http://dx.doi.org/10.1090/surv/138 Operator Theory in Function Spaces Second Edition Mathematical I Surveys I and I Monographs I Volume 138 I Operator Theory in Function Spaces I Second Edition I Kehe Zhu EDITORIAL COMMITTEE Jerry L. Bona Ralph L. Cohen Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 47-02, 30-02, 46-02, 32-02. For additional information and updates on this book, visit www.ams.org/bookpages/surv-138 Library of Congress Cataloging-in-Publication Data Zhu, Kehe, 1961- Operator theory in function spaces / Kehe Zhu ; second edition. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 138) Includes bibliographical references and index. ISBN 978-0-8218-3965-2 (alk. paper) 1. Operator theory. 2. Toeplitz operators. 3. Hankel operators. 4. Functions of complex variables. 5. Function spaces. I. Title. QA329.Z48 2007 515/724—dc22 2007060704 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissionOams.org. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 To my family: Peijia, Peter, and Michael Contents Preface to the Second Edition xi Preface to the First Edition XV Chapter 1. Bounded Linear Operators 1 1.1. Bounded Operators on Banach Spaces 1 1.2. Bounded Operators on Hilbert Spaces 5 1.3. Compact Operators on Hilbert Spaces 10 1.4. Schatten Class Operators 18 1.5. Notes 30 1.6. Exercises 30 Chapter 2. Interpolation of Banach Spaces 33 2.1. Interpolation Spaces 33 2.2. Complex Interpolation 36 2.3. W Spaces and Schatten Classes 39 2.4. The Marcinkiewicz Interpolation Theorem 41 2.5. Notes 45 2.6. Exercises 46 Chapter 3. Integral Operators on IP Spaces 47 3.1. Holder's Inequalities 47 3.2. Hilbert-Schmidt Integral Operators 49 3.3. Schur's Theorem 52 3.4. Integral Operators on the Unit Disk 54 3.5. Notes 61 3.6. Exercises 62 Chapter 4. Bergman Spaces 65 4.1. The Mobius Group 65 4.2. The Bergman Metric 66 4.3. Bergman Spaces 72 4.4. Kernel Functions and Related Projections 76 4.5. Atomic Decomposition 87 4.6. Notes 94 4.7. Exercises 97 vii Vlll CONTENTS Chapter 5. Bloch and Besov Spaces 101 5.1. The Bloch Space 101 5.2. The Little Bloch Space 108 5.3. Analytic Besov Spaces 114 5.4. Notes 125 5.5. Exercises 128 Chapter 6. The B erezin Transform 133 6.1. The Berezin Transform 133 6.2. The Case of Weighted Bergman Spaces 135 6.3. The Berezin Transform of Functions 141 6.4. Notes 159 6.5. Exercises 160 Chapter 7. Toeplitz Operators on the Bergman Space 163 7.1. Toeplitz Operators 163 7.2. Carleson Type Measures 166 7.3. Toeplitz Operators in Schatten Classes 173 7.4. Toeplitz Operators with Bounded Symbols 189 7.5. Commuting Toeplitz Operators with Harmonic Symbols 197 7.6. Notes 202 7.7. Exercises 204 Chapter 8. Hankel Operators on the Bergman Space 207 8.1. BMO in the Bergman Metric 207 8.2. VMO in the Bergman Metric 219 8.3. Bounded Hankel Operators 221 8.4. Compact Hankel Opeators 225 8.5. Schatten Class Hankel Operators 227 8.6. Hankel Operators on the Unweighted Bergman Space 233 8.7. Little Hankel Operators 244 8.8. Notes 248 8.9. Exercises 250 Chapter 9. Hardy Spaces and BMO 253 9.1. #p Spaces 253 9.2. Carