exists a collection {Sw }w∈W of subsets of X such that (a) Sw ∈ ∆w for each w ∈ W , (b) Sw ⊂ Sv if v < w, (c) Aπ = AπSw for each w ∈ W . The proof of this theorem is similar in spirit to the proof of Theorem 4.48. We therefore only indicate how the collection {Sw }w∈W is defined by transfinite induction. For w = 1, let S1 = X. If w has no predecessor and Sv has Sv . If w has a predecessor v and Sv been defined for v < w, then Sw := v<w

has been defined, then, by Theorem 1.35(d), AπSv = max dist(AπSv , ES ) : S ∈ ∆w , S ⊂ Sv and Sw is defined by AπSv = dist(AπSv , ESw ). An immediate consequence of the above theorem is that Aπ = max AπS ∀ w ∈ W S∈∆w

∀ A ∈ alg T (L∞ ).

(4.25)

(c) The result in (a) can now be refined as follows. Let a, b ∈ L∞ and suppose for each maximal antisymmetric set S for C + H ∞ either a|S ∈ H ∞ |S or b|S ∈ H ∞ |S. Then (4.24) holds. To see this, apply the argument of part (a) to show that TSπ (ab) − TSπ (a)TSπ (b) = 0 for each S in question and then apply Proposition 4.42(b) and (4.25).


203

(d) (The Axler-Chang-Sarason-Volberg theorem). Let a, b ∈ L∞ . Then (4.24) holds if and only if alg (H ∞ , a) ∩ alg (H ∞ , b) ⊂ C + H ∞ . This is the final solution of the compactness problem for the quasicommutators T (ab) − T (a)T (b). Here alg (H ∞ , f ) denotes the smallest closed subalgebra of L∞ containing H ∞ and f . In connection with this criterion notice the following well known fact (see, e.g., Douglas [162]): If A is a closed subalgebra of L∞ containing H ∞ , then either A = H ∞ or C + H ∞ ⊂ A. (e) Open problems. Establish an Axler-Chang-Sarason-Volberg theorem for harmonic approximation or stable convergence, i.e., find necessary and sufficient conditions for

or

{kλ (ab) − (kλ a)(kλ b)}π ∈ N

(4.26)

π T (kλ (ab)) − T (kλ a)T (kλ b) ∈ J

(4.27)

to hold. In the original edition of this book we wrote that a reasonable conjecture would be that (4.26) is true (say, for the approximate identity generated by the Poisson kernel) if and only if alg (H ∞ , a) ∩ alg (H ∞ , b) ⊂ C + H ∞ and

alg (H ∞ , a) ∩ alg (H ∞ , b) ⊂ C + H ∞ .

In 1998, Gorkin and Zheng [240] proved that the “if” part of this conjecture is true for the Poisson kernel and gave counterexamples showing that the “only if” portion is not true for the Poisson kernel. One of their counterexamples is a = (B + B)/2, b = (B − B)/2, where B is an infinite Blaschke product. Theorem 3.23 and Lemma 4.25 in conjunction with Theorem 1.35(c) imply that (4.26) and (4.27) are valid if either a|Xξ ∈ C|Xξ or b|Xξ ∈ C|Xξ for each ξ ∈ M (QC). Trying to refine this result to maximal antisymmetric sets for C + H ∞ immediately leads to the following problem. Can transfinite localization be applied to harmonic approximation or staπ π (T HJ ) in place of T π ? We suspect ble convergence, i.e., to Kπ (Hπ ) or T KJ that in these cases localization with respect to fibers over QC (Corollary 3.69 and Theorem 4.26) is the final stage. In order to support this, we remark that we do not know any good analogue of formula (2.20) which forms the basis for the “deleting of H ∞ symbols.” See also the end of the Section 4.77. Find a criterion for H(a)H()b) to be a trace class operator on H 2 (or, more generally, to be in Cp (H 2 )). This problem is of interest in connection with the theory of Toeplitz determinants (see 10.12, 10.27, 10.61, 10.63). We now use the transfinite induction approach to determine the norm of local Hankel operators, since the knowledge of this norm will enable us to

204


employ Lemma 4.35 to establish criteria for the invertibility of local Toeplitz operators. However, it is necessary to modify some of the above arguments. The reason is that not every bounded Hankel operator belongs to alg T (L∞ ). To show this is the content of the following proposition. 4.51. Proposition. Let f ∈ C be any function such that f (−1) = f (1) = 0. If a ∈ L∞ and H(a) ∈ alg T (L∞ ), then af ∈ C + H ∞ . In particular, if a ∈ L∞ has a jump discontinuity at some point in T \ {±1}, then H(a) ∈ / alg T (L∞ ). Proof. Suppose H(a) ∈ alg T (L∞ ). If ϕ ∈ C, then T (ϕ) commutes modulo compact operators with every operator in alg T (L∞ ) and hence T π (ϕ) belongs to the center of alg T π (L∞ ). Thus, T (ϕ)H(a) − H(a)T (ϕ) ∈ C∞ (H 2 )

∀ ϕ ∈ C.

But T (ϕ)H(a) − H(a)T (ϕ) equals H(ϕa) − H(ϕ)T () a) − H(aϕ) ) + T (a)H(ϕ) ) (this can be verified by using the P ’s and Q’s as in the proof of Proposition 2.14), and since H(ϕ) and H(ϕ) ) are compact for ϕ ∈ C, it follows that H(a(ϕ − ϕ)) ) must be compact for every ϕ ∈ C. So Theorem 2.54 shows that a(ϕ − ϕ) ) ∈ C + H ∞ and thus af (ϕ − ϕ) ) ∈ C + H ∞ for every ϕ ∈ C. Let S be any maximal antisymmetric set for C + H ∞ . Because C ⊂ C + H ∞ , there is a τ ∈ T such that S ⊂ Xτ . If τ = ±1, then there exists a ϕ ∈ C such that ϕ(τ ) = ϕ(τ ) ) and we conclude that af |S ∈ C + H ∞ |S. If τ = ±1, then obviously af |S = 0|S ∈ C + H ∞ |S. Thus, by Corollary 1.23, af ∈ C + H ∞ . It remains to notice that functions in H ∞ and thus in C + H ∞ cannot have jumps. 4.52. The algebra generated by singular integral operators. (a) Let A ∞ denote the direct product L∞ N ×N × LN ×N . On defining α (a, b) := (α a, α b), (a, b) + (c, d) := (a + c, b + d), (a, b)(c, d) = (ac, bd), (a, b)∗ := (a∗ , b∗ ), and , (a, b)A := max aL∞ , bL∞ N ×N N ×N we make A become a C ∗ -algebra. (b) It will be convenient to denote the multiplication operator on L2N generated by f ∈ L∞ N ×N simply by f . The mapping σ given by σ : A → L(L2N ),

(a, b) → aP + bQ

(recall that P := diag (P, . . . , P ), Q := diag (Q, . . . , Q)) is a submultiplicative embedding. To see this note first that " " " " " " " " (ajk P + bjk Q)" = "U −n (ajk P + bjk Q) U n " " j

j

k

k

" " " " " " " " =" (ajk U −n P U n + bjk U −n QU n )" ≥ " ajk ", j

k

j

k


205

because U −n P U n → I and U −n QU n → 0 strongly. It can be shown analogously that " " " " " " " " (ajk P + bjk Q)" ≥ " bjk ". " j

j

k

k

Since, in particular, aP + bQ ≥ max{a, b}, it follows that Ker σ = {0} and that Im σ is closed. Finally, we have " " " " " " " " (ajk , bjk ) " = "σ ajk , bjk " "σ j

j

k

j

k

k

" " " " " " " " " " " " =" ajk P + bjk Q" ≤ " ajk " + " bjk " j

k

j

k

j

j

k

k

" " " " " " " " ≤ 2" (ajk P + bjk Q)" = 2" σ(ajk , bjk )", j

j

k

k

which shows that σ is 2-submultiplicative. (c) The algebra alg σ(A) contains QaP (= (0 · P + 1 · Q)(a · P + 0 · Q)) and P aQ for every a ∈ L∞ N ×N . Note that QaP and P aQ can be identified with H() a) and H(a), respectively (see 2.10, 2.15, and also 4.36). (d) The collection C∞ (L2N ) of all compact operators on L2N is a subset of the quasicommutator ideal Qσ (A). Indeed, by Lemma 3.47 it suffices to consider the case N = 1, the operators (0 · P + 1 · Q) − (0 · P + χ−n Q)(0 · P + χn Q) = χ−n P χn Q, (1 · P + 0 · Q) − (χn P + 0 · Q)(χ−n P + 0 · Q) = χn Qχ−n P

−1

n−1 2 take k∈Z fk χk ∈ L into k=−n fk χk and k=0 fk χk , respectively, and therefore the operator f → (f, χk )χm belongs to Qσ (A) for all k, m ∈ Z. The rest is as in the proof of Proposition 4.5. (e) By (d) and Theorem 3.52, the mapping σ π : A → alg σ π (A) := alg σ(A)/C∞ (L2N ), (a, b) → (aP + bQ)π := aP + bQ + C∞ (L2N ) is a submultiplicative quasi-embedding whose kernel is {(a, b) ∈ A : aP + bQ ∈ C∞ (L2N )}. But if aP + bQ ∈ C∞ (L2N ), then P aP = P (aP + bQ)P and QbQ = Q(aP + bQ)Q are in C∞ (L2N ), which implies that a = b = 0 (Lemma 4.4(d)). Thus, σ π is actually an embedding. (f) For F a closed subset of X = M (L∞ ), define RπF = closidalg σπ (A) (aP + bQ)π : a, b ∈ L∞ N ×N , a|F = b|F = 0 .

206


Theorem 3.52 and Lemma 3.59 give that the mapping σFπ : A → alg σFπ (A) := alg σ π (A)/RπF , (a, b) → (aP + bQ)πF := (aP + bQ)π + RπF is a submultiplicative quasi-embedding. Put JFπ = closidalg σπ (A) (P aP )π : a ∈ L∞ N ×N , a|F = 0 (note that P aP ∈ alg σ(A)). (g) The equality P π RπF P π = JFπ holds. The inclusion “⊃” is obvious. We prove the reverse inclusion. By Lemma 3.46, it is enough to consider the case N = 1. Let n n A= (ak P + bk Q), B = (ck P + dk Q), k=1

k=1

∞

∞

where ak , bk , ck , dk ∈ L , and let f, g ∈ L be such that f |F = g|F = 0. We must show that (P A(f P + gQ)BP )π is in JFπ . Put A1 =

n−1

(ak P + bk Q),

k=1

B1 =

n

(ck P + dk Q).

k=2

We have (P A(f P + gQ)BP )π = (P A(P f P + Qf P + P gQ + QgQ)BP )π ,

(4.28)

and (P AP f P BP )π is clearly in JFπ . Let us write C π ≡ Dπ in case C π − Dπ is in JFπ . Then (P AQf P BP )π = = = π (P AP gQBP ) ≡ ≡

(P A1 (an P + bn Q)Qf P BP )π (P A1 bn Qf P BP )π ≡ (P A1 bn f P BP )π (P A1 Qbn f P BP )π , (P AP gBP )π = (P AP (gc1 P + gd1 Q)B1 P )π (P AP gd1 QB1 )π ,

and it follows by induction with respect to n that the second and third terms in (4.28) always belong to JFπ . Finally, using this we get (P AQgQBP )π = (P AQgBP )π − (P AQgP BP )π ≡ (P AQgBP )π = (P AQgc1 P B1 P )π + (P AQgd1 QB1 P )π ≡ (P AQgd1 QB1 P )π , and so again by induction with respect to n we conclude that the fourth term in (4.28) also belongs to JFπ .


207

π ∞ (h) Define alg τ π (L∞ N ×N ) := algalg σ π (A) {(P aP ) : a ∈ L }. Then the π π π π ∞ equality P alg σ (A)P = alg τ (LN ×N ) holds. The inclusion “⊃” is again trivial. To show the opposite inclusion, assume N = 1 (Lemma 3.47) and let B and B1 be as in (g). Because P BP = P c1 P B1 P + P d1 QB1 P , it suffices to show that (P f P BP )π and (P f QBP )π belong to alg τ π (L∞ N ×N ) for every . This is readily verified for n = 1 and since f ∈ L∞ N ×N

(P f P BP )π = (P f P c1 P B1 P )π + (P f P d1 QB1 P )π , (P f QBP )π = (P f Qc1 P B1 P )π + (P f Qd1 QB1 P )π = (P f c1 P B1 P )π − (P f P c1 P B1 P )π +(P f d1 QB1 P )π − (P f P d1 QB1 P )π , the assertion for general n follows by induction with respect to n. π (k) Let F be a closed subset of X and let a ∈ L∞ N ×N . Then TF (a) is left π ∞ (right, resp. two-sided) invertible in alg TF (LN ×N ) if and only if (P aP + Q)πF is so in alg σFπ (A). To see this suppose, e.g., that TFπ (a) is left-invertible. Thus, there exists ∞ an operator B ∈ alg τ (L∞ N ×N ) := algalg σ(A) {P aP : a ∈ LN ×N } such that π π π π B (P aP ) − P ∈ JF . Taking into account that B = P BP , it is easy to deduce from (g) that

(P BP + Q)π (P aP + Q)π − I π = B π (P aP )π − P π belongs to JFπ ⊂ P π RπF P π ⊂ RπF , i.e., that (P aP + Q)π is left invertible in alg σFπ (A). On the other hand, if there is a B π in the algebra alg σ π (A) such that π B (P aP + Q)π − I π ∈ RπF , then (P BP )π (P aP )π − P π ∈ P π RπF P π , and so (g) and (h) imply that TFπ (a) is left-invertible in alg TFπ (L∞ N ×N ). (l) To every operator A ∈ alg T (L∞ N ×N ) there corresponds in a natural way an element (P AP )π ∈ alg τ π (L∞ N ×N ). If F is a closed subset of X and ), then A ∈ alg T (L∞ N ×N (P AP )π + RπF alg σπ (A) = Aπ + JFπ alg T π (L∞ . N ×N ) Indeed, Aπ + JFπ alg T π (L∞ = (P AP )π + JFπ alg σπ (A) N ×N ) = (P AP )π + P π RπF P π alg σπ (A)

(by (g))

= P π (P AP )π P π + P π RπF P π alg σπ (A) ≤ (P AP )π + RπF alg σπ (A)

(since P π = P = 1)

= (P AP )π + JFπ alg σπ (A)

(because JFπ ⊂ RπF )

= Aπ + JFπ alg T π (L∞ . N ×N )


208

∞ Recall that, for a closed subset F of X, for a ∈ L∞ N ×N , and B ⊂ LN ×N , we defined distF (a, B) := inf max a(x) − b(x)L(CN ) . b∈B x∈F

4.53. Theorem. Let a ∈ L∞ N ×N and define the partitions ∆w (w ∈ W ) of X as in 4.41. If w ∈ W and S ∈ ∆w , then ∞ (QaP )πS alg σSπ (A) = distS (a, CN ×N + HN ×N ).

(4.29)

∞ Proof. Choose h ∈ CN ×N + HN ×N so that ∞ (a − h)|S ≤ distS (a, CN ×N + HN ×N ) + ε.

Let IS := {g ∈ L∞ N ×N : g|S = 0}. Because (QaP )πS = (Q(a − h)P )πS (since (QhP )πS = 0) ≤ inf (Q(a − h − g)P )π ≤ inf a − h − gL∞ N ×N g∈IS

g∈IS

=: a − hL∞ = (a − h)|S N ×N /IS

(proof of Theorem 3.61)

and ε > 0 can be chosen arbitrarily, it follows that for each w ∈ W and each S ∈ ∆w in (4.29) the inequality “≤” holds. That actually equality holds will be proved by transfinite induction. (i) For w = 1 this is Theorem 4.34. (ii) Suppose that w has no predecessor and that (4.29) holds for all S ∈ ∆v with v < w. Let S ∈ ∆w . The definition of ∆w shows that there is a family Sv . Given {Sv }v<w of sets Sv ∈ ∆v such that Su ⊂ Sv for v 0 there is a v0 < w such that / dist (QaP )π , clos RπSv ≥ dist (QaP )π , RπSv0 − ε. v<w

Thus, ∞ ∞ distS (a, CN ×N + HN ×N ) ≤ distSv0 (a, CN ×N + HN ×N )

= (QaP )πSv0 (induction hypothesis) / = dist QaP )π , RπSv0 ≤ dist (QaP )π , clos RπSv + ε v<w

≤ dist (QaP )π , RπS + ε = (QaP )πS + ε, π where the last “≤” results from the inclusion clos RSv ⊂ RπS , which can v<w π be proved in the same way as Lemma 4.43 (of course, actually clos RSv v<w

is equal to RπS ). Letting ε go to zero, we arrive at the desired inequality.


209

(iii) Now suppose w has a predecessor v and (4.29) holds for all S ∈ ∆v . Let S0 ∈ ∆w . There is an Sv ∈ ∆v such that S0 ⊂ Sv . Let A denote the operator T (a∗ a) − T (a∗ )T (a) ∈ alg T (L∞ N ×N ). Recall that, by Lemma 4.47(a), M (QSv ) can be identified with {S ∈ ∆w : S ⊂ Sv }. Now let ε > 0 and put U (S0 ) := S ∈ ∆w : S ⊂ Sv , AπS < AπS0 + ε . From Theorem 1.35(b) applied in the same setting as in the proof of Theorem 4.48 we deduce that the mapping M (QSv ) → R+ , S → AπS is upper semi-continuous. Therefore U (S0 ) is an open subset of M (QSv ). Thus, there is a ϕ ∈ QSv such that ϕ|S0 = 1, ϕ|(M (QSv )\U (S0 )) = 0, 0 ≤ ϕ ≤ 1. Choose a function f ∈ C + H ∞ so that f |Sv = ϕ. If S ∈ U (S0 ), then f |S AπS 1/2 < (AπS0 + ε)1/2

(4.30)

and if S ∈ M (QSv ) \ U (S0 ), then f |S AπS 1/2 = 0.

(4.31)

Now define g ∈ L∞ N ×N as g = f IN ×N . Then ∞ ∞ distS0 (a, CN ×N + HN ×N ) = distS0 (ag, CN ×N + HN ×N ) ∞ ≤ distSv (ag, CN ×N + HN ×N ) (because S0 ⊂ Sv )

= (QagP )πSv (induction hypothesis) " π "1/2 (C ∗ -norm property) = " (QagP )∗ (QagP ) Sv " = (P ga∗ QagP )πSv 1/2

(P gQ)π = (QgP )π = 0 = (P gP a∗ QaP gP )πSv 1/2 " "1/2 π = " P gP (P a∗ QaP )P gP + RπSv " " "1/2 π = " P gP (P a∗ aP − P a∗ P aP )P gP + RπSv " = T π (g)Aπ T π (g) + JSπv 1/2 = TSπv (g)AπSv TSπv (g)1/2 .

(by 4.52(l)) (4.32)

Now Theorem 1.35(d) applied in the same context as in the proof of Theorem 4.48 gives that (4.32) equals max TSπ (g)AπS TSπ (g)1/2 : S ∈ ∆w , S ⊂ Sv (4.33) ≤ max f |S AπS 1/2 : S ∈ ∆w , S ⊂ Sv . Taking into account (4.30) and (4.31) we see that (4.33) is not greater than (AπS0 + ε)1/2 and since ε > 0 can be chosen arbitrarily, we get

210

4 Toeplitz Operators on H 2 ∞ π 1/2 distS0 (a, CN ×N + HN . ×N ) ≤ AS0

But

" "1/2 π AπS0 1/2 = " T (a∗ a) − T (a∗ )T (a) + JSπ0 " = (P a∗ aP − P a∗ P aP )π + RπS0 1/2 (by 4.52(l)) " π "1/2 = (P a∗ QaP )πS0 1/2 = " (QaP )∗ (QaP ) S0 " = (QaP )πS0 .

Remark. If w > 1 and S ∈ ∆w , then S is contained in some fiber Xξ over ξ ∈ M (QC) (Proposition 4.42(a)). Thus, in that case C|S ∼ = C and hence, for , a ∈ L∞ N ×N ∞ (QaP )πS alg σSπ (A) = distS (a, HN ×N ). 4.54. Corollary. Let u ∈ GL∞ N ×N be unitary-valued, let B be C, QC, or C + H ∞ , and let F be a maximal antisymmetric set for B. Then TFπ (u) is ∞ left resp. right invertible in alg TFπ (L∞ N ×N ) if and only if distF (u, HN ×N ) < 1 ∞ resp. distF (u, HN ×N ) < 1. Proof. If B = QC or B = C +H ∞ , then there is a w ∈ W such that F belongs to ∆w . So it remains to apply 4.52(k), Lemma 4.35, and the remark in 4.53. The case B = C can be reduced to the case B = QC by using Theorem 3.70 (whose proof shows that the statement of the theorem is also true for onesided invertibility) along with Theorem 3.5. Instead of Theorem 3.5 one can also apply Theorem 1.35(d) in the spirit of the proof of Theorem 3.70. ∞ We finally show that distF (u, HN ×N ) is equal to some other quantity, a fact that will be needed later.

4.55. The algebras HF∞ . Let F be a weak peak set for H ∞ . Then, by 1.28, H ∞ |F is a closed subalgebra of L∞ |F and hence, HF∞ := f ∈ L∞ : f |F ∈ H ∞ |F is a closed subalgebra of L∞ . Clearly, HF∞ = H ∞ + IF

where

IF := g ∈ L∞ : g|F = 0 .

If F is a maximal antisymmetric set for C + H ∞ , then, we mentioned this in 1.28, F is a weak peak set for C + H ∞ and thus for H ∞ . So in that case HF∞ makes a sense. If F = Xτ (τ ∈ T) or F = Xξ (ξ ∈ M (QC)), then F is a peak set for C + H ∞ (recall 2.81) and thus for H ∞ . So we now consider the ∞ ∞ and HX . algebras HX τ ξ 4.56. Proposition. Let F be a weak peak set for H ∞ and let a ∈ L∞ N ×N . Then ∞ inf max a(x) − h(x)L(CN ) distF (a, HN ×N ) := ∞ h∈HN ×N x∈F

is equal to

dist(a, (HF∞ )N ×N ) :=

inf

∞) f ∈(HF N ×N

a − f L∞ . N ×N

4.5 Local Toeplitz Operators

211

∞ ∞ Proof. Let m := distF (a, HN ×N ). For each ε > 0, there is an h ∈ HN ×N such that (a − h)|F < m + ε and there is an open set U ⊃ F such that sup a(x) − h(x) < m + ε. Since F is a weak peak set for H ∞ , there exists a x∈U

peak set P for H ∞ such that F ⊂ P ⊂ U . Thus, (a − h)|P < m + ε. Choose ∞ ϕ ∈ H ∞ so that ϕ|P = 1 and |ϕ(x)| < 1 for x ∈ X \ P , and define g ∈ HN ×N n as g = ϕIN ×N . There exists an n ∈ Z+ such that g (x)(a(x) − h(x)) < ε = 1, it follows that for all x in the (compact) set X \ U , and since gL∞ N ×N n g n (a − h)L∞ < m + ε. But g (a − h) = a − h + f with some f ∈ L∞ N ×N N ×N satisfying f |F = 0, whence dist(a, (HF∞ )N ×N ) ≤ a − h + f L∞ < m + ε. N ×N On the other hand, if we let m := dist(a, (HF∞ )N ×N ), then for each ε > 0 there < m + ε, and since h = f + g with is an h ∈ (HF∞ )N ×N such that a − hL∞ N ×N ∞ f ∈ HN and g|F = 0, we arrive at the inequality (a − f )|F < m + ε. ×N

4.5 Local Toeplitz Operators Let B be C, QC, or C + H ∞ and let S denote the collection of the maximal antisymmetric sets for B. Let a ∈ L∞ . We know from Theorem 2.83 that T (a) is Fredholm if and only if T (a) is S-restricted invertible for all S ∈ S. On the other hand, Theorem 4.12 (for B = C or QC) and Corollary 4.49(a) (for B = QC or C + H ∞ ) tell us that T (a) is Fredholm if and only if the local operators TSπ (a) are invertible for all S ∈ S. The conclusion is that T (a) is S-restricted invertible for all S ∈ S if and only if TSπ (a) is invertible for all S ∈ S. The question we are interested in here is as follows: given an individual F ∈ S, is it true that T (a) is F -restricted invertible if and only if TFπ (a) is invertible? We shall show that the answer is yes. Note that the “only if” part is trivial. 4.57. Lemma. Let F be a closed subset of X = M (L∞ ), let a ∈ L∞ , and suppose a(x) = 0 for x ∈ F . Then there exists a b ∈ GL∞ such that b|F = a|F . Proof. Let U be a clopen (:= simultaneously closed and open) neighborhood of F such that a(x) = 0 for x ∈ U (recall that X is totally disconnected). Then the characteristic function χU of U is (the Gelfand transform of a function) in L∞ . Put b = aχU + 1 − χU . Thus b|F = a|F and the function c ∈ L∞ given by c(x) = 1/a(x) for x ∈ U and c(x) = 1 for x ∈ X \ U is the inverse of b. 4.58. Proposition. Let F be a closed subset of X and let a ∈ L∞ . Then T (a) is F -restricted left, right, or two-sided invertible (resp. TFπ (a) is left, right, twosided invertible) if and only if there is a b ∈ GL∞ such that b|F = a|F and T (b/|b|) (resp. TFπ (b/|b|)) has the corresponding property.


212

Proof. If T (a) is F -restricted left, right, or two-sided invertible, then a(x) = 0 for x ∈ F due to Theorem 2.30. The same conclusion can be drawn from the left, right, or two-sided invertibility of TFπ (a) by using Corollary 3.63. Hence, by the preceding lemma, there is a b ∈ GL∞ such that b|F = a|F . We saw in the proof of Proposition 2.19 that b/|b| factors as hbh with h ∈ GH ∞ . Since T (b/|b|) = T (h)T (b)T (h), all assertions of the proposition follow at once. 4.59. The Chang-Marshall theorem. A closed subalgebra of L∞ is called a Douglas algebra if it contains H ∞ . If B is an arbitrary subset of L∞ , then algL∞ (B, H ∞ ) is clearly a Douglas algebra and vice versa, every Douglas algebra is of this form. The following remarkable fact was conjectured by Douglas and proved by Chang and Marshall: Every Douglas algebra A is of the form A = algL∞ (B, H ∞ ), where B is some collection of inner functions. Given a Douglas algebra A define ΣA := {ϕ ∈ H ∞ : ϕ inner , ϕ ∈ A}. Then, by the Chang-Marshall theorem, A = algL∞ (ΣA , H ∞ ). The algebras HF∞ defined in 4.55 are obviously Douglas algebras. So the Chang-Marshall theorem implies that HF∞ = algL∞ (ΣF , H ∞ ) where ΣF := ϕ ∈ H ∞ : ϕ inner , ϕ ∈ HF∞ . Thus, HF∞ = clos

n

hi ϕi : hi ∈ H ∞ , ϕi ∈ ΣF ,

i=1

and because h1 ϕ1 + h2 ϕ2 = (h1 ϕ2 + h2 ϕ1 )ϕ1 ϕ2 , we have HF∞ = closL∞ hϕ : h ∈ H ∞ , ϕ ∈ H ∞ , ϕ inner, ϕ ∈ HF∞ .

(4.34)

Actually the full strength of the Chang-Marshall theorem is not required for our purposes, since all what we need is equality (4.34), i.e., the ChangMarshall theorem for the special case that A = HF∞ . For this case the theorem was proved by Axler [12] by employing techniques that are simpler than those required for the general case. 4.60. Clancey-Gosselin sets. A closed subset F of X = M (L∞ ) will be called a Clancey-Gosselin set (briefly a CG-set) if it has the following properties: (a) F is a weak peak set for H ∞ . (b) ϕ ∈ H ∞ , ϕ inner, ϕ ∈ HF∞ , implies that ϕ is constant on F . In other words, the CG-sets are those weak peak sets for H ∞ for which ΣF |F ∼ = C. Clancey and Gosselin [139] showed that the fibers Xτ (τ ∈ T), the fibers Xξ (ξ ∈ M (QC)), and the maximal antisymmetric sets for C +H ∞ have property


213

(b). The simplest case is that in which F is a maximal antisymmetric set for C + H ∞ : if ϕ ∈ H ∞ is inner and ϕ|F ∈ H ∞ |F , then (ϕ + ϕ)|F ∈ H ∞ |F and (1/i)(ϕ − ϕ)|F ∈ H ∞ |F are real-valued, so the antisymmetry property of F implies that (ϕ + ϕ)|F and (ϕ − ϕ)|F are constant, and so ϕ|F must also be constant. The verification of (b) for the fibers Xτ and the fibers Xξ is not trivial. It requires a series of ingredients from the theory of function algebras, so its proof must be omitted here. The extension result stated in Proposition 4.62 below may serve as a motivation for the introduction of CG-sets. 4.61. Lemma. Let F be a CG-set and let a ∈ L∞ . Then distF (a, GH ∞ ) = dist(a, GHF∞ ).

(4.35)

Proof. Choose an f ∈ GHF∞ so that a − f < dist(a, GHF∞ ) + ε. By virtue of (4.34) we may assume that f = gϕ with g ∈ H ∞ , ϕ ∈ HF∞ , ϕ inner. Since ϕ ∈ GHF∞ (ϕ−1 = ϕ ∈ H ∞ ⊂ HF∞ ), it follows that g ∈ GHF∞ . Write g = ψh with ψ inner and h ∈ GH ∞ (see 1.41). Then ψ = gh−1 ∈ GHF∞ , hence ψ = ψ −1 ∈ HF∞ . Because F is a CG-set, we conclude that ψ|F and ϕ|F are constant. Without loss of generality assume ψ|F = ϕ|F = 1 (otherwise write f = (cg)(c ϕ), g = (dψ)(dh)). Thus, (a − h)|F = (a − ψhϕ)|F ≤ a − ψhϕ∞ = a − f ∞ and letting ε go to zero we obtain “≤” in (4.35). We now prove the opposite inequality. Let ε > 0 and let h0 ∈ GH ∞ satisfy (a − h0 )|F < distF (a, GH ∞ ) + ε/2. Then let V0 be a clopen neighborhood of F such that (4.36) (a − h0 )|V0 < distF (a, GH ∞ ) + ε. n Vi , where each Vi is clopen and Since X \ V0 is compact, we have X \ V0 = i=1

there is an xi ∈ Vi such that max |a(xi ) − a(y)| < y∈Vi

ε 2

∀ i = 1, . . . , n.

(4.37)

Put hi = a(xi ) if a(xi ) if a(xi ) = 0 and let hi = ε/2 if a(xi ) = 0. Let χVi n denote the characteristic function of Vi and put h = i=0 hi χVi . Because because h|F = h |F , h is even in HF∞ . The each Vi is clopen, h is in L∞ , and 0

n −1 −1 −1 −1 inverse of h is clearly h = i=0 hi χVi , and since h |F = h−1 0 |F , h belongs to HF∞ . Thus, h ∈ GHF∞ . We have dist(a, GHF∞ ) ≤ a − h∞ = max (a − h)|Vi , i=0,...,n

and because of (4.36) and the inequalities (a − b)|Vi < ε for i = 1, . . . , n (resulting from (4.37)), it follows that dist(a, GHF∞ ) < distF (a, GH ∞ ) + ε.

214


4.62. Proposition. Let F be a CG-set and suppose a function u ∈ L∞ is unimodular on F (i.e., |u(x)| = 1 for x ∈ F ). If distF (u, H ∞ ) < 1 (resp. distF (u, GH ∞ ) < 1), then there exists a unimodular function v ∈ GL∞ (i.e., |v(x)| = 1 for all x ∈ X) such that v|F = u|F and dist(v, H ∞ ) < 1 (resp. dist(v, GH ∞ ) < 1). Proof. Due to Lemma 4.57 there is a w ∈ GL∞ such that w|F = u|F , and since w/|w| also coincides with u on F , it can be a priori assumed that |u(x)| = 1 for all x ∈ X. Suppose distF (u, H ∞ ) < 1. Then, by Proposition 4.56, dist(u, HF∞ ) < 1 and so (4.34) implies that there are a function g ∈ H ∞ and an inner function ϕ ∈ H ∞ such that ϕ ∈ HF∞ and u − gϕ∞ < 1. Since F is a CG-set, ϕ|F is constant, say ϕ|F = 1. Put v = ϕu. Then v|F = u|F , |v(x)| = |ϕ(x)||u(x)| = 1 for x ∈ X, and because v − g∞ = ϕu − g∞ = u − ϕg∞ < 1, it follows that dist(v, H ∞ ) < 1. Now suppose distF (u, GH ∞ ) < 1. Lemma 4.61 shows that u−f ∞ < 1 for some f ∈ GHF∞ . By (4.34) and an argument used in the proof of Lemma 4.61, we may assume that f = ψhϕ with h ∈ GH ∞ , ϕ and ψ inner, ϕ and ψ in HF∞ . Since F is a CG-set, it may be assumed that ϕ|F = ψ|F = 1. Thus, if we let v = ϕψu, then v|F = u|F , |v(x)| = 1 for x ∈ X, and since v − h∞ = ϕψu − h∞ = u − ψhϕ∞ = u − f ∞ < 1, we finally see that dist(v, GH ∞ ) < 1. 4.63. Theorem (Clancey/Gosselin). Let B be C, QC, or C + H ∞ and let F be a maximal antisymmetric set for B. Let a ∈ L∞ . Then the following are equivalent: (i) T (a) is F -restricted left (resp. right) invertible; (ii) TFπ (a) is left (resp. right) invertible; (iii) we have a(x) = 0 for all x ∈ F and distF (a/|a|, H ∞ ) < 1 (resp. distF (a/|a|, H ∞ ) < 1). Proof. We only consider the case of left-invertibility. (i) =⇒ (ii). Obvious. (ii) =⇒ (iii). By Proposition 4.58, there is a b ∈ GL∞ such that b|F = a|F and TFπ (b/|b|) is left-invertible. From Corollary 4.54 we deduce that distF (a/|a|, H ∞ ) = distF (b/|b|, H ∞ ) < 1. (iii) =⇒ (i). Lemma 4.57 shows that there is a b ∈ GL∞ such that b|F = a|F . Clearly, distF (b/|b|, H ∞ ) < 1. Now Proposition 4.62 yields the


215

existence of a unimodular function v ∈ GL∞ such that v|F = (b/|b|)|F and dist(v, H ∞ ) < 1. By Theorem 2.20(a), T (v) is left-invertible, and hence T (b/|b|) is F -restricted left-invertible. It remains to apply Proposition 4.58. 4.64. Theorem (Clancey/Gosselin). Assume B is C, QC, or C +H ∞ and F is a maximal antisymmetric set for B. Let a ∈ L∞ . Then the following are equivalent: (i) T (a) is F -restricted invertible; (ii) T (a) is F -restricted left-invertible and F -restricted right-invertible; (iii) TFπ (a) is invertible; (iv) a(x) = 0 for x ∈ F , distF (a/|a|, H ∞ ) < 1, and distF (a/|a|, H ∞ ) < 1; (v) a(x) = 0 for x ∈ F and distF (a/|a|, GH ∞ ) < 1. Proof. (i) =⇒ (ii) =⇒ (iii). Obvious. (ii) ⇐= (iii) ⇐⇒ (iv). Theorem 4.63. (v) =⇒ (i). First choose a b ∈ GL∞ satisfying b|F = a|F (Lemma 4.57). Then apply Proposition 4.62 to deduce that there is a unimodular v ∈ GL∞ such that v|F = (b/|b|)|F = (a/|a|)|F and dist(v, GH ∞ ) < 1. Thus, by Theorem 2.20(c), T (v) ∈ GL(H 2 ), which implies the F -restricted invertibility of T (b/|b|), and Proposition 4.58 then gives that of T (a). (ii) =⇒ (v). Again choose b ∈ GL∞ so that b|F = a|F , put u = b/|b|, and deduce from Proposition 4.58 that T (u) is both F -restricted left-invertible and F -restricted right-invertible. So, by Theorem 4.63 and Proposition 4.56, dist(u, HF∞ ) < 1,

dist(u, HF∞ ) < 1.

(4.38)

Now (4.34) implies that there are an h ∈ H ∞ and an inner function ϕ such that ϕ ∈ HF∞ and u − hϕ∞ < 1. We show that hϕ ∈ GHF∞ , which, by Lemma 4.61, will complete the proof. Since F is a CG-set, it can be assumed that ϕ|F = 1. Write h = ψg with an inner function ψ and an outer function g ∈ H ∞ (see 1.41(a)). Because 1 − u ϕψg∞ = u − hϕ∞ < 1,

(4.39)

it follows that g ∈ GL∞ and hence g ∈ GH ∞ (1.41(g)). Also because of (4.39), TFπ (u ϕψg) = TFπ (u)IFπ TFπ (g)TFπ (ψ) (recall that ϕ|F = 1) is invertible. From (4.38) and Theorem 4.63 we get the invertibility of TFπ (u), and since g ∈ GH ∞ , TFπ (g) is also invertible. The conclusion is that TFπ (ψ) must also be invertible. So Theorem 4.63 in conjunction with Proposition 4.56 shows that there is an f ∈ HF∞ such

216


that ψ − f ∞ < 1. Hence 1 − ψf ∞ < 1 and since ψf ∈ HF∞ , we obtain that actually ψf ∈ GHF∞ . Thus, ψ = f (ψf )−1 ∈ HF∞ and since ψ ∈ H ∞ ⊂ HF∞ , it follows that ψ ∈ GHF∞ . Finally, taking into account that ϕ ∈ GHF∞ (ϕ−1 = ϕ ∈ H ∞ ⊂ HF∞ ) and that g ∈ GH ∞ we deduce that hϕ = ψgϕ ∈ GHF∞ , as desired. 4.65. Corollary. Let B be C, QC, or C + H ∞ and let S denote the family of the maximal antisymmetric sets for B. Then if S ∈ S and a ∈ L∞ , $ $ sp T π (f ) = sp T (f ), (4.40) sp TSπ (a) = f ∈a+IS

f ∈a+IS

where IS = {g ∈ L∞ : g|S = 0}. Proof. Clearly, if f ∈ a + IS , then sp T (f ) ⊃ sp T π (f ) ⊃ sp TSπ (f ) = sp TSπ (a) and thus (4.40) holds with the two “=” replaced by “⊂”. It remains to show that sp T (f ) is contained in sp TSπ (a), i.e., that the invertibilf ∈a+IS

ity of TSπ (a − λ) (λ ∈ C) implies the existence of an f ∈ a + IS such that T (f − λ) is invertible. But this is equivalent to saying that the invertibility of TSπ (a − λ) implies the S-restricted invertibility of T (a − λ), and this was proved in Theorem 4.64. Remark. As an immediate consequence of Theorem 4.12 (B = C or QC) and Corollary 4.49(a) (B = QC or B = C + H ∞ ) we have that / sp TSπ (a), sp T π (a) = S∈S

i.e., the essential spectrum of a Toeplitz operator is the union of all its “local spectra.” The above theorem lies deeper and involves the following identification of the local spectrum: the spectrum of a local Toeplitz operator TSπ (a) is precisely the common part of the essential spectra of all Toeplitz operators whose symbols coincide on S with a|S. 4.66. Open problems. (a) What can be said for matrix symbols about the connection between the invertibility of local Toeplitz operators and restricted invertibility? (b) Find an analogue of Theorem 4.64 (well, say with B = C or B = QC) for harmonic approximation or stable convergence, i.e., for the case that T π π is replaced by K or T KJ . Note that it is again the lack of an analogue of formula (2.20) and the observation that will be made in 4.77 which complicate the things and require other techniques (the “again” is because of 4.50(e)).

4.6 Symbols with Specific Local Range

217

4.6 Symbols with Specific Local Range 4.67. Theorem. Let ξ ∈ M (B), where B is C or QC. Let a ∈ L∞ and assume the set a(Xξ ) is contained in some straight line segment. Then sp Tξπ (a) = conv a(Xξ ). Proof. If a(Xξ ) is a singleton, Corollary 3.64 applies. Let conv a(Xξ ) = [z1 , z2 ], where z1 , z2 ∈ C and z1 = z2 . Put b = 2(z2 − z1 )−1 [a − (z1 + z2 )/2]. Then {−1, 1} ⊂ b(Xξ ) ⊂ [−1, 1] and it is clear that sp Tξπ (a) = [z1 , z2 ] if and only if sp Tξπ (b) = [−1, 1]. From Corollary 3.64 (or Theorem 4.24) we deduce that {−1, 1} ⊂ sp Tξπ (b) ⊂ [−1, 1]. Let µ ∈ (−1, 1) and assume µ ∈ / sp Tξπ (b). By Proposition 4.58, there is a ∞ c ∈ GL such that c|Xξ = (b − µ)|Xξ and Tξπ (c/|c|) ∈ GL(H 2 ). Hence, by Corollary 4.54, (4.41) distXξ (c/|c|, H ∞ ) < 1. But the range of c/|c| on Xξ is the doubleton {−1, 1}. So Lemma 2.90 shows that (4.41) is impossible and this contradiction completes the proof. 4.68. Open problems. Let F be a maximal antisymmetric set for C, QC, or C + H ∞ and let {Kλ }λ∈Λ be an approximate identity whose index set Λ is connected. Is it true that the local spectra sp TFπ (a),

sp {kλ a}πF ,

sp {T (kλ a)}πF

(4.42)

are connected for every a ∈ L∞ ? There is only one case in which we know that the answer is yes: if τ ∈ T = M (C), then sp {kλ a}πτ is connected for every a ∈ L∞ (Proposition 3.73(b) and Theorem 3.76(c)). We do not know any symbol a ∈ L∞ for which any of the local spectra (4.42) is disconnected. However, there are certain classes of symbols for which the connectedness of some of the local spectra (4.42) is known. For instance, if ξ ∈ M (C) or ξ ∈ M (QC) and if a(Xξ ) is contained in some straight line segment, then, by Theorems 4.67 and 4.24, π sp TX (a) = sp {T (kλ a)}πXξ = conv a(Xξ ) ξ

are connected; from Theorem 4.24 we also know that sp {kλ a}πXξ ⊂ conv a(Xξ ) and we conjecture that “⊂” can be replaced by equality. This conjecture is supported by the fact that equality holds for B = C (Corollary 3.78(a)) or for B = QC and a ∈ P QC (Theorem 3.79).

218


4.69. Lemma. Suppose A is a C ∗ -algebra with identity element. Suppose a = diag (a1 , . . . , aM ) ∈ AN ×N is a block-diagonal matrix each block ai (i = 1, . . . , M ) of which is upper-triangular with equal entries bi on the diagonal. Then a ∈ GAN ×N if and only if bi ∈ GA for i = 1, . . . , M . Proof. Due to the Gelfand-Naimark theorem 1.26(b) it can be assumed that A is a C ∗ -subalgebra of L(H), where H is some Hilbert space. Then a may be thought of as an operator in L(HN ). It is easily seen that the invertibility of a on HN implies that each diagonal block ai is an invertible operator on the direct sum of the corresponding number of copies of H. Consideration of the south-east entry of ai shows that bi must be onto, while consideration of the north-west entry of ai yields that bi is one-to-one. 4.70. Theorem. Let B = C or B = QC, let a ∈ L∞ N ×N , and suppose for each ξ ∈ M (B) the set conv a(Xξ ) is a (possibly degenerate) straight line segment. Then 2 ) ⇐⇒ a is locally sectorial over B. T (a) ∈ Φ(HN In that case {kλ a} is bounded away from zero on conv Λ for every approximate identity {Kλ }λ∈Λ and Ind T (a) = −ind {det kλ a}. Proof. In the scalar case (N = 1) the Fredholm criterion is immediate from Theorem 4.12 in conjunction with Theorem 4.67. In the general case we are by virtue of Corollary 4.13(b) and Corollary 4.30 left with the proof of the implication “=⇒”. 2 ). Then, by Theorem 4.12, Tξπ (a) is invertible for So assume T (a) ∈ Φ(HN each ξ ∈ M (B). Fix ξ ∈ M (B) and let conv a(Xξ ) = µE + (1 − µ)F : µ ∈ [0, 1] . In particular, there are x1 , x2 ∈ Xξ such that a(x1 ) = E and a(x2 ) = F . Due to Theorem 2.93, both E and F are invertible matrices. There exists an invertible matrix D such that J := D−1 E −1 F D is in Jordan canonical −1 −1 E aD. Then b(x) is an upper-triangular from. Define b ∈ L∞ N ×N as b := D matrix for each x ∈ Xξ , and if we let bii (i = 1, . . . , N ) denote the diagonal entries of b, then conv bii (Xξ ) = [1, λi ], where λi is an eigenvalue of J (note that bii (x1 ) = 1 and bii (x2 ) = λi ). The invertibility of Tξπ (a) implies that Tξπ (b) is invertible, and hence, by Lemma 4.69, Tξπ (bii ) is invertible for each i. Since sp Tξπ (bii ) = [1, λi ] (Theorem 4.67), it follows that the origin does not belong to any of the line segments [1, λi ]. Thus, det(µE + (1 − µ)F )(det E)−1 = det(µI + (1 − µ)J) =

n

(µ + (1 − µ)λi ) = 0

i=1

for all µ ∈ [0, 1]. Now Theorem 3.4 completes the proof.


219

Remark. The preceding theorem reduces the Fredholm theory of Toeplitz operators with (P2 C)N,N or (P2 QC)N,N symbols (or, more generally, with symbols whose range on each fiber over C or QC is contained in some line segment) to the question of finding conditions for the local sectoriality of such symbols. This is one reason for the due attention we paid to local sectoriality in Chapter 3. In particular, recall Theorem 3.4 (along with Proposition 3.2), Proposition 3.11 (together with Theorem 3.9), Corollary 3.82, and Corollary 3.85. 4.71. Pn C symbols. Things are more complicated for symbols taking n ≥ 3 values on some fiber Xτ (τ ∈ T). The knowledge we have about Toeplitz operators generated by such symbols is by no means comparable with the knowledge one has in the case of P2 C or even P2 QC symbols. In accordance with 2.89, a function a ∈ L∞ is said to belong to Pn C if its essential range consists of at most n distinct points. Clearly, if T is divided into three pairwise disjoint measurable subsets A, B, C of positive measure and if a, b, c are pairwise distinct complex numbers, then ϕ = aχA + bχB + cχC

(4.43)

belongs to P3 C \ P2 C and every ϕ ∈ P3 C \ P2 C is of this form. Let ∆(a, b, c) denote the triangle which is the closed convex hull of the three points a, b, c. We know from Theorems 2.30 and 2.33 that {a, b, c} ⊂ sp T π (ϕ) ⊂ ∆(a, b, c), and the question is which points of the triangle ∆(a, b, c) are in sp T π (ϕ), the essential spectrum of T (ϕ). There must exist a “sufficiently large” supply of such points, since sp T π (ϕ) is a connected set (2.35(b)). Using Theorem 2.74 it is easy to produce symbols ϕ ∈ P3 C ∩ P C such that sp T π (ϕ) consists of two or three sides of that triangle. Moreover, the same theorem shows that, for ϕ ∈ P3 C ∩ P C, an interior point of that triangle can never belong to sp T π (ϕ). More exotic symbols in P3 C can be obtained by putting ϕ = p ◦ ω, where p ∈ P3 C ∩ P C and ω ∈ H ∞ is an inner function. Suppose T (p) ∈ Φ(H 2 ). If Ind T (p) = 0, so that T (p) ∈ GL(H 2 ), then T (p ◦ ω) is invertible by Theorem 2.20. However, if Ind T (p) = 0, then by using Theorem 2.64 it is not difficult to show that T (p ◦ ω) is Fredholm if and only if ω is a finite Blaschke product. In particular, if we let p(eiθ ) be a, b, c for θ ∈ (0, 2π/3), θ ∈ (2π/3, 4π/3), θ ∈ (4π/3, 2π), respectively, and if ω is not a finite Blaschke product, then sp T (p ◦ ω) = sp T π (p ◦ ω) = ∆(a, b, c). Some sufficiently interesting Pn C (or even Pn C) functions are contained in the class LCS(T◦ ) we shall define in the next section. Finally, note that if a ∈ Pn C, then for each τ ∈ T there exists an aτ in Pn C such that a|Xτ = aτ |Xτ . This can be shown as follows. Let τ ∈ T and


220

a(Xτ ) = {v1 , . . . , vm } (m ≤ n) with vi = vj for i = j. Choose an ε > 0 so that the disks Di with center vi and radius ε (i = 1, . . . , m) are pairwise disjoint. m Di . By virtue of Proposition 2.79(a), there is a U ∈ Uτ such that a(U ) ⊂ i=1

m −1 Put Ui = U ∩ a (Di ). The function aτ := v1 χT\U + i=1 vi χUi belongs to Pn C, and once more using Proposition 2.79(a) it is easy to check that (a − aτ )(Xτ ) = {0}, whence a|Xτ = aτ |Xτ . The preceding observation in conjunction with Theorem 4.12 shows that the determination of the essential spectrum of operators with Pn C symbols can be reduced to the identification of the local spectrum of operators with Pn C symbols. 4.72. The class LCS(T◦ ). Let T◦ be the punctured circle T \ {−1} and let LCS(T◦ ) denote the class of all functions a ∈ GL∞ with the following property: there is an ε > 0 such that for each τ ∈ T◦ there are a subarc Uτ of T◦ containing τ and a cτ ∈ C, |cτ | = 1, such that Re (cτ a(t)) ≥ ε for almost all t ∈ Uτ . The LCS is for locally C-sectorial. An obvious modification of the argument used to prove the implication (iii) =⇒ (iv) of Theorem 3.9 shows that every function a ∈ LCS(T◦ ) can be written as a = cs, where c ∈ CU (T◦ ) (recall 2.25) and s ∈ GL∞ is sectorial (on T). Choose any b ∈ CR(T◦ ) satisfying c = eib and define a# ∈ C(R) by a# (x) = b((i−x)/(i+x)) for x ∈ R. Let a = s1 eib = s2 eib with b1 , b2 ∈ CR(T◦ ) and s1 , s2 sectorial (on T). There are γ1 , γ2 ∈ C such that Re (γ1 s1 ) ≥ ε, Re (γ2 s2 ) ≥ ε a.e. on T, which implies that there is a δ > 0 with the property that + γ s + γ eib1 + + + + + 2 2 2 + arg = + + arg + + < π − δ, ib γ1 e 2 γ1 s1 whence −π + δ < arg

γ 2

γ1

+ b1 − b2 < π − δ

on

T◦ .

Thus, any two functions a# only differ by a function of the form γ + v, where γ ∈ R and vL∞ (R) < π. So it makes a correct sense to say that a# (±∞) is equal to +∞ or −∞ or that a# be bounded from above or below at ±∞ (see 2.25). 4.73. Theorem. Let a ∈ LCS(T◦ ). π (a) If a# (+∞) = +∞ and a# is bounded from above at −∞, then T−1 (a) = is not invertible.

π (a) TX −1

(b) Suppose a# = ϕ + η, where η ∈ L∞ (R), ϕ is monotonuous on (−∞, 0) π π (a) = TX (a) is invertible, we and (0, ∞), and ϕ(±∞) = +∞. Then if T−1 −1 have a# (x) = O(log |x|) as |x| → ∞.


221

Proof. We have a = cs, where c = eib ∈ CU (T◦ ) and s ∈ GL∞ is sectorial. If π (a) is invertible, τ ∈ T◦ , then Tτπ (a) = c(τ )Tτπ (s) is invertible. Hence, if T−1 2 then T (a) ∈ Φ(H ) by Theorem 4.12. In that case there is an n ∈ Z such that T (χn a) ∈ GL(H 2 ) and we deduce from Theorem 2.23 and from 1.48(g) that the argument of χn a/|χn a| belongs to BM O. But the argument of χn a/|χn a| is equal to b plus the arguments of χn and s/|s|, i.e., it differs from b merely by a function in L∞ . Consequently, b ∈ BM O. The proof of 2.26(c) shows that under the hypothesis (a) the function b π (a) is not invertible under this cannot belong to BM O, which proves that T−1 hypothesis. Finally, if the hypothesis (b) is satisfied, then the argument of 2.26(d) gives that b((i − x)/(i + x)) = O(log |x|) as |x| → ∞, which implies the assertion of part (b) of the present theorem. 4.74. Application to P3 C. The preceding theorem can be applied to get full information about the spectra of Toeplitz operators generated by certain Pn C symbols. To illustrate this it suffices to consider the case n = 3. Choose real numbers θn (n ∈ Z) so that −π < . . . < θ−2 < θ−1 < θ0 < θ1 < θ2 < . . . < π and θn → −π, θn → π as n → ∞. Then put tn = eiθn . Let ϕ be of the form (4.43) and suppose each of the sets A, B, C is the union of some subarcs of the form (tn , tn+1 ). By Theorem 4.67, for τ ∈ T◦ the local spectrum sp Tτπ (ϕ) is either one of the points a, b, c (τ = tn ) or one of the line segments [a, b], [b, c], [c, a] (τ = tn ). So the only interesting part of π (ϕ). It is clear that for each λ ∈ C which does not belong sp T π (ϕ) is sp T−1 to the boundary of the triangle ∆(a, b, c) the function ϕ − λ is in LCS(T◦ ). It is also easy to determine the behavior of (ϕ − λ)# . The mapping T◦ → R, t → i(1 − t)/(1 + t) takes the sets A, B, C into certain subsets of R which will be denoted by A, B, C, respectively, too. Theorem 4.73(a) now gives the following. If the location of A, B, C on R is of the type

(in that case (ϕ − λ)# (−∞) = −∞ and (ϕ − λ)# (+∞) = +∞ for λ in the interior of ∆(a, b, c)) or, e.g., of the type

(in that case (ϕ−λ)# is bounded away from above at −∞ and (ϕ−λ)# (+∞) = +∞ for λ in the interior of ∆(a, b, c)), then π sp T−1 (ϕ) = ∆(a, b, c).

222


Theorem 4.73(b) can be applied to the situation where the sets A, B, C are located as follows:

In that case (ϕ−λ)# (±∞) = +∞ and (ϕ−λ)# (x) is monotonically increasing as |x| → ∞ for each λ in the interior of ∆(a, b, c). If n is sufficiently large, then the increment of (ϕ − λ)# when t is running through (tn , tn+3 ) or (t−n , t−n−3 ) equals 2π. Thus, in the case under consideration Theorem 4.73(b) together with a simple computation gives the following: if at least one point of the π (ϕ), then there are interior of the triangle ∆(a, b, c) does not belong to sp T−1 constants C > 0 and 0 < q < 1 such that dist(tn , −1) ≤ Cq |n| for all n ∈ Z. In other words, if, in the case at hand, there do not exist C > 0 and 0 < q < 1 π (ϕ) = ∆(a, b, c). such that dist(tn , −1) ≤ Cq |n| for all n ∈ Z, then sp T−1 Note that in all the above situations in which we were able to determine the spectrum of a Toeplitz operator generated by a symbol of the form (4.43) that spectrum was either the whole (possibly degenerate) triangle ∆(a, b, c) or consisted of two or three of its sides. A while we conjectured that the same is true for every P3 C symbols; however, we shall see that this is not so in general (Proposition 4.78). 4.75. Proposition. Let ϕ = aχA + bχB + cχC ∈ P3 C. Then either

(a, b) ⊂ sp T π (ϕ)

or

(a, b) ∩ sp T π (ϕ) = ∅.

In other words, a side of the triangle ∆(a, b, c) either entirely belongs to sp T π (ϕ) or, apart from the endpoints, entirely belongs to the complement of sp T π (ϕ). The same is true with sp T π (ϕ) replaced by sp T (ϕ). Proof. We first prove the proposition for sp T (ϕ). If ∆(a, b, c) is a line segment, the assertion follows from 2.36. So suppose ∆(a, b, c) is not a line segment and without loss of generality assume a and b are real, a < 0, b > 0, and Im c > 0. Let µ ∈ (a, b). Then, by Proposition 2.19, µ ∈ sp T (ϕ) ⇐⇒ 0 ∈ sp T (ϕ − µ)/|ϕ − µ| . The essential range of (ϕ − µ)/|ϕ − µ| consists of the two points −1 and 1 and of a point c(µ) lying on the upper half T+ of the unit circle. Thus, what we must show is the following: if c1 , c2 ∈ T+ and if 0 ∈ sp T (χB − χA + c1 χC ), then 0 ∈ sp T (χB − χA + c2 χC ). So suppose 0 ∈ sp T (χB − χA + c1 χC ). Then Wolff’s result 2.37 implies that there are zn ∈ D such that 2 2 χ2 B (zn ) − χ A (zn ) + c1 χ C (zn ) → 0 as

n → ∞.

(4.44)

2 2 2 Because χ2 A, χ B, χ C are real-valued, we deduce from (4.44) that χ C (zn ) → 0 as n → ∞, and therefore


223

|χ2 2 2 2 2 2 2 B (zn )− χ A (zn )+c2 χ C (zn )| ≤ |χ B (zn )− χ A (zn )+c1 χ C (zn )|+|c2 −c1 |χ C (zn ) also tends to zero as n → ∞. Again applying 2.37 we see that 0 belongs to sp T (χB − χA + c2 χC ), as desired. To get the assertion for the essential spectrum, let µ ∈ (a, b) and assume µ∈ / sp T π (ϕ). Then T (ϕ − µ) is Fredholm. Let κ := Ind T (ϕ − µ). There is an ε > 0 such that T (ϕ − µ + zχB ) ∈ Φ(H 2 ),

Ind T (ϕ − µ + zχB ) = κ

whenever z ∈ C and |z| < ε. Since among these z’s there is a z0 such that µ∈ / ∆(a, b +z0 , c), which implies that T (ϕ−µ+z0 χB ) is invertible, we deduce that κ must be zero. Consequently, by Corollary 2.40, T (ϕ − µ) is invertible and hence µ ∈ / sp T (ϕ). From what has been proved above we obtain that (a, b) ∩ sp T (ϕ) = ∅, whence (a, b) ∩ sp T π (ϕ) = ∅. 4.76. Open problem. Is the preceding proposition true with sp T π (ϕ) replaced by sp Tτπ (ϕ) (τ ∈ T)? In this connection it would be interesting to know whether Wolff’s result 2.37 has a local analogue: for a ∈ L∞ , λ belonging to the boundary of conv a(Xτ ), and τ ∈ T, is it true that λ ∈ sp Tτπ (a) ⇐⇒ λ ∈ ClH (a, τ )? 4.77. Harmonic extension of P3 C functions. The harmonic extension of Pn C functions has a very nice geometric interpretation. Let u be the conformal mapping of the unit disk D onto the upper half plane Π given by 1−z . u : D → Π, z → i 1+z Then u−1 is given by u−1 : Π → D,

ζ →

i−ζ . i+ζ

Note that u extends to a continuous function on clos D \ {−1} which maps T◦ = T \ {−1} onto R and −1 into the point at infinity. Given a bounded interval I on R we denote by ωI (ζ) the angle at which I is seen from ζ ∈ Π. Also let ω(x,∞) (ζ) := lim ω(x,y) (ζ) and define ω(−∞,x) (ζ) similarly. Finally y→∞

set ω(−∞,x)∪(y,∞) (ζ) := ω(−∞,x) (ζ) + ω(y,∞) (ζ). If E is a subarc of T, then u(E) is an interval on R, which is of the form (−∞, x) ∪ (y, ∞) in case E contains the point −1. The harmonic extension of the characteristic function χE at z ∈ D is then given by χ2 E (z) =

1 ωu(E) (u(z)). π

This is well known and can be verified without difficulty, e.g., using the representation of the harmonic extension via Poisson’s integral.

224


Thus, if ϕ is of the form (4.43), where each of the sets A, B, C is a (possibly countable) union of subarcs of T, and if we maintain our convention to denote a set on T and its image on R under the mapping u by the same symbol, then, for z ∈ D, (4.45) π ϕ(z) = aωA (u(z)) + bωB (u(z)) + cωC (u(z)), where, for I = In , each In being an interval and the union being possibly n

countable, ωI (ζ) is defined as n ωIn (ζ); note that the latter series always converges. In the case where {Kλ }λ∈(0,∞) is the approximate identity generated by the Poisson kernel, we write {hr a}r∈(0,1) instead of {kλ a}λ∈(0,∞) . We have seen that Toeplitz operators and the harmonic extension have many features in common. Example: if τ ∈ T, then a(Xτ ) ⊂ sp Tτπ (a) ⊂ conv a(Xτ ),

a(Xτ ) ⊂ sp {hr a}πτ ⊂ conv a(Xτ ).

But we have also seen that there are some decisive differences (see, e.g., the open problems 4.50, 4.66, 4.68). To study Toeplitz operators we have made frequent and essential use of Propositions 2.19 and 2.32. However, such an argument cannot be applied to harmonic extension: there exist functions ϕ ∈ P3 C ∩ GL∞ such that {hr ϕ} ∈ G(alg H(L∞ ))

but

{hr (ϕ/|ϕ|)}π ∈ / G(alg Hπ (L∞ )).

(4.46)

A ϕ satisfying (4.46) is ϕ = M χA + e2πi/3 χB + e−2πi/3 χC , where the location of (the images under the mapping u of) the sets A, B, C on R is as follows:

Here the length of an interval belonging to A is 1 and the lengths of the intervals forming B and C are 2. Using the above geometric interpretation of the harmonic extension (we omit the technical details) one can show that for and (ϕ/|ϕ|)(D) look as follows: sufficiently large M ∈ R+ the sets ϕ(D)

Thus, Theorem 3.76(a) implies that {hr ϕ} is invertible and that {hr (ϕ/|ϕ|)} is not invertible. One can show that (ϕ/|ϕ|)(u−1 (ζ)) → 0 as Im ζ → ∞ (since, for Im ζ sufficiently large, ωA (ζ) ≈ ωB (ζ) ≈ ωC (ζ) ≈ π/3) and so it follows that even {hr (ϕ/|ϕ|)}π is not invertible (Theorem 3.76(b)). 4.78. Proposition. There exist ϕ = aχA + bχB + cχC ∈ P3 C such that both (a, c) and (b, c) entirely belong to the complement of sp T (ϕ).


225

Proof. Let a, b, c be any complex numbers such that ∆(a, b, c) is not a line segment and let (the images under the mapping u of) the sets A, B, C be located on R as follows:

that is, C = (−∞, 0), A =

An , B =

n∈Z

Bn , Bn = (22n , 22n+1 ), An =

n∈Z

(22n+1 , 22n+2 ). By virtue of Proposition 4.75, in order to conclude that (a, c)∩ sp T (ϕ) = ∅, it suffices to show that µ := (a + c)/2 is not in sp T (ϕ). Assume the contrary, i.e., assume µ ∈ sp T (ϕ). Then, due to 2.37, there are zn ∈ D such that ϕ(z n ) → µ as n → ∞. Let a =: µ + δ, c =: µ − δ, b =: µ + γ. Thus, γ 2 ϕ(z n ) − µ = δ χ2 2 A (zn ) − χ C (zn ) + χ B (zn ) , δ and since χ2 2 2 A, χ B, χ C are real-valued and Im (γ/δ) = 0, we deduce that χ2 B (zn ) → 0,

χ2 2 A (zn ) − χ C (zn ) → 0 (n → ∞).

This and (4.45) show that there is a ζ0 ∈ Π such that ωB (ζ0 ) < ε

(4.47)

and −ε < ωA (ζ0 ) − ωC (ζ0 ) < ε, where tan(2ε) = 1/4 and thus 7◦ < ε < 8◦ . Since ωA + ωB + ωC = 180◦ , we have 90◦ − 2ε < ωC (ζ0 ) < 90◦ + 2ε.

(4.48)

From (4.48) we see that ζ0 lies in the angular sector S := {ζ ∈ Π : | arg ζ − 90◦ | < 2ε}. For n ∈ Z, let Tn denote the trapezium Tn := {ζ ∈ S : 22n ≤ Im ζ < 22n+2 }. There is obviously an n ∈ Z such that ζ0 ∈ Tn . We claim that ωBn (ζ0 ) > ε, which is a contradiction to (4.47). Let ζ1 and ζ2 denote the left upper and left lower vertex of the trapezium Tn , respectively. Thus (recall that tan(2ε) = 1/4) ζ1 = 22n + i22n+2 ,

ζ2 = 22n−2 + i22n .

It is clear that ωBn (ζ0 ) > min{ωBn (ζ1 ), ωBn (ζ2 )}. But ωBn (ζ1 ) = α1 − α2 , where tan α1 = 3/4 and tan α2 = 2/4, whence ωBn (ζ1 ) > 9◦ ; also ωBn (ζ2 ) = β1 − β2 , where tan β1 = 9/4 and tan β2 = 5/4, whence ωBn (ζ2 ) > 14◦ . This proves our claim. It can be shown analogously that (b, c) ∩ sp T (ϕ) = ∅, which completes the proof.

226


Remark. There are even symbols ϕ = aχA + bχB + cχC ∈ P3 C such that the vertices a, b, c are the only points on the boundary of the triangle ∆(a, b, c) which belong to sp T (ϕ). Thus, in that case sp T (ϕ) \ {a, b, c} (and hence sp T π (ϕ) \ {a, b, c}) is a certain connected set entirely lying in the interior of the triangle ∆(a, b, c). This happens, for instance, if the sets A, B, C are located on R in a Cantor-set type position as follows:

The proof is of the same kind as that given for the above proposition, although the technical details are more complicated.

4.7 Toeplitz Algebras 4.79. Theorem. Let B be a C ∗ -subalgebra of L∞ satisfying C ⊂ B ⊂ QC. Then the mappings SmbT π , SmbKπ , SmbT KπJ given by (3.34) are isometric π (BN ×N ), restar-isomorphisms of alg T π (BN ×N ), alg Kπ (BN ×N ), alg T KJ spectively, onto BN ×N : π alg T π (BN ×N ) ∼ (BN ×N ) ∼ = alg Kπ (BN ×N ) ∼ = alg T KJ = BN ×N .

Proof. Propositions 4.5, 3.51(a), and 4.21 show that 2 QT (BN ×N ) = C∞ (HN ), QK (BN ×N ) = NNB×N , QT K (BN ×N ) = JNB×N , (4.49)

respectively. Let i stand for T , K, or T K. We know that in each case Ker i = {0}. So Corollary 3.44 combined with Theorem 3.52 implies that σi = Smbiπ is a star-isomorphism of alg iπ (BN ×N ) onto BN ×N . Finally, from 1.26(e) we deduce that Smbiπ is an isometry. Remark. Thus, for A = BN ×N and B as in the above theorem, the mappings Φπ and Ψ π introduced in 4.19 are isometric star-isomorphisms. Note π (A) is isometrically star-isomorphic to alg T (A). This follows that alg T KM from the fact that the kernel of the mapping Φ defined in 4.18 coincides with MB N ×N , which, on its hand, follows from combining the representation (4.11), Proposition 4.21(a) and Proposition 4.4(d). The following proposition shows in what a sense the preceding theorem is best possible. 4.80. Proposition. Let B be a C ∗ -subalgebra of L∞ . (a) If QT (B) ⊂ C∞ (H 2 ), or QK (B) ⊂ N , or QT K (B) ⊂ J , then necessarily B ⊂ QC. (b) If there exists a bijective linear mapping of B onto alg T π (B), or onto π (B), then necessarily B ⊂ QC. alg Kπ (B), or onto alg T KJ

4.7 Toeplitz Algebras

227

Proof. (a) We first show that B ⊂ QC whenever QT (B) ⊂ C∞ (H 2 ). Thus, suppose QT (B) ⊂ C∞ (H 2 ). If a ∈ B, then a ∈ B, and therefore, by (2.18), H(a)H() a) = T (aa) − T (a)T (a) ∈ C∞ (H 2 ).

If a(t) = n∈Z an tn , then a(t) = n∈Z an t−n (t ∈ T), and hence H() a) = H(a)∗ . But if H(a)H(a)∗ ∈ C∞ (H 2 ), then (since L(H 2 )/C∞ (H 2 ) is a C ∗ algebra) H(a)2ess = H(a)H(a)∗ ess = 0, whence H(a) ∈ C∞ (H 2 ). Theorem 2.54 now gives that a ∈ C + H ∞ . The preceding argument applied to a in place of a shows that a ∈ C + H ∞ . Thus, a ∈ QC. That B ⊂ QC if QK (B) ⊂ N is an immediate consequence of Proposition 3.51(c). Finally, if T (kλ a · kλ a) − T (kλ a)T (kλ a) = K + Cλ with K ∈ C∞ (H 2 ) and Cλ → 0 as λ → 0, then passage to the strong limit λ → ∞ gives T (aa) − T (a)T (a) = K, which, by what has been proved above, implies that a ∈ QC. Hence B ⊂ QC if QT K (B) ⊂ J . (b) Let i be T , K, or T K. Corollary 3.44 together with the fact that Ker i = {0} tells us that B is algebraically isomorphic to alg i(B)/Qi (B). Hence, from the hypothesis we deduce that alg i(B)/Qi (B) is isomorphic as a linear space to alg i(B)/J, where J is one of the ideals C∞ (H 2 ), N B , J B . In particular, the zero elements of both spaces must be equal. It follows that Qi (B) = J and part (a) completes the proof. 4.81. Theorem. Let B = C + H ∞ and let the approximate identity be generated by the Poisson kernel. Then the mappings SmbT π , SmbHπ , and SmbT HπJ are isometric isomorphisms of alg T π (BN ×N ), alg Hπ (BN ×N ), and π (BN ×N ), respectively, onto BN ×N . alg T HJ Proof. The three equalities in (4.49) hold with B = C + H ∞ and K = H by virtue of Proposition 4.5, the remark in 3.51, and Proposition 4.21(b), respectively. So we conclude as in the proof of Theorem 4.79 that the corresponding algebras are homeomorphically isomorphic. That the corresponding isomorphisms are actually isometries can now be deduced from 4.10, Proposition 3.54, and 4.19(c), respectively. We finally show how Theorem 4.79 can be “localized.” 4.82. Theorem. Let B be a C ∗ -subalgebra of L∞ satisfying C ⊂ B ⊂ QC, π . Then let F be a closed subset of M (B), and let iπ stand for T π , Kπ , or T KJ π SmbiπF is an isometric star-isomorphism of alg iF (BN ×N ) onto (B|F )N ×N . Proof. Corollary 3.44 in conjunction with 1.26(e) shows that σiπF is an isometric star-isomorphism of alg iπF (BN ×N )/QiπF (BN ×N ) onto BN ×N /Ker iπF . From Theorem 4.79 we deduce that Qiπ (BN ×N ) = {0}, and therefore QiπF (BN ×N ) is also the zero ideal. We saw in 3.60 that Ker iπF equals IF := {b ∈ BN ×N :

228


b|F = 0} and when proving Theorem 3.61 we established the isometric starisomorphism BN ×N /IF ∼ = (B|F )N ×N . Finally, since QiπF (BN ×N ) = {0}, we may identify σiπF and SmbiπF . Our next concern are the algebras generated by P C or P QC symbols. 4.83. Proposition. The algebras alg T π (P QC),

alg Kπ (P QC),

π alg T KJ (P QC)

are commutative. π Proof. The commutativity of alg T KJ (P QC) will follow once we have shown that

{T (kλ a)T (kλ b) − T (kλ b)T (kλ a)} ∈ J

∀ a, b ∈ P QC0 .

(4.50)

In view of Lemma 4.25 it suffices to verify (4.50) for a, b ∈ P C0 . Moreover, there is no loss of generality in assuming that a and b have at most one discontinuity, say at the point τ ∈ T. Let χ ∈ P C0 denote any function which is continuous on T \ {τ } and satisfies χ(τ − 0) = 0, χ(τ + 0) = 1. Then there are α, β ∈ C and f, g ∈ C such that a = αχ + f , b = βχ + g. Hence, T (kλ a)T (kλ b) − T (kλ b)T (kλ a) = α[T (kλ χ)T (kλ g) − T (kλ g)T (kλ χ)] +β[T (kλ f )T (kλ χ) − T (kλ χ)T (kλ f )] +[T (kλ f )T (kλ g) − T (kλ g)T (kλ f )] and so Lemma 4.25 implies (4.50). The commutativity of alg T π (P QC) can be shown in the same way or π (P QC) by using the fact that Φπ is an can be derived from that of alg T KJ algebraic homomorphism. Finally, there is nothing to prove for alg Kπ (P QC). π , and let B be either C 4.84. Preliminaries. Let iπ stand for T π , Kπ , or T KJ π or QC. The maximal ideal space of alg i (P B) (whose commutativity results from the preceding proposition) will be denoted by NP B ; the possible dependence of NP B on iπ is suppressed in this notation. By the Gelfand-Naimark theorem 1.26(a), alg iπ (P B) is isometrically star-isomorphic to C(NP B ). Because alg iπ (B) is a C ∗ -subalgebra of alg iπ (P B) that is isometrically starisomorphic to B ∼ = C(M (B)) (Theorem 4.79), we regard B as a C ∗ -subalgebra π B = Mξ (alg iπ (P B)) for of alg i (P B). So the definition of the fibers NP ξ ξ ∈ M (B) makes acorrect sense. By 1.27(b), these fibers are nonempty and B = NP we have NP B = ξ . For ξ ∈ M (B), put ξ∈M (B)

Jξ := closidalg iπ (P B) iπ (f ) : f ∈ B, f (ξ) = 0 .

(4.51)

So alg iπ (P B)/Jξ coincides with the local algebra alg iπξ (P B) = alg iπXξ (P B) as it was defined in 3.60 and 3.66.


229

Throughout what follows suppose the index set Λ of the approximate identity {Kλ }λ∈Λ is connected. Also let χτ always denote the characteristic function of the arc (τ, τ eiπ/2 ). 4.85. Proposition. The local algebras alg iπξ (P C) and alg iπξ (P QC) are singly generated. For τ ∈ T, alg iπτ (P C) is generated by iπτ (χτ ), and sp iπτ (χτ ) = [0, 1]. The Gelfand map Γτ : alg iπτ (P C) → C[0, 1] is given by π Γτ iτ (a) (µ) = (1 − µ)a(τ − 0) + µa(τ + 0) for a ∈ P C. If ξ ∈ Mτ (QC) \ Mτ0 (QC), then alg iπξ (P QC) is isometrically isomorphic to the complex field C and for a ∈ P QC the isomorphism Γξ is given by Γξ iπξ (a) = a(ξ, 0) and Γξ iπξ (a) = a(ξ, 1) for ξ ∈ Mτ− (QC) and ξ ∈ Mτ+ (QC), respectively. If ξ ∈ Mτ0 (QC), then alg iπξ (P QC) is generated by iπξ (χτ ), and sp iπξ (χτ ) = [0, 1]. In this case the Gelfand map Γξ : alg iπξ (P QC) → C[0, 1] is for a ∈ P QC given by π Γξ iξ (a) (µ) = (1 − µ)a(ξ, 0) + µa(ξ, 1). Proof. Every a ∈ P C is of the form a = αχτ + g, where α ∈ C and g ∈ P C is continuous at τ . Hence, iπτ (a) = αiπτ (χτ ) + g(τ ), and it follows that alg iπτ (P C) is generated by iπτ (χτ ). That the spectrum of iπτ (χτ ) is [0, 1] follows from Theorem 4.67 for iπ = T π , from Corollary 3.78 for iπ = Kπ , and from Theorem 4.24 π . Thus, by 1.19, M (alg iπτ (P C)) can be identified with [0, 1]. For for iπ = T KJ µ ∈ [0, 1], denote the multiplicative linear functional on alg iπτ (P C) which sends iπτ (χτ ) into µ also by µ. Then π Γτ iτ (a) (µ) = αµ iπτ (χτ ) + g(τ ) = αµ + g(τ ) = (1 − µ)a(τ − 0) + µa(τ + 0), as desired. The algebra alg iπξ (P QC) is generated by the elements iπξ (a), where a

n ranges through P QC0 . Every such a is of the form a = αχτ q + i=1 pi qi , where α ∈ C, q ∈ QC, pi ∈ P C0 is continuous at τ , and qi ∈ QC. Hence, iπξ (a) = αq(ξ)iπξ (χτ ) + i pi (τ )qi (ξ), and what results is that alg iπξ (P QC) is generated by iπξ (χτ ). If ξ ∈ Mτ− (QC) \ Mτ0 (QC), then χτ (Xξ ) = {0} by Theorem 3.36(a), (c), and thus sp iπξ (χτ ) = {0} by Theorem 3.61. So 1.19 specializes to give that alg iπξ (P QC) is isometrically isomorphic to C = C({0}) and that, for a as above, the isomorphism is given by π Γξ iξ (a) (0) = α · 0 · q(ξ) + pi (τ )qi (ξ) pi (ξ, 0)qi (ξ, 0) = a(ξ, 0). = αχτ (ξ, 0)q(ξ, 0) + A simple continuity argument now shows that (Γξ iπξ (a))(0) equals a(ξ, 0) for all a ∈ P QC. The situation is the same for ξ ∈ Mτ+ (QC) \ Mτ0 (QC).


230

Now suppose ξ ∈ Mτ0 (QC). Then, by Theorem 3.36(a), (d), χτ (Xξ ) is the doubleton {0, 1}. So sp iπξ (χτ ) = [0, 1]: this is Theorem 4.67 for iπ = T π , this π then Theorem 4.24 follows from Theorem 3.79 for iπ = Kπ , and for iπ = T KJ applies. If µ ∈ [0, 1], we let again µ denote the multiplicative linear functional on alg iπξ (P QC) which assumes the value µ at iπξ (χτ ). Thus, for a as above,

pi (τ )qi (ξ) Γξ iπξ (a) (µ) = αµ iπξ (χτ ) q(ξ) + = αµq(ξ) + pi (τ )qi (ξ) = (1 − µ)a(ξ, 0) + µa(ξ, 1),

because a(ξ, 0) = pi (τ )qi (ξ) and a(ξ, 1) = αq(ξ)+ pi (τ )qi (ξ). Since P QC0 is dense in P QC, it follows that (Γξ iπξ (a))(µ) equals (1 − µ)a(ξ, 0) + µa(ξ, 1) for all a ∈ P QC. 4.86. Theorem. The maximal ideal space NP C of alg iπ (P C) is the cylinder T × [0, 1] and the Gelfand map Γ : alg iπ (P C) → C(NP C ) is for a ∈ P C given by π Γ i (a) (τ, µ) = (1 − µ)a(τ − 0) + µa(τ + 0), (τ, µ) ∈ T × [0, 1]. Proof. Let hτ denote the canonical homomorphism of the algebra alg iπ (P C) onto the algebra alg iπτ (P C) = alg iπ (P C)/Jτ . If (τ, µ) ∈ T × [0, 1], then due to the preceding proposition the mapping given for a ∈ P C by iπτ (a) → (1 − µ)a(τ − 0) + µa(τ + 0) extends to a multiplicative linear functional vτ,µ on alg iπτ (P C), and thus vτ,µ ◦ hτ is in NP C . Therefore, if we identify (τ, µ) with vτ,µ ◦ hτ , then π Γ i (a) (τ, µ) = (vτ,µ ◦ hτ )(iπ (a)) = vτ,µ (iπτ (a)) = (1 − µ)a(τ − 0) + µa(τ + 0). C Now suppose v ∈ NP C . Then there is a τ ∈ T such that v ∈ NP (recall τ 4.84). From (4.51) it is obvious that v(Jτ ) = {0}. Hence, the mapping

u : alg iπ (P C)/Jτ → C,

hτ c → v(c)

is well defined (i.e., v(c1 ) = v(c2 ) whenever hτ c1 = hτ c2 ) and is a multiplicative linear functional. Since alg iπ (P C)/Jτ = alg iπτ (P C), it follows from the preceding proposition that there is a µ ∈ [0, 1] such that v(iπ (a)) = u(iπτ (a)) = (1 − µ)a(τ − 0) + µa(τ + 0) ∀ a ∈ P C.

4.87. Theorem. The maximal ideal space NP QC of alg iπ (P QC) can be identified with a (proper) subset of M (QC) × [0, 1]: 0 0 NP QC = M− (QC) × {0} ∪ M 0 (QC) × [0, 1] ∪ (M+ (QC) × {1} , where


/ 0 Mτ± (QC) \ Mτ0 (QC) , M± (QC) :=

M 0 (QC) :=

τ ∈T

/

231

Mτ0 (QC).

τ ∈T

The Gelfand map Γ : alg iπ (P QC) → C(NP QC ) follows: π Γ i (a) (ξ, 0) = a(ξ, 0) π Γ i (a) (ξ, 1) = a(ξ, 1) π Γ i (a) (ξ, µ) = (1 − µ)a(ξ, 0) + µa(ξ, 1)

is given for a ∈ P QC as for

0 ξ ∈ M− (QC),

for

0 ξ ∈ M+ (QC),

for

ξ ∈ M 0 (QC).

Proof. We denote the canonical homomorphism of the algebra alg iπ (P QC) onto the algebra alg iπξ (P QC) = alg iπ (P QC)/Jξ by hξ . Let ξ ∈ Mτ− (QC) \ Mτ0 (QC). Then, by Proposition 4.85, the mapping given for a ∈ P QC by iπξ (a) → a(ξ, 0) extends to an isometric isomorphism Γξ of alg iπξ (P QC) onto C. It follows that Γξ ◦hξ is in NP QC , and if we identify Γξ ◦ hξ with (ξ, 0), then π Γ i (a) (ξ, 0) = (Γξ ◦ hξ )(iπ (a)) = Γξ iπξ (a) = a(ξ, 0) for a ∈ P QC. We have an analogous situation for ξ ∈ Mτ+ (QC) \ Mτ0 (QC). Now let ξ ∈ Mτ0 (QC). Then, again by Proposition 4.85, the mapping defined for a ∈ P QC by iπξ (a) → (1 − µ)a(ξ, 0) + µa(ξ, 1) extends to a multiplicative linear functional vξ,µ on alg iπξ (P QC). So vξ,µ ◦ hξ is in NP QC , and on identifying vξ,µ ◦ hξ with (ξ, µ) we have π Γ i (a) (ξ, µ) = (vξ,µ ◦ hξ )(iπ (a)) = vξ,µ iπξ (a) = (1 − µ)a(ξ, 0) + µa(ξ, 1). QC Finally, let v ∈ NP , where ξ ∈ M (QC). Then v(Jξ ) = {0} by virtue of ξ (4.51). This implies that the mapping

u : alg iπ (P QC)/Jξ → C,

hξ c → v(c)

is well defined and is a multiplicative linear functional. Taking into account that alg iπ (P QC)/Jξ = alg iπξ (P QC) and applying Proposition 4.85 we con0 (QC) or v(iπ (a)) = a(ξ, 1) clude that either v(iπ (a)) = a(ξ, 0) with ξ ∈ M− 0 π with ξ ∈ M+ (QC) or v(i (a)) = (1 − µ)a(ξ, 0) + µa(ξ, 1) with ξ ∈ M 0 (QC) and µ ∈ [0, 1] for all a ∈ P QC. Remark. The two preceding theorems show that NP B = M (alg iπ (P B)) does not depend on iπ . In particular, the algebras alg T π (P B), alg Kπ (P B), π (P B) are isometrically star-isomorphic to each other. alg T KJ

232


4.88. The Gelfand topologies on NP C and NP QC . The Gelfand topology on NP B is the coarsest (weakest) topology that makes Γ (iπ (a)) continuous for every a ∈ P B0 . It is therefore standard routine to check that this topology can be described as follows. The space NP C . An open neighborhood base of (τ, µ), where τ ∈ T and 0 < µ < 1, is formed by the sets {τ } × (µ − ε, µ + ε),

0 < ε < min{µ, 1 − µ}.

The sets [τ, τ eiε ) × (1 − ε, 1] ∪ (τ, τ eiε ) × [0, 1 − ε] ,

0 < ε < 1,

form an open neighborhood base of (τ, 1), and the sets −iε (τ e , τ ] × [0, ε) ∪ (τ e−iε , τ ) × [ε, 1] , 0 < ε < 1, form an open neighborhood base of (τ, 0). The space NP QC . First recall Remark 3 of 3.36. For ξ ∈ Mτ± (QC)\Mτ0 (QC), let V ± (ξ) be the family of all sets V ∈ V(ξ) satisfying V = Vτ ∪ Vτ± . Then, for ξ ∈ Mτ+ (QC) \ Mτ0 (QC), the sets 0 1 (Vτ × {1}) ∪ (Vτ+ × [0, 1]) ∩ NP QC , V ∈ V + (ξ), form an open neighborhood base of (ξ, 1). For ξ ∈ Mτ− (QC) \ Mτ0 (QC), the sets 1 0 (Vτ × {0}) ∪ (Vτ− × [0, 1]) ∩ NP QC , V ∈ V − (ξ), form an open neighborhood base of (ξ, 0). The sets {ξ} × (µ − ε, µ + ε),

0 < ε < min{µ, 1 − µ},

form an open neighborhood base of the point (ξ, µ) ∈ Mτ0 (QC) × (0, 1). For ξ ∈ Mτ0 (QC), the sets 0 1 (Vτ × (1 − ε, 1]) ∪ (Vτ+ × [0, 1 − ε]) ∩ NP QC , V ∈ V(ξ), 0 < ε < 1, form an open neighborhood base of (ξ, 1), and the sets 0 1 (Vτ × [0, ε)) ∪ (Vτ− × [ε, 1]) ∩ NP QC , V ∈ V(ξ),

0 < ε < 1,

form an open neighborhood base of (ξ, 0). 4.89. Corollary. Let B = C or B = QC. Then the mappings Φπ and Ψ π are π (P BN ×N ) onto alg T π (P BN ×N ) and isometric star-isomorphisms of alg T KJ alg Kπ (P BN ×N ), respectively. Each of these algebras is via ΓN ×N , Γ being the Gelfand map, isometrically star-isomorphic to [C(NP B )]N ×N . Thus, π alg T π (P BN ×N ) ∼ (P BN ×N ) ∼ = alg Kπ (P BN ×N ) ∼ = alg T KJ = [C(NP B )]N ×N .


Proof. Immediate from the remark in 4.87.

233

We now describe the structure of Toeplitz algebras generated by C or QC and a characteristic function. 4.90. Definition. Let E be a measurable subset of T and let χE denote the characteristic function of E. Define CE := algL∞ (χE , C),

QCE := algL∞ (χE , QC).

Note that both CE and QCE are C ∗ -algebras. 4.91. Lemma. Let B = C or B = QC. Then BE = f χE + g : f ∈ B, g ∈ B . Proof. Clearly, it suffices to show that f χE + g : f ∈ B, g ∈ B = hχE + gχE c : h ∈ B, g ∈ B is closed (here E c := T \ E). The mapping B → L∞ , b → bχE is an algebraic star-isomorphism, and therefore its image, the set χE B := {bχE : b ∈ B}, is closed by 1.26(e). It follows analogously that χE c B is closed. Now let a ∈ L∞ and suppose there are hn , gn such that a − hn χE − gn χE c ∞ → 0 as

n → ∞.

Then aχE − hn χE ∞ → 0 as n → ∞, and since χE B is closed, there is an h ∈ B with aχE = hχE . It can be shown similarly that aχE c = gχE c with some g ∈ B, which completes the proof. π } and B ∈ {C, QC}. Then alg iπ (BE ) 4.92. Lemma. Let iπ ∈ {T π , Kπ , T KJ is commutative.

Proof. In view of the preceding lemma it is enough to verify that iπ (a)iπ (χE ) = iπ (aχE ) for every a ∈ B. But this follows from Proposition 2.14 and Theorem 2.54 for iπ = T π , from Theorem 3.23 for iπ = Kπ , and from Lemma 4.25 π . for iπ = T KJ π } and B ∈ {C, QC}. Then, for 4.93. Proposition Let iπ ∈ {T π , Kπ , T KJ π ξ ∈ M (B), the local algebras alg iξ (BE ) are singly generated by iπξ (χE ). For τ ∈ T, we have sp iπτ (χE ) = conv χE (Xτ ), i.e.,

sp iπτ (χE ) = {1}

if

χE (Xτ ) = {1},

= {0}

if

χE (Xτ ) = {0},

if

χE (Xτ ) = {0, 1}.

sp iπτ (χE )

sp iπτ (χE ) = [0, 1] For ξ ∈ M (QC), we have

sp Tξπ (χE ) = sp T Kξπ (χE ) = conv χE (Xξ ). Remark. We have not been able to prove that sp Kξπ (χE ) = conv χE (Xξ ) for ξ ∈ M (QC). In this connection recall 4.68.

234


Proof. Lemma 4.91 implies that alg iπξ (BE ) is generated by iπξ (χE ). The identification of sp iπτ (χE ) as conv χE (Xτ ) follows from Theorem 4.67 for iπ = T π , from Corollary 3.78(a) for iπ = Kπ , and then from Theorem 4.24 for π iπ = T KJ . Finally, Theorem 4.67 and subsequent application of Theorem 4.24 give the last assertion of the present proposition. π . Then the 4.94. Theorem. Let B be C or QC and let iπ be T π , Kπ , or T KJ π maximal ideal space M (alg i (BE )) can be identified with the subset / / 0 1 {ξ} × M alg iπξ (BE ) ∼ {ξ} × sp iπξ (χE ) = ξ∈M (B)

ξ∈M (B)

of M (B) × C, and the Gelfand map / π Γ : alg i (BE ) → C

π {ξ} × M alg iξ (BE )

ξ∈M (B)

is for f χE + g ∈ BE (f, g ∈ B) given by π Γ i (f χE + g) (ξ, w) = f (ξ)w iπξ (χE ) + g(ξ). Proof. By Lemma 4.91, B can be identified with the subset {0·χE +g : g ∈ B} of BE , and, accordingly, on identifying B with alg iπ (B) we may regard B as a C ∗ -subalgebra of alg iπ (BE ). So, as in 4.84, alg iπξ (BE ) = alg iπ (BE )/Jξ with Jξ := closidalg iπ (BE ) iπ (f ) : f ∈ B, f (ξ) = 0 . Now the same reasoning as in the proof of Theorem 4.86 (or 4.87) completes the proof. 4.95. The algebra alg {T (P C), H(P C)}. Given a closed subalgebra A of L∞ let alg T H(A) denote the smallest closed subalgebra of L(H 2 ) containing all Toeplitz and all Hankel operators with symbol in A: alg T H(A) := alg {T (A), H(A)} := alg T (a), H(a) : a ∈ A . If C ⊂ A, then C∞ (H 2 ) ⊂ alg T H(A) (Proposition 4.5). Denote the quotient algebra alg T H(A)/C∞ (H 2 ) by alg T H π (A), and for B ∈ alg T H(A) let B π denote the coset B + C∞ (H 2 ). Since H(c) is compact for every c ∈ C, we have alg T H(C) = alg T (C),

alg T H π (C) = alg T π (C) ∼ = C.

The purpose of what follows is to analyze the C ∗ -algebra alg T H(P C). From Proposition 4.51 we know that alg T H(P C) is strictly larger that alg T (P C). c}, where ) c(t) := c(1/t) = c(t) (t ∈ T). It is easily Let Cs := {c ∈ C : c = ) seen that Cs is a C ∗ -subalgebra of C and that the maximal ideal space of Cs is homeomorphic to the closed upper half-circle with its usual topology,


235

M (Cs ) = T+ := {t ∈ T : Im t ≥ 0}. The Gelfand map Γ : Cs → C(T+ ) is of course given by (Γ c)(τ ) = c(τ ). If a ∈ L∞ and c ∈ Cs , then T π (a)T π (c) = T π (c)T π (a) and c) − T π (a)H π () c) = H π (ac) H π (a)T π (c) = H π (a) π π π = H (ca) = T (c)H (a) + H π (c)T π () a) = T π (c)H π (a) (Proposition 2.14). Consequently, if we identify a function c ∈ Cs with the coset (= essential Toeplitz operator) T π (c), then Cs may be viewed as a closed subalgebra of the center of alg T H π (P C). For τ ∈ T+ , let Jτπ denote the smallest closed two-sided ideal of the algebra alg T H π (P C) containing the set {T π (c) : c ∈ Cs , c(τ ) = 0}, put alg T Hτπ (P C) = alg T H π (P C)/Jτπ , and for B ∈ alg T H(P C) denote the coset B π + Jτπ by Bτπ . Theorem 1.35(a) implies that, for B ∈ alg T H(P C), / / π spess B = sp Bτπ = sp Bτπ ∪ sp B−1 ∪ sp B1π τ ∈T+

(4.52)

τ ∈T◦ +

where T◦+ := T+ \ {−1, 1}. Let a, b ∈ P C and τ ∈ T◦+ . It can be verified without difficulty that a|Xτ ∪ Xτ = b|Xτ ∪ Xτ =⇒ Tτπ (a) = Tτπ (b),

Hτπ (a) = Hτπ (b).

(4.53)

On the other hand, if a, b ∈ P C and τ ∈ {−1, 1}, a|Xτ = b|Xτ =⇒ Tτπ (a) = Tτπ (b),

Hτπ (a) = Hτπ (b).

(4.54)

Note that if τ ∈ T◦+ , then Xτ ∪ Xτ is the fiber of M (L∞ ) over τ ∈ M (Cs ): Xτ ∪ Xτ = x ∈ M (L∞ ) : f (x) = f (τ ) ∀ f ∈ Cs . 4.96. Lemma. Let τ ∈ T+ and let χE be the characteristic function of any arc E one endpoint of which is τ . Then sp Tτπ (χE ), the spectrum of Tτπ (χE ) in alg T Hτπ (P C), is equal to the segment [0, 1]. Proof. Let θ ∈ P C be any function which coincides with χE on some (sufficiently small) arcs Uτ τ and Uτ τ and is continuous and takes values in A := {z ∈ C : |z − 1/2| = 1/2, Im z ≥ 0} on T \ (Uτ ∪ Uτ ). We know that the spectrum of T π (θ) in alg T π (P C) is [0, 1] ∪ A and so, by 1.16(b) or 1.26(d), the spectrum of T π (θ) in alg T H π (P C) also equals [0, 1] ∪ A. Using (4.52)–(4.54) it is easily seen that sp Ttπ (θ) = {θ(t), θ(t)} for t ∈ T+ \ {τ }, and (4.53) immediately gives that sp Tτπ (θ) = sp Tτπ (χE ). Hence, by (4.52), {θ(t), θ(t)} ∪ sp Tτπ (χE ), which shows that sp Tτπ (χE ) = [0, 1]. [0, 1] ∪ A =

t =τ


236

4.97. Theorem. Let τ ∈ {−1, 1} and let χτ be the characteristic function of the arc (τ, τ eiπ/2 ). The algebra alg T Hτπ (P C) is singly generated by Tτπ (χτ ) and the spectrum of Tτπ (χτ ) is the segment [0, 1]. If a ∈ P C and µ ∈ [0, 1], then the Gelfand transform of Tτπ (a) and Hτπ (a) at µ is given by Γτ Tτπ (a) (µ) = a(τ + 0)µ + a(τ − 0)(1 − µ), 6 Γτ Hτπ (a) (µ) = −iτ [a(τ + 0) − a(τ − 0)] µ(1 − µ). Proof. Every function a ∈ P C can be written in the form a = λωτ + c, where λ = a(τ +0)−a(τ −0), c ∈ P C is continuous at τ and satisfies c(τ ) = a(τ −0), and ωτ is the characteristic function of the arc (τ, τ eiπ ), i.e., of the half-circle following the point τ . Consequently, by (4.54), Tτπ (a) = λTτπ (ωτ ) + c(τ ),

Hτπ (a) = λHτπ (ωτ )

(note that Hτπ (c) = c(τ )Hτπ (1) = 0). Proposition 2.14 gives that T (ωτ ) = T (ωτ2 ) = T (ωτ )T (ωτ ) + H(ωτ )H(1 − ωτ ) = T (ωτ )T (ωτ ) − H(ωτ )H(ωτ ), whence [H(ωτ )]2 = −T (ωτ )(I − T (ωτ )).

(4.55)

The n-th Fourier coefficient (ω1 )n of ω1 equals (ω1 )n = Hence, if f (t) =

1 2π

j≥0

(H(ω1 )f, f ) =

(

π

e−inθ dθ =

0

1 2πi

1

xn−1 dx −1

(n ≥ 1).

fj tj (t ∈ T) is in PA , then

(ω1 )j+k+1 fj fk =

j,k≥0

=

(

1 2πi

(

( 1 1 fj fk xj+k dx 2πi −1 j,k≥0

1

−1

f j xj

j≥0

fk xk dx.

k≥0

It follows that iH(ω1 ) is a positive operator and therefore we may deduce from (4.55) that 0 11/2 H(ω1 ) = −i T (ω1 ) I − T (ω1 ) . A similar argument yields the equality 0 11/2 . H(ω−1 ) = i T (ω−1 ) I − T (ω−1 ) Since Tτπ (ωτ ) = Tτπ (χτ ) (see (4.54)) and sp Tτπ (χτ ) = [0, 1] (Lemma 4.96), all assertions of the theorem now follow straightforwardly.


237

4.98. Definitions. (a) Suppose τ ∈ T◦+ . Let ψτ be the characteristic function of the arc (τ, τ ), i.e., of the arc {eiθ : θ0 < θ < 2π − θ0 }, where τ = eiθ0 . .τ . Furthermore, let ϕτ ∈ C be any function such that Note that ψτ2 = ψτ = ψ .τ = 1. Put 0 ≤ ϕτ ≤ 1, ϕτ (τ ) = 1, ϕτ (τ ) = 0, ϕτ + ϕ q := qτ := Tτπ (ψτ ) + Hτπ (ψτ ),

p := pτ := Tτπ (ϕτ )

and let e := eτ denote the identity element of alg T Hτπ (P C). (b) Let A be a C ∗ -algebra. Given a finite subset {a1 , . . . , ak } of A let C (a1 , . . . , ak ) denote the smallest C ∗ -subalgebra of A containing the set {a1 , . . . , ak }. An element a ∈ A is called selfadjoint if a = a∗ and is said to be an idempotent if a2 = a. ∗

4.99. Lemma. Let τ ∈ T◦+ . The elements p and q are selfadjoint idempotents and alg T Hτπ (P C) = C ∗ (p, q, e). Moreover, pqp = Tτπ (ψτ ϕτ ),

pq(e − p) = Hτπ (ψτ ϕτ ),

.τ ), (e − p)qp = Hτπ (ψτ ϕ

(e − p)q(e − p) = Tτπ (ψτ ϕ .τ ).

(4.56) (4.57)

The spectrum of pqp in alg T Hτπ (P C) is [0, 1]. Proof. It is clear that p = p∗ and q = q ∗ . Proposition 2.14 shows that T (ψτ ) + H(ψτ ) T (ψτ ) + H(ψτ ) = T (ψτ )T (ψτ ) + H(ψτ )H(ψτ ) + H(ψτ )T (ψτ ) + T (ψτ )H(ψτ ) = T (ψτ ) + H(ψτ ). Thus T (ψτ ) + H(ψτ ) is a projection and therefore q 2 = q. Since Tτπ (ϕτ )Tτπ (ϕτ ) = Tτπ (ϕ2τ ) (Proposition 2.14) and Tτπ (ϕ2τ ) = Tτπ (ϕτ ) (see (4.53)), we deduce that p2 = p. We now prove (4.56), (4.57). A few application of Proposition 2.14 gives T π (ϕτ )T π (ψτ )T π (ϕτ ) = T π (ϕ2τ ψτ ), .τ ) = H π (ϕτ ψτ ϕ .τ ), T π (ϕτ )H π (ψτ )T π (ϕτ ) = T π (ϕτ )H π (ψτ ϕ whence, by (4.53), .τ ) = Tτπ (ϕ2τ ψτ ) = Tτπ (ϕτ ψτ ). pqp = Tτπ (ϕ2τ ψτ ) + Hτπ (ϕτ ψτ ϕ The remaining three equalities can be proved analogously.


238

From (4.53) we obtain that Tτπ (ϕτ ψτ ) = Tτπ (χE ), where E is some subarc of the arc (τ, −1). So Lemma 4.96 implies that sp Tτπ (ϕτ ψτ ) = [0, 1], i.e., sp (pqp) = [0, 1]. It remains to show that alg T Hτπ (P C) = C ∗ (p, q, e). Let a ∈ P C and write a in the form λψτ + c, where λ = a(τ + 0) − a(τ − 0) and c ∈ P C is continuous at τ and satisfies c(τ ) = a(τ − 0). It follows that Tτπ (aϕτ ) = λTτπ (ψτ ϕτ ) + Tτπ (cϕτ ) = pqp + c(τ )p. Writing a = κψτ + d, where κ = a(τ − 0) − a(τ + 0) and d ∈ P C is continuous at τ and satisfies d(τ ) = a(τ + 0), we obtain that Tτπ a(1 − ϕτ ) = κTτπ ψτ (1 − ϕτ ) + Tτπ d(1 − ϕτ ) .τ ) + Tτπ d(1 − ϕτ ) = κTτπ (ψτ ϕ = κ(e − p)q(e − p) + d(τ )(e − p). Thus Tτπ (a) = [a(τ + 0) − a(τ − 0)]pqp + [a(τ − 0) − a(τ + 0)](e − p)q(e − p) +a(τ − 0)p + a(τ + 0)(e − p).

(4.58)

It can be shown similarly that Hτπ (a) = [a(τ + 0) − a(τ − 0)]pq(e − p) + [a(τ − 0) − a(τ + 0)](e − p)qp. (4.59) This completes the proof.

4.100. Two projections theorem. Let A be a C ∗ -algebra with identity e and let p, q ∈ A be selfadjoint idempotents such that the spectrum of pqp is [0, 1]. Let C2×2 [0, 1] denote the C ∗ -algebra of all continuous C2×2 -valued functions on [0, 1], let e be the identity element of C2×2 [0, 1], and define p, q ∈ C2×2 [0, 1] by 6 10 µ µ(1 − µ) 6 p(µ) = , q(µ) = (µ ∈ [0, 1]). 00 µ(1 − µ) 1 − µ Then C ∗ (p, q, e) is isometrically star-isomorphic to C ∗ (p, q, e) and the isomorphism takes p, q, e into p, q, e, respectively. Proof. See Halmos [264] (also recall 1.26(b)). 4.101. Theorem. If τ ∈

T◦+ ,

then the mapping

Γτ : Tτπ (a) + Hτπ (a) 6 a(τ + 0)µ + a(τ − 0)(1 − µ) [b(τ + 0) − b(τ − 0)] µ(1 − µ) → 6 [b(τ − 0) − b(τ + 0)] µ(1 − µ) a(τ − 0)(1 − µ) + a(τ + 0)µ


239

extends to an isometric star-isomorphism of alg T Hτπ (P C) onto C ∗ (p, q, e), where p, q, e are as in the previous theorem. Proof. Lemma 4.99 and Theorem 4.100 show that the algebras alg T Hτπ (P C) and C ∗ (p, q, e) are isometrically star-isomorphic. From (4.58) and (4.59) we deduce that the isomorphism takes Tτπ (a) and Hτπ (b) into

[a(τ + 0) − a(τ − 0)]µ + a(τ − 0)

0

0

[a(τ − 0) − a(τ + 0)](1 − µ) + a(τ + 0)

and

0

6 [b(τ − 0) − b(τ + 0)] µ(1 − µ) respectively.

6 [b(τ + 0) − b(τ − 0)] µ(1 − µ)

,

0

Theorems 4.97 and 4.101 along with (4.52) yield a Fredholm criterion for operators in alg T H(P C). We confine ourselves to stating a consequence of these theorems for the spectral theory of Hankel operators. If c ∈ C, then sp H(c) = spess H(c) = {0}. Indeed, since H(c) is compact, we have spess H(c) = {0}, and if λ = 0, then −1/λ − (1/λ2 )H(c) is the inverse of H(c)−λI (note that (I −P cQ)−1 = I +P cQ). The following result describes the essential spectrum of Hankel operators with P C symbols. 4.102. Corollary (Power). For b ∈ P C and τ ∈ T, put bτ :=

1 [b(τ + 0) − b(τ − 0)]. 2

Then H(b)ess = max |bτ |, τ ∈T

spess H(b) = [0, ib−1 ] ∪ [0, −ib1 ] ∪

(4.60) /

6 6 − i bτ bτ , i bτ bτ .

(4.61)

τ ∈T◦ +

Proof. Notice that λI − H(b) = λT (1) − H(b), and combine Theorems 4.97 and 4.101 with Theorem 1.35(d) to get (4.60) and with Theorem 1.35(a) (or (4.52)) to get (4.61).

240


4.8 The Role of the Harmonic Extension We conclude this chapter by discussing some questions on the connection between Fredholmness, local sectoriality, harmonic approximation, and stable convergence. 4.103. Special symbol classes. Let {Kλ }λ∈Λ be any approximate identity whose index set Λ is connected. We have proved that for a ∈ P2 C or a ∈ P QC the following implications hold.

Here B = C for a ∈ P2 C and B = QC for a ∈ P QC; “iπ (a) is invertible” means invertibility in alg iπ (L∞ ). The function a = 1/2 − χE , where E is any subarc of T, is in GL∞ but neither {kλ a}π nor T π (a) nor {T (kλ a)}πJ are invertible. In case {Kλ } is generated by the Poisson kernel we write {hr a} in place of {kλ a}. If a ∈ C + H ∞ , then the following implications are valid:

It would be interesting to know a function in G(C + H ∞ ) which is not locally sectorial over C + H ∞ .

4.8 The Role of the Harmonic Extension

241

4.104. L∞ symbols. For a ∈ L∞ , we established the following implications.

We have not been able to show that the implication (*) cannot be reversed. However, it turns out that none of the remaining implications can be reversed. To see this it suffices to show that there exist b, c ∈ L∞ such that T π (b) is invertible but {hr b}π is not invertible , {hr c}π is invertible but T π (c) is not invertible.

(4.62) (4.63)

Douglas’ symbol. This is a symbol b satisfying (4.62). It can be constructed as follows. Define b0 ∈ C by b0 (eiθ ) = e2iθ

(0 < θ < π),

b0 (eiθ ) = e−2iθ

(π < θ < 2π).

(4.64)

So T (b0 ) is invertible and b0 (0), the harmonic extension of b0 at the point 0 (= zeroth Fourier coefficient), is zero. Let ω be an infinite Blaschke product and put b := b0 ◦ ω. Then T (b) is invertible by Proposition 2.20(c). However, (b0 ◦ ω)(z) = b0 (ω(z)) is zero whenever ω(z) is zero, that is, 0 ∈ ClH (b, T). Hence, by Theorem 3.76(b), {hr b}π is not invertible. Wolff ’s symbol. This is a symbol c which satisfies (4.63).

Its construction ∞ is as follows. Let M ≥ 4 be an integer. For x ∈ (0, 1) let x = j=1 εj (x)M −j be the base M expansion of x, put p(x) := min{j : εj (x) = 0 or M − 1}, and define g by 2 ++ θ ++ g(eiθ ) = log 2 − + p + + − 1 log(1 − 2M ), θ ∈ (−π, π). M π *1

∞ Since 0 p(x) dx = n=1 2n(M − 2)n−1 M −n < ∞, we have g ∈ L1 . Then let c = ei)g , where g) refers to the conjugate function of g. Wolff [581] showed that for M sufficiently large, c satisfies (4.63). The proof is rather complicated and therefore omitted here. Neither the Douglas symbol nor the Wolff symbol is locally sectorial over C+H ∞ , since otherwise {hr b}π or T π (c) were invertible. It is easy to construct

242


symbols which are locally sectorial over QC but not over C: if we let ω be as in 2.80, then a = e2πi Im ω is in GQC and thus locally sectorial over QC, but there is a τ ∈ T such that a(Xτ ) = T (Proposition 2.79(a)), so that a is not locally sectorial over C. Thus, we have a symbol a for which {kλ a}π , T π (a), {T (kλ a)}πJ are invertible but which is not locally sectorial over C. The function ϕ constructed in 4.77 has the following property: {hr ϕ} is invertible, but ϕ is not locally sectorial over QC. Indeed, if ϕ were locally sectorial over QC, then so also were ϕ/|ϕ|, and this would imply that {hr (ϕ/|ϕ|)}π is invertible, a contradiction to (4.46). We finally show that symbols whose harmonic extensions is bounded sufficiently far away from zero in D (resp. near T) generate invertible (resp. Fredholm) Toeplitz operators. 4.105. Lemma. If a ∈ L∞ , then distL∞ (a, H ∞ ) ≤ Ba∗ ,

distL∞ (a, H ∞ ) ≤ Ba∗

with some absolute constant B; here a∗ refers to the BMO “norm” introduced in 1.47. Proof. We have distL∞ (a, H ∞ ) = inf ϕ∞ : ϕ ∈ L∞ , ϕ − a ∈ H ∞ ≤ inf ϕ∞ : ϕ ∈ L∞ , P ϕ = P a (because P ϕ = P a =⇒ ϕ − a ∈ H ∞ ) = inf u + v∞ : u, v ∈ L∞ , P (u + v) = P a ≤ inf u∞ + v∞ : u, v ∈ L∞ , P (u + v) = P a ≤ inf u∞ + v∞ : u ∈ H ∞ , v ∈ L∞ , u + P v = P a ≤ inf u∞ + v∞ : u ∈ L∞ , v ∈ L∞ , u + P v = P a , the last inequality resulting from the observation that if u is a function in L∞ and u + P v = P a, then u is in H ∞ . Hence, by 1.48(l) and (k), distL∞ (a, H ∞ ) ≤ β1 P aBM O ≤ β1 β2 aBM O with some absolute constants β1 , β2 . Thus, distL∞ (a, H ∞ ) = distL∞ (a − a0 , H ∞ ) ≤ β1 β2 a − a0 BM O = β1 β2 a − a0 ∗ = β1 β2 a∗ . The proof for distL∞ (a, H ∞ ) is analogous.

4.106. Lemma. Let {Kλ }λ∈Λ be any approximate identity. Then there exist constants D1 and D2 depending only on K and Λ such that for all unimodular a ∈ L∞ ,

4.8 The Role of the Harmonic Extension

243

1/2 a∗ ≤ D1 sup sup 1 − |kλ,t a|2 , λ∈Λ t∈T

1/2 M0 (a) := lim Mδ (a) ≤ D2 lim sup sup 1 − |kµ,t a|2 . δ→0

λ→∞ µ>λ t∈T

Proof. In the proof of Proposition 3.51(c) we established the inequality 1/2 kλ,t (|a − kλ,t |) ≤ kλ,t (|a|2 ) − |kλ,t a|2 . Thus, if |a| = 1 a.e., then Theorem 3.21 implies that 1/2 a∗ ≤ D1 aK ≤ D1 sup sup 1 − |kλ,t a|2 . λ∈Λ t∈T

Moreover, when proving Proposition 3.51(c) we also observed that 1/2 M2π/λ (a) ≤ D2 sup sup 1 − |kµ,t a|2 , µ>λ t∈T

where D2 depends only on K and Λ. This gives the second inequality of the lemma at once. 4.107. Theorem. Let {Kλ }λ∈Λ be any approximate identity. There are constants δG = δG (K, Λ) and δΦ = δΦ (K, Λ) depending only on K and Λ which have the following property: (a) If a ∈ L∞ is unimodular and |kλ,t a| ≥ δG for all λ ∈ Λ and all t ∈ T, then T (a) ∈ GL(H 2 ). (b) If a ∈ L∞ is unimodular and |kλ,t a| ≥ δΦ for sufficiently large λ ∈ Λ and all t ∈ T, then T (a) ∈ Φ(H 2 ). Proof. (a) Let B as in Lemma 4.105 and let D1 be the constant appearing 2 1/2 ) < 1. Lemmas 4.105 and in Lemma 4.106. Choose δG so that BD1 (1 − δG 4.106 then give that 1/2 distL∞ (a, H ∞ ) ≤ BD1 sup sup 1 − |kλ,t a|2 λ∈Λ t∈T

and so T (a) is right-invertible by Proposition 2.20(a). The same argument applies to show that T (a) is left-invertible. (b) Let A2 be the constant occurring in 1.48(f) and let B, D2 be as in Lemmas 4.105 and 4.106. Choose δΦ so that A2 BD2 (1 − δΦ2 )1/2 < 1. Since H(ϕ) is compact for ϕ ∈ QC = L∞ ∩ V M O, we have distL∞ (a, C + H ∞ ) = H(a)ess (Theorem 2.54) ≤ inf H(a − ϕ) : ϕ ∈ L∞ ∩ V M O = inf distL∞ (a − ϕ, H ∞ ) : ϕ ∈ L∞ ∩ V M O (Theorem 2.11) ∞ (Lemma 4.105) ≤ inf Ba − ϕ∗ : ϕ ∈ L ∩ V M O ∞ ≤ inf Ba − ϕBM O : ϕ ∈ L ∩ V M O = B distBM O (a, L∞ ∩ V M O). (4.65)

244


Let d := distBM O (a, V M O). Thus, there are u, v ∈ C and ϕ ∈ BM O such that a = u+P v +ϕ and ϕBM O < d+ε (1.48(l)). Choose v1 ∈ P and v2 ∈ C so that v = v1 + v2 and P v2 BM O ≤ P L(L∞ ,BM O) v2 ∞ < ε (1.48(k)). So a = u+P v1 +P v2 +ϕ with u+P v1 ∈ L∞ ∩V M O and P v2 +ϕBM O < d+2ε. Hence, (4.66) distBM O (a, L∞ ∩ V M O) ≤ distBM O (a, V M O). Combining (4.65), (4.66), and 1.48(f) we arrive at the inequality distL∞ (a, C + H ∞ ) ≤ BA2 M0 (a). Now Lemma 4.106 implies that distL∞ (a, C + H ∞ ) < 1 and Theorem 2.75 gives that T (a) ∈ Φ− (H 2 ). It can be shown analogously that T (a) ∈ Φ+ (H 2 ). Remark. Tolokonnikov [518], who was the first to establish the existence of δG , showed that δG ≤ 45/46 if {Kλ }λ∈(0,∞) is generated by the Poisson kernel. In Nikolski [368], it is shown that even δG ≤ 23/24 in this case.

4.9 Notes and Comments 4.1–4.6. The approach is due to the authors (B¨ ottcher, Silbermann [106], Silbermann [483]), for the results we refer to Nikolski [366] (4.1 for p = 2), Simonenko [492] (4.2), Devinatz, Shinbrot [155] (4.2), Clancey [136] (4.3, 4.4), Böttcher, Silbermann [106] (4.3, 4.4), Brown, Halmos [125] (4.4(d)), Douglas [160] (4.6). The proof of 4.5 uses an argument of Coburn [142]. 4.7–4.8. Parts (a) are from Douglas [160], parts (b) are results of the authors and are published here for the first time. 4.10–4.13. Local Toeplitz operators are an invention of Douglas [161]. The basic equality (4.8) was established by Axler [12] for N = 1 and by Silbermann [483] in the matrix case. Douglas [160], [161] stated the results of 4.12 and 4.13 for B = QC. Corollary 4.13 for B = C is Simonenko’s [492]. 4.14–4.26. Silbermann [481], [483]. 4.27–4.30. This is the approach of Silbermann [481], [482], [483]. That the index of operators in alg T (P QC) (N = 1) can be computed via the harmonic extension (i.e., as in Theorem 4.28) was first shown by Sarason [456]. Corollary 4.30 was obtained by Faour [198] (using other methods) for the following two cases: (i) B = C and N = 1, (ii) a ∈ P2 CN,N . In all these works {Kλ } was the approximate identity generated by the Poisson kernel (i.e., the harmonic extension). For arbitrary approximate identities the results were first proved in B¨ ottcher, Silbermann [112]. That the index of a Toeplitz operator can be expressed via the FejérCesaro means of the symbol is of importance in connection with Gohberg,


245

Lerer, Rodman [234], where explicit formulas for the partial indices of rational matrix functions are given (also see Chapter 1 of Clancey, Gohberg [138]). In Böttcher [70] (see also the remark in 8.53) it is shown that there exist matrix functions a ∈ W2×2 such that the partial indices of hr a (resp. σn a) are zero for all r ∈ (0, 1) (resp. n = 2, 3, . . .) but the partial indices of a itself are not all equal to zero. This shows that even for very smooth a the kernel and cokernel dimensions of the block Toeplitz operator T (a) cannot be recovered from the kernel and cokernel dimensions of T (hr a) and T (σn a). 4.31. These arguments, though originated by the approach of B¨ ottcher [70], are published here for the first time. 4.32–4.34. Parrott [376], Power [403], [404]. 4.35–4.39. The results of Pousson [398] and Rabindranathan [409] are, respectively, matrix and operator-valued analogues of the Widom-Devinatz criterion in the form 2.22. The statements of 4.36 and 4.37 can be found in Clancey, Gohberg [138]; also see Devinatz, Shinbrot [155]. Corollary 4.39 is new (but see also Böttcher [69]). 4.40. See Sierpinski [472], for example. 4.41–4.49. This is (the matrix-case version of) Axler’s method [12]. 4.50. Result (a) goes back to Sarason [453]. The “if” part of (d) was proved by Axler, Chang, Sarason [15] and the “only if” of (d) is Volberg’s [549]. A discussion and a proof of (d) is also in Nikolski [366]. See also Sarason [457] and Power [404]. A series of new interesting results on compactness of commutators and quasicommutators of Toeplitz and Hankel operators were obtained in recent work by Gorkin, Gu, Guo, and Zheng. In Zheng [589] it is proved that the quasicommutator T (f g) − T (f )T (g) is compact on H 2 if and only if H(f )kz H(g)kz → 0 as |z| → 1; here kz denotes the normalized reproducing kernel in H 2 for point evaluation at z. Gorkin and and Zheng [241] proved an analogous characterization for the commutator T (f )T (g) − T (g)T (f ). The above two results were extended to the block case in Gu, Zheng [256]. Martinez-Avenda˜ no [348] found necessary and sufficient conditions guaranteeing that T (f )H(g) = H(g)T (f ). Inspired by this work, Guo and Zheng [259] found criteria for the compactness of T (f )H(g) − H(g)T (f ). Recently Guo and Zheng [260] also established necessary and sufficient conditions for a product of Toeplitz operators T (f1 ) . . . T (fm ) (m ≥ 2) to be a compact perturbation of a Toeplitz operator. The approach of all these works is based on Littlewood-Paley theory and some inequalities involving the Luzin area integral and certain maximal functions. In connection with the open problems in 4.50(e) notice that Volberg and Ivanov [550] found necessary conditions in terms of martingales for H(a)H()b) to be in Cp (H 2 ) (p ≥ 2). On the other hand, Rochberg and Semmes [435, Section 7] obtained some estimates for the s-numbers of H(a)H()b).

246


This is perhaps also the right place to mention the paper Gu [254]. An open question by Douglas [162] is as follows: If X is a bounded linear operator on H 2 such that X − T ∗ (θ)XT (θ) is a compact operator for every inner function θ, does it follow that X = T (a) + K with a ∈ L∞ and a compact operator K? Gu’s main result says that the answer is in the affirmative provided the twice occurring “compact” is replaced by “finite rank.” A theorem by Davidson [146] says that if X is a bounded linear operator on H 2 and XT (h) − T (h)X is a compact operator for every h ∈ H ∞ , then X = T (a) + K with a compact operator K and a function a ∈ L∞ for which H(ψ) is a compact operator. As a corollary of his main result, Gu shows that this theorem remains valid if the thrice occurring “compact” is replaced with “finite rank.” For commuting finite Toeplitz matrices, normality of finite Toeplitz matrices, and finite Toeplitz matrices with Toeplitz inverses see Gu, Patton [255] and the references cited there. 4.51–4.53. The observation made in 4.51 is due to S. Axler. Theorem 4.53 was stated (but not proved) in Clancey, Gosselin [139]. Douglas [161] proved this theorem for w = 2, that is, for the case where S is a fiber Xξ (ξ ∈ M (QC)). The proof given here is based on Douglas’ argument. 4.54–4.65. Clancey, Gosselin [139]. For the Chang-Marshall theorem see Sarason [457] and Garnett [211]. 4.67. Clancey [135], Clancey, Morrel [140], Douglas [160], [161] for B = C and Silbermann [483] for B = QC. 4.70. For several classes of symbols (P C, P2 C, P QC) this theorem had been known before it was explicitly stated in this form in Silbermann [483]. We are grateful to I. M. Spitkovsky for pointing out an error in our original proof. 4.71–4.78. Böttcher [68]. 4.79–4.94. The derivation of all these results rests on the observation that the corresponding local algebras are singly generated, a fact which was pointed out in Silbermann [483] for the first time. Due to this observation the proofs given here are essentially simpler than the original ones. For Theorem 4.79 (i = T, B = C) see Mikhlin [360], Gohberg [217], Coburn [142]. Theorems 4.79 and 4.81 were established by Douglas [162], [160] for i = T . Theorem 4.86 for i = T is due to Gohberg and Krupnik [229], and Theorem 4.87 was obtained by Sarason [456] for i = T and i = H; these authors also described the Gelfand topologies of NP C and NP QC . Theorem 4.94 is new, although results of this type (i = T ) are also in Douglas [161]. It is clear that Theorem 4.94 also holds in the matrix case and that the index of a Fredholm Toeplitz operator with symbol a ∈ (BE )N ×N equals −ind {det(kλ a)}. All statements concerning the cases i = T K and i = K are due to Silbermann [481], [482], [483]. The possibility of such results being valid was suggested by the paper B¨ ottcher, Silbermann [105]. For a detailed discussion of algebras generated by Toeplitz


247

operators and for further results along these lines see also Nikolski [366], [368], [369]. 4.95–4.102. For extensions of the two projections theorem (Theorem 4.100) that are suitable for the study of operators acting on Banach spaces and having massive local spectra see, e.g., Finck, Roch, Silbermann [202], [203], Gohberg, Krupnik [233]. An N -projections version was elaborated in B¨ ottcher, Gohberg, Karlovich, Krupnik, Roch, Silbermann, Spitkovsky [79] (many projections required many authors!). The application of this result to singular integral operators on Lebesgue spaces with Muckenhoupt weights over composed Carleson curves is illustrated in B¨ ottcher, Karlovich [92]. For Corollary 4.102 and its generalization to P QC see Power [399], [400], [401]. Theorem 4.101 and the approach presented here are from Silbermann [484]. In this paper, 4.101 is even proved for P QC symbols. Also see Roch, Silbermann [425]. 4.103–4.107. That there are symbols satisfying (4.62) was discovered by Douglas [160, pp. 13–14]. At the same time Douglas asked whether T (c) is invertible whenever {hr c} is bafz. Wolff [581] has shown that the answer is negative by constructing the symbol mentioned in the text. Theorem 4.107(a) was established by Tolokonnikov [518] and independently also in Wolff [581] (where it is attributed to A. Chang) for the case that {Kλ } is generated by the Poisson kernel.

5 Toeplitz Operators on H p

5.1 General Theorems Some questions for Toeplitz operators on H p have already answered in the preceding chapters. In particular, we settled the Fredholm theory for the oper∞ ators in algL(HNp ) T (CN ×N + HN ×N ). Also notice the localization result stated in Theorem 2.96. However, many questions still remain open and it is only a small number of them which will be answered in this chapter. 5.1. Conventions. Throughout the present chapter we suppose that p and q are real numbers satisfying 1 < p < ∞ and 1/p + 1/q = 1. When considering Lp or H p with a weight w, we always suppose that w is (Lebesgue) measurable on T, w ≥ 0 almost everywhere on T, w ∈ Lp , w−1 ∈ Lq , and that w satisfies the Hunt-Muckenhoupt-Wheeden condition (Ap ) (see 1.46). So the Toeplitz operator T (a) is bounded on H p (w) for a ∈ L∞ . 5.2. The Hartman-Wintner and Coburn theorems. The HartmanWintner spectral inclusion theorem (2.30 and 2.93) continues to hold for the p p spaces H p (w): if a ∈ L∞ N ×N and T (a) ∈ Φ+ (HN (w)) or T (a) ∈ Φ− (HN (w)), ∞ as the one given for the spaces H p . then a ∈ GLN ×N . The proof is the same * Note that under the pairing (f, g) = T f g dm for f ∈ Lp (w) and g ∈ Lq (w−1 ), we may think of Lq (w−1 ) as the dual space of Lp (w); in particular, this shows that U n converges weakly to zero on Lp (w) as |n| → ∞. Coburn’s theorem (2.38 and 2.40) also remains valid for the spaces H p (w): if a ∈ L∞ , then T (a) ∈ GL(H p (w)) ⇐⇒ T (a) ∈ Φ(H p (w)) and Ind T (a) = 0. This can be proved in the same way as for H p . Note that by virtue of the inequality (

( T

|f | dm ≤

T

1/p ( |f |p wp dm

w−q dm T

1/q


250

H p (w) is contained in H 1 , so that the F. and M. Riesz theorem 1.40(b) is applicable to the spaces H p (w). Finally, notice that Ind T (χk ) = −k (k ∈ Z) in H p (w). 5.3. Rochberg’s invertibility criterion. This is an extension of the WidomDevinatz criterion 2.23 to Toeplitz operators on H p (w). Let a ∈ L∞ . Then T (a) ∈ GL(H p (w)) if and only if a ∈ GL∞ and a/|a| = ei(c+)y) , where c ∈ R and y(∈ BM O) is a real-valued function with the property that we−y/2 satisfies the Hunt-Muckenhoupt-Wheeden condition (Ap ), i.e., sup I

1 |I|

(

p −py/2

w e

1/p dm

I

1 |I|

( w

−q qy/2

e

1/q dm

< ∞.

I

Here y) refers to the conjugate function of y. Proof. For a proof see Rochberg [433]. Let us show that this reduces to the Widom-Devinatz criterion in case p = 2 and w = 1. By the Helson-Szeg˝o theorem, e−y/2 satisfies (A2 ) if and only if −y/2 = f + g) with f, g ∈ L∞ real-valued and g∞ < π/4. Thus, y = −2f − 2) g , hence c + y) = −2f)+ 2g + const = u ) + v + const with u, v ∈ L∞ real-valued and v∞ = 2g∞ < π/2. We now prove a theorem which provides an alternative criterion for Toeplitz operators on H p (w) in terms of a certain factorization of the symbol. 5.4. Definition. A function a ∈ GL∞ is said to admit a Wiener-Hopf factorization in Lp (w) if it can be represented in the form a = a− χκ a+ , where κ ∈ Z, a− ∈ Lp− (w),

q −1 a−1 ), − ∈ L− (w

and sup I

a+ ∈ Lq+ (w−1 ),

p a−1 + ∈ L+ (w)

1 −1 a wLp (I) a+ w−1 Lq (I) < ∞, |I| +

(5.1)

(5.2)

the supremum over all subarcs I of T. In view of the Hunt-Muckenhoupt-Wheeden theorem 1.46, under the condition that (5.1) holds, (5.2) may be replaced by the requirement that p ∞ and a−1 + P (a+ ϕ) ∈ L (w) for all ϕ ∈ L a−1 + P (a+ ϕ)p,w ≤ cp,w ϕp,w

∀ ϕ ∈ L∞

(5.3)

with some constant cp,w depending only on p and w. Here L∞ may be replaced by any of its subsets which is dense in Lp (w). 5.5. Theorem (Simonenko). Let a ∈ L∞ . Then T (a) ∈ Φ(H p (w)) if and only if a ∈ GL∞ and if a admits a Wiener-Hopf factorization a = a− χκ a+ in Lp (w). In that case Ind T (a) = −κ.

5.1 General Theorems

251

Proof. First suppose a ∈ GL∞ admits the Wiener-Hopf factorization a = a− a+ in Lp (w) (i.e. assume κ = 0). Then Ker T (a) = {0}. Indeed, if T (a)ϕ = 0 1 for ϕ ∈ Lp+ (w), then a− a+ ϕ = g− ∈ Lp− (w), hence a+ ϕ = a−1 − g− ∈ L− , and 1 since a+ ϕ ∈ L+ , it follows that a+ ϕ = 0, which implies that ϕ = 0. We now show that Im T (a) = H p (w). By (5.3), the mapping H ∞ → H p (w),

−1 −1 −1 ϕ → a−1 ϕ + P a− ϕ = a+ P a+ a

extends to a bounded operator A on H p (w). For ϕ ∈ H ∞ , −1 −1 −1 T (a)Aϕ = P a− a+ a−1 + P a− ϕ = P a− P a− ϕ = ϕ − P a− Qa− ϕ = ϕ,

and since both T (a) and A are bounded, it follows that T (a)Aϕ = ϕ for all ϕ ∈ H p (w). Thus Im T (a) = H p (w), and we have proved that T (a) is invertible. If a = a− χκ a+ , then either T (a) = T (a− a+ )T (χκ )

or T (a) = T (χκ )T (a− a+ ),

and because T (a− a+ ) has just been proved to be invertible, we deduce that T (a) ∈ Φ(H p (w)) and that Ind T (a) = Ind T (χκ ) = −κ. Now suppose that T (a) ∈ Φ(H p (w)) and Ind T (a) = −κ. Then, by 5.2, a ∈ GL∞ and T (b) ∈ GL(H p (w)), where b = aχ−κ . Let ϕ+ ∈ Lp+ (w) and ψ+ ∈ Lq+ (w−1 ) denote the solutions of the equations P (bϕ+ ) = 1, P (bψ+ ) = 1. So bϕ+ = 1 + g− and bψ+ = 1 + h+ with g− ∈ Lp− (w) and h+ ∈ Lq+ (w−1 ), hence ψ+ (1 + g− ) = bϕ+ ψ+ = ϕ+ (1 + h+ ) and since H 1 ∩ H 1 = C, we obtain that ψ+ (1 + g− ) = ϕ+ (1 + h+ ) = c = const. By the F. and M. Riesz theorem, ϕ+ = 0 and ψ+ = 0 a.e. on T. In particular, 1 + h+ = bψ+ = 0 a.e. on T, and hence c = 0. Put a+ = ϕ−1 + and a− = 1 − g− . p −1 Then a = a− χκ a+ and a− = 1 − g− ∈ L− (w), a− = c−1 ψ+ ∈ Lq− (w−1 ), p a+ = c−1 (1 + h+ ) ∈ Lq+ (w−1 ), a−1 + = ϕ+ ∈ L+ (w). −1 −1 ∞ If f ∈ H , then P ba+ P a− f = f . Since T (b) is invertible, we deduce that −1 −1 (b) f p,w ∀ f ∈ H ∞ . a−1 + P a− f p,w ≤ T ∞ Moreover, because P a−1 − f = 0 for f ∈ H , we even have −1 −1 a−1 (b) P f p,w + P a− f p,w ≤ T

and thus

∀f ∈P

−1 (b) P b∞ gp,w a−1 + P a+ gp,w ≤ T

for all g ∈ b−1 P := {h ∈ Lp (w) : bh ∈ P}. But b−1 P is clearly dense in Lp (w) and so (5.2) holds by what was said at the end of Section 5.4.

252


Remark. The preceding theorem extends to the case of matrix-valued symbols as follows. p p Let a ∈ L∞ N ×N . Then T (a) ∈ GL(HN (w)) (resp. Φ(HN (w))) if and only if a admits a factorization a = a− a+ (resp. a = a− da+ , d being of the form d = diag (χκ1 , . . . , χκN ) with κj ∈ Z for all j), where

a− ∈ [Lp− (w)]N ×N ,

q −1 a−1 )]N ×N , − ∈ [L− (w

a+ ∈ [Lq+ (w−1 )]N ×N ,

p a−1 + ∈ [L+ (w)]N ×N

p and the following holds: a−1 + P a+ ϕ ∈ HN (w) for all ϕ ∈ PN and

a−1 + P (a+ ϕ)p,w ≤ cp,w ϕp,w

∀ ϕ ∈ PN .

p If T (a) ∈ Φ(HN (w)), then

dim Ker T (a) = −

κj ,

dim Coker T (a) =

κj 0

For a proof see Simonenko [496], Clancey, Gohberg [137], or Litvinchuk, Spitkovsky [340]. Note that the above proof with only minor modifications can be used to prove the part of this result concerned with invertibility. 5.6. Convention. We have always assumed that the norm on L2N is defined by ( ( N 2 2 f L2 := |fj (t)| dm = f (t)2CN dm, N

T

T

j=1

but so far we have not specified a norm on LpN or LpN (w). Henceforth suppose the norm on LpN (w) is given by f pLp (w) := N

( N T

p/2 |fj (t)|2

( wp dm =

j=1

T

f (t)pCN wp dm.

(5.4)

∞ Recall that the norm on L∞ N ×N was defined as aLN ×N := M (a)L(L2N ) and that = ess sup a(t)L(CN ) = max a(x)L(CN ) . aL∞ N ×N

t∈T

x∈X

The choice of the norm (5.4) is motivated by the following proposition. 5.7. Proposition. If the norm on LpN (w) is given by (5.4), then M (a)L(LpN (w)) = aL∞ N ×N

∀ a ∈ L∞ N ×N .

(5.5)

5.2 Khvedelidze Weights

253

Proof. In (5.5) the inequality “≥” holds for every norm on LpN (w) that is equivalent to the norm (5.4). Indeed, we then have aL∞ ≤ CM (a)L(LpN (w)) N ×N with some constant C > 0 (see, e.g., (4.3)), thus a = an 1/n ≤ C 1/n M (an )1/n ≤ C 1/n M (a), and letting n go to infinity, we get the “≥” in (5.5). The reverse inequality is a consequence of the particular choice of the p norm: for a ∈ L∞ N ×N and f ∈ LN (w), one has ( ( p p p af LN (w) = a(t)f (t)CN w dm ≤ a(t)pL(CN ) f (t)pCN wp dm T T ( p p p f (t)CN w dm = apL∞ f pLp (w) . ≤ aL∞ N ×N

N ×N

T

N

5.2 Khvedelidze Weights 5.8. Definition. A Khvedelidze weight is a function on T of the form (t) =

n

|t − tj |µj

(t ∈ T),

(5.6)

j=1

where t1 , . . . , tn are pairwise distinct points on T and µ1 , . . . , µn are real numbers. 5.9. Theorem. Let 1 < p < ∞ and 1/p + 1/q = 1. Suppose is a weight of the form (5.6). Then P is bounded on Lp () if and only if −1/p < µj < 1/q for j = 1, . . . , n. Proof. Since −1/p < µj ∀ j ⇐⇒ ∈ Lp and µj < 1/q ∀ j ⇐⇒ −1 ∈ Lq , the “only if” part results from the Hunt-Muckenhoupt-Wheeden theorem 1.46. However, there is a simple direct argument to prove this part of the above (and of the Hunt-Muckenhoupt-Wheeden) theorem: if P ∈ L(Lp ()), then S = 2P − I ∈ L(Lp ()), where S is the Cauchy singular integral operator (1.15), hence A := SM (χ1 ) − M (χ1 )S ∈ L(Lp ()), and thus B := A−1 belongs to L(Lp ); but ( (t) −1 (τ )ϕ(τ ) dτ, (Bϕ)(t) = πi T whence ∈ Lp and −1 ∈ Lq . Now suppose that ∈ Lp and −1 ∈ Lq . The boundedness of P will follow once we have shown that the weight satisfies the Hunt-MuckenhouptWheeden condition (Ap ): there is a constant M independent of I such that Lp (I) −1 Lq (I) ≤ M |I| for all arcs I of length |I| ≤ δ.

254


Choose δ so small that the arcs (tj − 2δ, tj + 2δ) (j = 1, . . . , n) are pairwise disjoint and then choose M1 , M2 so that |(t)| ≤ M1 ,

|

−1

(t)| ≤ M1

if

t∈T\

n /

(tj − δ, tj + δ),

j=1

|(t)| |t − tj |−µj ≤ M2 ,

|−1 (t)| |t − tj |µj ≤ M2

t ∈ (tj − 2δ, tj + 2δ).

if

If I ∩ (tj − δ, tj + δ) = ∅ for all j, then Lp (I) −1 Lq (I) ≤ M12 |I|1/p+1/q = M12 |I|, and if I overlaps with (tj − δ, tj + δ), then Lp (I) −1 Lq (I) ≤ M22

1/p (

( |t − tj |pµj dm

I 2 M2 M3 |I|µj +1/p |I|−µj +1/q

|t − tj |−qµj dm

1/q

I

≤ = M22 M3 |I|, * with M3 arising when I |t − tj |α dm is replaced by an integral of the type *b |x − x0 |α dx. a 5.10. Convention. In what follows the letter always denotes a Khvedelidze weight satisfying the conditions of the preceding theorem. Sometimes we shall say “let be a Khvedelidze weight on Lp ” to mean that is a weight of the form (5.6) which satisfies −1/p < µj < 1/q for j = 1, . . . , n. 5.11. The norm of the Cauchy singular integral operator. Let t ∈ T, 1 1. Then SL(LpN (|t−τ |µ )) = cot

π , 2r

where r = max{p, q, (1/p+µ)−1 , (1/q −µ)−1 }. A proof is in in Krupnik’s book [329].

5.3 Locally p, -Sectorial Symbols 5.12. Definitions. Let a be a matrix function in GL∞ N ×N , let F be a closed subset of X = M (L∞ ), and let 2 ≤ r < ∞. For r > 2, Sr will denote the sector {z ∈ C : |Im z| < (tan π/r)Re z}, and S2 will refer to the right open half-plane {z ∈ C : Re z > 0}. The unit sphere in CN will be denoted by SCN , i.e., SCN = {z ∈ CN : zCN = 1}. The numerical range of a matrix d ∈ CN ×N (operator d ∈ L(CN )) is defined by W (d) := {(dz, z) : z ∈ SCN }. If f, g ∈ CN ×N are selfadjoint (f = f ∗ , g = g ∗ ), then f ≥ g (resp. f > g) will mean that (f z, z) ≥ (gz, z) (resp. (f z, z) > (gz, z)) for all z ∈ SCN . In

5.3 Locally p, -Sectorial Symbols

255

case g = αI (α ∈ R), we shall write f ≥ α and f > α instead of f ≥ αI and f > αI, respectively. The matrix function a is said to be r-sectorial on F if there are c, d ∈ GCN ×N such that W (ca(x)d) ⊂ Sr for all x ∈ F . Note that W (ca(x)d) ⊂ Sr (r ≥ 2) if and only if π Re (ca(x)dz, z) > cos |(ca(x)dz, z)| ∀ z ∈ SCN . (5.7) r It is easily seen that each of the two conditions π ca(x)d Re (a(x)d) > cos r and max I − ca(x)dL(CN ) < sin x∈F

(5.8)

π r

(5.9)

is sufficient for (5.7) to hold. Also notice that, for r > 2, (5.7) is equivalent to π |Im (ca(x)dz, z)| < tan Re (ca(x)dz, z) ∀ z ∈ SCN . (5.10) r Since {(ca(x)dz, z) : x ∈ F, z ∈ SCN } is compact, the matrix function a is 2-sectorial on F if and only if there are c, d ∈ GCN ×N and ε > 0 such that Re (ca(x)d) ≥ ε for all x ∈ F , i.e., if and only if a is (analytically) sectorial on F in the sense of Definition 3.1. It is clear that a scalar-valued function a ∈ GL∞ is r-sectorial on F if and only if a(F ) is contained in some open angular sector spanned by an angle whose vertex is the origin and whose size is 2π/r. Furthermore, if r = 2 or if N = 1, then (5.7) and (5.8) are equivalent. A connection between (5.7) and (5.9) will be established in Lemma 5.14. Now allow r to take values in (1, ∞). Given two points z1 , z2 in the complex plane, let Ar (z1 , z2 ) denote the circular arc from the points of which the line segment [z1 , z2 ] is seen at the angle 2π/ max{r, s} (1/r + 1/s = 1) and which lies on the right (resp. left) of the straight line passing first z1 and then z2 if 2 < r < ∞ (resp. 1 < r < 2). For r = 2, Ar (z1 , z2 ) is nothing but the line segment [z1 , z2 ] itself. Ar (z1 , z2 ) is thought of as being oriented from z1 to z2 . Note that Ar (z1 , z2 ) has the parametric representation z(µ) = z1 + (z2 − z1 )σr (µ),

0 ≤ µ ≤ 1,

where σr (µ) = µ for r = 2 and sin(θµ) exp(iθµ) σr (µ) = , sin θ exp(iθ)

θ := π

1 1 − s r

for r = 2. In what follows let [r, s] refer to the segment [min{r, s}, max{r, s}]. Finally, let Or (z1 , z2 ) denote the (closed) lentiform domain between Ar (z1 , z2 ) and As (z1 , z2 ):

256


Or (z1 , z2 ) =

/

Aν (z1 , z2 ).

ν∈[r,s]

If a ∈ L∞ and conv a(F ) is the line segment [z1 , z2 ], then it is clear that a is r-sectorial on F (r ≥ 2) if and only if 0 ∈ / Or (z1 , z2 ). It is not too difficult to show that a ∈ GL∞ N ×N is r-sectorial on a fiber Xτ = Mτ (L∞ ) (τ ∈ T) if and only if there are a neighborhood U ∈ Uτ and matrices c, d ∈ GCN ×N such that W (ca(t)d) ⊂ Sr for almost all t ∈ U . Matrix functions which are r-sectorial on the whole space X will be called r-sectorial on T. Now let be a Khvedelidze weight on Lp of the form (5.6) and let B be a closed subalgebra of L∞ containing C. A matrix function a ∈ GL∞ N ×N is said to be locally p, -sectorial over B if it has the following property: for each τ ∈ T, a is rτ -sectorial on each maximal antisymmetric set for B which is contained in the fiber Xτ , and rτ is given by ! max{p, q} for τ ∈ T \ {t1 , . . . , tn }, rτ = max{p, q, (1/p + µj )−1 , (1/q − µj )−1 } for τ = tj . Note that a matrix function is locally 2, 1-sectorial (i.e., p = 2, ≡ 1) over B if and only if it is (analytically) locally sectorial over B in the sense of Definition 3.1. 5.13. Lemma. Let v ∈ L∞ N ×N and let F be a closed subset of X. Suppose v is positive definite on F , that is v(x) = v ∗ (x) ≥ ε > 0 for all x ∈ F . Then ∞ ∗ there is an h ∈ GHN ×N such that v(x) = h (x)h(x) for all x ∈ F . Proof. The mapping X × SCN → C, (x, z) → Re (v(x)z, z) is continuous. Hence, there is a clopen neighborhood U ⊂ X such that Re v(x) ≥ ε/2 for all ∗ x ∈ U . Put f = χU (Re v) + (1 − χU )I. Then f ∈ GL∞ N ×N and f = f ≥ ε/2 2 ∗ on X. Therefore T (f ) ∈ GL(HN ) (Corollary 4.2) and thus f = g k with 2 ∗ g ±1 , k±1 ∈ HN ×N (see the remark after Theorem 5.5). Since f = f , we have ∗ ∗ ∗ −1 ∗ −1 = c ∈ CN ×N , that is, g = ck and so g k = k g, hence (k ) g = gk f = k∗ ck. Clearly, c = c∗ . Because f ∈ GL∞ N ×N , there is a τ ∈ T such that k(τ ) ∈ GCN ×N . Consequently, for z ∈ CN and y = k−1 (τ )z, (cz, z) = (ck(τ )y, k(τ )y) = (f (τ )y, y) ≥

ε ε y2 ≥ k(τ )−2 z2 . 2 2

Thus, c is positive definite, and so c = e∗ e for some e ∈ GCN ×N . If we put h = ek, then f = h∗ h. Consideration of the diagonal entries of f and h∗ h ∞ −1 = h−1 (h−1 )∗ , it follows shows that actually h ∈ HN ×N . Similarly, since f −1 ∞ that h ∈ HN ×N . Because f (x) = v(x) for x ∈ F , we are done. 5.14. Lemma. Let a ∈ GL∞ N ×N be r-sectorial on a closed subset F of X. Then ∞ ∞ a can be represented in the form a = f bg, where f, g ∈ GHN ×N , b ∈ GLN ×N , and π (5.11) max I − b(x)L(CN ) < sin . x∈F r


257

Proof. If a matrix function is r-sectorial on a closed subset of X, then it is s-sectorial for all s ∈ [2, r + ε), where ε is sufficiently small. Thus, we may suppose that r > 2. Choose c, d ∈ GCN ×N so that (5.10) is fulfilled for all x ∈ F , and put v := Re (cad). Then v satisfies the hypothesis of the preceding ∞ lemma, and hence v(x) = h∗ (x)h(x) for all x ∈ F with some h ∈ GHN ×N . ∗ −1 −1 Let ω := (h ) cadh . Clearly Re ω(x) = (h∗ (x))−1 v(x)h−1 (x) = I

∀ x ∈ F.

If z ∈ CN and x ∈ F , then + + + + + Im ω(x)z, z + = +Im ca(x)dh−1 (x)z, h−1 (x)z + π < tan Re ca(x)dh−1 (x)z, h−1 (x)z r π π −1 z2 . (h (x))∗ v(x)h−1 (x)z, z = tan = tan r r Consequently, for x ∈ F , the spectrum of the normal matrix I + iIm ω(x) is contained in the interior of the line segment whose endpoints are 1−i tan(π/r) and 1+i tan(π/r). Put b := (cos2 (π/r))ω. Then, again for x ∈ F , the spectrum of b(x) = (cos2 (π/r))(I + iIm ω(x)) is a subset of the open disk with center 1 and radius sin(π/r). Hence, the spectral radius and thus the norm of the normal matrix I − b(x) is less than sin(π/r). From the compactness of F we deduce that (5.11) holds. If we put f := (cos2 (π/r))−1 c−1 h∗ and g = hd−1 , then a = f bg is the desired representation. 5.15. Lemma. Let B ∈ L(CN ). For s ∈ (2, ∞), put ω = (cos(π/s))−1 . Then I − BL(CN ) ≤ sin

π s

(5.12)

if and only if I + ωB ∈ GL(CN )

and

(I + ωB)−1 (I − ωB)L(CN ) ≤ tan

π . 2s

(5.13)

Proof. If (5.12) is satisfied, then ω(1 + ω)−1 (I − B) ≤ tan(π/(2s)) < 1, and since I + ωB = (1 + ω)[I − ω(1 + ω)−1 (I − B)], (5.12) implies the invertibility of I + ωB. Now note that for A ∈ L(CN ) the equality A2 = sup{(AA∗ z, z) : z ∈ CN , z = 1} holds and that therefore A2 ≤ M 2 if and only if AA∗ ≤ M 2 . Thus, π ⇐⇒ ω −2 + BB ∗ ≤ B + B ∗ s ⇐⇒ (ω + 1)(I − ωB)(I − ωB ∗ ) ≤ (ω − 1)(I + ωB)(I + ωB ∗ ) π ω−1 = tan2 ⇐⇒ (I + ωB)−1 (I − ωB)(I − ωB ∗ )(I + ωB ∗ )−1 ≤ ω+1 2s ⇐⇒ (5.13). (5.12) ⇐⇒ (I − B)(I − B ∗ ) ≤ sin2

258


5.16. Theorem. Let 1 < p < ∞, −1/p < µ < 1/q, t ∈ T, (t) := |t − τ |µ , and put r = max{p, q, (1/p + µ)−1 , (1/q − µ)−1 }. p ∞ (a) If b ∈ L∞ N ×N and I − bLN ×N < sin(π/r), then T (b) ∈ GL(HN ()). p (b) If a ∈ GL∞ N ×N is r-sectorial on T, then T (a) ∈ GL(HN ()). p Proof. (a) By formula (2.29), the invertibility of T (b) on HN () is equivalent p to the invertibility of bP + Q on LN () (the multiplication operator M (ϕ) will be simply denoted by ϕ). There is an s > r such that I −b ≤ sin(π/s). Put ω = (cos(π/s))−1 . Then I +ωb(x) ∈ GL(CN ) for all x ∈ X by Lemma 5.15, and thus I +ωb ∈ GL∞ N ×N . This in turn implies that M (I + ωb) ∈ GL(LpN ()). Now write bP + Q in the form

bP + Q =

1 (I + ωb)[I − (I + ωb)−1 (I − ωb)S](ω −1 P + Q), 2

where S = I − 2Q is the singular integral operator on LpN (). Since ωP + Q is the inverse of ω −1 P +Q, it remains to show that D := I −(I +ωb)−1 (I −ωb)S is invertible. Due to 5.11, SL(LpN ()) = cot(π/(2r)). Because I−b ≤ sin(π/s), we deduce from Lemma 5.15 that " " "(I + ωb)−1 (I − ωb)" ∞ ≤ tan π . LN ×N 2s Consequently, by Proposition 5.7, I − DL(LpN ()) ≤ tan

π π cot < 1. 2s 2r

(b) Immediate from Lemma 5.14 and part (a). 5.17. Theorem. Let be a Khvedelidze weight on Lp and let a ∈ GL∞ N ×N be locally p, -sectorial over a C ∗ -subalgebra B between C and QC. Then T (a) p ()). is in Φ(HN Proof. First note that it suffices to consider the case B = QC. Then notice that p p replaced by HN (); the proof is almost Theorem 2.96 remains true with HN literally the same. Hence, it is enough to show that for each ξ ∈ M (QC) there p exists an aξ ∈ L∞ N ×N such that aξ |Xξ = a|Xξ and T (aξ ) ∈ Φ(HN ()). Let ξ ∈ Mτ (QC) (τ ∈ T). Put τ (t) := 1 if τ ∈ T \ {t1 , . . . , tn } and τ (t) := |t − tj |µj if τ = tj . Since a is r-sectorial on Xξ (r depends on the τ ∞ above which ξ lies as in 5.12), we have a = f bg, where f, g ∈ GHN ×N , b ∈ , and I − b(x) < sin(π/r) for x ∈ X (Lemma 5.14). Let U GL∞ ξ 1 ⊂ X be N ×N a (sufficiently small) clopen neighborhood of Xξ . A little thought shows that U1 ⊂ X can be chosen so that Vτ := {t ∈ T : U1 ∩ Xt = ∅} has the following property: the restriction Lp (Vτ , τ ) of Lp (τ ) to Vτ is equal to the restriction Lp (Vτ , ) of Lp () to Vτ . Then let U2 ⊂ U1 be a clopen neighborhood of Xξ


259

such that I − b(x) < sin(π/r) for x ∈ U2 , and put bξ := χU2 b + (1 − χU2 )I. ∞ Clearly, bξ ∈ GL∞ N ×N , bξ |Xξ = b|Xξ , and I − bξ LN ×N < sin(π/r). If we set p ()). aξ = f bξ g, then aξ |Xξ = a|Xξ . So it remains to show that T (aξ ) ∈ Φ(HN p This will follow once we have proved that T (bξ ) ∈ Φ(HN ()). p (τ )). By construcFrom Theorem 5.16(a) we know that T (bξ ) ∈ GL(HN tion, we have the following direct sums: ·

LpN (τ ) = LpN (Vτ , τ ) + LpN (Vτc , τ ), ·

LpN () = LpN (Vτ , τ ) + LpN (Vτc , ) (Vτc := T \ Vτ ). Let R1 denote the projection of LpN () onto LpN (Vτ , τ ) parallel to LpN (Vτc , ) and let R2 := I − R1 . Put A := bξ P + Q. Then A = R1 A + R2 A = R1 A + R2 bξ P + R2 Q = R1 A + R2 P + R2 Q (since bξ |Vτc = I) = R1 A + R2 = (R1 A + R2 )(R1 + R2 ) = R1 AR1 + R1 AR2 + R2 = (I + R1 AR2 )(R1 AR1 + R2 ).

(5.14)

Since A ∈ GL(LpN (τ )) and I + R1 AR2 ∈ GL(LpN (τ )) (the inverse is I − R1 AR2 ), it follows from (5.14) that R1 AR2 + R2 ∈ GL(LpN (τ )), hence R1 AR1 is invertible on R1 LpN (τ ) = R1 LpN (), and thus R1 AR1 + R2 is in GL(LpN ()). Again by (5.14), this implies that A ∈ GL(LpN ()), whence p ()). T (bξ ) ∈ GL(HN 5.18. Proposition. Let be a Khvedelidze weight on Lp and let a ∈ GL∞ N ×N be locally p, -sectorial over C. In addition, suppose at least one of the following three conditions is satisfied: (a)

∞ a ∈ CN ×N + HN ×N ,

(b)

≡ 1,

(c)

N = 1.

Then Indp, T (a) = Ind2 T (a), where Indp, T (a) and Ind2 T (a) refer to the p 2 () and HN , respectively. index of T (a) as an operator on HN Remark. A matrix function which is locally p, -sectorial over C is necessarily 2 locally sectorial over C in the sense of Definition 3.1. Thus, T (a) ∈ Φ(HN ). Also recall that Ind2 T (a) = −ind {kλ a} (Corollary 4.30). 2 ∞ Proof. (a) Because T (a) ∈ Φ(HN ), it follows that a−1 ∈ CN ×N + HN ×N and so the argument of the proof of Theorem 2.94 can be applied.

(b) By the hypothesis, a is r-sectorial on each fiber Xτ (τ ∈ T), where r = max{p, q}. In a similar way as this was done in the proof of the implication (iii) =⇒ (iv) of Theorem 3.9, one can show that a = ϕs, where ϕ is in GCN ×N and s ∈ GL∞ N ×N is r-sectorial on T. Hence T (a) = T (ϕ)T (s) + K, where K p 2 ) and C∞ (HN ). Thus, is in both C∞ (HN

260


Indp T (a) = Indp T (ϕ) + Indp T (s) = Indp T (ϕ) (Theorem 5.16(b)) = Ind2 T (ϕ) (Theorem 2.94) = Ind2 T (ϕ) + Ind2 T (s) = Ind2 T (a).

(Corollary 4.2)

(c) A moment’s thought reveals that T (a) is homotopic to a Toeplitz operator T (a0 ) with piecewise constant symbol a0 through Toeplitz operators the symbols of which are locally p, -sectorial (and in particular 2, 1-sectorial) over C. So Indp, T (a) = Indp, T (a0 ) and Ind2 T (a) = Ind2 T (a0 ) and the assertion follows from the P C-theory (Proposition 5.39). 5.19. Remark. I. M. Spitkovsky turned our attention to the following fact: p if a ∈ L∞ N ×N , then T (a) ∈ Φ(HN ()) and Indp, T (a) = κ if and only if p T (aϕ) ∈ Φ(HN ) and Indp T (aϕ) = κ, where ϕ is a certain appropriately chosen function in P C (see 5.60–5.62). The results of Shneiberg [471] imply r ) for all r ∈ [p, 2]. that Indp T (aϕ) = Ind2 T (aϕ) whenever T (aϕ) is in Φ(HN This provides a possibility of computing Indp, T (a) in case assumption (b) of Proposition 5.18 is not satisfied. The attempt of applying a result like Theorem 4.28 leads to the following problem. Let Bp, be the Banach algebra of all (generalized) sequences p p ()) such that there exists an A ∈ L(HN ()) {Aλ }λ∈Λ of operators Aλ ∈ L(HN p q ∗ ∗ −1 with Aλ → A strongly on HN () and Aλ → A strongly on HN ( ) as λ → ∞. Let Ip, denote the closed two-sided ideal of Bp, consisting of the p ()) and Cλ L(HNp ()) → 0 sequences of the form {L+Cλ }, where L ∈ C∞ (HN as λ → ∞. Finally, let {Kλ }λ∈Λ be an approximate identity. Is it true that {T (kλ a)}+Ip, is in G(Bp, /Ip, ) whenever a ∈ GL∞ N ×N is locally p, -sectorial over C (or over QC)? 5.20. Theorem. Let u ∈ GL∞ N ×N be unitary-valued, and let and r be as in Theorem 5.16. ∞ (a) If distL∞ (u, HN ×N ) < sin(π/r), then the operator T (u) is leftN ×N p invertible on HN (). ∞ (b) If distL∞ (u, CN ×N + HN ×N ) < sin(π/r), then the operator T (u) is N ×N p left-Fredholm on HN ().

(c) The assertions (a) and (b) remain true if H ∞ is replaced by H ∞ and “left” is replaced by “right.” ∞ Proof. (a) Choose h ∈ HN ×N so that u − h < sin(π/r). Then

I − h∗ u = u∗ u − h∗ u ≤ u∗ − h∗ = u − h < sin

π . r

Thus, by Theorem 5.16(a), T (h∗ u) = T (h∗ )T (u) is invertible, which implies that T (u) is left-invertible.


261

∞ (b) There are an n ≥ 0 and h ∈ HN ×N such that uχn I − h < sin(π/r). Hence, by virtue of part (a), T (uχn I) = T (u)T (χn I) is left-invertible, and because T (χn I) is Fredholm, it follows that T (u) is left-Fredholm.

(c) Take adjoints.

We conclude with two theorems on scalar Toeplitz operators which can be viewed as H p -analogues of Theorem 2.85 and Corollary 2.22, respectively. 5.21. Theorem. Let be a Khvedelidze weight on Lp and let a ∈ L∞ be locally p, -sectorial over C + H ∞ . Then T (a) ∈ Φ(H p ()). Proof. Fix τ ∈ T and consider the C ∗ -algebra L∞ |Xτ ∼ = C(Xτ ). This algebra contains (C + H ∞ )|Xτ = H ∞ |Xτ as a closed subalgebra (see 2.81). It can be checked straightforwardly that each antisymmetric set for C + H ∞ that is contained in Xτ is an antisymmetric set for (C + H ∞ )|Xτ (as subalgebra of C(Xτ )) and that, conversely, each antisymmetric set for (C + H ∞ )|Xτ is an antisymmetric set for C +H ∞ . Consequently, the maximal antisymmetric sets for C + H ∞ which are contained in Xτ are just the maximal antisymmetric sets for (C + H ∞ )|Xτ . Suppose S is any maximal antisymmetric set for (C + H ∞ )|Xτ . If a is rτ -sectorial on S, then so also is ϕ := a/|a|, and it is readily seen that then distXτ (ϕ, C) < sin(π/rτ ). Now Theorem 1.22 (in the setting Y = Xτ and B = (C + H ∞ )|Xτ ) can be applied to see that distS (ϕ, H ∞ ) < sin(π/rτ ). Thus, there is an hτ ∈ H ∞ such that |ϕ(x)−hτ (x)| < sin(π/rτ ) for all x ∈ Xτ , and using Proposition 2.79 we conclude that there is an open neighborhood Uτ ⊂ T of τ such that |ϕ(t) − hτ (t)| < sin(π/rτ ) a.e. on Uτ . Let τ (t) := 1 if τ ∈ T \ {t1 , . . . , tn } and let τ (t) = |t − tj |µj if τ = tj . Assume Uτ is small enough, so that Lp (Uτ , ) = Lp (Uτ , τ ). Define bτ ∈ L∞ by bτ (t) = ϕ−1 (t)hτ (t) for t ∈ Uτ and bτ (t) = 1 for t ∈ T \ Uτ . Since |1 − bτ (t)| = |ϕ(t) − hτ (t)| < sin

π rτ

for

t ∈ Uτ ,

is rτ -sectorial on T, too. So T (b−1 bτ is rτ -sectorial on T, and hence b−1 τ τ ) p is in GL(H (τ )) by Theorem 5.16(b). The argument used in the proof of p Theorem 5.17 shows that T (b−1 τ ) is even in GL(H ()). ∞ Now choose any gτ ∈ GL so that gτ |Uτ = hτ |Uτ , and let fτ ∈ C be any function such that supp fτ ⊂ Uτ and fτ ≡ 1 in some open neighborhood of τ . Then −1 −1 −1 T (gτ−1 )T −1 (b−1 (bτ )T (ϕτ fτ ) + K1 τ )T (ϕτ )T (fτ ) = T (gτ )T −1 −1 = T (gτ )T −1 (b−1 τ )T (bτ hτ fτ ) + K1 −1 = T (gτ−1 )T −1 (b−1 τ )T (bτ )T (hτ )T (fτ ) + K2

= T (gτ−1 hτ )T (fτ ) + K2 = T (gτ−1 hτ fτ ) + K3 = T (fτ ) + K3 ,

262


where K1 , K2 , K3 ∈ C∞ (H p ()). The conclusion is that T (ϕτ ) + C∞ (H p ()) is Mπτ -invertible from the left in L(H p ())/C∞ (H p ()) (recall 1.30), with Mπτ defined similarly as in the proof of Theorem 2.68. Now it is an easy matter to apply Theorem 1.32 to obtain that T (ϕ) is leftFredholm on H p (). It can be shown analogously that T (ϕ) is right-Fredholm on H p (). Thus, T (ϕ) = T (a/|a|) is in Φ(H p ()), and Proposition 2.32 (for H p () in place of H p ) completes the proof. 5.22. Theorem (Krupnik). Let 1 < p < ∞ and 1/p + 1/q = 1, and let a ∈ L∞ . Then the following are equivalent: (i) T (a) ∈ GL(H p ) and T (a) ∈ GL(H q ); (ii) T (a) ∈ GL(H r ) for all r ∈ [p, q]; (iii) a = heu+iv , where h ∈ GH ∞ , u and v are real-valued functions in L , and v∞ < π/ max{p, q}. ∞

Proof. (ii) =⇒ (i). Trivial. (iii) =⇒ (ii). In view of Propositions 2.31 and 2.32 it suffices to show that T (eiv ) ∈ GL(H r ) for all r ∈ [p, q]. But eiv is obviously r-sectorial on T and so Theorem 5.16 gives the assertion. (i) =⇒ (iii). Without loss of generality assume p ≥ 2 and |a| = 1. Theorem 5.5 shows that a = a− a+ = b− b+ with p a− , b−1 − ∈ L− ;

q a−1 − , b− ∈ L− ;

q a+ , b−1 + ∈ L+ ;

p a−1 + , b+ ∈ L+

−1 −1 −1 q 1 1 and P ∈ L(Lp (|a−1 + |)) ∩ L(L (|b+ |)). Since b+ a+ = b− a− is in L+ ∩ L− = C −1 −1 and so equals a constant γ ( = 0, by 1.40(b)), we have b+ = γa+ , hence P is in L(Lq (|a−1 + |)), and thus, by 1.46, w+) y , |a−1 + |=e

w, y ∈ L∞ real-valued,

y∞
0, and let A and B denote the set of all points ξ in M (QC) such that distXξ (a, H ∞ ) ≥ ε and distXξ (b, H ∞ ) ≥ ε, respectively. The sets A and B are disjoint (by the hypothesis) and closed (due to the upper semi-continuity of the mapping M (QC) → R, ξ → distXξ (a, H ∞ )). Hence, there is a ϕ ∈ QC such that 0 ≤ ϕ ≤ 1, ϕ|A = 0, and ϕ|B = 1. Put ψ := 1 − ϕ. Then, by (2.18), T (ab) − T (a)T (b) = T (aϕb) − T (a)T (ϕ)T (b) + T (aψb) − T (a)T (ψ)T (b) ) + T (a)H(ψ)H()b) = H(aϕ)H()b) + H(a)H(ϕ)T ) (b) + H(a)H(ψb) = H(aϕ)H()b) + H(a)H(ψ))b) + K, where K ∈ C∞ (H p (w)), since H(ϕ) ) and H(ψ) are compact on H p (w) (note ∞ that ϕ, ) ψ ∈ C + H ). Because distXξ (aϕ, H ∞ ) < ε for all ξ ∈ M (QC), it follows from Theorem 1.22 (with B = C + H ∞ ) that distL∞ (aϕ, C + H ∞ ) < ε. As in the proof of Theorem 2.54 one can see that there is a constant cp,w such that H(f )Φ(H p (w)) ≤ cp,w dist(f, C + H ∞ ) ∀ f ∈ L∞ . Consequently, H(aϕ)Φ(H p (w)) ≤ εcp,w . It can be shown analogously that H(ψ))b)Φ(H p (w)) ≤ εcp,w . Thus H(aϕ)H()b) + H(a)H(ψ))b)Φ(H p (w)) ≤ εcp,w cp,w (b∞ + a∞ ), where cp,w is from the estimate H(f )L(H p (w)) ≤ cp,w f ∞ . As ε > 0 can be chosen arbitrarily small, it follows that T (ab) − T (a)T (b) − K is compact. 5.33. Corollary. If a, b ∈ P C have no common points of discontinuity on T, then T (ab) − T (a)T (b) is compact on H p (w). Proof. If τ ∈ T, then either a|Xτ ∈ C|Xτ = C|Xτ or b|Xτ ∈ C|Xτ = C|Xτ , so that the previous theorem applies. 5.34. Theorem. The algebra alg Tp,w (P QC) is commutative. Proof. It suffices to prove that T (aϕ)T (bψ) − T (bψ)T (aϕ) ∈ C∞ (H p (w)) whenever a, b ∈ P C0 and ϕ, ψ ∈ QC. Because T (aϕ)T (bψ) − T (ϕ)T (ψ)T (a)T (b),

T (bψ)T (aϕ) − T (ϕ)T (ψ)T (b)T (a)

are compact, it remains to show that T (a)T (b) − T (b)T (a) ∈ C∞ (H p (w))

5.5 P C Symbols

269

for every a, b ∈ P C0 . We may clearly assume that a and b have at most one discontinuity, and in view of the preceding corollary it can be assumed that a and b have the jump at the same point of T. Then a = λb + c with λ ∈ C and c ∈ C, and hence T (a)T (b) − T (b)T (a) = λ[T (c)T (b) − T (b)T (c)] ∈ C∞ (H p (w)).

5.35. Definitions. Henceforth the argument arg z of a number z ∈ C \ {0} will be always chosen so that arg z ∈ (−π, π]. For β ∈ C and τ ∈ T, define ϕβ,τ ∈ P C0 as ϕβ,τ (t) := exp{ i β arg(−t/τ )}

(t ∈ T).

The dependence of ϕβ,τ on τ will be usually suppressed, that is, we shall write ϕβ in place of ϕβ,τ . It is readily seen that ϕβ has at most one discontinuity, namely a jump at τ , and that ϕβ (τ + 0) = e−πiβ and ϕ(τ − 0) = eπiβ . Let a ∈ P C0 and denote the points of discontinuity of a by t1 , . . . , tm . If a(tj ± 0) = 0 for all j = 1, . . . , m, then there are βj ∈ C such that a(tj − 0) = exp(2πiβj ) a(tj + 0) and thus a = bϕβ1 ,t1 . . . ϕβm ,tm

(5.19)

with some continuous function b ∈ C. Next, for t ∈ T \ {τ }, define + τ τ ++ τ β + ξβ (t) := ξβ,τ (t) := 1 − , := exp β log +1 − + + iβ arg 1 − t t t + t t ++ t β + ηβ (t) := ηβ,τ (t) := 1 − := exp β log +1 − + + iβ arg 1 − . τ τ τ The following basic identity can be verified straightforwardly: ϕβ (t) = ξ−β (t)ηβ (t) ∀ t ∈ T \ {τ }.

(5.20)

Note that ξβ (resp. ηβ ) is the limit on the unit circle T of that branch of the function (1 − τ /z)β (resp. (1 − z/τ )β ) which is analytic for |z| > 1 (resp. |z| < 1) and takes the value 1 at z = ∞ (resp. z = 0). Also notice that obviously ξα (t)ξβ (t) = ξα+β (t),

ηα (t)ηβ (t) = ηα+β (t) ∀ t ∈ T \ {τ }.

We have for t ∈ T \ {t0 }, + t0 t0 ++ + |ξβ,t0 (t)| = exp Re β log +1 − + − i Im β arg 1 − = |t − t0 |Re β b(t), t t

270


where b ∈ GL∞ , and therefore, if (t) = weight, ξβ ∈ Lp () ⇐⇒ ηβ ∈ Lp () ⇐⇒ −

5n j=0

|t − tj |µj is a Khvedelidze

1 1 < Re β + µ0 < . p q

(5.21)

Clearly, ξβ ∈ Lp () ⇐⇒ ξβ ∈ Lp− (), ηβ ∈ Lp () ⇐⇒ ηβ ∈ Lp+ (). (5.22) 5n 5.36. Lemma. Let (t) = j=0 |t − tj |µj be a Khvedelidze weight and let β ∈ C. Then the following are equivalent: (i) T (ϕβ,t0 ) ∈ Φ(H p ()) and Ind T (ϕβ,t0 ) = −κ; (ii) κ − 1/q < Re β − µ0 < κ + 1/p; (iii) 0 ∈ / Ar (ϕβ (t0 − 0), ϕβ (t0 + 0)), where r = (1/p + µ0 )−1 , and the index of the closed continuous and naturally oriented curve obtained from the range of ϕβ by filling in the arc Ar (ϕβ (t0 − 0), ϕβ (t0 + 0)) is equal to κ. Proof. (ii) ⇐⇒ (iii). Straightforward. (ii) =⇒ (i). Put γ = β − κ. Then −1/q < Re γ − µ0 < 1/p. Hence, by (5.21), (5.22), ξ−γ ∈ Lp− (),

−1 ξ−γ ∈ Lq− (−1 ),

ηγ ∈ Lq+ (−1 ),

ηγ−1 ∈ Lp+ (),

and since |ηγ−1 | is also a Khvedelidze weight, it follows from Theorem 5.9 that P ∈ L(Lp (|ηγ−1 |)). Thus, by (5.20) and Theorem 5.5, T (ϕγ ) = T (ξ−γ ηγ ) is in GL(H p ()). Because T (ϕβ ) equals T (ϕγ )T (χκ ) or T (χκ )T (ϕγ ), we conclude that T (ϕβ ) ∈ Φ(H p ()) and Ind T (ϕβ ) = Ind T (χκ ) = −κ. (i) =⇒ (ii). There is a k ∈ Z such that k − 1/q < Re β − µ0 ≤ k + 1/p. If Re β − µ0 < k + 1/p, then k = κ by what has just been proved and we are done. So assume Re β − µ0 = k + 1/p. The hypothesis (i) implies that T (ϕβ1 ) and T (ϕβ2 ) are Fredholm of index κ whenever β1 , β2 ∈ C are sufficiently close to β. But if we let k−

1 1 1 < Re β1 − µ0 < k + < Re β2 − µ0 < k + 1 + , q p p

then, again, by what has already been proved, Ind T (ϕβ1 ) = k, Ind T (ϕβ2 ) = k + 1, which is a contradiction. 5.37. Definitions. Let 1 < p < ∞ and let be a Khvedelidze weight of the form (5.6). For a ∈ P CN ×N , define ap, : T × [0, 1] → CN ×N by ap, (t, µ) := 1 − σ(t, µ) a(t − 0) + σ(t, µ)a(t + 0), (t, µ) ∈ T × [0, 1],

5.5 P C Symbols

271

where σ(t, µ) := σp (µ) for t ∈ T \ {t1 , . . . , tn } and σ(t, µ) := σ(1/p+µj )−1 (µ) for t = tj , and σr (µ) is defined as in 5.12. The range of det(ap, ) is a continuous closed and naturally oriented curve. If N = 1, it is obtained from the (essential) range of a by filling in the arcs Ar(τ ) (a(τ − 0), a(τ + 0)) for each τ ∈ T at which a has a jump; here r(τ ) := p for τ ∈ T \ {t1 , . . . , tn } and r(τ ) := (1/p + µj )−1 for τ = tj . If the curve does not pass through the origin, its winding number with respect to the origin will be denoted by ind det(ap, ). Note that, in general, det(ap, ) = (det a)p, and ind det(ap, ) = ind (det a)p, . For a ∈ (P C0 )N ×N , we have ind det(ap, ) = ind (det(ap, ) ◦ ωS(det a) ), where ωS(det a) is defined as in 2.73 and the latter “ind ” refers to the index as it was defined in 2.41. Finally, if a ∈ P CN ×N and det(ap, )(t, µ) = 0 for all (t, µ) ∈ T × [0, 1], then ind det(ap, ) = lim ind det(anp, ), where {an } n→∞

is any sequence of functions in (P C0 )N ×N such that anp, (t, µ) = 0 for all → 0 as n → ∞. (t, µ) ∈ T × [0, 1] and a − an L∞ N ×N 5.38. Lemma. Let a, b ∈ P C0 have no common points of discontinuity on T and suppose ap, and bp, do not vanish on T × [0, 1]. Then (ab)p, = ap, bp, and ind (ab)p, = ind ap, + ind bp, . Proof. The equality (ab)p, = ap, bp, is obvious and the index formula can be shown as follows: ind (ab)p, = ind (ab)p, ◦ ωS(ab) 0 1 = ind (ap, bp, ) ◦ ωS(ab) = ind ap, ◦ ωS(ab) bp, ◦ ωS(ab) = ind ap, ◦ ωS(ab) + ind bp, ◦ ωS(ab) = ind ap, ◦ ωS(a) + ind bp, ◦ ωS(a) = ind ap, + ind bp, . 5.39. Proposition. Let a ∈ P C0 and let be a Khvedelidze weight. Then T (a) ∈ Φ(H p ()) ⇐⇒ ap, (t, µ) = 0 ∀ (t, µ) ∈ T × [0, 1]. If T (a) ∈ Φ(H p ()), then Ind T (a) = −ind ap, . Proof. Suppose ap, does not vanish on T × [0, 1]. Then a can be written in the form (5.19). In view of Corollary 5.33 and Lemma 5.38 we have T (a) − T (ϕβ1 ) . . . T (ϕβm )T (b) ∈ C∞ (H p ()),

(5.23)

ap, = (ϕβ1 )p, . . . (ϕβm )p, bp, ,

(5.24)

ind ap, = ind b +

m

ind (ϕβj )p, .

(5.25)

j=1

From (5.24) we deduce that (ϕβj )p, (t, µ) = 0 for all (t, µ) ∈ T × [0, 1]. Thus, by Lemma 5.36, T (ϕβj ) ∈ Φ(H p ()) and Ind T (ϕβj ) = −ind (ϕβj )p, . Theorem 5.31(b) implies that T (b) ∈ Φ(H p ()) and that Ind T (b) = −ind b. So

272


(5.23) shows that T (a) ∈ Φ(H p ()) and Atkinson’s theorem combined with (5.25) gives the index formula Ind T (a) = −ind ap, . Once the index formula has been proved, the usual perturbation argument (see the proof of Theorem 2.74) shows that ap, (t, µ) = 0 for all (t, µ) in T × [0, 1] if T (a) ∈ Φ(H p ()). 5.40. Proposition. Let be a Khvedelidze weight, let a ∈ P C, let τ ∈ T, and define r(τ ) as in 5.37. Then τ τ τ = spP C Tp, (a) = spL∞ Tp, (a) sp Tp, (a) + Zp, = sp τp, T (a) = Ar(τ ) a(τ − 0), a(τ + 0) . Proof. Let A and B denote the arcs (τ e−iπ/2 , τ ) and (τ, τ eiπ/2 ), respectively. Choose fτ , gτ ∈ P C0 so that fτ |A = gτ |A = 0, fτ |B = gτ |B = 1, fτ and gτ are continuous on T \ {τ }, R(fτ ) ⊂ Ar(τ ) (0, 1), and R(gτ ) ⊂ As (0, 1), where s = r(τ ). The preceding proposition gives that spΦ(H p ()) T (fτ ) = Ar(τ ) (0, 1),

spΦ(H p ()) T (gτ ) = Ar(τ ) (0, 1) ∪ As (0, 1).

From 1.16(b) we deduce that spalg Tp, (P C) Tp, (fτ ) = spalg Tp, (L∞ ) Tp, (fτ ) = Ar(τ ) (0, 1). Since each of the “local” spectra is obviously contained in the corresponding τ ), “global” spectrum, it follows that each of the spectra sp(Tp, (fτ ) + Zp, τ τ τ spP C Tp, (fτ ), spL∞ Tp, (fτ ), sp p, T (fτ ) is a subset of Ar(τ ) (0, 1). τ (fτ ) contains Ar(τ ) (0, 1). Theorem 5.29(b) imLet us show that spP C Tp, plies that / t spΦ(H p ()) T (gτ ) ⊂ spalg Tp, (P C) T (gτ ) = spP C Tp, (gτ ) =

/

t∈T

t τ spP C Tp, (gτ ) ∪ spP C Tp, (gτ ).

(5.26)

t =τ

Since gτ is continuous on T \ {τ }, the first union in (5.26) equals As (0, 1) τ (gτ ), and (Corollary 5.26). Hence Ar(τ ) (0, 1) must be contained in spP C Tp, τ τ because Tp, (gτ ) = Tp, (fτ ) (Proposition 5.25), it results that Ar(τ ) (0, 1) ⊂ τ (fτ ). spP C Tp, τ (fτ ), It can be shown analogously that Ar(τ ) (0, 1) is contained in spL∞ Tp, τ τ sp(Tp, (fτ ) + Zp, ), sp p, T (fτ ). Thus, for a = fτ the proposition is proved. If a is any function in P C, then a = cfτ + g with fτ as above, c ∈ C, and g ∈ P C continuous at τ . Therefore, by Proposition 5.25 and Corollary 5.26, τ τ (a) = c spP C Tp, (fτ ) + g(τ ) spP C Tp, = cAr(τ ) (0, 1) + g(τ ) = Ar(τ ) (a(τ − 0), a(τ + 0)),

and equally for the other three spectra.

5.5 P C Symbols

273

5.41. Proposition. Let be a Khvedelidze weight, let bjk be finitely many m 5n functions in P C, and put A = j=1 k=1 T (bjk ). Then if τ ∈ T, τ ) = spP C Aτp, = spL∞ Aτp, sp (Ap, + Zp, = (bjk )p, (τ, λ) : λ ∈ [0, 1] . j

k

Proof. Each bjk can be written in the form bjk = cjk χτ +gjk , where cjk ∈ C, χτ is the characteristic function of the arc (τ, τ eiπ/2 ), and gjk ∈ P C is continuous at τ . Hence, by Proposition 5.25 and Corollary 5.26, τ τ cjk Tp, (χτ ) + gjk (τ ) + Zp, sp (Ap, + Zp, ) = sp =

j

j

k

τ cjk sp Tp, (χτ ) + Zp, + gjk (τ )

k

(here we applied the spectral mapping theorem) cjk Ar(τ ) (0, 1) + gjk (τ ) (Proposition 5.40) = j

k

cjk σr(τ ) (λ) + gjk (τ ) : λ ∈ [0, 1] = = =

j

k

j

k

j

k

0

1 1 − σr(τ ) (λ) bjk (τ − 0) + σr(τ ) (λ)bjk (τ + 0) : λ ∈ [0, 1]

(bjk )p, (τ, λ) : λ ∈ [0, 1] .

The same argument applies to spP C Aτp, and spL∞ Aτp, . 5.42. Definition. For A ∈ algp,w T (P CN ×N ), let det A ∈ algp,w T (P C) denote the determinant defined by (1.6). Since alg Tp,w (P C) is commutative (Theorem 5.34), any determinant of A resulting by reordering the factors in the terms of the sum (1.6) differs from that one only by a compact operator. 5.43. Theorem. Let be a Khvedelidze weight on Lp . Let A=

r s

T (ajk ),

ajk ∈ P CN ×N

j=1 k=1

and det A =

m n

T (bjk ),

bjk ∈ P C.

j=1 k=1 p ()) if and only if Then A ∈ Φ(HN m n

(bjk )p, (t, λ) = 0 ∀ (t, λ) ∈ T × [0, 1].

j=1 k=1

274


p Proof. Theorems 5.34 and 1.14(c) show that A ∈ Φ(HN ()) if and only if p det A ∈ Φ(H ()). So it remains to apply Theorem 5.29(a) and Proposition 5.41.

5.44. Proposition. Let be a Khvedelidze weight on Lp and let τ ∈ T. τ τ (a) The algebra alg Tp, (P C) is singly generated by Tp, (χτ ), where χτ is iπ/2 the characteristic function of the arc (τ, τ e ). τ (b) The maximal ideal space M (alg Tp, (P C)) is homeomorphic to the segment [0, 1] (equipped with the topology inherited from the Euclidean R) and the τ (P C) → C([0, 1]) is for a ∈ P C given by Gelfand map Γ : alg Tp, τ Γ Tp, (a) (λ) = 1 − σr(τ ) (λ) a(τ − 0) + σr(τ ) (λ)a(τ + 0),

where r(τ ) and σr(τ ) are defined as in 5.37 and 5.12. Proof. (a) If a ∈ P C, then a = cχτ + g, where c ∈ C and g ∈ P C is continuous τ τ (a) = cTp, (χτ )+g(τ ), at τ . Hence, by Proposition 5.25 and Corollary 5.26, Tp, τ τ and it follows that alg Tp, (P C) is generated by Tp, (χτ ). τ (b) From Proposition 5.40 we know that spP C Tp, (χτ ) = Ar(τ ) (0, 1) and τ (P C)) is homeomorphic to Ar(τ ) (0, 1) (see 1.19) and thus therefore M (alg Tp, to [0, 1]. For λ ∈ [0, 1] let λ denote the multiplicative linear functional on τ τ (P C) which sends Tp, (χτ ) into (1 − σr(τ ) (λ)) · 0 + σr(τ ) (λ) · 1. Since alg Tp, every a ∈ P C can be written in the form cχτ + g as in (a), we have τ τ Γ Tp, (a) (λ) = λ(Tp, (a)) = 1 − σr(τ ) (λ) a(τ − 0) + σr(τ ) (λ)a(τ + 0)

for every a ∈ P C.

5.45. Theorem. Suppose is a Khvedelidze weight on Lp . Then the maximal ideal space M (alg Tp, (P C)) is homeomorphic to the cylinder T × [0, 1] equipped with an exotic topology. For a function a ∈ P C the Gelfand map Γ : alg Tp, (P C) → C(T × [0, 1]) is given by Γ Tp, (a) (τ, λ) = ap, (τ, λ). (5.27) Proof. Let πτ denote the canonical homomorphism of the algebra alg Tp, (P C) τ (P C). onto the algebra alg Tp, τ (P C), then vτ ◦ πτ is If vτ is a multiplicative linear functional on alg Tp, clearly in M (alg Tp, (P C)). Thus, by the preceding proposition, T × [0, 1] can be identified with a subset of M (algp, (P C)) and (5.27) holds. Now let v ∈ M (alg Tp, (P C)). Since alg Tp, (C) is a closed subalgebra of alg Tp, (P C) and M (alg Tp, (C)) = T (Theorem 5.31), there must exist a τ ∈ T such that v(Tp, (f )) = f (τ ) for all f ∈ C. This implies that τ ) = {0}. Consequently, there exists a multiplicative linear functional vτ v(Jp, τ τ on alg Tp, (P C) = alg Tp, (P C)/Jp, such that v = vτ ◦ πτ . The conclusion is that T × [0, 1] equals M (alg Tp, (P C)).

5.5 P C Symbols

275

Remark. The Gelfand topology on M (alg Tp, (P C)) = T × [0, 1] coincides with the topology on T × [0, 1] described in 4.88. 5.46. Theorem. Let be a Khvedelidze weight on Lp . Then the Shilov boundary of the maximal ideal space M (alg Tp, (P C)) coincides with the whole maximal ideal space M (alg Tp, (P C)). Proof. According to 1.20(b) we must show that for each (τ0 , λ0 ) ∈ T × [0, 1] and each open neighborhood U ⊂ T × [0, 1] of (τ0 , λ0 ) there exists an operator A ∈ algp, T (P C) such that sup |(Γ Ap, )(t, λ)| > sup |(Γ Ap, )(t, λ)|. (t,λ)∈U

(5.28)

(t,λ)∈U /

First suppose λ0 ∈ (0, 1). Then U can be assumed to be of the form U = {(τ0 , λ) : |λ − λ0 | < ε}, where ε < min{λ0 , 1 − λ0 }. Recall that r(τ0 ) := p / {t1 , . . . , tn } and r(τ0 ) := (1/p + µj )−1 for τ0 = tj . In either case for τ0 ∈ there is a ν ∈ (−1/2, 1/2) satisfying (1/2 + ν)−1 = r(τ0 ). Let 0 denote the Khvedelidze weight |t − τ0 |ν . Since alg T2,0 (P C) is a C ∗ -algebra, there is an 2, 5 A ∈ alg2,0 T (P C) which satisfies (5.28) with

0 in place of p, . We may clearly assume that A is a finite product-sum j k T (ajk ) with ajk ∈ P C0 and thus that A is in algp, T (P C). Because for (τ0 , λ) ∈ U both (Γ Ap, )(τ0 , λ) and (Γ A2,0 )(τ0 , λ) are equal to 1 − σr(τ0 ) (λ) ajk (τ0 − 0) + σr(τ0 ) (λ)ajk (τ0 + 0) , j

k

it follows that A fulfills (5.28). Now suppose λ0 = 0. Choose a z0 ∈ D so that Ar(τ0 ) (1, z0 ) ∩ T = {1} and let a ∈ P C0 be any function with the following properties: a(τ0 − 0) = 1, a(τ0 + 0) = z0 , a is continuous on T \ {τ0 }, and |a(t)| < 1 for t = τ0 . It is easily seen that for each neighborhood U of (τ0 , 0) the inequality + + + + sup + Γ Tp, (a) (t, λ)+ = 1 > sup + Γ Tp, (a) (t, λ)+ (t,λ)∈U

(t,λ)∈U /

holds. The situation is analogous for λ0 = 1. 5.47. Theorem. Let be a Khvedelidze weight on Lp and let A belong to algp, T (P CN ×N ). Then p A ∈ Φ(HN ()) ⇐⇒ Γ (det A)p, (t, λ) = 0 ∀ (t, λ) ∈ T × [0, 1]. Proof. In view of Theorems 5.34 and 1.14(c) we have p ()) ⇐⇒ det A ∈ Φ(H p ()). A ∈ Φ(HN

So the implication “⇐=” of the theorem is immediate from Theorem 5.45. To get the opposite implication assume (Γ (det A)p, )(t0 , λ0 ) = 0 for some


276

(t0 , λ0 ) ∈ T × [0, 1]. Then Theorem 5.46 in conjunction with 1.20(c) gives that (det A)p, is a topological divisor of zero in alg Tp, (P C) and thus in the Calkin algebra L(H p ())/C∞ (H p ()). Consequently, det A cannot be in Φ(H p ()). 5.48. Index computation. (a) Proposition 5.39 says that Ind T (a) equals −ind ap, whenever a ∈ P C0 and T (a) ∈ Φ(H p ()). The same is true for a ∈ P C. Indeed, if a ∈ P C and T (a) ∈ Φ(H p ()), then ap, does not vanish on T × [0, 1] (Theorem 5.43), so there is an ε > 0 such that T (b) ∈ Φ(H p ()), Ind T (b) = Ind T (a), and ind bp, = ind ap, whenever b ∈ P C0 , a − b∞ < ε (the equality Ind T (b) = Ind T (a) is a consequence of 1.12(d)), and since it is already known that Ind T (b) = −ind bp, , it follows that Ind T (a) = −ind ap, . p ()), then Ind T (a) = −ind det(ap, ) (b) If a ∈ (P C0 )N ×N , T (a) ∈ Φ(HN (note that det(ap, ) = 0 on T × [0, 1] by Theorem 5.43). To see this recall first the following well known fact:

If a ∈ (P C0 )N ×N ∩ GL∞ N ×N , then a can be written in the form a = ϕbψ, where ϕ and ψ are in GCN ×N and b ∈ (P C0 )N ×N ∩ GL∞ N ×N is an uppertriangular matrix function. A proof is, e.g., in Clancey, Gohberg [138, Chap. VIII, Lemma 2.2]. Now let a ∈ (P C0 )N ×N and suppose det(ap, ) = 0 on T × [0, 1]. Write a in the form a = ϕbψ as above and note that ap, = ϕbp, ψ. Hence det(bp, ) and thus bjp, (b1 , . . . , bN are the diagonal entries of b) do not vanish on T × [0, 1]. It follows that T (b) and T (bj ) are Fredholm, and since Ind T (ϕ) = −ind det ϕ,

Ind T (ψ) = −ind det ψ,

and Ind T (b) =

N

Ind T (b ) = − j

j=1

N

ind bjp,

j=1

= −ind

N

bjp,

(because ind fp, + ind gp, = ind fp, gp, )

j=1

= −ind det(bp, ), we have Ind T (a) = Ind (T (ϕ)T (b)T (ψ) + compact operator) = Ind T (ϕ) + Ind T (b) + Ind T (ψ) (Atkinson) = −ind (det ϕ · det(bp, ) · det ψ) = −ind det(ϕbp, ψ) = −ind det(ap, ). (c) The same argument as in (a) shows that Ind T (a) = −ind det(ap, ) p ()). whenever a ∈ P CN ×N and T (a) ∈ Φ(HN

5.6 P2 C Symbols

277

r 5 s

m 5n (d) Now let A = i=1 k=1 T (ajk ) and det A = i=1 k=1 T (bjk ) with ajk ∈ P CN ×N and bjk ∈ P C. One can show that there exists a c ∈ P CM ×M , where M = N (mn + n + 1), such that p bjk = det c, IndHNp () A = IndHM () T (c) j

k

(see Krupnik [329, Theorems 1.7 and 2.4]). This and (c) imply that Ind A = −ind det(cp, ) = −ind (bjk )p, . j

k

(e) Finally, if A is any operator in algp, T (P CN ×N ) which is Fredholm p () and An ∈ algp, T (P CN ×N ) are any finite product-sums such that on HN p ()) for all n large enough A−An L(HNp ()) → 0, then, by 1.12(d), An ∈ Φ(HN and Ind A = −ind Γ (det A)p, := − lim ind Γ (det An )p, . n→∞

5.6 P2 C Symbols 5.49. Definitions. Let a ∈ (P2 C)N,N . Denote by Ψ (a) the set of all points τ ∈ T such that a|Xτ = b|Xτ for some b ∈ P CN ×N and set a(τ, 0) := b(τ − 0), a(τ, 1) := b(τ + 0) in that case. Put Ψ c (a) = T \ Ψ (a), and for τ ∈ Ψ c (a) let a(τ, 0) and a(τ, 1) simply denote the two elements of the set a(Xτ ). Given α, β ∈ CN ×N define 0 1 AN r (α, β) := λ ∈ C : det (1 − σr (µ))α + σr (µ)β − λI = 0 ∀ µ ∈ [0, 1] , where σr is as in 5.12. For 1 < p < ∞ put / OpN (α, β) := AN r (α, β) r∈[p,q]

1 p

+

1 =1 . q

N In the case N = 1 the sets AN r (α, β) and Op (α, β) are nothing else than the arcs and lentiform domains introduced in 5.12. Finally if ≡ 1 we denote spτp, T (a) by spτp T (a).

5.50. Theorem (Spitkovsky). If a ∈ (P2 C)N,N , then , N Ap (a(τ, 0), a(τ, 1)) for τ ∈ Ψ (a), τ spp T (a) = OpN (a(τ, 0), a(τ, 1)) for τ ∈ Ψ c (a).

(5.29)

Proof. A proof of this theorem is in Spitkovsky [503]. We confine ourselves to the proof of the above theorem for a special but sufficiently large class of scalar-valued P2 C functions. Note that the inclusion spτp T (a) ⊂ Op (a(τ, 0), a(τ, 1)) (scalar case!) immediately results from Corollary 5.28.

278


5.51. Definition. For p > 0 let H p denote the set of all functions f which are analytic in D and satisfy ( 2π |f (reiθ )|p dθ < ∞. sup r∈(0,1)

0

For 1 ≤ p < ∞ this definition agrees with Definition 1.39. If f ∈ H p (p > 0), then the nontangential limits f (eiθ ) exists almost everywhere and define a * 2π function in Lp , that is, 0 |f (eiθ )|p dθ < ∞ (see Duren [178, Theorem 2.2]). Therefore functions in H p may be identified with their boundary values on T. 5.52. Lemma. Let f ∈ H 1/2 and suppose that f (z) = 0 for all z ∈ D and that f is real-valued and nonnegative on T. Then f is constant in D. Proof. Since f does not vanish in D, there is a function g which is analytic in D and satisfies g 2 = f . Because g is in H 1 and is real-valued on T, it follows that g is constant in D. Hence f is constant in D, too. Note that every function a ∈ P2 C can be written in the form a = αχE + βχE c , where α, β ∈ C, E is a measurable subset of T, and E c := T \ E. 5.53. Proposition. If a = αχE + βχE c ∈ P2 C \ C, then spL(H p ) T (a) = Op (α, β). Proof. Theorem 5.16(b) shows that spL(H p ) T (a) ⊂ Op (α, β). So let us prove the reverse inclusion. Without loss of generality suppose that 1 < p < 2. Assume 0 ∈ Op (α, β) and T (a) ∈ GL(H p ). Then 0 ∈ Op (α/|α|, β/|β|) and T (a/|a|) ∈ GL(H p ), and therefore it can be assumed that α = 1 and β = eiθ , where 2π/q ≤ θ < π. From Theorem 5.5 we deduce that a = bc−1 , where b ∈ H q , b−1 ∈ H p and c ∈ H q , c−1 ∈ H p (note that actually b and c are outer: if b = hω with h ∈ H q outer and ω inner then ω = hb−1 ∈ H 1 , whence ω = const). Because bc = bc−1 |c|2 , the argument of bc takes only the values 2kπ and θ+2kπ (k ∈ Z) on T. Let v be the real-valued function on T given by 0 ≤ v ≤ θ,

v ≡ arg(bc) mod 2π,

let v) be the conjugate function of v, and put ϕ := e−)v+i v . It is clear that ϕ(z) = 0 for z ∈ D, and Theorem V, D, 1◦ in Koosis [316] implies that ϕ and ϕ−1 are in H r for all r < π/θ. Put ψ := (bc)ϕ−1 . Obviously, ψ(z) = 0 for z ∈ D. By the choice of v, ψ is real-valued and nonnegative on T. Since b, c ∈ H q ⊂ H 2π/θ , we have bc ∈ H π/θ ⊂ H 1 . Finally, because ϕ−1 ∈ H 1 , it follows that ψ ∈ H 1/2 . So Lemma 5.52 shows that ψ is some positive constant, ψ(0) say, in D. Put := 2π/θ. We have bc = ϕψ and eiv = 1 a.e. on T, hence b(t)c(t) = ψ(0) ϕ(t) = ψ(0) e−)v(t) a.e. on T.

5.6 P2 C Symbols

279

Thus, (bc) is nonnegative on T. Clearly, (b(z)c(z)) = 0 for z ∈ D, and since bc ∈ H π/θ , the function (bc) belongs to H 1/2 . Therefore again Lemma 5.52 can be applied to deduce that (bc) is constant in D, and hence bc itself must be constant in D. The conclusion is that the argument of bc and thus the argument of bc−1 is constant on T, which contradicts our assumption that a be not constant. 5.54. Definition. A function a = αχE +βχE c ∈ P2 C will be said to be regular if for each open subarc U of T at least one of the sets E ∩ U and E c ∩ U has a nonempty interior. For instance, if both E and E c are (possibly countable) unions of subarcs of T, then a is regular. Also notice that a is regular if E or E c is a Cantor set. A function b ∈ P2 C will be called locally regular if for each τ ∈ T there exists a regular function aτ ∈ P2 C such that b|Xτ = aτ |Xτ . p 5.55. Lemma. Let 1 < p < 2, a ∈ L∞ N ×N , and suppose T (a) ∈ Φ(HN ) and 2 T (a) ∈ Φ(HN ). Then Indp T (a) ≥ Ind2 T (a). p q 2 Proof. Since HN ⊃ HN ⊃ HN , we conclude that αp (T (a)) ≥ α2 (T (a)) and α2 (T (a∗ )) ≥ αq (T (a∗ )), which gives the assertion at once.

5.56. Proposition. Let a = αχE + βχE c be a regular function in P2 C and suppose Ψ c (a) is not empty. Then spΦ(H p ) T (a) = Op (α, β). Proof. From Proposition 5.53 we deduce that spΦ(H p ) T (a) ⊂ Op (α, β). To prove the opposite inclusion suppose 1 < p < 2, 0 ∈ Op (α, β) \ [α, β] and / [α, β], we have T (a) ∈ GL(H 2 ). The preceding T (a) ∈ Φ(H p ). Since 0 ∈ lemma therefore shows that Ind T (a) = m ≥ 0. Let τ ∈ Ψ c (a). In view of the hypothesis that a be regular we may assume that there are points t1 , t2 , . . . ∈ T such that tn ≺ tn+1 for all n, tn → τ as n → ∞, a is identically α on the arcs (t2k , t2k+1 ) and takes the value β on a subset of positive measure of each of the arcs (t2k+1 , t2k+2 ). Define b1 ∈ P2 C / (t3 , t4 ). Put q := a/b1 . by b1 (t) = α for t ∈ (t3 , t4 ) and b1 (t) = a(t) for t ∈ Due to Theorem 5.32 we have T (a) = T (b1 )T (q) + K = T (q)T (b1 ) + L with K, L ∈ C∞ (H p ). Therefore T (b1 ) and T (q) belong to Φ(H p ) and Ind T (b1 ) = Ind T (a) − Ind T (q). Lemma 5.55 together with Proposition 5.53 gives that Ind T (q) ≥ 1, and thus Ind T (b1 ) ≤ m − 1. On repeating this construction we finally arrive at a function bm+1 ∈ P2 C \ C whose essential range is the set {α, β} and which has the property that T (bm+1 ) ∈ Φ(H p ) and Ind T (bm+1 ) < 0. By Lemma 5.55 this is impossible. Thus Op (α, β) \ [α, β] is a subset of spΦ(H p ) T (a), and therefore spΦ(H p ) T (a) = Op (α, β). 5.57. Theorem Let a ∈ P2 C be locally regular. Then (5.29) holds.

280


Proof. For τ ∈ Ψ (a) this follows from Propositions 5.25 and 5.40. So let τ ∈ Ψ c (a) and let b ∈ L∞ be any function satisfying b|Xτ = a|Xτ . We must show that Op (a(τ, 0), a(τ, 1)) is a subset of spΦ(H p ) T (b). This shows that Op (a(τ, 0), a(τ, 1)) is contained in sp τp T (a), and as the reverse inclusion results from Corollary 5.28, the assertion of the theorem follows. Thus, assume T (b) ∈ Φ(H p ) but let the origin lie in the interior of Op (a(τ, 0), a(τ + 0)). Choose a regular function c ∈ P2 C so that c|Xτ = τ ) is invertible (Propoa|Xτ = b|Xτ . Then Tp (c) + Zpτ (:= Tp,1 (c) + Zp,1 sition 5.25 and Theorem 5.29(a)). From Theorem 1.32(c) we deduce that Tp (c) + Zpτ is invertible for all t in some open arc U := (τ − δ, τ + δ). Because c is regular, there are open arcs V := (τ − δ1 , τ − δ2 ) ⊂ (τ − δ, τ ) and W := (τ + δ3 , τ + δ4 ) ⊂ (τ, τ + δ) such that c|V and c|W are constant. Define d ∈ P2 C by d(t) = c|V for t ∈ (−τ, τ − δ2 ), d(t) = c|W for t ∈ (τ + δ3 , −τ ), d(t) = c(t) for t ∈ (τ − δ2 , τ + δ3 ). Then Tp (d) + Zpτ is invertible for all t ∈ T (recall that the origin has been supposed to be not on the boundary of Op (a(τ, 0), a(τ + 0))). Theorem 5.29(a) now implies that T (d) ∈ Φ(H p ), and therefore, by Proposition 5.56, 0∈ / Op (d(τ, 0), d(τ, 1)) = Op (a(τ, 0), a(τ, 1)). This contradicts our assumption that the origin be an inner point of the set Op (a(τ, 0), a(τ, 1)). Thus, all inner points of Op (a(τ, 0), a(τ + 0)) belong to spΦ(H p ) T (b) and hence this spectrum contains Op (a(τ, 0), a(τ, 1)). 5.58. Index computation. Let a ∈ P2 C and T (a) ∈ Φ(H p ). We claim that the set Ψ0 (a) := τ ∈ Ψ (a) : 0 ∈ Op (a(τ, 0), a(τ, 1)) is finite. Indeed, if τ ∈ Ψ0 (a), then τ ∈ Ψ (a) and therefore the (essential) values taken by a on the right (resp. left) of τ are close to a(τ, 1) (resp. a(τ, 0)) and thus close to each other. It follows that there is an open subarc U (τ ) of T containing τ such that U (τ ) \ {τ } ⊂ T \ Ψ0 (a). In other words, the points in Ψ0 (a) are isolated points of T. Theorem 5.50 implies that a is p-sectorial on / Ψ0 (a). So the “p-sectorial version” of Theorem 3.9 (see 5.12) Xt for each t ∈ implies that T \ Ψ0 (a) is open and thus that Ψ0 (a) is closed. It results that Ψ0 (a) is finite, as desired. We now construct b ∈ P2 C as follows. Choose an ε > 0 so that for each τ ∈ Ψ0 (a) the union of (τ − ε, τ ) and (τ, τ + ε) is a subset of T \ Ψ0 (a) and that the origin is in Op (τ1 , τ2 ) whenever τ1 ∈ (τ − ε, τ ), τ2 ∈ (τ, τ + ε), τ ∈ Ψ0 (a). Then let b ∈ P2 C be any function which is continuous on [τ − ε, τ + ε] and satisfies b(τ − ε) = a(τ, 0), b(τ + ε) = a(τ, 1), b [τ − ε, τ + ε] ⊂ Ap (a(τ, 0), a(τ, 1)) for each τ ∈ Ψ0 (a) and equals a on T \ [τ − ε, τ + ε]. Theorem 5.50 τ ∈ Ψ0 (a)

shows that b is locally p, 1-sectorial (i.e., is identically 1) over C, and thus

5.7 Fisher-Hartwig Symbols

281

Indp T (b) = Ind2 T (b) = −ind {kλ b} (Proposition 5.18) One can show that Indp T (a) = Indp T (b). Thus, Indp T (a) = −ind {kλ b}, a formula which is not very good but better than nothing. 5.59. Open problem. Establish an index formula for (block) Toeplitz operators on H p (or H p ()) with P QC symbols. In this connection (and in connection with 5.58, too) it would be interesting to know whether the harmonic extension hr a can be replaced by something, hr,p a say, so that hr,p a ∈ C and Ind T (a) = −ind {hr,p a} for every a ∈ P C (P2 C, P QC, P2 QC) generating a Fredholm operator T (a).

5.7 Fisher-Hartwig Symbols 5.60. Definitions. A Fisher-Hartwig symbol is a function of the form a(t) = b(t)

m

|t − tj |2αj ϕβj ,tj (t)

(t ∈ T),

(5.30)

j=1

where (i) b ∈ C and b(t) = 0 for t ∈ T; (ii) t1 , . . . , tm are pairwise distinct points on T; (iii) αj ∈ C and Re αj > −1/2 for j = 1, . . . , m; (iv) βj ∈ C for j = 1, . . . , m and ϕβj ,tj is given as in 5.35. Condition (iii) ensures that a is a function in L1 . Denote the function |t−tj |2αj by ωαj = ωαj ,tj . Then due to 5.35, ωαj = ξαj ηαj ,

ϕβj = ξ−βj ηβj ,

and therefore the function (5.30) can be written in the form a(t) = b(t)

m j=1

tj 1− t

δj

t 1− tj

γj

(t ∈ T),

(5.31)

where δj = αj − βj and γj = αj + βj . Note that ωαj = ωRe αj ωi Im αj , and that ωRe αj has a zero (Re αj > 0) or a pole (Re αj < 0) at t = tj , while ωi Im αj has a discontinuity of oscillating type at t = tj if Re αj = 0 and Im αj = 0. For βj = 0, ϕβj has a jump discontinuity at t = tj and ϕβj (tj − 0) = exp(2πiβj ). ϕβj (tj + 0) H p spaces with Khvedelidze weight are a natural terrain for the study of Toeplitz operators with Fisher-Hartwig symbols.

282


5.61. Lemma. Let ν ∈ C, τ ∈ T, and let be a Khvedelidze weight on Lp . If |t − τ |Re ν (t) is also a Khvedelidze weight on Lp , then T (ξν,τ ) : H p (|t − τ |Re ν (t)) → H p ((t)) is bounded and invertible and the inverse is T (ξ−ν,τ ). The same is true with ξ±ν,τ replaced by η±ν,τ . * Proof. If ϕ ∈ H p (|t − τ |Re ν (t)), then T |ξν ϕ|p p dm equals ( τ exp − p Im ν arg 1 − |t − τ |pRe ν |ϕ(t)|p p (t) dm t T ( ≤ exp(πp Im ν) |ϕ(t)|p (|t − τ |Re ν (t))p dm, T

which gives the boundedness of T (ξν ). It can be shown similarly that T (ξ−ν ) : H p ((t)) → H p (|t − τ |Re ν (t)) is bounded. Since, for ∈ H p (|t − τ |Re ν (t)), P (ξ−ν P (ξν ϕ)) = P (ξ−ν ξν ϕ) − P (ξ−ν Q(ξν ϕ)) = ϕ, it follows that T (ξ−ν ) is the inverse of T (ξν ). The proof for T (ην ) is analogous. 5.62. Theorem. Let a be given by (5.30) and suppose that, in addition to (iii), Re αj < 1/2 for all j = 1, . . . , n. Choose µj ∈ R and κj ∈ Z so that |µj |
, j+1 2 and it results that

6.1 Multipliers on Weighted p Spaces N

|a−k |p ≤ 2

k=0

N

|a−k |p

k=0

j+1−k j+1

289

pµ ≤ 2C,

i.e., {a−k }k∈Z+ ∈ p , as desired. A simple computation shows that An e0 − M (a)e0 pp =

+ µ +p + n + 1 − k ++ |a−k |p ++1 − + n+1 k=1 + µ +p ∞ + n + 1 + k ++ p+ + |ak | +1 − + . n+1 n

(6.2)

k=0

Let ε > 0 be arbitrarily given. Because + µ +p + + +1 − n + 1 − k + ≤ 1 + + n+1 and

+ µ +p µ p + + +1 − n + 1 + k + ≤ 1 + n + 1 + k + + n+1 n+1 ≤ (1 + (1 + k)µ )p ≤ 2p (k + 1)pµ

and {an }n∈Z ∈ p,p 0,µ , there is an N = N (ε) such that (6.2) is not greater than + + µ +p µ +p N N + + n + 1 − k ++ n + 1 + k ++ ε p+ 1 − + |a−k |p ++1 − + |a | k + + + (6.3) 2 n+1 n+1 k=1

k=0

for all n ≥ N . But if n ∈ Z+ is large enough, then (6.3) is smaller than ε, and this finally shows that (6.2) goes to zero as n → ∞. Now let µ < 0. Since T (a) ∈ L(q|µ| ), we have {a−k }k∈Z+ = T (a)e0 ∈ q|µ| ⊂ p|µ| (recall that p ≥ 2). Put C := T (a)pL(pµ ) and notice that ∞

|ak |p (k + j + 1)−p|µ| = T (a)ej pp,µ ≤ Cej pp,µ = C(j + 1)−p|µ|

k=−j

and therefore

∞ k=0

|ak |p

j+1 j+k+1

p|µ| ≤C

for all j ∈ Z+ . If there would exist an N ∈ Z+ such that N k=0

|ak |p > 4C,

6 Toeplitz Operators on p

290

then for all j ≥ N satisfying

j+1 N +j+1

p|µ| >

1 2

we would have N k=0

|ak |p

j+1 j+k+1

p|µ|

1 |ak |p > 2C, 2 N

≥

k=0

which is a contradiction. Thus, {ak }k∈Z+ ∈ . Since An e0 − M (a)e0 pp again equals (6.2) and + µ +p + + +1 − n + 1 + k + ≤ 1 + + n+1 and + µ +p |µ| p + + n+1 +1 − n + 1 − k + ≤ 1 + + + n+1 n+1−k p

≤ (1 + (1 + k)|µ| )p ≤ 2(k + 1)p|µ| , and because we have seen that {an }n∈Z ∈ p,p |µ|,0 , the same argument as for µ ≥ 0 can be applied to show that (6.2) converges to zero as n → ∞. (c) If T (a) ∈ L(pµ ), then aM p ≤ T (a)Φ(pµ ) . Proof. Without loss of generality assume µ ≥ 0. Let K be any operator in C∞ (p ). The restriction of the mapping Λ defined in the preceding proof to pµ is an isometric isomorphism of pµ onto p and will be denoted by Λ, too. Since V (±n) L(p ) = 1, we have V (−n) Λ(T (a) + K)Λ−1 V n xp ≤ T (a) + KL(pµ ) xp

∀x ∈ 0 .

(6.4)

If x ∈ 0 , then T (a)x ∈ pµ ⊂ p . A straightforward computation gives that V (−n) ΛT (a)Λ−1 V n ej − T (a)ej pp,µ (j = 0, 1, 2, . . .) equals j k=0

+ + µ +p µ +p ∞ + + n + j + 1 − k ++ n + j + 1 + k ++ p+ + |a−k | +1 − |ak | +1 − + + + n+j+1 n+j+1 p

k=0

and a similar reasoning as in the previous proof shows that this converges to zero as n → ∞. Because V n → 0 weakly on p , we deduce from 1.1(f) that V (−n) ΛKΛ−1 V n → 0 strongly on p . Thus, passage to the limit n → ∞ in (6.4) leads to T (a)xp ≤ T (a) + KL(pµ ) xp

∀ x ∈ 0 .

It follows that T (a)L(p ) ≤ T (a) + KL(pµ ) , and as K can be chosen arbitrarily, we obtain that T (a)L(p ) ≤ T (a) + KΦ(pµ ) . Equality (2.9) completes the proof.

6.1 Multipliers on Weighted p Spaces

291

(d) If µ > 1/q, then Mµp = F pµ ⊂ W . If µ < −1/p, then Mµp = F q−µ ⊂ W . Proof. Let µ > 1/q. It is not difficult to see that then −qµ (|k| + 1)(|n − k| + 1) sup (|n| + 1)qµ =: Ap,µ < ∞. n∈Z

k∈Z

If a ∈ Mµp and x ∈ pµ (Z), then, by H¨ older’s inequality, + +p + ++ + p M (a)xp,µ = an−k xk + (|n| + 1)pµ + + + n k p p pµ pµ ≤ |an−k | |xk | (|n − k| + 1) (|k| + 1) n

k

×

p/q −qµ

(|n − k| + 1)

−qµ

(|k| + 1)

(|n| + 1)pµ

k

≤

Ap−1 µ,p

n

|an−k |p |xk |p (|n − k| + 1)pµ (|k| + 1)pµ

k

p p = Ap−1 (6.5) p,µ ap,µ xp,µ .

Also note that M (a)e0 pp,µ = k |ak |p (|k| + 1)pµ . This and (6.5) give

ap,µ ≤ aMµp ≤ A1/q p,µ ap,µ ,

(6.6)

which shows that Mµp = F pµ . From property (a) we now deduce that Mµp = F q−µ in case µ < −1/p. Finally, the inclusions F pµ ⊂ W (µ > 1/q) and older’s inequality. F q−µ ⊂ W (µ < −1/p) can be easily verified using H¨ (e) Suppose µ = 1/q or µ = −1/p and let a ∈ Mµp . Then a cannot have jump discontinuities, i.e., if τ ∈ T and the finite limits a(τ − 0) and a(τ + 0) exist, then a(τ − 0) = a(τ + 0). Proof. Let µ = 1/q. Assume the finite limits a(τ ±0) exist. Then (see Zygmund [591, Chap. II, Theorem 8.13]) lim

n→∞

a(τ − 0) − a(τ + 0) (Sn ) a)(τ ) = , log n π

where (Sn ) a)(τ ) denotes the n-th partial sum of the Fourier series of the conjugate function ) a at τ . But n 1/p n 1/q n a)(τ ) ≤ |ak | ≤ |ak |p (|k| + 1)pµ (|k| + 1)−qµ (Sn ) k=−n

≤ M (a)e0 p,µ

k=−n

n

1/q (|k| + 1)−qµ

k=−n

k=−n

≤ const aMµp (log n)1/q ,

292


whence a(τ − 0) = a(τ + 0). Property (a) gives the assertion for µ = −1/p. (f) Let −1/p < µ < 1/q, and suppose a is in L∞ and has finite total variation V1 (a). Then a ∈ Mµp and there is a constant cp,µ depending only on p and µ such that aMµp ≤ cp,µ aL∞ + V1 (a) . Proof. This can be proved in the same way as for µ = 0 (see 2.5(f)). Note that the restriction −1/p < µ < 1/q comes from the discrete Hunt-MuckenhouptWheeden theorem 1.49. (g) Mµp is a Banach algebra under the norm aMµp := M (a)L(pµ (Z)) . Proof. The same arguments as in the proof of 2.5 apply. 6.3. Remark. A decisive distinction between Toeplitz operators on p spaces with and without weight is the failure of the Brown-Halmos theorem 2.7 for weighted spaces: If µ = 0 and the Toeplitz operator T (a) is bounded on pµ , then a need not belong to Mµp . Indeed, if µ > 0 and a(t) =

∞

a−n t−n (t ∈ T),

n=0

∞

|a−n | < ∞,

n=0

∞ then T (a) = n=0 a−n V (−n) ∈ L(pµ ) because V (−n) L(pµ (Z)) = 1; on the other hand, since ∞

M (a)e0 pp,µ =

|a−n |p (n + 1)pµ ,

n=0

L(pµ (Z))

we have M (a) ∈ / whenever a ∈ / F pµ ; finally, if m ∈ Z satisfies the inequality m > (1/µ)(1 + 1/q), then a(t) :=

∞ −km t k=1

k2

(t ∈ T)

belongs to W \ F pµ . 6.4. Open problem. Is it true that T (a) ∈ L(pµ ) ⇐⇒ M (a) ∈ L(p,p 0,µ )? Note that the implication “⇐=” is trivial. 6.5. Theorem. Let 1 < p < ∞ and µ ∈ R. If T (a) ∈ L(pµ ) and T (a) ∈ Φ(pµ ), then a ∈ GM p .

6.2 Continuous Symbols

293

Proof. In view of 2.5(b) we may assume that p ≥ 2. Let Λ be defined as in the proof of 6.2(b). If T (a) ∈ L(pµ ) and T (a) ∈ Φ(pµ ), then a ∈ M p (see 6.2(b)) and clearly ΛP M (a)P Λ−1 ∈ Φ(p ). As in the proof of Theorem 2.30 one can see that there are K ∈ C∞ (p ) and δ > 0 such that U −n ΛP M (a)P Λ−1 U n xp + P KP U n xp + δU −n QU n xp ≥ δxp (6.7) for all x ∈ p (Z). The arguments of the proof of 6.2(b) and Theorem 2.30 show that passage to the limit n → ∞ in (6.7) gives that M (a)xp ≥ δxp for all x ∈ p (Z). Now Proposition 2.29(b) implies that M (a) ∈ GL(p (Z)) and Proposition 2.28(c) finally shows that a ∈ GM p . Remark. It is clear that the above proof also works in the matrix case. Thus, p if T (a) ∈ L((pµ )N ) and T (a) ∈ Φ((pµ )N ), then a ∈ GMN ×N . 6.6. Theorem. Let 1 < p < ∞ and µ ∈ R. (a) If a ∈ Mµp ∩ GM p , then the kernel of T (a) in pµ or the kernel of T (a) in q−µ is trivial. (b) If a ∈ Mµp , T (a) ∈ Φ(pµ ), and Ind T (a) = 0, then T (a) ∈ GL(pµ ). Proof. (a) By 6.2(a), we may assume that µ ≥ 0. Let T (a)x+ = 0 and T (a)y+ = 0, where x+ ∈ pµ , y+ ∈ q−µ , and y+ = 0. A similar reasoning as in the proof of Theorem 2.38(b) leads to the equality M (a)x+ = 0. Because x+ ∈ pµ ⊂ p and a−1 ∈ M p , it follows that x+ = M (a−1 )M (a)x+ = 0. (b) Immediate from part (a) and Theorem 6.5.

6.2 Continuous Symbols 6.7. Definitions. For 1 < p < ∞ and µ ∈ R, let Cp,µ denote the closure of the Laurent polynomials in Mµp , i.e., Cp,µ := closMµp P. It is clear that Cp,µ is a closed subalgebra of Mµp . Inequality (6.6) implies that Cp,µ = F pµ ( = Mµp ) for µ > 1/q or µ < −1/p. Also notice that Cp,µ ⊂ Cp ⊂ C due to 6.2(b). Define M p,µ (p = 2 and µ = 0) as the collection of all a ∈ L∞ which belong to Mµ)p) for all p) and µ ) in some neighborhood of p and µ (depending on 2,µ a), respectively. We let M and M p,0 (p = 2 and µ = 0) refer to the set ∞ 2 ) and p) in some neighborhood of of all a ∈ L which are in Mµ) and M p) for µ µ and p, respectively. Finally, let M 2,0 := L∞ . p,µ p,µ p,µ p In what follows we shall write MN ×N , CN ×N , N instead of (Mµ )N ×N , p (Cp,µ )N ×N , and (µ )N . 6.8. Proposition. Let 1 < p < ∞ and µ ∈ R. (a) The maximal ideal space M (Cp,µ ) of Cp,µ is T and the Gelfand map is given by Γ : Cp,µ → C(T), (Γ a)(τ ) = a(τ ). (b) Cp,µ = closMµp (C ∩ M p,µ ).

294


Proof. (a) See the proof of Proposition 2.46(a). (b) First note that Lemma 2.44 remains valid for spaces with weight: if a ∈ Mµp , then σn aMµp ≤ aMµp for all n ≥ 0; the proof is the same as the one for spaces without weight. Now let a ∈ C ∩ M p,µ and suppose p = 2 and µ = 0. Then, by 6.2(b), a ∈ M0p+ε , where ε > 0 (resp. ε < 0) for p > 2 (resp. p < 2). The Riesz-Thorin interpolation theorem gives γ M (a − σn a)M p ≤ M (a − σn a)1−γ M p+ε M (a − σn a)M 2 ,

(6.8)

where γ ∈ (0, 1) is some constant. By what was said above, the first factor in (6.8) remains bounded as n → ∞, and since M 2 = L∞ and a ∈ C, the second p , where ε > 0 (resp. factor in (6.8) goes to zero as n → ∞. Because a ∈ Mµ+ε ε < 0) for µ > 0 (resp. µ < 0), application of the Stein-Weiss interpolation theorem leads to M (a − σn a)δM p , M (a − σn a)Mµp ≤ M (a − σn a)1−δ Mp µ+ε

(6.9)

where δ ∈ (0, 1) is some constant. The first factor in (6.9) remains bounded by what was said above and the second converges to zero by what has already been proved. The conclusion is that a ∈ Cp,µ . The proof can now be finished as in 2.45. 6.9. Definition. Let 1 < p < ∞ and µ ∈ R. Define p,µ p,µ p,µ alg T (CN ×N ) := algL(N ) T (f ) : f ∈ CN ×N , T (f ) : f ∈ PN ×N . algp,µ T (P) := algL(p,µ N ) It is clear that the second algebra is contained in the first algebra, and from the definition of Cp,µ it is immediately seen that the two algebras are actually p equal to each other. In the case µ = 0 we write alg T (CN ×N ) instead of p,0 alg T (CN ×N ). If f ∈ Cp,µ , then H(f ) ∈ C∞ (pµ ). This can be proved as for µ = 0 (Theorem 2.47(a)). Consequently, if fjk is a finite collection of functions in Cp,µ , then, by (2.18), T (fjk ) − T fjk ∈ C∞ (pµ ). j

k

j

k

The same reasoning as in the proof of Proposition 4.5 shows that C∞ (p,µ N ) is p,µ p,µ ). Therefore, alg T (C ) equals a subset of alg T (CN ×N N ×N p,µ p,µ T (f ) + K : f ∈ CN closL(p,µ ×N , K ∈ C∞ (N ) N ) T (f ) + K : f ∈ PN ×N , K ∈ C∞ (p,µ = closL(p,µ N ) . N )

(6.10)


295

6.10. Remark. Let a be the function constructed in Remark 6.3. Then n " " " " a−k χ−k " "T (a) − T

L(p µ)

k=0

≤

∞

|a−k | = o(1)

(n → ∞),

k=n+1

hence T (a) ∈ algp,µ T (P), and thus T (a) ∈ alg T (Cp,µ ), although a is not in Cp,µ (µ > 0)! 6.11. Definition. Put p,µ p,µ p,µ alg T π (CN ×N ) := alg T (CN ×N )/C∞ (N ), p,µ π and for A ∈ alg T (CN ×N ) let A denote the coset of the quotient algebra containing A. It is readily verified that π p,µ p,µ p,µ alg T π (CN ×N ) = closL(N )/C∞ (N ) T (g) : g ∈ PN ×N

and that alg T π (Cp,µ ) is commutative. 6.12. Theorem. Let 1 < p < ∞ and µ ∈ R. (a) The maximal ideal space M (alg T π (Cp,µ )) is T and for T (a) in the algebra alg T (Cp,µ ) the Gelfand transform is given by (Γ T π (a))(τ ) = a(τ ). (b) The Shilov boundary of M (alg T π (Cp,µ )) coincides with the whole space M (alg T π (Cp,µ )). p,µ (c) If A ∈ alg T (CN ×N ), then π A ∈ Φ(p,µ N ) ⇐⇒ (Γ (det A) )(τ ) = 0

∀ τ ∈ T.

In that case Ind A = −ind (Γ (det A)π ). Proof. (a) Let v ∈ M (alg T π (Cp,µ )) and put τ = v(T π (χ1 )). Theorem 6.5 implies that T ⊂ sp T π (χ1 ) and from Proposition 6.8(a) we deduce that / T, then T π ((χ1 − λ)−1 ) is the inverse of T π (χ1 − λ)). sp T π (χ1 ) ⊂ T (if λ ∈ π Hence sp T (χ1 ) = T and we have τ ∈ T. It follows that v(T π (g)) = g(τ ) for all g ∈ P. If T (a) ∈ alg T (Cp,µ ), then there are gn ∈ P such that T (a) − T (gn )L(pµ ) → 0 as n → ∞, whence a − gn L∞ → 0 (6.2(b)) and so v(T π (a)) = lim v(T π (gn )) = lim gn (τ ) = a(τ ). n→∞

n→∞

On the other hand, if τ ∈ T, then, by 6.2(c), |g(τ )| ≤ g∞ ≤ T π (g) for all g ∈ P, and therefore (recall what was said at the end of Section 6.9) the mapping vτ : T π (g) → g(τ ) (g ∈ P) extends to a multiplicative linear functional on alg T π (Cp,µ ). That the Gelfand topology on T is the topology inherited from the inclusion T ⊂ C can be checked straightforwardly.

296


(b) This follows from 1.20(a) along with the observation that T (f ) is in alg T (Cp,µ ) for every f ∈ C ∞ . (c) The commutativity of alg T π (Cp,µ ) and Theorem 1.14(c) imply that p A ∈ Φ(p,µ N ) ⇐⇒ det A ∈ Φ(µ ).

If Γ (det A)π = 0 on T, then (det A)π ∈ G(alg T π (Cp,µ )) and thus det A belongs to Φ(pµ ). If (Γ (det A)π )(τ0 ) = 0 for some τ ∈ T, then, by (b) and 1.20(c), / Φ(pµ ). det Aπ is a topological divisor of zero and therefore det A ∈ We are left with the index formula. Without loss of generality assume µ ≥ 0 (otherwise consider adjoints). Again from what was said at the end of Section 6.9 we deduce that there are gn ∈ PN ×N and Kn ∈ C∞ (p,µ N ) such that → 0 (n → ∞). (6.11) A − T (gn ) − Kn L(p,µ N ) Hence, there is an n0 such that T (gn ) ∈ Φ(p,µ N ) and Indp,µ A = Indp,µ T (gn ) for all n ≥ n0 . Clearly, T (gn ) ∈ Φ(pN ). We claim that Indp,µ T (gn ) = Indp T (gn ). Because pµ ⊂ p and q−µ ⊃ q , we have Indp,µ T (gn ) = αp,µ (T (gn )) − αq,−µ (T (gn∗ )) ≤ αp (T (gn )) − αq (T (gn∗ )) = Indp T (gn ). The operator T (gn−1 ) is a regularizer of T (gn ). Thus, analogously, Indp,µ T (gn−1 ) ≤ Indp T (gn−1 ). Because Ind T (gn−1 ) = −Ind T (gn ) (see 1.12(c)), we arrive at the desired equality Indp,µ T (gn ) = Indp T (gn ). This and Corollary 4.8(c) give that Indp,µ A = −ind (det gn ) (n ≥ n0 ). But it is easily seen from (6.11) that Γ (det A)π − det gn L∞ → 0 (n → ∞), which implies the asserted index formula. Remark. Part (c) is true for p = 1 and Γ (det A)π replaced by det SmbT (A). This can be proved using an index perturbation argument. We finally mention some (trivial) consequences of a (deep) result of Zafran which show that, to put it mildly, the theory of Toeplitz operators with symbols in C ∩ M p is substantially more complicated than the corresponding theory for symbols in Cp (no weight is involved!). Note that C ∩ M p is obviously a closed subalgebra of M p . 6.13. Definition. A function F : [−1, 1] → C is said to operate from C ∩ M p into M p if F ◦ a ∈ M p whenever a ∈ C ∩ M p and a(T) ⊂ [−1, 1]. 6.14. Theorem (Zafran). If p = 2 and F : [−1, 1] → C operates from C∩M p into M p , then F is the restriction of an entire function to [−1, 1].

6.3 Piecewise Continuous Symbols

297

Proof. For a proof see Zafran [585]. 6.15. Corollary. By identifying τ ∈ T with the multiplicative linear functional vτ : C ∩ M p → C,

a → a(τ ),

T may be viewed as a subset of the maximal ideal space M (C ∩M p ) of C ∩M p , but if p = 2 then M (C ∩ M p ) is strictly larger than T. Proof. The functional vτ is clearly linear and multiplicative on C ∩ M p . Since a∞ ≤ aM p for a ∈ M p , we have |a(τ )| ≤ a∞ ≤ aM p for every a ∈ C ∩ M p , which implies that vτ is continuous. Now assume p = 2 and M (C ∩ M p ) = T. Then a − i is in G(C ∩ M p ) whenever a ∈ C ∩ M p is real-valued, and hence the function F (x) := 1/(x − i) operates from C ∩ M p into M p . But this contradicts Theorem 6.14, since F (z) := 1/(z − i) (z ∈ C) is not entire. In connection with the following corollary see Proposition 2.32 and Theorems 2.42 and 2.47 (and also the remark in 2.29). 6.16. Corollary. Let p = 2. / M p. (a) There exist a ∈ C ∩ M p such that |a| ∈ (b) There exist real-valued a ∈ C ∩ M p such that the spectrum of a in M p , i.e. the spectrum of M (a) in L(p (Z)), is not contained in R. Proof. (a) It is immediate from Theorem 6.14 that F (x) := |x| does not operate from C ∩ M p into M p . (b) If spM p a would be a subset of R for every real-valued a ∈ C ∩M p , then F (x) := 1/(x − i) would operate from C ∩ M p into M p , which is impossible by virtue of Theorem 6.14.

6.3 Piecewise Continuous Symbols 6.17. Definitions. Recall how ϕβ = ϕβ,τ , ξδ = ξδ,τ , ηγ = ηγ,τ were defined in 5.35. The functions ξδ and ηγ are in L1 if and only if Re δ > −1 and Re γ > −1. In that case their Fourier coefficients are δ n (6.12) ξδ,−n = (−τ ) (n ≥ 0), ξδ,n = 0 (n > 0), n γ (6.13) ηγ,n = (−1/τ )n (n ≥ 0), ηγ,−n = 0 (n > 0). n On defining T (ξδ ) and T (ηγ ) as the Toeplitz matrices (ξδ,j−k )∞ j,k=0 and with ξ and η given by (6.12) and (6.13), respectively, T (ξδ ) (ηγ,j−k )∞ δ,n γ,n j,k=0 and T (ηγ ) make sense for all δ, γ ∈ C.

298


The function ξδ ηγ is in L1 if and only if Re (γ + δ) > −1. In that case we let T (ξδ ηγ ) denote the Toeplitz matrix ((ξδ ηγ )j−k )∞ j.k=0 , where (ξδ ηγ )n refers to the n-th Fourier coefficient of ξδ ηγ . The computation of these Fourier coefficients is our first concern. 6.18. Lemma. If Re (γ + δ) > −1, then the n-th Fourier coefficient of ξδ ηγ is equal to Γ (1 + γ + δ) (6.14) (−1/τ )n Γ (γ − n + 1)Γ (δ + n + 1) in case neither γ − n + 1 nor δ + n + 1 is a nonpositive integer and is equal to zero in case γ − n + 1 or δ + n + 1 is a nonpositive integer. Proof. Suppose first that neither γ nor δ is an integer. Choose an integer κ so that −1 < Re γ + κ ≤ 0. Then Re δ − κ > −1. Write ξδ (t)ηγ (t) as ξ(t)η(t)(−τ /t)κ with ξ(t) := (1 − τ /t)δ−κ and η(t) := (1 − t/τ )γ+κ (t ∈ T). The Fourier coefficients of ξ and η are δ−κ γ+κ , ηn = (−1/τ )n (n ≥ 0), ξ−n = (−τ )n n n ξn = η−n = 0 (n > 0). There exist p, q ∈ (1, ∞) such that 1/p + 1/q = 1, Re γ + κ > −1/p, and Re δ − κ > −1/q. Consequently, ξ ∈ Lq and η ∈ Lp , and this ensures (see Zygmund [591, Chap. IV, Theorem 8.7]) that for n ≥ 0 the n-th Fourier coefficient of ξη is n

(ξη)n = (−1/τ )

∞ γ+κ δ−κ

n+j j ⎤ ⎡ ∞ (−γ + κ + n) (−δ + κ) γ + κ j j ⎦, ⎣1 + = (−1/τ )n n j!(n + 1)j j=0

j=1

where (x)j := x(x + 1) . . . (x + j − 1). The sum in square brackets is nothing else than the hypergeometric series F (−γ − κ + n, −δ + κ; n + 1; 1), which converges just for Re (n + 1 − (−γ − κ + n) − (−δ + κ)) = Re (γ + δ + 1) > 0 and has the sum Γ (n + 1)Γ (1 + γ + δ) Γ (1 + γ + κ)Γ (δ + n − κ + 1) (see Whittaker, Watson [554, 14.11]). From Γ (1 + γ + κ) γ+κ = n Γ (n + 1)Γ (γ − n + κ + 1) we obtain that (ξη)n is (−1/τ )n times


Γ (1 + γ + δ) . Γ (γ − n + κ + 1)Γ (δ + n − κ + 1)

299

(6.15)

Repeating these arguments one can see that (ξη)n is (−1/τ )n times the expression (6.15) for n < 0, too. Finally, since (ξδ ηγ )n = (−τ )κ (ξη)n+κ , we arrive at (6.14). Now let γ be an integer. Then ∞ τ δ+γ τ γ δ + γ = ξδ (t)ηγ (t) = 1 − (−τ )j−γ tγ−j . − j t t j=0

δ+γ Hence (ξδ ηγ )n = 0 for n ≥ γ + 1 and (ξδ ηγ )n = (−1/τ ) for n ≤ γ, γ−n which coincides with (6.14) for δ ∈ Z as well as for δ ∈ / Z. The case where δ is an integer can be treated analogously. n

6.19. Definition. For α ∈ C, put n µ(α) n := (−1)

−1 − α n

and let Mα denote the diagonal matrix (operator) (α)

(α)

(α)

Mα = diag (µ0 , µ1 , µ2 , . . .). 6.20. Theorem (Duduchava/Roch). Let γ, δ ∈ C \ {−1, −2, . . .} and suppose that Re (γ + δ) > −1. Then T (ηγ )Mγ+δ T (ξδ ) = Γγ,δ Mδ T (ξδ ηγ )Mγ ,

(6.16)

where Γγ,δ := Γ (1 + γ)Γ (1 + δ)/Γ (1 + γ + δ). Proof. By computing the ik entry of both sides of (6.16) (using Lemma 6.18 for the right-hand side) one sees that (6.16) will follow as soon as one has shown that

min{i,k}

j=0

j

(−1)

γ i−j

−1 − γ − δ j

δ k−j

=

δ+i k

γ+k i

for all i, k ∈ Z+ . Without loss of generality assume i ≥ k. Then i = k + m with m ≥ 0 and what we must prove is that k j (−1) j=0

γ k+m−j

−1 − γ − δ j

δ k−j

=

Let F (a, b; c; x) denote the hypergeometric function

δ+k+m k

γ+k k+m

.

300


F (a, b; c; x) := 1 +

∞ (a)j (b)j j=1

j!(c)j

xj .

A well known formula of Gauss (see Whittaker, Watson [554, 14.4]) says that F (a, b; c, x) = (1 − x)c−a−b F (c − a, c − b; c; x).

(6.17)

After putting a = γ+ 1,b = δ + m + 1, c = m + 1 and multiplying both sides γ of this formula by we get m γ γ 1−γ−δ F (γ+1, δ+m+1; m+1; x) = (1−x) F (−γ+m, −δ; m+1; x). m m Now expand both sides of this equality into apower series to x. with respect γ+k δ+m+k k The coefficient of x on the left-hand side is . Taking n+k k into account that ∞ −1 − γ − δ −1−γ−δ j (1 − x) = (−1) xj , j j=0 ∞ γ δ γ xl , F (−γ + m, −δ; m + 1; x) = m+l l m l=0

the coefficient of xk on the right-hand side is seen to be k −1 − γ − δ γ δ j (−1) j m+k−j k−j j=0

and so the proof is complete. 6.21. Lemma. Let K be a compact subset of C\{−1, −2, −3, . . .}. Then there exists a constant cK depending only on K such that Re α Re α c−1 ≤ |µ(α) ∀ n ∈ Z+ ∀ α ∈ K. n | ≤ cK (n + 1) K (n + 1) 5 (α) n Proof. We have |µn | = k=1 |1 + α/k|. It is well known that the infinite 5∞ −α/k converges uniformly on K to e−Cα /Γ (α + 1), product k=1 (1 + α/k)e where C := 0.577 . . . is Euler’s constant. Hence, if n0 ∈ Z+ is large enough, then + + −Cα + + + + n + e + α ++ ++ −α(1+ 11 +...+ n1 ) ++ 1 ++ e−Cα ++ ++ + + ≤ 1 + ≤ 2 e + + ++ + + + + 2 Γ (α + 1) k + Γ (α + 1) + k=1

for all n ≥ n0 and all α ∈ K. Taking into account that 1 1 + . . . + = C + log(n + 1) + o(1) (n → ∞) 1 n it is now not difficult to complete the proof.


301

6.22. Corollary. If α ∈ C \ {−1, −2, −3, . . .}, 1 < p < ∞, and µ ∈ R, then Mα is a bounded and (boundedly) invertible operator from pµ onto p−µ−Re α . Proof. Immediate from the preceding lemma.

We are now prepared to begin with the study of Toeplitz operators with P C symbols on pµ . 6.23. Proposition. Let β ∈ C \ Z, 1 < p < ∞, µ ∈ R. Then T (ϕβ ) ∈ L(pµ ) ⇐⇒ −

1 1 −1/p.

6.24. Proposition. Let β ∈ C, 1 < p < ∞, −1/p < µ < 1/q. Then the following are equivalent: (i) T (ϕβ ) ∈ Φ(pµ ) and Ind T (ϕβ ) = −κ. (ii) κ − 1/p < Re β + µ < κ + 1/q. (iii) 0 ∈ / Ar (ϕβ (τ − 0), ϕβ (τ + 0)), where r := (1/q − µ)−1 , and the index of the closed continuous and naturally oriented curve obtained from the range of ϕβ by filling in the arc Ar (ϕβ (τ − 0), ϕβ (τ + 0)) is equal to κ. Proof. (ii) ⇐⇒ (iii). Straightforward. (ii) =⇒ (i). Put α = β − κ. Then |Re α| < 1. There is nothing to prove for α = 0; so assume α = 0. Let Aα denote the matrix T (η−α )T (ξα ) and let ajk (α) and ϕj−k (α) denote the jk entry of Aα and T (ϕα ), respectively. If Re α = 0, then ξ±α ∈ H ∞ and η±α ∈ H ∞ . Therefore, since ϕα = ξ−α ηα , we have for x ∈ 2 ∼ = H 2, T (ϕα )Aα x = T (ξ−α )T (ηα )T (η−α )T (ξα )x = x, Aα T (ϕα )x = T (η−α )T (ξα )T (ξ−α )T (ηα )x = x. This implies that for Re α = 0 both T (ϕα )Aα and Aα T (ϕα ) are equal to the identity matrix. We want to show that the same is true for all α’s in question. It is easily seen that each ajk (α) and each ϕj−k (α) is an analytic function in the punctured stripe S := {α ∈ C : |Re α| < 1, α = 0}. We claim that for each j ∈ Z+ and each k ∈ Z+ the series

302

6 Toeplitz Operators on p ∞

ϕj−n (α)ank (α)

∞

and

n=0

ajn (α)ϕn−k (α)

(6.18)

n=0

converge uniformly on compact subsets of S. This will imply that the entries of T (ϕα )Aα and Aα T (ϕα ) are analytic in S, which together with the above result for Re α = 0 shows that T (ϕα )Aα = I and Aα T (ϕα ) = I for all α ∈ S. To prove our claim choose r ∈ (1, ∞) so that |Re α| < 1 − 1/r an let s satisfy 1/r + 1/s = 1. From Theorem 6.20 (with δ = α, γ = −α) we deduce that (−α) . ank (α) = Γ−α,α µ(α) n ϕn−k (−α)µk Hence, n

≤

(−α)

|ϕj−n (α)ank (α)| = |Γ−α,α µk

|ϕj−n (α)µ(α) n ϕn−k (−α)|

n

(−α) |Γ−α,α µk |

|

n

1/s |ϕj−n (α)|

s

1/r r |µ(α) n ϕn−k (−α)|

,

n

and since ϕl (β) = π sin(πβ) τ −l /(β − l), Lemma 6.22 gives our claim. We now prove that the matrix Aα generates a bounded operator on pµ : by virtue of Theorem 6.20, Aα = Γ−α,α Mα T (ϕ−α )M−α , we have M−α ∈ L(pµ , pµ+Re α ), Mα ∈ L(pµ+Re α , pµ ) (Corollary 6.22) and T (ϕ−α ) ∈ L(pµ+Re α ) (Proposition 6.23 and hypothesis (ii)), whence Aα ∈ L(pµ ). Thus, we have proved that Aα ∈ L(pµ ) and that Aα T (ϕα ) = T (ϕα )Aα = I. The conclusion is that T (ϕα ) ∈ GL(pµ ). Since T (ϕβ ) = T (ϕα )T (χκ ) (κ ≥ 0) or T (ϕβ ) = T (χκ )T (ϕα ) (κ ≤ 0) and since T (χκ ) is Fredholm with index −κ, we deduce that T (ϕβ ) ∈ Φ(pµ ) and Ind T (ϕβ ) = −κ. (i) =⇒ (ii). This can be proved using the same perturbation argument as in the proof of Lemma 5.36. 6.25. Definitions. Let P K denote the collection of all piecewise constant functions on T having only a finite number of jumps. For 1 < p < ∞ and −1/p < µ < 1/q, define P Cp,µ as the closure in Mµp of P K, that is, P Cp,µ := closMµp P K (that P K is a subset of Mµp follows from 6.2(f)). Note that obviously T (a) ∈ L(pµ ) for a ∈ P Cp,µ . It is clear that P Cp,µ is a closed subalgebra of Mµp . Also notice that P C2,0 = P C. The following proposition shows that P Cp,µ contains sufficiently many interesting functions.

m 6.26. Proposition. If ϕ = i=1 gi fi with gi ∈ P K and fi ∈ Cp,µ , then ϕ is in P Cp,µ . Proof. It suffices to show that χ1 ∈ P Cp,µ , where χ1 (t) = t (t ∈ T). One can easily see that there are functions gn ∈ P K such that χ1 − gn L∞ → 0 as n → ∞ and V1 (χ1 − gn ) ≤ M with some constant M > 0. Both χ1 and all


303

the functions gn belong to Mµp for all p ∈ (1, ∞) and µ ∈ (−1/p, 1/q). The Riesz-Thorin interpolation theorem gives γ χ1 − gn M p ≤ χ1 − gn 1−γ M p±ε χ1 − gn L∞

(6.19)

(we identify M (f ) with f ), where ε > 0 (resp. ε < 0) for p > 2 (resp. p < 2) and γ ∈ (0, 1) is some constant. The inequality in 6.2(f) shows that the first factor in (6.19) remains bounded while the second factor goes to zero as n → ∞. Now we apply the Stein-Weiss interpolation theorem to get χ1 − gn δM p , χ1 − gn Mµp ≤ χ1 − gn 1−δ Mp µ±ε

(6.20)

where ε > 0 (resp. ε < 0) for µ > 0 (resp. µ < 0) and δ ∈ (0, 1) is some constant. The first factor in (6.20) again remains bounded (by 6.2(f)) and the second factor has just been shown to converge to zero as n → ∞. Thus, χ1 − gn Mµp → 0 as n → ∞ and so χ1 ∈ P Cp,µ . 6.27. Open problem. Once Proposition 6.8(b) has been proved it is not difficult to show that P C0 ∩ M p,µ is contained in P Cp,µ , and this gives that P Cp,µ = closMµp (P C0 ∩ M p,µ ). We have not been able to show that P C ∩ M p,µ is a subset of P Cp,µ and therefore we must raise the inclusion P C ∩M p,µ ⊂ P Cp,µ and the resulting equality P Cp,µ = closMµp (P C ∩M p,µ ) as an open question (even for µ = 0). 6.28. Proposition. Let 1 < p < ∞ and −1/p < µ < 1/q. The maximal ideal space of P Cp,µ is T × {0, 1} and the Gelfand map Γ : P Cp,µ → C(T × {0, 1}) is given by (Γ f )(τ, 0) = f (τ − 0),

(Γ f )(τ, 1) = f (τ + 0).

Proof. From 6.2(b) and 2.5(d) we deduce that (τ, 0) and (τ, 1) belong to M (P Cp,µ ). Conversely, let v ∈ M (P Cp,µ ). The preceding proposition shows that Cp,µ is a closed subalgebra of P Cp,µ , and since M (Cp,µ ) = T (Proposition 6.8(a)), v belongs to some fiber Mτ (P Cp,µ ) over τ ∈ T. Every function f ∈ P K can be written as f = cχτ +g, where c ∈ C, χτ is the characteristic function of the arc (τ, τ eiπ/2 ), and g ∈ P K is constant on some arc (τ e−iδ , τ eiδ ). The spectrum of χτ in P Cp,µ is clearly the doubleton {0, 1}. To see that v(g) = g(τ ), let ϕ ∈ C ∞ ⊂ Cp,µ be any function satisfying ϕ(τ ) = 1 and ϕg ∈ C ∞ and note that v(g) = v(g)ϕ(τ ) = v(g)v(ϕ) = v(gϕ) = (gϕ)(τ ) = g(τ ). Hence, either v(f ) = f (τ −0) or v(f ) = f (τ +0) for every f ∈ P K and thus for all f ∈ P Cp,µ , which implies that Mτ (P Cp,µ ) is the doubleton {(τ, 0), (τ, 1)}. Remark. The Gelfand topology on M (P Cp,µ ) coincides with that of M (P C) described in Proposition 3.24.

304


6.29. Proposition. Let a, b ∈ Mµp (1 < p < ∞, µ ∈ R) and suppose for each τ ∈ T there are an open neighborhood Uτ ⊂ T and a function fτ ∈ Cp,µ such that a|Uτ = fτ |Uτ or b|Uτ = fτ |Uτ . Then T (ab) − T (a)T (b) is in C∞ (pµ ). Proof. Choose a finite collection {Uτi } of neighborhoods with the property required in the proposition such that their union is T. Let i ψi = 1 be a subordinate smooth partition of unity. It is obvious that each ψi can be written as ψi = ϕ2i where ϕi is also smooth. Then, by (2.18), 0 1 T (aϕ2i b) − T (a)T (ϕ2i )T (b) T (ab) − T (a)T (b) = i

0 1 T (aϕ2i b) − T (a)T (ϕi )T (ϕi )T (b) − T (a)H(ϕi )H(ϕ)i )T (b) = i

0 T (aϕ2i b) − T (aϕi )T (ϕi b) + T (aϕi )H(ϕi )H()b) = i

1 +H(a)H(ϕ)i )T (ϕi b) − H(a)H(ϕ)i )H(ϕi )H(b) − T (a)H(ϕi )H(ϕ)i )T (b) . Since H(ϕi ) and H(ϕ)i ) are in C∞ (pµ ), we have T (ab) − T (a)T (b) −

0 1 T (aϕ2i b) − T (aϕi )T (ϕi b) ∈ C∞ (pµ ). i

But

T (aϕ2i b) − T (aϕi )T (ϕi b) = H(aϕi )H(ϕ)i)b) ∈ C∞ (pµ ),

because by our hypothesis at least one of the functions aϕi and ϕ)i)b belongs to Cp,µ and thus generates a compact Hankel operator on pµ . 6.30. Proposition. If a, b ∈ P Cp,µ (1 < p < ∞, −1/p < µ < 1/q), then T (a)T (b) − T (b)T (a) ∈ C∞ (pµ ). Proof. Since a and b can be approximated by piecewise constant functions in the norm of Mµp as closely as desired, we may suppose that a, b ∈ P K. Moreover, because every function in P K can be written as a finite sum of piecewise smooth (C ∞ ) functions each of which has at most one jump discontinuity, we may assume that a and b are piecewise smooth and have at most one jump. In view of the preceding proposition it is enough to consider the case that a and b have the jump at the same point of T. Then a = λb + c with λ ∈ C and c ∈ Cp,µ . So 0 1 T (a)T (b) − T (b)T (a) = λ T (c)T (b) − T (b)T (c) 0 1 0 1 = λ T (c)T (b) − T (cb) + λ T (bc) − T (b)T (c) , which is in C∞ (pµ ) by the preceding proposition.


305

6.31. Definition. Let 1 < p < ∞ and −1/p < µ < 1/q. For a ∈ P C, define ap,µ : T × [0, 1] → C by ap,µ (t, λ) := 1 − σr (λ) a(t − 0) + σr (λ)a(t + 0) (t ∈ T, λ ∈ [0, 1]), where r := (1/q − µ)−1 and σr is as in 5.12. The range of ap,µ is a continuous closed curve with a natural orientation; it is obtained from the (essential) range of a by filling in the arcs Ar (a(τ − 0), a(τ + 0)), r = (1/q − µ)−1 , for each τ ∈ T at which a has a jump. If the curve does not pass through the origin, its winding number with respect to the origin will be denoted by ind ap,µ .

n 6.32. Proposition. Let a = i=1 gi fi , where the functions gi are piecewise constant and the functions fi are continuously differentiable on T. Then T (a) ∈ Φ(pµ ) ⇐⇒ ap,µ (t, λ) = 0

∀ (t, λ) ∈ T × [0, 1].

If T (a) ∈ Φ(pµ ), then Ind T (a) = −ind ap,µ . Proof. If ap,µ does not vanish on T × [0, 1], then a can be written in the form (5.19): a = ϕβ1 . . . ϕβm b. Since continuously differentiable functions have finite −1 total variation, 6.2(f) implies that a ∈ M p,µ . Hence, b = aϕ−1 β1 . . . ϕβm is in p,µ C ∩M and thus, by Proposition 6.8(b), in Cp,µ . Now one can proceed as in the proof of Proposition 5.39. 6.33. Remark. Let 2 < p < ∞. If a is as in the preceding proposition, then T (a) ∈ L(p ) and T (a) ∈ L(H p ). For deciding of whether T (a) is Fredholm on p or H p we must fill in an arc Ar (a(τ − 0), a(τ + 0)) for each τ ∈ T at which a has a jump and then look whether the curve obtained in this way does pass through the origin or not. In the H p case the r is p and in the p case r equals q. Thus, although the angle at which the line segment [a(τ − 0), a(τ + 0)] is seen from that arc equals 2π/p in both cases, in the H p case the arc must be drawn on the right whereas in the p case it must be drawn on the left of the segment [a(τ − 0), a(τ + 0)]. Naturally, a similar observation can be made for 1 < p < 2. To express this circumstance analytically, note that ap,0 = aq,1 (6.31 and 5.37). 6.34. Definitions. Let 1 < p < ∞ and −1/p < µ < 1/q. Put p,µ p,µ p,µ alg T (P CN ×N ) := algL(N ) T (f ) : f ∈ P CN ×N , T (g) : g ∈ P KN ×N . algp,µ T (P KN ×N ) := algL(p,µ N ) These two algebras are easily seen to be equal to each other. The compact p,µ operators on p,µ N belong to alg T (P CN ×N ) (see 6.9). Define p,µ p,µ p,µ alg T π (P CN ×N ) := alg T (P CN ×N )/C∞ (N ), p,µ π denote the coset containing A ∈ alg T (P CN ×N ) by A , and notice that

306


π p,µ p,µ p,µ alg T π (P CN ×N ) = algL(N )/C∞ (N ) T (g) : g ∈ RN ×N , where R can be P K or P Cp,µ . By Proposition 6.30, the algebra alg T π (P Cp,µ ) is commutative. For τ ∈ T, let Jτπ denote the smallest closed two-sided ideal p,µ of alg T π (P CN ×N ) containing the set π T (f ) : f = diag (ϕ, . . . , ϕ), ϕ ∈ B, ϕ(τ ) = 0 , where B may be P, C ∞ , or Cp,µ . It is clear that Jτπ does not depend on the particular choice of B. Finally, define p,µ p,µ π π alg Tτπ (P CN ×N ) := alg T (P CN ×N )/Jτ , p,µ let Aπτ refer to the coset containing Aπ ∈ alg T π (P CN ×N ), and observe that p,µ π the algebra alg Tτ (P CN ×N ) is generated by the set {Tτπ (g) : g ∈ RN ×N }, p,µ π π where R is P K or P Cp,µ . For A in alg T (P CN ×N ), let spp,µ A and spp,µ Aτ p,µ p,µ π π π π denote the spectrum of A and Aτ in alg T (P CN ×N ) and alg Tτ (P CN ×N ), respectively. p,µ 6.35. Proposition. If a, b ∈ P CN ×N and a|Xτ = b|Xτ for some τ ∈ T, then p,µ π π Tτ (a) = Tτ (b). If A ∈ alg T (P CN ×N ), then / spp,µ Aπτ . (6.21) spp,µ Aπ = τ ∈T

Proof. To prove the first assertion we must show that T π (a) ∈ Jτπ whenever a ∈ P Cp,µ and a|Xτ = 0. Take such an a, let ε > 0 be an arbitrarily given number, and choose f ∈ P K so that a − f Mµp < ε. There is an open arc U = (τ e−iδ , τ eiδ ) such that |a(t)| < ε a.e. on U (Proposition 2.79) and f has at most one jump discontinuity in U . So 6.2(b) and 2.5(d) give that |f (t)| < 2ε on U . Now choose ϕ ∈ C ∞ so that ϕ(τ ) = 1, supp ϕ ⊂ U , and ϕ is monotonous on (τ e−iδ , τ ) and (τ, τ eiδ ). Then, by Proposition 6.29, T π (f ) − T π (f ϕ) = T π (f )T π (1 − ϕ) ∈ Jτπ . Since f ϕL∞ < 2ε and V1 (f ϕ) < 4ε, we deduce from 6.2(f) that T π (f ϕ) < 6cp,µ ε. Because dist(T π (a), Jτπ ) < dist(T π (f ), Jτπ ) + ε ≤ T π (f ϕ) + ε, it follows that dist(T π (a), Jτπ ) = 0, i.e., T π (a) ∈ Jτπ , as desired. p,µ Now let us prove (6.21). Put A := alg T π (P CN ×N ) and π B := D = diag (Aπ , . . . , Aπ ) : A ∈ alg T (Cp,µ ) . The algebra B is a closed subalgebra of the center of A (Proposition 6.30). From Theorem 6.13(a) we know that M (B) = T. Since, by (6.10), closidA Dπ ∈ B : (Γ Dπ )(τ ) = 0 = closidA T π (a) : f = diag (ϕ, . . . , ϕ), ϕ ∈ Cp,µ , ϕ(τ ) = 0 and this is nothing else than Jτπ , equality (6.21) can be immediately derived from Theorem 1.35(a).


307

6.36. Proposition. If a ∈ P Cp,µ , then

spp,µ Tτπ (a) = A(1/q−µ)−1 a(τ − 0), a(τ + 0) .

Proof. This can be proved in the same way as Proposition 5.40; note that the functions f and g occurring there can be chosen continuously differentiable on T \ {τ }, so that Proposition 6.32 can occupy the place of Proposition 5.39. Remark. There are pµ versions of the definitions and results in 5.24, 5.25, 5.26, 5.27, 5.28 and of Theorem 5.29 (although some troublesome but immaterial technical complications arise). In particular, pµ analogues of Propositions 5.40 and 5.41 (B = C) can be established. We want not to tire the reader with these things. 6.37. Proposition. (a) The algebra alg Tτπ (P Cp,µ ) is singly generated by Tτπ (χτ ), where χτ is the characteristic function of the arc (τ, τ eiπ/2 ). (b) The maximal ideal space M (alg Tτπ (P Cp,µ )) is homeomorphic to the segment [0, 1] (with its usual topology) and for a ∈ P Cp,µ the Gelfand map Γ : alg Tτπ (P Cp,µ ) → C([0, 1]) is given by π Γ Tτ (a) (λ) = 1 − σr (λ) a(τ − 0) + σr (λ)a(τ + 0), where r := (1/q − µ)−1 and σr is as in 5.12. Proof. See the proof of Proposition 5.44. 6.38. Theorem. (a) The maximal ideal space M (alg T π (P Cp,µ )) is homeomorphic to the cylinder T × [0, 1] equipped with the topology described in 4.88 and the Gelfand map Γ : alg T π (P Cp,µ ) → C(T × [0, 1]) is for a ∈ P Cp,µ given by (Γ T π (a))(τ, λ) = ap,µ (τ, λ), where ap,µ is as in 6.31. (b) The Shilov boundary of M (alg T π (P Cp,µ )) coincides with the whole maximal ideal space M (alg T π (P Cp,µ )). Proof. The proofs are the same as those of Theorems 5.45 and 5.46.

p,µ alg T (P CN ×N ).

Then 6.39. Theorem. Let A ∈ p,µ π A ∈ Φ(N ) ⇐⇒ Γ (det A) (t, λ) = 0 ∀ (t, λ) ∈ T × [0, 1]. Proof. The same arguments as in the proof of Theorem 5.47 apply.

6.40. Index computation. It can be shown that the index of a Fredholm p,µ π operator A ∈ alg T (P CN ×N ) is given by Ind A = −ind Γ (det A) . To verify this one can proceed as in 5.48. Finally notice that Theorem 6.6(a) and Proposition 6.28 give that dim Kerp,µ T (a) = max{Indp,µ T (a), 0}, dim Cokerp,µ T (a) = max{−Indp,µ T (a), 0} for every a ∈ P Cp,µ such that T (a) ∈ Φ(pµ ).

308


6.4 Analytic Symbols In Chapter 5 we saw that H p spaces with Khvedelidze weight can be advantageously used to study Toeplitz generated by Fisher-Hartwig sym5operators m bols, i.e., symbols of the form b j=1 ξδj ,tj ηγj ,tj . However, Theorem 5.9 forced us into restricting ourselves to symbols with small “size” of the singularities (that is, with small real parts of the exponents δj and γj ). The spaces p with weight enable us to treat Toeplitz operators generated by symbols with only one Fisher-Hartwig singularity but with large “size” of the singularity. One of the purposes of what follows is to establish an invertibility theory for the Toeplitz operators T (ξδ ηγ b) with Re γ + Re δ ≥ 0, Re γ > −1, and Re δ > −1. The n-th Fourier coefficient of ηα,τ (recall 5.35) is α (−1/τ )n = O(n−1−Re α ) (n → ∞) = τ −n µ(−1−α) n n (Lemma 6.21), hence ηα ∈ W and thus ηα ∈ Hp∞ := H ∞ ∩ M p (1 ≤ p < ∞) for all α ∈ C with Re α > 0. Our first concern is to show that ηα is in Hp∞ even in the case Re α = 0. 6.41. Theorem (Vinogradov). Let a ∈ L∞ and let {an }n∈Z denote its Fourier coefficients sequence. If there exists a constant A > 0 such that |an − an−k | ≤ A, |an − an+k | ≤ A |n|≥2k

|n|≥2k

for all k ∈ N, then a ∈ M p for all p ∈ (1, ∞). This result was established by Vinogradov [545]. He first showed the discrete analogue of Lemma 2.2 of Hörmander [288] (the proof in the discrete case is the same as the one in the continuous situation) and observed that then the above theorem can be proved in an analogous fashion as its continuous version (Theorem 2.2 of H¨ ormander [288]). 6.42. Theorem (Vinogradov). If Re α = 0 then ηα ∈ Hp∞ for all values p ∈ (1, ∞). Proof. Because ηα ∈ H ∞ , it suffices by the previous theorem to show that there is a constant A > 0 such that n≥2k |ηα,n − ηα,n−k | ≤ A for all k ∈ N. α n Using that ηα,n = (−1/τ ) it is easy to see that n ηα,n − ηα,n−k = (−1/τ )n−k

α n−k

3 1−

1+α n−k+1

4 1+α ... 1 − −1 . n

If z1 , . . . , zk are complex numbers and |zj | ≤ r for all j, then

6.4 Analytic Symbols

309

|(1 + z1 ) . . . (1 + zk ) − 1| ≤ |z1 + . . . + zk | + |z1 z2 + z1 z3 + . . . + zk−1 zk | + . . . + |z1 z2 . . . zk | k k rk = (1 + r)k − 1 ≤ k(1 + r)k−1 r. ≤ kr + r2 + . . . + k 2 Thus n≥2k

k−1 ++ α ++ |1 + α| |1 + α| + + , |ηα,n − ηα,n−k | ≤ + n − k +k 1 + n − k + 1 n−k+1 n≥2k

+ + + + cα α + + + n−k +≤ n−k+1

and since

(Lemma 6.21) and k−1 k−1 |1 + α| |1 + α| ≤ 1+ ≤ e|1+α| 1+ n−k+1 k−1

it follows that n≥2k |ηα,n − ηα,n−k | is not greater than cα e|1+α| k

n≥2k

1 ≤ cα e|1+α| k (n − k + 1)2

(

∞ k

(n ≥ 2k)

dx = cα e|1+α| . x2

6.43. Further results about analytic multipliers. All results listed in this section are essentially due to Vinogradov and Verbitsky. (a) Theorem 6.42 is also an immediate consequence of the following much more general theorem. Let a be a function which is analytic and bounded in some region of the form π z ∈ C : |z| < r, | arg(z − 1)| > δ r > 1, 0 ≤ δ < . (6.22) 2 Then the restriction a|T is in Hp∞ for all p ∈ (1, ∞). (b) Let b be a Blaschke product of the form (1.13). If the sequence {αn } of its zeros can be divided into a finite number of subsequences {αnk } having the property that |1 − αnk | < 1, (6.23) sup |1 − αnk+1 | k then b ∈ Hp∞ for all p ∈ (1, ∞). Note that the zeros αn are allowed to approach 1 tangentially. (c) Let b be a Blaschke product of the form (1.13). Then the following are equivalent:


310

(i) The sequence of the zeros {αn } is contained in some Stolz angle {z ∈ D : |1 − z| ≤ c(1 − |z|)}

(c > 1)

and can be divided into a finite number of subsequences {αnk } each of which satisfies (6.23). (ii) b is the restriction to D (or T) of a function which is analytic and bounded in some region of the form (6.22). (iii)

sup

|θb (eiθ )| < ∞.

θ∈[−π,π]

(d) Let b be a Blaschke product of the form (1.13). Then the following are equivalent: (iv) The sequence {αn } of the zeroth of b can be divided into a finite number of subsequences {αnk } which satisfy (6.23). (v) The variation of b(eiθ ) on the sets {π/2k+1 ≤ |θ| ≤ π/2k } is uniformly bounded with respect to k ∈ Z+ . Notice that each of the conditions (i)–(v) implies that b ∈ Hp∞ for all p ∈ (1, ∞). (e) That no conditions relating only to the moduli of the zeros of a Blaschke product b can guarantee that b is in Hp∞ (p = 2) is seen from the following result. Let {rn }∞ n=1 be any sequence of real numbers such that 0 < r1 ≤ r 2 ≤ . . . < 1

and

∞

(1 − rn ) < ∞.

n=1

Then there exists a sequence {ξn }∞ T, such that the Blaschke product n=1 , ξn ∈ M p. with the zeros {rn ξn } does not belong to 1 0. Denote the image of pµ (µ > −1/p) under the operator T (ξα ) by Rµp (α). On defining a norm in Rµp (α) by yRµp (α) := T (ξ−α )ypµ (note that, by Proposition 6.44, T (ξα ) is one-to-one on pµ ) we make Rµp (α) become a Banach space and T (ξα ) become an isometric isomorphism of pµ onto Rµ (α). Let Dµp (α) denote the linear set of all sequences x = {xn }∞ n=0 of complex numbers such that T (ηα )x ∈ pµ (µ < 1/q). Since T (ηα )x = 0 can only occur if x = 0, through xDµp (α) := T (ηα )xpµ a norm is defined in Dµp (α) and T (ηα ) is an isometric isomorphism of the Banach space Dµp (α) onto pµ . Finally, in case Re α = 0 put Dµp (α) = Rµp (α) = pµ . Now Proposition 6.45 can be formulated as follows. 6.47. Proposition. Let Re α ≥ 0. (a) If µ > −1/p, then the space pµ+Re α is continuously and densely embedded in the space Rµp (α). (b) If µ < 1/q, then the space Dµp (α) is continuously and densely embedded in the space pµ−Re α . Proof. The continuity of the embeddings is equivalent to Proposition 6.45 and their density can be verified straightforwardly.

314


6.48. Theorem. Let Re δ ≥ 0, Re γ ≥ 0, and −1/p < µ < 1/q. Suppose b ∈ Mµp and T (b) belongs to GL(pµ ). Then the operator T (η−γ )T −1 (b)T (ξ−δ ) belongs to L(pµ+Re δ , pµ−Re γ ), the operator T (ξδ ηγ b) is a (boundedly) invertible operator in L(Dµp (γ), Rµp (δ)), and the restriction of its inverse T −1 (ξδ ηγ b) to pµ+Re γ coincides with T (η−γ )T −1 (b)T (ξ−δ ). Proof. Because T (ξδ ηγ b) = T (ξδ )T (b)T (ηγ ), all assertions follow from the def inition of Dµp (γ) and Rµp (δ) and from Proposition 6.45. 6.49. Theorem. (a) Let Re γ + Re δ ≥ 0 and −1 < Re γ < 0. If |Re γ| −

1 1 λ, µ < 1/q, f ∈ F q−µ ;

(iii) −1/q < λ, µ > 1/q, f ∈ F qλ ; (iv) −1/q > λ, µ > 1/q, f ∈ F qmax{λ,−µ} .


315

If p = q = 2, then these conditions ensure that H(f ) is Hilbert-Schmidt from 2µ into 2λ . Proof. The operator H(f ) is compact from pµ into qλ (Hilbert-Schmidt in case p = q = 2) if ∞ n=0

H(f )en qq (n λ

−µq

+ 1)

=

∞

|fn |

n=0

q

n

(n − k + 1)λq (k + 1)−µq < ∞,

k=0

and so the assertion can be easily proved by straightforward estimation of the n sums k=0 (n − k + 1)λq (k + 1)−µq . 6.51. Lemma. Let 1 −1, and let ε > 0 be a real number which can be chosen as small as desired. (a) Each of the following conditions is sufficient for H(ξδ ηγ )H(f ) to be compact from pµ into pλ : (i)

µ ≤ 1/q, −1/p ≤ λ − σ < 1/q, f ∈ F q1/p+λ−σ−µ+ε ;

(ii)

µ ≤ 1/q, −1/p > λ − σ,

f ∈ F q−µ+ε ;

(iii) µ > 1/q, −1/p < λ − σ < 1/q, f ∈ F q1/p+λ−σ−1/q+ε . (b) Each of the following conditions is sufficient for H(f )H(ξδ ηγ ) to be compact from pµ into pλ : (i)

λ ≥ −1/p, −1/p < µ + σ ≤ 1/q, f ∈ F p1/q−µ−σ+λ+ε ;

(ii)

λ ≥ −1/p,

µ + σ > 1/q, f ∈ F pλ+ε ;

(iii) λ < −1/p, −1/p < µ + σ < 1/q, f ∈ F p1/q−µ−σ+1/p+ε . (c) If p = q = 2, then in (a) and (b) “compact” can be replaced by “trace class.” Proof. To prove (a) and (c) for the operator H(ξδ ηγ )H(f ) it suffices to choose an appropriate number τ and then to apply Lemma 6.50 and the fact that (ξδ ηγ )n = O(1/|n|1+Re γ+Re δ ) (|n| → ∞) (resulting from Lemma 6.18) to H(f ) : pµ → qτ and H(ξδ ηγ ) : qτ → pλ . The assertion for H(f )H(ξδ ηγ ) follows by taking adjoints. 6.52. Definition. Given a real number x define (x)◦ as (x)◦ := max{0, x}.

316


6.53. Proposition. Let Re α > 0, b+ ∈ H ∞ , b− ∈ H ∞ . (a) If µ > −1/p and b− ∈ F 1(−µ)◦ , then T (b− ) ∈ L(Rµp (α)). (b) If µ < 1/q and b+ ∈ F 1(µ)◦ , then T (b+ ) ∈ L(Dµp (α)). (c) If µ > −1/p and b+ ∈ F 1(µ)◦ ∩ F ps , where s > max{1/q, Re α + µ}, then T (b+ ) ∈ L(Rµp (α)). (d) If µ < 1/q and b− ∈ F 1(−µ)◦ ∩ F pr , where r > max{1/p, Re α − µ}, then T (b− ) ∈ L(Dµp (α)). Proof. First notice that T (b) ∈ L(Rµp (α)) ⇐⇒ T (ξ−α )T (b)T (ξα ) ∈ L(pµ ), T (b) ∈ L(Dµp (α)) ⇐⇒ T (ηα )T (b)T (η−α ) ∈ L(pµ ). If b− ∈ F 1(−µ)◦ , then T (b− ) ∈ L(pµ ), since the norms of V (−n) on pµ are O(1) for µ ≥ 0 and O(n|µ| ) for µ ≤ 0. This and the equality T (b− )T (ξα ) = T (ξα )T (b− ) give (a). Let us prove (c). Due to Proposition 2.14, T (ξ−α )T (b+ )T (ξα ) = T (b+ ) − T (ξ−α )H(b+ )H(ξ. α ). If b+ ∈ F 1(µ)◦ , then T (b+ ) ∈ L(pµ ). Using Lemma 6.51 one can show that p p p H(b+ )H(ξ. α ) ∈ L(µ , µ+Re α ) if b+ ∈ F s with s given as in the propop p sition. So Proposition 6.47(a) gives H(b+ )H(ξ. α ) ∈ L(µ , Rµ (α)) and thus p T (ξ−α )H(b+ )H(ξ. α ) ∈ L(µ ), as desired. Finally, (b) and (d) follow from (a) and (c) by passing to the adjoint: T (ηα )T (b± )T (η−α ) ∈ L(pµ ) ⇐⇒ T (ξ−α )T (b± )T (ξα ) ∈ L(q−µ ). Before proceeding to Toeplitz operators on Rµp (α) and Dµp (α) whose symbols are not necessarily analytic or anti-analytic we state a result which is 1,1 . also of interest by itself. Recall that W denotes the Wiener algebra F 0,0 6.54. Theorem (D. Horbach). If 1 ≤ r < ∞, 1 ≤ s < ∞, 0 ≤ α < ∞, 0 ≤ β < ∞, then W ∩ F r,s α,β is an algebra under pointwise multiplication. Proof. Because f g = (1/2)[(f + g)2 − f 2 − g 2 ], it suffices to show that a2 is r,s in F r,s α,β whenever a is in W ∩ F α,β . Let {an } be the Fourier coefficients sequence of a and put bn = |an |. We have, for n ≥ 0, + ∞ + ∞ s + + + + aj an−j + ≤ bj bn−j + + + j=−∞ j=−∞ −1 s ∞ n = bj bn−j + bj bn−j + bj bn−j j=−∞

j=n+1

j=0


∞

≤

b−j bn+j +

j=1

=2

s

n−[n/2]

bn+j b−j + 2

j=1

∞

s

∞

317

bj bn−j

j=0

s

n−[n/2]

b−j bn+j +

j=1

bj bn−j

.

(6.25)

j=0

If yj , zj are nonnegative real numbers and s ≥ 1, then

s

≤

yj zj

j

yj

s−1

j

yj zjs .

j

Indeed, the function f (x) = xs is convex and so Jensen’s inequality gives

s

yj zjs y z y z y f (zj )

j j

j j ≤

j =f =

. yj yj yj yj Thus, (6.25) is not greater than 2s

∞

s−1

n−[n/2]

b−j +

j=1

≤ 2s as−1 W

bj

j=0 ∞

∞

b−j bsn+j +

j=1

b−j bsn+j +

j=1

n−[n/2]

n

bj bsn−j

j=0

bn−j bsj .

j=[n/2]

It follows that N n=0

+ ∞ +s + + + (n + 1) + aj an−j + ≤ 2s as−1 W (σ1 + σ2 ), + + βs +

j=−∞

where σ1 :=

N

(n + 1)βs

n=0

∞

b−j bsn+j ,

σ2 :=

N

(n + 1)βs

n=0

j=1

n j=[n/2]

For σ1 we have σ1 ≤

N ∞

b−j (n + j + 1)βs bsn+j

n=0 j=1

=

∞ j=1

b−j

N

(n + j + 1)βs bsn+j ≤ aW asF r,s

n=0

and σ2 can be estimated as follows:

α,β

bn−j bsj .

318


σ2 ≤

N n

bn−j 2βs (j + 1)βs bsj

(since n + 1 ≤ 2(j + 1))

n=0 j=[n/2]

≤

∞ N

bn−j 2βs (j + 1)βs bsj

n=0 j=0

= 2βs

∞

(j + 1)βs bsj

N

bn−j ≤ 2βs asF r,s aW .

n=0

j=0

α,β

As N can be chosen arbitrarily, we arrive at the inequality ∞

(n + 1)βs |(a2 )n |s ≤ 2s (1 + 2βs )asW asF r,s . α,β

n=0

It can be shown analogously that ∞

(n + 1)αr |(a2 )−n |r ≤ 2r (1 + 2αr )arW arF r,s . α,β

n=0

6.55. Corollary. Let r, s, α, β be as in the preceding theorem. Suppose a function b ∈ W ∩F r,s α,β does not vanish on T and ind b = 0. Then b has a logarithm r,s in W ∩ F α,β and if we let G(b) := exp(log b)0 and ,∞ ±n (log b)±n t (t ∈ T), (6.26) b± (t) := exp n=1

then b = G(b)b− b+ and

b±1 ±

∈ W ∩ F r,s α,β .

Proof. Theorem 6.54 shows that W ∩ F r,s α,β is a Banach algebra with the ), where a constant c can be chosen so that norm a := c(aW + aF r,s α,β ab ≤ a b. It is easy to see that the maximal ideal space of this algebra is T. Hence 2.41(e) implies that b has a logarithm in W ∩ F r,s α,β and this gives the remaining assertions of the corollary immediately. 6.56. Theorem. Let Re α > 0 and suppose b ∈ C does not vanish on T and ind b = 0. 1,p (a) If µ > −1/p and b ∈ F 1,1 (−µ)◦ ,(µ)◦ ∩ F 0,s , where s is as in Proposition 6.53(c), then T (b) ∈ GL(Rµp (α)). 1,1 p,1 (b) If µ < 1/q and b ∈ F (µ) ◦ ,(−µ)◦ ∩ F r,0 , where r is as in Proposition 6.53(d), then T (b) ∈ GL(Dµp (α)). ±1 Proof. (a) Corollary 6.55 implies that b = b− b+ , where b±1 − and b+ satisfy the hypotheses of Proposition 6.53(a) and (c), respectively. This shows that −1 p T (b) = T (b− )T (b+ ) is in L(Rµp (α)) and that T (b−1 + )T (b− ) ∈ L(Rµ (α)) is the inverse of T (b).

(b) The proof is analogous.


319

6.57. Toeplitz operators on p,±∞ . Let p,+∞ (1 ≤ p < ∞) denote the linear space

∞

pµ . In this section µ stands for an integer. It is clear that

µ=0

p,+∞ regarded as a set does not depend on the value p ∈ [1, ∞). Define a metric on p,+∞ by d(x, y) :=

∞ 1 x − ypµ 2µ 1 + x − ypµ µ=0

(x, y ∈ p,+∞ ).

(6.27)

This metric makes p,+∞ into a Fréchet space and for different values of p the spaces p,+∞ are homeomorphic to each other. We let p,−∞ refer to the dual space of p,+∞ and think of p,−∞ as being provided with the strong ∞ topology. Note that p,−∞ = p−µ . The topologies on the corresponding µ=0

spaces of vector-valued sequences p,+∞ and p,−∞ can be introduced in a N N natural way. into p,±∞ is said to be A linear and bounded operator A of p,±∞ N N /Im A are finiteFredholm if its image is closed and both Ker A and p,±∞ N dimensional. In that case the index Ind±∞ A is defined by Ind±∞ A := dim Ker A − dim(p,±∞ /Im A). N ∞ If a ∈ CN ×N :=

∞ µ=0

F [pµ (Z)]N ×N , then T (a) is obviously bounded on

p,±∞ . Proposition 6.45(a) and Corollary 6.55 imply that if a ∈ C ∞ is of the N form a = bξα1 ,τ1 . . . ξαm ,τm with τj ∈ T, αj ∈ N, b ∈ C ∞ , b(t) = 0 for t ∈ T, and ind b = 0, then 1 −1 T (b−1 + )T (b− )T (ξ−α1 ,τ1 ) . . . T (ξ−αm ,τm ) G(b) is bounded on p,+∞ and is an (the) inverse of T (a). It turns out that these Toeplitz operators are the only invertible ones on p,+∞ . This is the consequence of the following result, whose sufficiency part is due to Pr¨ ossdorf [405] and whose necessity portion was proved in Silbermann [473]. p,±∞ ∞ Let a ∈ CN ×N . Then T (a) is Fredholm on N ×N if and only if det a has at most finitely many zeros of integral order on T.

If det a ∈ C ∞ has at most finitely many zeros of integral order on T, then det a can be written in the form γj m t τj δj det a(t) = b(t) (t ∈ T), (6.28) 1− 1− τj t j=1 where τ1 , . . . , τm are pairwise distinct points on T, γj ∈ Z+ , δj ∈ Z+ , and b ∈ C ∞ does not vanish on T. Notice that such a representation is not unique

320


(the sums γj + δj , however, are determined uniquely). If det a is of the form (6.28), then Ind+∞ T (a) = −

m

γj − ind b,

Ind−∞ T (a) = −

j=1

m

δj − ind b.

j=1

Moreover, one can show that T (a) is invertible on p,±∞ (scalar case!) if and only if T (a) is Fredholm in p,±∞ and has index zero. In particular, if a ∈ C ∞ , then T (a) is invertible on p,+∞ resp. p,−∞ if and only if a(t) = b(t)

m τj γj 1− t j=1

resp. a(t) = b(t)

δ m t j , 1− τj j=1

where τj ∈ T, γj resp. δj are nonnegative integers, b ∈ C ∞ does not vanish on T, and ind b = 0.

6.5 Notes and Comments 6.2. Almost all these results were established by Verbitsky [538]. 6.5–6.6. Under somewhat stronger hypotheses, such results are known from the work of Duduchava, Verbitsky, and Krupnik. 6.7–6.12. The theory of Toeplitz operators with continuous symbols on weighted p spaces was developed in the work of Duduchava [167], [169], Gohberg, Krupnik [232], Verbitsky, Krupnik [543]. The approach presented here is the authors’. 6.13–6.16. Zafran [585]. See also Nikolski [365] and Peller, Khrushchev [390, Section 3.4]. A naturally arising question is as follows: is T dense in M (C∩M p ) or has T a “corona”? 6.20. This formula was established by Duduchava [169] for the case γ + δ = 0. Because there is no p analogue of Theorem 5.5, the theory of Toeplitz operators on the spaces p with piecewise continuous symbols differs from the corresponding theory for the spaces H p significantly (although, curiously, the final results are almost the same in both cases). Duduchava’s formula T (ηβ )T (ξβ ) = Γβ,−β M−β T (ϕβ )Mβ was just the discovery on the basis of which the p theory of Toeplitz operators with P C symbols could be developed. When studying Toeplitz determinants with Fisher-Hartwig symbols we were led to the problem of proving that the finite section method (on 2 ) is applicable to the operator T −1 (ϕα )T (ϕβ )T −1 (ϕ−α ) with |Re α| < 1/2, |Re β| < 1/2. Since we had been unable to prove this in full generality, we asked Steffen Roch to try his hands. He proved what we wanted and, as a byproduct or, more precisely, as the key observation for his proof, he discovered


321

formula (6.16). It was published in B¨ ottcher, Silbermann [108], [110] for the first time. The original proofs of Duduchava and Roch were very complicated; the proof given here and in our paper [110] is due to the authors. 6.23–6.40. Propositions 6.29 and 6.30, Theorems 6.38(a) and 6.39, and the results of 6.40 go back to Duduchava [167], [169], Gohberg, Krupnik [232], and Verbitsky, Krupnik [543]. The approach presented here is new. This concerns in particular Propositions 6.36 and 6.37. The proof of Proposition 6.29 is the one given in B¨ ottcher, Silbermann [106], and Theorem 6.38(b) was established in B¨ ottcher [64]. The theory of Toeplitz operators with piecewise continuous symbols on p with general (discrete) Muckenhoupt weight is elaborated in Böttcher, Seybold [101]. 6.41–6.43. Theorems 6.41 and 6.42 are Vinogradov’s [545] (the luxurious proof of Theorem 6.42 given here is the author’s). 6.43(a) and 6.43(b) for the case that αn → 1 nontangentially are in Vinogradov [545], 6.43(b), (c), (d) can be found in Verbitsky [541], [542], 6.43(e), (f) were established in Vinogradov [547], and 6.43(g) is contained in Verbitsky [541], [542]. 6.44–6.49. A major part of these results is well known (see Pr¨ ossdorf [406] and Pr¨ ossdorf, Silbermann [407]), the presentation follows B¨ ottcher, Silbermann [111]. The difficult part of Proposition 6.44 (Re α = 0) and Theorems 6.48 and 6.49 were first obtained in the latter paper. Both the result and the proof of Proposition 6.45 are Pomp’s [397]; we have already remarked in the text that Pomp’s result is very nontrivial. 6.50–6.53. The tedious Lemmas 6.50 and 6.51 were established in Böttcher, Silbermann [111]. Results like Proposition 6.53 are in Pr¨ ossdorf [406] and Pr¨ ossdorf, Silbermann [407]. 6.54. The first result along these lines go back to Hirschman [281] and Krein 2,2 ∞ ∩ F 1/2,1/2 are [323], who showed, respectively, that W ∩ F 2,2 1/2,1/2 and L algebras. Since that time it has been noticed (but as far as we know not published) by several people that W ∩ F r,s α,β is an algebra for certain values of α, β, r, s. In particular, in B¨ ottcher, Silbermann [106] we pointed out that this is so if either r = s = 2, α ≥ 0, β ≥ 0 or r > 1, s > 1, 1/r + 1/s = 1, α > 1/s, β > 1/r. In 1984, during an examination in mathematical analysis, we asked Detlef Horbach, a gifted second year student of ours, the question of whether W ∩ F r,s α,β is an algebra. A few weeks later he reported to us the proof given in the text. It is certainly typical that such “naive” proofs can only be found by students but not by “professionals.” A. Karlovich [300] generalized Theorem 6.54 to the setting of Orlicz spaces with weights wn satisfying w2n ≤ Cwn . 6.56. Böttcher, Silbermann [111]. 6.57. The study of Toeplitz operators on spaces of generalized functions was originated by Cherski [133]. Dybin and Karapetyants [181] were the first to

322


show that a Toeplitz operator is Fredholm on p,±∞ if its symbol is in C ∞ and has at most finitely many zeros of integral order on T. Independently, Pr¨ ossdorf [405] established this result (for the matrix case) and developed a systematic theory of singular integral equations and convolution equations with “degenerate” symbols on locally convex vector spaces. Silbermann [473] finally showed that the symbol of a Toeplitz operator which is Fredholm on p,±∞ N ×N can have at most finitely many zeros of integral order. For more about this topic see Pr¨ ossdorf [406]. Some developments of the matter (e.g. symbols with countably many zeros) are illuminated in the books by Dybin [179] and Dybin, Grudsky [180].

7 Finite Section Method

7.1 Basic Facts 7.1. Projection methods. Let X and Y be Banach spaces and A ∈ L(X, Y ). A projection method is a method for the approximate solution of the equation Ax = y

(7.1)

which can be described as follows. Let {Pn } and {Rn } be sequences of projections Pn ∈ L(X) and Rn ∈ L(Y ) with the property Pn → IX and Rn → IY strongly, and let An : Pn X → Rn Y be certain given bounded operators. For example, one can take An = Rn APn |Pn X. We shall frequently identify An with An Pn and may therefore regard An as an element of L(X, Y ). Now consider the equation An xn = Rn y,

xn ∈ Pn X.

(7.2)

We write A ∈ Π{X, Y ; An } if (i) there exists an n0 such that for each y ∈ Y equation (7.2) has a unique solution xn ∈ Pn X for all n ≥ n0 ; (ii) xn converges in the norm of X to a solution x ∈ X of equation (7.1). If X = Y and Pn = Rn , then Π{X, Y ; An } will be abbreviated to Π{X; An } and if there is no fear of confusion even to Π{An }. In case An = Pn APn |Pn X we shall write Π{X, Y ; Pn } and Π{X; Pn } (or even Π{Pn }) in place of Π{X, Y ; Pn APn } and Π{X, X; Pn APn }, respectively. 7.2. Algebraization and essentialization. Let X be a Banach space and ∞ = D∞ (X) let {Pn }∞ n=0 be a sequence of projections in L(X). We let D ∞ denote the collection of all sequences {An }n=0 of operators An : Pn X → Pn X such that sup An Pn L(X) < ∞. On defining n≥0

α{An } := {αAn },

{An } + {Bn } := {An + Bn },

{An }{Bn } := {An Bn },

324


and {An } := sup An Pn L(X) we make D∞ into a Banach algebra. If X is n≥0

a Hilbert space, D∞ is a C ∗ -algebra with the involution {An }∗ := {A∗n }. Let D = D(X) denote the (closed) subalgebra of D∞ consisting of all sequences {An } ∈ D∞ for which there exists an A ∈ L(X) such that An → A strongly on X. Let G = G(X) refer to the set of all sequences {An } ∈ D∞ with An Pn L(X) → ∞ as n → ∞. It is easy to see that G is a closed two-sided ideal of both D∞ and D. For {An } ∈ D∞ , let {An }πG stand for the coset {An } + G. If A ∈ L(X), if the projections Pn converge strongly to I on X and if Pn = 1 for all n, then {Pn APn }πG = AL(X) .

(7.3)

It is clear that in (7.3) “≤” holds. On the other hand, if {Cn } is any element of G, then, by 1.1(e), A ≤ lim inf Pn APn + Cn ≤ sup Pn APn + Cn = {Pn APn } + {Cn }, n→∞

n

which implies the “≥” in (7.3). 7.3. Proposition. Suppose Pn → I and An Pn → A strongly as n → ∞. Then the following are equivalent. (i) A ∈ Π{X; An }. (ii) A ∈ GL(X), An ∈ GL(Pn X) for all sufficiently large n (n ≥ n0 , say), and sup A−1 n Pn L(X) < ∞. n≥n0

(iii) A ∈ GL(X), An ∈ GL(Pn X) for all sufficiently large n, and A−1 n Pn converges to A−1 strongly on X as n → ∞. (iv) {An }πG ∈ G(D/G). (v) A ∈ GL(X) and {An }πG ∈ G(D∞ /G). Proof. (i) =⇒ (ii). It is clear that An ∈ GL(Pn X) for all n ≥ n0 . From 1.1(e) we deduce that sup A−1 n Pn =: C < ∞. Hence n≥n0

Pn x ≤ CAn Pn x ∀ x ∈ X

∀ n ≥ n0 .

(7.4)

Passage to the limit n → ∞ in (7.4) gives the inequality x ≤ CAx. Thus, Ker A = {0} and since the definition of Π{X; An } involves that Im A = X, we have A ∈ GL(X). (ii) =⇒ (iii). Again (7.4) holds and so, for n ≥ n0 and y ∈ X, −1 −1 A−1 y ≤ A−1 y + Pn A−1 y − A−1 y n Pn y − A n Pn y − Pn A −1 ≤ CPn y − An Pn A y + Pn A−1 y − A−1 y. (7.5)

7.1 Basic Facts

325

−1 Since Pn → I and An Pn → A strongly, it follows that A−1 y for n Pn y → A each y ∈ X.

(iii) =⇒ (iv). Suppose An ∈ GL(Pn X) for n ≥ n0 . Put Bn = Pn for n < n0 ∞ and Bn = A−1 n Pn for n ≥ n0 . From 1.1(e) we obtain that {Bn } ∈ D , and −1 −1 since An Pn converges strongly to A , we actually have {Bn } ∈ D. Because {Bn }{An } − {Pn } ∈ G and {An }{Bn } − {Pn } ∈ G, it results that {Bn }πG is the inverse of {An }πG . (iv) =⇒ (v). Obvious. (v) =⇒ (i). Let {Bn }πG be the inverse of {An }πG . Then Bn An = Pn + Cn and An Bn = Pn + Dn with certain {Cn } and {Dn } in G. There is an n0 such that Cn < 1/2 and Dn < 1/2 for all n ≥ n0 . Thus Pn + Cn and Pn + Dn are in GL(Pn X) for all n ≥ n0 and the norms of their inverses are uniformly bounded by 1/(1 − 1/2) = 2. This implies that An ∈ GL(Pn X) for all n ≥ n0 and that sup A−1 n Pn ≤ 2 sup Bn =: C < ∞. Consequently, again (7.4) n≥n0

n≥n0

and thus (7.5) holds, which shows that A ∈ Π{X; An }. 7.4. Corollary. If Pn converges strongly to I, then Π{X; Pn } is an open subset of L(X). Proof. This is immediate from the equivalence (i) ⇐⇒ (iv) of the preceding proposition applied to An = Pn APn |Im Pn . 7.5. The finite section method. Let Pn (n = 0, 1, 2, . . .) denote the projections acting on p,µ N by the rule Pn : {x0 , x1 , x2 , , . . .} → {x0 , x2 , . . . , xn , 0, 0, . . .}.

(7.6)

Here xk ∈ CN . It is clear that Pn L(p,µ = 1 and that Pn converges strongly N ) p,µ to I on N for all p ∈ [1, ∞) and µ ∈ R. p (w) by Define Pn (n = 0, 1, 2, . . .) on HN Pn :

∞ k=0

ϕk χk →

n

ϕk χk

(ϕk ∈ CN ).

(7.7)

k=0

Here and throughout what follows we suppose that the assumptions 5.1 are p (w)) satisfied. Because Pn ϕ = ϕ−χn+1 P (χ−n−1 ϕ), it follows that Pn ∈ L(HN and that sup Pn L(HNp (w)) < ∞. Moreover, since Pn ϕ − ϕHNp (w) → 0 as n≥0

n → ∞ for each polynomial ϕ ∈ PA , we deduce from 1.1(d) that Pn → I p (w). strongly on HN p Let X be p,µ N or HN (w), let the projections Pn be given by (7.6) or (7.7), and let A ∈ L(X). If A ∈ Π{X; Pn }, then the finite section method is said to be applicable to A on X. We are mainly interested in the case where A is a bounded Toeplitz operator T (a) on X. Note that the operators Pn T (a)Pn : Pn X → Pn X may

326


be identified with the finite block Toeplitz matrices Tn (a) := (aj−k )nj,k=0 (where ai is the i-th matrix Fourier coefficient of a). Thus, by Proposition 7.3, T (a) ∈ Π{X; Pn } if and only if T (a) is invertible on X, if the matrices Tn (a) are invertible for all sufficiently large n, and if Tn−1 (a)Pn → T −1 (a) strongly on X as n → ∞. Throughout the rest of the chapter suppose 1 < p < ∞ (if it is not explicitly stated otherwise). Our first concern is to establish a formula which plays the same role in the theory of the finite section method for Toeplitz operators as identity (2.18) in their Fredholm theory. 7.6. The operator Wn . Define the linear operators Wn (n = 0, 1, 2, . . .) on p p,µ N and HN (w) by Wn : {x0 , x1 , x2 , . . .} → {xn , xn−1 , . . . , x0 , 0, 0, . . .}, ∞ n Wn : ϕk χk → ϕn−k χk , k=0

k=0

p respectively. It is clear that Wn ∈ L(p,µ N ). We have Wn L(N ) = 1 and it is p,µ easily seen that sup Wn L(N ) < ∞ if and only if µ = 0. Because (Wn ϕ)(t)

n≥0

equals tn (Pn ϕ)(1/t) (t ∈ T), the equality p Wn L(HNp (w)) = Pn L(HNp (w),H ) N (w))

(7.8)

holds; here w(t) ) := w(1/t). Since Pn ϕHNp (w) ≤ M (n, p, N, w)Pn ϕHNp (w) )

p ∀ ϕ ∈ HN (w) )

(note that any two norms on a finite-dimensional space are equivalent to each p p other) and Pn is bounded on HN (w) ) (by 7.5), we obtain that Wn ∈ L(HN (w)). Using (7.8) it is not difficult to see that sup Wn L(HNp (w)) < ∞ if and only if n

H p (w) ) is continuously embedded in H p (w). In particular, sup Wn L(HNp ) < ∞. n

The following identities are extremely important and can be verified straightforwardly: Wn2 = Pn ,

Wn Pn = Pn Wn = Wn ,

Wn Tn (a)Wn = Tn () a),

where, as usual, ) a(t) = a(1/t) (t ∈ T). Also notice that Wn converges weakly p (1 < p < ∞). to zero on pN as well as HN Finally, throughout what follows let Qn := I − Pn , where I is the identity p operator on p,µ N or HN (w).

7.1 Basic Facts

327

p,µ 7.7. Proposition. Let a, b ∈ L∞ N ×N resp. a, b ∈ MN ×N . Then

Pn T (a)Qn T (b)Pn = Wn H() a)H(b)Wn , Tn (ab) = Tn (a)Tn (b) + Pn H(a)H()b)Pn + Wn H() a)H(b)Wn .

(7.9) (7.10)

Proof. We have Pn T (a)Qn T (b)Pn = Wn (Wn T (a)Qn )(Qn T (b)Wn )Wn and since, with V (±n) := T (χ±n I), Wn T (a)Qn = Pn H() a)V (−n−1) ,

Qn T (b)Wn = V n+1 H(b)Pn ,

(7.11)

identity (7.9) follows. Formula (2.18) gives Tn (ab) = Pn T (ab)Pn = Pn T (a)T (b)Pn + Pn H(a)H()b)Pn = Pn T (a)Pn T (b)Pn + Pn T (a)Qn T (b)Pn + Pn H(a)H()b)Pn and now (7.10) results from (7.9). 7.8. Toeplitz-adapted algebraization and essentialization. For a Toeplitz operator to be Fredholm means to have an inverse modulo the ideal C∞ of compact operators. According to (2.18) we have T (a)T (b) = T (ab) − H(a)H()b),

(7.12)

and there are many cases in which the product of Hankel operators in (7.12) is known to be compact. Then T (a)T (b) equals T (ab) modulo C∞ and from this fact at long last all what we know about the Fredholm theory of Toeplitz operators follows. In particular, the applicability of local principles essentially rests on (7.12) and the circumstance that H(f ) ∈ C∞ for f ∈ C + H ∞ . The analogue of (7.12) for finite Toeplitz matrices is formula (7.10): Tn (a)Tn (b) = Tn (ab) − Pn H(a)H()b)Pn − Wn H() a)H(b)Wn .

(7.13)

In view of Proposition 7.3 the applicability of the finite section method to T (a) is equivalent to the existence of the inverse {An } ∈ D of {Tn (a)} ∈ D modulo the ideal G. Looking for a connection between G and (7.13) we observe that the ideal G is, in a sense, too small: sequences of the form {Pn KPn + Wn LWn }

(K, L ∈ C∞ )

(7.14)

do, in general, not belong to G. Thus, in order to develop a theory of the finite section method in analogy to the Fredholm theory, it would be desirable to have an ideal J that contains all sequences of the form (7.14). But there is no such ideal in D. This algebra is, again in a certain sense, too large. We therefore shall construct a smaller algebra possessing, on the one hand,

328


such an ideal and containing, on the other hand, sufficiently many interesting elements, in particular, all elements of the form {Tn (a)}. p (note that no weight is allowed), let Pn and Wn Let X be either pN or HN be as in 7.5 and 7.6, and define D∞ , D, G as in 7.2. Given An ∈ L(Pn X) define )n ∈ L(Pn X) as A )n := Wn An Wn . Let S = S(X) denote the subset of D∞ A ) ∈ L(X) consisting of all {An } ∈ D∞ for which there exist A ∈ L(X) and A such that An Pn → A,

A∗n Pn → A∗ ,

)n Pn → A, ) A

)∗n Pn → A∗ . A

Here “→” denotes strong convergence and the asterisk refers to the adjoint p with the operator whose operator on X ∗ . On identifying an operator B on HN ◦

·

p matrix representation with respect to the decomposition LpN = (H pN )− + HN I 0 p ∗ q is , we may identify (HN ) with HN (recall 2.39). It is easy to see that 0B S is a closed subalgebra of D, i.e., S itself is a Banach algebra. Obviously, G ⊂ S. If K ∈ C∞ (X), then, by virtue of 1.1(f), {Pn KPn } and {Wn KWn } are in S(X) (note that Pn∗ = Pn , Wn∗ = Wn ). The identity a) implies that {Tn (a)} ∈ S(X) whenever T (a) ∈ L(X). Wn Tn (a)Wn = Tn () Let J = J (X) denote the collection of all elements {An } ∈ D∞ (X) of the form {An } = {Pn KPn + Wn LWn + Cn }, where K ∈ C∞ (X), L ∈ C∞ (X), {Cn } ∈ G(X). Clearly, J is a subset of S.

7.9. Proposition. J is a closed two-sided ideal of S. Proof. We first show that J is closed. If {Bn } = {Pn KPn +Wn LWn +Cn } ∈ J then, by 1.1(e), K ≤ lim inf Bn and L ≤ lim inf Bn . Thus, if n→∞

n→∞

∞ (j) {A(j) Pn + Wn L(j) Wn + Cn(j) n }j=1 = Pn K

∞ j=1

⊂J

is a Cauchy sequence, then {K (j) } and {L(j) } are Cauchy sequences in C∞ . Consequently, there exist K and L in C∞ such that K − K (j) → 0 and L − L(j) → 0 as j → ∞. But if ai = ki + li + ci and {ai }, {ki }, {li } are (j) Cauchy sequences, then so also is {ci }. Hence, {Cn }∞ j=1 is a Cauchy sequence (j)

in G and thus, {Cn } − {Cn } → 0 as j → ∞ for some {Cn } ∈ G. It follows that as j → ∞, {A(j) n } → {Pn KPn + Wn LWn + Cn } ∈ J which proves that J is closed. Now let {An } = {Pn KPn + Wn LWn + Cn } ∈ J and {Bn } ∈ S. Then Bn An = Bn Pn KPn + Bn Wn LWn + Bn Cn )n Pn LWn + Bn Cn = Bn Pn KPn + Wn B ) = Pn BKPn + Wn BLW n + Pn (Bn Pn − B)KPn )n Pn − B)LW ) +Wn (B n + Bn Cn

7.1 Basic Facts

329

and the Banach space version of 1.3(d) with p = ∞ implies that {Bn An } ∈ J . It can be shown similarly that {An Bn } ∈ J . p , put 7.10. Definitions. For X = pN or X = HN

SJπ = SJπ (X) := S(X)/J (X) and denote the coset {An } + J by {An }πJ . If A ∈ L(X), {An } ∈ S(X), and An Pn → A strongly on X, then the strong limit s- lim Wn An Wn will be n→∞

denoted by W{An } (A). If An = Pn APn , then W{An } (A) will be abbreviated to W(A). 7.11. Theorem. Let A ∈ L(X) and let {An } ∈ S(X) be any sequence such that An Pn → A strongly on X. Then A ∈ Π{X; An } ⇐⇒ A ∈ GL(X), W{An } (A) ∈ GL(X), {An }πJ ∈ GSJπ (X). If A ∈ Π{X; An } and if {Bn }πJ ∈ SJπ (X) is the inverse of {An }πJ in SJπ (X), then π Bn + Pn (A−1 − B)Pn + Wn [W{An } (A)]−1 − W{Bn } (B) Wn G (7.15) is the inverse of {An }πG in D(X)/G(X); here B := s- lim Bn . n→∞

Proof. Suppose A ∈ Π{An }. Then A ∈ GL(X) by Proposition 7.3. Let us ) := W{A } (A) ∈ GL(X). Again from Proposition 7.3 we deduce prove that A n ) that An := Wn An Wn is invertible for all sufficiently large n (n ≥ n0 ) and that −1 )−1 sup A n Pn = sup Wn An Wn

n≥n0

n≥n0

≤ sup Wn 2 A−1 n Pn =: C < ∞.

(7.16)

n≥n0

)n Pn ϕ for all ϕ ∈ X, and passing to the limit n → ∞ Hence Pn ϕ ≤ CA ) we obtain ϕ ≤ CAϕ for all ϕ ∈ X. It can be shown analogously that ∗ ) ϕ for all ϕ ∈ X ∗ . This implies that A ) ∈ GL(X). ϕ ≤ CA ) ∈ Π{A )n }, A∗ ∈ Π{A )∗n }, Using (7.16) and Proposition 7.3 we see that A ∗ ∗ ) ) and A ∈ Π{An }. Let {Bn } ∈ D be the inverse of {An } modulo G. Then Bn An = Pn + Cn with {Cn } ∈ G and thus, )n = (Pn + Wn Cn Wn )A )−1 )−1 , B n Pn → A Bn∗ = (A∗n )−1 (Pn + Cn∗ )Pn → (A∗ )−1 , )n∗ = (A )∗n )−1 (Pn + Wn Cn∗ Wn )Pn → (A )∗ )−1 B (strong convergence). Therefore {Bn } ∈ S and it follows that {Bn }πJ is the inverse of {An }πJ , that is, {An }πJ ∈ GSJπ .

330


) are invertible on X and {An }π is invertible in S π . Now suppose A and A J J Then there is a sequence {Bn } ∈ S such that An Bn = Pn + Pn KPn + Wn LWn + Cn ,

(7.17)

where K and L are in C∞ (X) and {Cn } ∈ G. Passage to the limit n → ∞ gives AB = I + K, and if we multiply (7.17) by Wn from the left and the right and )B ) = I + L, where then pass to the limit n → ∞, we arrive at the equality A −1 −1 ) ) ) =: T ∈ C∞ (X). B := W{An } (B). Hence A − B =: R ∈ C∞ (X) and A − B Put Bn := Bn + Pn RPn + Wn T Wn . Then {Bn } ∈ S and An Bn = Pn + Pn (K + An Pn R)Pn + Wn (L + An Pn T )Wn + Cn ) )Wn + Cn = Pn + Cn , = Pn + Pn (K + AR)Pn + Wn (L + AT where {Cn } ∈ G. It can be shown analogously that {Bn An } − {Pn } ∈ G. This gives the remaining assertion of the theorem. Our next objective is to show that {Pn APn } is in S(H p ) (resp. S(p )) if A is in algL(H p ) T (L∞ ) (resp. algL(p ) T (M p )). We also want to compute W(A) = s- lim Wn AWn for these cases. n→∞

p or pN and let LT (X) denote the collection 7.12. Definitions. Let X be HN of all operators A ∈ L(X) for which the four strong limits

T (A) := s- lim V (−n−1) AV n+1 ,

W(A) := s- lim Wn AWn ,

H(A) := s- lim V (−n−1) AWn ,

K(A) := s- lim Wn AV n+1

n→∞

n→∞

n→∞

n→∞

exist and are in L(X). The “LT ” is for “like Toeplitz.” 7.13. Theorem. LT (X) is a closed subalgebra of L(X). If A, B ∈ LT (X), then T (AB) = T (A)T (B) + H(A)K(B),

(7.18)

W(AB) = W(A)W(B) + K(A)H(B),

(7.19)

H(AB) = H(A)W(B) + T (A)H(B),

(7.20)

K(AB) = W(A)K(B) + K(A)T (B).

(7.21)

Proof. It is clear that LT (X) is a linear set. To see that LT (X) is an algebra, it suffices to verify identities (7.18)–(7.21). But these follow from the obvious equality I = Pn + Qn = Wn2 + V n+1 V (−n−1) : V (−n−1) ABV n+1 = V (−n−1) AWn Wn BV n+1 +V (−n−1) AV n+1 V (−n−1) BV n+1 → H(A)K(B) + T (A)T (B) strongly as n → ∞,

7.1 Basic Facts

331

and analogously for (7.19)–(7.21). It remains to show that LT (X) is closed. From 7.5 and 7.6 we know that M := sup V (−n) , V n , Pn , Wn < ∞. n≥0

Hence, by 1.1(e), T (A), W(A), H(A), K(A) ≤ M 2 A ∀ A ∈ LT (X).

(7.22)

Now let Ak ∈ LT (X), A ∈ L(X), and suppose A−Ak → 0 as k → ∞. From (7.22) we deduce that {T (Ak )} is a Cauchy sequence in L(X). Consequently, there is a B ∈ L(X) such that B − T (Ak ) → 0 as k → ∞. Thus, if x ∈ X, then V (−n) AV n x − Bx ≤ V (−n) AV n x − V (−n) Ak V n x +V (−n) Ak V n x − T (Ak )x + T (Ak )x − Bx,

(7.23)

and given ε > 0 there is a k0 = k0 (ε) such that the first and the third terms on the right of (7.23) are smaller than ε/2 for k = k0 and all n ∈ Z+ , and then one can find an n0 = n0 (k0 , ε) such that the second term on the right of (7.23) becomes smaller than ε/2 for k = k0 and n ≥ n0 . Hence, s- lim V (−n) AV n n→∞ exists and equals B. It can be shown similarly that the remaining three limits occurring in 7.12 also exist for A. In accordance with 3.43 (also recall Proposition 4.1 and Corollary 4.3) we let SmbT denote the continuous extension of the mapping given by SmbT :

n m

T (ajk ) →

j=1 k=1

n m

ajk

j=1 k=1

p p to algL(HNp ) T (L∞ N ×N ) (resp. algL(N ) T (MN ×N )). Note that SmbT is a continp ∞ uous algebraic homomorphism onto LN ×N (resp. MN ×N ).

7.14. Corollary. We have p algL(HNp ) T (L∞ N ×N ) ⊂ LT (HN ),

p p algL(pN ) T (MN ×N ) ⊂ LT (N ).

(7.24)

p p If A is in algL(HNp ) T (L∞ N ×N ) resp. algL(N ) T (MN ×N ), then {Pn APn } belongs p p to S(HN ) resp. S(N ) and

T (A) = T (SmbT A),

W(A) = T ((SmbT A))),

(7.25)

H(A) = H(SmbT A),

K(A) = H((SmbT A))).

(7.26)

Proof. Because V (−n) T (a)V n = T (a), V

(−n−1)

Wn T (a)Wn = Pn T () a)Pn → T () a),

T (a)Wn = H(a)Pn → H(a),

Wn T (a)V n+1 = Pn H() a) → H() a),

332


it follows that every bounded Toeplitz operator belongs to LT (X), and since LT (X) is a closed algebra, we arrive at the inclusions (7.24). If A is in ∗ ∞ q algL(HNp ) T (L∞ N ×N ), then A belongs to algL(HN ) T (LN ×N ). So (7.24) shows p q that the limits W(A) and W(A∗ ) exist and belong to L(HN ) and L(HN ), p respectively, which implies that {Pn APn } ∈ S(HN ). The proof is analogous for pN . From the computations at the beginning of this proof we see that (7.25) and (7.26) hold for A = T (a). Identities (7.18)–(7.21)

and (2.18)–(2.19) then 5m n show that (7.25) and (7.26) are true for A of the form j=1 k=1 T (ajk ), and from the continuity of SmbT , T , W, H, K (recall (7.22)) we obtain (7.25) and (7.26) for the general case. The following proposition provides a further important tool for the study of the finite section method. 7.15. Proposition. Let X1 and X2 be linear spaces, A : X1 → X2 a linear and invertible operator, P1 : X1 → X1 and P2 : X2 → X2 linear projections, and put Q1 = I − P1 and Q2 = I − P2 . Then P2 AP1 : Im P1 → Im P2 is invertible if and only if Q1 A−1 Q2 : Im Q2 → Im Q1 is invertible. In that case P1 (P2 AP1 )−1 P2 = P1 A−1 P2 − P1 A−1 Q2 (Q1 A−1 Q2 )−1 Q1 A−1 P2 .

(7.27)

Proof. (Kozak). P2 AP1 P1 A−1 P2 − P1 A−1 Q2 (Q1 A−1 Q2 )−1 Q1 A−1 P2 = P2 AP1 A−1 P2 − P2 AP1 A−1 Q2 (Q1 A−1 Q2 )−1 Q1 A−1 P2 = P2 AP1 A−1 P2 − P2 AA−1 Q2 (Q1 A−1 Q2 )−1 Q1 A−1 P2 +P2 AQ1 A−1 Q2 (Q1 A−1 Q2 )−1 Q1 A−1 P2 = P2 AP1 A−1 P2 − 0 + P2 AQ1 A−1 P2 = P2 . 7.16. Corollary. Suppose X1 , X2 are Banach spaces and A ∈ L(X1 , X2 ) is (boundedly) invertible. Let Pn1 ∈ L(X1 ) and Pn2 ∈ L(X2 ) be projections converging strongly to the identity operator on X1 and X2 , respectively, and put Q1n = I − Pn1 , Q2n = I − Pn2 . Then A ∈ Π{X1 , X2 ; Pn2 APn1 } if and only if Q1n A−1 Q2n : Q2n X2 → Q1n X1 is invertible for all sufficiently large n (n ≥ n0 , say) and if sup (Q1n A−1 Q2n )−1 Q1n L(X1 ,X2 ) < ∞. n≥n0

Proof. Immediate from Propositions 7.3 and 7.15. 7.17. Corollary. Let X be a Banach space, let Pn ∈ L(X) be projections, and suppose Pn → I strongly on X. If A ∈ Π{X; Pn }, K ∈ C∞ (X), and A + K ∈ GL(X), then A + K ∈ Π{X; Pn }. Proof. A ∈ Π{Pn } implies that A ∈ GL(X). We have (A + K)−1 − A−1 = −(A + K)−1 KA−1 =: L ∈ C∞ (X).

7.1 Basic Facts

333

By the preceding corollary, Qn A−1 Qn : Qn X → Qn X is invertible for all sufficiently large n and the norms of the inverses are uniformly bounded. Hence Qn (A + K)−1 Qn = Qn A−1 Qn I + (Qn A−1 Qn )−1 Qn LQn , and because Qn LQn converges uniformly to zero as n → ∞, it follows that Qn (A + K)−1 Qn : Qn X → Qn X is invertible for all n large enough and that the norms of the inverses are uniformly bounded (by Neumann’s series expansion). Once more applying the previous corollary we get the assertion. Note that the hypotheses of the preceding two corollaries are satisfied for p (w) and X = X1 = X2 = p,µ X = X1 = X2 = H N N and Pn as in 7.5. The following corollary is another curious consequence of Corollary 7.16. 7.18. Corollary. We have p p −1 (a) ∈ Π{HN (w); Pn }, a ∈ L∞ N ×N , T (a) ∈ GL(HN (w)) =⇒ T p p −1 a ∈ MN (a) ∈ Π{pN ; Pn }. ×N , T (a) ∈ GL(N ) =⇒ T

Proof. Because V (−n−1) V n+1 = I,

V n+1 V (−n−1) = Qn ,

V (−n−1) T (a)V n+1 = T (a),

we have Qn T (a)Qn V n+1 T −1 (a)V (−n−1) = Qn , V n+1 T −1 (a)V (−n−1) Qn T (a)Qn = Qn , whence (Qn T (a)Qn )−1 Qn = V n+1 T −1 (a)V (−n−1) , and since sup V (±n) L(X) < ∞

n≥0

p (w) or X = pN , the assertion follows from Corollary 7.16. for X = HN

Open problem. Is the above result true for the space p,µ N ? We finally establish some connection between the Toeplitz operators generated by the matrix functions a, ) a, a∗ , a−1 . Recall that ) a(t) := a(1/t),

a∗ (t) := a(t)

(t ∈ T),

∗ where

denotes transposition. For N = 1, a is also denoted by a. Thus, if a = n∈Z an χn , then ) a= a−n χn , a∗ = a∗−n χn . n∈Z

n∈Z

334


p ∞ 7.19. Proposition. Let X = HN (w) (resp. p,µ N ) and let a ∈ GLN ×N (resp. p,µ q ∗ −1 a ∈ GMN ) (resp. X ∗ = q,−µ ). ×N ). Put X = HN (w N

(a) T (a) ∈ GL(X) ⇐⇒ T (a∗ ) ∈ GL(X ∗ ). (b) T () a) ∈ GL(X) ⇐⇒ T (a−1 ) ∈ GL(X ∗ ). (c) If X = H p (N = 1, w = 1) or X = pµ (N = 1), then T () a) ∈ GL(X) ⇐⇒ T (a) ∈ GL(X). (d) There exist a ∈ P2×2 such that T (a) ∈ GL(H22 ) but T () a) ∈ / GL(H22 ). (e) T (a) ∈ Φ(X) ⇐⇒ T (a∗ ) ∈ Φ(X ∗ ).

IndX T (a) = −IndX ∗ T (a∗ ).

(f) T () a) ∈ Φ(X) ⇐⇒ T (a−1 ) ∈ Φ(X).

IndX T () a) = IndX T (a−1 ).

(g) If X = H p (N = 1, w = 1) or X = pµ (N = 1), then T () a) ∈ Φ(X) ⇐⇒ T (a) ∈ Φ(X).

IndX T () a) = IndX T (a).

Proof. (a), (e) This is obvious for X = p,µ N , and the arguments of the proof p (w). of Lemma 2.39 can be used for X = HN (b) If T () a) ∈ GL(X) resp. T (a−1 ) ∈ GL(X), then T (a)−H(a)T −1 () a)H() a) −1 a−1 )T −1 (a−1 )H(a−1 ) is the inverse of T (a−1 ) resp. T () a). resp. T () a ) − H() This can be easily verified using Proposition 2.14. (c) This follows from the identity T () a) = W T (a)W , where W is given by (W ϕ)(t) = ϕ(1/t) for ϕ ∈ H p and (W x)n = xn for {xn } ∈ pµ . (d) Let a = (aij )2i,j=1 , where a11 = χ1 , a12 = 1, a21 = 0, a22 = χ−1 . Then a = g− dg+ , where a = h− h+ and ) 1 0 t 1 h− (t) = , , h+ (t) = t−1 1 −1 0 −1 t 1 01 t 0 g− (t) = (t) = , , d(t) = , g + 10 1 0 0 t−1 ∞ and h , g ∈ GH ∞ , we deduce that Since h− , g− ∈ GH2×2 + + 2×2

T (a) ∈ GL(H22 ),

dim Ker T () a) = dim Coker T () a) = 1.

(f) If R and S are regularizers of T () a) and T (a−1 ), respectively, then, a−1 )SH(a−1 ) are by Proposition 2.14, T (a) − H(a)RH() a) and T () a−1 ) − H() −1 a), respectively. Let us prove the index equality. regularizers of T (a ) and T () By 1.12(a), there is a regularizer R of T () a) such that I − T () a)R ∈ C0 (X),

I − RT () a) ∈ C0 (X).

7.2 C + H ∞ Symbols

335

Hence, by Proposition 2.14, T (a−1 ) T (a) − H(a)RH() a) = I − H(a−1 )H() a) + H(a−1 )T () a)RH() a) −1 = I − H(a ) I − T () a)R H() a) ∈ I + C0 (X), −1 −1 T (a) − H(a)RH() a) T (a ) = I − H(a)H() a ) + H(a)RT () a)H() a−1 ) = I − H(a) I − RT () a) H() a−1 ) ∈ I + C0 (X). So 1.12(b) gives that Ind T (a−1 ) equals 0 1 0 1 tr H(a) I − RT () a) H() a−1 ) − tr H(a−1 ) I − T () a)R H() a) 1 0 1 0 a)R H() a)H(a−1 ) (by 1.4(b)) = tr I − RT () a) H() a−1 )H(a) − tr I − T () 0 1 0 1 = tr I − RT () a) I − T () a−1 )T () a) − tr I − T () a)R I − T () a)T () a−1 ) 1 0 1 0 = tr T () a)R − RT () a) + tr T () a) I − RT () a) T () a−1 ) 1 0 a) + tr T () a)T () a−1 ) − tr T () a−1 )T () a) −tr I − RT () a) T () a−1 )T () = tr [T () a)R − RT () a)] which is equal to Ind T () a), again by 1.12(b). (g) If T () a) resp. T (a) is in Φ(X), then there is an n ∈ Z such that T () aχn ) resp. T (aχn ) is in GL(X). So the assertion can be derived from (c). Remark. Note that there is a close connection between (b) and Proposition 7.15.

7.2 C + H ∞ Symbols p ∞ 7.20. Theorem. (a) Let a ∈ CN ×N + HN ×N and K ∈ C∞ (HN ). Then p p p T (a) + K ∈ Π{HN ; Pn } ⇐⇒ T (a) + K ∈ GL(HN ), T () a) ∈ GL(HN ).

(b) Let a ∈ (Cp + Hp∞ )N ×N and K ∈ C∞ (pN ). Then T (a) + K ∈ Π{pN ; Pn } ⇐⇒ T (a) + K ∈ GL(pN ), T () a) ∈ GL(pN ). (c) Under the hypothesis of (a) or (c), if T (a) + K ∈ Π{Pn } then π Pn (T (a) + K)−1 Pn + Wn T −1 () a) − T () a−1 ) Wn G (7.28) is the inverse of {Pn (T (a) + K)Pn }πG . Proof. We apply Theorem 7.11 with A = T (a)+K and An = Pn (T (a)+K)Pn . Thus, if A ∈ Π{Pn }, then both A and W(A) = s- lim Wn (T (a) + K)Wn = T () a) n→∞

336


must be invertible. Conversely, suppose A and W(A) are invertible. To get that A ∈ Π{Pn }, it remains to show that {An }πJ is in GSJπ . Theorem 2.94 implies that a is invertible in (C + H ∞ )N ×N resp. (Cp + Hp∞ )N ×N , and therefore H() a) and H() a−1 ) are compact. Hence, by (7.10), π π π π Pn (T (a) + K)Pn J Pn T (a−1 )Pn J = Pn T (a)Pn J Pn T (a−1 )Pn J π = Pn − Pn H(a)H() a−1 )Pn − Wn H() a)H(a−1 )Wn J = {Pn }πJ and it can be shown equally that the coset {Pn T (a−1 )Pn }πJ is a left inverse of the coset {Pn (T (a) + K)Pn }πJ . Thus, {An }πJ ∈ GSJπ . Finally, (c) results from the fact that (7.15) is the inverse of {An }πJ . p 7.21. Remark. Let a ∈ L∞ a) N ×N (resp. MN ×N ) and suppose T (a) and T () are invertible. It can be shown that then 0 1 Pn T −1 (a)Pn + Wn T −1 () a) − T () a−1 ) Wn Pn T (a)Pn = Pn − Pn T −1 (a) − T (a−1 ) Qn T (a)Pn − Wn T −1 () a) − T () a−1 ) Qn T () a)Wn .

In the C + H ∞ case the operators T −1 (a) − T (a−1 ) and T −1 () a) − T () a−1 ) are compact and so this identity immediately gives the implications “⇐=” of the above theorem for K = 0. Using Corollary 7.16 one can treat Toeplitz operators with QCN ×N symbols on H p with weight. p (w)). Then 7.22. Theorem. Let a ∈ QCN ×N and K ∈ C∞ (HN p p p T (a) + K ∈ Π{HN (w); Pn } ⇐⇒ T (a) + K ∈ GL(HN (w)), T () a) ∈ GL(HN (w)).

Proof. Suppose T (a) + K ∈ Π{Pn }. Then T (a) + K is invertible (Proposition 7.3), and Theorem 5.31(b) implies that a ∈ GQCN ×N . Hence, by (2.18) p (w), and the compactness of Hankel operators with QC symbols on HN (T (a) + K)−1 = T (a−1 ) + L with

p L ∈ C∞ (HN (w)).

(7.29)

p p By virtue of Corollary 7.16, Qn (T (a) + K)−1 Qn : Qn HN (w) → Qn HN (w) is invertible for all sufficiently large n and the norms of the inverses are uniformly bounded. Since 0 −1 1 Qn T (a−1 )Qn = Qn (T (a) + K)−1 Qn I − Qn (T (a) + K)−1 Qn Qn LQn p and Qn LQn → 0 as n → ∞, we deduce that Qn T (a−1 )Qn : Qn HN (w) → p p n+1 : HN (w) → Qn HN (w) is invertible for all sufficiently large n. Because V p (w) is an isometric isomorphism, it follows that T (a−1 ) is invertible Q n HN and Proposition 7.19(b) yields the invertibility of T () a).


337

Now suppose T (a) + K and T () a) are invertible. Then T (a−1 ) is invertible p p −1 (w) → Qn HN (w) is invertible for all and, consequently, Qn T (a )Qn : Qn HN −1 −1 n ≥ 0 and the norms of the inverses are equal to T (a )L(HNp (w)) . Again (7.29) holds and hence, 0 1 Qn (T (a) + K)−1 Qn = Qn T (a−1 )Qn I + (Qn T (a−1 )Qn )−1 Qn LQn = Qn T (a−1 )Qn (I + Cn ) with Cn L(HNp (w)) → 0 as n → ∞. It follows that Qn (T (a) + K)−1 Qn : p p (w) → Qn HN (w) is invertible for all sufficiently large n and that the Q n HN norms of the inverses are uniformly bounded. Corollary 7.16 completes the proof. p,µ 7.23. Lemma. Let 1 0 independent of x ∈ pN . Proof. Let Λ be as in the proof of 6.2(c) and denote ΛIN ×N by Λ, too. It is eas−1 ∈ Π{pN ; Pn }. ily seen that T (a) ∈ Π{p,µ N ; Pn } if and only if A := ΛT (a)Λ p a) on N . This will imply We claim that Wn AWn converges strongly to T () (7.30), since if A ∈ Π{pN ; Pn }, we have for every x ∈ pN , Pn x = (Wn AWn )−1 Wn AWn x = Wn (Pn APn )−1 Wn Wn AWn x ≤ sup (Pn APn )−1 Pn Wn AWn x, n

which gives (7.30) as n → ∞. To prove that Wn AWn → T () a) strongly on pN we may assume that N = 1. Since the operators Wn AWn are uniformly bounded, it suffices to show that (Pn T () a)Pn − Wn AWn )ej pp → 0 (n → ∞)

(7.31)

for all j ∈ Z+ . A simple computation shows that for n > j the left-hand side

n−j (n) of (7.31) is equal to k=−j bk , where (n)

bk

:= |a−k |p |1 − (n + 1 − j − k)µ (n + 1 − j)−µ |p . (n)

For k = 0, . . . , n − j we have bk ≤ |a−k |p (1 + 1µ )p = 2p |a−k |p . Hence,

n−j (n) given any ε > 0 there is an L = L(ε) such that < ε/2 for k=n+1 bk all n ≥ j + L + 1, and then one can find an M = M (j, L, ε) such that

L (n) < ε/2 for all n > M . As ε > 0 can be chosen arbitrarily we get k=−j bk (7.31).

338


7.24. Proposition. Let 1 ≤ p < ∞ and let µ, λ ≥ 0. Suppose A is an operator p,λ π which belongs to both GL(p,µ N ) and GL(N ). If {Pn APn }G is invertible in p,µ p,λ π ∞ p,λ D∞ (p,µ N )/G(N ), then {Pn APn }G is invertible in D (N )/G(N ). Proof. For simplicity let N = 1; it is easily seen that the following proof also works for N > 1. We first state two simple estimates: if δ, γ are real numbers such that 0 ≤ δ ≤ γ, then Pn xp,γ ≤ (n + 1)γ−δ Pn xp,δ

∀ x ∈ pδ ,

(7.32)

Qn xp,δ ≤ (n + 1)

∀x∈

(7.33)

δ−γ

Qn xp,γ

pγ .

Indeed, Pn xpp,γ = ≤ Qn xpp,δ = ≤

n

(k + 1)γp |xk |p =

k=0 n k=0 ∞

k+1 n+1

γp n k+1 k=0

δp

k=n+1

k+1 n+1

(n + 1)γp |xk |p

(n + 1)γp |xk |p = (n + 1)(γ−δ)p Pn xpp,δ ,

(k + 1)δp |xk |p =

k=n+1 ∞

n+1

γp

δp ∞ k+1 (n + 1)δp |xk |p n+1

k=n+1

(n + 1)δp |xk |p = (n + 1)(δ−γ)p Qn xpp,γ .

Put An = Pn APn |Im Pn and suppose {An }πG is invertible in the quotient algebra D∞ (pµ )/G(pµ ). Then, for x ∈ pλ and all sufficiently large n, −1 −1 Pn x + Pn A−1 Pn xp,λ A−1 n Pn xp,λ = An Pn x − Pn A −1 ≤ A−1 Pn xp,λ + A−1 L(pλ ) Pn xp,λ . n Pn x − Pn A

(7.34)

First assume µ ≤ λ. Then −1 A−1 Pn xp,λ n Pn x − Pn A −1 ≤ (n + 1)λ−µ A−1 Pn xp,µ n Pn x − Pn A

(by (7.32))

≤ C(n + 1)λ−µ Pn x − Pn APn A−1 Pn xp,µ p (because sup A−1 n Pn L(µ ) < ∞)

n

= C(n + 1)λ−µ Pn AQn A−1 Pn xp,µ = C(n + 1)λ−µ AL(pµ ) (n + 1)µ−λ A−1 L(pµ ) Pn xp,λ

(by (7.33)).

From this and (7.34) we get the invertibility of {An }πG in D∞ (pλ )/G(pλ ). Now let µ ≥ λ. Then


339

−1 A−1 Pn xp,λ n Pn x − Pn A

= A−1 (APn − Pn APn )A−1 n Pn xp,λ = A−1 Qn APn A−1 n Pn xp,λ ≤ A−1 L(pµ ) (n + 1)λ−µ AL(pµ ) A−1 n Pn xp,µ

(by (7.33))

and since A−1 n Pn xp,µ ≤ CPn xp,µ

(because {An }πG ∈ G(D∞ (pµ )/G(pµ )))

≤ C(n + 1)µ−λ Pn xp,λ

(by (7.32)),

we deduce from (7.34) that {An }πG is invertible in D∞ (pλ )/G(pλ ).

Remark. The above proposition together with Proposition 7.3 gives the following implication: p,µ p,λ p,λ A ∈ GL(p,µ N ), A ∈ Π{N ; Pn }, A ∈ GL(N ) =⇒ A ∈ Π{N ; Pn }. p,µ p,µ 7.25. Theorem. Let 1 < p < ∞, µ ∈ R, a ∈ CN ×N , and K ∈ C∞ (N ). Then the following are equivalent:

(i) T (a) + K ∈ Π{p,µ N ; Pn }; (ii) T (a) + K ∈ GL(p,µ a) ∈ GL(p,µ N ), T () N ); (iii) T (a) + K ∈ GL(p,µ a) ∈ GL(pN ). N ), T () Proof. (i) =⇒ (iii). Proposition 7.3 gives the invertibility of T (a) + K on p,µ N . Hence det a does not vanish on T and ind det a = 0 by 6.12(c). From 6.2(b) and 6.12(c) we deduce that T () a) is Fredholm of index zero on pN , and Lemma 7.23 then gives the invertibility of T () a) on pN . (ii) ⇐⇒ (iii). Without loss of generality assume µ ≥ 0. Theorem 6.12(c) p implies that T () a) is Fredholm of index zero on both p,µ N and N . Since the p,µ p a) in N , it follows that kernel of T () a) in N is contained in the kernel of T () p if it is invertible on . Analogously, because the T () a) is invertible on p,µ N N a) in qN is a subset of the kernel of T ∗ () a) in q,−µ , the invertibility kernel of T ∗ () N p implies the invertibility of T () a ) on . of T () a) on p,µ N N (ii)+(iii) =⇒ (i). It suffices again to consider the case µ ≥ 0. From 7.19(b) we deduce that T (a−1 ) is invertible on both pN and p,µ N . Corollary 7.18 shows that A := T −1 (a−1 ) ∈ Π{pN ; Pn }. Combining this with Propositions 7.3 and 7.24 we obtain that A ∈ Π{p,µ N ; Pn }. Finally, since T (a) + K = T −1 (a−1 ) − H(a)H() a−1 )T −1 (a−1 ) + K, we see that T (a) + K differs from A only by a compact operator, and so Corollary 7.17 implies that T (a) + K ∈ Π{p,µ N ; Pn }.

340


Remark. Theorem 7.25 also holds for p = 1. Open problem. Extend Theorem 7.20 to spaces with weight. 7.26. alg T F (AN ×N ). Let A be a closed subalgebra of L∞ resp. M p containing C resp. Cp and put A = AN ×N . The norm of a ∈ A is defined as the norm of the multiplication operator M (a) on L2N resp. pN (Z). (a) The mapping given by p T F : (A ⊂ L∞ N ×N ) → S(HN ),

a → {Tn (a)}

is a submultiplicative embedding, and the mapping p p T F : (A ⊂ MN ×N ) → S(N ),

a → {Tn (a)}

is a 1-submultiplicative isometry. To see this for the first mapping notice that aL∞ ≤ c1 T (a)L(HNp ) N ×N

(by (4.6))

≤ c1 lim inf Tn (a)L(HNp )

(by 1.1(e))

n→∞

≤ c1 sup Tn (a)L(HNp ) = c1 {Tn (a)}S(HNp ) ,

(7.35)

n≥0

which implies that Im T F is closed and that Ker T F = {0}, and also take into account that " " " " " " " " ajk " ≤ c a " Tn "T " 2 jk p p j

k

" " " " ≤ c3 " T (ajk )" j

k

S(HN )

p L(HN )

" " " " Tn (ajk )" ≤ c3 lim inf " n→∞

j

k

" " " " Tn (ajk )" ≤ c3 sup " n≥0

j

k

j

k

L(HN )

(by Proposition 4.1) (by 1.1(e))

p L(HN )

p L(HN )

" " " " =" {Tn (ajk )}" j

k

p S(HN )

,

which gives the submultiplicativity. The proof for the p case is analogous (note that in this case one can take c1 = 1 in (7.35) and that (7.35) can be continued by “≤ aMNp ×N ”). (b) Thus Theorem 3.42 gives that ·

alg T F (A) = T F (A) + QT F (A). Define the mapping LD by p p LD : D(HN or pN ) → L(HN or pN ),

{An } → s- lim An . n→∞


341

It is clear that LD = 1 and that LD (alg T F (A)) is a subset of alg T (A). The mappings ST F and SmbT F (recall 3.43) which are given at finite product-sums (correctly) by ST F : {Tn (ajk )} → Tn ajk , j

SmbT F :

j

k

j

{Tn (ajk )} →

k

j

k

ajk

k

can be easily verified to be representable as ST F = T F ◦ SmbT ◦ LD |alg T F (A),

(7.36)

SmbT F = SmbT ◦ LD |alg T F (A).

(7.37)

p Sometimes, in order to indicate that the underlying space is HN resp. pN , we shall write algS(HNp ) T F (A) resp. algS(pN ) T F (A) instead of alg T F (A).

(c) Let ∆ denote the restriction of LD to alg T F (A). The analogue of the “upper half” of the diagram 4.18(f) looks as follows:

Unfortunately we do not know any analogue of the “lower half” of that diagram. It should be mentioned here that it was the attempt to search for this analogue which led us first to the study of the algebra alg K(A), where {Kλ }λ∈Z+ is generated by the Fejér kernel, and then to the observation that some important results on Toeplitz operators which can be expressed in terms of the harmonic extension remain valid if the Abel-Poisson means are replaced by an arbitrary approximate identity. 7.27. Proposition. (a) If B is a closed subalgebra of C + H ∞ containing C, then p ). QT F (BN ×N ) = J (HN (b) If B is a closed subalgebra of Cp + Hp∞ containing Cp , then QT F (BN ×N ) = J (pN ). Proof. It suffices to consider the case N = 1. If a, b ∈ B, then H() a) and H()b) are compact and therefore, by Proposition 7.7, a)H(b)Wn } ∈ J , {Tn (ab) − Tn (a)Tn (b)} = {Pn H(a)H()b)Pn + Wn H() which shows that QT F (B) ⊂ J . So we are left with the opposite inclusion.

342


Suppose we had shown that J is a subset of the algebra alg T F (B). Then if {An } = {Pn KPn + Wn LWn + Cn } (K and L compact, {Cn } in G), we have, by (7.36)–(7.37) and Proposition 4.5, ST F {An } = (T F ◦ SmbT ◦ LD ){An } = (T F ◦ SmbT )(K) = 0, and it results that J ⊂ QT F (B). Thus let us prove that J ⊂ alg T F (B). To see that {Pn KPn } ∈ alg T F (B) for every compact operator K, it is sufficient to show that {Pn T (a1 ) . . . T (am )Pn } is in alg T F (B) for every finite collection a1 , . . . , am of functions from C resp. Cp (Proposition 4.5). This is + trivial for m = 1. Let m = 2. If a1 is a Laurent polynomial, then a1 = a− 1 + a1 − + ∞ ∞ with a1 ∈ H and a1 ∈ H , hence + {Pn T (a1 )T (a2 )Pn } = {Pn T (a− 1 )T (a2 )Pn + Pn T (a1 )T (a2 )Pn } + = {Tn (a− 1 a2 ) + Tn (a1 )Tn (a2 )} ∈ alg T F (B).

Since every function a1 in C resp. Cp can be approximated as closely as desired in the norm of L∞ reps. M p by Laurent polynomials, it follows that {Pn T (a1 )T (a2 )Pn } is in alg T F (B) for every a1 , a2 ∈ C resp. Cp . The assertion for general m ≥ 1 can be proved analogously by induction. In a similar fashion one can show that {Wn LWn } belongs to T F (B) for every L ∈ C∞ . It remains to prove that {Cn } ∈ alg T F (B) for every {Cn } ∈ G. For i, j ∈ Z+ , let Kij denote the finite-rank operators on H p resp. p whose ma∞ trix representation with respect to the standard bases {χn }∞ n=0 resp. {en }n=0 ∞ is given by (δir δjs )r,s=0 (δkl the Kronecker delta). A little thought shows that the inclusion G ⊂ alg T F (B) will follow as soon as we have proved that {Cn } belongs to alg T F (B), where {Cn } is any sequence with Cn = 0 for n = n0 and Cn0 = Pn0 Kij Pn0 (0 ≤ i, j ≤ n0 ). Since the operators Kij and Kn0 −j,n0 −j have finite rank, from what has already been proved we deduce that {Pn Kij Pn } and {Wn Kn0 −j,n0 −j Wn } are in alg T F (B). Because Pn Kij Pn · Wn Kn0 −j,n0 −j Wn equals Pn0 Kij Pn0 for n = n0 and 0 for n = n0 , we arrive at the conclusion that {Cn } is also in alg T F (B). 7.28. alg J FGπ (AN ×N ) and alg T FGπ (AN ×N ). Let A and A be as in 7.26. (a) The preceding proposition implies that both G and J are closed twosided ideals of alg T F (A) and that ST F (G) = ST F (J ) = {0} ⊂ G ⊂ J . Hence, by Theorem 3.52, the mappings T FGπ : A → alg T F (A)/G, a → {Tn (a)}πG := {Tn (a)} + G, T FJπ : A → alg T F (A)/J , a → {Tn (a)}πJ := {Tn (a)} + J are submultiplicative quasi-embeddings and ·

alg T F (A)/G = alg T FGπ (A) = T FGπ (A) + QT FGπ (A), ·

alg T F (A)/J = alg T FJπ (A) = T FJπ (A) + QT FJπ (A).


343

If {Tn (a)} ∈ J , then T (a) = s- lim Tn (a) must be compact and so a = 0. It n→∞ follows that the mappings T FGπ and T FJπ are actually embeddings. (b) We have {Tn (a)}πJ SJπ (pN ) = aMNp ×N

p ∀a ∈ MN ×N .

(7.38)

The inequality “≤” in (7.38) is obvious. If K and L are compact and {Cn } is in G, then {Tn (a) + Pn KPn + Wn LWn + Cn } = sup Tn (a) + Pn KPn + Wn LWn + Cn n

≥ lim inf Tn (a) + Pn KPn + Wn LWn + Cn n→∞

≥ T (a) + K ≥ T (a)Φ(pN ) = aMNp ×N , where the last equality results from Propositions 4.4(d) and 4.1(b). This gives “≥” in (7.38). Hence, in the p case T FJπ is an isometry. (c) Since ∆ (recall 7.26(c)) maps J into (even onto) the set of all compact operators, the quotient mapping ∆π : alg T FJπ (A) → alg T π (A) is well defined. So we arrive at the following analogue of the “upper half” of the diagram 4.19(g):

Theorem 7.91 will imply that ∆π is not one-to one in case A = L∞ . However, the following theorem (which may be viewed as the “finite section analogue” of Corollary 4.7 and Theorems 4.79 and 4.81) shows that ∆π is an isomorphism if A is a closed algebra between C and C + H ∞ . 7.29. Theorem. (a) Let B be a closed subalgebra of C + H ∞ containing C. Then the mapping SmbT FJπ given by (3.34) is a homeomorphic algebraic isomorphism of the algebra algSJπ (HNp ) T FJπ (BN ×N ) onto BN ×N . (b) If B is a closed subalgebra of Cp + Hp∞ containing Cp , then the mapping SmbT FJπ given by (3.34) is an isometric algebraic isomorphism of algSJπ (pN ) T FJπ (BN ×N ) onto BN ×N .

344


Proof. Combining Corollary 3.44 and Proposition 7.27 we see that SmbT FJπ is a homeomorphic isomorphism. That SmbT FJπ is an isometry in the case (b) results from (7.38). 7.30. Corollary. Let A=

r s

T (ajk ) + K,

An =

j=1 k=1

r s

Tn (ajk ) + Pn KPn ,

j=1 k=1

p where ajk belong to (C + H ∞ )N ×N resp. (Cp + Hp∞ )N ×N and K ∈ C∞ (HN ) p resp. K ∈ C∞ (N ). Then A ∈ Π{An } if and only if both A and

W{An } (A) :=

r s

T (a. jk )

j=1 k=1

are invertible. Proof. Clearly, An Pn → A and Wn Tn (ajk ) Wn + Wn KWn Wn An Wn = j

k

Wn Tn (ajk )Wn + Wn KWn = j

=

k

j

Tn (a. jk ) + Wn KWn →

k

j

T (a. jk ).

k

So Theorem 7.11 gives 5 the “only if” part. On the other hand, if A is invertible, then SmbT A = j k ajk is invertible in (C +H ∞ )N ×N resp. (Cp +Hp∞ )N ×N (Corollary 4.8), hence {An }πJ is invertible in SJπ (Theorem 7.29), and Theorem 7.11 completes the proof.

7.3 Locally Sectorial Symbols 7.31. Localization. Let F be a closed subset of X = M (L∞ ), let A be a closed subalgebra of L∞ containing C, and put A = AN ×N . In this section 2 . we always assume that the underlying space is HN (a) In accordance with 3.58 define JF0 := closidalg T FJπ (A) {Tn (a)}πJ : a ∈ A, a|F = 0 . Lemma 3.59 and Theorem 3.52 imply that T FFπ : A → alg T FFπ (A) := alg T FJπ (A)/JF0 , a → {Tn (a)}πF := {Tn (a)}πJ + JF0

7.3 Locally Sectorial Symbols

345

is a 1-submultiplicative quasi-embedding whose kernel is {a ∈ A : a|F = 0}. If F is a fiber Xβ (β ∈ M (B), B being a C ∗ -subalgebra of L∞ with identity), π and {An }πXβ will be abbreviated to T Fβπ and {An }πβ , respectively. then T FX β (b) It is not difficult to see that ∆π (recall 7.28(c)) maps JF0 into closidalg T π (A) T π (a) : a ∈ A, a|F = 0 . So the quotient mapping ∆πF of ∆π can be naturally defined and we arrive at the following analogue of the “upper half” of the diagram 4.23(c):

(c) The arguments used to prove Theorem 4.24 also yield the following (local) spectral inclusions: sp (a|F ) ⊂ sp TFπ (a) ⊂ sp {Tn (a)}πF , sp {Tn (a)}πF ⊂ conv a(F ) if N = 1. (d) Theorem 3.61 specializes to give that {Tn (a)}πF = a|F for all a ∈ L∞ N ×N . (e) Let B be a C ∗ -algebra between C and QC. By virtue of Theorem 7.29(b) (p = 2), B is isometrically star-isomorphic to alg T FJπ (BIN ×N ) and may therefore be identified with a C ∗ -subalgebra of alg T FJπ (L∞ N ×N ). If c ∈ QC and a ∈ L∞ , then, by (7.10), {Tn (ca) − Tn (c)Tn (a)} = {Pn H(c)H() a)Pn + Wn H() c)H(a)Wn } ∈ J (7.39) and hence Theorem 3.67 gives sp {An }πJ = sp {An }πβ for every {An } in alg T F (L∞ N ×N ).

β∈M (B)

7.32. Theorem. Let B be a C ∗ -algebra between C and QC and let K be 2 ). Suppose a ∈ L∞ in C∞ (HN N ×N satisfies at least one of the following three conditions: (i) For each β ∈ M (B) there exists a bβ ∈ L∞ N ×N such that a|Xβ = bβ |Xβ 2 and T (bβ ) ∈ Π{HN ; Pn }; (ii) a is locally sectorial over B; (iii) for each β ∈ M (B) the set a(Xβ ) is contained in some straight line segment (which may depend on β). 2 2 ; Pn } if and only if T (a) + K ∈ GL(HN ) and Then T (a) + K ∈ Π{HN 2 ). T () a) ∈ GL(HN

346


Proof. Let a satisfy (i). From Theorem 7.11 we deduce that {Tn (bβ )}πJ and thus {Tn (bβ )}πβ is invertible for each β ∈ M (B). In view of 7.31(d) we have {Tn (bβ )}πβ = {Tn (a)}πβ and so 7.31(e) implies the invertibility of {Tn (a)}πJ . The assertion now follows from Theorem 7.11. If a satisfies (ii), then Corollary 3.62, 7.31(e), and Theorem 7.11 give the assertion. Finally, let a have the property (iii). If T (a) + K is invertible, then T (a) is Fredholm, and so Theorem 4.70 shows that a must be locally sectorial over QC. This reduces the things to the case that a satisfies (ii). π π (P C) and alg T FJ (P QC). Again suppose the underlying 7.33. alg T FJ 2 space is H . Once (7.39) has been established the same arguments as in the proof of Proposition 4.83 show that alg T FJπ (P QC) is commutative. From the spectral inclusions 7.31(c) and Theorem 4.67 we obtain that sp {Tn (χτ )}πβ = sp Tβπ (χτ ), where τ ∈ T, χτ is the characteristic function of the arc (τ, τ eiπ/2 ), and β ∈ M (C) or β ∈ M (QC). This implies the following.

(a) Proposition 4.85 and Theorems 4.86 and 4.87 are true with iπ = T FJπ . (b) If B is C or QC, then ∆π is an isometric star-isomorphism of the algebra alg T FJπ (P BN ×N ) onto the algebra alg T π (P BN ×N ) and the algebra alg T FJπ (P BN ×N ) is isometrically star-isomorphic to the algebra [C(NP B )]N ×N . (c) Lemma 4.92, Proposition 4.93, and Theorem 4.94 are true with iπ = T FJπ . Since alg T FJπ (P QCN ×N ) ∼ = alg T π (P QCN ×N ), the same reasoning as in the proof of Corollary 7.30 gives the following result.

r 5 s (d) Let A = j=1 k=1 T (ajk ) + K, where ajk ∈ P QCN ×N and K be in

r 5 s 2 2 ). Put An = j=1 k=1 Tn (ajk ) + Pn KPn . Then A ∈ Π{HN ; An } if C∞ (HN

r 5 s and only if both A and W{An } (A) := j=1 k=1 T (a. jk ) are invertible on the 2 space HN . 7.34. Open problems. (a) Is T (a) in Π{H p ; Pn } if a ∈ L∞ is locally psectorial (p > 2) over a C ∗ -algebra B between C and QC and both T (a) and T () a) are invertible on H p ? We conjecture that the answer is yes. Note that we have not been able to answer the question even for B = C, i.e., for symbols which are (globally) p-sectorial on T (in this case T (a) and T () a) are automatically invertible). (b) What can be said about the applicability of the finite section method to Toeplitz operators on H 2 generated by symbols that are locally sectorial over C + H ∞ ?

7.4 P C Symbols: p Theory

347

7.4 P C Symbols: p Theory In the following we assume that 1 < p < ∞, 1/p + 1/q = 1, −1/p < µ < 1/q. 7.35. Lemma. Let a ∈ M p , suppose {Tn (a)}πJ is invertible in SJπ (p ), and assume at least one of the conditions (i) T (a) and T () a) are left-invertible, (ii) T (a) and T () a) are right-invertible is satisfied. Then T (a) ∈ Π{p ; Pn }. Proof. For the sake of definiteness, assume (i) is fulfilled. Because {Tn (a)}πJ is invertible, there are {Bn } ∈ S(p ), K and L in C∞ (p ), and {Cn } ∈ G(p ) such that (7.40) Bn Tn (a) = Pn + Pn KPn + Wn LWn + Cn . Let X and Y be left inverses of T (a) and T () a), respectively, and put Bn := Bn − Pn KXPn − Wn LY Wn . A computation similar to that in the proof of Theorem 7.11 shows that Bn Tn (a) = Pn + Cn with {Cn } ∈ G(p ). It follows that Tn (a) is invertible for all sufficiently large n and that Bn (Pn + Cn )−1 is the inverse. Since Bn (Pn + Cn )−1 converges strongly to B, we conclude that {Tn (a)}πG is in G(D/G), and so Proposition 7.3 gives the assertion. 7.36. Proposition. (a) If a ∈ M p and T (a) ∈ Π{p ; Pn }, then T (a) ∈ Π{r ; Pn } for all r ∈ [p, q] and, in particular, T (a) ∈ GL(r ) for all r ∈ [p, q]. (b) If a ∈ M p and {Tn (a)}πJ is invertible in SJπ (p ), then {Tn (a)}πJ is in for all r ∈ [p, q], T (a) is in Φ(r ) for all r ∈ [p, q], and the index of T (a) on r does not depend on r ∈ [p, q]. GSJπ (r )

Proof. (a) Let C denote the mapping given on r by C : {xn } → {xn }. If A is in L(r ), then CAC is also in L(r ) and we have CAC = A. Let T (a) ∈ Π{p ; Pn }. From Proposition 7.3 we know that there is an M such that Tn−1 (a)Pn L(p ) ≤ M for all n ≥ n0 . Hence, Tn−1 (a)Pn L(q ) = Tn−1 (a)Pn L(p ) = CWn Tn−1 (a)Wn CL(p ) ≤ Tn−1 (a)Pn L(p ) ≤ M for all n ≥ n0 , and the Riesz-Thorin interpolation theorem gives Tn−1 (a)Pn L(r ) ≤ M

∀ n ≥ n0

∀ r ∈ [p, q].

(7.41)

In view of Proposition 7.3 it remains to show that T (a) ∈ GL(r ) for all r ∈ [p, q]. From (7.41) we deduce that Tn−1 (a)Pn L(s ) ≤ M for all n ≥ n0 and s ∈ [p, q]. Thus, if x ∈ r , y ∈ s (1/r + 1/s = 1), and n ≥ n0 , then Pn xr ≤ M T (a)Pn xr ,

Pn ys ≤ M T (a)Pn ys

348


and passage to the limit n → ∞ gives xr ≤ M T (a)xr ,

ys ≤ M T (a)ys ,

from which we infer that T (a) ∈ GL(r ). (b) Passage to the strong limit n → ∞ in (7.40) implies that T (a) ∈ Φ(p ). After multiplying (7.40) from both sides by Wn and then passing to the strong limit n → ∞ we see that T () a) ∈ Φ(p ). Choose m ∈ Z so that p π T (aχm ) ∈ GL( ). Since {Tn (aχm )}J equals {Tn (a)}πJ {Tn (χm )}πJ , it results that {Tn (aχm )}πJ is also invertible. By virtue of Theorem 2.38 the kernel or the cokernel of the (Fredholm) operator T () aχ−m ) is trivial. Hence, T (aχm ) and T () aχ−m ) are either simultaneously left-invertible or simultaneously rightinvertible. So Lemma 7.35 can be applied to deduce that T (aχm ) ∈ Π{p ; Pn }. Now part (a) gives that T (aχm ) ∈ Π{r ; Pn } for all r ∈ [p, q], which implies that T (a) ∈ Φ(r ) and Ind T (a) = m for all r ∈ [p, q] and that {Tn (a)}πJ = {Tn (aχm }πJ {Tn (χ−m )}πJ is invertible in SJπ (r ) for all r ∈ [p, q] (Theorem 7.11). 7.37. Theorem (Verbitsky/Krupnik). Let β ∈ C. Then the following are equivalent: (i) T (ϕβ ) ∈ Π{pµ ; Pn }; (ii) T (ϕβ ) ∈ GL(pµ ), T (. ϕβ ) ∈ GL(p ); (iii) −1/p < Re β + µ < 1/q, −1/q < Re β < 1/p. Proof. (ii) ⇐⇒ (iii): Proposition 6.24. (i) =⇒ (ii). Without loss of generality assume 0 ≤ µ < 1/q. Let Λ be as in the proof of 6.2(c) (see also 7.23) and put An := Pn ΛT (ϕβ )Λ−1 Pn . We shall prove that {An } ∈ S(p ). In the proof of Lemma 7.23 we established that ϕβ ) strongly on p , so that Theorem 7.11 can be applied to Wn An Wn → T (. deduce that T (. ϕβ ) ∈ GL(p ). To prove that {An } ∈ S(p ), it remains to show that Wn A∗n Wn → T ϕ .β strongly on q . As in the proof of Lemma 7.23, we have n−j (n) "q " ∗ " "(Pn T ϕ .β Pn − Wn An Wn )ej q = ck ,

(7.42)

k=−j

+ µ +q + +q + n+1−j + . + +1 − + −k n+1−j−k + + +

0 (n) It is clear that k=−j ck → 0 as n → ∞. Since + ϕ .β −k + is not larger than

n−j (n) a constant times 1/k, the sum k=1 ck can be estimated by the integral 4q 3 µ ( n−j n+1−j 1 − 1 dx, xq n+1−j−x 1 (n) ck

+ =+ ϕ .β


349

and substituting x = (n + 1 − j)y one sees that this integral can be estimated by q ( 1 1 − (1 − y)µ 1 (n + 1 − j)1−q dy. (7.43) y (1 − y)µq 0 If µq < 1, the integral in (7.43) is finite, and hence (7.43) and thus (7.42) goes to zero as n → ∞. (iii) =⇒ (i). Because −1/p < Re β + µ < 1/q, the operator T (ϕβ ) is invertible on pµ . Theorem 6.20 (δ = −β, γ = β) gives Γ−β,β M−β T (ϕβ )Mβ = T (ηβ )T (ξ−β ).

(7.44)

Taking into account that Pn T (ηβ ) = Pn T (ηβ )Pn , T (ξ−β )Pn = Pn T (ξ−β )Pn we obtain from (7.44) that Γ−β,β Pn M−β Pn Tn (ϕβ )Pn Mβ Pn = Tn (ηβ )Tn (ξ−β ), and it follows that det Tn (ϕβ ) = 0 for all n ∈ Z+ . Hence, by Proposition 7.15, Qn T −1 (ϕβ )Qn is invertible on Qn pµ for all n ∈ Z+ . But (7.44) implies that Qn T −1 (ϕβ )Qn = Γ−β,β Qn Mβ T (ϕ−β )M−β Qn , (α)

(α)

and therefore, if we put Mα,n := diag (µn+1 , µn+2 , . . .), Mβ,n T (ϕ−β )M−β,n is invertible on the weighted p space ∞ p pµ p pn := x = {xk }∞ : x := (n + k + 1) |x | < ∞ k k=0 p,n k=0

for all n ∈ Z+ . Moreover, we have −1 −1 −1 (Qn T −1 (ϕβ )Qn )−1 Qn L(pµ ) = Γ−β,β M−β,n T −1 (ϕ−β )Mβ,n L(pn ) −1 −1 = M−β,n M−β T (ϕβ )Mβ Mβ,n L(pn ) (by (7.42)) + + + µ(β) µ(−β) + + k k + −1 −1 M−β )T (ϕβ )(M−β,n M−β )−1 L(pn ) ≤ sup + (β) + (M−β,n (−β) + + k µn+k µn+k −1 −1 ≤ c4 (M−β,n M−β )T (ϕβ )(M−β,n M−β )−1 L(pn )

(Lemma 6.21)

≤ c6 T (ϕβ )L(prn ) , where prn is the weighted p space ∞ (n + k + 1)(µ+α)p p p |x | < ∞ , prn := x = {xk }∞ k k=0 : xp,rn := (k + 1)αp k=0

with α := Re β (again use Lemma 6.21). Without loss of generality assume µ + α ≥ 0; otherwise consider adjoints. Then

350


(n + k + 1)(µ+α)p ≤ d (n + 1)(µ+α)p + (k + 1)(µ+α)p and hence, for x ∈ prn ,

T (ϕβ )xpp,rn ≤ d T (ϕβ )xpp,−α (n + 1)(µ+α)p + T (ϕβ )xpp,µ ≤ d T (ϕβ )pL(p ) xpp,−α (n + 1)(µ+α)p + T (ϕβ )pL(pµ ) xpp,µ −α p ≤ d T (ϕβ )L(p ) xpp,rn + T (ϕβ )pL(pµ ) xpp,rn . −α

Since −1/q < α < 1/p and −1/p < µ < 1/q, we deduce from Proposition 6.23 that T (ϕβ )L(p−α ) < ∞, T (ϕβ )L(pµ ) < ∞, and thus sup (Qn T −1 (ϕβ )Qn )−1 Qn L(pµ ) < ∞. Corollary 7.16 completes the proof.

n

Remark. Notice that in the case µ = 0 T (ϕβ ) ∈ Π{p ; Pn } ⇐⇒ T (ϕβ ) ∈ GL(p ), T (. ϕβ ) ∈ GL(p ) ⇐⇒ T (ϕβ ) ∈ GL(p ), T (ϕβ ) ∈ GL(q ) ⇐⇒ T (ϕβ ) ∈ GL(r ) ∀ r ∈ [p, q] ⇐⇒ |Re β| < min{1/p, 1/q}. 7.38. Lemma. (a) If a, b ∈ M p and if for each τ ∈ T there are an open arc Uτ ⊂ T and a function fτ ∈ Cp such that a|Uτ = fτ |Uτ or b|Uτ = fτ |Uτ , then {Tn (ab)}πJ = {Tn (a)}πJ {Tn (b)}πJ . (b) If a, b ∈ P Cp , then {Tn (a)}πJ {Tn (b)}πJ = {Tn (b)}πJ {Tn (a)}πJ . Proof. Using formula (7.10) this can be shown in the same way as Propositions 6.29 and 6.30.

r 7.39. Proposition. Let a = i=1 gi fi , where the functions gi are piecewise constant and the functions fi are continuously differentiable on T. Then T (a) ∈ Π{p ; Pn } ⇐⇒ T (a) ∈ GL(p ), T () a) ∈ GL(p ). Proof. The implication “=⇒” is an immediate consequence of Theorem 7.11. On the other hand, if T (a) and T () a) are invertible on p , then a can be written in the form (5.19), a = ϕβ1 . . . ϕβm b, and one has |Re βi | < min{1/p, 1/q} for all i and b ∈ Cp (see the proof of Proposition 6.32). Hence, by Theorems 7.37, 7.20(a), and 7.11, {Tn (ϕβi )}πJ and {Tn (b)}πJ are invertible in SJπ (p ). Lemma 7.38(a) implies that {Tn (a)}πJ = {Tn (ϕβ1 )}πJ . . . {Tn (ϕβm )}πJ {Tn (b)}πJ , and therefore {Tn (a)}πJ ∈ GSJπ (p ). It remains to apply Theorem 7.11.


351

7.40. Localization. For τ ∈ T, let Jτ0 denote the smallest closed two-sided p ideal of the algebra algSJπ (pN ) T FJπ (P CN ×N ) containing the set {Tn (f )}πJ : f = diag (ϕ, . . . , ϕ), ϕ ∈ B, ϕ(τ ) = 0 , where B may be P, C ∞ , or Cp (Jτ0 does not depend on the particular choice of B, see also 6.34). Define p p π 0 alg T Fτπ (P CN π (p ) T FJ (P C ×N ) := algSJ N ×N )/Jτ , N

(7.45)

denote the coset of this algebra containing {An }πJ by {An }πτ , and for {An } in p π π π alg T F (P CN ×N ) let spp {An }τ and spp {An }J refer to the spectrum of {An }τ π and {An }J as element of the algebra (7.45) and as element of the algebra p algSJπ (pN ) T FJπ (P CN ×N ), respectively. Lemma 7.38(b) implies that alg T FJπ (P Cp ) is commutative. A similar reasoning as in the proof of Proposition 6.35 shows that p π π a, b ∈ P CN ×N , a|Xτ = b|Xτ =⇒ {Tn (a)}τ = {Tn (b)}τ

and that spp {An }πJ =

/

spp {An }πτ

(7.46) (7.47)

τ ∈T p for every {An } ∈ alg T F (P CN ×N ).

7.41. Proposition. If a ∈ P Cp , then spp {Tn (a)}πτ = Op (a(τ − 0), a(τ + 0)), / spp {Tn (a)}πJ = Op (a(τ − 0), a(τ + 0)).

(7.48) (7.49)

τ ∈T

Proof. By virtue of (7.46) we may assume that τ is the only point of discontinuity of a, that a is as in Proposition 7.39, and that R(a) is the arc Ap (a(τ − 0), a(τ + 0)). Propositions 6.32 and 7.25 together with Theorem 7.11 give the inclusion spSJπ (p ) {Tn (a)}πJ ⊂ Op (a(τ − 0), a(τ + 0)),

(7.50)

and Propositions 7.36(b) and 6.32 yield that the reverse inclusion in (7.50) also holds. Since alg T FJπ (P Cp ) is a closed subalgebra of SJπ (p ), we can apply 1.16(b) to deduce that the spectrum of {Tn (a)}πJ in alg T FJπ (P Cp ) equals Op (a(τ − 0), a(τ + 0)). Again using (7.46) we obtain that spp {Tn (a)}πt = {a(t)} ∈ Ap (a(τ − 0), a(τ + 0)) for t = τ . Thus, by (7.47), Op (a(τ − 0), a(τ + 0)) ⊃ spp {Tn (a)}πτ ⊃

/

Ar (a(τ − 0), a(τ + 0))

r∈(p,q)

and since a spectrum is always closed, equality (7.48) follows. Equality (7.49) results from (7.48) and (7.47).

352


p p 7.42. Theorem. Let a ∈ P CN ×N and K ∈ C∞ (N ). Then

a) ∈ GL(pN ). T (a) + K ∈ Π{pN ; Pn } ⇐⇒ T (a) + K ∈ GL(pN ), T () Proof. The implication “=⇒” is a consequence of Theorem 7.11. So we are left with the opposite implication. First let N = 1. If T (a) + K and T () a) are invertible on p , then T (a) ∈ p q Φ( ), T (a) ∈ Φ( ), and Indp T (a) = Indq T (a) = 0. From 6.39 and 6.40 (with Op (a(τ − 0), a(τ + 0)), and so (7.49) and A = T (a)) we infer that 0 ∈ / τ ∈T

Theorem 7.11 give the assertion. Now let N > 1 and suppose T (a) + K and T () a) are invertible on pN . We π π p must show that {Tn (a)}J is invertible in SJ (N ). In view of (7.47) it suffices p to show that {Tn (a)}πτ is invertible in alg T Fτπ (P CN ×N ) for each τ ∈ T. Let aτ denote the matrix function which equals a(τ + 0) on (τ, τ eiπ ) and a(τ − 0) on (τ e−iπ , τ ). Using 6.39 and 6.40 it is not difficult to see that T (aτ ) ∈ Φ(pN ),

T (aτ ) ∈ Φ(qN ),

Indp T (aτ ) = Indq T (aτ ) = 0.

{Tn (aτ )}πJ

is invertible; then 7.40 will imply the invertibilWe shall prove that ity of {Tn (a)}πτ . Write aτ in the form ϕbψ as in 5.48(b). Due to Lemma 7.38(b) it is enough to show that {Tn (b)}πτ is invertible. Since b = ϕ−1 aτ ψ −1 , we have T (b) ∈ Φ(pN ),

T (b) ∈ Φ(qN ), Indp T (b) = Indq T (b).

N Consequently, as Ind T (b) = j=1 Ind T (bj ) (bj are the diagonal entries of b) and Indp T (b) ≤ Indq T (b) (resp. Indp T (b) ≥ Indq T (b)) if p ≤ q and thus pN ⊂ qN (resp. p ≥ q and thus pN ⊃ qN ), it follows that Indp T (bj ) = Indq T (bj ) for each j. Hence, T (bj ) ∈ Φ(r ) for all r ∈ [p, q] and all j = 1, . . . , N . Choose integers mj so that T (bj χmj ) ∈ GL(r ) for all r ∈ [p, q]. From (7.49) we obtain the invertibility of {Tn (bj χmj )}πJ and thus the invertibility of {Tn (bj )}πJ itself in alg T FJπ (P Cp ). Thus π p {Bn }πJ := diag Tn (b1 ), . . . , Tn (bN ) J ∈ G(alg T FJπ (P CN ×N )). Therefore {Tn (b)}πJ may be written in the form {Bn }πJ {I + Xn }πJ , where p N π π {Xn }πJ in alg T FJπ (P CN ×N ) has the property that {Xn }J = {0}J . But this implies that {I + Xn }πJ is invertible, since if xN = 0, then N N I = (I + x − x)N = (−1)N xN + (I + x)k (−x)N −k k = (I + x) =

N

k=1

N k=1

N k

N k

k=1

(I + x)k−1 (−x)N −k

k−1

(I + x)

N −k

(−x)

which shows that I + x is invertible.

(I + x),

7.5 P C Symbols: H p Theory

353

7.43. Proposition. (a) The algebra alg T Fτπ (P Cp ) is singly generated by {Tn (χτ )}πτ , where χτ is the characteristic function of the arc (τ, τ eiπ/2 ). (b) The maximal ideal space of alg T Fτπ (P Cp ) is homeomorphic to Op (0, 1) (with the topology inherited from C) and the Gelfand map Γ : alg T Fτπ (P Cp ) → C(Op (0, 1)) is for a ∈ P Cp given by Γ {Tn (a)}πτ (x) = (1 − x)a(τ − 0) + xa(τ + 0)

x ∈ Op (0, 1) .

Proof. Similar arguments as in the proof of Proposition 5.44 apply. 7.44. Theorem. The maximal ideal space of alg T FJπ (P Cp ) can be identified with T×Op (0, 1) (equipped with an exotic topology). For a ∈ P Cp , the Gelfand transform Γ : alg T FJπ (P Cp ) → C(T × Op (0, 1)) is given by Γ {Tn (a)}πJ (τ, x) = (1 − x)a(τ − 0) + xa(τ + 0) τ ∈ T, x ∈ Op (0, 1) . Proof. Proceed as in the proof of Theorem 5.45. 7.45. Toeplitz operators on p with weight. The following result is proved in detail in Roch, Silbermann [426, Part I]. If a ∈ (M p,µ ∩ P C)N ×N and K ∈ C∞ (p,µ N ), then p,µ T (a) + K ∈ Π{p,µ a) ∈ GL(pN ). N ; Pn } ⇐⇒ T (a) + K ∈ GL(N ), T ()

7.5 P C Symbols: H p Theory In what follows we always assume that 1 < p < ∞, 1/p + 1/q = 1. 7.46. Proposition. (a) If a ∈ L∞ and T (a) ∈ Π{H p ; Pn }, then T (a) is in Π{H r ; Pn } and thus T (a) is in GL(H r ) for all r ∈ [p, q]. (b) If a ∈ L∞ and {Tn (a)}πJ ∈ GSJπ (H p ), then {Tn (a)}πJ is in GSJπ (H r ) for all r ∈ [p, q], T (a) is in Φ(H r ) for all r ∈ [p, q], and the index of T (a) on H r does not depend on r ∈ [p, q]. Proof. The same reasoning as in the proof of Proposition 7.36 can be applied. 7.47. Marcinkiewicz’ multiplier theorem. Let λ0 , λ1 , λ2 , . . . be complex numbers such that |λn | ≤ M,

2n+1 −1 k=2n

|λk − λk+1 | ≤ M

(n = 0, 1, 2, . . .).

354


Then the operator Λ:

∞ k=0

fk χk →

∞

λ k f k χk

k=0

belongs to L(H p ) and ΛL(H p ) ≤ Ap M , where Ap is some constant depending only on p. Proof. For a proof see Zygmund [591, Vol. II, Chap. XV, Theorem 4.14]. 7.48. Theorem (Verbitsky). Let β ∈ C. Then the following are equivalent: (i) T (ϕβ ) ∈ Π{H p ; Pn }; (ii) T (ϕβ ) ∈ GL(H p ), T (. ϕβ ) ∈ GL(H p ); (iii) |Re β| < min{1/p, 1/q}. Proof. (ii) ⇐⇒ (iii): Lemma 5.36. (i) =⇒ (ii): Theorem 7.11. (iii) =⇒ (i). Without loss of generality assume α := Re β ≥ 0. As in the proof of Theorem 7.37 we arrive at the equality (Qn T −1 (ϕβ )Qn )−1 Qn L(H p ) −1 −1 −1 = Γ−β,β M−β,n T −1 (ϕ−β )M−β,n L(H p )

(7.51)

(note that Qn H p is isometrically isomorphic to H p ). Let Λn denote the operator diag ((n + 1)α , (n + 2)α , . . .) and write −1 −1 −1 Γ−β,β M−β,n T −1 (ϕ−β )M−β,n −1 −1 −1 −1 = Γ−β,β M−β,n Λ−1 (ϕ−β )M−β,n n (Λn − Λ0 )T −1 −1 +M−β,n Λ−1 n Λ0 M−β T (ϕβ )Mβ M−β,n .

(7.52)

Using the Marcinkiewicz multiplier theorem we first show that −1 sup M−β,n Λ−1 n L(H p ) < ∞. n

In what follows cβ (resp. cβ,p ) denotes a constant depending only on β (resp. β and p) but not necessarily the same at each occurrence. From Lemma 6.21 + (−β) −1 + we obtain that sup + µn+k (n+k +1)−α + ≤ cβ < ∞. So, by 7.47, it remains n,k

to verify that σmn :=

2m+1 −1

+ (−β) −1 + (−β) −1 + µ (n + k + 1)−α − µ (n + k + 2)−α + ≤ cβ n+k

n+k+1

k=2m

for all m, n ∈ Z+ . Taking into account Lemma 6.21 we get (the sums from k = 2m to k = 2m+1 − 1)

7.5 P C Symbols: H p Theory

σmn

355

+ (−β) + (−β) α α+ +µ ≤ cβ n+k+1 (n + k + 2) − µn+k+1 (n + k + 1) + + (−β) + ++ + β α α+ ++ 1 − +µ = cβ − (n + k + 1) (n + k + 2) n+k + + n+k+1 0 1 ≤ cβ (n + k + 1)−α (n + k + 2)α − (n + k + 1)α (n + k + 1)−α (n + k + 2)α (n + k + 1)−1 +cβ

and since (n + k + 2)α − (n + k + 1)α ≤ α(n + k + 1)α−1 , it follows that σmn ≤ cβ

m+1

2 1 1 ≤ cβ = cβ < ∞. n+k+1 k m k=2

Thus, sup σmn ≤ cβ , as desired. It can be shown analogously that m,n

Λ0 M−β L(H p ) ≤ cβ,p , Since, by Lemma 6.21, 2m+1 −1

(β) |(µn+k )−1

k=2m

−

−1 sup Mβ Mβ,n L(H p ) ≤ cβ,p . n

(β) sup |(µn+k )−1 | k (β) (µn+k+1 )−1 |

≤ cβ n−α and

≤ cβ

2m+1 −1

(n + k + 1)−α−1 ≤ cβ n−α ,

k=2m

−1 we conclude from 7.47 that Mβ,n L(H p ) ≤ cβ,p n−α . Finally consider ∞ Λn − Λ0 = diag (n + k + 1)α − (k + 1)α k=0 .

Because, for fixed n, the sequence {(n+k +1)α −(k +1)α }∞ k=0 is monotonically decreasing, we have (n + k + 1)α − (k + 1)α ≤ cα nα and 2m+1 −1

|(n + k + 1)α − (k + 1)α − (n + k + 2)α + (k + 2)α |

k=2m

= (n + 2m + 1)α − (2m + 1)α − [(n + 2m+1 + 1)α − (2m+1 + 1)α ] ≤ (n + 2m + 1)α − (2m + 1)α ≤ cα,p nα . Hence, again by 7.47, Λn − Λ0 L(H p ) ≤ cα,p nα . The above estimates together with the fact that T (ϕ−β ) ∈ GL(H p ) and T (ϕβ ) ∈ L(H p ) show that (7.52) and thus (7.51) is uniformly bounded for n ≥ 0, which by virtue of Corollary 7.16 gives the assertion. 7.49. Definition. For τ ∈ T, let Jτ0 be the smallest closed two-sided ideal of algSJπ (H p ) T FJπ (P C) containing the set {{Tn (f )}πJ : f ∈ C, f (τ ) = 0}, put algp T Fτπ (P C) := algSJπ (H p ) T FJπ (P C)/Jτ0 , let {Tn (a)}πτ denote the coset {Tn (a)}πJ + Jτ0 , and let spp {Tn (a)}πτ and spp {Tn (a)}πJ refer to the spectrum of {Tn (a)}πτ and {Tn (a)}πJ in algp T Fτπ (P C) and algSJπ (H p ) T FJπ (P C), respectively.

356


7.50. Theorem. (a) If a ∈ P C and τ ∈ T, then spp {Tn (a)}πτ = Op (a(τ − 0), a(τ + 0)), / spp {Tn (a)}πJ = Op (a(τ − 0), a(τ + 0)). τ ∈T

algp T Fτπ (P C)

(b) The algebra is singly generated by {Tn (χτ )}πτ , where χτ is the characteristic function of the arc (τ, τ eiπ/2 ). The maximal ideal space of algp T Fτπ (P C) is homeomorphic to Op (0, 1) (with the topology inherited from C) and for a ∈ P C the Gelfand transform is given by Γ {Tn (a)}πτ (x) = (1 − x)a(τ − 0) + xa(τ + 0) (x ∈ Op (0, 1)). (c) The algebra algSJπ (H p ) T FJπ (P C) is commutative. Its maximal ideal space can be identified with T×Op (0, 1) (the topology is exotic) and for a ∈ P C the Gelfand transform is given by Γ {Tn (a)}πJ (τ, x) = (1 − x)a(τ − 0) + xa(τ + 0) τ ∈ T, x ∈ Op (0, 1) . p 7.51. Theorem. Let a ∈ P CN ×N and K ∈ C∞ (HN ). Then p p p T (a) + K ∈ Π{HN ; Pn } ⇐⇒ T (a) + K ∈ GL(HN ), T () a) ∈ GL(HN ).

The two preceding theorems can be proved similarly as their p analogues and their “Fredholm counterparts.” 7.52. Open problems. (a) Establish a criterion for the applicability of the finite section method to Toeplitz operators with P C symbols on H p with Khvedelidze weight. The case p = 2 will be settled by Corollary 7.75. (b) Is Theorem 7.51 true for a ∈ P QCN ×N ? Even the case N = 1 is of interest. For p = 2 the answer is known to be affirmative (Theorem 7.32(iii)). (c) Extend Propositions 7.36 and 7.46 to the matrix case. 7.53. Gohberg-Krupnik localization. For τ ∈ T, let Mτp := {Tn (ϕ)}πJ ∈ SJπ (H p ) : ϕ ∈ C, 0 ≤ ϕ ≤ 1,

ϕ is identically 1 in some open neighborhood of τ ,

put Fp :=

τ ∈T

Mτp , and denote the commutant of Fp in SJπ (H p ) by Com Fp .

Note that Com Fp is a closed subalgebra of SJπ (H p ) containing alg T FJπ (L∞ ). Theorem 7.20 implies that {Tn (ϕ)}πJ is invertible in SJπ (H p ) if ϕ ∈ C and ϕ ≥ ε > 0 on T. This can be used to prove that {Mτp }τ ∈T is a covering system of bounded localizing classes in SJπ (H p ). Let Zτp denote the collection of all elements in Com Fp which are Mτp -equivalent to zero from the left and the right, and observe that Zτp is a closed two-sided ideal in Com Fp . For

7.6 Operators from algL(H 2 ) T (P C)

357

{An }πJ ∈ Com Fp let spp,τ {An }πJ refer to the spectrum of {An }πJ + Zτp as element of Com Fp /Zτp . It is not difficult to see that a, b ∈ L∞ , a|Xτ = b|Xτ =⇒ {Tn (a)}πJ + Zτp = {Tn (b)}πJ + Zτp , and Theorem 1.32(b) gives that / spSJπ (HNp ) {An }πJ = spp,τ {An }πJ

∀ {An }πJ ∈ Com Fp .

(7.53)

(7.54)

τ ∈T

7.54. Theorem. Let A=

r s

T (ajk ),

An =

j=1 k=1

r s

Tn (ajk ),

j=1 k=1

W{An } (a) =

r s

T (a. jk ),

j=1 k=1

where ajk ∈ P C, and let K ∈ C∞ (H p ). Then A + K ∈ Π{H p ; An + Pn KPn } if and only if A + K ∈ GL(H p ), W{An } ∈ GL(H p ), and A ∈ Φ(H r ) for all r ∈ [p, q]. Proof. First suppose A + K ∈ Π{H p ; An + Pn KPn }. Theorem 7.11 along with the computation in the proof of Corollary 7.30 shows that A + K and W{An } (A) are invertible on H p . Theorem 7.11 also implies that {An }πJ is in / spp,τ {An }πJ for each τ ∈ T. Using (7.53), GSJπ (H p ). Hence, by (7.54), 0 ∈ (7.54) one can verify as in the proof of Proposition 7.41 that, if a ∈ P C, spp,τ {Tn (a)}πJ is Op (a(τ − 0), a(τ + 0)), and then the same argument as in the proof of Proposition 5.41 shows that spp,τ {An }πJ is equal to the spectrum of {An }πτ in algp T Fτπ (P C) (recall 7.49 and 7.50). The conclusion is that {An }πτ is invertible and so Theorems 7.50 and 5.43 can be combined to obtain that A ∈ Φ(H r ) for all r ∈ [p, q]. To get the “if” part of the present theorem it suffices in view of Theorem 7.11 to show that {An }πJ is in GSJπ (H p ) if A ∈ Φ(H r ) for all r ∈ [p, q]. But this follows from Theorems 7.50 and 5.43. Remark. Also recall 7.33(d).

7.6 Operators from algL(H2 ) T (P C) Theorem 7.20 solves the problem of the applicability of the finite section ∞ method to operators in algL(HN2 ) T (CN ×N + HN ×N ), since every operator in ∞ this algebra can be written in the form T (a) + K with a ∈ CN ×N + HN ×N 2 and K ∈ C∞ (HN ) (Corollary 4.7). However, we shall arrive at situations (e.g., when investigating the finite section method for T (a) with a ∈ P C on H 2 ()) in which the necessity emerges to check whether the finite section method is applicable to operators belonging to algL(H 2 ) T (P C). Note that not every

358


operator in this algebra is the sum of a Toeplitz operator and a compact operator.

r 5s For simplicity, let A = j=1 k=1 T (ajk ), where ajk ∈ P C. The question we are interested in reads: When is A ∈ Π{2 ; Pn APn }? Notice that this the following question: When is A ∈ Π{2 ; An }, where An =

r is5not s j=1 k=1 Tn (ajk )? The latter question is answered by 7.33(d) (and for ajk ∈ P C and H p as underlying space by Theorem 7.54). 7.55. Lemma. If 0 < α < 1, 0 < β < 1, α + β > 1, and if m, n are integers with m = n, then +−α + +−β ++ + + + +k − m + 1 + +k − n + 1 + ≤ c|m − n|1−α−β (7.55) + + + 2 2+ k∈Z

where c is some constant independent of m and n. Proof. Without loss of generality assume m < n. Then the function +−α + +−β + + 1 ++ ++ 1 ++ + f (x) := +x − m + + +x − n + + 2 2 is monotonically increasing on the intervals (−∞, m+1/2) and (d, n+1/2), and it is monotonically decreasing on the intervals (m + 1/2, d) and (n + 1/2, ∞), where d = [(n + 1/2)α + (m + 1/2)β]/(α + β). Hence, the sum in (7.55) is not larger than +−α + +−β ( ∞ + +−α + +−β + + + + 1 ++ ++ 1 ++ 1 ++ 1 ++ + + + c1 +m−n+ + + +m − n + + + +x − m + 2 + +x − n + 2 + dx. 2 2 −∞ The substitution y = (x − m + 1/2)/(m − n) in the integral gives the assertion. 7.56. Lemma. (a) If, for all k and j in Z+ , −γ + + 1 +k − j + |bjk | ≤ c j + + 2 or

−γ + + 1 +k − j + |bjk | ≤ c k + + 2

+γ−1 1 ++ 2+

(7.56)

+γ−1 1 ++ , 2+

(7.57)

where 0 < γ < 1/2 and c is some constant that does not depend on k and j, 2 then (bjk )∞ j,k=0 defines a bounded operator on . (b) If bjk = 0 for j < k and |bjk | ≤ c(j + 1)−α (j − k + 1)−β (k + 1)−γ

for

j ≥ k,

where α + β + γ ≥ 1, α + β > 1/2, β < 1, and c is some constant independent 2 of k and j, then (bjk )∞ j,k=0 generates a bounded operator on .


359

∞ 2 Proof. Let B = (bjk )∞ j,k=0 and x = {xk }k=0 ∈ .

(a) Suppose (7.56) is fulfilled. Then +∞ +2 ∞ + + + + bjk xk + Bx22 = + + + j=0 k=0 −2γ ∞ 1 ≤ c2 j+ 2 j=0 ∞ 2 |xk |(k + 1)δ 1 × |k − j + 1/2|(1−γ)/2 |k − j + 1/2|(1−γ)/2 (k + 1)δ k=0 −2γ ∞ 1 2 ≤c j+ 2 j=0 ∞ ∞ |xk |2 (k + 1)2δ 1 × . |k − j + 1/2|1−γ |k − j + 1/2|1−γ (k + 1)2δ k=0

k=0

If δ ∈ (γ/2, (1 − γ)/2), then, by Lemma 7.55, −2γ γ−2δ ∞ ∞ 2 2δ |x | (k + 1) 1 1 k 2 Bx2 ≤ c1 j+ j+ 2 |k − j + 1/2|1−γ 2 j=0 k=0 +γ−1 −γ−2δ + ∞ ∞ + + 1 +j − k + 1 + = c1 |xk |2 (k + 1)2δ , j+ + 2 2+ j=0 k=0

whence, again by Lemma 7.55, Bx22 ≤ c2

∞

|xk |2 (k + 1)2δ (k + 1)−2δ = c2 x22 .

k=0

Passage to the adjoint yields the assertion for the case that (7.57) is satisfied. (b) We have Bx22 ≤ c2

∞

⎛ (j + 1)−2α ⎝

j=0

≤c

2

∞

−2α

(j + 1)

j=0

×

j

∞

⎞2 (j − k + 1)−β (k + 1)−γ |xk |⎠

j=0 j

−2(β+ε)

(j − k + 1)

k=0

(j − k + 1) (k + 1)

k=0

Let ε > −1/2, δ > −1/2. Then

2ε

2δ

.

−2(γ+δ)

(k + 1)

|xk |

2

360

7 Finite Section Method j

(j − k + 1)2ε (k + 1)2δ ≤ c1 (j + 1)2δ+2ε+1

k=0

and so Bx22 ≤ c2 = c2

j ∞ (j + 1)−2α+2δ+2ε+1 (j − k + 1)−2(β+ε) (k + 1)−2(γ+δ) |xk |2 j=0

k=0

∞

∞

(k + 1)−2(γ+δ) |xk |2

k=0

(j + 1)−2α+2δ+2ε+1 (j − k + 1)−2(β+ε) .

j=k

If 0 < 2α − 2δ − 2ε − 1 < 1, 0 < 2(β + ε) < 1, 2α + 2β − 2δ − 1 > 1, then, by Lemma 7.55, Bx22 ≤ c3 = c3

∞

(k + 1)−2(γ+δ) |xk |2 (k + 1)−2(α+β−δ−1)

k=0 ∞

|xk |2 (k + 1)−2(α+β+γ−1) ≤ c3 x22 .

k=0

It is not difficult (but tedious) to see that there exist ε and δ with the properties required above, which completes the proof. (γ)

7.57. Lemma. Let µn then

be given as in 6.19. If 0 < Re γ ≤ δ < 1 and j ≥ k,

(γ)

(γ)

|µj − µk | ≤ c(j − k)δ (k + 1)Re γ−δ , + + + 1 1 ++ + + (−γ) − (−γ) + ≤ c(j − k)δ (k + 1)Re γ−δ , +µ µk + j

(7.58) (7.59)

where c does not depend on k and j. (γ)

Proof. We have µn = Γ (γ + n + 1)/(Γ (γ + 1)Γ (n + 1)). Hence, using the formula ( ∞ xα−1 dx Γ (α)Γ (β) (Re α > 0, Re β > 0) = α+β (1 + x) Γ (α + β) 0 and Lemma 6.21 we get, for j > k, + + + + ( + 1 + 1 1 ++ ++ Γ (1 + γ) ∞ xγ−1 1 + − dx++ + (γ) − (γ) + = + γ k+1 j+1 +µ Γ (γ) (1 + x) (1 + x) (1 + x) 0 µj + k + + + + ( j j + ∞ + + Γ (γ + 1)Γ (s) + xγ dx + + + + = c1 ≤ c1 + + Γ (γ + s + 1) + (1 + x)γ+s+1 + 0 s=k+1

≤ c2

j s=k+1

s=k+1

j + µ(γ) + + s + s−1−σ ≤ c4 (k −σ − j −σ ), + ≤ c3 + s s=k+1


361

where σ := Re γ, and thus + + + + 1 1 + + (γ) (γ) (γ) (γ) |µj − µk | ≤ |µj | |µk | + (γ) − (γ) + ≤ c5 (j σ − k σ ). +µ µj + k

(7.60)

Since j σ − k σ ≤ (j − k)σ and j σ − k σ ≤ σ(j − k)kσ−1 (0 < σ < 1), we obtain 1−δ

δ−σ

j σ − k σ = (j − k) 1−σ (j σ − k σ ) 1−σ ≤ c6 (j − k)σ k σ−δ .

(7.61)

Now (7.58) results from (7.60) and (7.61). The proof of (7.59) is analogous. 7.58. Lemma. Let γ, δ ∈ C \ {−1, −2, . . .}. Then there exists a number c = 0 such that the operator I − cMγ−1 Mγ+δ Mδ−1 is compact on 2 . Proof. The n-th diagonal entry dn of the operator is (γ+δ)

1−c

µn

(γ) (δ)

µn µn

=1−c

Thus, if we let c = 1/

n n (γ + δ + j)j γδ =1−c 1− . (γ + j)(δ + j) (γ + j)(δ + j) j=1 j=1 j=1 1 −

5∞

which implies compactness.

γδ , then dn → 0 as n → ∞, (γ + j)(δ + j)

In what follows the functions ξα , ηα , ϕα , etc., are always assumed to have the (possible) discontinuity at the same point τ ∈ T, i.e., ξα = ξα,τ , ηα = ηα,τ , ϕα = ϕα,τ , etc. 7.59. Proposition. If γ ∈ C and δ ∈ C, then Mγ T (ξγ )Mδ T (ξδ ) = Mδ T (ξδ )Mγ T (ξγ ), T (ηγ )Mγ T (ηδ )Mδ = T (ηδ )Mδ T (ηγ )Mγ .

(7.62) (7.63)

Proof. It suffices to verify (7.62), since (7.63) results from (7.62) by transponation. Fix δ ∈ C \ Z. We prove that (7.62) holds for all γ ∈ C \ Z satisfying Re γ > Re δ. Since the entries of the matrices in (7.62) are analytic functions of γ and δ, it then follows that (7.62) is true for all γ, δ ∈ C. After computing the j, j + n entry of both sides of (7.62) one sees that the following identity must be verified: −1 n γ δ+j+k δ δ+j (−1)k (−1)n−k k j+k n−k j k=0 −1 n γ+j δ γ+j+k γ k n−k = (−1) (−1) . (7.64) j k j+k n−k k=0

Formula (6.17) with a = δ + j + 1, b = −γ, c = j + 1 gives

362


F (δ + j + 1, −γ; j + 1; x) = (1 − x)γ−δ F (−δ, γ + j + 1; j + 1; x)

(7.65)

(note that Re (γ − δ) > 0). On multiplying (7.65) by (1 − x)δ and computing the coefficient of xn on both sides of the resulting equality one gets (7.64). 7.60. Proposition. Let γ, δ ∈ C and suppose −

1 1 < Re γ < , 2 2

−

1 1 < Re δ < , 2 2

−

1 1 < Re (γ + δ) < . 2 2

Then each of the following operators is bounded on 2 : A = Mγ+δ T (ξγ+δ )T (ξ−γ )Mγ−1 T (ξ−δ )Mδ−1 , −1 A−1 = Mδ T (ξδ )Mγ T (ξγ )T (ξ−γ−δ )Mγ+δ ,

B = Mγ−1 T (η−γ )Mδ−1 T (η−δ )T (ηγ+δ )Mγ+δ , −1 B −1 = Mγ+δ T (η−γ−δ )T (ηδ )Mδ T (ηγ )Mγ .

Proof. We only prove the boundedness of A. That the remaining three operators are bounded can be shown analogously. If Re δ = 0, we have A = [Mγ+δ T (ξδ )Mγ−1 ][T (ξ−δ )Mδ−1 ] and each bracket is a bounded operator by virtue of Corollary 6.22 and Proposition 6.44. Since, by Proposition 7.59, T (ξ−γ )Mγ−1 T (ξ−δ )Mδ−1 = T (ξ−δ )Mδ−1 T (ξ−γ )Mγ−1 ,

(7.66)

the case Re γ = 0 can be reduced to the case Re δ = 0. Now suppose Re (γ + δ) = 0. In this case it suffices to show that the operator C = T (ξ−γ )Mγ−1 T (ξ−δ )Mδ−1 is bounded. Let Re δ > 0. We have C = C1 C2 + C3 , where C1 = T (ξ−γ )Mγ−1 − Mγ−1 T (ξ−γ ),

C2 = T (ξ−δ )Mδ−1 ,

C3 = Mδ−1 T (ξ−γ−δ )Mγ−1 . Corollary 6.22 and Propositions 6.44, 6.45 immediately imply that C2 and C3 −γ (γ) (γ) are bounded. The j, k entry cjk of C1 is (1/µk − 1/µj ). Let α j−k satisfy |Re γ| < α < |Re γ| + 1/2. Then, by Lemmas 6.21 and 7.57, |cjk | ≤ c(j − k + 1)−1−|Re γ| (j − k + 1)α (k + 1)|Re γ|−α and so Lemma 7.65(b) gives the boundedness of C1 . The case Re δ < 0 can be reduced to the case Re δ > 0 by taking into account (7.66). Thus, we may assume that Re γ = 0, Re δ = 0, Re (γ + δ) = 0. First suppose Re δ > 0. Write A = (A1 A2 + A3 + cI)A4 , where


363

A1 = (Mγ+δ T (ξδ ) − T (ξδ )Mγ+δ )M−γ , −1 −1 A2 = cM−γ Mγ+δ T (ξ−δ ), −1 A3 = Mγ+δ T (ξδ )[Mγ−1 Mδ−1 Mγ+δ − cI]Mγ+δ T (ξ−δ ),

A4 = T (ξδ )Mδ T (ξ−δ )Mδ−1 . Here c is chosen so that Mγ−1 Mδ−1 Mγ+δ − cI is in C∞ (2 ) (Lemma 7.58). We first show that A1 is bounded. Let ajk (j ≥ k) denote the jk entry of the operator A1 : 3 4 δ δ (γ+δ) (γ+δ) (−γ) ajk = µj − µk µk . j−k j−k Suppose Re (γ + δ) > 0 and choose α so that Re δ + |Re γ| < α < Re δ + 1/2. Then, by Lemmas 6.21 and 7.57, 1 α 1 −1−Re δ (k + 1)−Re γ j − k + (k + 1)Re (γ+δ)−α |ajk | ≤ c j − k + 2 2 1 −1−Re δ+α = c j−k+ (k + 1)Re δ−α 2 and so Lemma 7.56(b) gives the boundedness of A1 . If Re (γ + δ) < 0, choose α so that |Re γ| + Re δ < α < |Re γ| + 1/2. Then, again by Lemmas 6.21 and 7.57, + + + + 1 1 δ + + (−γ) (γ+δ) (γ+δ) µk µj |ajk | = + µk − (γ+δ) + (γ+δ) + j−k + µk µj 1 −1−Re δ+α ≤ c(j + 1)Re (γ+δ) j − k + (k + 1)−Re γ−α 2 and Lemma 7.56(b) again implies that A1 is in L(2 ). Lemma 6.21 shows that the jk entry ajk (j ≥ k) of A2 admits the estimate + ++ + 1 1 −1+Re δ 1 −δ + + (j + 1)−Re δ . |ajk | = +c (−γ) (γ+δ) +≤c j−k+ j−k + + µ 2 µ j

j

Hence, by Lemma 7.56(a), A2 ∈ L(2 ). The operator A4 can be written as [T (ξδ )Mδ − Mδ T (ξδ )]T (ξ−δ )Mδ−1 + I and therefore its boundedness can be proved as for the operators C1 and C2 considered above. We are left with A3 . Let D = diag (dn )∞ n=0 denote the operator Mγ−1 Mδ−1 Mγ+δ − cI. Since dn = 1 −

∞ 1− j=n

γδ (γ + j)(δ + j)

−1 (Lemma 7.58),

364


it is easily seen that dn = O(1/n) as n → ∞. Because A3 = c−1 Mγ+δ T (ξδ )DM−γ A2 , it suffices to show that Mγ+δ T (ξδ )DM−γ is bounded. We shall prove that this operator is even Hilbert-Schmidt. Indeed, for the jk entry ajk (j ≥ k) we have + + + (γ+δ) δ (−γ) + ajk = ++µj dk µk ++ j−k ≤ c(j + 1)Re γ+Re δ (j − k + 1)−1−Re δ (k + 1)−1−Re γ ≤ c(j + 1)Re γ−1 (k + 1)−1−Re γ

∞ and thus j,k=0 |ajk |2 < ∞. This settles the proof for the case Re δ > 0. If Re δ < 0, write A = (A1 A2 + A3 + cI)A4 with A1 = cMγ+δ T (ξδ )M−γ , −1 −1 −1 A2 = M−γ (Mγ+δ T (ξ−δ ) − T (ξ−δ )Mγ+δ ), −1 A3 = Mγ+δ T (ξδ )[Mγ−1 Mδ−1 Mγ+δ − cI]Mγ+δ T (ξ−δ ),

A4 = T (ξδ )Mδ T (ξ−δ )Mδ−1 and then proceed in analogy to the case Re δ > 0. 7.61. Lemma. If α, γ, δ ∈ C and |Re α| < 1/2, then there is a c ∈ C \ {0} such that Mα T (ξα )(I − cMδ−1 Mδ+γ Mγ−1 )T (ξ−α )Mα−1 is Hilbert-Schmidt on 2 . Proof. The arguments we have used in the preceding proof to show that A3 is Hilbert-Schmidt also apply in the case at hand. 7.62. Proposition. Let β1 , . . . , βm ∈ C, suppose |Re βk |
f − pC[0,1] = A − p(D) x∈[0,1]

and it follows that p(D) is invertible. Theorem 7.68 now gives that p(D) is in Π{H 2 ; Pn }. Put a := {Pn APn }πG = {Pn f (D)Pn }πG ,

b := {Pn p(D)Pn }πG .

Thus, b is invertible in D/G. We claim that a − b < b−1 −1 . This will show that a is also invertible in D/G, as desired. By virtue of (7.3) and (7.77) we have a − b = f (D) − p(D) = f − pC[0,1] .

(7.79)

Since D = D∗ , it follows that bb∗ = b∗ b and, thus, that (b−1 )(b−1 )∗ = (b−1 )∗ (b−1 ). Consequently, b−1 is equal to the spectral radius (b−1 ) of b−1 . Because λ∈ / sp b ⇐⇒ b − λ ∈ G(D/G) ⇐⇒ p(D) − λ ∈ Π{H 2 ; Pn } (by 7.3) ⇐⇒ p(D) − λ ∈ GL(H 2 ), W(p(D)) − λ ∈ GL(H 2 ) (by 7.68) we obtain from (7.74) and (7.76) that sp b = p(x) : x ∈ [0, 1] ∪ (1 − λ)p(0) + λp(1) : λ ∈ [0, 1] . The invertibility of b along with the spectral mapping theorem implies that sp b−1 = (sp b)−1 , hence ! # 1 1 −1 , max (b ) = max max , x∈[0,1] |p(x)| λ∈[0,1] |(1 − λ)p(0) + λp(1)| and thus, because b−1 −1 = 1/(b−1 ), ! # −1 −1 b = min min |p(x)|, min |(1 − λ)p(0) + λp(1)| . x∈[0,1]

(7.80)

λ∈[0,1]

Since p(0) = f (0) and p(1) = f (1), we deduce from (7.78) and (7.79) that a − b < min |(1 − λ)p(0) + λp(1)|.

(7.81)

λ∈[0,1]

By (7.78), we have |f (x)| − |p(x)| ≤ |f (x) − p(x)| < (1/2) min |f (x)|, whence x∈[0,1]

|f (x)|−(1/2) min |f (x)| < |p(x)|, thus (1/2) min |f (x)| < min |p(x)|, and x∈[0,1]

x∈[0,1]

x∈[0,1]

so (7.78) and (7.79) give a − b
−1,

Re δ > −1

(recall what was said before 6.41). More precisely, we shall construct pairs of spaces pr and ps such that (i) the equation T (ξδ ηγ b)x = y has a unique solution x ∈ ps for each y ∈ pr ; (ii) there exists a constant c such that xps ≤ cypr for all y ∈ pr ; (iii) Tn (ξδ ηγ b) is invertible for all sufficiently large n, and for each y ∈ pr , Tn−1 (ξδ ηγ b)Pn y − xps → 0 as

n → ∞.

It is the nature of the matter to distinguish three cases (recall Theorems 6.48 and 6.49). (a) Re γ ≥ 0 and Re δ ≥ 0. (b) Re δ ≥ 0 and −1 < Re γ < 0; putting β = −γ and ν = γ + δ we have ξδ ηγ = ξν ϕ−β with Re ν ≥ 0 and 0 < Re β < 1. (c) Re γ ≥ 0 and −1 < Re δ < 0; then ξδ ηγ = ϕβ ην with β = −δ, ν = γ + δ and thus, 0 < Re β < 1, Re ν ≥ 0. Throughout the following let 1 1. Then Tn (ξδ ηγ ) is invertible for all n ≥ 0. Proof. Multiply equality (6.16) from the left and from the right by Pn and take into account that Pn T (ηγ ) = Pn T (ηγ )Pn ,

T (ξδ )Pn = Pn T (ξδ )Pn .

(7.87)

What results is Tn (ηγ )Pn Mγ+δ Pn Tn (ξδ ) = Γγ,δ Pn Mδ Pn Tn (ξδ ηγ )Pn Mγ Pn , and this gives the assertion at once.

(7.88)

7.8 Fisher-Hartwig Symbols: pµ Theory

379

If Re γ ≥ 0 and Re δ ≥ 0, then T (ξδ ηγ ) is bounded and invertible as operator from Dµp (γ) onto Rµp (δ) for every µ such that −1/p < µ < 1/q, and its inverse T −1 (ξδ ηγ ) is in L(pµ+Re δ , pµ−Re γ ) (Theorem 6.48). 7.78. Proposition. Let Re γ ≥ 0, Re δ ≥ 0, −1/p < µ < 1/q. Then Tn−1 (ξδ ηγ )Pn y − T −1 (ξδ ηγ )ypµ−Re γ → 0

(n → ∞)

(7.89)

for each y ∈ pµ+Re δ , and if λ > 0 and µ + λ < 1/q, then as n → ∞, Tn−1 (ξδ ηγ )Pn − T −1 (ξδ ηγ )L(pµ+λ+Re δ ,pµ−Re γ ) = O

1 . nλ

(7.90)

Proof. From (7.87), (7.88) we obtain that −1 T −1 (ξδ ηγ )Pn = Γγ,δ Mγ T (ξ−δ )Mγ+δ Pn T (η−γ )Mδ

and from Theorem 6.20 we know that −1 T −1 (ξδ ηγ ) = Γγ,δ Mγ T (ξ−δ )Mγ+δ T (η−γ )Mδ .

Hence, Tn−1 (ξδ ηγ )Pn y − T −1 (ξδ ηγ )yp,µ−Re γ −1 = |Γγ,δ | Mγ T (ξ−δ )Mγ+δ Qn T (η−γ )Mδ yp,µ−Re γ −1 ≤ cT (ξ−δ )Mγ+δ Qn T (η−γ )Mδ yp,µ

(by Corollary 6.22)

−1 cMγ+δ Qn T (η−γ )Mδ yp,µ+Re δ

≤ ≤ cQn T (η−γ )Mδ yp,µ−Re γ

(by Proposition 6.45) (by Corollary 6.22).

Here and throughout the following c denotes a constant independent of n but not necessarily the same at each occurrence. Again by Corollary 6.22 and Proposition 6.45, T (η−γ )Mδ yp,µ−Re γ ≤ cMδ yp,µ ≤ cyp,µ+Re δ , hence T (η−γ )Mδ y ∈ pµ−Re γ , and since Qn converges strongly to zero on pµ−Re γ , we get (7.89). To obtain (7.90) note that Qn T (η−γ )Mδ yp,µ−Re γ ≤ cn−γ Qn T (η−γ )Mδ yp,µ−Re γ+λ ≤ cn−γ Mδ yp,µ+λ (we use that µ + λ < 1/q) ≤ cn−λ yp,µ+λ+Re γ . Now consider T (ξν ϕ−β ) = T (ξν )T (ϕ−β ) with Re ν ≥ 0, 0 < Re β < 1. First let ν = 0. Then Proposition 6.24 shows that T (ϕ−β ) ∈ GL(pµ ) whenever Re β − 1/p < µ < 1/q.

380


7.79. Lemma. Let A ∈ GL(pµ ) and suppose An := Pn APn |Im Pn is invertible. Then, for y ∈ pµ , −1 −1 p p A−1 ypµ ≤ 1 + A−1 ypµ . n Pn y − A n Pn L(µ ) AL(µ ) Qn A Proof. −1 −1 y ≤ A−1 y + Qn A−1 y A−1 n Pn y − A n Pn y − Pn A −1 ≤ An Pn Pn y − Pn APn A−1 y + Qn A−1 y −1 ≤ A−1 y + Qn A−1 y. n Pn Pn AQn A

7.80. Proposition. If 0 < Re β < 1/q and Re β − 1/p < µ < 1/q, then Tn−1 (ϕ−β )Pn y − T −1 (ϕ−β )ypµ → 0

(n → ∞)

(7.91)

for each y ∈ pµ , and if λ > 0 and µ + λ < Re β + 1/q, then, as n → ∞, Tn−1 (ϕ−β )Pn − T −1 (ϕ−β )L(pµ+λ ,pµ ) = O

1 . nλ

(7.92)

Proof. That (7.91) holds is a consequence of Theorem 7.37. From Lemma 7.79, (7.91), and Proposition 6.23 we infer that Tn−1 (ϕ−β )Pn y − T −1 (ϕ−β )yp,µ ≤ cQn T −1 (ϕ−β )yp,µ ≤ cn−λ Qn T −1 (ϕ−β )yp,µ+λ for all n large enough. Using (6.16), Proposition 6.23, and Corollary 6.22 we obtain Qn T −1 (ϕ−β )yp,µ+λ = Qn Γ−β,β M−β T (ϕβ )Mβ yp,µ+λ ≤ cT (ϕβ )Mβ yp,µ+λ−Re β ≤ cMβ yp,µ+λ−Re β ≤ cyp,µ+λ , which gives (7.92).

Now let Re ν ≥ 0. Then, by virtue of Propositions 6.24 and 6.47, the operator T (ξν ϕ−β ) = T (ξν )T (ϕ−β ) is bounded and invertible from pµ onto Rµp (ν) and its inverse T −1 (ξν ϕ−β ) is in L(pµ+Re ν , pµ ). 7.81. Proposition. Let Re ν ≥ 0 and 0 < Re β < 1. If Re β < 1/q and Re β − 1/p < µ < 1/q, then Tn−1 (ξν ϕ−β )Pn y − T −1 (ξν ϕ−β )ypµ → 0

(n → ∞)

(7.93)

for each y ∈ pµ+Re ν , and if λ > 0 and µ + λ < Re β + 1/q, then, as n → ∞, Tn−1 (ξν ϕ−β )Pn − T −1 (ξν ϕ−β )L(pµ+λ+Re ν ,pµ ) = O

1 . nλ

(7.94)


381

Proof. We have Pn − Pn T −1 (ξν ϕ−β )Pn T (ξν ϕ−β )Pn = Pn T −1 (ξν ϕ−β )Qn T (ξν ϕ−β )Pn −1 = Pn T −1 (ξν ϕ−β )Mν+β T (η−β )Qn T (ηβ )Mν+β Qn T (ξν ϕ−β )Pn −1 = Pn T −1 (ξν ϕ−β )Mν+β T (η−β )Qn T (ηβ )Mν+β T (ξν ϕ−β )Pn −1 −Pn T −1 (ξν ϕ−β )Mν+β T (η−β )Qn T (ηβ )Mν+β Pn T (ξν ϕ−β )Pn ,

and since, by (6.16) and (7.87), Qn T (ηβ )Mν+β T (ξν ϕ−β )Pn −1 −1 = cQn T (ηβ )Mν+β Mν+β T (η−β )Mν T (ξν+β )M−β Pn −1 = cMν Qn T (ξν+β )Pn M−β = 0,

we arrive at the formula Tn−1 (ξν ϕ−β )Pn − Pn T −1 (ξν ϕ−β )Pn −1 T (η−β )Qn T (ηβ )Mν+β Pn . = −Pn T −1 (ξν ϕ−β )Mν+β

(7.95)

−1 Here T −1 (ξν ϕ−β )Mν+β ∈ L(pµ−Re β , pµ ) and the term in braces equals

0 −1 10 10 1 −1 T (ϕ−β )M−β M−β T (ξ−β )Qn T (ηβ )Mβ Pn Mβ−1 Mν+β .

(7.96)

The operator in the first brackets is in L(pµ , pµ−Re β ), the operator in the third brackets belongs to L(pµ+Re ν , pµ ), and the operator in the middle brackets is Pn M−β T (ξ−β )Qn T (ηβ )Mβ Pn + Qn M−β T (ξ−β )Qn T (ηβ )Mβ Pn = Pn T −1 (ϕ−β )Pn − Tn−1 (ϕ−β )Pn + Qn T −1 (ϕ−β )Pn = T −1 (ϕ−β )Pn − Tn−1 (ϕ−β )Pn and thus, by (7.91), it converges strongly to zero on pµ . We now deduce from (7.95) that Tn−1 (ξν ϕ−β )Pn y − Pn T −1 (ξν ϕ−β )Pn yp,µ → 0 (n → ∞) for each y ∈ pµ+Re ν , and since Tn−1 (ξν ϕ−β )Pn y − T −1 (ξν ϕ−β )yp,µ ≤ Tn−1 (ξν ϕ−β )Pn y − Pn T −1 (ξν ϕ−β )Pn yp,µ +Pn T −1 (ξν ϕ−β )Qn yp,µ + Qn T −1 (ξν ϕ−β )yp,µ ,

(7.97)

and Qn converges strongly to zero on pµ+Re ν and pµ , we obtain (7.93). Due to (7.92) the L(pµ+λ , pµ ) norm of the operator in the middle brackets of (7.96) is O(1/nλ ), and because the operator in the third brackets of (7.96) belongs to L(pµ+λ+Re ν , pµ+λ ), we get from (7.95) that

382


Tn−1 (ξν ϕ−β )Pn − Pn T −1 (ξν ϕ−β )Pn L(pµ+λ+Re ν ,pµ ) = O

1 nλ

as n → ∞. Finally, since Qn T −1 (ξν ϕ−β )yp,µ ≤ cn−λ Qn T −1 (ξν ϕ−β )yp,µ+λ ≤ cn−λ T −1 (ξν ϕ−β )yp,µ+λ ≤ cn−λ yp,µ+λ+Re ν and T −1 (ξν ϕ−β )Qn yp,µ ≤ cQn yp,µ+Re ν ≤ cn−λ yp,µ+λ+Re ν , inequality (7.97) yields estimate (7.94). Out next objective is to extend the results hitherto obtained for the “pure singularity” ξδ ηγ to symbols which still involve a “regular part” b. As usual, this will be done by applying a perturbation argument (see, e.g., Pr¨ ossdorf, Silbermann [407]). The following theorem is just what is needed in our situation. 7.82. Theorem. Suppose (a) X, Y, Z, U are Banach spaces and Pn (n = 0, 1, 2, . . .) are projections defined and bounded on each of the spaces X, Y, Z, U ; (b) Z ⊂ Y and X ⊂ U , the embeddings being continuous; (c) A ∈ L(X, Y ) is (boundedly) invertible; (d) the operators An := Pn APn ∈ L(Pn X, Pn Y ) are invertible for all −1 zU → 0 as n → ∞; sufficiently large n and, for each z ∈ Z, A−1 n Pn z − A (e) T ∈ C∞ (U, Z); (f) A + T ∈ L(X, Y ) is (boundedly) invertible. Then (g) the operators Pn (A + T )Pn ∈ L(Pn X, Pn Y ) are invertible for all n large enough; (h) (Pn (A + T )Pn )−1 Pn z − (A + T )−1 zU → 0 (n → ∞) ∀ z ∈ Z; (k) there is a constant c independent of n and z such that (Pn (A + T )Pn )−1 Pn z − (A + T )−1 zU −1 −1 ≤ cA−1 zU + cA−1 T L(U ) zZ . n Pn z − A n Pn T − A Proof. Obviously, I + A−1 T ∈ GL(X): the inverse is (A + T )−1 A. We claim that I +A−1 T is also in GL(U ). Since A−1 T ∈ C∞ (U ), it follows that I +A−1 T is Fredholm on U and has index zero there. Thus we must show that it has a trivial kernel. Let (I + A−1 T )u = 0 for some u ∈ U . Then u = −A−1 T u,


383

hence u ∈ X, hence Au = −T u, hence (A + T )u = 0, and this gives u = 0, as desired. −1 T From (d) and (e) we conclude that I + A−1 n Pn T converges to I + A uniformly on U . Therefore, by what has been proved in the preceding paragraph, I + A−1 n Pn T is in GL(U ) for all sufficiently large n, say n ≥ n0 . Let Bn ∈ L(U ) denote the inverse: Bn + Bn A−1 n Pn T = I,

Bn + A−1 n Pn T Bn = I.

(7.98)

The second equality in (7.98) implies that Bn ∈ L(X). It also implies that Pn Bn Pn = Bn Pn . Thus, for y ∈ Y , −1 −1 Pn (A + T )Pn Bn A−1 n Pn y = An Bn An Pn y + Pn T Bn An Pn y −1 = An Bn A−1 n Pn y + (An − An Bn )An Pn y = An A−1 n Pn y = Pn y

and, for x ∈ X, −1 −1 Bn A−1 n Pn Pn (A + T )Pn x = Bn An Pn An x + Bn An Pn T Pn x = Bn Pn x + (I − Bn )Pn x = Pn x.

It results that Pn (A + T )Pn ∈ L(Pn X, Pn Y ) is invertible for all n ≥ n0 and that −1 −1 −1 An Pn ∈ L(Pn Y, Pn X) Bn A−1 n Pn = (I + An Pn T ) is the inverse. Now let z ∈ Z and n ≥ n0 . Since (I + A−1 T )−1 A−1 ∈ L(Y, X) is the inverse of A + T ∈ L(X, Y ), we get (Pn (A + T )Pn )−1 Pn z − (A + T )−1 Pn zU −1 ≤ ((I + A−1 − (I + A−1 T )−1 )A−1 n Pn T ) n Pn zU −1 −1 −1 +(I + A T ) (A−1 P z − A z) n U n −1 −1 −1 −1 ≤ (I + A−1 P T ) − (I + A T ) n L(U ) An Pn L(Z,U ) zZ n −1 +(I + A−1 T )−1 L(U ) A−1 zU . n Pn z − A −1 But (I + A−1 − (I + A−1 T )−1 L(U ) is not larger than n Pn T ) −1 T L(U ) (I + A−1 T )−1 2L(U ) A−1 n Pn T − A

−1 T 1 − (I + A−1 T )−1 L(U ) A−1 n Pn T − A L(U ) −1 and since A−1 T L(U ) → 0 as n → ∞, the proof is complete. n Pn T − A

7.83. Conventions. Here and throughout Sections 7.84–7.85 and 7.87–7.89 we shall assume that the “regular part” b is a function with (at least) absolutely convergent Fourier series which does not vanish on

T and has index zero. So 2.41(e) implies that b has a logarithm log b = n∈Z (log b)n χn in

384


W = F 1,1 0,0 . If we define G(b), b− , b+ as in Corollary 6.55, then b = G(b)b− b+ . For what follows in this chapter we may without loss of generality assume that G(b) = 1. Let ε0 > 0 denote a real number which can be chosen as small as desired but remains fixed throughout the following. Given a real number x we define (x + 0) := ε0

if

x ≤ 0,

(x + 0) := x if

x > 0.

Also recall how (x)◦ was defined in 6.52. If Re γ ≥ 0, Re δ ≥ 0, and T (b) ∈ GL(pµ ) (−1/p < µ < 1/q), then T (ξδ ηγ b) is an invertible operator in L(Dµp (γ), Rµp (δ)) and its inverse T −1 (ξδ ηγ b) belongs to L(pµ+Re δ , pµ−Re γ ) (Theorem 6.48). 7.84. Theorem. Let Re γ ≥ 0, Re δ ≥ 0, −1/p < µ < 1/q. (a) If q,p b ∈ F 1,1 (Re γ−µ)◦ ,(µ+Re δ)◦ ∩ F 1/p+(Re γ+0),1/q+(Re δ+0) ,

then Tn (ξδ ηγ b) is invertible for all sufficiently large n and if y ∈ pµ+Re δ , then Tn−1 (ξδ ηγ b)Pn y − T −1 (ξδ ηγ b)ypµ−Re γ → 0 (n → ∞). (b) If λ > 0 and µ + λ < 1/q, and if 1,1 q,p b ∈ F (Re γ−µ)◦ ,(µ+λ+Re δ)◦ ∩ F 1/p+λ+(Re γ+0),1/q+λ+(Re δ+0) ,

then, as n → ∞, Tn−1 (ξδ ηγ b)Pn − T −1 (ξδ ηγ b)L(pµ+λ+Re δ ,pµ−Re γ ) = O

1 . nλ

Proof. (a) We apply Theorem 7.82 with X = Dµp (γ),

Y = Rµp (δ),

Z = pµ+Re δ ,

U = pµ−Re γ .

The projections Pn are clearly bounded on Z and U , and it is easy to prove that they are bounded on X and Y (see the remark following the proof). Proposition 6.47 shows that Z and X are continuously embedded in Y and U , respectively. Put A = T (b+ )T (ξδ ηγ )T (b− ),

A + T = T (b− )T (ξδ ηγ )T (b+ ).

Note that A + T = T (ξδ ηγ b). From Theorem 6.54 and Proposition 6.53 it can be deduced that T (b± ) ∈ GL(X) and T (b± ) ∈ GL(Y ). Consequently, both A and A + T are bounded and invertible from X to Y . Since −1 −1 (Pn APn )−1 Pn = T (b−1 − )Pn Tn (ξδ ηγ )Pn T (b+ )

(7.99)


385

−1 (recall 7.77) and T (b−1 − ) ∈ L(U ), T (b+ ) ∈ L(Z), we infer from Proposition 7.78 that the hypothesis (d) of Theorem 7.82 is satisfied. Using Proposition 2.14 we get

T = T (b+ )H(ξδ ηγ )H(b− ) +H(b+ )H(ξ)δ η. γ )T (b− ) . +H(b+ )T (ξ)δ η. γ )H(b− ).

(7.100)

Lemma 6.51 implies that p p ). H(ξδ ηγ )H(b. − ), H(b+ )H(ξδ η γ ) ∈ C∞ (µ−Re γ , µ+Re δ ),

and Lemma 6.50 shows that H(b+ ) ∈ C∞ (pτ , pµ+Re δ ),

H(b− ) ∈ C∞ (pµ−Re γ , pτ ),

where τ := µ + 1/p − 1/q. The Toeplitz operators still occurring in (7.100) q are bounded on the corresponding spaces (in particular, T (ξ)δ η. γ ) ∈ L(τ ) by Proposition 6.44), and so it follows that T ∈ C∞ (U, Z). Thus, Theorem 7.82 can be applied and its conclusions (g) and (h) give the assertion. (b) Under the stronger restrictions imposed on the smoothness of b, all arguments of the proof of part (a) remain true with µ + Re δ replaced by µ + λ + Re δ. Hence, combining (7.99) and (7.90) we get 1 (Pn APn )−1 Pn − A−1 L(pµ+λ+Re δ ,pµ−Re γ ) = O λ n and, consequently, (Pn APn )−1 Pn T − A−1 T L(pµ−Re γ ) ≤ cn−λ T L(pµ−Re γ ,pµ+λ+Re δ ) = O

1 . nλ

Conclusion (k) of Theorem 7.82 finishes the proof. Remark. We emphasize that we do not assert the uniform boundedness of the projections Pn on X and on Y . It can be shown that, for µ > −1/p, sup Pn L(Rµp (α)) < ∞ ⇐⇒ Re α
0 is a real number which can be chosen as small as desired and c denotes a constant which does not depend on k, j, n. 7.88. Theorem. Let Re γ ≥ 0, Re δ ≥ 0, and A = T (ξδ ηγ b). If p and q are any real numbers such that 1 < p < ∞, 1/p + 1/q = 1, and if q,p b ∈ F 1,1 1/p+Re γ,1/q+Re δ ∩ F 1+1/p+Re γ,1+1/q+Re δ , −1 )jk | is not larger than then |(A−1 n )jk − (A

c(k + 1)1/q+Re δ−ε/2 (j + 1)1/p+Re γ−ε/2 n−1+ε . Proof. Apply Theorem 7.84(b) with µ = −1/p + ε/2 and λ = 1 − ε: −1 −1 |(A−1 ek , ej )| ≤ A−1 ek p,µ−Re γ ej q,−µ+Re γ n Pn ek − A n Pn ek − A −1 −1 p p ≤ An Pn − A L(µ+λ+Re δ ,µ−Re γ ) ek p,µ+λ+Re δ ej q,−µ+Re γ .

7.89. Theorem. Put A = T (ξν ϕ−β b). (a) Let Re ν ≥ 0 and 0 < Re β < 1. If q is any real number such that Re β < 1/q < 1, if 1/p + 1/q = 1, and if 1,1 q,p b ∈ F 1/p,1/q+Re ν ∩ F 1+1/p−Re β,1+1/q+Re ν−Re β , −1 )jk | is not larger than then |(A−1 n )jk − (A

c(k + 1)1/q+Re ν−ε/2 (j + 1)1/p−Re β−ε/2 n−1+Re β+ε . (b) Let Re ν = 0 and 0 < Re β < 1. If q is any real number such that q ≤ 2, Re β < 1/q < 1, if 1/p + 1/q = 1, and if b ∈ F 11/q+Re β , then −1 )jk | is not larger than |(A−1 n )jk − (A c(k + 1)1/q+Re β−ε/2 (j + 1)1/p−Re β−ε/2 n−1+ε .

7.9 Invertibility Versus Finite Section Method

389

Proof. (a) Theorem 7.85(b) with µ = Re β − 1/p + ε/2, λ = 1 − Re β − ε. (b) Put µ = Re β − 1/p + ε/2. Lemma 7.79 gives −1 −1 p p A−1 yp,µ ≤ (1 + A−1 yp,µ n Pn y − A n Pn L(µ ) AL(µ ) )Qn A

≤ cQn A−1 yp,µ

(7.85(a), 6.44, 6.23).

Put λ = 1 − ε. Then again by Propositions 6.23 and 6.44, Qn A−1 yp,µ ≤ n−λ Qn A−1 yp,µ+λ −1 ≤ cn−λ T (b−1 (ϕ−β )T (ξ−ν )T (b−1 + )T − )yp,µ+λ ≤ cn−λ yp,µ+λ . Thus

1 nλ and now one can proceed as in the proof of Theorem 7.88. Tn−1 (ξν ϕ−β b)Pn − T −1 (ξν ϕ−β b)L(pµ+λ ,pµ ) = O

7.90. Toeplitz operators on p,+∞ . Recall the definitions and results . We say that A of 6.57. Let A be a linear and bounded operator on p,+∞ N ; P } if there is an n such that for each y ∈ p,+∞ the belongs to Π{p,+∞ n 0 N N equations Pn APn x = Pn y have a unique solution xn ∈ Im Pn for all n ≥ n0 and if xn converges in the topology of p,+∞ to a solution x ∈ p,+∞ of the N N p,+∞ denote the collection of all linear and bounded equation Ax = y. Let CN having the following properties: operators K on p,+∞ N ; (i) K is defined on the whole space pN and maps pN into p,+∞ N (ii) K ∈ C∞ (pN , p,µ N ) for all µ ∈ Z+ . p,+∞ ∞ ∞ . Finally for a ∈ CN Example: if c ∈ CN ×N , then H(c) ∈ CN ×N , define ) a as usual by ) a = a(1/t) (t ∈ T). The following result was established by Böttcher [61] (for N = 1 see also Gorodetsky [244]). p,+∞ ∞ . Then T (a) + K ∈ Π{p,+∞ ; Pn } if and Let a ∈ CN ×N and K ∈ CN N p,+∞ only if T (a) + K is invertible on N and T () a) is invertible on p,−∞ . N

7.9 Invertibility Versus Finite Section Method We now assume that the underlying space is H 2 . In 7.28(c) we observed that there is a natural continuous algebraic homomorphism ∆π of alg T FJπ (A) onto alg T π (A). Moreover, we showed that ∆π is even an isomorphism if A is a closed algebra between C and C + H ∞ or if A = P C or A = P QC. 7.91. Theorem. There exist {An } ∈ alg T F (L∞ ) such that Aπ is invertible in alg T π (L∞ ), where A := s- lim An , but {An }πJ is not invertible in n→∞

alg T FJπ (L∞ ). In particular, ∆π is not an isomorphism between alg T FJπ (L∞ ) and alg T π (L∞ ).

390


Proof. If {An } ∈ alg T F (L∞ ) and if {An }πJ is invertible, then {Wn An Wn } also belongs to alg T F (L∞ ) and {Wn An Wn }πJ is invertible, too. Hence, since ∆π is an algebraic homomorphism, the invertibility of {An }πJ implies that W{An } (A) = s- lim Wn An Wn is Fredholm. Therefore, in order to prove the n→∞

theorem it suffices to find A ∈ Φ(H 2 ) and {An } ∈ alg T F (L∞ ) such that / Φ(H 2 ). An → A strongly and W{An } (A) ∈ ∞ is an infinite Blaschke product, put A = I and Suppose b ∈ H An = Tn (b)Tn (b). Then {An } ∈ alg T F (L∞ ), An → A strongly, and W{An } (A) = T ()b)T ()b). Assume T ()b)T ()b) ∈ Φ(H 2 ). Then T ()b) ∈ Φ+ (H 2 ), and because T ()b)T ()b−1 ) = I, we have T ()b) ∈ Φ− (H 2 ) and thus T ()b) ∈ Φ(H 2 ). But Theorem 2.65 shows that T ()b) cannot be Fredholm, since the harmonic (= analytic) extension of an infinite Blaschke product is not bounded away from zero near the unit circle T. The following theorem shows that there exist even a ∈ L∞ such that the previous theorem is true with An = Tn (a). 7.92. Theorem (Treil). There exists a function a ∈ L∞ such that T (a) is / Π{H 2 ; Pn }. invertible on H 2 but T (a) ∈ Proof. We shall construct three sequences {ak }, {nk }, {fk } (k = 0, 1, 2, . . .): . {ak } consists of unimodular functions ak = ei(ϕk +ψk ) ∈ L∞ (ϕk , ψk are real∞ valued functions in L and the tilde will always refer to the conjugation operator); {nk } is a sequence of positive integers satisfying nk < nk+1 ; {fk } is constituted by polynomials fk ∈ Im Pnk satisfying fk 2 = 1. The construction is required to provide ak , ϕk , ψk , nk , fk which fulfil the following conditions: (i) Tnk (ak )Pnk − Tnk (ak+1 )Pnk L(H 2 ) ≤ 1/2k ; (ii) Tnk (ak )fk 2 ≤ 1/2k ; (iii) ∃ α, β > 0 : ϕk ∞ ≤ α < π/2, ψk ∞ ≤ β < ∞; (iv) supp ϕk ∩ supp ψk = ∅; (v) ϕk − ϕk+1 2 < 1/2k , ψk − ψk+1 2 < 1/2k . Condition (v) implies that there are ϕ, ψ ∈ L2 such that ϕk − ϕ2 → 0, ψk − ψ2 → 0 as k → ∞. Taking into account (iii) we see that actually ) ϕ, ψ ∈ L∞ and ϕ∞ ≤ α, ψ∞ ≤ β. Hence, if we define a = ei(ϕ+ψ) , then . T (a) is invertible due to Theorem 2.23. Since ak = ei(ϕk +ψk ) converges in the ) .k − ψ ) 2 → 0 by the continuity of L2 -norm to a = ei(ϕ+ψ) (also note that ψ 2 the conjugation operator on L ), we deduce from 1.1(d) that T (ak ) converges strongly to T (a) on H 2 . This observation combined with (i) and (ii) gives that

7.9 Invertibility Versus Finite Section Method

391

∞ " " " " Tnk (a)fk 2 = " Pnk T (aj+1 ) − T (aj ) fk + Pnk T (ak )fk "

≤

2

j=k ∞

" " "Pn T (aj+1 ) − T (aj ) fk " + Pn T (ak )fk 2 k k 2

j=k ∞ 1 1 1 ≤ + k < k−2 , 2j 2 2 j=k

from which it is easily seen that T (a) ∈ / Π{H 2 ; Pn }. . We now construct the sequences {ak } = {ei(ϕk +ψk ) }, {nk }, {fk }. Condition (iv) is needed to carry out the construction. Let b0 be the function defined by (4.64). Since T (b0 ) is invertible, there exist c ∈ R and real-valued functions u, v ∈ L∞ such that b0 = ei(u+)v+c) ,

u∞ ≤ α
0, the ε-pseudospectrum of an operator A ∈ L(X) on a Banach space X is defined by spε,X A = λ ∈ C : (A − λI)−1 L(X) ≥ 1/ε . This definition has several modifications and generalizations. One of them is so-called structured pseudospectra (also called spectral value sets). In this context one is given two operators B, C ∈ L(X) and one defines −1 BL(X) ≥ 1/ε . spB,C ε,X A = λ ∈ C : C(A − λI) Clearly, spε,X A is just spI,I ε,X A. The following theorem provides us with alternative characterizations of structured pseudospectra. 7.98. Theorem. If X is a Hilbert space or if X is a Banach space and at least one of the operators B and C is compact, then / spB,C sp (A + BKC), (7.108) ε,X A = K≤ε

the union over all K ∈ L(X) with the given norm constraint. Proof. To prove that the right-hand side of (7.108) is contained in the lefthand side, it suffices to show that if A is invertible, CA−1 B < 1/ε, and K ≤ ε, then A+BKC is invertible. So let A be invertible, CA−1 B < 1/ε, and K ≤ ε. Then CA−1 BK < 1 and hence I + CA−1 BK is invertible. Since I + M N is invertible if and only if so is I + N M (in which case (I + M N )−1 = I − M (I + N M )−1 N ), we conclude that I + A−1 BKC and thus also A + BKC = A(I + A−1 BKC) are invertible, as desired.

7.10 Pseudospectra

397

In order to prove that the left-hand side of (7.108) is a subset of the righthand side, suppose A is invertible and CA−1 B = 1/δ > 0. We show that there is a K ∈ L(X) such that K = δ and A + BKC is not invertible. The operator A + BKC = A(I + A−1 BKC) is invertible if and only if so is I + A−1 BKC, which, by the I + M N versus I + N M trick employed above, is equivalent to the invertibility of I + KCA−1 B. Assume first that X is a Hilbert space. Abbreviate CA−1 B to S and put K = −δ 2 S ∗ . Then K = δ and I + KCA−1 B = I − δ 2 S ∗ S. The spectral radius of the positive semi-definite operator S ∗ S coincides with its norm, that is, with S ∗ S = S2 = 1/δ 2 . It follows that 1/δ 2 ∈ sp S ∗ S and hence that 0 ∈ sp (I − δ 2 S ∗ S). If X is a Banach space and B or C is compact, then CA−1 B is compact. Consequently, there is a u ∈ X such that u = 1 and CA−1 Bu = 1/δ. By the Hahn-Banach theorem, there exists a functional ϕ ∈ X ∗ such that ϕ = 1 and ϕ(CA−1 Bu) = CA−1 Bu = 1/δ. Let K ∈ L(X) be the rankone operator defined by Kx = −δϕ(x)u. Clearly, K ≤ δ. We have BKCA−1 Bu = B(−δϕ(CA−1 Bu)u) = −δϕ(CA−1 Bu)Bu = −δ(1/δ)Bu = −Bu

(7.109)

Put y = A−1 Bu. If y = 0, then CA−1 Bu = Cy = 0, which contradicts the inequality CA−1 Bu = 1/δ > 0. Thus, y = 0. From (7.109) we infer that BKCy = −Bu = −Ay, whence (A+BKC)y = 0. This implies that A+BKC is not invertible. Open problem. Is Theorem 7.98 true for Banach spaces without the extra assumption that one of the operators B and C be compact? 7.99. Norm of the resolvent. The modulus of a nonconstant analytic function cannot be locally constant. This is no longer true for operator-valued analytic functions. Indeed, the function λ0 f : C → L(C2 ), λ → 01 is analytic but f (λ)2 = max{|λ|, 1} is constant for |λ| ≤ 1. However, if f (λ) is not an arbitrary operator-valued analytic function but the resolvent of an operator, f (λ) = (A − λI)−1 , things are different. In this case f (λ) cannot be locally constant in two important situations. (a) Andrzej Daniluk proved in 1994 that if X is a Hilbert space and A is in L(X), then (A − λI)−1 cannot be locally constant. (b) In B¨ ottcher, Grudsky, Silbermann [87] it is shown that if X is LpN (Ω, dµ) with 1 < p < ∞ and A ∈ L(X), then (A − λI)−1 is nowhere locally constant. Full proofs of these two results are also in B¨ ottcher, Grudsky [86].

398


Open problem. Is the norm (A − λI)−1 nowhere locally constant for an operator A ∈ L(X) on an arbitrary Banach space X? 7.100. Limiting sets. Let {En }∞ n=1 be a sequence of sets En ⊂ C. The (uniform) limiting set lim inf En is defined as the set of all λ ∈ C for which there are λ1 ∈ E1 , λ2 ∈ E2 , . . . such that λn → λ as n → ∞, and the (partial ) limiting set lim sup En is the set of all λ ∈ C for which there exist natural numbers n1 < n2 < . . . and λnk ∈ Enk such that λnk → λ as k → ∞. If all En and E are compact, then the two equalities lim inf En = lim sup En = E are equivalent to the convergence of En to E in the Hausdorff metric, which means that d(En , E) → 0 with d(A, B) := max max dist (a, B), max dist (b, A) . a∈A

b∈B

This is a result of Hausdorff [268, Section 2.8]. A full proof is also in Hagen, Roch, Silbermann [263, Sections 3.1.1 and 3.1.2]. p 7.101. Theorem. If {An } is in algp T F (CN ×N ) (1 < p < ∞) or in alg2 T F (P CN ×N ) (p = 2), then

lim inf spε,pN An = lim sup spε,pN An = spε,pN A ∪ spε,pN W(A). Here spε,pN An refers to the ε-pseudospectrum of An as an operator on Im Pn with the p norm. Proof. We abbreviate spε,pN to spε . We first prove that spε A ⊂ lim inf spε An .

(7.110)

If λ ∈ sp A, then (An − λI)−1 p → ∞ by Proposition 7.3. Consequently, (An − λI)−1 p ≥ 1/ε for all n ≥ n0 , which implies that λ ∈ spε An for all n ≥ n0 . Thus, λ is in the right-hand side of (7.110). Now suppose that λ ∈ spε A \ sp A. Then (A − λI)−1 p ≥ 1/ε. Let U ⊂ C be any open neighborhood of λ. From 7.99 we deduce that there is a point µ ∈ U such that (A − µI)−1 p > 1/ε (since otherwise (A − zI)−1 p would be constant in U ). Hence, we can find a natural number k0 such that (A − µI)−1 p ≥

1 ε − 1/k

∀ k ≥ k0 .

As U was arbitrary, we can therefore find µ1 , µ2 , . . . such that µk ∈ spε−1/k A and µk → λ. Since −1 Bp (7.111) B −1 p = inf y =0 yp

7.10 Pseudospectra

399

for every invertible operator B and (A − µk I)−1 p ≥ 1/(ε − 1/k), it follows that 1 inf (A − µk I)yp ≤ ε − . k yp =1 Thus, there are yk ∈ pN such that yk p = 1 and (A−µk I)yk p < ε−1/(2k). We have (An − µk I)Pn yk p → (A − µk I)yk p ,

Pn yk p → yk p = 1

as n → ∞. Consequently, 1 (An − µk I)Pn yk p n0 (k). Again by (7.111), 1 −1 1 (An − µk I)−1 p > ε − > 3k ε and thus µk ∈ spε An for all n > n0 (k). This implies that λ = lim µk belongs to the right-hand side of (7.110). Repeating the above reasoning with Wn An Wn and W(A) in place of An and A we get spε W(A) ⊂ lim inf spε Wn An Wn , and since Wn is an isometry on Im Pn , we have spε Wn An Wn = spε An . At this point we have proved that spε A ∪ spε W(A) ⊂ lim inf spε An , and we are left with the inclusion lim sup spε An ⊂ spε A ∪ spε W(A).

(7.112)

Let λ ∈ / spε A ∪ spε W(A). Then 1 1 , (W(A) − λI)−1 p < , ε ε and from Theorem 7.96 we deduce that 1 1 ∀ n ≥ n0 (An − λI)−1 p < − δ < ε ε (A − λI)−1 p
0. If |µ − λ| is sufficiently small, then An − µI is invertible together with An − λI and (An − µI)−1 p ≤

(An − λI)−1 p . 1 − |µ − λ| (An − λI)−1 p

(7.114)

Let |µ − λ| < εδ(1/ε − δ)−1 . Then (7.113) and (7.114) give (An − µI)−1 p
0 is a real number and b is a sufficiently smooth function with values in (0, ∞). Then Tn (a) is positive definite and hence invertible for all n ≥ 0. There is a continuous function Gα : [0, 1]2 → (0, ∞) such that 0 1 lim n1−2α Tn−1 (a) nx,ny =

n→∞

1 Gα (x, y) b(t0 )

(7.115)

uniformly in (x, y) ∈ [0, 1]2 ; here z denotes the integral part of z. A weakened version of this result was in principle already established by Widom [560], [561], showed that the integral operator on L2 (0, 1) with the ker0 who 1 1−2α −1 Tn (a) nx,ny converges uniformly to the integral operator with nel n the kernel Gα (x, y)/b(t0 ). In the form cited here, (7.115) is due to Rambour, Seghier [412], [413], [414]. See also B¨ ottcher [77] and B¨ ottcher, Grudsky [86]. We remark that if α is a natural number, then Gα (x, y) is Green’s kernel for the boundary value problem (−1)α u(2α) (x) = v(x) on [0, 1] and u(k) (0) = u(k) (1) = 0 for k = 0, 1, . . . , α − 1. Different questions on the behavior of the entries of the inverses of Toeplitz ¨ matrices are considered in Strohmer [509] and Deift, Ostensson [150]. 7.91–7.93. We do not know who was the first to raise the question on whether the finite section method is applicable to every Toeplitz operator T (a) in ottcher, Silbermann [106, p. 76] and GL(H 2 ), but it is an old question. In B¨ Silbermann [480] we formulated this question and conjectured that the answer be yes. We even conjectured that the algebras alg T π (L∞ ) and alg T FJπ (L∞ ) are isomorphic to each other. However, we soon realized that the two algebras are not isomorphic (Theorem 7.91), which then made us become thoroughly convinced in that the answer to the first question is also negative. By the way, this conviction had also been held by the colleagues in Kishinev for a long time. We then tried to find a symbol in P3 C which generates an invertible Toeplitz operator to which the finite section method is not applicable, but our endeavor was not (and has not yet been) crowned with success. The state of affairs was fortunately altered by Treil’s [524] result 7.92, which is undoubtedly one of the most significant achievements in this field. Notice that Treil’s theorem 7.92 is a “positive” result in the following sense: if it would have turned out that the finite section method were applicable to every invertible Toeplitz operator, then a major part of all earlier work were for nothing; Treil’s result a-posteriorily justifies this work. The material of 7.92 and 7.93 is from Treil [525]. The following result is shown in B¨ ottcher, Grudsky [80].


405

Suppose b ∈ C, T (b) ∈ GL(H 2 ), and the zeroth Fourier coefficient b0 of b is zero. If a ∈ L∞ is given by t+1 a(t) = b exp (t ∈ T), t−1 / Π{H 2 ; Pn }. then T (a) ∈ GL(H 2 ) but T (a) ∈ Notice that a and b are related by the formula x−i a (7.116) = b(eix ) (x ∈ R). x+i √ √ The function b(t) = ( 3−t)2 ( 3−1/t)−2 satisfies all hypotheses and delivers the symbol 2 √ x−i 3 − eix (x ∈ R) a = √ x+i 3 − e−ix satisfying T (a) ∈ GL(H 2 ) but T (a) ∈ / Π{H 2 ; Pn }. Furthermore, the symbol x−i a = eif (x) (x ∈ R) x+i defined by f (x) = π −

8 π

cos 3x cos 5x + + . . . cos x + 32 52

can also be represented in the form (7.116) with b ∈ C, T (b) ∈ GL(H 2 ), b0 = 0 (simply let b be the function given by (4.62)). This symbol is almost periodic and reveals that we need not look for symbols a with T (a) ∈ GL(H 2 ) but T (a) ∈ / Π{H 2 ; Pn } in the entire abyss of L∞ . 7.94–7.103. Trefethen and Embree’s book [523] is an excellent source on all aspects of pseudospectra. The philosophy is that some basic phenomena for an operator can be understood by considering the norm (A−λI)−1 . If A is a normal Hilbert space operator, then (A−λI)−1 = 1/dist (λ, sp A), and hence knowledge of the spectrum alone is sufficient to understand (A − λI)−1 . However, for nonnormal Hilbert space operators, or more generally, for Banach space operators, the information contained in the spectrum does not provide a precise description of (A − λI)−1 . In that case pseudospectra do a perfect job, since they encode all information about (A − λI)−1 on the one hand and do this in a visual manner (as subsets of the plane) on the other. The Toeplitz operators Tn (a) and T (a) are normal in rare cases only, and hence pseudospectra are expected to tell us more on Toeplitz operators than spectra. The pioneering works on pseudospectra of Toeplitz matrices are due to Henry Landau [331], [332], [333] and Reichel and Trefethen [418]. They essentially established Corollary 7.103 for smooth symbols and for p = 2 by using different methods. Corollary 7.103 for P CN ×N (p = 2) was first proved

406


p in B¨ ottcher [73] (by C ∗ -algebra methods), and for CN ×N (1 < p < ∞) it is from Grudsky, Kozak [253] and B¨ ottcher, Grudsky, Silbermann [87]. Theorems 7.95, 7.96, 7.101 are from B¨ ottcher [73], Roch, Silbermann [430] in the ottcher, Grudsky, Silbermann [87] 2 case and from Grudsky, Kozak [253], B¨ for p spaces. See also Beam, Warming [51]. For B = C = I, Theorem 7.98 is in principle already in Trefethen [521], [522]. In the general case, this theorem is due to Gallestey, Hinrichsen, Pritchard [207], [208], Hinrichsen, Kelb [278], Hinrichsen, Pritchard [279]. Our proof is based on ideas of Gallestey, Hinrichsen, Pritchard [208] and follows Böttcher, Grudsky [82]. The question whether the norm of the resolvent may be locally constant was posed by one of us (A.B.) during a Banach semester in Warsaw in 1994. A few months later, Andrzej Daniluk of Cracow sent us a proof of the result of 7.99(a). In B¨ ottcher, Grudsky, Silbermann [87] we proved 7.99(b) for X = LpN (Ω, dµ). The fact that (A − λI)−1 is nowhere locally constant is equivalent to saying that the pseudospectra spε,X A do not jump as ε varies continuously. Thus, 7.99(a) tells us that the usual pseudospectra of Hilbert space operators cannot jump. In contrast to this, structured pseudospectra of Hilbert space operators can jump. An example is given in B¨ ottcher, Grudsky [82]. There it is shown that if U = L(χ1 ) is the forward shift on 2 (Z) and P1 is the projection that leaves x0 and x1 unchanged and replaces xj with zero for 1 ,P1 j ∈ / {0, 1}, then the structured pseudospectrum spP ε,2 (Z) U does jump. The > √ precise result is as follows. Put ε0 = (3 − 5)/2. There is a continuous and strictly monotonically increasing function h : [ε0 , ∞) → [0, ∞) such that h(ε0 ) = 0, h(∞) = ∞, and ⎧ for 0 < ε < ε0 , ⎨ {λ ∈ C : |λ| = 1} 1 ,P1 {λ ∈ C : 1 ≤ |λ| ≤ 1 + h(ε)} for ε0 ≤ ε < 1, spP U = 2 ε, (Z) ⎩ {λ ∈ C : 0 ≤ |λ| ≤ 1 + h(ε)} for 1 ≤ ε. 1 ,P1 Thus, spP ε,2 (Z) U jumps at ε = 1. Results on pseudospectra of Toeplitz operators (= infinite Toeplitz matrices) T (a) can be found in B¨ ottcher, Grudsky [86, Section 7.4], B¨ ottcher, Grudsky, Silbermann [87], and B¨ ottcher, Silbermann [116, Section 3.6]. There in particular the following is shown. If a ∈ L∞ , then

spL(2 ) T (a) + ε D ⊂ spε,2 T (a) ⊂ conv R(a) + ε D, and each of the two inclusions may be proper. Here D is the closed unit disk. If a ∈ L∞ and spL(2 ) T (a) is convex, then spε,2 T (a) = conv R(a) + ε D. Chapter 14 of B¨ ottcher, Grudsky [86] contains results on structured pseudospectra of finite and infinite Toeplitz matrices which were obtained in joint work with Mark Embree, Viatcheslav Sokolov, and Marko Lindner.


407

Another issue concerns the speed of convergence of the pseudospectra spε,2 Tn (a) to spε,2 T (a). If a ∈ L∞ is rational, a ∈ R, and ind (a−λ) = 0, then Tn−1 (a − λ)2 increases exponentially fast (see Reichel, Trefethen [418] and Böttcher, Grudsky [81]). Consequently, the inequality Tn−1 (a − λ)2 ≥ 1/ε is already satisfied for moderately sized n and it follows that the convergence of ottcher, Embree, Trefethen [78] it spε,2 Tn (a) to spε,2 T (a) is very fast. In B¨ was observed that if a ∈ P C, then Tn−1 (a−λ)2 may grow only polynomially. This implies that the convergence of spε,2 Tn (a) to spε,2 T (a) may be spectacularly slow (and almost invisible on the computer’s screen). In B¨ ottcher, Grudsky [83] it was shown that such a slow convergence is generic even within the class of continuous symbols. We remark that the proofs of [78] and [83] make essential use of formulas for Toeplitz determinants, which have FisherHartwig symbols in [78] and symbols with nonvanishing index in [83]. The numerical range (= Hausdorff range = field of values) of Toeplitz matrices behaves as nicely as the pseudospectrum. Roch [423] showed that if A is a bounded linear operator on 2 , then the numerical range of the finite sections Pn APn |Im Pn converges to the closure of the numerical range of A in the Hausdorff metric. For a ∈ L∞ (T), the numerical range of T (a) was determined by Klein [313]. Theorem 1 of [313] says that if the Toeplitz operator has a non-constant symbol and is normal, so that the spectrum is a closed interval [γ, δ] ⊂ C, then the numerical range is the corresponding open interval (γ, δ). Theorem 2 of [313] states that if the Toeplitz operator is not normal, then its numerical range is the interior of the convex hull of its spectrum. Polynomial numerical hulls are another alternative to spectra. They were independently invented by Nevanlinna [364] and Greenbaum [248]. For polynomial numerical hulls of Toeplitz matrices we refer to Faber, Greenbaum, and Marshall [197], Burke, Greenbaum [128], and Chapter 8 of B¨ ottcher, Grudsky [86]. Theorem 7.96 shows in particular that if a ∈ P C (scalar case) and T (a) is invertible on 2 , then Tn−1 (a)2 converges to T −1 (a)2 . This implies that the condition number κ(Tn (a)) := Tn (a)2 Tn−1 (a)2 converges to κ(T (a)) := T (a)2 T −1 (a)2 , which is a finite number. If T (a) is not invertible, then κ(Tn (a)) grows to infinity. Results on the speed of the growth can be found in B¨ ottcher, Grudsky [81], [86]. When working with structured matrices it is frequently reasonable to replace the usual condition numbers by so-called structured condition numbers. See D.J. Higham and N.J. Higham [276], N.J. Higham [277], and Rump [445] for more on this topic. Special attention to the Toeplitz structure is paid in Böttcher, Grudsky [85], [86, Chapter 13], Graillath [247], Rump [445]. In [85] it is shown that structured condition numbers of Toeplitz matrices with rational symbols are rarely better (that is, essentially smaller) than the usual condition numbers.

8 Toeplitz Operators over the Quarter-Plane

8.1 Function Classes on the Torus 8.1. p (Z2 ) and p (Z2++ ). Let 1 ≤ p < ∞ and let Ω be a subset of Zk (k = 1, 2 . . .). Define p (Ω) := x = {xj }j∈Ω : xpp := |xj |p < ∞ . j∈Ω

If Ω1 and Ω2 are subsets of Z and if x = {xj } ∈ p (Ω1 ) and y = {yk } ∈ p (Ω2 ) then x ⊗ y ∈ p (Ω1 × Ω2 ) is defined by (x ⊗ y)jk = xj yk (j ∈ Ω1 , k ∈ Ω2 ). Note that obviously x ⊗ yp = xp yp . Given subsets A ⊂ p (Ω1 ) and B ⊂ p (Ω2 ), let A " B refer to the subset of p (Ω1 × Ω2 ) consisting of all finite sums i xi ⊗ yi (xi ∈ p (Ω1 ), yi ∈ p (Ω2 )) and let A ⊗ B denote the closure of A " B in the norm of p (Ω1 × Ω2 ) (if both Ω1 and Ω2 are finite sets, then A " B = A ⊗ B). It can be easily verified that p (Z2 ) = p (Z) ⊗ p (Z),

p (Z2++ ) = p ⊗ p ,

where Z2++ := Z+ × Z+ and p := p (Z+ ). Also note that p (Z2 ) = 0 (Z) ⊗ 0 (Z), p (Z2++ ) = 0 (Z+ )⊗0 (Z+ ), where 0 refers to the sequences with finite support. Let pN (Ω) denote the space of all column-vectors x = (x1 , . . . , xN ) whose components are in p (Ω) and define a norm in pN (Ω) by (x1 , . . . , xN ) pp := x1 pp + . . . + xN pp . Note that one can also think of pN (Ω) as the space of CN -valued sequences

p 1/p (see 2.92). over Ω with finite norm j∈Ω xj CN Again let Ω1 , Ω2 ⊂ Z. For x = (x1 , . . . , xN ) ∈ pN (Ω1 ) and y ∈ p (Ω2 ) define x ⊗ y ∈ pN (Ω1 × Ω2 ) as (x1 ⊗ y, . . . , xN ⊗ y). It is readily seen that x ⊗ ypN (Ω1 ×Ω2 ) = xpN (Ω1 ) yp (Ω2 ) .

410


Given subsets A ⊂ pN (Ω1 ) and B ⊂ p (Ω2 ), we let A " B stand

for the subset of pN (Ω1 × Ω2 ) the elements of which are the finite sums i xi ⊗ yi (xi ∈ pN (Ω1 ), yi ∈ p (Ω2 )) and we denote the closure of A " B in pN (Ω1 × Ω2 ) by A ⊗ B. In particular, pN (Z2 ) = pN (Z) ⊗ p (Z),

pN (Z2++ ) = pN ⊗ p .

In analogy to the preceding paragraph one can define x ⊗ y for x ∈ p (Ω1 ) and y ∈ pN (Ω2 ), A " B and A ⊗ B for A ⊂ p (Ω1 ) and B ⊂ pN (Ω2 ). We have, for example, pN (Z2++ ) = pN ⊗ p = p ⊗ pN = (p ⊗ p )N . 8.2. Lp (T2 ) and H p (T2 ). Let T2 denote the torus T × T. We define Lp (T2 ) (1 ≤ p < ∞) as the Banach space of all (classes of) measurable functions f on T2 for which ( 2π ( 2π 1 p f p := |f (eiθ , eiψ )|p dθ dψ < ∞. (2π)2 0 0 The m, n Fourier coefficient of f ∈ L1 (T2 ) is given by ( 2π ( 2π 1 fm,n := f (eiθ , eiψ )e−imθ e−inψ dθ dψ, (2π)2 0 0 and H p (T2 ) is defined as the (obviously closed) subspace of Lp (T2 ) consisting of all f ∈ Lp (T2 ) for which fm,n = 0 unless m ≥ 0 and n ≥ 0. For f, g ∈ Lp define f ⊗ g ∈ Lp (T2 ) by (f ⊗ g)(s, t) := f (s)g(t) (s, t ∈ T). p Clearly, f ⊗ gp = f p gp . If A and

B are subsets of L , we let A " B denote the collection of all finite sums i fi ⊗ gi (fi ∈ A, gi ∈ B), and A ⊗ B will denote the closure of A " B in Lp (T2 ). The n, n partial sum sn,n f of the Fourier series of a function f ∈ Lp (T2 ) is sn,n f = fjk χj ⊗ χk , |j|≤n |k|≤n

and it is well known that f − sn,n f p → 0 as n → ∞ (1 ≤ p < ∞). This implies that Lp (T2 ) = P ⊗ P and H p (T2 ) = PA ⊗ PA , and thus that 2 ) = H p ⊗ H p. Lp (T2 ) = Lp ⊗ Lp and H p (T

Notice that the mapping fij χi ⊗χj → {fij } is an isometric isomorphism of L2 (T2 ) onto 2 (Z2 ) as well as of H 2 (T2 ) onto 2 (Z2++ ). p (T2 ) are the spaces of column-vectors f = (f 1 , . . . , f N )

LpN (T2 ) and HN with components in Lp (T2 ) and H p (T2 ), respectively. The norm in these spaces is given by (f

1

, . . . , f N ) pp

:=

( ( N T

T

k=1

p/2 |f (s, t)| k

2

dms dmt .

8.1 Function Classes on the Torus

411

For f = (f 1 , . . . , f N ) ∈ LpN and g ∈ Lp , define f ⊗ g = (f 1 ⊗ g, . . . , f N ⊗ g) ,

g ⊗ f = (g ⊗ f 1 , . . . , g ⊗ f N ) .

It can be easily verified that f ⊗ gp = g ⊗ f p = f p gp . Given sets p " B (resp. B " A) denote the collection of all A ⊂ LpN and

B ⊂ L , let A

finite sums i fi ⊗ gi (resp i gi ⊗ fi ) with fi ∈ A, gi ∈ B, and let A ⊗ B (resp. B ⊗ A) denote the closure of A " B (resp. B " A) in LpN (T2 ). Thus, we have LpN (T2 ) = LpN ⊗ Lp = Lp ⊗ LpN ,

p p p HN (T2 ) = HN ⊗ H p = H p ⊗ HN .

p 8.3. Tensor products of operators on H p and p . Let Y be LpN or HN , let Z be Lp or H p , and let A ∈ L(Y ) and B ∈ L(Z). For a finite sum h = i fi ⊗ gi (fi in PN resp. (PA )N , gi in P resp. PA ) define (A ⊗ B) fi ⊗ gi := Afi ⊗ Bgi . (8.1) i

i

We claim that (A ⊗ B)hY ⊗Z ≤ AL(Y ) BL(Z) hY ⊗Z .

(8.2)

To see this notice first that (( " "p " " (A ⊗ I)hp = (Afi )(s)gi (t)" dms dmt " CN

( ( " i "p " " = gi (t)fi (s)" dms dmt "A CN

( ( "i "p " " p ≤ A fi (s)gi (t)" dms dmt = Ap hp , " i

CN

observe that, similarly, (I ⊗ B)h ≤ B h, and thus (A ⊗ B)h = (A ⊗ I)(I ⊗ B)h ≤ A B h, as desired. From (8.2) we deduce that A ⊗ B extends to a bounded operator on Y ⊗ Z, which will be denoted by A ⊗ B, too. If we choose f ∈ Y, g ∈ Z so that f = g = 1 and Af ≥ (1 − ε)A, Bg ≥ (1 − ε)B, then (A ⊗ B)(f ⊗ g) = Af ⊗ Bg = Af Bg ≥ (1 − ε)2 A B. Consequently, A ⊗ BL(Y ⊗Z) = AL(Y ) BL(Z) . pN (Z)

pN (Z+ ),

(8.3)

In case Y is or Z is p (Z) or p (Z+ ), A ∈ L(Y ) and B ∈ L(Z), we define (A ⊗ B)h for finite sums h = i fi ⊗ gi (fi in 0N (Z) resp. 0N (Z+ ), gi in 0 (Z) resp. 0 (Z+ )) again by (8.1). A similar reasoning as above shows

412


that (8.2) holds and that, therefore, A ⊗ B extends to a bounded operator Y ⊗ Z, which satisfies (8.3). Analogously one can define A ⊗ B for the case that A acts on scalar-valued and B acts on vector-valued functions or sequences. Examples. (a) If Ω ⊂ Z2 , then pN (Ω) may be viewed as a subspace of pN (Z2 ). Let PΩ denote the projection of pN (Z2 ) onto pN (Ω) parallel to pN (Z2 \ Ω). It is clear that PΩ has norm 1. Let P be the canonical projection of pN (Z) onto pN (Z+ ). Then for Ω = Z+ × Z, Ω = Z × Z+ , Ω = Z2++ = Z+ × Z+ , the projection PΩ equals P ⊗ I, I ⊗ P , and P++ := P ⊗ P , respectively. Here P ∈ L(pN (Z)) and I ∈ L(p (Z)) if pN (Z+ × Z) is identified with pN (Z+ ) ⊗ p (Z), while P ∈ L(p (Z)) and I ∈ L(pN (Z)) if pN (Z+ × Z) is identified with p (Z+ ) ⊗ pN (Z). p (b) Let P : LpN → HN (1 < p < ∞) denote the Riesz projection (see 1.42). From what was said above we infer that P++ := P ⊗ P is bounded on LpN (T2 ) p (T2 ) = P++ LpN (T2 ). (1 < p < ∞) and has norm P 2L(Lp ) . Clearly, HN

m p ∞ or ci ∈ MN (c) Let b = i=1 ci ⊗ di , where ci ∈ L∞ N ×N and di ∈ L ×N p and di ∈ M . Then the operators M (ci ) ⊗ M (di ), M (ci ) ⊗ T (di ), i

i

T (ci ) ⊗ M (di ),

i

T (ci ) ⊗ T (di )

i

can be identified with the operators (I ⊗ I)b(I ⊗ I), (I ⊗ P )b(I ⊗ P )|Im (I ⊗ P ), (P ⊗ I)b(P ⊗ I)|Im (P ⊗ I), (P ⊗ P )b(P ⊗ P )|Im (P ⊗ P ), respectively. Here b refers to the operator of multiplication by (or convolution with) b, which will be defined precisely in the next section. (d) If αi ∈ C, Ai ∈ L(Y ), and I denotes the identity operator on Z, then m i=1

Ai ⊗ αi I =

m

αi Ai ⊗ I.

i=1

8.4. Multiplication operators. Let L∞ (T2 ) denote the C ∗ -algebra of all (classes of) measurable and essentially bounded functions on T2 . If a is in L∞ (T2 ), then the operator M2 (a) : Lp (T2 ) → Lp (T2 ),

ϕ → aϕ

(8.4)

is obviously bounded. It is called the multiplication operator on Lp (T2 ) with symbol a. The arguments of the proof of Proposition 2.2 can be used to show the following. If A ∈ L(Lp (T2 )) (1 < p < ∞) and if there are complex


413

numbers amn (m, n ∈ Z) such that A has the matrix representation (ai−k,j−l ) with respect to the basis {χm ⊗ χn }m,n∈Z in Lp (T2 ), then there exists an a ∈ L∞ (T2 ) such that A = M2 (a). In that case {amn } is the Fourier coefficient sequence of a and M2 (a)L(Lp (T2 )) = aL∞ (T2 ) . Denote the set of sequences {γmn }m,n∈Z with finite support by 0 (Z2 ). Let a ∈ L1 (T2 ) have Fourier coefficients sequence {amn }m,n∈Z . For x in 0 (Z2 ) define ai−k,j−l xkl (i, j ∈ Z). (a ∗ x)ij = k,l∈Z

Let M p (T2 ) (1 ≤ p < ∞) denote the collection of all a ∈ L1 (T2 ) with the following property: if x ∈ 0 (Z2 ), then a ∗ x ∈ p (Z2 ) and # ! a ∗ xp : x ∈ 0 (Z2 ), x = 0 < ∞. sup xp If a ∈ M p (T2 ), then the mapping 0 (Z2 ) → p (Z2 ), x → a ∗ x extends to a bounded operator M2 (a) : p (Z2 ) → p (Z2 ),

x → a ∗ x,

(8.5)

which is referred to as the multiplication operator on p (Z2 ) with symbol a. From what was said above we know that M 2 (T2 ) = L∞ (T2 ) and it is easy to show that M 1 (T2 ) coincides with W (T2 ), the algebra of all functions on T2 with absolutely convergent Fourier series (see 2.5). It can be proved as in the one-dimensional case (see 2.5) that, for 1 < r < p < 2, 1/r + 1/s = 1, 1/p + 1/q = 1, W (T2 ) ⊂ M r (T2 ) = M s (T2 ) ⊂ M p (T2 ) = M q (T2 ) ⊂ L∞ (T2 ), the embeddings being continuous, and that M p (T2 ) is a Banach algebra under the norm a := M2 (a)L(p (Z2 )) . p 2 2 For a ∈ L∞ N ×N (T ) resp. a ∈ MN ×N (T ) the multiplication operators p p 2 2 M2 (a) on LN (T ) resp. N (Z ) are defined in the natural manner. We introp 2 2 duce norms on L∞ N ×N (T ) resp. MN ×N (T ) by setting 2 = M2 (a)L(L2 (T2 )) , aL∞ N ×N (T ) N

aMNp ×N (T2 ) = M2 (a)L(pN (Z2 )) .

From 1.29(a) we infer that 2 = a∞ := ess sup a(s, t)L(C ) . aL∞ N N ×N (T )

(s,t)∈T2

8.5. Tensor products of subalgebras of M p . If a and b are in M p , then a ⊗ b is in M p (T2 ). This and the equality a ⊗ bM p (T2 ) = aM p bM p follow p from 8.3. Let A and B be two

closed subalgebras of M . Define A " B as the collection of all finite sums i ai ⊗ bi (ai ∈ A, bi ∈ B) and let A ⊗ B denote the closure of A " B in M p (T2 ). It is clear that A ⊗ B is a closed subalgebra of

414


M p (T2 ). The collection of all functions a ⊗ χ0 (a ∈ A) is a closed subalgebra of A ⊗ B and it will be identified with A. Hence, if ω ∈ M (A ⊗ B) is a multiplicative linear functional on A ⊗ B, then ω|A and ω|B belong to M (A) and M (B), respectively. 8.6. Proposition. Let A and B be closed subalgebras of M p having the following property: for each open subset of the maximal ideal space there exists a nonzero element of the algebra whose Gelfand transform is supported in this subset. Then the mapping ϕ : M (A ⊗ B) → M (a) × M (B),

ω → (ω|A, ω|B)

is a homeomorphism of M (A ⊗ B) onto M (a) × M (b), where the latter space is provided with the product topology. Proof. To show that ϕ is one-to-one, suppose ω1 |A = ω2 |A = α and ω1 |B = ω2 |B = β. Then ω1 ai ⊗ b i = ai (α)bi (β) = ω2 ai ⊗ b i i

i

i

for every finite sum i ai ⊗bi = i (ai ⊗χ0 )(χ0 ⊗bi ), and since these sums are dense in A⊗B, it follows that ω1 = ω2 . It is easily seen that ϕ is continuous and because M (A ⊗ B) is compact, ϕ(M (A ⊗ B)) is a closed subset of the product M (A)×M (B). Assume ϕ(M (A⊗B)) is not equal to M (A)×M (B). Then there are open sets U ⊂ M (A) and V ⊂ M (B) such that (U ×V )∩ϕ(M (A⊗B)) = ∅. Choose nonzero a ∈ A and b ∈ B so that supp a ⊂ U and supp b ⊂ V . Then ω(a ⊗ b) = 0 for all ω ∈ M (A ⊗ B), so the spectrum of a ⊗ b in A ⊗ B is {0}, and thus the spectrum of a ⊗ b in L∞ (T2 ) also equals {0}. Since the spectrum in L∞ (T2 ) is the essential range, it follows that either a = 0 or b = 0 , which is a contradiction. Thus ϕ is onto. 8.7. Subalgebras of M p (T2 ). (a) The previous proposition applies to A = B = Cp (1 ≤ p < ∞). Note that Cp ⊗ Cp coincides with Cp (T2 ), the closure in M p (T2 ) of P(T2 ) = P " P. So the fact that M (Cp ⊗ Cp ) = T2 could also be proved using the reasoning of the proof of Proposition 2.46(a). (b) From Proposition 6.28 we see that if U is any open subset of M (P Cp ), there is a nonzero a ∈ P K such that the support of the Gelfand transform of a is entirely contained in U . So the preceding proposition gives that M (P Cp ⊗ P Cp ) = (T × {0, 1}) × (T × {0, 1}). We claim that for each a ∈ P Cp ⊗ P Cp the four limits a(σ ± 0, τ ± 0) =

lim

θ → θ0 ± 0 ψ → ψ0 ± 0

a(eiθ , eiψ )

(8.6)


415

exist and are finite for each (σ, τ ) = (eiθ0 , eiψ0 ) ∈ T2 . Choose an ∈ P Cp " P Cp so that a − an M p (T2 ) → 0. It is clear that the limits ln := an (σ − 0, τ − 0) exist and that |ln − lm | ≤ an − am ∞ . Hence {ln }n∈Z+ is a Cauchy sequence and, consequently, there is an l ∈ C such that |ln − l| → 0 as n → ∞. We have |a(eiθ , eiψ ) − l| ≤ |a(eiθ , eiψ ) − an (eiθ , eiψ )| + |an (eiθ , eiψ ) − ln | + |ln − l|. (8.7) Given any ε > 0, there is an n0 such that the first and the third terms on the right of (8.7) are smaller than ε/3 for n = n0 and then one can find a δ = δ(ε, n0 ) such that the second term is smaller than ε/3 whenever θ ∈ (θ0 − δ, θ0 ), ψ ∈ (ψ0 − δ, ψ0 ). This implies that the limit a(σ − 0, τ − 0) exists and equals l. The proof is the same for the remaining three limits in (8.6). Now it is obvious that, for (σ, τ ) ∈ T2 , the Gelfand transform of a function a ∈ P Cp ⊗ P Cp is given by a(σ, 0; τ, 0) = a(σ − 0, τ − 0), a(σ, 1; τ, 0) = a(σ + 0, τ − 0),

a(σ, 0; τ, 1) = a(σ − 0, τ + 0), a(σ, 1; τ, 1) = a(σ + 0, τ + 0).

(c) If A is a C ∗ -subalgebra of L∞ = M 2 (e.g., A = C, P C, QC, P QC, CE , QCE , L∞ ), then A satisfies the hypothesis of Proposition 8.6. Thus, A ⊗ A is a C ∗ -subalgebra of L∞ (T2 ) which

is isometrically star-isomorphic to the C ∗ -algebra C(M (A) × M (A)). If i ci ⊗ di ∈ A ⊗ A and x, y ∈ M (A), then ci ⊗ di (x, y) = ci (x)di (y) (x, y ∈ M (A)). i

i ∞

(d) We want to show that L ⊗ L∞ does not coincide with L∞ (T2 ). Define χ ∈ L∞ (T2 ) by χ(s, t) = χU (st−1 ) (s, t ∈ T), where χU ∈ P C is the characteristic function of the upper half circle. Assume χ ∈ L∞ ⊗ L∞ . Then

(n) (n) there are bn := i ci ⊗di ∈ L∞ "L∞ such that χ−bn ∞ → 0 as n → ∞.

(n) (n) Put an := i ci di . Then an ∈ L∞ , and because obviously an − am L∞ ≤ bn − bm L∞ (T2 ) , there exists a function a ∈ L∞ with a − an ∞ → 0 as n → ∞. For (s, t) ∈ T2 we have |χ(s, t) − a(t)| ≤ |χ(s, t) − bn (s, t)| + |bn (s, t) − an (t)| + |an (t) − a(t)|. (8.8) There is an n0 such that the first and the third terms on the right of (8.8) are smaller than 1/8 for n = n0 and almost all (s, t) ∈ T2 and t ∈ T, respectively. From writing ci (teih ) − ci (t) di (t) bn0 (teih , t) − an0 (t) = i

we get

416


( |χ(te , t) − a(t)| dm ≤ 2π ih

T

1 1 + 8 8

*

+

i

( di ∞

T

|ci (teih ) − ci (t)| dm,

|c(teih , t) − c(t)| dm = 0 for every c ∈ L1 , it follows that there * is a δ > 0 such that ω(h) := T |χ(teih , t) − a(t)| dm < π for all h ∈ (−δ, δ). But from the definition of χ we obtain that ( ω(0 + 0) := lim ω(h) = |a(t)| dm, h→0+0 (T ω(0 − 0) := lim ω(h) = |1 − a(t)| dm, and since lim

h→0 T

h→0−0

*

*

T

*

and because T |a(t)| dm + T |1 − a(t)| dm ≥ T dm = 2π it is impossible that both ω(0 + 0) and ω(0 − 0) are smaller than π. This contradiction shows that χ is not in L∞ ⊗ L∞ . (e) Let A and B be closed subalgebras of M p . For functions a ∈ AN ×N p 2 and b ∈ BN ×N define a ⊗ b ∈ MN Let ×N (T ) by (a ⊗ b)(s, t) = a(s)b(t).

AN ×N " BN ×N denote the collection of all finite sums of the form i ai ⊗ bi (ai ∈ AN ×N , bi ∈ BN ×N ) and denote the closure of AN ×N " BN ×N in p 2 MN ×N (T ) by AN ×N ⊗ BN ×N .

The collection of all finite sums i ai ⊗ bi , where ai is in AN ×N and bi in BN ×N is of the form bi = diag (ci , . . . , ci ) with ci ∈ B (of course, one can also think of bi as a scalar-valued function) will be denoted by AN ×N " B. p 2 We let AN ×N ⊗ B refer to the closure of AN ×N " B in MN ×N (T ). Similarly A " BN ×N and A ⊗ BN ×N are defined. It is not difficult to verify that AN ×N " BN ×N = AN ×N " B = A " BN ×N = (A " B)N ×N . The same is therefore true with " replaced by ⊗. We merely

introduced AN ×N " B for the following reason: in the sequel, when writing i ai ⊗ bi ∈ AN ×N " B we shall always assume that bi refers to a diagonal matrix function whose entries on the diagonal are equal to each other (or, equivalently, bi is a scalar-valued function). ∞ (f) Finally, let H ∞ (T2 ) = H++ (T2 ) denote the space of all functions a in 2 L (T ) whose Fourier coefficients amn are zero if m < 0 or n < 0. The spaces ∞ ∞ ∞ ∞ (T2 ), H−+ (T2 ), H+− (T2 ) are defined analogously. Note that H±± (T2 ) H−− ∞ 2 are closed

subalgebras of L (T ). a given by If a = m,n≥0 amn χm ⊗ χn is in H ∞ (T2 ), then the function m n a z w is holomorphic and bounded in D × D. Con a(z, w) = mn m,n≥0 versely, if b is a holomorphic and bounded function in D × D, then there is an a. a ∈ H ∞ (T2 ) such that b = ∞

8.8. Tensor products of operators

on L∞ . Abbreviate L∞ N ×N to Y and f ⊗ g ∈ Y " Y define (A ⊗ B)h as let A, B ∈ L(Y ). For a finite sum h = i i i

Af ⊗ Bg . Then i i i

8.2 Elementary Properties of Quarter-Plane Operators

417

(A ⊗ B)hY ⊗Y ≤ AL(Y ) BL(Y ) hY ⊗Y . Indeed, if we let X = M (L∞ ), " " " " fi ⊗ Bgi " (I ⊗ B)hY ⊗Y = " i

Y ⊗Y

" " " " = max " fi (x)(Bgi )(y)" x,y∈X

i

L(CN )

(see 8.7(c))

" " " " fi (x)gi (y)" = max "B x,y∈X

≤ BL(Y )

L(CN )

i

" " " " max " fi (x)gi (y)"

x,y∈X

L(CN )

i

= BL(Y ) hY ⊗Y

(again by 8.7(c)),

which implies the asserted inequality as in 8.3. Hence, A ⊗ B extends to a bounded operator on Y ⊗ Y , which will be denoted by A ⊗ B, too. Finally, the argument given in 8.3 also shows that A ⊗ BL(Y ⊗Y ) = AL(Y ) BL(Y ) . ∞ ∞ ∞ For A ∈ L(L∞ N ×N ) and B ∈ L(L ), one can define A⊗B ∈ L(LN ×N ⊗L ) in a similar fashion. It can be shown as above that ∞ = AL(L∞ BL(L∞ ) . A ⊗ BL(L∞ N ×N ⊗L ) N ×N )

(8.9)

Example. Suppose {Kλ }λ∈Λ is an approximate identity. For λ ∈ Λ, define kλ ∈ L(L∞ N ×N ) by kλ (ajk ) := (kλ ajk ), where kλ ajk is as in 3.14. Thus, if λ and µ are in Λ, then the operator given by ∞ kλ ⊗ kµ : L∞ fi ⊗ gi → kλ fi ⊗ kµ gi , N ×N " LN ×N → CN ×N " CN ×N , i

i

∞ extends to an operator in L(L∞ N ×N ⊗ LN ×N , CN ×N ⊗ CN ×N ), for which the equality kλ ⊗ kµ = kλ kµ holds.

8.2 Elementary Properties of Quarter-Plane Operators 2 8.9. Toeplitz operators over the quarter-plane. For a ∈ L∞ N ×N (T ) the p 2 Toeplitz operator T2 (a) with symbol a on HN (T ) (1 < p < ∞) is the (obviously bounded) operator acting by the rule p p (T2 ) → HN (T2 ), T2 (a) : HN

ϕ → P++ (aϕ),

(8.10)

p 2 and for a ∈ MN ×N (T ) (1 ≤ p < ∞) the Toeplitz operator T2 (a) with symbol p 2 a on N (Z++ ) is the (clearly bounded) operator given by

418


T2 (a) : pN (Z2++ ) → pN (Z2++ ),

ϕ → P++ (a ∗ ϕ).

(8.11)

Sometimes these operators are also called two-dimensional Toeplitz operators. If p = 2, then the operators defined by (8.10) and (8.11) are unitarily equivalent through the isomorphism 2 (T2 ) → 2N (Z2++ ), ϕij χi ⊗ χj → {ϕij }. HN

p p 2 2 Clearly, if ϕ = ϕij χ

i ⊗ χj ∈ HN (T ) resp. ϕ = {ϕij } ∈ N (Z++ ) (ϕij ∈ CN ), then T2 (a)ϕ = ψij χi ⊗ χj resp. T2 (a)ϕ = {ψij }, where ψij = ai−k,j−l ϕkl (i, j ∈ Z+ ) k,l∈Z+

and amn ∈ CN ×N is the m, n Fourier coefficient of a. p ∞ ∞ p Suppose

a = i ci ⊗ di belongs to LN ×N " L resp. MN ×N " M . Then T2 (a) = i T (ci ) ⊗ T (di ) (recall Example (c) in 8.3, but also 8.7(d)). p 2 Every function a ∈ MN ×N (T ) can be written as a(s, t) =

aij si tj =

i,j

bi (t) :=

si bi (t) =

i j

aij t ,

cj (s) :=

j

tj cj (s),

j

aij s

i

(s, t ∈ T).

i

For i ∈ Z+ , let Hi and Ki denote the subspaces Hi := pN ({i} × Z+ ) and Ki := pN (Z+ × {i}) of pN (Z2++ ). Then ·

·

·

·

·

·

pN (Z2++ ) = H0 + H1 + H2 + . . . = K0 + K1 + K2 + . . .

(8.12)

and a simple computation shows that T2 (a) has the matrix representation ∞ (T (bj−k ))∞ j,k=0 and (T (cj−k ))j,k=0 with respect to the first and second decomposition in (8.12), respectively. Thus, a Toeplitz operator over the quarterplane can be interpreted as a one-dimensional Toeplitz operator with operator entries that are themselves one-dimensional Toeplitz operators. The following proposition is the two-dimensional analogue of formula (2.20), but it does not even nearly play the same role in the two-dimensional theory as formula (2.20) does in the one-dimensional situation. p 2 ∞ 2 8.10. Proposition. (a) Suppose a−− , a++ ∈ MN ×N (T ) ∩ HN ×N (T ) and p b ∈ MN ×N (T2 ). Then

T2 (a−− ba++ ) = T2 (a−− )T2 (b)T2 (a++ ). ∞ (T2 )]N ×N , then (b) If a±∓ , b±∓ ∈ [M p (T2 ) ∩ H±∓

T2 (a+− b+− ) = T2 (a+− )T2 (b+− ),

T2 (a−+ b−+ ) = T2 (a−+ )T2 (b−+ ).

8.2 Elementary Properties of Quarter-Plane Operators

419

Proof. (a) Abbreviate M2 (f ) to f . Because P++ a−− ba++ P++ equals P++ a−− P++ bP++ a++ P++ + P++ a−− (I − P++ )ba++ P++ +P++ a−− P++ b(I − P++ )a++ P++ +P++ a−− (I − P++ )b(I − P++ )a++ P++ , and since a−− and a++ leave Im (I −P++ ) and Im P++ , respectively, invariant, and since P++ (I − P++ ) = (I − P++ )P++ = 0, we get the asserted formula. (b) The proof goes similarly. We now state three theorems about two-dimensional Toeplitz operators whose proofs can be carried out using the same arguments as in the proofs of their one-dimensional analogues. 2 8.11. Theorem. (a) If 1 < p < ∞ and a ∈ L∞ N ×N (T ), then

c1 a∞ ≤ T2 (a)Φ(HNp (T2 )) ≤ T2 (a)L(HNp (T2 )) ≤ c2 a∞ ,

(8.13)

with certain positive constants c1 and c2 independent of a. p 2 (b) If 1 ≤ p < ∞ and a ∈ MN ×N (T ), then

T2 (a)Φ(pN (Z2++ )) = T2 (a)L(pN (Z2++ )) = aMNp ×N (T2 ) .

(8.14)

Proof. (a) The second and the third inequalities are obvious. To see that c1 a∞ ≤ T2 (a)ess proceed as in the proofs of Propositions 4.1(a) and p p (T2 )), then, because V n ⊗ V n → 0 weakly on HN ⊗ Hp 4.4(d): if K ∈ C∞ (HN (−n) (−n) n n and (V ⊗V )T2 (a)(V ⊗ V ) = T2 (a), we have T2 (a) ≤ lim inf (V (−n) ⊗ V (−n) )(T2 (a) + K)(V n ⊗ V n ) n→∞ ≤ sup P++ U −n ⊗ U −n T2 (a) + K V n ⊗ V n n

= P++ T2 (a) + K, whence T2 (a) ≤ P++ T2 (a)ess , and since T2 (a) ≥ cP++ M2 (a)P++ = c(U −n ⊗ U −n )P++ (U n ⊗ U n )M2 (a)(U −n ⊗ U −n )P++ (U n ⊗ U n ) ≥ cM2 (a) ≥ c1 a∞ (note that (U −n ⊗ U −n )P++ (U n ⊗ U n ) → I strongly on LpN ⊗ Lp ), we get the inequality c1 a∞ ≤ T2 (a)ess . (b) This follows from the arguments of part (a) along with the equality P++ = 1.

420


2 8.12. Toeplitz operators over the half-plane. For a ∈ L∞ N ×N (T ) resp. p 2 a ∈ MN ×N (T ) the (obviously bounded) operators p p ⊗ Lp → HN ⊗ Lp , T+· (a) : HN

ϕ → (P ⊗ I)(M2 (a)ϕ),

p HN

ϕ → (I ⊗ P )(M2 (a)ϕ),

T·+ (a) : Lp ⊗ T+· (a) : T·+ (a) :

→ Lp ⊗

pN (Z+ × Z) pN (Z × Z+ )

→ →

p HN ,

pN (Z+ × Z), pN (Z × Z+ ),

x → (P ⊗ I)(M2 (a)x), x → (I ⊗ P )(M2 (a)x)

are called Toeplitz operators over the half plane. If we set Hi := pN ({i} × Z), Ki := pN (Z+ × {i}), then the operator T+· (a) ∞ has the matrix representations (M (bj−k ))∞ j,k=0 , (T (cj−k ))j,k=−∞ with respect to the decompositions ·

·

·

·

·

·

pN (Z+ × Z) = H0 + H1 + . . . = . . . + K−1 + K0 + K1 + . . . , where b and c are as in 8.9. 2 8.13. Theorem. (a) If 1 < p < ∞ and a ∈ L∞ N ×N (T ), then

M2 (a) ∈ Φ± (LpN (T2 ))

⇐⇒ M2 (a) ∈ GL(LpN (T2 )) 2 ⇐⇒ a ∈ GL∞ N ×N (T ),

p p T2 (a) ∈ Φ± (HN ⊗ H p ) =⇒ T+· (a) ∈ GL(HN ⊗ H p) 2 =⇒ a ∈ GL∞ N ×N (T ), p 2 T+· (a) ∈ Φ± (HN ⊗ H p ) =⇒ a ∈ GL∞ N ×N (T ). p 2 (b) If 1 ≤ p < ∞ and a ∈ MN ×N (T ), then

M2 (a) ∈ Φ(pN (Z2 ))

⇐⇒ M2 (a) ∈ GL(pN (Z2 )) p 2 ⇐⇒ a ∈ GMN ×N (T ),

T2 (a) ∈ Φ(pN (Z2++ ))

=⇒ T+· (a) ∈ GL(pN (Z+ × Z)) p 2 =⇒ a ∈ GMN ×N (T ),

p 2 T+· (a) ∈ Φ(pN (Z+ × Z)) =⇒ a ∈ GMN ×N (T ).

Proof. The assertions for the multiplication operators can be proved similarly as their scalar-valued one-dimensional analogues (2.28 and 2.29). The proof of Theorem 2.30 with an appropriate replacements of U ±n by I ⊗ U ±n and U ±n ⊗ I gives the implication concerning Toeplitz operators. 2 2 8.14. Theorem. Suppose a ∈ L∞ N ×N (T ) is sectorial on T , i.e., there are c, d ∈ GCN ×N and ε > 0 such that Re (ca(s, t)dz, z) ≥ εz2 for all z ∈ CN 2 and almost all (s, t) ∈ T2 . Then T2 (a) ∈ GL(HN (T2 )).


421

Proof. See the proof of Corollary 3.62. 8.15. Corollary. If a ∈ L∞ (T2 ), then R(a) ⊂ spΦ(H 2 (T2 )) T2 (a) ⊂ spL(H 2 (T2 )) T2 (a) ⊂ conv R(a), where R(a) = spL∞ (T2 ) (a) is the essential range of a. Proof. Combine Theorems 8.13 and 8.14.

8.3 Continuous Symbols 8.16. Definitions. (a) Let x ∈ X = M (L∞ ) and let Γx denote the operator in L(L∞ ) which assigns the constant function a(x) to a function a ∈ L∞ . From 8.8 we know that the operator Γx ⊗ I is well defined and bounded ∞ ∞ 1 ⊗ L∞ ⊗ L∞ on L

N ×N . For a ∈ L N ×N , define

ax := (Γx ⊗ I)a. Thus, if ∞ ∞ 1 a = i ci ⊗ di ∈ L " LN ×N , then ax (t) = i ci (x)di (t) (t ∈ T). Moreover, we have 2 ≤ aL∞ ∀ a ∈ L∞ ⊗ L∞ (8.15) a1x L∞ N ×N . N ×N N ×N (T ) 1 If A is a C ∗ -subalgebra of L∞ and a ∈ A ⊗ L∞ N ×N , then ax is the same ∞ function in LN ×N for all x in the fiber Xα (α ∈ M (A)). This function will ∞ be denoted by a1α . For example, if a = i ci ⊗ di ∈ P C " LN ×N , then 1 2 aτ ±0 (t) = i ci (τ ± 0)di (t) (t ∈ T). Finally, define ax as (I ⊗ Γx )a, and for all 2 2 a ∈ L∞ N ×N ⊗ A and α ∈ M (A) define aα as ax , where x is any point in Xα .

(b) Now let a = i ci ⊗ di ∈ P " P. If τ ∈ T, then a1τ ∈ P and

a1τ ∞ = max |a(τ, t)| ≤ a∞ . t∈T

We also have ( · W being the Wiener norm) + + + ++ + + + a1τ W = ci (τ )(di )n + = (ci )m (di )n τ m + + + n

n

m

i + i+ + ++ + + + m = τ (ci )m (di )n + ≤ (ci )m (di )n + + + n

m

" " " " =" ci ⊗ d i " i

n,m

i W ⊗W

i

= aW ⊗W .

Hence, I ⊗ M (a1τ )L(p ⊗p ) ≤ M2 (a)L(p ⊗p ) for p = 1, 2, and the RieszThorin interpolation theorem combined with passage to adjoints extends this p to all values p ∈ [1, ∞). So it is clear that a1τ ∈ CN ×N is well defined for p p a ∈ C ⊗ CN ×N and that a1τ MNp ×N ≤ caMNp ×N (T2 )

p 2 ∀ a ∈ CN ×N (T ),

422


with some c independent of a. Of course, the matrix function a2τ given by a2τ (t) = a(t, τ ) has similar properties as a1τ .

p (c) Assume a = i ci ⊗ di ∈ P K " MN ×N (recall 6.25). We claim that there is a constant c depending only on p and N such that a1α MNp ×N ≤ caMNp ×N (T2 )

∀ α = (τ, j) ∈ T × {0, 1} = M (P Cp ), (8.16)

where a1(τ,0) (t) := i ci (τ − 0)di (t) and a1(τ,1) (t) := i ci (τ + 0)di (t). To prove this claim we may confine ourselves to the case N = 1. Put bi = ci − ci (τ − 0)χ0 . Then there exists an ε > 0 such that bi (t) = 0 whenever arg τ − ε < arg t < arg τ . Choose u ∈ P K so that u(t) = 1 for t satisfying arg τ − ε/2 < arg t < arg τ and u(t) = 0 otherwise. So bi (t)u(t) = 0 for all t ∈ T. We have 1 = u∞ ≤ uM p ≤ cp u∞ + V1 (u) ≤ cp (1 + 2) = 3cp and hence, " " " " ci (τ − 0)di " "

Mp

" " " " ≤ uM p " ci (τ − 0)di "

" "i " " = "u ⊗ ci (τ − 0)di " " " i " " =" ci u ⊗ d i "

M p (T2 )

i

" " " " ≤ 3cp " ci ⊗ d i " i

Mp

M p (T2 )

i " " " " =" ci (τ − 0)u ⊗ di "

" i " " " =" ci ⊗ di (u ⊗ χ0 )"

M p (T2 )

i

M p (T2 )

M p (T2 )

.

As the same arguments applies to τ + 0 in place of τ − 0, we get our claim. p 1 Thus, for α ∈ M (P Cp ) and a ∈ P Cp ⊗ MN ×N the matrix function aα is well p defined and (8.16) holds. A similar statement is valid for a ∈ MN ×N ⊗ P Cp and a2α . (d) For a ∈ GC(T2 ) define the mapping i1 by i1 : T → Z, τ → ind a2τ . From 2.41(b) we know that i1 is continuous, and therefore i1 must be constant. Let ind1 a denote this constant value and define ind2 a analogously. It can be shown that the abstract index group of C(T2 ) (see Douglas [162, 2.10]) is isomorphic to Z2 . A function a ∈ GC(T2 ) belongs to the connected component of GC(T2 ) containing the identity if and only if ind1 a = ind2 a = 0. Also notice the following fact: if a ∈ GC(T2 ) and f1 , f2 are continuous mappings of T into T, then ind a(f1 (t), f2 (t)) = (ind1 a)(ind f1 ) + (ind2 a)(ind f2 ).

(8.17)

8.17. Definitions. Suppose {Y, Z} ⊂ {H p , Lp } (1 < p < ∞) or {Y, Z} ⊂ {p (Z+ ), p (Z)} (1 ≤ p < ∞). Given subset A ⊂ L(YN ) and B ⊂ L(Z) define


423

A " B as the collection of all finite sums i Ai ⊗ Bi (Ai ∈ A, Bi ∈ B) and let A ⊗ B denote the closure of A " B in L(YN ⊗ Z). It is easy to see that C∞ (YN ) ⊗ L(Z), L(YN ) ⊗ C∞ (Z), and C∞ (YN ) ⊗ C∞ (Z) are closed twosided ideals of L(YN ) ⊗ L(Z). It is clear that C∞ (YN ) ⊗ C∞ (Z) is a subset of C∞ (YN ⊗ Z). Since the (finite-rank) projections Pn defined in 7.5 converge strongly to I on both YN and Z and since each finite-rank operator on YN ⊗ Z is readily seen to be in C∞ (YN ) ⊗ C∞ (Z), it results that C∞ (YN ) ⊗ C∞ (Z) is actually equal to C∞ (YN ⊗ Z). Put Lπ1 (YN ⊗ Z) := L(YN ) ⊗ L(Z)/C∞ (YN ) ⊗ L(Z), Lπ2 (YN ⊗ Z) := L(YN ) ⊗ L(Z)/L(YN ) ⊗ C∞ (Z), Lπ12 (YN ⊗ Z) := L(YN ) ⊗ L(Z)/C∞ (YN ) ⊗ C∞ (Z), and for A ∈ L(YN ) ⊗ L(Z) let Aπ1 , Aπ2 , Aπ12 denote the coset in the corresponding quotient algebra containing A. Notice that the above definitions can also be made for the case that Z is provided with the subscript N , i.e., that A ⊂ L(Y ) and B ⊂ L(ZN ). 8.18. Lemma. Let A ∈ L(YN ) ⊗ L(Z) and suppose Aπ1 ∈ GLπ1 (YN ⊗ Z) and Aπ2 ∈ GLπ2 (YN ⊗ Z). Then Aπ12 is in GLπ12 (YN ⊗ Z) and if B1π and C2π are the inverses of Aπ1 and Aπ2 , respectively, then (B + C − BAC)π12 is the inverse of Aπ12 . Proof. If BA = I + K1 and CA + I + K2 with K1 ∈ C∞ (YN ) ⊗ L(Z) and K2 ∈ L(YN ) ⊗ C∞ (Z), then (B + C − BAC)A = I − K1 K2 , and a little thought shows that K1 K2 ∈ C∞ (YN ) ⊗ C∞ (Z). Before turning to Toeplitz operators over the quarter-plane we state two propositions on Toeplitz operators over the half-plane. The first provides a sufficient condition for the invertibility of a half-plane operator for a large class of symbols, and the second shows that this condition is even necessary for the operator to be Fredholm in case the symbol is continuous. 8.19. Proposition. Let A be a C ∗ -subalgebra of L∞ containing the constants resp. A ∈ {Cp , P Cp } and let H p (1 < p < ∞) resp. p (1 ≤ p < ∞ for A = Cp and 1 < p < ∞ for A = P Cp ) be the underlying space. Let b ∈ (L∞ ⊗ A)N ×N resp. b ∈ (M p ⊗ A)N ×N . If T (b2α ) is invertible for each α ∈ M (A), then T+· (b) is invertible in p L(HN ) ⊗ L(Lp )

resp.

L(pN (Z+ )) ⊗ L(p (Z))

p and, consequently, invertible in L(HN ⊗ Lp ) resp. L(pN (Z+ ) ⊗ p (Z)).

Proof. We only consider the H p case; apart from some technical details (such as in the proof of Theorem 2.69), the p case can be treated similarly. For α ∈ M (A), let Nα denote the collection of all ϕ ∈ A ∼ = C(M (A)) such that 0 ≤ ϕ ≤ 1 and ϕ is identically 1 in some neighborhood of α (depending on

424


ϕ). Put Mα = {I ⊗ M (ϕ) : ϕ ∈ Nα }, where I refers to the identity operator p . It can be easily verified that {Mα }α∈M (A) is a covering system of on HN p p that T+· (b) commutes with every localizing classes in L(HN

) ⊗ L(L ) and operator in Mα . If b = i ci ⊗ di ∈ L∞ N ×N " A and ϕ ∈ Nα , then α

[T+· (b) − T (b2α ) ⊗ I] [I ⊗ M (ϕ)] = T (ci ) ⊗ M (di ) − T (ci ) ⊗ di (α)I [I ⊗ M (ϕ)] i

=

i

1 0 T (ci ) ⊗ M di − di (α)χ0 .

i

This implies that T+· (b) and T (b2α ) ⊗ I are Mα -equivalent from the left. Lemma 7.70 shows that this is also true for b ∈ L∞ N ×N ⊗ A. The right equivalence can be proved analogously. To complete the proof is remains to apply Theorem 1.32(a). 8.20. Proposition. Let B be a C ∗ -algebra between C and QC resp. B = Cp and let H p (1 < p < ∞) resp. p (1 < p < ∞) be the underlying space. If b ∈ (B ⊗ B)N ×N , then the following are equivalent. (i) T+· (b) is Fredholm. (ii) T+· (b) is invertible. p (iii) T+· (b) is invertible in L(HN ) ⊗ L(Lp ) resp. L(pN (Z+ )) ⊗ L(p (Z)).

(iv) T (b2β ) is invertible for each β ∈ M (B). Proof. Again we only consider the H p case. The implication (iv) =⇒ (iii) results from the preceding proposition and the implications (iii) =⇒ (ii) =⇒ (i) are trivial. So suppose (i) holds. From Theorem 8.13 we obtain that b p ). It remains to show that is in G(B ⊗ B)N ×N , and hence T (b2β ) ∈ Φ(HN Ker T (b2β ) = Ker T ∗ (b2β ) = {0}. Define Nβ (β ∈ M (B)) as in the previous proof and put p Mπβ = I ⊗ M (ϕ) + C∞ (HN ⊗ Lp ) : ϕ ∈ Nβ . Then {Mπβ }β∈M (B) is a covering system of localizing classes in the algebra p p p ⊗ Lp )/C∞ (HN ⊗ Lp ), T+· (b) + C∞ (HN ⊗ Lp ) commutes with Mπβ and L(HN β p ⊗ Lp ). So is Mπβ -equivalent from the left and the right to T (b2β ) ⊗ I + C∞ (HN p p ⊗Lp ), Theorem 1.32(a) implies that there are Ai ∈ L(HN ⊗Lp ), Ki ∈ C∞ (HN and ϕi ∈ B (i = 1, 2) such that A1 T (b2β ) ⊗ I I ⊗ M (ϕ1 ) = I ⊗ M (ϕ1 ) + K1 , (8.18) 2 I ⊗ M (ϕ2 ) T (bβ ) ⊗ I A2 = I ⊗ M (ϕ2 ) + K2 . (8.19)


425

We show that (8.18) implies that Ker T (b2β ) = {0}. Taking the adjoint of (8.19) we then obtain analogously that Ker T ∗ (b2β ) = {0}. Assume T (b2β )f = 0. Then (8.18) gives that f ⊗ M (ϕ1 )U n χ0 = −K1 (I ⊗ U n )(f ⊗ χ0 ) ∀ n ∈ Z+ p and since I ⊗ U n converges weakly to zero on HN ⊗ Lp and K1 is compact, it n follows that f ⊗ M (ϕ1 )U χ0 → 0 as n → ∞. But

f ⊗ M (ϕ1 )U n χ0 = f M (ϕ1 )U n χ0 = f ϕ1 , hence f ϕ1 = 0 and thus f = 0, as desired.

Remark 1. See also Corollary 8.80. Remark 2. If b ∈ WN ×N (T2 ) and T+· (b) is invertible on 1N (Z+ × Z), then T (a2τ ) is in GL(1N (Z+ )) for each τ ∈ T. To see this localize as in the proof of Proposition 8.19 to get the above equalities (8.18), (8.19) with K1 = K2 = 0. 8.21. Theorem. (a) Let B be a C ∗ -algebra between C and QC, let a be in (B ⊗ B)N ×N , and let 1 < p < ∞. Then p p T2 (a) ∈ Φ(HN (T2 )) ⇐⇒ T (a1β ), T (a2β ) ∈ GL(HN )

∀ β ∈ M (B).

(b) Let a ∈ (Cp ⊗ Cp )N ×N and 1 ≤ p < ∞. Then T2 (a) ∈ Φ(pN (Z2++ )) ⇐⇒ T (a1τ ), T (a2τ ) ∈ GL(pN )

∀ τ ∈ T.

(c) Under the hypotheses of (a) or (b), if T2 (a) is Fredholm then T+· (a) and T·+ (a) are invertible and −1 −1 −1 −1 P++ T+· (a) + T·+ (a) − T+· (a)P++ M2 (a)P++ T·+ (a) P++ is a regularizer of T2 (a). Proof. (a), (c) Theorem 8.13 and Proposition 8.20 give the implication “=⇒”. Conversely, suppose T (a1β ) and T (a2β ) are invertible for all β ∈ M (B). From Proposition 8.20 (Proposition 8.19) we deduce that T+· (a) is invertible and p ) ⊗ L(Lp ). We have that its inverse is in L(HN −1 (P ⊗ P )T+· (a)(P ⊗ P )T2 (a) = (P ⊗ P )[(P ⊗ I)a(P ⊗ I)]−1 (P ⊗ I)a(P ⊗ I)(P ⊗ P )

−(P ⊗ P )[(P ⊗ I)a(P ⊗ I)]−1 (P ⊗ Q)a(P ⊗ P ). (8.20)

The first term on the right equals P ⊗ P . If a = i ci ⊗ di ∈ QCN ×N " QC, then p P ci P ⊗ Qdi P ∈ L(HN ) " C∞ (Lp ), (P ⊗ Q)a(P ⊗ P ) = i

426


and since QCN ×N ⊗ QC is the closure of QCN ×N " QC, we conclude that p ) ⊗ C∞ (Lp ) for every a ∈ QCN ×N ⊗ QC. It follows (P ⊗ Q)a(P ⊗ P ) is in L(HN p ) ⊗ C∞ (Lp ). A that the second term on the right of (8.20) belongs to L(HN similar reasoning for T·+ (a) in place of T+· (a) and Lemma 8.18 yield the implication “⇐=” and part (c). (b), (c) Proceed as in the H p case and take into account Remark 2 of 8.20. 8.22. Corollary. Let a ∈ C(T2 ) resp. a ∈ Cp (T2 ). Then T2 (a) is Fredholm on H p (T2 ) (1 < p < ∞) resp. p (Z2++ ) (1 ≤ p < ∞) if and only if a(s, t) = 0

∀ (s, t) ∈ T2 ,

ind1 a = ind2 a = 0.

(8.21)

If T2 (a) is Fredholm, then Ind T2 (a) = 0. Proof. It remains to prove that Ind T2 (a) = 0 if (8.21) is fulfilled. Since the functions satisfying (8.21) are just the functions belonging to the connected component of GC(T2 ) containing the identity and since the index is constant on each connected component of GC(T2 ) (Theorem 8.11(a)), we obtain the equality Ind T2 (a) = 0 in the H p case. To see that the same is true in the p case note that there is a b ∈ P(T2 ) such that Ind T2 (a) = Ind T2 (b) and b ∈ GC(T2 ), ind1 b = ind2 b = 0. Remark. Sazonov [458] showed that if a ∈ C(T2 ) and T2 (a) is normally solvable on H 2 (T2 ), then either a(s, t) = 0 for all (s, t) ∈ T2 or a vanishes identically. He also proved that if a ∈ GC(T2 ), then there exists an operator K ∈ C∞ (H 2 (T2 )) such that T2 (a) + K is normally solvable on H 2 (T2 ). Open problem. Let a ∈ QC ⊗ QC and suppose T2 (a) ∈ Φ(H 2 (T2 )). Is Ind T2 (a) equal to zero? 8.23. Factorable symbols. (a) Assume a = a−− a−+ a+− a++ , where a±± 2 ∞ 2 and a−1 ±± are in C(T ) ∩ H±± (T ) (here and in the following “±±” means one of the four pairs “+−”, “++”, “−+”, “−−”, that is, the second sign does not depend on the first; equality of pairs is understood as usual). In that case T+· (a) and T·+ (a) are invertible on the corresponding H p spaces. The inverses are −1 −1 −1 −1 T+· (a) = (P ⊗ I)a−1 ++ a+− (P ⊗ I)a−+ a−− (P ⊗ I), −1 −1 −1 −1 T·+ (a) = (I ⊗ P )a−1 ++ a−+ (I ⊗ P )a+− a−− (I ⊗ P ).

This can be readily verified by a direct calculation (take into account that, for example, (Q ⊗ I)a+· (P ⊗ I) = 0 if the Fourier coefficient sequence of a+· is supported in the right half-plane Z+ × Z). (b) Thus, under the hypothesis of (a), Theorem 8.21(c) provides a regularizer of T2 (a) which can be explicitly computed. Another (somewhat simpler) regularizer was discovered by Strang [508]:


427

−1 −1 R = P++ T+· (a) + T·+ (a) − M2 (a−1 ) P++ . Let us prove this. Write P+− = P ⊗ Q, P−+ = Q ⊗ P , P−− = Q ⊗ Q. Then −1 −1 P++ T+· (a)P++ aP++ = P++ − P++ T+· (a)P+− aP++ −1 −1 −1 = P++ − P++ a−1 ++ a+− (P++ + P+− )a−+ a−− P+− aP++ , −1 −1 P++ T·+ (a)P++ aP++ = P++ − P++ T·+ (a)P−+ aP++ −1 −1 −1 = P++ − P++ a−1 ++ a−+ (P++ + P−+ )a+− a−− P−+ aP++ ,

−P++ a−1 P++ aP++ = −P++ + P++ a−1 P+− aP++ + P++ a−1 P−+ aP++ + P++ a−1 P−− aP++ , adding these equalities we arrive at RT2 (a) = P++ + P++ a−1 P−− aP++ −1 −1 −1 +P++ a−1 ++ a+− (P−+ + P−− )a−+ a−− P+− aP++ −1 −1 −1 +P++ a−1 ++ a−+ (P+− + P−− )a+− a−− P−+ aP++ ,

and since P++ ϕP−− , P+− ϕP−+ , P−+ ϕP+− , P−− ϕP++ are compact whenever ϕ ∈ C ⊗ C, it follows that RT2 (a) − P++ is compact, as desired. (c) The results of (a) and (b) remain valid for Toeplitz operators on p 2 ∞ 2 spaces if one requires that a±± and a−1 ±± are in Cp (T ) ∩ H±± (T ). (d) If a ∈ W (T2 ), a(s, t) = 0 for all (s, t) ∈ T2 , and ind1 a = ind2 a = 0, then a admits a factorization a = a−− a−+ a+− a++ with a±± and a−1 ±± in ∞ (T2 ). This follows from the fact that under these assumptions W (T2 ) ∩ H±± a is in the connected component of GW (T2 ) containing the identity, so that a = exp b with some b ∈ W (T2 ) (see 1.16(a)), and therefore a±± = exp(P±± b) are the factors of the wanted factorization. (e) If a ∈ (C ⊗ C)N ×N is sufficiently smooth (e.g., if the second partial derivatives satisfy a H¨ older condition) and if T+· (a) and T·+ (a) are invertible on H 2 , then a = a−· a+· = a·− a·+ , −1 where the functions a±· , a−1 ±· and a·± , a·± are smooth and belong to the spaces ∞ 2 ∞ 2 [H±· (T )]N ×N and [H·± (T )]N ×N , respectively. Under these conditions T2 (a) 2 (T2 )) and the index of T2 (a) equals is in Φ(HN

−1 1 −1 −1 tr [S1 , a−· ]a−1 −· [S2 , a·− ]a·− − tr a+· [S1 , a+· ]a·+ [S2 , a·+ ] . 4 This formula is due to Duduchava [172]. Here S1 = S ⊗ I, S2 := I ⊗ S (S := 2P −I being the Cauchy singular integral operator), [A, B] := AB−BA, and if 2 (T2 )) is an integral operator with sufficiently smooth matrix-kernel K ∈ C1 (HN

428


k(s1 , s2 ; t1 , t2 ) = (kij (s1 , s2 ; t1 , t2 ))N i,j=1 (the operators in the parentheses of the above expression are of this kind), then tr K =

N (( j=1

T2

kjj (t1 , t2 ; t1 , t2 ) dt1 dt2 .

sn −tm (f) If a(s, t) = (n, m ∈ Z+ ), then T2 (a) ∈ Φ(H22 (T2 )) but t−m s−n Ind T2 (a) = −nm. This can be verified with the help of the formula in (e). Another proof of this fact is in Douglas [160, pp. 45–46]. Also see Coburn, Douglas, Singer [145].

8.4 The Invertibility Problem Corollary 8.22 provides us with an effective criterion for deciding whether a given Toeplitz operator over the quarter-plane is Fredholm. However, it is far from easy to decide whether such an operator is invertible. In the onedimensional scalar case this question can be answered by computing the index (Corollary 2.40). A half-plane operator is invertible if and only if it is Fredholm (Proposition 8.20), and thus it has always index zero. We also know that the index of a quarter-plane operator with scalar-valued continuous symbol is always zero, but we shall see that there exist such operators which are Fredholm but not invertible. The purpose of what follows is to list some classes of symbols for which the invertibility problem for the corresponding Toeplitz operators is solved (also see 8.45(e)). ∞ (T2 )]N ×N and 1 < p < ∞. Then the 8.24. Theorem. (a) Let a ∈ [H±± following are equivalent. p (i) T2 (a) ∈ Φ(HN (T2 )). p (ii) T2 (a) ∈ GL(HN (T2 )). ∞ (iii) a ∈ G[H±± (T2 )]N ×N . ∞ (iv) det a ∈ GH±± (T2 ).

(v)

inf (z,w)∈D×D

|(det a)(z ±1 , w±1 )| > 0.

p ∞ 2 (T2 )]N ×N ∩ MN (b) Let a ∈ [H±± ×N (T ) and 1 ≤ p < ∞. Then the conditions p (T2 ) by pN (Z2++ ) in (i), (ii) (i)–(v) are equivalent provided one replaces HN p 2 and adds “and a ∈ GMN ×N (T )” in (iii)–(v).

Proof. (a) The equivalences (iii) ⇐⇒ (iv) ⇐⇒ (v) can be easily verified (recall 8.7(f)). The implications (iii) =⇒ (ii) =⇒ (i) are trivial. So assume (i) holds, ∞ (T2 )]N ×N (the other cases and for the sake of definiteness let a = a+− ∈ [H+−

8.4 The Invertibility Problem

429

∞ 2 can be treated similarly). By Theorem 8.13, b = a−1 +− ∈ LN ×N (T ) and also p q p ∗ q T+· (a+− ) ∈ GL(HN ⊗ L ) and T·+ (a+− ) ∈ GL(L ⊗ HN ). Hence, there exist ϕ+· ∈ (H p ⊗ Lp )N ×N and ψ·+ ∈ (Lq ⊗ H q )N ×N such that

a+− ϕ+· = (P ⊗ I)(a+· ϕ+· ) = IN ×N ,

a∗+− ψ·+ = (I ⊗ P )(a∗+− ψ·+ ) = IN ×N ,

∗ p p q q therefore b = a−1 +− = ϕ+· = ψ·+ ∈ (H ⊗ L )N ×N ∩ (L ⊗ H )N ×N , which ∞ 2 implies that b ∈ [H+− (T )]N ×N .

Remark. Thus, if a is either of the form a = a−− a−+ a++ or of the form ∞ (T2 )]N ×N , then T2 (a) is invertible on a = a−− a+− a++ with a±± ∈ G[H±± p −1 −1 −1 2 HN (T ) and the inverse is T2 (a) = T2 (a−1 ++ )T2 (a∓± )T2 (a−− ). 8.25. Toeplitz operators with analytic symbols on 2,∞ (Z2++ ). Let 2,∞ (Z2++ ) be the linear space of all sequences x = {xjk }∞ 2,∞ ++ := j,k=0 for which x2m :=

∞

|xjk |2 (j + 1)2m (k + 1)2m < ∞ ∀ m ∈ Z+ .

j,k=0 2,∞ On defining a metric in analogy to (6.27) we make ++ into a Fréchet space. ∞ 2 If a ∈ C ∞ (T2 ), then T2 (a) is obviously bounded on 2,∞ ++ . Put C±± (T ) := ∞ (T2 ). The following results are due to Gorodetsky [243]. C ∞ (T2 ) ∩ H±± ∞ (T2 ), then (a) If a++ ∈ C++ 2,∞ 2 2 T (a++ ) ∈ Φ(2,∞ ++ ) ⇐⇒ T (a++ ) ∈ GL(++ ) ⇐⇒ T (a++ ) ∈ GL( (Z++ )). ∞ ∞ (T2 ) and λ ∈ T, define a− (b) For a function a+− ∈ C+− λ ∈ C (T) by := a+− (λ, t) (t ∈ T). Then the following are equivalent.

a− λ (t)

(i) T (a+− ) ∈ Φ(2,∞ ++ ). (ii) T (a+− ) ∈ GL(2,∞ ++ ). (iii) a+− (z, ∞) = 0 for all |z| ≤ 1 and the operator T (a− λ ) is invertible on 2,∞ (Z+ ) for all λ ∈ T. (iv) a+− (z, ∞) = 0 for all |z| ≤ 1, and for each λ ∈ T, the function a− λ has (z) = 0 for |z| > 1. at most finitely many zeros of integral order on T and a− λ Examples: If a(s, t) = 1 − sn t−m (n, m ∈ Z+ ) and b(s, t) = 2 − (1 + s)t−1 , 2,∞ . then T2 (a) and T2 (b) are invertible on ++ ∞ ∞ (T2 ) and λ ∈ T, define a− (c) For a function a−− ∈ C−− λ ∈ C (T) by = a−− (t, λt) (t ∈ T). Then the following are equivalent.

a− λ (t)

(i) T2 (a−− ) ∈ Φ(2,∞ ++ ). 2,∞ (ii) T2 (a−− ) ∈ GL(++ ).

430


j k (iii) a−− (∞, ∞) = 0, and if j,k≥0 bjk z w is the series expansion of the −1 , w−1 ) in a neighborhood of (0, 0), then function a−1 −− (z |bjk | ≤ C(j + 1)m (k + 1)m

∀ j, k ≥ 0,

where C > 0 and m > 0 are certain constants independent of j and k. 2,∞ (Z+ )) for all λ ∈ T. (iv) T (a− λ ) ∈ GL(

(v) For each λ ∈ T the function a− λ has at most finitely many zeros of integral (z) = 0 for |z| > 1. order on T and a− λ Examples: 1 − s−n t−m (n, m ∈ Z+ ) and 2 − (1 + s−1 )t−1 generate invertible 2,∞ . Toeplitz operators on ++ 8.26. Lemma. If A is invertible on Y and Fredholm of index zero on Z and if Z ⊂ Y or Z ∗ ⊂ Y ∗ , then A is invertible on Z. Proof. If Z ⊂ Y (resp. Z ∗ ⊂ Y ∗ ), then the kernel of A in Z (resp. the kernel of A∗ in Z ∗ ) is contained in the kernel of A in Y (resp. the kernel of A∗ in Y ∗ ). The next theorem shows that Toeplitz operators over the quarter-plane with, in a sense, almost analytic symbols are invertible if and only if they are Fredholm. 8.27. Theorem (Malyshev/Douglas). Let b++ be in C(T2 )∩H ∞ (T2 ) resp. Cp (T2 ) ∩ H ∞ (T2 ), let (α, β) ∈ D × D, and let a(s, t) = (s − α)−1 (t − β)−1 b++ (s, t)

(s, t ∈ T).

Then T2 (a) is invertible on H p (T2 ) (1 < p < ∞) resp. p (Z2++ ) (1 ≤ p < ∞) if and only if (8.21) holds. This result was established by Malyshev [344] for α = β = 0 and p = 1, was then generalized to the case (α, β) ∈ D × D and p = 2 by Douglas [160], and was explicitly stated in the present form by Duduchava [174]. A proof for p = 2 is in Douglas [160, pp. 47–48]. Lemma 8.26 and Corollary 8.22 extend this result to other values of p. 8.28. Theorem (Osher). Let a, b, c ∈ W and g(s, t) = a(s)b(t) + c(t)

(s, t ∈ T).

Then T2 (g) is invertible on H p (T2 ) (1 < p < ∞) resp. p (Z2+ ) (1 ≤ p < ∞) if and only if (8.21) is satisfied. Proof. In view of Corollary 8.22 and Lemma 8.26 it suffices to consider the case p = 2. So assume T2 (g) ∈ Φ(2 (Z2++ )). Note that the operator T2 (g) = T (a) ⊗ T (b) + I ⊗ T (c) has the matrix representation A = (bj−k T (a) + cj−k I)∞ j,k=0 with respect to the second decomposition in (8.12).


431

We claim that for each µ ∈ sp T (a) the function µb + c does not vanish on T and has index zero. For µ ∈ a(T) this is immediate from (8.21). Let µ0 ∈ sp T (a) \ a(T) and suppose µ0 b(t0 ) + c(t0 ) = 0 for some t0 ∈ T. If b(t0 ) = 0, then c(t0 ) = 0, and this is impossible. Hence b(t0 ) = 0 and thus, µ0 = −c(t0 )/b(t0 ). It follows that inds (a(s) − µ0 ) = inds (a(s) + c(t0 )/b(t0 )) = inds (a(s)b(t0 ) + c(t0 )) = 0 (again by (8.21)), which is impossible for µ0 ∈ sp T (a). This proves our claim. Thus, for each µ ∈ sp T (a) we have a (uniquely determined) factorization µb(t) + c(t) = d− (µ, t)d+ (µ, t)

(t ∈ T),

d− (µ, ∞) = 1,

∞ and d±1 (µ, ·) ∈ W ∩H ∞ . Hence T (µb+c) ∈ GL(1 ), where d±1 − (µ, ·) ∈ W ∩H + −1 and since d+ (µ, t) = [T −1 (µb + c)χ0 ](t), it follows that, for each fixed t ∈ T, the four mappings sp T (a) → C, µ → d±1 ± (µ, t) are continuous and that

max µ∈sp T (a)

d±1 ± (µ, ·)W ≤ M < ∞.

Consequently, by 1.19 and 1.26(a), b(t)T (a) + c(t)I = d− (T (a), t)d+ (T (a), t), d−1 ± (T (a), t)d± (T (a), t) = I

(t ∈ T).

If we write d± (µ, t) =

n d± n (µ)t ,

d−1 ± (µ, t) =

n∈Z±

n e± n (µ)t ,

n∈Z±

then A = D− D+ and D± E± = E± D± = I, where ∞ ∞ D± := d± , E± := e± j−k T (a) j−k T (a) j,k=0

.

j,k=0

± Since n d± n (T (a)) = n dn C(sp T (a)) ≤ M and the same is also true for ± ± en in place of dn , each of the matrices D± , E± represents a bounded operator, from which we infer that T2 (g) is invertible. p 2 8.29. Homogeneous symbols. A function a ∈ MN ×N (T ) is said to be p 2 homogeneous if there exists a b ∈ MN ×N (T ) such that a(s, t) = b(st−1 ) for 8.7(d) is homogeneous. (s, t) ∈ T2 . Note that the function χ constructed in

−1 n ) be homogeneous. If b(t) = Let a(s, t) = b(st n∈Z bn t , then a(s, t) =

n −n 2 (s, t ∈ T). For n ∈ Z+ , put Ωn = {(j, k) ∈ Z++ : j + k = n}, n∈Z bn s t ·

·

·

and let En = pN (Ωn ). Then pN (Z2++ ) = E0 + E1 + E2 + . . . and it is easily seen that En is an invariant subspace of T2 (a). If x ∈ pN (Ωn ) and y = T2 (a)x, then

432


yi,n−i =

ai−j,n−i−k xjk =

(j,k)∈Ωn

n

bi−j xj,n−j .

j=0

Consequently, T2 (a) has the diagonal matrix representation diag (T0 (b), T2 (b), T2 (b), . . .) ·

·

·

with respect to the decomposition pN (Z2++ ) = E0 + E1 + E2 + . . . (here Tn (b) is defined as in 7.5). A first consequence of this representation is that the functions a given by p p 2 a(s, t) = b(st−1 ) (s, t ∈ T) is in MN ×N (T ) whenever b ∈ MN ×N (T). The following theorem provides some much more interesting consequences of this representation. p 8.30. Theorem (Douglas/Howe). Let a(s, t) = b(st−1 ), where b ∈ MN ×N (1 ≤ p < ∞). Then

T2 (a) ∈ Φ(pN (Z2++ )) ⇐⇒ T (b) ∈ Π{pN ; Pn } and T2 (a) ∈ GL(pN (Z2++ )) ⇐⇒ T (b) ∈ Π{pN ; Pn } and det Tn (b) = 0 ∀ n ∈ Z+ . If T2 (a) is Fredholm, then Ind T2 (a) = 0. There exist homogeneous functions a ∈ P(T2 ) such that T2 (a) is Fredholm but not invertible on p (Z2++ ). p invertible for all n ≥ n0 , then Proof. n } and Tn (b) is If T (b) ∈ Π{N ; P−1 (b), . . . is clearly a regularizer of T2 (a) diag P0 , P1 , . . . , Pn0 −1 , Tn0 (b), Tn−1 0 +1 (recall Proposition 7.3). Now suppose T2 (a) is Fredholm. Then there exists an n0 ∈ Z+ such that Tn (b) is invertible for all n ≥ n0 , since otherwise the kernel of T2 (a) would have infinite dimension. Assume there is a sequence {nk } ⊂ Z+ such (b)Pnk → ∞ as k → ∞. Without loss of generality assume that Tn−1 k (b)P ≥ k 2 (take a subsequence if necessary). Then one can find Tn−1 n k k that yk = Tnk (b)xk , xk = 1, yk ≤ 1/k2 . Put xk , y k ∈

nIm Pnk such

∞ wn = k=0 yk , w = k=0 yk . It is clear that wn ∈ Im T2 (a), wn → w, but w ∈ / Im T2 (a). The conclusion is that T2 (a) is not normally solvable and this is a contradiction. Thus, sup Tn−1 (b)Pn =: M < ∞. We have n≥n0

Pn x ≤ Tn−1 (b)Pn Tn (b)Pn x for all x ∈ pN and n ≥ n0 . Passage to the limit n → ∞ gives x ≤ M T (b)x for all x ∈ pN . Since Tn−1 (b)Pn L(pN ) = Tn−1 (b∗ )Pn L(XN ) , where X = q for 1 < p < ∞ and X = c0 for p = 1, it follows analogously that zX ≤ M T (b)zX for all z ∈ X. This proves that T (b) is invertible, and now Proposition 7.3 yields that T (b) is in Π{pN ; Pn }.


433

The implication concerning invertibility and the fact that Ind T2 (a) = 0 are now obvious. Finally, let 3 3 2 b(t) = 16t2 − 36t + 27t−1 = 16t−1 t + . t− 4 2 Then b(t) = 0 for t ∈ T and ind b = 0, so that T (b) ∈ Π{p ; Pn }. However, T0 (b) = 0. 8.31. Toeplitz operators with kernels supported in a half-plane. Let γ and δ be real numbers and suppose (γ, δ) = (0, 0). Put Πγ,δ := (x, y) ∈ R2 : γx + δy ≥ 0 . Given a function a ∈ W (T2 ) define supp a as the support of the Fourier

coefficients sequence, i.e., if a = ajk χj ⊗ χk , then supp a := (j, k) ∈ Z2 : ajk = 0 . Finally, let

Wγ,δ := a ∈ W (T2 ) : supp a ⊂ Πγ,δ .

Note that Wγ,δ is a closed subalgebra of W (T2 ). Theorem 8.35 will provide an invertibility criterion for Toeplitz operators over the quarter-plane with symbols in Wγ,δ . To prove

this theorem we need some lemmas. We shall always assume that a = γj+δk≥0 ajk χj ⊗ χk and

that b0 is defined by b0 = γj+δk=0 ajk χj ⊗ χk . In particular, if the ascent of the line γx + δy = 0 is irrational, then b0 (s, t) = a00 for all (s, t) ∈ T2 . 8.32. Lemma. Let γ and δ be positive integers and let a ∈ Wγ,δ . If 1 ≤ p < ∞ and T2 (a) ∈ GL(p (Z2++ )), then Ker T2 (b0 ) = {0}. Proof. Without loss of generality assume the largest common divisor of γ and δ equals 1. Put Ωn := (j, k) ∈ Z2++ : γj + δk = n , En = p (Ωn ). Of course, it may happen that Ωn = ∅. In that case we let En = {0}. We ·

·

then have p (Z2++ ) = E0 + E1 + . . . and it is easily seen that the matrix representation of T2 (a) with respect to this decomposition is lower triangular: ⎛ ⎞ B00 ⎜ ⎟ ⎜ B10 B11 ⎟ ⎜ ⎟ (8.22) ⎜ ⎟. ⎜ B20 B21 B22 ⎟ ⎝ ⎠ .. .. .. . . . . . . The corresponding representation of T2 (b0 ) is of diagonal form and results from (8.22) by putting B10 = B20 = B21 = . . . = 0. This observation gives the assertion straightforwardly.

434


8.33. Lemma. Let a ∈ W (T2 ) and suppose there are µ, λ ∈ (0, 1) such that aµ,λ := ajk µj λk χj ⊗ χk is also in W (T2 ). If 1 ≤ p < ∞ and Ker T2 (aµ,λ ) = {0} in p (Z2++ ), then Ker T2 (a) = {0} in p (Z2++ ). Proof. Define p (µ, λ) as the Banach space of all sequences x = {xjk }∞ j,k=0 for which µjp λkp |xjk |p < ∞ (1 ≤ p < ∞), xpp := j,k≥0

x∞ := sup µj λk |xjk | : (j, k) ∈ Z2++ < ∞ (p = ∞) and let Λ be the isometric isomorphism Λ : p (µ, λ) → p (Z2++ ),

{xjk } → {µj λk xjk }.

It is easy to verify that T2 (aµ,λ ) = ΛT2 (a)Λ−1 . Now assume T2 (a)x = 0 for some nonzero x ∈ p (Z2++ ). Since p (Z2++ ) is embedded in p (µ, λ), we have 0 = y := Λx ∈ p (Z2++ ) and thus, T2 (aµ,λ )y = ΛT2 (a)Λ−1 Λx = ΛT2 (a)x = 0, which is impossible if Ker T2 (aµ,λ ) = {0} in p (Z2++ ).

8.34. Lemma. Let γ and δ be nonzero real numbers and let a ∈ Wγ,δ . Suppose a does not vanish on T2 and ind1 a = ind2 a = 0. Then (a) b0 does not vanish on T2 and ind1 b0 = ind2 b0 = 0; (b) a ∈ GWγ,δ . Proof. Without loss of generality assume γ > 0 and δ > 0. (a) First divisor is 1.

suppose γ and δ are integers whose largest common

∞ Put bn = γj+δk=n ajk χj ⊗ χk and, for µ ∈ clos D, define gµ := n=0 µn bn . If |µ| = 1, then gµ (s, t) = a(µγ s, µδ t). Hence, if we think of s and t as being fixed and of g as a function of µ, we have gµ (s, t) = 0 for µ ∈ T and indµ gµ (s, t) = γ ind1 a + δ ind2 a = 0 (see (8.16)). This implies that gµ (s, t) = 0 for all (s, t) ∈ T2 and µ ∈ clos D. Therefore, a = g1 and b0 = g0 belong to the same connected component of GC(T2 ), which gives the assertion immediately. Now suppose the ascent of the straight line γx+δy = 0 is irrational. There is an N ∈ Z+ such that the N, N -th partial sum sN N a of the Fourier series of a is in GC(T2 ) and satisfies ind1 sN N a = ind2 sN N a = 0. Application of the above homotopy argument to sN N a shows that a00 = 0.


435

(b) Since the maximal ideal space of W (T2 ) can be identified with T2 , it follows that d := a−1 ∈ W (T2 ). Again let us first consider the case that γ and δ are integers. Define gµ as above and recall that

we have proved that gµ (s, t) = 0 for all (s, t) ∈ T2 and µ ∈ clos D. Put dn = γj+δk=n djk χj ⊗ χk . If |µ| = 1, then d(µγ s, µδ t) =

1 1 = . a(µγ s, µδ t) gµ (s, t)

2 Consequently, for fixed (s, t) ∈ T

, d(µγ s, µδ t) is an analytic function of µ in γ δ D. Thus, because d(µ s, µ t) = n∈Z µn dn (s, t), we conclude that dn = 0 for n < 0, i.e., that d ∈ Wγ,δ . We are left with the case that the ascent of the straight line γx + δy = 0 is irrational. Let sN N a be as in the proof of part (a) and put rN a := a − sN N a. Then ∞ r a n N a−1 = (sN N a)−1 (−1)n . s N Na n=0

From what was proved in part (a) we know that (sN N a)−1 ∈ Wγ,δ , and since obviously rN a ∈ Wγ,δ , it results that d = a−1 ∈ Wγ,δ . 8.35. Theorem. Let (γ, δ) ∈ R2 \ {(0, 0)} and a ∈ Wγ,δ . Put S = (x, y) ∈ R2 : x ≥ 0, y ≥ 0, x2 + y 2 > 0 and suppose 1 ≤ p < ∞. (a) If the intersection of the straight line γx + δy = 0 and S is not empty, then T2 (a) ∈ GL(p (Z2++ )) ⇐⇒ T2 (a) ∈ Φ(p (Z2++ )). (b) If the intersection of the straight line γx + δy = 0 and S is empty, then T2 (a) ∈ GL(p (Z2++ )) ⇐⇒ T2 (a) ∈ Φ(p (Z2++ )) and T2 (b0 ) ∈ GL(p (Z2++ )). Proof. (a) Suppose T2 (a) is Fredholm. Then, by Corollary 8.22, a ∈ GW (T2 ) and ind1 a = ind2 a = 0. If γ = 1 and δ = 0, then the function a admits a factorization a = a+− a++ as in 8.23(d), and so Proposition 8.10 implies −1 that T2 (a−1 ++ )T2 (a+− ) is the inverse of T2 (a). The case γ = 0 and δ = 1 can be settled analogously. Thus, let γ < 0 and δ > 0. Lemma 8.34 shows that a ∈ GWγ,δ and that b0 ∈ GW (T2 ), ind1 b0 = ind2 b0 = 0. We want to show that a is even in the connected component of GWγ,δ containing the identity. First suppose the straight line γx + δy = 0 has rational ascent. A similar reasoning as in the proof of Lemma 8.34(a) shows that a and b0 are in the same connected component of GWγ,δ . Without loss of generality assume the largest common divisor of γ and δ is 1. Then

436


b0 (s, t) =

ajk sj tk =

σ(t) :=

a−δn,γn s−δn tγn .

n∈Z

γj+δk=0

Put

a−δn,γn tn

(t ∈ T).

(8.23)

n∈Z

There are α, β ∈ Z such that αγ + βδ = 1. Hence, a−δn,γn (tαγ+βδ )n = b0 (t−β , tα ). σ(t) = n∈Z

It follows that σ does not vanish on T and that ind σ = 0 (by (8.16)). Consequently, there is a homotopy σµ of σ1 = σ to σ0 = 1 through GC(T2 ). Let τµ (s, t) := σµ (s−δ tγ ). Then τµ ∈ Wγ,δ ∩ GC(T2 ), ind1 τµ = ind2 τµ = 0, and hence τµ ∈ GWγ,δ (Lemma 8.34(b)). Since τ0 = 1 and τ1 (s, t) = a−δn,γn s−δn tγn = b0 (s, t), n∈Z

we conclude that b0 and thus also a is in the connected component of Wγ,δ containing the identity. If the ascent of γx + δy = 0 is irrational, then choose a number N so large that a and sN N a are in the same connected component of GWγ,δ and that ind1 sN N a = ind2 sN N a = 0, and then proceed as in the case where γ and δ are integers to show that sN N a and a00 are in the same connected component of GWγ,δ . Thus, in either case a belongs to the connected component of GWγ,δ containing the identity. Therefore a = exp f with f ∈ Wγ,δ (1.16(a)) and since ∞ (T2 ), we deduce from Propof = f−− + f−+ + f++ ; with f±± ∈ W (T2 ) ∩ H±± −1 −1 −1 sition 8.10 that T2 (a++ )T2 (a−+ )T2 (a−− ) (a±± = exp f±± ) is the inverse of T2 (a). The case γ > 0 and δ < 0 is analogous. (b) Without loss of generality assume γ > 0 and δ > 0; otherwise consider adjoints. Suppose T2 (a) is Fredholm and T2 (b0 ) is invertible. For ∈ (0, 1), put µ = γ and λ = δ . We have (recall Lemma 8.33) ajk µj λk sj tk aµ,λ (s, t) := γj+δk≥0

= b0 (s, t) +

ajk γj+δk sj tk = b0 (s, t) + b (s, t),

γj+δk>0

where b W (T2 ) → 0 as → 0. Hence, since T2 (b0 ) is invertible, the operators T2 (aµ,λ ) are invertible for all sufficiently small , so Lemma 8.33 gives that Ker T2 (a) = {0}, which implies that T2 (a) is invertible (Corollary 8.22). Conversely, suppose now that the operator T2 (a) is invertible. If the straight line δx + γy = 0 has irrational ascent, then b0 (s, t) = a00 = 0


437

(Lemma 8.34(a)) and thus T2 (b0 ) = a00 I is invertible. So let γ and δ be integers. From Lemma 8.32 we deduce that Ker T2 (b0 ) = {0} and Lemma 8.34(a) along with Corollary 8.22 implies that T2 (b0 ) is Fredholm with index zero. Hence, T2 (b0 ) is invertible. Remark. If the straight line γx + δy = 0 has irrational ascent, then the requirement in (b) that T2 (b0 ) be invertible is redundant, since T2 (b0 ) = a00 I = 0 (Lemma 8.34(a)). In case the ascent of γx + δy = 0 is rational, the invertibility of T2 (b0 ) may be replaced by the condition that det Tn (σ) = 0 for all n ∈ Z+ , where σ is given by (8.23). This follows from the representation (8.22), Lemma 8.34(a), Corollary 8.22, and Theorem 8.30. 8.36. Convolutions over angular sectors in Z2 . A subset W0 of R2 is called an angular sector in R2 with vertex (0, 0) if W0 is of the form W0 = (λx, λy) ∈ R2 : λ ∈ [0, ∞), (x, y) ∈ Ψ , where Ψ is a closed connected subset of T = {(x, y) ∈ R2 : x2 + y 2 = 1} containing at least two points. An angular sector in Z2 with vertex (0, 0) is a set of the form K = W0 ∩ Z2 , where W0 is angular sector in R2 with vertex (0, 0). Given an angular sector K in Z2 with vertex (0, 0) and a function a ∈ Cp (T2 ) (1 ≤ p < ∞) define the operator TK (a) on p (K) by TK (a) : p (K) → p (K), {xij }(i,j)∈K → ai−k,j−l xkl . (k,l)∈K

(i,j)∈K

The following result is due to Simonenko [495]. For TK (a) to be in Φ(p (K)) it is necessary and sufficient that (a) a ∈ GC(T2 ) if W0 = R2 ; (b) a ∈ GC(T2 ) and (ind1 a, ind2 a) ∈ ∂W0 ∩ Z2 (∂W0 being the boundary of W0 ) if W0 is a half-plane; (c) a ∈ GC(T2 ) and (ind1 a, ind2 a) = (0, 0) in the remaining cases. Note that a ∈ Cp (T2 ) ∩ GC(T2 ) implies that a ∈ GCp (T2 ) (8.7(a)). In the cases (a) and (b) Fredholmness yields invertibility and in the case (c) Fredholmness yields that

the index is zero. Now suppose a = ajk χj ⊗ χk ∈ Wγ,δ , where (γ, δ) ∈ R2 \ {(0, 0)}, and

2 put b0 = γj+δk=0 ajk χj ⊗ χk . Also assume that W0 is neither R nor a half-plane. Put W0 = W0 \ {(0, 0)} if the opening of W0 is less than π and W0 = clos (R2 \ W0 ) \ {(0, 0)} if the opening of W0 is larger than π. The following invertibility criterion was proved in B¨ ottcher [62]. (d) If the intersection of the straight line γx+δy = 0 and W0 is not empty, then

438


TK (a) ∈ GL(p (K)) ⇐⇒ TK (a) ∈ Φ(p (K)). (e) If the intersection of the straight line γx + δy = 0 and W0 is empty, then TK (a) ∈ GL(p (K)) ⇐⇒ TK (a) ∈ Φ(p (K)) and TK (b0 ) ∈ GL(p (K)). If the ascent of γx + δy = 0 is irrational, then the condition that TK (b0 ) be in GL(p (K)) is redundant. In case γ and δ are nonzero integers, it can be replaced by the requirement that det Tn () = 0 for all n belonging to a certain countable set N depending on γ, δ, W0 . Here = σ (resp. = 1/σ) if the opening of W0 is smaller (resp. larger) than π, and σ is given by (8.23). Examples: (i) N = Z+ if W0 = {2x − y ≥ 0, −x + y ≥ 0} and γ = δ = 1; (ii) N = {2k + [k/2] : k ∈ Z+ } if W0 = {x + 3y ≥ 0, 2x + y ≥ 0} and γ = δ = 1; (iii) N = {k + [2k/3] : k ∈ Z+ } if W0 = {x + 3y ≥ 0, 2x + y ≥ 0} and γ = 1, δ = 2. In (ii) and (iii), [x] denotes the largest integer n satisfying n ≤ x.

8.5 Bilocal Fredholm Theory The possibility of constructing a regularizer for Fredholm Toeplitz operators over the quarter-plane with (quasi) continuous symbols enabled us to avoid local techniques, except for Theorem 8.20, where we localized over the maximal ideal space M (B) of a C ∗ -algebra B between C and QC. Local techniques are a natural and powerful tool for the treatment of operators with discontinuous symbols. Duduchava [172] was the first to show how local methods can be used to establish a Fredholm criterion for operators with symbols from the algebra P C ⊗ P C on 2 (Z2++ ). He localized over the maximal ideal space M (P C) = T × {0, 1}; this reduced the problem to the study of “local representatives” of the form T (a) ⊗ T (b) + I ⊗ T (c) with a, b, c ∈ P C, and he succeeded to overcome the difficulties arising when investigating such “rather complicated” local representatives. In our paper [105], we pointed out that the things can be substantially simplified by localizing over the maximal ideal space M (alg T π (P C)) = T × [0, 1]; then the local representatives take the extremely simple form T (a) ⊗ I, where a ∈ P C. In what follows we shall show that localization over M (alg T π (A)) leads to a fairly simple Fredholm theory of the operators T2 (a) with a ∈ A ⊗ A, provided A satisfies some additional conditions. Note that M (alg T π (A)) = M (A) in case A is a C ∗ -algebra between C and QC. Moreover, since the techniques

8.5 Bilocal Fredholm Theory

439

employed in the following do not cause essential complications when passing from the “pure” Toeplitz operators T2 (a) (a ∈ A ⊗ A) to operators from alg T (A) ⊗ alg T (A) we shall without delay turn to the study of operators belonging to alg T (A) ⊗ alg T (A). Throughout the following let 1 < p < ∞ and 1/p + 1/q = 1. 8.37. Definitions. (a) Let A be a C ∗ -subalgebra of L∞ containing C and put AπN ×N := algL(HNp )/C∞ (HNp ) T π (AN ×N ).

AN ×N := algL(HNp ) T (AN ×N ),

Abbreviate A1×1 and Aπ1×1 to A and Aπ , respectively. Suppose Aπ is comπ mutative and the Shilov boundary of the maximal ideal space NA p := M (A ) A π A coincides with the whole space Np . Let Γp : A → C(Np ) denote the Gelfand map. For example, one can take A = C, QC, P C, P QC if 1 < p < ∞ and A = CE , QCE if p = 2. Notice that the proof of ∂S M (alg Tp (P QC)) = ottcher, Spitkovsky M (alg Tp (P QC)) for p = 2 is based on the results from B¨ [120] and the arguments of the proof of Theorem 5.46.

1 (b) Given B = i Ci ⊗ Di ∈ A " AN ×N and ν ∈ NA p define Bp,ν ∈ AN ×N by 1 Bp,ν := (Γp Ciπ )(ν)Di . (8.24) If B = T2 (b) = where

For B =

i

i 1 T (ci ) ⊗ T (di ) with ci ∈ A, d ∈ AN ×N , then Bp,ν = T (b1p,ν ), Γp T π (ci ) (ν)di . (8.25) b1p,ν := i

Ci ⊗ Di ∈ AN ×N " A and b = i ci ⊗ di ∈ AN ×N " A we define 2 Bp,ν := Ci (Γp Diπ )(ν)I, b2p,ν := ci Γp Tiπ (di ) (ν)I.

i

i

i

If A is a C ∗ -algebra between C and QC, then NA p is naturally homeomorphic to M (A) (Theorems 4.79 and 5.31) and so in this case the function defined by (8.25) is just the function defined in 8.16(a). (c) Put p BN ×N := algL(pN ) T (P CN ×N ),

p BπN ×N := algL(pN )/C∞ (pN ) T π (P CN ×N ),

let B = B1×1 , Bπ := Bπ1×1 , denote the maximal ideal space of Bπ by Np , and π let Γ

p : B → C(Np ) be the Gelfand map. For B = i Ci ⊗ Di ∈ B " BN ×N , p 1 1 , ν ∈ N , define B b = i ci ⊗ di ∈ P Cp " P CN p p,ν and bp,ν by (8.24) and ×N 2 2 (8.25). Bp,ν and bp,ν are defined similarly. 8.38. Proposition. Let A resp. B be as in the previous section and let H p resp. p be the underlying space (1 < p < ∞).

440


(a) If E1 , . . . , Em are in A resp. B and if there is a ν in NA p resp. Np such π l r that (Γp Ei )(ν) = 0 for i = 1, . . . , m, then there exist Un , Un (n = 1, 2, . . .) in A resp. B such that Unl = Unr = 1 and m

Unl Ei → 0

and

i=1

m

Ei Unr → 0

as

n → ∞.

i=1

p resp. pN , (b) If B is in AN ×N resp. BN ×N and if B is not invertible on HN then there exist Un (n = 1, 2, . . .) in AN ×N resp. BN ×N such that Un = 1 and Un B → 0 or BUn → 0 as n → ∞.

Proof. We only consider the H p case, as the proof for the p case is analogous. (a) By virtue of 1.20(c) there are Vj ∈ A such that Vjπ = 1 and → 0 as j → ∞ for all i. Hence, there exist Kij ∈ C∞ (H p ) and Cij ∈ A such that

Vjπ Eiπ

Vj Ei = Kij + Cij ,

Cij → 0 (j → ∞, ∀ i).

Let Pk denote the projections defined in 7.5 and put Qk := I − Pk , M := sup Qk L(H p ) . We have Qk Vj Ei = Qk Kij +Qk Cij . Given any n ∈ N there is a k

j0 such that Qk Cij0 < 1/(2n) for all k and i, and then one can find a k0 such that Qk0 Kij0 < 1/(2n) for all i (1.3(d)). It follows that Qk0 Vj0 Ei < 1/n for all i. Because Pk0 Vj0 is a finite-rank operator, we have Qk0 Vj0 = Vj0 − Pk0 Vj0 ≥ Vjπ0 = 1. Thus, if we let Unl = Qk0 Vj0 /Qk0 Vj0 (notice that k0 and j0 depend on n), then Unl = 1 and Unl Ei < 1/(nQk0 Vj0 ) ≤ 1/n. The existence of Unr can be shown analogously. (b) Let B = (Bij )N i,j=1 with Bij ∈ A. First suppose B is not Fredholm. Then B π is not invertible in AπN ×N and so (det B)π cannot be invertible in π Aπ . Consequently, there is a ν ∈ NA p such that (Γp (det B) )(ν) = 0. This implies that there exists an ω0 = (ω0ij ) ∈ CN ×N \ {0} satisfying N π ω0 (Γp B π )(ν) := ω0 (Γp Bij )(ν) i,j=1 = 0. From part (a) we infer that there are Vn ∈ A such that " " π "Vn bij − (Γp Bij )(ν)I " → 0 as n → ∞ for all i, j and Vn = 1 for all n. If we put Un = ω0 Vn , then Un AN ×N ≥ δ max ω0ij Vn A = δ max |ω0ij | =: ε > 0. i,j

i,j


441

Since ω0 Vn = Vn ω0 , we get " " Un B ≤ ω0 "Vn B − (Γp B π )(ν)I " + Vn ω0 (Γp B π )(ν)I " " ≤ ω0 ∆ max "Vn B − (Γp B π )(ν)I " = o(1) (n → ∞). i,j

Thus, Un = Un /Un has the desired properties. p . Let Y Now suppose B is Fredholm but Ker B ∗ = {0}, i.e., Im B = HN p p be any finite-dimensional subspace of HN such that HN decomposes into the ·

p onto Y parallel direct sum Im B + Y and let PY denote the projection of HN to Im B. Then PY ≥ 1 and PY B = 0. Hence, if we let Un := PY /PY for all n, then Un has the required properties. Finally suppose B is Fredholm but Ker B = {0}. Choose any closed sub·

p p space Z of HN such that HN = Z + Ker B and let PZ be the projection of p HN onto Ker B parallel to Z. Again PZ ≥ 1, but now we have BPZ = 0. So Un := PZ /PZ is as we wanted.

8.39. Lemma. (a) Let A be as in 8.37(a), let B ∈ A " AN ×N and ν ∈ NA p. Then 1 L(HNp ) ≤ BΦ(HNp (T2 )) . Bp,ν (b) If B ∈ B " BN ×N and ν ∈ Np , then 1 Bp,ν L(pN ) ≤ BΦ(pN (Z2++ )) .

m Proof. (a) Let B = i=1 Ci ⊗ Di with Ci ∈ A, Di ∈ AN ×N and let L be p 2 any compact

soperator on HN (T ).p Let ε > 0 pbe arbitrarily given. There exist operators j=1 Kj ⊗ Mj ∈ C∞ (H ) " C∞ (HN ) and R ∈ A such that L=

s

Kj ⊗ Mj + R,

R
0 and L ∈ C∞ (HN 1 Bp,ν ≤ Bess .

(b) The proof is the same. i 8.40. Corollary. The mappings B → Bp,ν (i = 1, 2) defined in 8.37 can be naturally extended to mappings of (A ⊗ A)N ×N onto AN ×N resp. (B ⊗ B)N ×N onto BN ×N .

Proof. Immediate from the preceding lemma.

Remark. From the results of 8.7(b) we deduce that for b ∈ (P C ⊗ P C)N ×N , p C 2 1 1 ν = (τ, λ) ∈ NP p , and HN (T ) as underlying space the function bp,ν = bp;τ,λ is given by b1p;τ,λ (t) = 1 − σp (λ) b(τ − 0, t) + σp (λ)b(τ + 0, t), and that for b ∈ (P Cp ⊗ P Cp )N ×N , ν = (τ, λ) ∈ Np , and pN (Z2++ ) as underlying space we have b1p;τ,λ (t) = 1 − σq (λ) b(τ − 0, t) + σq (λ)b(τ + 0, t), with σr as in 5.12 (also recall Remark 6.33). Our next objective is to prove Theorem 8.43, which provides necessary and sufficient conditions for an operator in (A ⊗ A)N ×N resp. (B ⊗ B)N ×N to be Fredholm. The following proposition settles the “necessity portion” and Section 8.42 prepares the proof of the “sufficiency part.” 8.41. Proposition. (a) Let A be as in 8.37(a) and let B ∈ (A ⊗ A)N ×N . If p p p 1 2 (T2 )), then Bp,ν ∈ GL(HN ) and Bp,ν ∈ GL(HN ) for all ν ∈ NA B ∈ Φ(HN p. (b) Let B be as in 8.37(c) and let B ∈ (B ⊗ B)N ×N . If B ∈ Φ(pN (Z2++ )), 1 2 ∈ GL(pN ) and Bp,ν ∈ GL(pN ) for all ν ∈ Np . then Bp,ν Proof. (a) Assume, contrary to what we want, there is a ν0 ∈ NA p such that p 1 is not invertible on H . There exists an ε > 0 with the property that Bp,ν N 0

m p (T2

)) whenever B − B < ε. Choose i=1 Ci ⊗ Di ∈ A " AN ×N B ∈ Φ(HN so that B − i Ci ⊗ Di < ε/2. Corollary 8.40 and Lemma 8.39(a) imply that " " " ε " " " 1 " " 1 − (Γp Ciπ )(ν0 ) ⊗ Di " = "Bp,ν − (Γp Ciπ )(ν0 )Di " < "I ⊗ Bp,ν 0 0 2 i i


443

(also see example (d) in 8.3). Hence " " " " 1 π C − − (Γ C )(ν )I ⊗ D "B − I ⊗ Bp,ν i p i 0 i " < ε. 0 i

m 1 Put Ei = Ci − (Γp Ciπ )(ν0 )I and B := I ⊗ Bp,ν + i=1 Ei ⊗ Di . Then B 0 p 1 is in Φ(HN (T2 )). From Proposition 8.38(b) we deduce that Bp,ν is a left or 0 right topological divisor of zero. For the sake of definiteness, assume there 1 → 0 as n → ∞. Let are Un in AN ×N such that Un = 1 and Un Bp,ν 0 p p 2 2 (T )) and K ∈ C (H (T )) satisfy B F = I + K, and choose F ∈ L(H ∞ N N

s p p L ⊗ M in C (H ) " C (H ) so that K = j j ∞ ∞ N j Lj ⊗ Mj + R with j=1 R < 1/3. Since (Γp Eiπ )(ν0 ) = 0 and Lπi = 0, Proposition 8.38(a) shows that there are Unl ∈ A such that Unl = 1 and Unl Ei → 0 (n → ∞, ∀ j),

Unl Lj → 0 (n → ∞, ∀ i).

It follows that 1 (Unl ⊗ Un )B F ≤ Un Bp,ν F + 0

m

Unl Ei Di F

i=1

is smaller than 1/3 for all sufficiently large n, while " " " " (Unl ⊗ Un )B F = "(Unl ⊗ Un ) I + Lj ⊗ Mj + R " ≥ 1−

j

Unl Lj Mj − R > 1 −

j

1 1 1 − = 3 3 3

if only n is large enough. This contradiction completes the proof. (b) The proof for the p case is analogous.

8.42. Bilocalization. Let A be as in 8.37(a). In accordance with 8.17 put Aπ1 := A ⊗ AN ×N /C∞ (H p ) ⊗ AN ×N . Let Uπ1 denote the closure in Aπ1 of the set of all elements of the form (B ⊗I)π1 , p . It is clear that Uπ1 is where B ∈ A and I is the identity operator on HN π contained in the center of A1 . Denote the maximal ideal space of Uπ1 by MA p. The mapping γ1 : Aπ → Uπ1 , C π → (C ⊗ I)π1 is well defined (C π = Dπ =⇒ (C ⊗I)π1 = (D⊗I)π1 ), it is obviously an algebraic homomorphism, and since (C ⊗ I)π1 = inf C ⊗ I + K : K ∈ C∞ (H p ) ⊗ AN ×N ≤ inf C ⊗ I + L ⊗ I : L ∈ C∞ (H p ) = inf C + L : L ∈ C∞ (H p ) = C π ,

444


it follows that γ1 is continuous. Hence, if ϕ is a multiplicative linear functional on Uπ1 , then ϕ ◦ γ1 is a multiplicative linear functional on Aπ , and because C π ∈ Ker (ϕ ◦ γ1 ) if and only if (C ⊗ I)π1 ∈ Ker ϕ, we deduce that the set γ1−1 (µ) ⊂ Aπ is a maximal ideal of Aπ whenever µ ⊂ Uπ1 is a maximal ideal 1 π of Uπ1 . For µ ∈ MA p , let Jµ denote the smallest closed two-sided ideal of A1 containing µ. From what was said above we infer that J1µ = closidAπ1 (C ⊗ I)π1 : Γp C π γ1−1 (µ) = 0 . (8.27) In a completely analogous fashion we define Aπ2 := AN ×N ⊗ A/AN ×N ⊗ C∞ (H p ), then Uπ2 and γ2 : Aπ → Uπ2 , and for µ in the maximal ideal space of Uπ2 the closed two-sided ideal of Aπ2 generated by µ can be shown to be of the form J2µ = closidAπ2 (I ⊗ D)π2 : Γp Dπ γ2−1 (µ) = 0 . (8.28) Also in accordance with 8.17 we let p Aπ12 = (A ⊗ A)N ×N /C∞ (HN (T2 )).

Bilocalization is nothing else than the following. To show that an operator p B ∈ (A ⊗ A)N ×N is Fredholm on HN (T2 )) it suffices to show that B1π is in π π π π is in GAπ12 ), GA1 and that B2 is in GA2 (Lemma 8.18 then implies that B12 π π and in order to show that Bi is in GAi it is enough to show that Biπ + Jiµ is invertible in Aπi /Jiµ for each µ ∈ MA p (Theorem 1.35(a)). Finally, in view of π equalities (8.27) and (8.28) we need not know anything about MA p = M (Ui ), A π we have merely to know what Np = M (A ) is. Notice that all the above definitions make sense if A is replaced by the algebra B introduced in 8.37(c) and at the same time H p is replaced by p . 8.43. Theorem. (a) Let A be as in 8.37(a) and B ∈ (A ⊗ A)N ×N . Then the following are equivalent: p (i) B ∈ Φ(HN (T2 )). π (ii) B12 ∈ GAπ12 . p p 1 2 (iii) Bp,ν ∈ GL(HN ), Bp,ν ∈ GL(HN ) ∀ ν ∈ NA p.

(b) Let B be as in 8.37(c) and B ∈ (B ⊗ B)N ×N . Then the following are equivalent: (i) B ∈ Φ(pN (Z2++ )). π (ii) B12 ∈ GBπ12 . 1 2 (iii) Bp,ν ∈ GL(pN ), Bp,ν ∈ GL(pN ) ∀ ν ∈ Np .


445

Proof. Proposition 8.41 gives the implications (i) =⇒ (iii). The implications (ii) =⇒ (i) are trivial. So suppose (iii) is satisfied, and for the sake of definiteπ ∈ GAπ12 . ness let us consider the H p case. We shall prove that B12 −1 A A Let µ ∈ Mp and put ν = γ1 (µ). Recall that ν ∈ Np . We claim that

If B =

1 )π1 ∈ J1µ . (B − I ⊗ Bp,ν

i

(8.29)

Ci ⊗ Di ∈ A " AN ×N , then 1 B − I ⊗ Bp,ν = Ei ⊗ Di = (Ei ⊗ I)(I ⊗ Di ), i

i

where Ei := Ci − (Γp Ciπ )(ν)I, and since (Γp Eiπ )(ν) = 0, we get (8.29). If B ∈ A ⊗ AN ×N , choose Bn ∈ A " AN ×N so that B − Bn → 0 as n → ∞. 1 − (Bn )1p,ν → 0 as n → ∞, and because Corollary 8.40 implies that Bp,ν (8.29) holds for Bn in place of B and J1µ is closed, it results that (8.29) is true for every B ∈ A ⊗ AN ×N . p 1 is in Φ(HN ). The commutativity of Aπ along with TheDue to (iii), Bp,ν 1 ∈ Φ(H p ). If there would exist a ν0 ∈ NA orem 1.14(c) show that det Bp,ν p 1 π such that (Γp (det Bp,ν ) )(ν0 ) = 0, then combining our assumption that A 1 π ∂S NA p = Np with 1.20(c), it would follow that (det Bp,ν ) is a topological π p p divisor of zero in A and thus in L(H )/C∞ (H ), which is clearly impos1 1 is Fredholm. Thus (det Bp,ν )π ∈ GAπ and therefore sible in case det Bp,ν 1 π π 1 and because (Bp,ν ) ∈ GAN ×N . So there is a regularizer R ∈ AN ×N of Bp,ν p 1 −1 1 belongs (Bp,ν ) − R ∈ C∞ (HN ) ⊂ AN ×N , it follows that the inverse of Bp,ν to AN ×N . Since 1 1 )−1 (I ⊗ Bp,ν ) = I ⊗ I, I ⊗ (Bp,ν we conclude from (8.29) that 1 B1π + J1µ = (I ⊗ Bp,ν )π1 + J1µ ∈ G(Aπ1 /J1µ ).

Recalling 8.42 we see that the proof is complete.

8.44. Toeplitz operators with P C ⊗ P C symbols. (a) Suppose that a ∈ (P C ⊗ P C)N ×N . Then the preceding theorem and the remark in 8.40 p (T2 )) if and only if give that T2 (a) is in Φ(HN 0 1 p T 1 − σp (λ) a(τ − 0, ·) + σp (λ)a(τ + 0, ·) ∈ GL(HN ), 1 0 p T 1 − σp (λ) a(·, τ − 0) + σp (λ)a(·, τ + 0) ∈ GL(HN ) for all (τ, λ) ∈ T × [0, 1]. Here σp (λ) is as in 5.12. (b) Analogously, if a ∈ (P Cp ⊗ P Cp )N ×N , then T2 (a) ∈ Φ(pN (Z2++ )) if and only if 0 1 T 1 − σq (λ) a(τ − 0, ·) + σq (λ)a(τ + 0, ·) ∈ GL(pN ), 1 0 T 1 − σq (λ) a(·, τ − 0) + σq (λ)a(·, τ + 0) ∈ GL(pN ) for all (τ, λ) ∈ T × [0, 1]. Here 1/p + 1/q = 1.

446


(c) After interpreting the function ap : T × [0, 1] → C (see 5.37) resp. ap : T × [0, 1] → C (see 6.31) for a ∈ P C resp. a ∈ P Cp as a function given on T (see 2.79) a similar reasoning as in the proof of Corollary 8.22 can be used to show that the index of a scalar Fredholm Toeplitz operator on H p (T2 ) resp. p (Z2++ ) with symbol from P C ⊗ P C resp. P Cp ⊗ P Cp is always zero. 8.45. Locally sectorial symbols. (a) Let A and B be C ∗ -algebras between C and L∞ . A function a ∈ L∞ ⊗ L∞ ∼ = C(X × X) is said to be locally sectorial over A ⊗ B if it is sectorial on Xα × Xβ for all (α, β) ∈ M (A) × M (B). (b) For a function a ∈ L∞ ⊗ L∞ the following are equivalent: (i) a is locally sectorial over C ⊗ C; (ii) each point σ ∈ T has an open neighborhood Uσ ⊂ T such that a is sectorial on Uσ × T, i.e., there are γ ∈ C (|γ| = 1) and ε > 0 with Re (γa(s, t)) ≥ ε for almost all

(s, t) ∈ Uσ × T;

(iii) there exist a sectorial function ϕ ∈ L∞ ⊗ L∞ and a smooth function c ∈ GC such that a(s, t) = ϕ(s, t)c(s) for almost all (s, t) ∈ T2 . (c) For a function a ∈ L∞ ⊗ L∞ the following are equivalent: (i) a is locally sectorial over C ⊗ C; (ii) each point (σ, τ ) ∈ T2 has open neighborhood Uστ ⊂ T2 such that a is sectorial on Uστ , i.e., there are γ ∈ C (|γ| = 1) and ε > 0 with Re (γa(s, t)) ≥ ε for almost all

(s, t) ∈ Uστ ;

(iii) there exist a sectorial function ϕ ∈ L∞ ⊗ L∞ and a smooth function c ∈ G(C ⊗ C) such that a(s, t) = ϕ(s, t)c(s, t) for almost all (s, t) ∈ T2 . The assertions (b) and (c) can be proved by arguments similar to those of 2.79 and 2.86. (d) Let a ∈ L∞ ⊗ L∞ be locally sectorial over C ⊗ C, choose ϕ and c as in (c), (iii), and define ind1 c and ind2 c as in 8.16(d). It is easily seen that the integers indj c (j = 1, 2) do not depend on the particular choice of c. We therefore denote these integers by indj a (j = 1, 2). (e) If a ∈ L∞ ⊗ L∞ is locally sectorial over C ⊗ C and ind1 a = 0, then T2 (a) is invertible on H 2 (T2 ). Proof. We have a(s, t) = ϕ(s, t)c(s), where ϕ is sectorial and c is a smooth invertible function of index zero. It follows that c can be written in the form c = c− c+ , where c− ∈ GH ∞ and c+ ∈ GH ∞ . Hence T2 (a) = T2 (c− ⊗ 1)T2 (ϕ)T2 (c+ ⊗ 1) (recall Proposition 8.10(a)), and so Theorems 8.14 and 8.24 imply the invert ibility of T2 (a).


447

(f) If a ∈ L∞ ⊗ L∞ is locally sectorial over C ⊗ C and ind1 a = ind2 a = 0, then T2 (a) is Fredholm on H 2 (T2 ).

∞ ∞ Proof.

We have a = ϕs, where ϕ = i di ⊗2 ei ∈ L ⊗ L is sectorial and c = j uj ⊗ vj ∈ C ⊗ C does not vanish on T and satisfies ind1 c = ind2 c = 0. Put A := L(H 2 )⊗L(H 2 )/L(H 2 )⊗C∞ (H 2 ), and for A ∈ L(H 2 )⊗L(H 2 ) let π A refer to the coset of A containing A. Denote by Rτ the collection of all f in C such that 0 ≤ f ≤ 1 and f is identically 1 in an open neighborhood of τ ∈ T, and define Mτ := {(I ⊗ T (f ))π : f ∈ Rτ }. Then {Mτ }τ ∈T forms a covering π system of localizing classes in A and∞T2 (a)∞commutes with every operator Mτ . For τ ∈ T, define aτ ∈ L ⊗ L by aτ (s, t) := ϕ(s, t)c(s, τ ). If in τ ∈T

f ∈ Rτ , then π π T2 (a) − T2π (aτ ) I ⊗ T (f ) π = T2π di uj ⊗ ei vj − di uj ⊗ ei vj (τ ) I ⊗ T (f ) i,j

=

T2π

i,j

di uj ⊗ ei vj − vj (τ ) f ,

i,j

which shows that T2π (a) is Mτ -equivalent to T2π (aτ ). Because, by (e), the operator T2 (aτ ) is invertible, the local principle 1.32(a) gives the invertibility of T2π (a). It can be shown analogously that T2 (a) + C∞ (H 2 ) ⊗ L(H 2 ) is invertible in the algebra L(H 2 ) ⊗ L(H 2 )/C∞ (H 2 ) ⊗ L(H 2 ). Lemma 8.18 completes the proof. 8.46. Sazonov’s results. Let now 1 < p < ∞ and A = algL(H p (T)) T (L∞ (T)),

A(2) = algL(H p (T2 )) T (L∞ (T2 )).

The bilocalization approach used above works perfectly for subalgebras of A⊗A. However, it is not applicable to A(2) since A⊗C∞ (H p ) and C∞ (H p )⊗A do not form ideals of A(2) . To go beyond tensor products, Sazonov [459], [460] studied the largest subalgebra of A(2) in which A ⊗ C∞ (H p ) and C∞ (H p ) ⊗ A are still ideals and obtained results for Toeplitz operators whose symbols need not belong to L∞ ⊗ L∞ . He was in particular able to consider symbols that are piecewise continuous with jumps along smooth curves. Another approach is in Sazonov [461]. There he replaced A ⊗ C∞ (H p ) and C∞ (H p ) ⊗ A by two ideals J1 and J2 of A(2) such that J1 J2 and J2 J1 are contained in C∞ (H p (T2 )) on the one hand and invertibility in A(2) /J1 and A(2) /J2 can be studied with the help of localization techniques on the other. Here are three beautiful results of Sazonov [461]. (a) Let a, b ∈ L∞ (T2 ) and suppose there are closed subsets F1 , G1 , F2 , G2 of T such that F1 ∩ G1 = ∅, F2 ∩ G2 = ∅, a ∈ C(T2 \ (F1 × F2 )), b ∈ C(T2 \ (G1 × G2 )). If T2 (a) + C∞ (H p (T2 )) and T2 (b) + C∞ (H p (T2 )) are

448


invertible in A(2) /C∞ (H p (T2 )), then T2 (ab) + C∞ (H p (T2 )) is also invertible in A(2) /C∞ (H p (T2 )). In the case p = 2 we obtain in particular that T2 (a), T2 (b) ∈ Φ(H 2 (T2 )) =⇒ T2 (ab) ∈ Φ(H 2 (T2 )). (b) Let b ∈ C(T2 ), a ∈ L∞ (T2 ), and suppose T2 (a) + C∞ (H p (T2 )) is invertible in A(2) /C∞ (H p (T2 )). Then T2 (ab) + C∞ (H p (T2 )) is invertible in A(2) /C∞ (H p (T2 )) if and only if b has no zeros on T2 and ind1 b = ind2 b = 0. (c) If a ∈ L∞ (T2 ) is locally p-sectorial in the sense that for each point τ ∈ T2 there exist an open neighborhood Uτ ⊂ T2 and a constant γτ ∈ C such that 1 1 − γτ aL∞ (U ) < , P 2L(H p (T)) then T2 (a) + C∞ (H p (T2 )) is invertible in A(2) /C∞ (H p (T2 )) if and only if the (well-defined) indices ind1 a and ind2 a are zero. Here P is the Riesz projection. By Hollenbeck and Verbitsky [286] (see the notes to 5.11), 1 P 2L(H p (T))

= sin2

π max{p, q}

1 p

+

1 =1 . q

8.6 P QC ⊗ P QC Symbols We now consider some problems on harmonic approximation and stable con2 (T2 ). vergence for Toeplitz operators with (P QC ⊗ P QC)N ×N symbols on HN Note that Theorem 8.43 involves a Fredholm criterion for these operators (on 2 (T2 )). HN 8.47. Definitions. Throughout the following (up to 8.54) suppose the underlying space is H 2 . Let Aπ refer to alg T π (P QC) and let N denote its maximal ideal space. Given b ∈ (P QC ⊗ P QC)N ×N and ν ∈ N define b1ν := b12,ν and b2ν := b22,ν in P QCN ×N through 8.37 and Corollary 8.40. 8.48. Harmonic approximation. Let Λ = (r0 , ∞), where r0 ∈ R+ . Define 2 A∞ N ×N (T ) as the collection of all sequences {aλµ }λ,µ∈Λ of continuous matrix functions aλµ ∈ CN ×N (T2 ) such that 2 < ∞. {aλµ } := sup aλµ L∞ N ×N (T )

(8.30)

λ,µ∈Λ

Provided with natural algebraic operations and the norm (8.30) the set 2 ∗ ∞ A∞ N ×N (T ) becomes a C -algebra. Let AN ×N be as in 3.39. Given subsets ∞ A andB of AN ×Nwe let A " B denote the collection of all sequences of the

i i i i form i aλ ⊗ bµ λ,µ∈Λ with {aλ } ∈ A, {bλ } ∈ B and the sum finite, and

8.6 P QC ⊗ P QC Symbols

449

2 we let A ⊗ B denote the closure of A " B in A∞ N ×N (T ). If {Kλ }λ∈Λ is an ∞ ∞ approximate identity and a ∈ (L ⊗ L )N ×N , then

{(kλ ⊗ kµ )a} ∈ (A ⊗ A)N ×N = AN ×N ⊗ AN ×N in view of Proposition 3.40 and 8.8. We shall abbreviate kλ ⊗ kµ to kλµ . Let NN ×N be as in 3.50. It is easily seen that (N ⊗ N )N ×N , (N ⊗ A)N ×N , and (A⊗N )N ×N are closed two-sided ideals of (A⊗A)N ×N . The quotient algebras π,2 π,12 (A ⊗ A)N ×N /(N ⊗ A)N ×N etc. will be denoted by Aπ,1 N ×N , AN ×N , AN ×N and π the cosets containing an element {aλµ } ∈ (A ⊗ A)N ×N by {aλµ }1 etc. It is easily seen that an analogue of Lemma 8.18 holds: if a ∈ (A ⊗ A)N ×N and π,2 π,12 π π aπ1 ∈ GAπ,1 N ×N , a2 ∈ GAN ×N , then a12 ∈ GAN ×N . ∞ ∞ Let a ∈ (L ⊗ L )N ×N . The sequence {kλµ a}λ,µ∈Λ is said to be bounded away from zero (bafz ) if there is a λ0 ∈ Λ such that kλµ a ∈ GCN ×N (T2 ) for all λ > λ0 and µ > λ0 and if 2 < ∞. sup (kλµ a)−1 L∞ N ×N (T )

λ,µ>λ0

A simple application of 1.26(d) shows that {kλµ a} is bafz if and only if {kλµ a}π12 ∈ GAπ,12 N ×N . If {kλµ a} is bafz, then the mapping i1 : (λ0 , ∞) × (λ0 , ∞) → Z,

(λ, µ) → ind1 (kλµ a)

is well defined and continuous. Since (λ0 , ∞)×(λ0 , ∞) is connected, this mapping takes a constant value on (λ0 , ∞) × (λ0 , ∞). This value will be denoted by ind1 {kλµ a}. The integer ind2 {kλµ a} is defined similarly. 8.49. Theorem. Let a ∈ (P QC ⊗ P QC)N ×N and let {Kλ }λ∈Λ be an approximate identity whose index set Λ is connected. Then the following are equivalent. (i) {kλµ a} is bafz. (ii) {kλ a1ν }, {kλ a2ν } are bafz for all ν ∈ N. (iii) a1ν and a2ν are locally sectorial over QC for all ν ∈ N. 2 2 (iv) T (a1ν ) ∈ Φ(HN ), T (a2ν ) ∈ Φ(HN ) for all ν ∈ N.

Proof. (ii) ⇐⇒ (iii): Corollary 3.82(b). (iii) ⇐⇒ (iv): Theorem 4.70. (i) =⇒ (ii). The proof is similar to the proof of Proposition 8.41. Assume there is a ν0 ∈ N such that {kλ a1ν0 }π is not in G(AN ×N /NN ×N ). There whenever a − b < ε. Choose exists an ε > 0 such that {kλµ b} is bafz

c ⊗ d ∈ P QC " P QC so that a − i N ×N i i i ci ⊗ di < ε/2. As in the proof of Proposition 8.41 we get " " " " ci − Γ2 T π (ci ) (ν0 ) ⊗ di " < ε. "a − χ0 ⊗ a1ν0 − i

450


Hence, if we put ei := ci − Γ2 T π (ci ) (ν0 ) and b := χ0 ⊗ a1ν0 + i ei ⊗ di , then QC ) and {kλµ b} is bafz. Note that {kλ a1ν0 }π is not in G(alg Kπ (P QCN ×N )/NNP ×N that, by the remark in 4.87, Γ {kλ ei }π (ν0 ) = Γ {kλ ci }π (ν0 ) − Γ2 T π (ci ) (ν0 ) = 0. (8.31) We claim that there exist qk := {(qk )λ } ∈ alg K(P QC) which have the properties (a) sup qk =: M < ∞, k

(b) f qk → 0 as k → ∞ for all f ∈ N , (c) χ0 − qk ∈ N for each k. Write 1 instead of χ0 and put {(qk )λ } = {kλ 1} − {kλ (2 + χk + χ−k )} − {kλ (1 + χk )}{kλ (1 + χ−k )}. 2 ∈ C(R) is From (3.5) we obtain that (qk )λ = K(k/λ) {kλ 1}, where K ∞ ≤ 1, (a) is obviously satisfied. Because K(0) given by (3.6). Since K = 1, it follows that (c) is fulfilled. Finally, if f = {fλ } ∈ N , then there is a λ0 such that fλ (qk )λ ≤ M fλ < ε for all λ > λ0 and all k (property (a)) and then one can find a k0 such that fλ (qk )λ < ε for all λ ≤ λ0 and all k > k0 (note that K(+∞) = 0), which gives (b). Replacing in the proof of 8.38(a) and in the proof of the first step of 8.38(b) C∞ by N P QC and the projections Qk by qk we get the following two statements (d) and (e). (d) If {kλ e1 }, . . . , {kλ er } are in alg K(P QC) and if there is a ν0 ∈ N such that (Γ {kλ ei }π )(ν0 ) = 0 for i = 1, . . . , r, then there are uln ∈ alg K(P QC) r such that uln = 1 and i=1 uln {kλ ei } → 0 as n → ∞. (e) There are un ∈ alg K(P QCN ×N ) such that un = 1,

un {kλ a1ν0 } → 0 (n → ∞).

Using this the proof can be finished as in 8.41. (ii) =⇒ (i). This can be proved applying the bilocal argument used in the proof of Theorem 8.43. Put Aπ1 = alg K(P QC) ⊗ alg K(P QCN ×N )/N P QC ⊗ alg K(P QCN ×N ), Uπ1 = closAπ1 {fλ ⊗ kµ χ0 }π1 : {fλ } ∈ alg K(P QC) . Here we identify alg K(P QC) with the subalgebra of alg K(P QCN ×N ) consisting of all elements of the form diag (gλ , . . . , gλ ), {gλ } ∈ alg K(P QC). As in 8.42, it is easily seen that to each ω ∈ M (Uπ1 ) there corresponds a ν ∈ M (alg Kπ (P QC)) ∼ = M (alg T π (P QC)) = N


451

such that the closed two-sided ideal of Aπ1 generated by ω is of the form J1ω = closidAπ1 {eλ ⊗ kµ χ0 }π1 : (Γ {eλ }π )(ν) = 0 .

If a = i ci ⊗ di ∈ P QC " P QCN ×N , then kλ ci ⊗ kµ di , {kλµ a} = i

{kλ χ0 ⊗

kµ a1ν }

=

kλ χ0 ⊗ Γ2 T π (ci ) (ν)kµ di

i

= Γ2 T π (ci ) (ν)kλ χ0 ⊗ kµ di , i

which shows that {kλµ a}π1 − {kλ χ0 ⊗ kµ a1ν }π1 =

{kλ ei ⊗ kµ χ0 }π1 {kλ χ0 ⊗ kµ di }π1 ,

i

where ei := ci − (Γ2 T π (ci ))(ν0 )χ0 is in J1ω (recall (8.31)). Using Corollary 8.40 one can easily verify that the same is true for every a ∈ P QC ⊗ P QCN ×N . Let KΛ refer to the algebra alg K(P QCN ×N ) for the case where the index set is Λ. Since {kµ a1ν } is bafz, we deduce from 1.26(d) that {kµ a1ν }π is invertible in alg Kπ (P QCN ×N ). Hence, there is a µ0 = µ0 (ν) such that {kµ a1ν }µ∈(µ0 ,∞) is invertible in K(µ0 ,∞) . The mapping N → P QCN ×N , ν → a1ν is continuous. This is obvious for a ∈ P QC "P QCN ×N and follows for a ∈ P QC ⊗P QCN ×N from Corollary 8.40. It results that {kµ aν }µ∈(µ0 ,∞) is in GK(µ0 ,∞) for all ν in some neighborhood of ν. Since N is compact, there exists a µ0 such that {kµ a1ν }µ∈(µ0 ,∞) is in GK(µ0 ,∞) for all ν ∈ N. So Theorem 1.35(a) shows that {kλ χ0 ⊗ kµ a1ν }π1 (λ in the original index set, µ in (µ0 , ∞)) is invertible in alg K(P QC) ⊗ K(µ0 ,∞) /N P QC ⊗ K(µ0 ,∞) . The same reasoning yields that {kλ a2ν ⊗kµ χ0 }π2 is invertible in K(λ0 ,∞) ⊗alg K(P QC)/K(λ0 ,∞) ⊗N P QC . From the analogue of Lemma 8.18 stated in 8.48 we finally obtain that {kλµ a}π12 is invertible in % P QC P QC N(λ K(λ0 ,∞) ⊗ K(µ0 ,∞) N ×N ⊗ N . ,∞) ,∞) (µ 0 0 N ×N

∞ 2 8.50. Stable convergence. Let BN ×N (T ) denote the collection of all se2 (T2 )) such that quences {Aλµ }λ,µ∈Λ (Λ = (r0 , ∞)) of operators Aλµ ∈ L(HN

{Aλµ } := sup Aλµ L(HN2 (T2 )) < ∞. λ,µ∈Λ

∞ 2 ∗ This norm and natural algebraic operations make BN ×N (T ) into a C ∞ ∞ , B , M , J be as in 4.15, 4.16. If A ⊂ B algebra. Let BN N ×N N ×N N ×N ×N N ×N ∞ ∞ 2 and B ⊂ B∞ or if A ⊂ B∞ and B ⊂ BN ×N we define A " B ⊂ BN ×N (T ) and ∞ 2 ∞ ∞ A ⊗ B ⊂ BN ×N (T ) in the usual way (see 8.7(e) and 8.17). If a ∈ L ⊗ LN ×N and {Kλ }λ∈Λ is an approximate identity, then

452


{T2 (kλµ )} ∈ B ⊗ BN ×N = (B ⊗ B)N ×N . The sets (M ⊗ B)N ×N , (J ⊗ B)N ×N etc. are closed two-sided ideals of the algebra (B ⊗ B)N ×N , the corresponding quotient algebras will be denoted M,1 J ,1 by BN ×N , BN ×N etc., and the cosets containing {Aλµ } ∈ (B ⊗ B)N ×N by 1 {Aλµ }M , {Aλµ }1J etc. The following analogue of Lemma 8.18 can be proved M,1 M,2 2 without difficulty: if a ∈ (B ⊗ B)N ×N and a1M ∈ GBN ×N , aM ∈ GBN ×N , M,12 then a12 M ∈ GBN ×N ; the same is valid with J in place of M. Now let a ∈ (L∞ ⊗ L∞ )N ×N . The sequence {T2 (kλµ a)}λ,µ∈Λ is said to 2 (T2 )) converge stably to T2 (a) if there is a λ0 ∈ Λ such that T2 (kλµ a) ∈ GL(HN for all λ > λ0 , µ > λ0 and sup T2−1 (kλµ a)L(HN2 (T2 )) < ∞.

λ,µ>λ0

The usual C ∗ -algebra arguments (see the proof of Proposition 4.17) give that the following are equivalent. (i) T2 (kλµ a) converges stably to T2 (a). M,12 (ii) {T2 (kλµ a)}12 M ∈ GBN ×N . p J ,12 (T2 )) and {T2 (kλµ a)}12 (iii) T2 (a) ∈ GL(HN J ∈ GBN ×N .

8.51. Theorem. Let a ∈ (P QC ⊗ P QC)N ×N and let {Kλ }λ∈Λ be an approximate identity with connected index set Λ. Then the following are equivalent. J ,12 (i) {T2 (kλµ a)}12 J ∈ GBN ×N .

(ii) {T2 (kλµ a)}12 J is invertible in the algebra alg T (P QC) ⊗ alg T (P QC) N ×N / J P QC ⊗ J P QC N ×N . 2 (iii) T2 (a) ∈ Φ(HN (T2 )).

Proof. (i) ⇐⇒ (ii). Apply 1.26(d) and 1.26(g) (also see the proof of Theorem 3.56). 12 (ii) =⇒ (iii). If {Bλµ }12 J is the inverse of {T2 (kλµ a)}J , then the operator B := s− lim Bλµ is a regularizer of T2 (a). λ,µ→∞

2 ) for all ν ∈ N. (iii) =⇒ (ii). Theorem 8.43 implies that T (a1ν ) ∈ GL(HN 1 From Theorem 4.26(b) we deduce that {T (kλ aν )} is invertible in the algebra alg T K(λ0 ,∞) (P QCN ×N ) for some λ0 = λ0 (ν). Now the proof can be finished by a reasoning similar to that in the proof of Theorem 8.49.

8.52. Corollary. Let the hypothesis of the preceding theorem be satisfied. Then 2 T2 (kλ,µ a) converges stably to T2 (a) if and only if T2 (a) ∈ GL(HN (T2 )). Proof. Immediate from the previous theorem and from what was said at the end of 8.50.


453

8.53. Theorem. Let a ∈ P QC ⊗ P QC and let {Kλ }λ∈Λ be an approximate identity the index set of which is connected. Then T2 (a) ∈ Φ(H 2 (T2 )) if and only if {kλµ a} is bafz and ind1 {kλµ a} = ind2 {kλµ a} = 0. Proof. Suppose that T2 (a) is Fredholm. Theorems 8.43 and 8.49 give that {kλµ a} is bafz. Theorem 8.51 shows that {T2 (kλµ a)}12 J is invertible. Hence, there are {Bλµ } ∈ B ⊗ B, K ∈ C∞ (H 2 (T2 )), {Cλµ } ∈ M ⊗ M such that Bλµ T2 (kλµ a) = I + K + Cλµ . If Cλµ < 1, then (I + Cλµ )−1 Bλµ T2 (kλµ a) = I + (I + Cλµ )−1 K and the conclusion is that T2 (kλµ a) ∈ Φ(H 2 (T2 )) for all λ > λ0 , µ > λ0 . Corollary 8.22 now implies that ind1 {kλµ a} = ind2 {kλµ a} = 0. Now suppose {kλµ a} is bafz and ind1 {kλµ a} = ind2 {kλµ a} = 0. From Theorem 8.49 we obtain that T (ajν ) ∈ Φ(H 2 ) (j = 1, 2) for all ν ∈ N. The mappings N → P QC, ν → ajν are continuous and N is connected (since, by 2.35(b), the essential spectrum of a Toeplitz operator is always connected). Therefore the mappings N → Z, ν → Ind T (ajν ) are constant. Let κj = Ind T (ajν ) and put b := (χκ1 ⊗ χκ2 )a. Then T (bjν ) ∈ GL(H 2 ) (Corollary 2.40) and thus, by Theorem 8.43, T2 (b) ∈ Φ(H 2 (T2 )). From what has already been proved we 0 so that 3ε < a∞ and then choose deduce that indj {kλµ b} = 0. Choose ε >

c ⊗ d ∈ P QC " P QC so that a − i i i i ci ⊗ di < ε. From 8.8 we get " " " " kλ χκ1 ci ⊗ kµ χκ2 di " < ε "kλµ (χκ1 ⊗ χκ2 )a − i

for all λ, µ and from 3.14 we infer that there is a λ0 such that " " " " kλ χκ1 ci ⊗ kµ χκ2 di − (kλ χκ1 )(kλ ci ) ⊗ (kµ χκ2 )(kµ di )" < ε " i

i

for all λ > λ0 and µ > λ0 . Thus, if λ > λ0 and µ > λ0 then kλµ (χκ1 ⊗ χκ2 )a − (kλ χκ1 ⊗ kµ χκ2 )kλµ a < 3ε and so 2.41(b) gives that indj kλµ b = indj (kλ χκ1 ⊗ kµ χκ2 )kλµ a = κj + indj kλµ a. Because indj kλµ b = indj kλµ a = 0, it results that κ1 = κ2 = 0. Consequently, T (a1ν ) and T (a2ν ) are invertible for all ν ∈ N and so Theorem 8.43 implies that T2 (a) ∈ Φ(H 2 (T2 )). / GL(H22 ) Remark. There exist matrix functions b ∈ W2×2 such that T (b) ∈ but T (hr b) ∈ GL(H22 ) for all r ∈ (0, 1). We discovered the existence of such matrix functions using the following result of Krupnik, Feldman [330].

454


n

then the matrix (cj−k )nj,k=0 is invertible if and only if χ−n−1 0 . T (b) ∈ GL(H22 ), where b = c χn+1 If c =

k=−n ck χk ,

Hence, if we choose c = χ−1 + 1 + χ1 and, accordingly, −2 t 0 b(t) = (t ∈ T), (8.32) t−1 + 1 + t t2 11 2 is not invertible, while T (hr b) ∈ GL(H22 ) then T (b) ∈ / GL(H2 ) because −2 −1 11 r r χ−2 0 2 −1 −2 −1 because hr b = r with cr = r χ−1 +r +r χ1 and r−1 r−2 c r χ2 is invertible. Now let a = b ⊗ χ0 , where b is given by (8.32). Then T2 (a) = T (b) ⊗ I is not in Φ(H22 (T2 )) since the operator T (b) is not invertible. On the other hand, {hrs a} = {hr b ⊗ χ0 } is obviously bafz and T2 (hrs a) = T (hr b) ⊗ I is invertible for all r, s ∈ (0, 1). This shows that the implication “{hrs a} bafz, 2 2 (T2 )) for r > r0 , s > r0 =⇒ T (a) ∈ Φ(HN (T2 ))”, which T (hrs a) ∈ Φ(HN might be a natural extension of Theorem 8.53 to the matrix case, is not true if N > 1. 8.54. Open problem. Is the index of a scalar Fredholm Toeplitz operator with P QC ⊗ P QC symbol always zero? Note that this question is even open for QC ⊗ QC symbols. Does there exist a two-dimensional analogue of Theorem 4.28?

8.7 Finite Section Method: Kozak’s Theory 8.55. Definitions. (a) An angular sector in R2 is a proper subset W of R2 (i.e., W = R2 ) which is of the form W = w + W0 , where w ∈ Z2 and W0 is an angular sector in R2 with vertex (0, 0) whose boundary consists of two half-lines with rational ascent (recall 8.36). (b) Two sets S1 , S2 ⊂ R2 are said to coincide locally at a point w ∈ R2 if there is an open neighborhood V ⊂ R2 of w such that S1 ∩ V = S2 ∩ V . A compact and connected subset U ⊂ R2 will be called a polygon in R2 if U locally coincides with an angular sector in R2 at each point of its boundary and if there are only finitely many points of the boundary of U , the vertices, at which U locally coincides with an angular sector in R2 whose opening is different from π. The collection of all vertices of U will be denoted by V(U ), and for v ∈ V(U ) we let Wv denote the angular sector in R2 locally coinciding with U at v. Note that V(U ) is necessarily a (finite) subset of Z2 . (c) Suppose 1 ≤ p < ∞. Given a subset S of Z2 and a matrix function p p 2 a ∈ MN ×N (T ) let TS (a) ∈ L(N (S)) denote the operator given by

8.7 Finite Section Method: Kozak’s Theory

TS (a) : pN (S) → pN (S),

{xij }(i,j)∈S →

455

ai−k,j−l xkl

(k,l)∈S

. (i,j)∈S

Sometimes it will be convenient to write TS (a) as PS aPS |Im PS (where PS denotes the canonical projection of pN (Z2 ) onto pN (S) and a is an abbreviation for M2 (a)). If S1 , S2 ⊂ Z2 and if there exists a v = (r, s) ∈ Z2 such that S2 = v + S1 , then the mapping τv : pN (S1 ) → pN (S2 ),

(τv x)ij = xi−r,j−s

is an isometric isomorphism. It is readily verified that TS2 (a) = τv TS1 (a)τv−1 (“translation invariance”). (d) Let U ⊂ R2 , Ω = U ∩ Z2 . For n ∈ N, define nΩ := {nu : u ∈ U } ∩ Z2 p p 2 2 (sic!). Given a ∈ MN ×N (T ) we shall write a ∈ Π{N (Z ); PnΩ } if TnΩ (a) is p in GL(N (nΩ)) for all sufficiently large n, say n ≥ n0 , and if −1 sup TnΩ (a)L(pN (nΩ)) < ∞.

(8.33)

n≥n0

If PnΩ converges strongly to the identity operator on pN (Z2 ) (equivalently, if the origin is in the interior of U ), then a ∈ Π{pN (Z2 ); PnΩ } if and only if M2 (a) ∈ Π{pN (Z2 ); PnΩ } in the sense of 7.1. This follows from Proposition 7.3 along with the fact that M2 (a) must be invertible if (8.33) holds (since TnΩ (a)PnΩ → M2 (a) strongly on pN (Z2 ) and TnΩ (a∗ )PnΩ → M2 (a∗ ) strongly on a predual of pN (Z2 ); cf. the proof of Theorem 8.30). Let W0 be an angular sector in R2 and suppose the only vertex of W0 is (0, 0). Let U ⊂ W0 be a polygon in R2 locally coinciding with W0 at (0, 0). Put K = W0 ∩ Z2 and Ω = U ∩ Z2 . Under these assumptions PnΩ converges strongly to the identity operator on pN (K). As in the preceding paragraph we obtain that TK (a) ∈ GL(pN (K)) if (8.33) is valid. This and Proposition 7.3 imply that a ∈ Π{pN (Z2 ); PnΩ } ⇐⇒ TK (a) ∈ Π{pN (K); PnΩ },

(8.34)

where TK (a) ∈ Π{pN (K); PnΩ } is understood in the sense of 7.1. 8.56. Proposition. Let U be a polygon in R2 , let Ω = U ∩ Z2 , and for p 2 v ∈ V(U ) put Kv = Wv ∩ Z2 . Then if a ∈ MN ×N (T ) (1 ≤ p < ∞), a ∈ Π{pN (Z2 ); PnΩ } =⇒ TKv (a) ∈ GL(pN (Kv )) ∀ v ∈ V(U ). Proof. Let v ∈ V(U ). Put Ωv = Ω−v and Kv0 = Kv −v. From 8.55(c) we obtain that a ∈ Π{pN (Z2 ); PnΩv }, and thus, by (8.34), TKv0 (a) ∈ GL(pN (Kv0 )). Again taking into account 8.55(c) we can conclude that TKv (a) ∈ GL(pN (Kv )).

456


8.57. Operators of local type. For U and V subsets of Z2 , define (U, V ) := inf{u − v : u ∈ U, v ∈ V }, and in case U = {u} is a singleton write (u, V ) in place of ({u}, V ). Given an operator A ∈ L(pN (Z2 )) (1 ≤ p < ∞) we define the function ϕA : R+ → R+ ∪ {0} by ϕA (t) := sup PU APV L(pN (Z2 )) : U, V ⊂ Z2 , (U, V ) > t . It is clear that 0 ≤ ϕA (t) ≤ A for all t ∈ R+ and that ϕA (t1 ) ≥ ϕA (t2 ) if t1 < t2 . An operator A ∈ L(pN (Z2 )) is said to be of local type if lim ϕA (t) = 0. t→∞

The collection of all operators of local type is denoted by Λ(pN (Z2 )). One can easily verify that Λ(pN (Z2 )) is a closed (linear) subspace of p L(N (Z2 )). Moreover, Λ(pN (Z2 )) is even a closed subalgebra of L(pN (Z2 )). Indeed, if (U, V ) > t and if we define W := {w ∈ Z2 : (w, V ) > t/2}, then (W, V ) > t/2 and (Z2 \ W, U ) > t/2, and because PU ABPV = PU APW BPV + PU APZ2 \W BPV , it follows that ϕAB (t) ≤ AϕB (t/2) + BϕA (t/2). p p 2 2 Examples. If a ∈ CN ×N (T ), then M2 (a) ∈ Λ(

N (Z )). To verify this it m p 2 suffices to show that M2 (a) ∈ Λ(N (Z )) if a = j,k=−m ajk χj ⊗ χk is a Laurent polynomial. But this is trivial: ϕM2 (a) (t) = 0 if t > 2m + 2. If S is p 2 any subset of Z2 and a ∈ CN ×N (T ), then

A = PS aPS + PZ2 \S := PS M2 (a)PS + PZ2 \S is of local type. This is obvious from what has just been proved together with the observation that PU APV = PU ∩S aPV ∩S + PU ∩(Z2 \S)∩V . 8.58. Proposition (Kozak/Simonenko). If A ∈ GL(pN (Z2 )) is of local type, then A−1 is also of local type. Proof. We show that if A ∈ GL(pN (Z2 )), then ϕ

A−1

A−1 2 A + 4A−1 2 ϕA (t) ≤ n

t 4n − 1

(8.35)

for all t ∈ R+ and all n ∈ N. This implies that lim ϕA−1 (t) = 0 if A is of local t→∞

type. Indeed, given any ε > 0 there is an n0 such that A−1 2 A/n0 < ε/2 and then we can find a t0 such that 4A−1 2 ϕA (t/(4n0 − 1)) < ε/2 for all t > t0 . Let U, V ⊂ Z2 and (U, V ) = r > t. Let h, r1 , r2 , r3 , r4 be any real numbers satisfying 0 ≤ r1 < r2 < r3 < r4 ≤ r,

0 < h < r2 − r1 ,

h < r3 − r 2 ,

h < r4 − r 3


and put U1 := w ∈ Z2 : r1 ≤ (w, U ) ≤ r3 ,

457

V1 := w ∈ Z2 : r2 ≤ (w, U ) ≤ r4 .

We claim that PU A−1 PV = −PU A−1 PU1 APV1 A−1 PV + ∆1 , where ∆1 ≤ 3A−1 2 ϕA (h). Put V := {w ∈ Z2 : (w, U ) < r2 }, U := {w ∈ Z2 : (w, U ) ≤ r3 }. We have PU A−1 PV = PU PV A−1 PV = PU A−1 PU APV A−1 PV + δ1 , where

δ1 = PU A−1 PZ2 \U APV A−1 PV ≤ A−1 2 ϕA (h),

because (Z2 \ U , V ) > h. Further, PU A−1 PU APV A−1 PV = −PU A−1 PU APZ2 \V A−1 PV , since PU A−1 PU AA−1 PV = 0. Finally, PU A−1 PU APZ2 \V A−1 PV = PU A−1 PU1 APV1 A−1 PV + δ2 , where δ2 ≤ PU A−1 PU \U1 APV1 A−1 PV + PU A−1 PU AP(Z2 \V )\V1 A−1 PV ≤ 2A−1 2 ϕA (h), because (U \ U1 , V1 ) > h, (U , (Z2 \ V ) \ V1 ) > h. Putting these things together we get our claim. Now let n be any positive integer. Put h := t/(4n − 1), l := r/(4n − 1), Ui := w ∈ Z2 : (4i − 4)l ≤ (w, U ) ≤ (4i − 2)l , Vi := w ∈ Z2 : (4i − 3)l ≤ (w, U ) ≤ (4i − 1)l , where i = 1, . . . , n. From what has just been proved (with r1 = (4i − 4)l, r2 = (4i − 3)l, r3 = (4i − 2)l, r4 = (4i − 1)l) we obtain that PU A−1 PV = −P A−1 PUi APVi A−1 PV + ∆i ,

(8.36)

with ∆i ≤ 3A−1 2 ϕA (h) for i = 1, . . . , n. Adding the n equalities (8.36) we arrive at the equality nPU A−1 PV = −

n i=1

PU A−1 PUi APVi A−1 PV +

n

∆i

i=1

= −P AP(U1 ∪...∪Un ) AP(V1 ∪...∪Vn ) A−1 PV + ∆ +

n i=1

∆i ,

458


where ∆ :=

n

PU A−1 PUi AP(V1 ∪...∪Vi−1 ∪Vi+1 ∪...∪Vn ) A−1 PV .

i=1

Since (Ui , Vj ) > h for i = j, it follows that ∆ ≤ nA−1 2 ϕA (h), and thus PU A−1 PV ≤

A−1 2 A + A−1 2 ϕA (h) + 3A−1 2 ϕA (h), n

which implies (8.35). 8.59. Theorem (Kozak). Let U be a polygon in R2 , let Ω = U ∩ Z2 and for p 2 v ∈ V(U ) put Kv = Wv ∩ Z2 . Then if a ∈ CN ×N (T ) (1 ≤ p < ∞), a ∈ Π{pN (Z2 ); PnΩ } ⇐⇒ TKv (a) ∈ GL(pN (Kv ))

∀ v ∈ V(U ).

Proof. In view of Proposition 8.56 we are left with the proof of the implication “⇐=”. Let V(U ) = {v1 , . . . , vm } and abbreviate Kvi to Ki . Choose subsets U1 , . . . , Um , V1 , . . . , Vm of R2 such that Ui ∩ Uj = ∅ (i = j), vi ∈ Ui ⊂ Vi ⊂ Wvi ∩ U,

U1 ∪ . . . ∪ Um = U,

(Ui , Wvi \ Vi ) > 0,

(Vi , (R2 \ Wvi ) ∩ U ) > 0.

The following picture shows how the Ui ’s can be chosen; the sets Vi can be taken of the form Vi = δ(Ui − vi ) + vi , where δ > 1 is sufficiently close to 1.

Put Ωi = Ui ∩ Z2 , ∆i = Vi ∩ Z2 , and in accordance with 8.55(d) let nΩ = {nw : w ∈ Ω} ∩ Z2 ,

nKi = {nw : w ∈ Wvi } ∩ Z2 ,

nΩi = {nu : u ∈ Ui } ∩ Z2 ,

n∆i = {nv : v ∈ Vi } ∩ Z2

(thus, although it may happen that Ωi = ∅, we have nΩi = ∅ if n is large enough). By assumption, TnKi (a) ∈ GL(pN (nKi )). Define Rn :=

m i=1

PnΩi (PnKi aPnKi )−1 Pn∆i .


459

Then Rn PnΩ aPnΩ equals m

PnΩi (PnKi aPnKi )−1 Pn∆i aPnKi PnΩ

i=1

+

m

PnΩi (PnKi aPnKi )−1 Pn∆i aPZ2 \nKi PnΩ

i=1

=

m

PnΩi −

i=1 m

+

m

PnΩi (PnKi aPnKi )−1 PnKi \n∆i aPnKi PnΩ

i=1

PnΩi (PnKi aPnKi )−1 Pn∆i aP(Z2 \nKi )∩nΩ .

(8.37)

i=1

m It is obvious that i=1 PnΩi = PnΩ . We have nKi = Ki + nvi , and so, taking into consideration 8.55(c), −1 PnΩi (PnKi aPnKi )−1 PnKi \n∆i = PnΩi τnvi (PKi aPKi )−1 τnv P i nKi \n∆i −1 = τnvi PnΩi −nvi (PKi aPKi )−1 P(nKi \n∆i )−nvi τnv i −1 = τnvi PnΩi −nvi (PKi aPKi + PZ2 \Ki )−1 P(nKi \n∆i )−nvi τnv . i

The example in 8.57 and Proposition 8.58 imply that (PKi aPKi + PZ2 \Ki )−1 is of local type and since nΩi − nvi , (nKi \ n∆i ) − nvi = (nΩi , nKi \ n∆i ) ≥ (nUi , Wvi \ clos nVi ) = n(Ui , Wvi \ Vi ) → ∞ (n → ∞), it follows that the second term (8.37) converges uniformly to zero as n → ∞. Finally, since a = M2 (a) is of local type and either (Z2 \ nKi ) ∩ nΩ = ∅ for all n or (n∆i , (Z2 \ nKi ) ∩ nΩ) ≥ (nVi , clos [(R2 \ nWvi ) ∩ nU ]) = (nVi , (R2 \ nWvi ) ∩ nU ) = n(Vi , (R2 \ Wvi ) ∩ U ) → ∞ (n → ∞), we conclude that the third term in (8.37) also goes uniformly to zero as n goes to infinity. Thus, Rn TnΩ (a) = PnΩ (I + Cn ) with Cn → ∞ as n → ∞, and if Cn < 1/2, then TnΩ (a) is invertible and −1 TnΩ (a)PnΩ ≤ (I + Cn )−1 Rn ≤ 2Rn ≤ 2

m

−1 TK (a)PKi . i

i=1

8.60. Remarks. (a) If U is convex, then one can take Vi = U for all i, which simplifies the preceding proof (the third term in (8.37) does not appear). (b) The previous theorem is a special case of the following more general result established by Kozak [317], [319] using local techniques.

460


Let U be a closed and connected (but not necessarily bounded) subset of R2 which coincides with an angular sector in R2 at each point of its boundary ∂U . Suppose the (possibly empty or infinite) set of all points on ∂U for which the opening of that angular sector is different from π is a subset of Z2 . Put p 2 Ω = U ∩ Z2 , and for v ∈ ∂U put Kv = Wv ∩ Z2 . Then if a ∈ CN ×N (T ) (1 ≤ p < ∞), a ∈ Π{pN (Z2 ); PnΩ } ⇐⇒ TKv (a) ∈ GL(pN (Kv )) ∀ v ∈ ∂U. Note that the proof of Theorem 8.59 also applies to “unbounded polygons” with a finite number of vertices. (c) Suppose Ω is the rectangle Ω = {(i, j) ∈ Z2 : 0 ≤ i ≤ k, 0 ≤ j ≤ l}, p 2 where k, l ∈ N, and let a ∈ CN ×N (T ) (1 ≤ p < ∞). Define a1 , a2 , a12 in p CN ×N (T2 ) by a1 (s, t) = a(s−1 , t),

a2 (s, t) = a(s, t−1 ),

a12 (s, t) = a(s−1 , t−1 )

(s, t ∈ T).

Then T2 (a) ∈ Π{pN (Z2++ ); PnΩ } if and only if each of the four operators T2 (a), T2 (a1 ), T2 (a2 ), T2 (a12 ) is invertible. This is an immediate consequence of Theorem 8.59 along with the (almost obvious) observation that TZ2±± (a) ∈ GL(pN (Z2±± )) ⇐⇒ T2 (a±± ) ∈ GL(pN (Z2++ )), where Z2±± := Z± × Z± , a++ := a, a−+ := a1 , a+− := a2 , a−− := a12 . In the scalar case, T2 (a2 ) is the transposed of T2 (a1 ) and T2 (a12 ) is the transposed of T2 (a). Thus, in that case T2 (a) ∈ Π{p (Z2++ ); PnΩ } ⇐⇒ T2 (a), T2 (a1 ) ∈ GL(p (Z2++ )) (take into account Corollary 8.22 and Lemma 8.26). (d) Although it is difficult to decide whether a convolution operator over an angular sector is invertible, the problem of the applicability of the finite section method is regarded as solved if it has been reduced to the invertibility problem for certain (explicitly given) convolution operators over angular sectors. The results of 8.36 yield the following effective criterion for a to be in Π{p (Z2 ); PnΩ } in case the kernel of a is supported in a half-plane. Let U be a polygon in R2 , let Ω = U ∩ Z2 , and for v ∈ V(U ) let Kv = Wv ∩Z2 . Put Wv = (Wv −v)\{(0, 0)} (resp. Wv = clos (R2 \(Wv −v))\{(0, 0)}) if the opening

of Kv is less (resp. larger) than π, and let Kv = Z2 ∩ clos Wv . Suppose aij χi ⊗ χj ∈ Wγ,δ , where (γ, δ) ∈ R2 \ {(0, 0)}, and put

a = b0 = γi+δj=0 aij χi ⊗ χj . Then a belongs to Π{p (Z2 ); PnΩ } if and only if (i) a ∈ GC(T2 ) and ind1 a = ind2 a = 0, (ii) TKv (b0 ) ∈ GL(p (Kv )) in case the intersection of the straight line γx + δy = 0 and Wv is not empty.

8.8 Finite Section Method: Bilocal Theory

461

Note that condition (ii) is redundant if the ascent of γx+δy = 0 is irrational and that it can be replaced by the requirement that the determinants of a certain family of finite Toeplitz matrices do not vanish if the ascent of γx + δy = 0 is rational. 8.61. Remark. In general, the operators M2 (a) is no longer of local type if a ∈ P Cp ⊗ P Cp . To see this, it suffices to show that there are a ∈ P C such that Pn T (a)Q2n L(2 (Z+ )) does not go to zero as n → ∞. Choose any a ∈ P C such that a−n = 1/n (n ≥ 1) and let 1 1 ϕn = √ , 0, 0, . . . . ,..., √ n+1 n+1 ? @A B n+1

Then ϕn = 1 and Pn T (a)Q2n ϕn 2 equals 2 2 1 1 1 1 1 + ... + + ... + + ... + n+1 n+1 2n + 1 2n + 1 3n + 1 2 2 1 1 1 n+1 3 + ... + (n + 1) ≥ > 0, ≥ log n+1 2n + 1 3n + 1 n+1 2 i.e., Pn T (a)Q2n L(2 (Z+ )) does not go to zero as n → ∞.

8.8 Finite Section Method: Bilocal Theory 8.62. Definitions. (a) Let Y be H p (1 < p < ∞) or p (1 < p < ∞) ∞ 2 and let Pn refer to the projections defined in 7.5. We define DN ×N (T ) as ∞ the collection of all sequences {Bn }n=0 of operators Bn ∈ L(Pn YN ⊗ Pn Y ) such that sup Bn (Pn ⊗ Pn )L(YN ⊗Y ) < ∞. Provided with natural algebraic n

∞ 2 operations and the norm {Bn } = sup Bn (Pn ⊗ Pn ) the set DN ×N (T ) n

becomes a Banach algebra. Let D∞ be as in 7.2. If A ⊂ D∞ (YN ) and B ⊂ ∞ 2 D∞ (Y ) or if A ⊂ D∞ (Y ) and B ⊂ D∞ (YN ), then A " B ⊂ DN ×N (T ) and ∞ 2 (T ) are defined in the usual way. A ⊗ B ⊂ DN ×N p 2 2 2 (b) For b ∈ L∞ N ×N (T ) resp. b ∈ M N ×N (T ) define Tn (b) ∈ L(Pn Y ⊗Pn YN )

∞ 2 as (Pn ⊗ Pn )T2 (b)(Pn ⊗ Pn ). If b = i ci ⊗ di ∈ L " L∞ N ×N , then Tn (b) =

∞ ∞ ⊗ LN ×N , then i Tn (ci ) ⊗ Tn (di ). Hence, if b ∈ L p {Tn2 (b)} ∈ S(H p ) ⊗ S(HN ) = (S(H p ) ⊗ S(H p ))N ×N p p (recall 7.8). A similar observation can be made for b ∈ M p ⊗ MN ×N and p in place of H .

(c) Let J = J (Y ) be the ideal of S = S(Y ) defined in 7.8. Then the sets (J ⊗ J )N ×N , (J ⊗ S)N ×N , (S ⊗ J )N ×N are closed two-sided ideals of

462


12 (S ⊗ S)N ×N . The corresponding quotient algebras will be denoted by SN ×N , 1 2 SN ×N , SN ×N , and the coset containing {Bn } ∈ (S ⊗ S)N ×N by {Bn }12 J , {Bn }1J , {Bn }2J , respectively.

(d) Let Wn be as in 7.6. If {Bn } ∈ S(Y ) ⊗ S(YN ), then the strong limits (as n → ∞) of (Pn ⊗ Pn )Bn (Pn ⊗ Pn ), (Pn ⊗ Wn )Bn (Pn ⊗ Wn ),

(Wn ⊗ Pn )Bn (Wn ⊗ Pn ), (Wn ⊗ Wn )Bn (Wn ⊗ Wn )

exist and are in L(Y ) ⊗ L(YN ). These limits will be denoted by W0 {Bn }, W1 {Bn }, W2 {Bn }, W12 {Bn }, respectively. If b belongs to L∞ ⊗ L∞ N ×N resp. p , then M p ⊗ MN ×N W0 {Tn2 (b)} = T2 (b), W2 {Tn2 (b)} = T2 (b2 ),

W1 {Tn2 (b)} = T2 (b1 ), W12 {Tn2 (b)} = T2 (b12 ),

p p where b1 , b2 , b12 in L∞ ⊗ L∞ N ×N resp. M ⊗ MN ×N are given by

b1 (s, t) = b(s−1 , t),

b2 (s, t) = b(s, t−1 ),

b12 (s, t) = b(s−1 , t−1 )

(8.38)

for s, t ∈ T. 8.63. Theorem. Let 1 < p < ∞, let Y be H p or p , let B ∈ L(Y ) ⊗ L(YN ), and let {Bn } ∈ S(Y ) ⊗ S(YN ) be any sequence such that Bn (Pn ⊗ Pn ) → B strongly on Y ⊗ YN . Then the following are equivalent. (i) B ∈ Π{Y ⊗ YN ; Bn }. (ii) B, W1 {Bn }, W2 {Bn }, W12 {Bn } are in GL(Y ⊗ Yn ) and {Bn }12 J is in ).

12 GSN ×N (Y

(iii) B, W1 {Bn }, W2 {Bn }, W12 {Bn } are in GL(Y ⊗ Yn ) and {Bn }1J is in 2 ), {Bn }2J is in GSN ×N (Y ).

1 GSN ×N (Y

Proof. (i) =⇒ (ii). Similar arguments as in the proof of the implication “=⇒” of Theorem 7.11 can be applied. (ii) =⇒ (i). Note that J ⊗ JN ×N coincides with the collection of all sequences {Dn }∞ n=0 of the form Dn = (Pn ⊗ Pn )K0 (Pn ⊗ Pn ) + (Wn ⊗ Pn )K1 (Wn ⊗ Pn ) +(Pn ⊗ Wn )K2 (Pn ⊗ Wn ) + (Wn ⊗ Wn )K12 (Wn ⊗ Wn ) + Cn , where K0 , K1 , K2 , K12 ∈ C∞ (Y ) ⊗ C∞ (YN ) = C∞ (Y ⊗ YN ) and Cn → 0 as n → ∞. Now suppose there is a sequence {Rn } ∈ S(Y ) ⊗ S(YN ) such that Rn Bn = Pn ⊗ Pn + Dn . Then (Wn ⊗ Pn )Rn (Wn ⊗ Pn )2 Bn (Wn ⊗ Pn ) = Pn ⊗ Pn + (Wn ⊗ Pn )Dn (Wn ⊗ Pn )


463

and passage to the strong limit n → ∞ gives R1 B1 = I + K1 , where R1 = W1 {Rn }, B1 = W1 {Bn }. It can be shown analogously that R0 B0 = I + K0 , R2 B2 = I + K2 , R12 B12 = I + K12 , where R0 = W0 {Rn } etc. Hence, T0 := B0−1 − R0 ,

T1 := B1−1 − R1 ,

T2 := B2−1 − R2 ,

−1 T12 := B12 − R12

are in C∞ (Y ⊗ YN ). Put Rn := Rn + (Pn ⊗ Pn )T0 (Pn ⊗ Pn ) + . . . + (Wn ⊗ Wn )T12 (Wn ⊗ Wn ). It can be shown as in the proof of Theorem 7.11 that Rn Bn = Pn ⊗ Pn + Cn with Cn → 0 as n → ∞ and this implies (i). (ii) ⇐⇒ (iii). The implication “=⇒” is trivial and the reverse implication can be proved by the same reasoning as in the proof of Lemma 8.18. 8.64. Proposition. Let 1 < p < ∞. Suppose b ∈ (L∞ ⊗ L∞ )N ×N resp. b ∈ (M p ⊗ M p )N ×N and p T2 (b) ∈ Π{HN (T2 ); Pn ⊗ Pn } resp. T2 (b) ∈ Π{pN (Z2++ ); Pn ⊗ Pn }.

(a) Then the four operators T2 (b), T2 (b1 ), T2 (b2 ), T2 (b12 ),

(8.39)

p with b1 , b2 , b12 given by (8.38), are invertible on HN (T2 ) resp. pN (Z2++ ).

(b) If, in addition, N = 1, then T2 (b) ∈ Π{H r (T2 ); Pn ⊗ Pn }

resp.

T2 (b) ∈ Π{r (Z2++ ); Pn ⊗ Pn }

for all r ∈ [p, q] (1/p + 1/q = 1) and, in particular, the operators (8.39) are invertible on H r (T2 ) resp. r (Z2++ ) for all r ∈ [p, q]. Proof. Part (a) is immediate from the preceding theorem. Since Tn2 (b1 ) = (Wn ⊗ Pn )Tn2 (b)(Wn ⊗ Pn ), Tn2 (b2 ) = (Pn ⊗ Wn )Tn2 (b)(Pn ⊗ Wn ), Tn2 (b12 ) = (Wn ⊗ Wn )Tn2 (b)(Wn ⊗ Wn ), the reasoning of the proof of Proposition 7.36(a) gives the statement of part (b). p 2 Remark. In the p case part (a) is even true for symbols in MN ×N (T ) (Proposition 8.56).

Open problem. Extend part (b) to the matrix case. In this connection recall 7.52(b) and see 8.72.

464


8.65. Bilocalization. (a) Let A be a C ∗ -algebra between C and L∞ and put SN ×N := algS(HNp ) T F (AN ×N ),

SπN ×N := algS(HNp )/J (HNp ) T FJπ (AN ×N )

(recall 7.26 and 7.28). Abbreviate S1×1 and Sπ1×1 to S and Sπ , respectively. Suppose Sπ is commutative, let RA p denote its maximal ideal space and Γp : Sπ → C(RA ) the Gelfand map. Define Aπ as in 8.37(a). If Sπ is comp π mutative, then so also is A (7.28(c)). Finally suppose the Shilov boundary π A of the maximal ideal space NA p of A coincides with the whole space Np . In the case p = 2 we can take A = C, QC, P C, P QC, CE , or QCE (see 7.33). If 1 < p < ∞, then A = C, QC, or P C has all the properties required above (see 5.46, 7.50). (b) Let Sπ1 , Sπ2 , Sπ12 denote the algebras S ⊗ SN ×N /J (H p ) ⊗ SN ×N , SN ×N ⊗ S/SN ×N ⊗ J (H p ), p p S ⊗ SN ×N /J (H p ) ⊗ J (HN ) = SN ×N ⊗ S/J (HN ) ⊗ J (H p ), respectively (note that, by virtue of Proposition 7.27, J (H p ) ⊂ S). The closure in Sπ1 of the set of all elements of the form {Bn ⊗ Pn }π1 , where {Bn } ∈ S, will ) π is a subset of the center of Sπ . ) π . It is easily seen that U be denoted by U 1 1 1 π .A . As in 8.42 one can verify ) by M Denote the maximal ideal space of U p 1 straightforwardly that the mapping )π, γ )1 : Sπ → U 1

{Cn }πJ → {Cn ⊗ Pn }π1

is a well-defined continuous algebraic homomorphism and that, therefore, ) π is a maximal ideal. For γ )1−1 (µ) ⊂ Sπ is a maximal ideal whenever µ ⊂ U 1 A 1 . ) µ ∈ Mp , let Jµ denote the smallest closed two-sided ideal of Sπ1 containing µ. Then −1 ) )1 (µ) = 0 . J1µ = closidSπ1 {Cn ⊗ Pn }π1 : Γp {Cn }πJ γ )π, γ )2 Analogously U 2 )2 , Jµ are defined. Now let b ∈ (A ⊗ A)N ×N . As in 8.42 we see that in order to show that Jiµ is in G(Si /) Jiµ ) {Tn2 (b)}π12 is in GSπ12 it suffices to show that {Tn2 (b)}πi + ) .A . (i = 1, 2) for all µ ∈ M p (c) The above definitions and statements can also be made for A = P Cp and p (1 < p < ∞) as the underlying space. The maximal ideal spaces of algL(p )/C∞ (p ) T π (P Cp ),

algS(p )/J (p ) T F (P Cp )

will be denoted by Np and Rp , respectively. We let Γp refer to the Gelfand map in either case. 8.66. Lemma. (a) Suppose A ∈ {C, QC, P C,

P QC, CE , QCE } if p = 2 and A ∈ {C, QC, P C} if 1 < p < ∞. Let b = i ci ⊗ di ∈ A " AN ×N and let ν ∈ RA p . Define


b1p,ν :=

Γp {Tn (ci )}πJ (ν)di .

465

(8.40)

i

Then T (b1p,ν )L(HNp ) ≤ cp T2 (b)Φ(HNp (T2 )) with some constant cp independent of b and ν.

p 1 (b) Let b = i ci ⊗ di ∈ P Cp " P CN ×N , ν ∈ Rp , and define bp,ν by (8.40). Then T (b1p,ν )L(pN ) ≤ cp T2 (b)Φ(pN (Z2++ )) , where cp is some constant that does not depend on b and ν. (c) Under the hypothesis of (a) or (b), the mapping b → b1p,ν can be natup rally extended to a mapping of A ⊗ AN ×N resp. P Cp ⊗ P CN ×N onto AN ×N p resp. P CN ×N . Proof. (a) If p = 2, then Sπ and Aπ are isometrically star-isomorphic in a A natural way and so RA p can be identified with Np . Hence b1p,ν =

Γp {Tn (ci )}πJ (ν)di = Γp T π (ci ) (ν)di

i

i

and Lemma 8.39(a) gives the assertion. Now let 1 < p < ∞ and for the sake of definiteness let A = P C. Recall C C and RP given in 5.45 and 7.50. If ν = (τ, x), where the descriptions of NP p p τ ∈ T and x ∈ Op (0, 1), then (1 − x)ci (τ − 0) + xci (τ + 0) di b1p,ν = i

= (1 − x)

Γp T π (ci ) (τ, 0)di + x Γp T π (ci ) (τ, 1)di . i

i

From Lemma 8.39(a) we know that " " " " Γp T π (ci ) (τ, k)di " ≤ T2 (b)ess "T

(k = 0, 1),

i

whence with cp =

T (b1p,ν ) ≤ |1 − x| + |x| T2 (b)ess ≤ cp T2 (b)ess max (|1 − x| + |x|).

x∈Op (0,1)

(b) The proof is analogous (note that instead of Lemma 8.39(b) one can also use what was said in 8.16(c)). (c) Immediate from (a) and (b). In the following we always assume that b1 , b2 , b12 are given by (8.38).

466


8.67. Theorem. Let A ∈ {C, QC, P C, P QC, CE , QCE }, let b ∈ (A ⊗ A)N ×N 2 and K ∈ C∞ (HN (T2 )). Then 2 T2 (b) + K ∈ Π{HN (T2 )); Pn ⊗ Pn } 2 (T2 )). ⇐⇒ T2 (b) + K, T2 (b1 ), T2 (b2 ), T2 (b12 ) ∈ GL(HN 2 Proof. Since Wn ⊗Pn , Pn ⊗Wn , Wn ⊗Wn converge weakly to zero on HN (T2 ), the implication “=⇒” results from 1.1(f) and Theorem 8.63. Again by Theorem 8.63, to get the reverse implication it suffices to show that π 2 Tn (b) + (Pn ⊗ Pn )K(Pn ⊗ Pn ) 12 = {Tn2 (b)}π12 ∈ GSπ12 .

.A , ν = γ Let µ ∈ M )1−1 (µ) ∈ RA 2 2 , and ) J1µ = closSπ1 {En ⊗ Pn }π1 : Γ2 {En }πJ (ν) = 0 .

(8.41)

By what was said in 8.65(b), it remains to show that {Tn2 (b)}π1 + ) J1µ is in π )1 G(S1 /Jµ ). We claim that π 2 J1µ . (8.42) Tn (b) − Pn ⊗ Tn (b12,ν ) 1 ∈ )

If b = i ci ⊗ di ∈ A " AN ×N , then Tn2 (b) − Pn ⊗ Tn (b12,ν ) = Tn (ei ) ⊗ Tn (di ) = (Tn (ei ) ⊗ Pn )(Pn ⊗ Tn (di )), i

i

ci −(Γ2 {Tn (ci )}πJ (ν)χ0

where ei := and since (Γ2 {Tn (ei )}πJ )(ν) = 0, we arrive at (8.42). Lemma 8.66(c) now shows that (8.42) holds for every b ∈ A⊗AN ×N . A A By virtue of 7.33 we may identify R

2 with N2 . From Theorem 8.43 we 1 2 deduce that T (b2,ν ) ∈ GL(HN ). If b = i ci ⊗ di ∈ A " AN ×N , then Γ2 T π (ci ) (ν)d)i = (b2 )12,ν , (b12,ν )) = i

and once more using Lemma 8.66(c) we see that (b12,ν )) = (b2 )12,ν for every b in 2 (T2 )), we therefore obtain that T ((b12,ν ))) beA⊗AN ×N . Since T2 (b2 ) ∈ Φ(HN 2 2 ; Pn }. longs to GL(HN ). Now Theorem 7.32(ii) implies that T (b12,ν ) ∈ Π{HN Thus, there is an {Rn } ∈ SN ×N (take into account 1.26(d)) such that Rn Tn (b12,ν ) = Pn + Cn , where Cn → 0 as n → ∞. Hence, (Pn ⊗ Rn )(Pn ⊗ Tn (b12,ν )) = Pn ⊗ Pn + Pn ⊗ Cn = Pn ⊗ Pn + Cn 1/2 Pn ⊗ Cn −1/2 Cn

(if Cn > 0)

from which we infer that 2 {Pn ⊗ Rn }{Pn ⊗ Tn (b12,ν )} − {Pn ⊗ Pn } ∈ J (H 2 ) ⊗ J (HN ).

But (8.42) and (8.43) imply that {Tn2 (b)}π1 + ) J1µ is invertible.

(8.43)


467

p 8.68. Lemma. Let 1 < p < ∞, 1/p + 1/q = 1 and a ∈ P CN ×N . Then

T (a) ∈ GL(pN ),

T (a) ∈ GL(qN ) =⇒ T (a) ∈ GL(sN ) ∀ s ∈ [p, q].

Proof. Taking into account that σq (λ) + σp (1 − λ) = 1 it is easily verified ap,0 (1/τ, 1 − λ) (recall 6.31). Consequently, by 6.39 and that aq,0 (τ, λ) = ) 6.40, T () a) is Fredholm with index zero on pN . Lemma 8.26 then implies that a) ∈ GL(qN ). T () a) is even invertible on pN . It follows analogously that T () p q Hence, T (a) is in both Π{N ; Pn } and Π{N ; Pn } (Theorem 7.42). Now the interpolation argument used in the proof of Proposition 7.36(a) yields that T (a) ∈ Π{sN ; Pn } and thus that T (a) ∈ GL(sN ) for all s ∈ [p, q]. 8.69. Theorem. Suppose 1 < p < ∞ and 1/p + 1/q = 1. p (a) Let b ∈ (P C ⊗ P C)N ×N and K ∈ C∞ (HN (T2 )). If p (T2 )), T2 (b) + K, T2 (b1 ), T2 (b2 ), T2 (b12 ) ∈ GL(HN r (T2 )) T2 (b), T2 (b1 ), T2 (b2 ) ∈ Φ(HN

then T2 (b) + K ∈

p Π{HN (T2 ); Pn

∀ r ∈ [p, q],

⊗ Pn }.

(b) Let b ∈ (P Cp ⊗ P Cp )N ×N and K ∈ C∞ (pN (Z2++ )). If T2 (b) + K, T2 (b1 ), T2 (b2 ), T2 (b12 ) ∈ GL(pN (Z2++ )), T2 (b), T2 (b1 ), T2 (b2 ) ∈ Φ(rN (Z2++ ))

∀ r ∈ [p, q],

then T2 (b) + K ∈ Π{pN (Z2++ ); Pn ⊗ Pn }. Proof. (a) On identifying the arc Ap (0, 1) with the corresponding boundary C C = T × Ap (0, 1) as a subset of RP = arc of Op (0, 1) we may regard NP p p T×Op (0, 1). For τ ∈ T and x = σr (λ) ∈ Op (0, 1) (⇐⇒ r ∈ [p, q], λ ∈ [0, 1]), we C by (τ, r, λ). The natural extension of the denote the point ν := (τ, x) ∈ RP p π PC C a mapping of SπN ×N into [C(RP Gelfand map Γp : S → C(Rp ) to

p )]N ×N will be denoted by Γp , too. For b = i ci ⊗ di in P CN ×N " P CN ×N , put Γp {Tn (ci )}πJ (τ, r, λ)di b1τ,r,λ := i

b2τ,r,λ

and define analogously. Using the identity σp (λ) + σq (1 − λ) = 1 we get 1 − σp (λ) ci (τ − 0) + σp (λ)ci (τ + 0) di b1τ,p,λ = i

1 − σp (λ) ) = ci (1/τ + 0) + σp (λ)) ci (1/τ − 0) di i

1 − σq (1 − λ) ) = ci (1/τ − 0) + σq (λ)) ci (1/τ + 0) di i

=

Γp {Tn () ci )}πJ (1/τ, q, 1 − λ)di = (b1 )11/τ,q,1−λ . i

(8.44)

468


If s ∈ [p, q], τ, η ∈ T, λ, θ ∈ [0, 1], then (with ar := ar,1 defined as in 5.37) (b1τ,p,λ )s (η, θ) = 1 − σs (θ) b1τ,p,λ (η − 0) + σs (θ)b1τ,p,λ (η + 0) = 1 − σs (θ) 1 − σp (λ) b(τ − 0, η − 0) + σp (λ)b(τ + 0, η − 0) +σs (θ) 1 − σp (λ) b(τ − 0, η + 0) + σp (λ)b(τ + 0, η − 0) = 1 − σp (λ) b2η,s,θ (τ − 0) + σp (λ)b2η,s,θ (τ + 0) = (b2η,s,θ )p (τ, λ). (8.45) From Lemma 8.66(c) we infer that biτ,r,λ (i = 1, 2) can be defined for all b ∈ P CN ×N ⊗ P CN ×N and that (8.44) and (8.45) hold for all functions b ∈ P CN ×N ⊗ P CN ×N . Finally, from (8.45) and Theorem 5.47 we obtain that s T (b1τ,p,λ ) ∈ Φ(HN ) ∀ τ ∈ T ∀ λ ∈ [0, 1]

⇐⇒

T (b2η,s,θ )

∈

p Φ(HN )

∀ s ∈ [p, q]

∀ η ∈ T ∀ λ ∈ [0, 1]

∀ s ∈ [p, q]. (8.46)

C .P C , ν = (τ, r, λ) = γ )1 Now let µ ∈ M )1−1 (µ) ∈ RP p p , and define Jµ by (8.41). 2 π 1 ) J1µ ). To get our assertion it suffices to show that {Tn (b)}1 + Jµ is in G(Sπ1 /) The same argument as in the proof of Theorem 8.67 gives that 2 π Tn (b) − Pn ⊗ Tn (b1τ,r,λ ) 1 ∈ ) J1µ .

Our first objective is to prove that s T (b1τ,r,λ ) ∈ GL(HN )

∀ s ∈ [p, q]

and

p T ((b1τ,r,λ ))) ∈ GL(HN ).

(8.47)

p q Since T2 (b) ∈ Φ(HN (T2 )) and T2 (b2 ) ∈ Φ(HN (T2 )), it follows that p q T (b2τ,p,λ ) ∈ GL(HN ) and T (b2τ,p,λ ) = T (b2 )21/τ,q,1−λ ∈ GL(HN )

(recall (8.44)) for all τ ∈ T and all λ ∈ [0, 1]. From the remark in 5.22 we s deduce that T (b2τ,p,λ ) ∈ GL(HN ) for all τ ∈ T, λ ∈ [0, 1], s ∈ [p, q]. Hence, by p 1 (8.46), T (bτ,s,λ ) ∈ Φ(HN ) for all s ∈ [p, q]. The mapping p [p, q] → L(HN ),

s → T (b1τ,s,λ )

p is easily seen to be continuous. Therefore the index of T (b1τ,s,λ ) on HN does p , one has Ind T (b1τ,s,λ ) = 0 not depend on s. Since T (b1τ,p,λ ) is invertible on HN p is zero. Because, for all s ∈ [p, q]. In particular, the index of T (b1τ,r,λ ) on HN r 2 1 r ). So by assumption, T2 (b) ∈ Φ(HN (T )), we see that T (bτ,r,λ ) ∈ GL(HN p 1 Lemma 8.26 can be used to deduce that T (bτ,r,λ ) ∈ GL(HN ). If we replace in the preceding reasoning p by q, we arrive at the conclusion that T (b1τ,r,λ ) q s ). Hence, by the remark in 5.22, T (b1τ,r,λ ) ∈ GL(HN ) for belongs to GL(HN all s ∈ [p, q]. Finally, the same reasoning with b2 in place of b gives that p ). The proof of (8.47) is complete. T ((b1τ,r,λ ))) = T ((b2 )1τ,r,λ ) is in GL(HN


469

Combining (8.47) with Theorems 5.45 and 7.50 yields that {Tn (b1ν )}πJ is C invertible in the algebra SπN ×N for each ν ∈ RP p . Let {Rn } ∈ SN ×N satisfy π 1 π π {Rn }J {Tn (bν )}J = {Pn }J . From (8.47) and the proof of Theorem 7.11 it is p seen that there are K, L ∈ C∞ (HN ) such that Rn Tn (b1ν ) := (Rn + Pn KPn + Wn LWn )Tn (b1ν ) = Pn + Cn , with Cn → 0 as n → ∞. Since {Pn KPn } and {Wn LWn } are in SN ×N , we have {Rn } ∈ SN ×N . Now the proof can be finished as in 8.67. (b) The proof is analogous. The only difference is that the argument based on the remark in 5.22 must be replaced by Lemma 8.68. 8.70. Proposition. Let 1 < p < ∞ and 1/p+1/q = 1. If b ∈ (QC ⊗QC)N ×N or b ∈ P C ⊗ P C (N = 1) (resp. b ∈ (Cp ⊗ Cp )N ×N or b ∈ P Cp ⊗ P Cp (N = 1)), then the Fredholmness of the four operators T2 (b), T2 (b1 ), T2 (b2 ), p r (T2 ) (resp. pN (Z2++ )) implies their Fredholmness on HN (T2 ) T2 (b12 ) on HN r 2 (resp. N (Z++ )) for all r ∈ [p, q]. p Proof. Let b ∈ (QC ⊗ QC)N ×N and T2 (b) ∈ Φ(HN (T2 )). We show that then p r 2 ) T2 (b) ∈ Φ(HN (T )) for all r ∈ [p, q]. Theorem 8.21 gives that T (biξ ) ∈ GL(HN r for i = 1, 2 and all ξ ∈ M (QC). Hence biξ ∈ GQCN ×N and so T (biξ ) ∈ Φ(HN ) r for all r ∈ [p, q] (Theorem 5.31). The index of T (biξ ) on HN does not depend on p r, since it is −ind {kλ det biξ }. Because T (biξ ) is invertible on HN , Lemma 8.26 i r implies that T (bξ ) ∈ GL(HN ) for all r ∈ [p, q]. Thus, by Theorem 8.21, r (T2 )). T2 (b) ∈ Φ(HN Now let b ∈ P C ⊗ P C and suppose the four operators are in Φ(H p (T2 )). We show that T2 (b) ∈ Φ(H r (T2 )) for all r ∈ [p, q]. From Theorem 8.43 we obtain that T (b1p,ν ) ∈ GL(H p ) and T ((b1p,ν ))) = T ((b2 )1p,ν ) ∈ GL(H p ) for all C 1 q ν ∈ NP p . So Proposition 7.19(a), (c) implies that T (bp,ν ) ∈ Φ(H ) for all PC 1 ν ∈ Np . Now Theorem 5.22 (or 5.47 and 5.48) gives T (bp,ν ) ∈ GL(H r ) for C 2 p all r ∈ [p, q] and all ν ∈ NP p . Using (8.46) we conclude that T (br,ν ) ∈ Φ(H ) C for all r ∈ [p, q] and all ν ∈ NP p . As in the proof of Theorem 8.69 one can 2 see that the index of T (br,ν ) on H p does not depend on r, and since T (b2p,ν ) is invertible, it results that the index of T (b2r,ν ) on H p is zero for all r ∈ [p, q]. So Corollary 2.40 yields the invertibility of T (b2r,ν ) on H p . Since T2 (a) is the transposed operator of T2 (a12 ), we conclude that T2 (b) and T2 (b2 ) are in Φ(H q (T2 )). So the same reasoning with q in place of p gives the invertibility of T (b2r,ν ) on H q . Now Theorem 5.22 (or 5.47 and 5.48) shows that C T (b2r,ν ) belongs to GL(H r ) for all ν ∈ NP p . It can be shown analogously that 1 r PC T (br,ν ) ∈ GL(H ) for all ν ∈ Np . Theorem 8.43 completes the proof. The proofs for the p case are analogous (the arguments based on Theorem 5.22 can be replaced by arguments using the index formula in 6.40 or Lemma 8.68).

470


r 8.71. Corollary. Let 1 < p < ∞, 1/p + 1/q = 1, and Y r = HN (T2 ) (resp. r r 2 Y = N (Z++ )). If b ∈ (QC ⊗ QC)N ×N or b ∈ P C ⊗ P C (N = 1) (resp. b ∈ (Cp ⊗ Cp )N ×N or b ∈ P Cp ⊗ P Cp (N = 1)) and K ∈ C∞ (Y p ), then the following are equivalent.

(i) T2 (b) + K ∈ Π{Y p ; Pn ⊗ Pn }. (ii) T2 (b) + K, T2 (b1 ), T2 (b2 ), T2 (b12 ) ∈ GL(Y p ). (iii) T2 (b) + K, T2 (b1 ), T2 (b2 ), T2 (b12 ) ∈ GL(Y p ) and T2 (b), T2 (b1 ), T2 (b2 ) ∈ Φ(Y r ) ∀ r ∈ [p, q]. Proof. (i) =⇒ (ii): Theorem 8.63. (ii) =⇒ (iii): Proposition 8.70. (iii) =⇒ (i): This follows from Theorem 8.69 for P C, Cp , P Cp . An obvious combination of the arguments used to prove Theorems 8.67 and 8.69 gives the desired implication for b ∈ (QC ⊗ QC)N ×N . 8.72. Open problems. (a) Is Theorem 8.69(a) true with P C replaced by the algebra P QC? (b) Let N > 1 and let b ∈ (P C ⊗ P C)N ×N (resp. b ∈ (P Cp ⊗ P Cp )N ×N ). Which of the implications (i) =⇒ (iii), (ii) =⇒ (iii), (ii) =⇒ (i) of the preceding theorem are true?

8.9 Higher Dimensions 8.73. Definitions. The notations, definitions, and statements of 8.1–8.9 extend to the case of k variables in a natural way. We now put pN (Zk+ ) = p (Z+ × . . . × Z+ ) = pN ⊗ p ⊗ . . . ⊗ p , while the p over a half-space will be denoted by pN (Z+ ×Zk−1 ). For a function p p k k k a ∈ L∞ N ×N (T ) resp. a ∈ MN ×N (T ), the multiplication operator on LN (T ) p p k resp. N (Z ) will be denoted by Mk (a) and the Toeplitz operator on HN (Tk ) resp. pN (Zk+ ) generated by a will be written as Tk (a). p (Tr ) ⊗ Lp (Tk−r ) For 1 ≤ r ≤ k, let Tr,k−r (a) denote the operator on HN p r k−r ) given by resp. N (Z+ × Z r k C C ϕ → P ⊗ I Mk (a)ϕ . j=1

j=r+1

In particular, Tk,0 (a) = Tk (a), T0,k (a) = Mk (a), T1,1 (a) = T+· (a). ∞ ∞ ∼ Let a ∈ (L∞ ⊗L∞ ⊗L∞ )N ×N = L∞ = [C(X ×X ×X)]N ×N . N ×N ⊗L ⊗L For x, y, z ∈ X, define a12 yz (x) = a(y, z, x),

a13 yz (x) = a(y, x, z),

a23 yz (x) = a(x, y, z),

a1z (x, y) = a(z, x, y), a2z (x, y) = a(x, z, y), a3z (x, y) = a(x, y, z)

8.9 Higher Dimensions

471

1 (also recall 8.16). Note that we may think of a12 yz , . . . and az , . . . as matrix ∞ ∞ 2 functions in LN ×N (T) and LN ×N (T ), respectively. Moreover, if, for example, ∞ a ∈ (L∞ ⊗ P C ⊗ P C)N ×N , then a23 yz ∈ LN ×N is the same function for all y ∈ Xα , z ∈ Xβ (α, β ∈ M (P C)); this function will be denoted by a23 αβ . These notations extend to the case of k variables in the natural manner. In analogy to 8.17 define

Lπ1 = L ⊗ L ⊗ L/C∞ ⊗ L ⊗ L,

Lπ2 = L ⊗ L ⊗ L/L ⊗ C∞ ⊗ L,

Lπ3 = L ⊗ L ⊗ L/L ⊗ L ⊗ C∞ ,

Lπ123 = L ⊗ L ⊗ L/C∞ ⊗ C∞ ⊗ C∞ ,

and for A ∈ L ⊗ L ⊗ L let Aπ1 , Aπ2 , Aπ3 , and Aπ123 denote the coset in the corresponding quotient algebra containing A. If A ∈ L ⊗ L ⊗ L and (B1 )π1 , (B2 )π2 , (B3 )π3 are the inverses of Aπ1 , Aπ2 , Aπ3 , then (B1 + B2 + B3 − B1 AB2 − B1 AB3 − B2 AB3 + B1 AB2 AB3 )π123 is the inverse of Aπ123 . This can be proved in

the same way as Lemma 8.18. Let A, A, NA i ⊗ Di ∈ A " (AN ×N ⊗ A) p be as in 8.37(a). For B = i C

1 1 π define B ∈ A ⊗ A by B = and ν ∈ NA N ×N p p,ν p,ν i (Γp Ci )(ν)Di . The same 1 reasoning as in the proof Lemma 8.39 shows that Bp,ν ≤ Bess , and this 1 is well defined for every B ∈ A ⊗ (AN ×N ⊗ A) = [A ⊗ implies that Bp,ν 2 3 and Bp,ν can be defined and these definitions A ⊗ A]N ×N . Analogously Bp,ν extend to the case of k dimensions in a natural fashion. Finally, what was said in 8.16(b)–(d), 8.37(c), and 8.39(b) extends in a natural way to the case of k variables, too. 8.74. Theorem. Let A be a C ∗ -subalgebra of L∞ containing the constants resp. A ∈ {Cp , P Cp } and let H p (1 < p < ∞) resp. p (1 ≤ p < ∞ for A = Cp and 1 < p < ∞ for A = P Cp ) be the underlying space. Let k k C C A resp. b ∈ M p (Tr ) ⊗ A , b ∈ L∞ (Tr ) ⊗ j=r+1

N ×N

j=r+1

N ×N

r+1,...,k is invertible for all (αr+1 , . . . , αk ) in where 1 ≤ r ≤ k − 1. If Tr aα r+1 ,...,αk [M (A)]k−r , then Tr,k−r (a) is invertible in p L(HN (Tr )) ⊗ L(Lp (Tk−r ))

resp.

L(pN (Zr+ )) ⊗ L(p (Zk−r ))

resp.

L(pN (Zr+ ) ⊗ p (Zk−r ))

and, consequently, also in p L(HN (Tr ) ⊗ Lp (Tk−r ))

Proof. We only consider the H p case. The choice of A implies that k C j=r+1

A∼ = C([M (A)]k−r )

472


(Proposition 8.6). For α = (αr+1 , . . . , αk ) ∈ [M (A)]k−r let Rα denote the k D A such that 0 ≤ ϕ ≤ 1 and ϕ is identically 1 in collection of all ϕ ∈ j=r+1

some open neighborhood of α. Put Mα = {I ⊗ Mk−r (ϕ) : ϕ ∈ Rα } and then proceed as in the proof of Proposition 8.19. k D ∞ B (k ≥ 2). 8.75. Corollary. Let B be C or QC and let a ∈ L ⊗ j=2 ∞ is locally p, 1-sectorial over QC for each (β2 , . . . , βk ) Suppose a2...,k β2 ,...,βk ∈ L k−1 . Let {Kλ }λ∈Λ be any approximative identity and put in [M (B)] 2,...,k ind a2,...,k β2 ,...,βk = ind kλ aβ2 ,...,βk .

(a) The integer ind a2,...,k β2 ,...,βk does not depend on the particular choice of k−1 ; it will therefore be simply denoted by ind1 a. (β2 , . . . , βk ) ∈ [M (B)] (b) The operator T1,k−1 (a) is left but not right (right but not left, resp. twosided) invertible on H p (T) ⊗ Lp (Tk−1 ) if and only if ind1 a < 0 (ind1 a > 0, resp. ind1 a = 0). 2,...,k Proof. (a) Notice that ind kλ a2,...,k β2 ,...,βk equals minus the index of T aβ2 ,...,βk on H 2 . Since the mapping [M (B)]k−1 → Z, (β2 , . . . , βk ) → Ind T a2,...,k β2 ,...,βk is obviously continuous, to get the assertion it suffices to show that M (B) is connected. But this is trivial for B = C and was established in 3.77 for B = QC. 2,...,k (b) Put κ := ind1 a and b(t1 , . . . , tk ) := t−κ 1 a(t1 , . . . , tk ). Then T bβ2 ,...,βk is Fredholm on H p (Theorem 5.16(c)) and has index zero (Proposition 5.18 for B = C and Section 5.19 for B = QC). Consequently, T b2,...,k β2 ,...,βk is in GL(H p ). So the previous theorem implies that T1,k−1 (b) is invertible. Since T1,k−1 (a) equals (V (−|κ|) ⊗ I)T1,k−1 (b) for κ ≤ 0 and T1,k−1 (b)(V κ ⊗ I) for κ ≥ 0, and since V (−|κ|) ⊗ I is invertible from the right but not from the left if κ < 0 and V κ ⊗ I is invertible from the left but not from the right if κ > 0, we get the assertion. Theorem 8.74 states sufficient conditions for the invertibility of Tr,k−r (a). Theorem 8.77 and Corollary 8.80 will show that these conditions are also necessary ones for certain symbol classes. p k k p 8.76. Theorem. Let k > 1, let a ∈ L∞ N ×N (T ) resp. MN ×N (T ), and let H p (1 < p < ∞) resp. (1 ≤ p < ∞) be the underlying space. If the operator Tr,k−r (a) is Fredholm, then the operator Ts,k−s (a) is invertible for all s < r. k The operator Mk (a) = T0,k (a) is invertible if and only if a ∈ GL∞ N ×N (T ) p k resp. a ∈ GMN ×N (T ).

8.9 Higher Dimensions

473

Proof. This theorem can be proved by the same arguments as in the proof of Theorem 8.13. 8.77. Theorem. Let B be a C ∗ -algebra between C and QC resp. B = Cp and let H p (1 < p < ∞) resp. p (1 < p < ∞) be the underlying space. If k D a∈ B and 1 ≤ r ≤ k − 1, then the following are equivalent. j=1

N ×N

(i) Tr,k−r (a) is Fredholm. (ii) Tr,k−r (a) is invertible. (iii) Tr,k−r (a) is invertible in p (Tr )) ⊗ L(Lp (Tk−r )) resp. L(pN (Zr )) ⊗ L(p (Zk−r )). L(HN

is invertible for all (βr+1 , . . . , βk ) ∈ [M (B)]k−r . (iv) Tr aβr+1,...,k r+1 ,...,βk Proof. The implications (iv) =⇒ (iii) =⇒ (ii) =⇒ (i) result from Theorem 8.74 and the implication (i) =⇒ (iv) can be proved as in 8.20. 8.78. Corollary. Let the hypotheses of the previous theorem be fulfilled. Then Tk (a) is Fredholm if and only if Tk−1 (ajβ ) is invertible for all j ∈ {1, . . . , k} and all β ∈ M (B). Proof. The proof uses the two preceding theorems and is similar to the proof of Theorem 8.21. Remark. This corollary is also valid for B = W and 1 as the underlying space (recall Remark 2 of 8.20). k D , B, N be as in 8.37(a), let B ∈ A 8.79. Theorem. Let A, A, NA p p s=1 N ×N k D p p resp. B ∈ B , and let H resp. (1 < p < ∞) be the underlying s=1

N ×N

space. Then the following are equivalent. (i) B is Fredholm. π π ∈ GAπ1,...,k resp. B1,...,k ∈ GBπ1,...,k . (ii) B1,...,k i is invertible for all i ∈ {1, . . . , k} and all ν ∈ NA (iii) Bp,ν p resp. ν ∈ Np . k k D D If B is not invertible, then there exist Un in A resp. B s=1

N ×N

such that Un = 1 and Un B → 0 or BUn → 0 as n → ∞.

s=1

N ×N

Proof. Again we only consider the H p case. (iii) =⇒ (ii). This can be proved by induction with respect to k. Theorem 8.43 is the assertion for k = 2. Let k = 3. Put

474


Aπ1 := A ⊗ AN ×N ⊗ A/C∞ (H p ) ⊗ AN ×N ⊗ A. Let Uπ1 denote the closure in the algebra Aπ1 of the set of all elements of the form (E ⊗ I ⊗ I)π1 , where E ∈ A, and let MA p denote the maximal ideal space π of U1 . It can be shown as in 8.42 that to each µ ∈ MA p there corresponds a π such that the closed two-sided ideal of A generated by µ is of the ν ∈ NA p 1 form J1µ = closidAπ1 (E ⊗ I ⊗ I)π1 : (Γp E π )(ν) = 0 . As in the proof of Theorem 8.43 one can verify that 1 (B − I ⊗ Bp,ν )π1 ∈ J1µ . 1 1 has a regularizer in AN ×N ⊗ A if Bp,ν is Theorem 8.43 implies that Bp,ν p 2 1 Fredholm on HN (T ). If Bp,ν is invertible, then the inverse differs from a regularizer only by a compact operator, and hence the inverse is in AN ×N ⊗A, 1 )−1 )π1 + J1µ is the inverse of B1π + J1µ , and so too. Thus, it follows that (I ⊗ (Bp,ν the “multilocal” version of what was said in 8.42 gives the wanted implication. Replacing in the above argument Theorem 8.43 by what has just been proved for k = 3 we get the implication for k = 4 etc.

(ii) =⇒ (i). Trivial. We now show that B is a topological divisor of zero if it is not invertible. For k = 1 this is Proposition 8.38(b). Let k = 2. First assume B is not even Fredholm. Then, by the implication (iii) =⇒ (i), there is a ν ∈ NA p such that 1 2 1 is not invertible (the proof is analogous for B in place of B B p,ν p,ν ). Choose

p,ν m C ⊗ D ∈ A " A so that B − C ⊗ D < 1/n. Lemma 8.39 i i i i i=1 i

N ×N 1 − i (Γp Ciπ )(ν)Di < 1/n, hence shows that Bp,ν " " 1 " " 1 − (Γp Ciπ )(ν)I ⊗ Di " < , "I ⊗ Bp,ν n i and so, if we let Ei := Ci − (Γp Ciπ )(ν)I, 1 + B = I ⊗ Bp,ν

Ei ⊗ Di + B ,

B
0), 0

0

while the approach of Pilidi [393], [394], a student of Simonenko, also applied to operators of the form ( ∞ ( ∞ ϕ(t, s) → cϕ(t, s) + k1 (t − τ )ϕ(τ, s) dτ + k2 (s − σ)ϕ(t, σ) dσ 0 ( ∞ ( ∞0 + K(t − τ, s − σ) dτ dσ (t, s > 0). 0

0

Theorem 8.43 was established by Duduchava [172] for p = 2 and A = B = P C and in B¨ ottcher [63], [64] in the form presented here. We remark that Duduchava’s method [172] does not extend to the case p = 2. In B¨ ottcher, Silbermann [105] we localized over M (alg T π (P C)) and made essential use of C ∗ -algebra techniques; the observation that one can also localize over algebras whose maximal ideal space coincides with the Shilov boundary and the approach presented here are due to B¨ ottcher [63], [67]. The proof of Theorem 8.38(b) uses an idea of Hartmut Wolf (private communication). Seybold [464] proved some sufficient conditions for Fredholmness on the space p (v) ⊗ p (w) of the operators T (a1 ) ⊗ T (b1 ) + . . . + T (ak ) ⊗ T (bk ) with aj ∈ P Cp,v and bj ∈ P Cp,w , where P Cp,v and P Cp,w are the sets of piecewise continuous multipliers on the spaces p (v) and p (w) with arbitrary discrete Muckenhoupt weights v and w. Under natural additional hypotheses these conditions are also necessary. 8.45. These results are from Böttcher [71]. 8.46. See Sazonov [459], [460], [461] for details. 8.47–8.54. These results, in particular Theorem 8.53, are new. The observation in the remark to Theorem 8.54 was first made in B¨ ottcher [70].


479

8.55–8.60. Theorem 8.59 was proved by Kozak [317]–[320] using local techniques (apart from the short communications [317], [318], Kozak did not publish proofs, and so his approach was known only to the people to whom his dissertation [319] was available; only in 1983 he published a part of his dissertation in [320]). The proof of Theorem 8.59 given here as well as the material of 8.57 and 8.58 are taken from Kozak, Simonenko [321]. For still another proof see Gorodetsky [244]. 8.61. The problem of whether Toeplitz operators with piecewise continuous symbols are of local type had been open for a long time. The surprisingly simple argument of 8.61, which shows that the answer is negative, was communicated to us by S. Roch. 8.62–8.72. These results were established in Böttcher, Silbermann [105] for p = 2 and symbols in P C ⊗ P C and in B¨ ottcher [63], [64], [67] for general p and symbols in P Cp ⊗ P Cp . The approach presented here is taken from Böttcher [64], [67]. 8.73–8.81. Note that it is in general not easy (or even trivial) to extend results on two-dimensional Toeplitz operators to higher dimensions. Results like 8.74 and 8.75(b) (for the case of local sectoriality over C) were established by Speck [499] using other methods. Theorem 8.77 and Corollary 8.78 (for B = C resp. B = Cp ) are essentially due to Simonenko [495] and Douglas, Howe [163]. Corollary 8.80 for p = 2 and A = P C was stated (but not proved) by Duduchava [171]. The proof given here is the author’s. For 8.81 see Kozak [317]–[320] and B¨ ottcher, Silbermann [105].

9 Wiener-Hopf Integral Operators

9.1 Basic Properties 9.1. Function spaces on the line. (a) Throughout this chapter we let Lp and Lp+ (1 ≤ p ≤ ∞) refer to the Lp spaces of Lebesgue measure on R and R+ , respectively. The Lp spaces on the unit circle will be denoted by Lp (T). The operator P defined by P : Lp → Lp+ , ϕ → ϕ|R+ is clearly bounded for 1 ≤ p ≤ ∞. (b) Let R˙ and R be the compactifications of R by means of the point ∞ and the two points ±∞, respectively. The space of continuous functions on R that have finite limits at −∞ and +∞ is C(R), and we have ˙ := a ∈ C(R) : a(−∞) = a(+∞) . C(R) Let C stand for the constant functions on R and C0 (R) for the continuous functions on R which vanish at ±∞. Notice that C, C0 (R), C(R) are closed subalgebras of L∞ = L∞ (R). (c) For ϕ ∈ L1 , let F ϕ denote the Fourier transform: ( ∞ (F ϕ)(x) := ϕ(t)eixt dt (x ∈ R). −∞

If ϕ ∈ L1 then F ϕ ∈ C0 (R) and F ϕL∞ ≤ ϕL1 . The collection of all functions of the form c + F ϕ with c ∈ C and ϕ ∈ L1 is denoted by W (R). It is well known that W (R) is a Banach algebra under the norm c + F ϕ := |c| + ϕL1 . It is usually called the Wiener algebra. √ (d) If ϕ ∈ L1 ∩ L2 , then F ϕ ∈ L2 and F ϕL2 = 2πϕL2 . Since L1 ∩ L2 is dense in L2 , F extends to a bounded operator of L2 onto L2 , which will √ also be denoted by F . We have F ϕL2 = 2πϕL2 for all ϕ ∈ L2 , and the inverse of F ∈ GL(L2 ) is given by

482


(F −1 ϕ)(t) =

1 (F ϕ)(−t) 2π

a.e. on R.

(e) The mapping U defined by √

U : L2 (T) → L2 (R),

(U ϕ)(x) :=

2 ϕ i+x

i−x i+x

(x ∈ R)

is easily seen to be an isometric isomorphism. Its inverse acts by the rule √ 1−t i 2 −1 2 2 −1 ϕ i U : L (R) → L (T), (U ϕ)(t) := (t ∈ T◦ ), 1+t 1+t where T◦ := T \ {−1}. For a ∈ L∞ (T), define a# = U# a by i−x # a (x) = (U# a)(x) := a (x ∈ R). i+x It is clear that U# is an isometric isomorphism between L∞ (T) and L∞ (R) ˙ as well as between C(T) and C(R). (f) Let U H 2 (T) (resp. U# H ∞ (T)) denote the image of H 2 (T) ⊂ L2 (T) (resp. H ∞ (T) ⊂ L∞ (T)) under the mapping U (resp. U# ), and let F L2+ denote the image of L2+ := {ϕ ∈ L2 : ϕ(x) = 0 for x < 0} under the operator F . The harmonic extension of a function ϕ ∈ Lp (1 ≤ p ≤ ∞) into the upper half-plane Im z > 0 is given by ( y 1 ∞ ϕ(t) dt (x ∈ R, y > 0). ϕ(x + iy) = π −∞ (x − t)2 + y 2 A function ϕ ∈ Lp is said to belong to H p (R) if its harmonic extension is an analytic function in {z ∈ C : Im z > 0}. One can show that the following equalities hold: U H 2 (T) = F L2+ = H 2 (R),

U# H ∞ (T) = H ∞ (R).

(9.1)

9.2. Multiplication operators. If a ∈ L∞ , then the operators defined by m(a) : L2 → L2 ,

ϕ → aϕ,

MR (a) : L2 → L2 ,

ϕ → F −1 m(a)F ϕ

are obviously bounded and their norms equal a∞ . Let M p (R) (1 ≤ p < ∞) denote the collection of all functions a ∈ L∞ having the following property: if ϕ ∈ L2 ∩ Lp , then MR (a)ϕ ∈ Lp and MR (a)ϕLp ≤ cp ϕLp with some constant cp independent of ϕ. If a ∈ M p (R), then MR (a) : L2 ∩ Lp → Lp extends to a bounded operator on Lp , which will also be denoted by MR (a) and which is called the multiplication operator (or the convolution operator) on Lp with symbol a. An operator A ∈ L(Lp ) is said to be translation invariant if Aτv = τv A for all v ∈ R, where τv ∈ L(Lp ) is defined by (τv ϕ)(t) := ϕ(t − v) (t ∈ R). If

9.1 Basic Properties

483

a ∈ M p (R) (1 ≤ p < ∞), then the operator MR (a) is translation invariant. Moreover, it can be shown that every translation invariant operator A ∈ L(Lp ) (1 ≤ p < ∞) is of the form A = MR (a) with some a ∈ M p (R) (this is an analogue of Proposition 2.2). Examples. (a) If a = c + F k ∈ W (R) (c ∈ C, k ∈ L1 ), then a ∈ M p (R) for all p ∈ [1, ∞) and MR (a) acts on Lp by the rule ( ∞ MR (a)ϕ (t) = cϕ(t) + k(t − s)ϕ(s) ds (t ∈ R). −∞

(b) For ξ ∈ R, define the function sgnξ ∈ L∞ by sgnξ (x) = 1 for x > ξ and sgnξ x = −1 for x < ξ. It can be shown that sgnξ ∈ M p (R) for all p ∈ (1, ∞) and that MR (sgnξ ) is given on Lp by −1 MR (sgnξ )ϕ (t) = πi

(

∞

−∞

eiξ(s−t) ϕ(s) ds s−t

(t ∈ R),

where the integral is understood in the Cauchy principal value sense. (c) For δ ∈ R, define ωδ ∈ L∞ by ωδ (x) = eiδx . Then ωδ ∈ M p (R) for all p ∈ [1, ∞) and if ϕ ∈ Lp , then MR (ωδ )ϕ (t) = ϕ(t − δ) (t ∈ R). (d) Let a ∈ L∞ (T) and define a# ∈ L∞ (R) as in 9.1(e). Then MR (a# ) ∈ L(L2 ) can be represented in the form MR (a# ) = F −1 U M (a)U −1 F, where U ±1 are as in 9.1(e) and M (a) is given on L2 (T) as in 2.1. 9.3. Basic properties of M p (R). Let 1 < p < ∞ and 1/p + 1/q = 1. For a ∈ L∞ define a ∈ L∞ by a(x) = a(x) (x ∈ R). (a) M 1 (R) = W (R), M 2 (R) = L∞ . (b) If a ∈ M p (R), then a ∈ M q (R) and the adjoint MR∗ (a) ∈ L(Lq ) of MR (a) ∈ L(Lp ) equals MR (a). (c) M p (R) = M q (R), and if a ∈ M p (R) then MR (a)L(Lp ) = MR (a)L(Lq ) . (d) If a ∈ M p (R), then a ∈ M r (R) for all r ∈ [p, q] and aL∞ ≤ MR (a)L(Lr ) ≤ MR (a)L(Lp ) ≤ aW (R) , γ MR (a)L(Lr ) ≤ a1−γ L∞ MR (a)L(Lp ) ,

where γ =

p|r − 2| . r|p − 2|

484


(e) If a ∈ L∞ has finite total variation V1 (a), then a ∈ M p (R) for all p ∈ (1, ∞) and MR (a)L(Lp ) ≤ cp aL∞ + V1 (a) , where cp := MR (sgn0 )L(Lp ) (recall example (b) in 9.2). (f) M p (R) is a Banach algebra under the norm aM p (R) := MR (a)L(Lp ) . 9.4. Wiener-Hopf integral operators. For a ∈ M p (R) (1 ≤ p < ∞), the Wiener-Hopf integral operator on Lp+ with symbol a is the operator W (a) defined by W (a) : Lp+ → Lp+ , ϕ → P MR (a)ϕ. Clearly, W (a)L(Lp+ ) ≤ aM p (R) . In particular, if a, sgnξ , ωδ are as in the examples (a)–(c) of 9.2, then ( ∞ W (a)ϕ (t) = cϕ(t) + k(t − s)ϕ(s) ds (t > 0), 0

(

1 ∞ eiξ(s−t) W (−sgnξ )ϕ (t) = ϕ(s) ds (t > 0), πi 0 s−t ! ϕ(t − δ) for max{δ, 0} < t, W (ωδ )ϕ (t) = 0 for 0 < t < max{δ, 0}. p p p For a ∈ MN ×N (R), the operator W (a) is defined on (L+ )N = LN (R+ ) in the natural manner.

9.5. General properties of Wiener-Hopf integral operators. (a) For 1 ≤ p < ∞, the mapping p p W : MN ×N (R) → L(LN (R+ )),

a → W (a)

is a 1-submultiplicative isometry. Proof. For v ∈ R, define τv as in 9.2. Then, as in the proof of Proposition 4.1, " " " " " " " " W (ajk )" ≥ " P MR (ajk )P " " j

j

k

k

" " " " =" (τ−v P τv )MR (ajk )(τ−v P τv )" j

k

" " " " ≥ "MR ajk " (τ−v P τv → I strongly as v → +∞) j

k

" " " " ajk ", ≥ "W j

k

which yields the 1-submultiplicativity. Because, in particular, W (a) ≥ MR (a) ≥ W (a), it follows that W (a) = MR (a) = aMNp ×N (R) .

9.1 Basic Properties

485

p p p (b) If a ∈ MN ×N (R), then W (a)L(LN (R+ )) = W (a)Φ(LN (R+ )) .

Proof. The proof of Proposition 4.4(a) with V (±n) replaced by P τ±v P applies. p (c) If 1 ≤ p < ∞ and a ∈ MN ×N (R), then p MR (a) ∈ Φ(LpN ) ⇐⇒ MR (a) ∈ GL(LpN ) ⇐⇒ a ∈ GMN ×N (R), p p W (a) ∈ Φ(LN (R+ )) =⇒ a ∈ GMN ×N (R).

Proof. Let A ∈ L(LpN ) be the inverse of MR (a). Multiplying the identity MR (a)τv = τv MR (a) from the left and the right by A we obtain that τv A = Aτv , i.e., that A is translation invariant. From what was said in 9.2 we p deduce that A = MR (b) with b ∈ MN ×N (R) and it is not difficult to show that p ab = ba = I. This proves that a ∈ GMN ×N (R) whenever MR (a) is invertible. Since τv ϕ ∈ Ker MR (a) for all v ∈ R if ϕ ∈ Ker MR (a), a reasoning similar to the one in the proof of Proposition 2.29 shows that MR (a) is invertible if it is Fredholm. Finally, the proof of Theorem 2.30 with the “shifts” U ±n replaced by the “translations” τ±v (v ∈ R) implies that MR (a) is invertible if W (a) is Fredholm. (d) Let 1 < p < ∞ and suppose a ∈ M p (R) does not vanish identically. Then the operator W (a) has a trivial kernel or a dense range in Lp+ . In particular, if W (a) ∈ Φ(Lp+ ) then W (a) ∈ GL(Lp+ ) if and only if Ind W (a) = 0. Proof. The proof is similar to the proof of Theorem 2.38(b). For details see Duduchava [173, Proposition 2.8]. √ −1 is an isometric isomorphism of L2N (R+ ) (e) The mapping (1/ 2π)U √ F −1 2 onto HN (T) whose inverse is 2πF U . If a ∈ L∞ N ×N , then −1 W (a) = F −1 U T (U# a)U −1 F,

1−t −1 a)(t) = a i where (U# (t ∈ T). 1+t Proof. Immediate from 9.1(e), (f) and example 9.2(d). Remark. This result shows that many questions concerning the invertibility and the Fredholm theory of Wiener-Hopf integral operators on L2N (R+ ) can 2 (T) ∼ be reduced to the corresponding questions for Toeplitz operators on HN = 2 N (Z+ ). p (f) Let a− , a+ ∈ (H ∞ (R) ∩ M p (R))N ×N and b ∈ MN ×N (R). Then

W (a− )W (b)W (a+ ) = W (a− ba+ ).

486


−1 −1 Proof. From 9.2(e), (f) we infer that U# a+ ∈ H ∞ (T) and U# a− ∈ H ∞ (T). So Proposition 2.14(b) and the above property (e) give that

W (a− )W (b)W (a+ )ϕ = W (a− ba+ )ϕ ∀ ϕ ∈ L2+ ∩ Lp+ , which by continuity extends to all ϕ ∈ Lp+ . 9.6. Hankel integral operators. Let J denote the flip operator on the space Lp (1 ≤ p ≤ ∞): (Jϕ)(x) := ϕ(−x) (x ∈ R). Put Q := I − P , i.e., for ϕ ∈ Lp (1 ≤ p ≤ ∞) define Qϕ ∈ Lp (R− ) by Qϕ = ϕ|R− . For a ∈ M p (R), a) on Lp+ the (obviously bounded) Hankel integral operators HR (a) and HR () (1 ≤ p < ∞) are defined by HR (a) : Lp+ → Lp+ , HR () a) : Lp+ → Lp+ ,

ϕ → P MR (a)QJϕ, ϕ → JQMR (a)P ϕ.

If a, b ∈ M p (R), then W (ab) = P MR (ab)P = P MR (a)(P 2 + QJJQ)MR (b)P = P MR (a)P P MR (b)P + P MR (a)QJJQMR (b)P = W (a)W (b) + HR (a)HR ()b). Let a = c + F k ∈ W (R). It is easily seen that ( ∞ HR (a)ϕ (t) = k(t + s)ϕ(s) ds (t > 0), 0 ( ∞ HR () a)ϕ (t) = k(−t − s)ϕ(s) ds (t > 0). 0

The kernel k(t) (t > 0) can be approximated in the L1 norm by continuous functions with finite support as closely as desired, and these functions can be (uniformly) approximated by functions of the form e−t p(t) (t > 0), where p is a polynomial, as closely as desired. Since, for n ∈ Z+ , the operator ( ∞ ϕ(t) → e−t−s (t + s)n ϕ(s) ds (t > 0) 0

has finite rank, it follows that HR (a), HR () a) are compact on Lp+ (1 ≤ p < ∞) whenever a ∈ W (R).

9.2 Continuous Symbols ˙ denote the closure of ˙ For 1 ≤ p < ∞, let Cp (R) 9.7. The algebra Cp (R). p ˙ W (R) in M (R). Note that Cp (R) is obviously a closed subalgebra of M p (R). ˙ = W (R) and C2 (R) ˙ = C(R). ˙ Since the maximal ideal space of Clearly, C1 (R)


487

˙ (Wiener-Lévy), a similar reasoning as in the proof W (R) is known to be R of Proposition 2.46(a) gives that for each p ∈ [1, ∞) the maximal ideal space ˙ can be identified with R˙ and that the Gelfand map Γ : Cp (R) ˙ → of Cp (R) ˙ ˙ C(R) is given by (Γ a)(x) = a(x). Because R is naturally homeomorphic to ˙ An analogous argument as in the T, we may define ind a for a ∈ GC(R). ˙ proof of Proposition 2.46(b) shows that the connected component of GCp (R) ˙ containing the identity consists precisely of the functions in GCp (R) with ind a = 0. ˙ and a ∈ M p (R) (1 ≤ p < ∞). Then 9.8. Lemma. Let c ∈ Cp (R) W (ac) − W (a)W (c) ∈ C∞ (Lp+ ). ˙ Proof. Immediate from what was said in 9.6 and from the definition of Cp (R). 9.9. The algebra alg W (Cp ). Let alg W (Cp ) (1 ≤ p < ∞) denote the ˙ smallest closed subalgebra of L(Lp+ ) containing the set {W (c) : c ∈ Cp (R)}. p We claim that C∞ (L+ ) ⊂ alg W (Cp ) if 1 0) and define ψn (n = 1, 2, . . .) as ψn := V n ψ0 . Then ∆k ψn = δkn ψn (δkn the Kronecker delta). For f ∈ Lp+ and g ∈ Lq+ (1/p + 1/q = 1) put ( ∞ (f, g) := f (t)g(t) dt. 0

Notice that (ψj , ψk ) = δjk . The functions ψ0 , ψ1 , ψ2 , . . . are nothing but the Laguerre functions on R+ and it is well known that the linear hull of {ψ0 , ψ1 , ψ2 , . . .} is dense in Lp+ for all p ∈ [1, ∞) (however, note that, by a result of Askey, Wainger [10], the Laguerre functions form a basis in Lp+ if and

n (n) only if 4/3 < p < 4). Hence, if ϕ ∈ Lp+ and j=0 αj ψj → ϕ (n → ∞), then ∆k ϕ = lim

n→∞

= lim

n→∞

(n)

n→∞

j

(n)

αj ∆k ψj = lim αk ψk (n)

αj (ψj , ψk )ψk = (ϕ, ψk )ψk ,

j

which implies that the operator ϕ → (ϕ, ψk )ψk is in alg W (Cp ). Since (ϕ, ψk )ψk+n = (ϕ, ψk )V n ψk = V n ∆k ϕ, (ϕ, ψk+n )ψk = (ϕ, ψk+n )V (−n) ψk+n = V (−n) ∆k+n ϕ,

488


it results that the operators ϕ → (ϕ, ψn )ψm are in alg W (Cp ) for all n, m ≥ 0. Using the density of lin{ψ0 , ψ1 , . . .} in Lp+ and Lq+ it is readily verified that the operator ϕ → (ϕ, f )g belongs to alg W (Cp ) for each f ∈ Lq+ and each g ∈ Lp+ , and because every compact operator on Lp+ can be uniformly approximated by finite-rank operators, we obtain that C∞ (Lp+ ) is contained in alg W (Cp ). Denote the coset of the quotient algebra alg W (Cp )/C∞ (Lp+ ) (1 < p < ∞) containing A by Aπ . Because the quotient algebra is generated by the set ˙ it will be denoted by alg W π (Cp ). Lemma 9.8 shows {W π (c) : c ∈ Cp (R)}, π that alg W (Cp ) is commutative. 9.10. Theorem. Let 1 < p < ∞. ˙ If (a) The maximal ideal space M (alg W π (Cp )) is homeomorphic to R. π ˙ then the Gelfand transform of W (c) is given by c ∈ Cp (R), ˙ (Γ W π (c))(x) = c(x) (x ∈ R). (b) ∂S M (alg W π (Cp )) = M (alg W π (Cp )). p (c) If A ∈ alg W (CN ×N ), then

A ∈ Φ(LpN (R+ )) ⇐⇒ Γ (det A)π (x) = 0

˙ ∀ x ∈ R.

If A is Fredholm, then Ind A = −ind Γ (det A)π . Proof. The proof is similar to the proofs of Corollary 4.8(b), (c) and Theorem 6.12. Note that part (c) is also true for p = 1 if one defines Γ (det A)π e.g. as the Gelfand transform of det A as operator on L2+ . 9.11. Remark. Notice that ˙ + H ∞ (R) = U# (C(T) + H ∞ (T)). C(R) Hence, taking into consideration 9.5(e), many results for Toeplitz operators on H 2 (T) with C(T) + H ∞ (T) symbols can be immediately translated into ˙ + H ∞ (R) results for Wiener-Hopf integral operators on L2 (R+ ) with C(R) symbols.

9.3 Piecewise Continuous Symbols In what follows let 1 < p < ∞ and 1/p + 1/q = 1. 9.12. The algebra P Cp (R). Let P K(R) be the collection of all piecewise constant functions on R having only a finite number of jumps and let P Cp (R) denote the closure of P K(R) in M p (R) (note that P K(R) ⊂ M p (R) due to


489

9.3(e)). A function a ∈ P Cp (R) possesses finite limits a(x ± 0) for each x ∈ R and the limits a(−∞) := lim a(x),

a(+∞) := lim a(x)

x→−∞

x→∞

exist and are finite, too. We remark that P C2 (R) = U# P C(T) (recall 2.79 and 9.1(e)) and that C(R) ⊂ P C2 (R). The algebra P C2 (R) will be denoted by P C(R). ˙ The maximal P Cp (R) is a closed subalgebra of M p (R) containing Cp (R). ˙ ideal space of P Cp (R) can be identified with R × {0, 1} (provided with an exotic topology), and the Gelfand transform of a ∈ P Cp (R) is given by (Γ a)(x, 0) = a(x − 0), (Γ a)(∞, 0) = a(+∞),

(Γ a)(x, 1) = a(x + 0) for (Γ a)(∞, 1) = a(−∞)

x ∈ R,

(see Proposition 6.28). 9.13. Definitions. Define γp ∈ C(R) by γp : R → C,

µ → coth

πi q

+ πµ .

We have γp (−∞) = −1 and γp (+∞) = +1. If µ runs from −∞ to +∞, then γp (µ) runs along a circular arc joining −1 and +1. From the points of this arc the segment [−1, 1] is seen under the angle 2π/ max{p, q}, and the arc is located on the right (resp. left) of [−1, 1] if 1 < p < 2 (resp. 2 < p < ∞). Note that γ2 (µ) = tanh(πµ). Put σp (µ) :=

πi 1 1 + coth + πµ (µ ∈ R). 2 2 p

Given a ∈ P Cp (R) define ap : R˙ × R → C by ap (x, µ) := 1 − σp (µ) a(x − 0) + σp (µ)a(x + 0), where, and this is a convention of significant importance, a(∞ − 0) := a(−∞),

a(∞ + 0) := a(+∞).

The range of ap is a continuous closed curve with a natural orientation. It is obtained from the (essential) range of a by filling in the arcs Aq (a(x − 0), a(x + 0)) for x ∈ R and by filling in the arc Ap (a(+∞), a(−∞)) if a has a jump at ∞ (for the definition of Ar (z1 , z2 ) see 5.12). If ap (x, µ) = 0 for all (x, µ) ∈ R˙ × R, this curve has a well-defined index (with respect to the origin), which will be denoted by ind ap .

490


9.14. Lemma. The function γp is in P Cp (R) and MR (γp ) acts on Lp by the rule ( 1 ∞ kp (s − t)ϕ(s) ds (t ∈ R), (MR (γp )ϕ)(t) = πi −∞ where kp (t) := exp(−t/p)/(1−exp(−t)) and the integral is taken in the Cauchy principal value sense. Proof. For a proof of this (nontrivial) result see Duduchava [173, pp. 24–25 and pp. 51–52]. In this connection note that example 9.2(b), though being “well known,” is also nontrivial; it is proved in the above reference on pp. 24–25. 9.15. Proposition. Let a ∈ P Cp (R). Then W (a) ∈ Φ(Lp+ ) ⇐⇒ ap (x, µ) = 0

∀ (x, µ) ∈ R˙ × R.

If W (a) ∈ Φ(Lp+ ), then Ind W (a) = −ind ap . ˙ let Nξ denote the Proof. Suppose ap does not vanish on R˙ × R. For ξ ∈ R, ˙ collection of all piecewise linear functions in C(R) which are identically 1 in some open neighborhood U of ξ, vanish outside some other open neighborhood ˙ V ⊃ U of ξ, and are linear on V \ U . From 9.3(e) we see that Nξ ⊂ Cp (R). p p p π Let A := L(L+ )/C∞ (L+ ) and for b ∈ M (R) let W (b) denote the coset W (b) + C∞ (Lp+ ). Put Mξ := {W π (c) : c ∈ Nξ }. Lemma 9.8 and Theorem 9.10 imply that {Mξ }ξ∈R˙ is a covering system of localizing classes in A, and since cL∞ = 1 and V1 (c) = 2 for c ∈ Nξ , we deduce from 9.3(e) that each Mξ is a bounded subset of A. ˙ Choose an ∈ P K(R) so that a − an M p (R) → 0 as n → ∞. For ξ ∈ R, ξ define an ∈ P K(R) by aξn (x) = an (ξ − 0) (x < ξ), a∞ n (x) = an (+∞) (x > 0),

aξn (x) = an (ξ + 0) (x > ξ) if a∞ n (x) = an (−∞) (x < 0).

ξ ∈ R,

Using Lemma 9.8 it can be easily verified that W π (aξn ) is Mξ -equivalent to W π (an ) from both the left and the right. Define aξ ∈ P K(R) by aξ (x) = a(ξ − 0) (x < ξ), a∞ (x) = a(+∞) (x > 0),

aξ (x) = a(ξ + 0) (x > ξ) if a∞ (x) = a(−∞) (x < 0).

ξ ∈ R,

Because aξ − aξn L∞ = max |a(ξ + 0) − an (ξ + 0)|, |a(ξ − 0) − an (ξ − 0)| , V1 (aξ − aξn ) = |a(ξ + 0) − an (ξ + 0) + an (ξ − 0) − a(ξ − 0)| go to zero as n → ∞, we obtain from 9.3(e) that aξ − aξn M p (R) → 0 as n → ∞. So Lemma 7.71 yields that W π (a) and W π (aξ ) are Mξ -equivalent from the left and from the right.


491

The function aξ can be written in the form aξ (x) = a+ − a− sgn(x − η), where a± := [a(ξ − 0) ± a(ξ + 0)]/2, and η = ξ for ξ ∈ R and η = 0 for ξ = ∞. Consequently, by example 9.2(b), W (aξ ) acts on Lp+ by the formula ( a− ∞ exp[iη(s − t)] W (aξ )ϕ (t) = a+ ϕ(t) + ϕ(s) ds (t > 0). πi 0 s−t A straightforward calculation shows that the mapping Λ given by Λ : Lp (R+ ) → Lp (R),

(Λϕ)(t) = e−t/p exp(iηe−t )ϕ(e−t )

is an isometry. Notice that Λ−1 : Lp (R) → Lp (R+ ),

(Λ−1 ϕ)(y) = y −1/p e−iηy ϕ(− log y).

It is also straightforward to check that ( a− ∞ ξ −1 ΛW (a )Λ ϕ (t) = a+ ϕ(t) + kp (s − t)ϕ(s) ds πi −∞

(t ∈ R),

where kp (t) = exp(−t/p)/(1 − exp(−t)). Hence, by Lemma 9.14, ΛW (aξ )Λ−1 = a+ I + a− MR (γp ) = MR [a(ξ − 0)(1 − σp ) + a(ξ + 0)σp ]. ˙ and µ ∈ R, it follows from 9.5(c) and 9.12 that Since ap (ξ, µ) = 0 for ξ ∈ R ξ −1 p ∈ GL(L ), whence W (aξ ) ∈ GL(Lp+ ). Thus Theorem 1.32(a) ΛW (a )Λ π gives that W (a) ∈ GA, i.e., that W (a) ∈ Φ(Lp+ ). Now a homotopy argument can be used to get the index formula and then a perturbation argument yields the implication “=⇒” (see Duduchava [173, pp. 47–49 and pp. 52–54] for details). 9.16. The algebra alg W (P Cp ). This is the smallest closed subalgebra of L(Lp+ ) containing the set {W (a) : a ∈ P Cp (R)}. From 9.9 we know that C∞ (Lp+ ) is a closed two-sided ideal of alg W (P Cp ). The quotient algebra will be denoted by alg W π (P Cp ) and for A ∈ alg W (P Cp ) let Aπ denote the coset A + C∞ (Lp+ ). 9.17. Theorem. (a) The algebra alg W π (P Cp ) is commutative, its maximal ideal space M (alg W π (P Cp )) can be identified with R˙ × R (equipped with an exotic topology), and the Gelfand transform of W π (a) (a ∈ P Cp (R)) is given by ˙ µ ∈ R). Γp W (a) (x, µ) = ap (x, µ) (x ∈ R, (b) ∂S M (alg W π (P Cp )) = M (alg W π (P Cp )). p (c) If A ∈ alg W π (P CN ×N ), then p A ∈ Φ(LN (R+ )) ⇐⇒ Γp (det A)π (x, µ) = 0

∀ (x, µ) ∈ R˙ × R.

If A is Fredholm, then Ind A = −ind Γp (det A)π . Proof. The proof is similar to the proof of the results of 6.38–6.40.

492


9.4 AP and SAP Symbols 9.18. Preliminaries. Let C(R) denote the collection of all continuous functions on R and put BC(R) := C(R) ∩ L∞ (R). Note that C(R) = U# C(T◦ ), BC(R) = U# C(T◦ ) ∩ L∞ (T) , ˙ ⊂ C(R) ⊂ BC(R) ⊂ C(R), C(R) each of the last three inclusions being proper. We now consider Wiener-Hopf operators on L2+ for certain classes of symbols contained in BC(R). By virtue of 9.5(e), each result stated below provides a result for Toeplitz operators on the space H 2 (T). Let alg W (L∞ ) denote the smallest closed subalgebra of L(L2+ ) containing the set {W (a) : a ∈ L∞ (R)}, note that C∞ (L2+ ) ⊂ alg W (L∞ ) (see 9.9), put alg W π (L∞ ) := alg W (L∞ )/C∞ (L2+ ), and for a ∈ L∞ (R) let W π (a) := W (a) + C∞ (L2+ ). Lemma 9.8 implies that alg W π (C2 ) is a closed subalgebra of the center of alg W π (L∞ ) and from Theorem 9.10 we know that ˙ For ξ ∈ R, ˙ let J π denote the smallest closed two-sided M (alg W π (C2 )) = R. ξ ˙ c(ξ) = 0}, put ideal of alg W π (L∞ ) containing the set {W π (c) : c ∈ C(R), π ∞ π ∞ π π alg Wξ (L ) := alg W (L )/Jξ , let Wξ (a) denote the coset W π (a) + Jξπ , and call Wξπ (a) a local Wiener-Hopf operator. The fiber of M (L∞ (R)) over ˙ will be denoted by Xξ . Clearly, if a, b ∈ L∞ (R) and ξ ∈ R˙ = M (C(R)) a|Xξ = b|Xξ , then Wξπ (a) = Wξπ (b). Now let a ∈ BC(R). Theorem 1.35(a) shows that / π sp Wξπ (a) = a(R) ∪ sp W∞ (a). spess W (a) = sp W π (a) = ˙ ξ∈R π Thus, the only interesting part of spess W (a) is sp W∞ (a). A function a ∈ BC(R) is invertible in BC(R) if and only if it is invertible in L∞ (R). In that case there is a real-valued function b ∈ C(R), which (although it is not determined uniquely) will be denoted by arg a, such that a = |a|eib . From Proposition 2.26 and 9.5(e) we obtain the following.

(a) If arg a(+∞) = +∞ and arg is bounded from above at −∞ then 0 π (a). belongs to sp W∞ (b) Suppose arg a = c + d, where c ∈ C(R) is monotonous on (−∞, 0) and (0, ∞), c(±∞) = +∞, and d ∈ BC(R). Then arg a(x) = O(log |x|) as π |x| → ∞ whenever 0 ∈ / sp W∞ (a). The purpose of what follows is to study certain classes of Wiener-Hopf operators with symbols that are not covered by (a), (b). 9.19. Almost periodic symbols. Let AP (R) denote the closure in L∞ (R) of the set of all finite linear combinations of functions of the form eiαx with

9.4 AP and SAP Symbols

493

α real. The functions in AP (R) are called almost periodic functions. Note that AP (R) ⊂ BC(R) and that AP (R) is a C ∗ -subalgebra of L∞ (R). Thus GAP (R) = AP (R) ∩ GL∞ (R). Some important properties of almost periodic functions can be recorded as follows. (a) If a ∈ GAP (R), then the three limits 1 [arg a(x) − arg a(−x)], x→+∞ 2x 1 [arg a(2x) − arg a(x)] ω ± (a) := lim x→±∞ x

ω(a) := lim

exist, are finite, and ω(a) = ω − (a) = ω + (a). The common value of these limits is referred to as the mean motion of a. (b) If a ∈ AP (R), then the three limits ( x ( 1 1 2x m(a) := lim a(t) dt, m± (a) := lim a(t) dt, x→+∞ 2x −x x→±∞ x x exist, are finite, and m(a) = m− (a) = m+ (a). The common value of these limits is called the Bohr mean value of a. (c) (Bohr’s theorem). Every a ∈ GAP (R) can be represented in the form a(x) = eiω(a)x eb(x) with b ∈ AP (R). Hence, the argument of a function in GAP (R) is a linear function plus a function in AP (R). (d) If a ∈ AP (R), then clos a(R) = a(X∞ ) and aL∞ = lim sup |a(x)| = lim sup |a(x)| = lim sup |a(x)| = max |a(y)|. |x|→∞

x→−∞

x→+∞

y∈X∞

Finally notice that Wiener-Hopf operators with almost periodic symbol arise in the study of “difference equations” on the half-line. Namely, if b in AP (R) is of the form bn eiαn x , |bn | < ∞, b(x) = n∈Z

n∈Z

L2+

then, by example 9.2(c), W (b) acts on by the rule bn ϕ(t − αn ) (t > 0), W (b)ϕ (t) =

(9.2)

n∈Z

where ϕ(t − αn ) := 0 for t − αn ≤ 0. 9.20. Theorem (Gohberg/Feldman/Coburn/Douglas). Let a ∈ AP (R). Then the following are equivalent. π (a) is invertible. (i) W∞

(ii) W (a) ∈ Φ(L2+ ). (iii) W (a) ∈ GL(L2+ ). (iv) a ∈ GAP (R) and ω(a) = 0.

494


π Proof. (i) =⇒ (iv). If W∞ (a) is invertible, then a(y) = 0 for all y ∈ X∞ (Corollary 3.63 or Theorem 4.24), so a ∈ GL∞ (R) (by 9.19(d)) and thus a ∈ GAP (R). Now 9.19(c) and 9.18(a) give that ω(a) = 0.

(iv) =⇒ (iii). Since arg a ∈ AP (R) (by 9.19(c)), there are “exponential polynomials” q(x) =

N

qn e−iβn x ,

p(x) :=

n=0

N

pn eiαn x

(βn , αn ≥ 0)

n=0

such that arg a = q+p+c with cL∞ < π/2. The operator W (eic ) is invertible since eic is sectorial. The operators W (eiq ), W (eip ) are invertible, because eiq , eip ∈ GH ∞ (R). This implies that W (a/|a|) = W (eiq )W (eic )W (eip ) is invertible (9.5(f)). From 9.5(e) and Proposition 2.19 we deduce that W (a) is invertible. (iii) =⇒ (ii) =⇒ (i). Trivial. Remark. A proof of the implication (ii) =⇒ (iv) which does not invoke 9.18(a) and 9.19(d) goes as follows. If W (a) ∈ Φ(L2+ ), then a ∈ GAP (R) due to 9.5(c). Consequently, by 9.19(c), a = ϕb, where ϕ(x) = eiω(a)x , b ∈ GAP (R), ω(b) = 0. Hence W (a) = W (ϕ)W (b) if ω(a) < 0 and W (a) = W (b)W (ϕ) if ω(a) > 0 (9.5(f)). From the implication (iv) =⇒ (iii) we deduce that W (b) is invertible and from (9.2) we see that dim Ker W (ϕ) = ∞ if ω(a) < 0 and dim Coker W (ϕ) = ∞ if ω(a) > 0, which completes the proof. ˙ If a ∈ AP (R) and c ∈ C0 (R), then 9.21. Symbols from AP (R) + C(R). clearly ac ∈ C0 (R). This shows that AP (R) + C0 (R) is an algebra. Using 9.19(d) it is easy to check that AP (R) + C0 (R) is a closed subset of L∞ (R). ˙ denote the smallest closed subalgebra of Thus, if we let alg (AP (R), C(R)) · ˙ = C + C0 (R), then L∞ (R) containing AP (R) and C(R) ·

˙ = AP (R) + C(R) ˙ = AP (R) + C0 (R) alg (AP (R), C(R)) (note that, by 9.19(d), the last sum is direct).

·

Let b ∈ AP (R), f ∈ C0 (R), and suppose b + f ∈ GL∞ (R). Since AP (R) + C0 (R) is a C ∗ -subalgebra of L∞ , there are (uniquely determined) c ∈ AP (R) and g ∈ C0 (R) such that (c + g)(b + f ) = 1, whence cb − 1 = −cf − gb − gf ∈ AP (R) ∩ C0 (R), ˙ that is, cb = 1. We therefore may write b + f = bh with h := 1 + cf ∈ GC(R). Define ω(b + f ) as ω(b) and define ind (b + f ) as ind (1 + cf ). It is easily seen that

9.4 AP and SAP Symbols

495

1 1 0 arg(b + f )(x) − arg(b + f )(−x) , x→∞ 2x

ω(b + f ) = lim ·

i.e., for a ∈ AP (R) + C0 (R) the number ω(a) can be computed even if the decomposition of a into the sum of a function in AP (R) and a function in C0 (R) is not explicitly given. ·

Wiener-Hopf operators with symbols in AP (R) + C0 (R) generate “integrodifference” operators: if b is as in 9.19 and f = F k (k ∈ L1 (R)), then ( ∞ W (b + f )ϕ (t) = an ϕ(t − αn ) + k(t − s)ϕ(s) ds (t > 0). 0

n∈Z

9.22. Theorem (Gohberg/Feldman/Coburn/Douglas). Let a belong to ·

the algebra AP (R) + C0 (R). Then W (a) ∈ Φ(L2+ ) if and only if a ∈ GL∞ (R) and ω(a) = 0. If W (a) ∈ Φ(L2+ ), then Ind W (a) = −ind a. Proof. Let a = b + f with b ∈ AP (R) and f ∈ C0 (R). We have π π spess W (b + f ) = (b + f )(R) ∪ sp W∞ (b + f ) = (b + f )(R) ∪ sp W∞ (b) = (b + f )(R) ∪ spess W (b)

(the last equality results from Theorem 9.20). Hence W (a) ∈ Φ(L2+ ) if and only if a ∈ GL∞ (R) and W (b) ∈ Φ(L2+ ), which by Theorem 9.20 is equivalent to saying that a ∈ GL∞ (R) and ω(a) := ω(b) = 0. If W (b + f ) ∈ Φ(L2+ ), then W π (b + f ) = W π (b)W π (1 + b−1 f ), and so Theorem 9.20 gives Ind W (b + f ) = −ind (1 + b−1 f ). 9.23. Semi-almost periodic symbols. Fix a function u+ in C(R) which has values in [0, 1] and satisfies u+ (−∞) = 0 and u+ (+∞) = 1, and put u− = 1 − u+ . Let SAP (R) denote the collection of all functions a ∈ BC(R) of the form a = u− a − + u + a + + a 0 ,

a± ∈ AP (R),

a0 ∈ C0 (R).

From 9.19(d) it is easily seen that a− , a+ , a0 are uniquely determined by a. It can be shown as in 9.21 that SAP (R) is a C ∗ -subalgebra of L∞ and that the mappings a → a− and a → a+ are (continuous) star-homomorphisms of SAP (R) onto AP (R). This has two immediate consequences: the first is that alg (AP (R), C(R)) = SAP (R) and the second is that a± ∈ GAP (R) whenever a ∈ GL∞ (R). ± Let a ∈ GSAP (R). Define ω ± (a) as ω(a± ). Then a± (x) = eiω (a)x eb± (x) with b± ∈ AP (R). From 9.19(a) it is easily seen that ω ± (a) can be directly obtained from a by the formula ω ± (a) = lim

x→±∞

1 [arg a(2x) − arg a(x)]. x

496


Put λ± (a) = em(b± ) , where m(b± ) is the Bohr mean value of b± . Now suppose ω + (a) = ω − (a) = 0. Choose any b ∈ C(R) such that a = eb . Since c := eb−u− b− −u+ b+ = (u− a− + u+ a+ + a0 )e−u− b− −u+ b+ ˙ and c(∞) = 1, it follows that there is a k ∈ Z such that b = is in GC(R) b0 + u− b− + u+ b+ − 2πik, which implies that λ± (a) can also be given by λ± (a) = em(b± ) = em

±

(b± )

= em

±

(b) 2πik

e

,

(9.3)

where (recall 9.19(b)) 1 m (b) = lim x→±∞ x ±

(

2x

b(t) dt. x

Again suppose ω − (a) = ω + (a) = 0. Then d := ae−u− (b− −m(b− ))−u+ (b+ −m(b+ ))

(9.4)

belongs to GC(R) and we have d(±∞) = λ± (a). Let ind a denote the index of the closed continuous and naturally oriented curve obtained from the range of d by filling in the line segment [λ+ (a), λ− (a)]. 9.24. Theorem (Sarason). Let a ∈ SAP (R). Then W (a) ∈ Φ(L2+ ) if and / [λ+ (a), λ− (a)]. In that only if a ∈ GL∞ (R), ω − (a) = ω + (a) = 0, and 0 ∈ case Ind A = −ind a. Proof. For a proof see Sarason [455]. Notice that for deciding whether or not 0 ∈ [λ+ (a), λ− (a)] we need not know the k in (9.3). We also remark that the necessity of the condition ω ± (a) = 0 results from 9.18(a), (b). Indeed, arg a is asymptotically equal to ω ± (a)x + Im b± (x) as x → ±∞, so 9.18(a) implies that ω − (a) and ω + (a) must have equal sign, in which case 9.18(b) gives ω ± (a) = 0. 9.25. Corollary. Let a± ∈ AP (R) and a := u− a− +u+ a+ . Then the following are equivalent. π (i) W∞ (a) is invertible.

(ii) ∃ a0 ∈ C0 (R) : W (a + a0 ) ∈ Φ(L2+ ). (iii) a± ∈ GL∞ (R), ω ± (a) = 0, 0 ∈ / [λ+ (a), λ− (a)]. Proof. (i) =⇒ (ii). Theorem 1.35(b) implies that there is an x0 ∈ R+ such that Wxπ (a) is invertible for all x ∈ R˙ \ (−x0 , x0 ). In particular, a(x) = 0 for |x| ≥ x0 . Choose any f ∈ GC[−x0 , x0 ] such that f (−x0 ) = a(−x0 ) and f (x0 ) = a(x0 ), and define a0 ∈ C0 (R) by a0 (x) = 0 for |x| ≥ x0 and a0 (x) = f (x) − a(x) for |x| ≤ x0 . Then a + a0 ∈ GL∞ (R), and since

9.5 Some Phenomena Caused by SAP Symbols

497

π spess W (a + a0 ) = (a + a0 )(R) ∪ sp W∞ (a),

it follows that W (a + a0 ) ∈ Φ(L2+ ). (ii) =⇒ (iii). Theorem 9.24. (iii) =⇒ (i). Construct a0 ∈ C0 (R) such that a+a0 ∈ GL∞ (R) in the same way as in the proof of the implication (i) =⇒ (ii). Theorem 9.24 shows that π π (a) = W∞ (a + a0 ) is invertible. W (a + a0 ) ∈ Φ(L2+ ), and therefore W∞ 9.26. Operators with SAP symbols on Lp+ . Let APp (R) denote the closure in M p (R) of the set of all finite linear combinations of functions of the form eiαx , where α is real. Fix u± ∈ P Cp (R) as in 9.23 and put ˙ ∩ C0 (R) . SAPp (R) = a = u− a− + u+ a+ + a0 : a± ∈ APp (R), a0 ∈ Cp (R) For a ∈ SAPp (R) ∩ GL∞ (R) define ω ± (a), b± , m(b± ), λ± (a) as in 9.23. If ω ± (a) = 0, then the function d given by (9.4) is in P Cp (R) ∩ GC(R) and satisfies d(±∞) = λ± (a). We let indp a denote the index of the closed continuous and naturally oriented curve obtained from the range of d by filling in the arc Ap (λ+ (a), λ− (a)). 9.27. Theorem (Duduchava/Saginashvili). Let 1 < p < ∞ and a be in SAPp (R). Then W (a) ∈ Φ(Lp+ ) if and only if a ∈ GL∞ (R), ω ± (a) = 0, and 0∈ / Ap (λ+ (a), λ− (a)). If W (a) ∈ Φ(Lp+ ), then Ind W (a) = −indp a. Proof. A proof is in Duduchava, Saginashvili [176] (and also in Section 19.5 of Böttcher, Karlovich, Spitkovsky [96]).

9.5 Some Phenomena Caused by SAP Symbols The consideration of SAP symbols may appear artificial at the first glance, but in fact it is such symbols that reveal a series of phenomena that remain in the dark when restricting oneself to C + H ∞ or P C symbols. We confine ourselves to citing four such phenomena and refer to Chapter 4 of B¨ ottcher, Karlovich, Spitkovsky [96] for details and full proofs. Throughout this section the underlying space is L2+ . 9.28. Fredholm criteria versus essential spectra. The Fredholm criteria for Wiener-Hopf operators with continuous or piecewise continuous symbols immediately yield descriptions of the essential spectra of these operators: ˙ for a ∈ C(R) ˙ and sp W (a) = a2 (R˙ × R) for a ∈ P C2 (R). spess W (a) = a(R) ess For almost periodic symbols, a ∈ AP (R), Theorem 9.20 gives spess W (a) = a(R) ∪ {λ ∈ / a(R) : a − λ ∈ / GAP (R) or ω(a − λ) = 0}, (9.5) which provides us with a still acceptable description of the essential spectrum. However, for symbols in SAP (R), Theorem 9.24 enables us to decide whether

498


W (a − λ) is Fredholm for a given point λ in the plane, but the problem of characterizing the set of all λ for which W (a − λ) is not Fredholm remains intricate. For a ∈ SAP (R), we divide spess W (a) into two disjoint parts. We define spAP W (a) := a(R) ∪ spess W (a− ) ∪ spess W (a+ ),

(9.6)

where a− and a+ are as in Section 9.23. The set spAP W (a) can be identified using (9.5) and (9.6). The interesting part of spess W (a) is the set spP C W (a) := spess W (a) \ spAP W (a). It is easily seen that spAP W (a) is compact. Hence C \ spAP W (a) is always open. We call a set X ⊂ C an analytic star if there are a point λ0 ∈ C, an open neighborhood U ⊂ C of λ0 , and a nonconstant analytic function f : U → C such that f (λ0 ) = 0 and X = {λ ∈ U : f (λ) = 0}. If f (λ0 ) = 0, then X ∩ V is an analytic arc through λ0 whenever V ⊂ C is a sufficiently small open neighborhood of λ0 . If f (λ0 ) = . . . = f (k−1) (λ0 ) = 0 and f (k) (λ0 ) = 0, then, for every sufficiently small open neighborhood V ⊂ C of the point λ0 , the set (X ∩ V ) \ {λ0 } is composed of 2k pairwise disjoint analytic arcs whose closures all contain the point λ0 . We call λ0 the center of the star and refer to the number of arcs in (X ∩ V ) \ {λ0 } as the valency of the star. (a) For a connected component Ω of the open set C \ spAP W (a) we have exactly one of the following three possibilities: (i) Ω ∩ spP C W (a) = ∅; (ii) Ω ⊂ spP C W (a); (iii) Ω ∩ spP C W (a) = ∅ and each point λ0 ∈ Ω ∩ spP C W (a) has an open neighborhood U ⊂ Ω such that U ∩ spP C W (a) is an analytic star. (b) Given any analytic star X, there exists a function a ∈ SAP (R) such that spP C W (a) ∩ U = X ∩ U for some open set U ⊂ C containing the center of the star. (c) Given any distinct points λ1 , . . . , λn in the complex plane and any even natural numbers 2k1 , . . . , 2kn , there exists an a ∈ SAP (R) such that λ1 , . . . , λn are all located in a single component Ω of C \ spP C W (a) and such that, for each j, the set spP C W (a) is an analytic star with the center λj and the valency 2kj in some neighborhood of λj . (d) Given any open bounded subset G of the complex plane whose boundary ∂ G is a Jordan curve, there exists an a ∈ SAP (R) such that spP C W (a) = G. (e) Let G ⊂ C be an open bounded set whose boundary is a Jordan curve and let f be a conformal map of D onto G. Then there exists an a ∈ SAP (R) such that spP C W (a) = f ((0, 1)). Part (a) is due to Sarason [455]. Parts (b) to (e), which show that (a) provides us with a complete description of the local geometry of the essential spectra,

9.5 Some Phenomena Caused by SAP Symbols

499

were established in Böttcher, Grudsky, Spitkovsky [91]. Full proofs are also in Böttcher, Karlovich, Spitkovsky [96]. 9.29. Discontinuity of the spectrum. Let {An }∞ n=1 be a sequence of operators An ∈ L(H) on some Hilbert space H. The limiting sets lim inf sp An and lim sup sp An are defined as in 7.100. Clearly, lim inf sp An ⊂ lim sup sp An .

(9.7)

Now suppose that the operators An ∈ L(H) converge uniformly to A ∈ L(H). It is easily seen that then lim sup sp An ⊂ sp A.

(9.8)

It is not difficult to find examples for which neither in (9.7) nor in (9.8) equality holds. Farenick and Lee [199] raised the question as to whether or not one has equalities in (9.7) and (9.8) provided the operators are Toeplitz operators on 2 . Equivalently, we may ask the question for Wiener-Hopf operators on L2+ . According to Hwang and Lee [290], Farenick and Lee conjectured that if an ∈ L∞ (R) and an − a∞ → 0 for some a ∈ L∞ (R), then lim inf sp W (an ) = sp W (a). Farenick and Lee [199] and Hwang and Lee ˙ + H ∞ (R), P QC(R), and [290] verified this equality for symbols an in C(R) ˙ The following result from B¨ AP (R) + C(R). ottcher, Grudsky, Spitkovsky [89] shows that the equality is nevertheless false in general. Notice that it is symbols in SAP (R) which produce the simplest counterexamples. There exist an and a in SAP (R) such that an − a∞ → 0 and sp W (an ) = spess W (an ) = T,

sp W (a) = spess W (a) = D.

As shown in Böttcher, Grudsky, Spitkovsky [89], one can even construct Wiener-Hopf operators for which both inclusions (9.7) and (9.8) are strict. There exist cn and c in L∞ (R) such that cn − c → 0 and sp W (c) = spess W (c) = D ∪ (2 + D), ! T ∪ (2 + T) if n is odd , sp W (cn ) = spess W (cn ) = T ∪ (2 + D) if n is even. In particular, lim inf sp W (cn ), lim sup sp W (cn ), and sp W (c) are three different sets. The proof of this result tells that it is true with functions cn and c that are continuous on R \ {0} and have discontinuities of semi-almost periodic type at 0 and ∞. 9.30. Frequency modulation can destroy Fredholmness. Fix a function a ∈ GL∞ (R). We say that b ∈ GL∞ (R) results from a by amplitude modulation if b(x) = µ(x)a(x) for x ∈ R with some positive function µ ∈ GL∞ (R).

500


Proceeding as in the proof of Proposition 2.19 one can show that if b results from a by amplitude modulation, then W (b) is Fredholm on L2+ if and only if so is W (a). Things change for frequency modulation. The function b ∈ GL∞ (R) is said to result from a by frequency modulation if b = a ◦ α (that is, b(x) = a(α(x)) for x ∈ R) with an orientation preserving homeomorphism α : R → R. The Fredholm criteria for Wiener-Hopf operators with continuous and piecewise continuous symbols imply that if a ∈ GP C2 (R) and b results from a by frequency modulation, then W (b) and W (a) are either simultaneously Fredholm or not. The situation is different for symbols beyond P C2 (R). There exist functions a ∈ GAP (R) and orientation preserving homeomorphisms α : R → R such that W (a) is Fredholm on L2+ but W (a ◦ α) is not even normally solvable. This result was established in B¨ ottcher, Grudsky, Spitkovsky [90] (a full proof is also in Böttcher, Karlovich, Spitkovsky [96, Theorem 4.24]). The proof is based on choosing α so that a ◦ α is a symbol in SAP (R) and on subsequently invoking Sarason’s criterion [455] (see Theorem 9.23 for Fredholmness and the notes to this theorem for normal solvability). 9.31. The index of the associated operator. In the scalar case (N = 1), W () a) ∈ L(L2+ ) is simply the transpose of W (a) ∈ L(L2+ ) and hence W (a) is Fredholm of index κ if and only if W () a) is Fredholm of index −κ. In case a is ˙ + H ∞ (R) or P C2 (R), we have the implications in C(R) W (a) ∈ Φ(L2+ ) =⇒ W () a) ∈ Φ(L2+ ), (9.9) 2 W (a) ∈ Φ(L+ ), W () a) ∈ Φ(L2+ ) =⇒ Ind W (a) = −Ind W () a). (9.10) Implication (9.9) is no longer true for almost periodic matrix symbols. (a) There exist a ∈ [AP (R)]2×2 such that W (a) is invertible but W () a) is not Fredholm. One can show that implication (9.10) remains true for symbols a belonging to [AP (R)]N ×N . It is again (matrix-valued) SAP symbols for which this implication begins to fail. (b) There are a ∈ [SAP (R)]2×2 such that W (a) is Fredholm of index 0 and W () a) is Fredholm of index 1. Part (a) is from Spitkovsky [500] and part (b) from B¨ ottcher, Grudsky, Spitkovsky [88]. More on this subject can be found in Chapter 11 of B¨ ottcher, Karlovich, Spitkovsky [96]. 9.32. The matrix case. There arise serious difficulties in the study of Wiener-Hopf operators with oscillating matrix symbols and many problems on such operators have not yet been solved in a satisfactory way, although remarkable progress has been made in recent years. The pioneering work in

9.6 Other Oscillating Symbols

501

this field is due to Karlovich and Spitkovsky [308], [309]. Even citing only the basic achievements would require the introduction of too many concepts and is therefore beyond the scope of this book. Readers interested in just this topic may consult the recent monograph B¨ ottcher, Karlovich, Spitkovsky [96]. This monograph contains almost everything that is presently known on Wiener-Hopf operators with symbols from [AP (R)]N ×N and [SAP (R)]N ×N , including the matrix versions of the Fredholm criteria by Sarason and by Duduchava and Saginashvili.

9.6 Other Oscillating Symbols 9.33. Piecewise almost periodic symbols. If p ∈ P C(R) and a ∈ AP (R), then pa = u− pa + u+ pa = u− (p(+∞) + p− )a + u+ (p(−∞) + p+ )a = u− p(+∞)a + u+ p(−∞)a + u− p− a + u+ p+ a = u− a− + u+ a+ + a0 , where a± ∈ AP (R),

a0 ∈ P C0 (R) := f ∈ P C(R) : f (±∞) = 0

(9.11)

(note that p− (+∞) = p+ (−∞) = 0). Let P AP (R) denote the collection of all functions of the form a = u− a− + u+ a+ + a0 , where a± , a0 satisfy (9.11). The above calculation shows that P AP (R) is an algebra. It can be seen as in 9.23 that a± are uniquely determined by a, that P AP (R) is a C ∗ -subalgebra of L∞ (R), that the mappings a → a± are star-homomorphisms of P AP (R) onto AP (R), and that alg (AP (R), P C(R)) = P AP (R). For a ∈ GP AP (R) one can define ω ± (a) and λ± (a) as in 9.23. Let P AP (T) denote the smallest closed subalgebra of L∞ (T) containing P C(T) and all functions of the form τ +t θα,τ (t) = exp α (α ∈ R, τ ∈ T). τ −t For τ ∈ T, define ∞

∞

Uτ : L (T) → L (R),

x−i (Uτ ϕ)(x) = ϕ τ x+i

(x ∈ R).

Note that Uτ is an isometric star-isomorphism of P AP (T) onto P AP (R). If ϕ ∈ GP AP (T), then Uτ ϕ = u− aτ− + u+ aτ+ + aτ0 ,

(9.12)

where aτ± ∈ GAP (R), aτ0 ∈ P C0 (R). Define ωτ± (ϕ) := ω ± (Uτ ϕ), λ± τ (ϕ) := λ± (Uτ ϕ).

502


9.34. Theorem. (a) Let a ∈ P AP (R). Then W (a) ∈ Φ(L2+ ) if and only if a ∈ GL∞ (R), ω ± (a) = 0, and 0∈ / [a(x − 0), a(x + 0)]

∀ x ∈ R,

0∈ / [λ+ (a), λ− (a)].

(b) Let ϕ ∈ P AP (T). Then T (ϕ) ∈ Φ(H 2 (T)) if and only if ϕ ∈ GL∞ (T), = 0 for all τ ∈ T, and

ωτ± (ϕ)

− 0∈ / [λ+ τ (ϕ), λτ (ϕ)]

∀ τ ∈ T.

Proof. (a) Let a = u− a− + u+ a+ + a0 , where a± , a0 satisfy (9.11). Since / π sp Wxπ (a) ∪ sp W∞ (a) spess W (a) = x∈R

=

/

π [a(x − 0), a(x + 0)] ∪ sp W∞ (u− a− + u+ a+ ),

x∈R

Corollary 9.25 gives the assertion. (b) Using an appropriate analogue of 9.5(e) (with Uτ in place of U# = U−1 ) we obtain that spess T (ϕ) = spess W (Uτ ϕ). If we write Uτ ϕ in the form (9.12), π π (Uτ ϕ) = W∞ (u− a− + u+ a+ ). then clearly W∞ π Because sp W∞ (Uτ ϕ) ⊂ spess W (Uτ ϕ), the “only if” portion results from Corollary 9.25 immediately. / To prove the “if” part suppose ϕ ∈ GL∞ (T), ωτ± (ϕ) = 0, and 0 ∈ − π [λ+ τ (ϕ), λτ (ϕ)]. Then W∞ (u− a− + u+ a+ ) is invertible due to Corollary 9.25. From the same corollary we infer that there exists a c0 ∈ C0 (R) such that W (u− aτ− + u+ aτ+ + c0 ) is Fredholm. This implies that T [Uτ−1 (u− aτ− + u+ aτ+ + c0 )] = T (ϕ − Uτ−1 aτ0 + Uτ−1 c0 ) is Fredholm, which in turn gives the invertibility of Tτπ (ϕ − Uτ−1 aτ0 + Uτ−1 c0 ) = Tτπ (ϕ). Since spess T (ϕ) =

τ ∈T

sp Tτπ (ϕ), we finally obtain that T (ϕ) is Fredholm.

9.35. Slowly oscillating symbols. For a function a defined on a set I let osc (a, I) denote the oscillation of a over I, that is, the supremum of |a(s)−a(t)| for s and t in I. Put SO(R) := a ∈ C(R) : osc (a, [−2x, −x] ∪ [x, 2x]) → 0 as x → ∞ and let BSO(R) := SO(R) ∩ L∞ (R). Since, for b a real-valued function in C(R), + + + |b(s) − b(t)| ++ ib(s) ib(t) + |e −e | = 2 +sin +, 2

9.6 Other Oscillating Symbols

503

it is easily seen that eib ∈ BSO(R) ⇐⇒ b ∈ SO(R). Note that, for example, (log |x|)γ (0 ≤ γ < 1) or log log |x| (|x| large) are in SO(R). It can be shown that (9.13) BSO(R) ⊂ U# (V M O(T) ∩ L∞ (T)) = U# QC(T). 9.36. Theorem. (a) Let a ∈ BSO(R). Then W (a) ∈ Φ(L2+ ) if and only if a ∈ GL∞ (R). (b) Let a ∈ alg (BSO(R), C(R)). Then W (a) ∈ Φ(L2+ ) if and only if a is in GL∞ (R) and lim inf dist 0, [a(x), a(−x)] > 0. x→∞

Proof. (a) This follows from (9.13). (b) Since, by (9.13), a ∈ U# P QC(T), this can be without difficulty derived from Proposition 3.84(c). 9.37. Theorem (Power). Let a ∈ alg (BSO(R), AP (R)). Then the following are equivalent. (i) W (a) ∈ Φ(L2+ ). (ii) a ∈ GL∞ (R) and the limit ω + (a) = lim

x→+∞

and is zero.

1 [arg a(2x) − arg(x)] exists x

1 [arg a(2x)−arg(x)] exists x→−∞ x

(iii) a ∈ GL∞ (R) and the limit ω − (a) = lim and is zero. Proof. For a proof see Power [402].

We only outline some interesting ingredients of his proof. The algebras [BSO, AP ] := alg (BSO(R), AP (R)), BSO := BSO(R), ˙ and so the fibers M∞ [BSO, AP ] and M∞ (BSO) AP := AP (R) contain C(R) are well defined. It can be shown that the mapping M∞ [BSO, AP ] → M∞ (BSO) × M (AP ),

x → (x|BSO, x|AP )

is a homeomorphism. Hence, if we think of a|M∞ [BSO, AP ] as a function in C(M∞ (BSO) × M (AP )), then for each x ∈ M∞ (BSO) a function ax ∈ AP can be defined. If a ∈ GL∞ (R), then ax ∈ GAP and so ω + (ax ) exists and is finite. The maximal ideal space of BSO is the union of R and M∞ (BSO), and one can show that M∞ (BSO) is the weak-star closure of R minus R (compare this with Lemma 3.31; Sarason [456] showed that there is a sense in which M∞ (BSO) can be identified with M10 (QC)). If {xα } is any net of positive real numbers which converges to x in M (BSO), then ω + (ax ) = lim α

1 [arg a(2xα ) − arg a(xα )]. xα

(9.14)

504


From (9.13) we see that BSO is contained in the center of alg W π (L∞ ). Therefore, if a ∈ G[BSO, AP ] then / sp Wxπ (a) ∪ a(R), spess W (a) = x∈M∞ (BSO)

and since one can show that Wxπ (a) = Wxπ (ax ), the assertion of the above theorem follows from (9.14) and Theorem 9.20. √ Finally, notice that the symbol ei( log |x|+cos x) (|x| large) is covered by the above theorem but by none of its predecessors.

9.7 Finite Section Method In the following suppose 1 < p < ∞ and 1/p + 1/q = 1. 9.38. Definitions. (a) For τ a positive real number, let Pτ ∈ L(LpN (R+ )) be the projection given by (Pτ ϕ)(t) = ϕ(t) (0 < t < τ ),

(Pτ ϕ)(t) = 0

(τ < t < ∞).

The image of Pτ can be identified with LpN (0, τ ). It is clear that Pτ = 1 and that Pτ → I strongly. Put Qτ = I − Pτ . (b) Given A ∈ L(LpN (R+ )) we shall write A ∈ Π{Pτ } if the operators Pτ APτ |LpN (0, τ ) are invertible for all sufficiently large τ (say, τ > τ0 ) and if sup (Pτ APτ )−1 Pτ L(LpN (0,τ )) < ∞.

τ >τ0

In that case the finite section method is said to be applicable to A on LpN (R+ ). Since Pτ → I strongly on LpN (R+ ) and LqN (R+ ), it results that −1 strongly on LpN (R+ ) whenever A ∈ GL(LpN (R+ )) and that A−1 τ Pτ → A A ∈ Πp {Pτ }. p p For a ∈ MN ×N (R), we denote Pτ W (a)Pτ |LN (0, τ ) by Wτ (a). (c) Define Rτ ∈ L(LpN (R+ )) by (Rτ ϕ)(t) = ϕ(τ − t) (0 < t < τ ),

(Rτ ϕ)(t) = 0 (t > τ ).

p We have Rτ = 1, Rτ2 = Pτ , Rτ Pτ = Pτ Rτ = Rτ . If a ∈ MN ×N (R), then p a), where ) a ∈ MN ×N (R) is given by ) a(x) := a(−x) for Rτ Wτ (a)Rτ = Wτ () x ∈ R.

(d) The following analogue of Proposition 7.7 can be verified as in the discrete case: Wτ (ab) = Wτ (a)Wτ (b) + Pτ HR (a)HR ()b)Pτ + Rτ HR () a)HR (b)Rτ .

9.7 Finite Section Method

505

p (e) Let DN denote the collection of all sequences {Aτ }τ ∈R+ of operators p Aτ ∈ L(LN (0, τ )) such that

{Aτ } := sup Aτ Pτ L(LpN (0,τ )) < ∞. τ >0

p This norm and natural algebraic operations make DN into a Banach alp ∗ such that gebra (C -algebra in the case p = 2). The set of all {Aτ } ∈ DN p Aτ Pτ → 0 as τ → ∞ is a closed two-sided ideal of DN and will be denoted p )τ ∈ L(Lp (0, τ )) by A )τ := Rτ Aτ Rτ . by GN . For Aτ ∈ L(LpN (0, τ )), define A N p p p consisting of all {Aτ } ∈ DN Let SN refer to the closed subalgebra of DN ) ∈ L(Lp (R+ )) such that for which there are A, A N

Aτ Pτ → A,

A∗τ Pτ → A∗ ,

)τ Pτ → A, ) A

)∗τ Pτ → A )∗ A

p (strong convergence) and let JNp be the closed two-sided ideal of SN whose p elements are the sequences {Aτ } ∈ DN of the form {Pτ KPτ + Rτ LRτ + Cτ }, p p . If a ∈ MN where K, L ∈ C∞ (LpN (R+ )) and {Cτ } ∈ GN ×N (R), then clearly p {Wτ (a)} ∈ SN . p p , denote the cosets {Aτ } + GN and {Aτ } + JNp by {Aτ }πG For {Aτ } ∈ SN π and {Aτ }J , respectively. p . Then 9.39. Theorem. Let A ∈ L(LpN (R+ )) and suppose {Pτ APτ } ∈ SN p ) A ∈ Πp {Pτ } if and only if A and A := s−lim Rτ ARτ are in GL(LN (R+ )) p and if {Pτ APτ }πJ is in G(SN /JNp ).

τ →∞

Proof. The proof is similar to the proof of Theorem 7.11. Note that Proposition 7.3 and Theorem 7.11 (which is more general than the discrete analogue of the above theorem) can be carried over to the continuous case completely. ˙ + H ∞ (R))N ×N and K ∈ C∞ (L2 (R+ )). 9.40. Theorem. (a) Let a ∈ (C(R) N Then W (a) + K ∈ Π2 {Pτ } ⇐⇒ W (a) + K, W () a) ∈ GL(L2N (R+ )). p p ˙ (b) Let a ∈ CN ×N (R) and K ∈ C∞ (LN (R+ )). Then

W (a) + K ∈ Πp {Pτ } ⇐⇒ W (a) + K, W () a) ∈ GL(LpN (R+ )). Proof. See the proof of Theorems 7.20 and 7.22.

9.41. Finite section method versus Galerkin method. We remarked in 9.5(e) that the Fredholm and invertibility theory of Wiener-Hopf operators on L2N (R+ ) is equivalent to the corresponding theory of Toeplitz operators on 2 (T) # 2N (Z+ ). The same in not true for the finite section method. HN

506


Let Ln denote the orthogonal projection of L2 (R+ ) onto the linear hull lin{ψ0 , ψ1 , . . . , ψn } of the first n + 1 Laguerre functions (see 9.8). Let A be in L(L2 (R+ )). If the operators An := Ln ALn |Im Ln are invertible for all sufficiently large n, say n ≥ n0 , and if sup A−1 n Ln L(L2 (R+ )) < ∞, then the n≥n0

Galerkin method (with respect to Laguerre functions) is said to be applicable to A and we write A ∈ Π2 {Ln }. If A ∈ Π2 {Ln }, then A ∈ GL(L2 (R+ )) and −1 A−1 strongly. Finally, define U as in 9.1(e) and recall (9.1). n Ln → A (a) We have U −1 F Ln F −1 U = Pn ,

U −1 F Pτ F −1 U = Pθτ ,

where Pn is as in 7.5, θτ is given by (7.106), and Pθτ is defined as in 7.93. Proof. To see this note that, by 9.8 and 9.2(c), Ln = I − V V n

(−n)

Pτ = I − Vτ V−τ ,

,

V

(±n)

:= W

V±τ := W (e±iτ x ).

x−i x+i

±n ,

(b) If a ∈ L∞ (R), then −1 W (a) ∈ Π2 {Ln } ⇐⇒ T (U# a) ∈ Π{H 2 (T); Pn }, −1 W (a) ∈ Π2 {Pτ } ⇐⇒ T (U# a) ∈ Π{H 2 (T); Pθτ }.

Proof. This is an immediate consequence of (a) and 9.5(e). In particular, Treil’s result 7.93(b) implies that there are a ∈ L∞ (R) such / Π2 {Pτ }. that W (a) ∈ GL(L2 (R+ )) but W (a) ∈ Thus, the theory of the Galerkin method for Wiener-Hopf operators on L2N (R+ ) is equivalent to the theory of the finite section method for Toeplitz 2 (T). The Galerkin method for Wiener-Hopf integral operators operators on HN p on L (R+ ) with symbol in W (R) was studied by Pomp [396]. Among other things he proved the following. (c) We have 0 1 W (a) ∈ Πp {Ln } ⇐⇒ W (a) ∈ GL(Lp+ )

∀ a ∈ W (R)

if and only if 4/3 < p < 4. In this connection note that (Askey, Wainger [10]) 4/3 < p < 4 ⇐⇒ Op (0, 1) is convex ⇐⇒ Ln → I strongly on Lp (R+ ). ˙ or P Cp (R) and put A := AN ×N . It can 9.42. Definitions. Let A be Cp (R) be shown as in 7.26 that the mapping

9.7 Finite Section Method p p W F : MN ×N (R) → SN ,

507

a → {Wτ (a)}

is a 1-submultiplicative isometry. The set JNA := JNp ∩ alg W F (A) is a closed two-sided ideal of alg W F (A) and the arguments of 7.28 give that the mapping W FJπ : A → alg W F (A)/JNA ,

A a → {Wτ (a)}A J := {Wτ (a)} + JN

is a 1-submultiplicative isometry and that alg W F (A)/JNA = alg W FJπ (A). 9.43. Theorem. The mapping W FJπ is an isometric algebraic isomorphism p p π π ˙ of CN ×N (R) onto alg W FJ (CN ×N ). In particular, alg W FJ (Cp ) is commu˙ and if a ∈ Cp (R) ˙ and tative, its maximal ideal space is homeomorphic to R, A A ˙ x ∈ R then the Gelfand transform of {Wτ (a)}J at x is (Γ {Wτ (a)}J )(x) = a(x). Proof. The proof is based on the arguments used to prove Theorem 7.29.

9.44. Proposition. Let a ∈ M p (R) and W (a) ∈ Πp {Pτ }. Then W (a) ∈ Πr {Pτ } and thus W (a) ∈ GL(Lr+ ) for all r ∈ [p, q]. Proof. Without loss of generality assume 1 0 such that Wτ (a) ∈ GL(Lpτ ) for all τ > τ0 . Let Bτ ∈ L(Lpτ ) denote the inverse: Bτ Wτ (a)ϕ = Wτ (a)Bτ ϕ = ϕ

∀ ϕ ∈ Lpτ .

(9.15)

If Wτ (a) ∈ GL(Lpτ ) then Wτ∗ (a) ∈ GL(Lqτ ), and since Wτ (a) = ZRτ Wτ (a)Rτ Z = ZRτ Wτ∗ (a)Rτ Z, it follows that Wτ (a) ∈ GL(Lq )

(9.16)

and that Dτ := ZRτ Bτ∗ Rτ Z is the inverse of Wτ (a) on Lqτ . We claim that Bτ ∈ L(Lq ). Indeed, for each ϕ ∈ Lqτ we can in view of (9.16) find a ψ ∈ Lqτ such that ϕ = Wτ (a)ψ. Since Lqτ ⊂ Lpτ (1 < p < 2), we deduce from (9.15) that Bτ ϕ = ψ ∈ Lqτ , i.e., Bτ maps Lqτ into itself. Application of the closed graph theorem now yields that Bτ ∈ L(Lq ). Thus both Bτ and Dτ are inverses of Wτ (a) on Lqτ , whence Bτ = Dτ . Consequently, Bτ L(Lqτ ) = ZRτ Bτ∗ Rτ ZL(Lqτ ) = Bτ∗ L(Lqτ ) = Bτ L(Lpτ ) and the Riesz-Thorin interpolation theorem gives that Bτ ∈ L(Lrτ ) and that Bτ L(Lrτ ) ≤ Bτ L(Lpτ ) for all r ∈ [p, q]. As Bτ is easily seen to be the inverse of Wτ (a) on Lrτ , we conclude that W (a) ∈ Πr {Pτ }.

508


9.45. Proposition. Let sgnξ be as in 9.2(b) and let α, β ∈ C. Then W (α + β sgnξ ) ∈ Πp {Pτ } ⇐⇒ 0 ∈ / Op (α − β, α + β). Proof. Let Λτ : Lp (0, τ ) → Lp (R+ ) denote the isometry (Λτ ϕ)(t) := τ 1/p e−t/p exp(−iξτ s)ϕ(τ e−t ) (t > 0). Note that

s −1/p (Λ−1 (0 < s < τ ). exp(iξs)ϕ − log τ ϕ)(s) = s t

A straightforward computation shows that ( 1 ∞ (Λτ Wτ (sgnξ )Λ−1 ϕ)(t) = kp (t − s)ϕ(s) ds τ πi 0

(t > 0),

where kp is as in 9.14. Hence, by Lemma 9.14, Wτ (α + β sgnξ ) = Λ−1 γp )Λτ τ W (α + β) () γp (x) := γp (−x)), and since, by Proposition 9.15, W (α + β) γp ) ∈ GL(Lp+ ) if and only if 0 ∈ / Op (α − β, α + β), the assertion follows at once. 9.46. Theorem. Let a ∈ P Cp (R). Then the following are equivalent. (i) W (a) ∈ Πp {Pτ }. (ii) W (a) ∈ Πr {Pτ } ∀ r ∈ [p, q]. (iii) W (a) ∈ GL(Lp+ ), W () a) ∈ GL(Lp+ ), W (a) ∈ Φ(Lr+ ) (iv) W (a) ∈ (v) The set

GL(Lp+ ),

W (a) ∈

GL(Lq+ ),

W (a) ∈

Φ(Lr+ )

∀ r ∈ [p, q]. ∀ r ∈ [p, q].

Op (a(x − 0), a(x + 0)), obtained from the range of a by

˙ x∈R

filling in the lentiform domains Op (a(x − 0), a(x + 0)) for each x at which a has a jump, does not contain the origin and the curve a2 , obtained from the range of a by filling in the line segments [a(x − 0), a(x + 0)] for each point of discontinuity, has index zero. Proof. (iii) ⇐⇒ (iv) ⇐⇒ (v). Proposition 9.13. (i) =⇒ (ii) =⇒ (iii). Proposition 9.44 and Theorem 9.39. ˙ (iii)+(v) =⇒ (i). LetπNξ (ξ ∈ R)be as in the proof of Proposition 9.15 and define Mξ := {Wτ (c)}J : c ∈ Nξ . It can be shown in the usual way that {Mξ }ξ∈R˙ is a covering system of bounded localizing classes in S p /J p and that {Wτ (a)}πJ is Mξ -equivalent from the left and the right to {Wτ (aξ )}πJ , where aξ is as in the proof of Proposition 9.15. The previous proposition implies that W (aξ ) ∈ Πp {Pτ }, whence {Wτ (aξ )}πJ ∈ G(S p /J p ). So Theorem 1.32(a) gives that {Wτ (a)}πJ ∈ G(S p /J p ) and Theorem 9.39 then completes the proof.


509

Remark. The implication a) ∈ CL(Lp+ ) =⇒ W (a) ∈ Πp {Pτ } (a ∈ P Cp (R)) W (a) ∈ GL(Lp+ ), W () is, in general, not true. The point of the matter is that, in contrast to the discrete situation, the invertibility of W (a) and W () a) on Lp+ does not automatically imply that W (a) is invertible (or even normally solvable) on Lr+ for r ∈ [p, q]. Example: the singular integral operator W (−sgn0 ) on the half-line is invertible on Lp+ if and only if p = 2 and it is not normally solvable on L2+ . ˙ define Nξ as in the proof of Proposition 9.15 9.47. Localization. For ξ ∈ R, and put p /JNp : ϕ = diag (f, . . . , f ), f ∈ Nξ . Mξ := {Wτ (ϕ)}πJ ∈ SN It is easily seen that {Mξ }ξ∈R˙ is a covering and overlapping system of lo p calizing classes in SN /JNp . Let Fp := Mξ and let Zpξ denote the closed ˙ ξ∈R

two-sided ideal of the commutant Com Fp consisting of all elements which are Mξ -equivalent to zero from the left and the right. If we let spp {Wτ (a)}0ξ refer to the spectrum of the coset {Wτ (a)}0ξ := {Wτ (a)}πJ +Zpξ in Com Fp /Zpξ , then /

spSNp /JNp {Wτ (a)}πJ =

spp {Wτ (a)}0ξ ,

(9.17)

˙ ξ∈R p for every a ∈ P CN ×N (R) (cf. Theorem 1.32(b) and Theorem 5.29(a)). ˙ let J P C denote the smallest closed two-sided ideal of Again for ξ ∈ R, ξ p alg W FJπ (P CN ×N ) containing the set

C ˙ {Wτ (ϕ)}P J : ϕ = diag (f, . . . , f ), f ∈ Cp (R), f (ξ) = 0 P Cp (R)

(recall 9.42; we abbreviate JN spectively). Put

P Cp (R)

, {Wτ (ϕ)}J

PC C to JN , {Wτ (ϕ)}P J , re-

p π PC alg W Fξπ (P CN ×N ) := alg W FJ (P CN ×N )/Jξ .

(9.18)

p For a ∈ P CN ×N (R), denote the spectrum of the coset C C PC {Wτ (a)}P := {Wτ (a)}P ξ J + Jξ C C in the algebra (9.18) by spp {Wτ (a)}P and let spp {Wτ (a)}P refer to the ξ ξ p PC π spectrum of {Wτ (a)}J as an element of alg W FJ (P CN ×N ). Theorem 1.35(a) (see also Theorem 5.29(b)) yields the equality / C C spp {Wτ (a)}P spp {Wτ (a)}P J = ξ . ˙ ξ∈R

510


A reasoning similar to the one in the proof of Proposition 6.35 shows that {Wτ (a)}0ξ = {Wτ (b)}0ξ ,

C C {Wτ (a)}P = {Wτ (b)}P ξ ξ

p ∞ whenever a, b ∈ P CN ×N (R) coincide on Xξ = Mξ (L (R)).

˙ Are the equalities 9.48. Open problems. Let a ∈ P Cp (R) and ξ ∈ R. C spp {Wτ (a)}0ξ = spp {Wτ (a)}P = Op (a(ξ − 0), a(ξ + 0)) ξ

true? We conjecture that the answer is yes, since we know that the second equality holds in the discrete case (Proposition 7.41). However, the proof of Lemma 7.35 (which is needed to prove Proposition 7.41) relies essentially on the fact that Im Pn is finite-dimensional, whereas Im Pτ has infinite dimension. Also note that Theorem 9.46 implies that both local spectra are contained in Op (a(ξ − 0), a(ξ + 0)). Theorem 9.39, Proposition 9.45, and 1.16(b) give that C spS p /J p {Wτ (sgnξ )}πJ = spp {Wτ (sgnξ )}P J = Op (−1, 1)

(9.19)

for all ξ ∈ R. Therefore the above question is equivalent to asking whether the equalities spp {Wτ (sgnξ )}0ξ = spp {Wτ (sgnξ )}0∞ = Op (−1, 1),

(9.20)

C spp {Wτ (sgnξ )}P ξ

(9.21)

=

C spp {Wτ (sgnξ )}P ∞

= Op (−1, 1)

are valid. One feels that the local spectra at ξ and ∞ must be equal to each other, which would imply (9.20) and (9.21) due to (9.19), but we have not been able to prove this. Fortunately we can show that at least one of the local spectra in (9.20) and at least one of the local spectra occurring in (9.21) equals Op (−1, 1) (although we do not know which of the two spectra has this property). Let us prove this for the spectra in (9.20). Contrary to what we want, assume both local spectra have a “hole.” Choose a function a ∈ P Cp (R) as follows:

Then W (a) and W () a) are invertible by Proposition 9.15, while {Wτ (a)}πJ is in p p G(S /J ) by our assumption. So Theorem 9.39 implies that W (a) ∈ Πp {Pτ }, which contradicts Theorem 9.46 and completes the proof.


511

p p 9.49. Theorem. Suppose a ∈ P CN ×N (R) and K ∈ C∞ (LN (R+ )). Then a) are in GL(LpN (R+ )) W (a) + K ∈ Πp {Pτ } if and only if W (a) + K and W () r and W (a) is in Φ(LN (R+ )) for all r ∈ [p, q].

Proof. Sufficiency. Let ξ ∈ R˙ and put a± := [a(ξ−0)±a(ξ+0)]/2. Without loss of generality assume a+ is the identity matrix and a− is in Jordan canonical form. Using Theorem 9.17(c) one can easily verify that 0 ∈ / Op (1−λj , 1+λj ) for all eigenvalues λi of a− whenever W (a) ∈ Φ(LrN (R+ )) for all r ∈ [p, q]. Let bξ be any function in P Cp (R) satisfying bξ (ξ ± 0) = ±1 and put aξ := a+ − a− bξ . Then a(ξ ± 0) = aξ (ξ ± 0) and hence, {Wτ (a)}0ξ = {Wτ (aξ )}0ξ . Think of {Wτ (aξ )}0ξ as an N × N matrix with entries in Com Fp /Zpξ . The diagonal entries of this matrix are of the form {Wτ (1−λj bξ )}0ξ , where λj is an eigenvalue of a− . Since Op (1 − λj , 1 + λj ) does not contain the origin, {Wτ (1 − λj bξ )}0ξ is invertible for all j and so {Wτ (aξ )}0ξ must also be invertible. From (9.17) p /JNp , and Theorem 9.39 yields we deduce that {Wτ (a)}πJ is invertible in SN the assertion. Necessity. Theorem 9.39 gives the invertibility of W (a) + K and W () a) on LpN (R+ ). If W (a) + K is in Πp {Pτ } but W (a) is not in Φ(LrN0 (R+ )) for some r0 ∈ [p, q], then there is a b ∈ P KN ×N (R) such that W (b) + K ∈ Πp {Pτ } and W (b) ∈ / Φ(LrN0 (R+ )). Therefore we may a priori assume that a ∈ P KN ×N (R). From 9.48 we know that at least one of the following is true: spp {Wτ (sgnξ )}0∞ = Op (−1, 1)

∀ξ∈R

(9.22)

spp {Wτ (sgnξ )}0ξ = Op (−1, 1)

∀ ξ ∈ R.

(9.23)

or Suppose (9.22) is valid. Put a± := [a(−∞) ± a(+∞)]/2 and without loss of generality assume a+ is the identity matrix and a− is in Jordan canonical form. Let λi denote the eigenvalues of a− . Choose any b∞ ∈ GCp (R) such that b∞ (±∞) = ±1 and put a∞ := a+ − a− b∞ . Then a(±∞) = a∞ (±∞). p /JNp ), we see Hence, {Wτ (a)}0∞ = {Wτ (a∞ )}0∞ and since {Wτ (a)}πJ ∈ G(SN 0 0 that {Wτ (a∞ )}∞ is invertible. The element {Wτ (a∞ )}∞ may be thought of as a block-diagonal matrix with entries in Com Fp /Zp∞ . This implies that each diagonal block is invertible, and because the entries on the diagonal of each such diagonal block are equal to each other, it follows that these entries are themselves invertible in Com Fp /Zp∞ . But these entries are of the form {Wτ (1 − λj b∞ )}0∞ and therefore, by our assumption (9.22), the origin does not belong to Op (1 − λj , 1 + λj ) for all j. What results is that 0 1 det 1 − σr (µ) a(+∞) + σr (µ)a(−∞) = 0 ∀ µ ∈ R ∀ r ∈ [p, q]. (9.24) Write Φ0 (Y ) for the collection of all Fredholm operators on Y with index zero. Taking into account (9.24) and that W (a) and W () a) are in Φ0 (LpN (R+ )) and using the Fredholm criteria and index formulas of 6.39, 6.40, 9.17 we obtain that

512


W (a), W () a) ∈ Φ0 (LpN (R+ )) −1 −1 −1 ⇐⇒ T (U# ) a), T (U# a) = T ((U# a))) ∈ Φ0 (pN ) −1 −1 ⇐⇒ T (U# a) ∈ Φ0 (pN ), T (U# a) ∈ Φ0 (qN )

(9.25)

−1 −1 (note that U# a ∈ P KN ×N (T)). But (9.25) implies that T (U# a) ∈ Φ0 (rN ) for all r ∈ [p, q] (see, e.g., Böttcher, Silbermann [106, p. 73]) and by inverting the argument leading to (9.25) we arrive at the conclusion that W (a) is in Φ0 (LrN (R+ )) for all r ∈ [p, q], as desired. If we suppose that (9.23) holds, then a similar reasoning as in the preceding case shows that 0 1 det 1 − σr (µ) a(ξ − 0) + σr (µ)a(ξ + 0) = 0 (9.26)

˙ From this and the fact that W (a) and W () a) are for all µ ∈ R, r ∈ [p, q], ξ ∈ R. p invertible on LN (R+ ) one can conclude as above that W (a) is in Φ0 (LrN (R+ )) for all r ∈ [p, q]. Remark. Notice that in the scalar case the condition (9.24) (resp. (9.26)) is equivalent to the statement that a may have a jump at ∞ (respectively, at ξ1 , . . . , ξk ∈ R) but that the origin does not belong to Op (a(+∞), a(−∞)) (resp. Op (ξl − 0, ξl + 0) for each l). We conclude the topic by stating the following result. 9.50. Theorem (Gohberg/Feldman). Let, for x ∈ R, b(x) = bn eiαn x , |bn | < ∞, f (x) = (F k)(x), n∈Z

k ∈ L1 (R).

n∈Z

Then W (b + f ) ∈ Πp {Pτ } if and only if W (b + f ) ∈ GL(Lp+ ). Proof. Put a = b + f and suppose W (a) is invertible. By virtue of Corollary 7.16 it suffices to prove that " " lim sup "(Qτ W −1 (a)Qτ )−1 Qτ "L(Lp ) < ∞. (9.27) τ →∞

Since

+

W −1 (a) = W (a−1 ) + W −1 (a)HR (a)HR () a−1 )

and W (a−1 ) is invertible together with W (a), (9.27) will follow once we have a−1 )Qτ L(Lp+ ) → 0 as τ → ∞. We have (see, e.g., Gohberg, shown that HR () Feldman [220, Chap. VII]) a representation ) a−1 = c + F l, where |cn | < ∞, l ∈ L1 (R). c(x) = cn eiβn x , Let dτ = cτ + F lτ , where

9.8 Operators over the Quarter Plane

cτ (x) =

! cn eiβn x ,

lτ (t) =

βn

Analysis of Toeplitz operators

Analysis of Toeplitz Operators

Analysis of Toeplitz Operators

Extreme Eigenvalues of Toeplitz Operators

The Spectral Theory of Toeplitz Operators

A similarity problem for Toeplitz operators

Analysis of Operators

Commutative algebras of Toeplitz operators on the Bergman space

Commutative Algebras of Toeplitz Operators on the Bergman Space

Commutative algebras of Toeplitz operators on the Bergman space

Commutative algebras of Toeplitz operators on the Bergman space

Spectral Analysis of Relativistic Operators

Spectral analysis of nonlinear operators

Spectral Analysis of Relativistic Operators

Spectral Analysis of Nonlinear Operators

Algebraic Methods for Toeplitz-like Matrices and Operators

The Analysis of Linear PD Operators. III, Pseudo-Differential Operators

Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators

The Analysis of Linear PD Operators. IV, Fourier Integral Operators

Harmonic analysis of operators on Hilbert space

Analysis, geometry and topology of elliptic operators

Harmonic Analysis of Operators on Hilbert Space

Spectral Properties of Banded Toeplitz Matrices

Direct methods of qualitative spectral analysis of singular differential operators

Methods of Modern Mathematical Physics. Analysis of Operators

0-1-Sequences of Toeplitz Type

Methods of modern mathematical physics. Analysis of operators

Spectral properties of banded Toeplitz matrices

Spectral Properties of Banded Toeplitz Matrices

Function spaces, differential operators and nonlinear analysis

Microlocal analysis for differential operators: An introduction

Analysis of Toeplitz operators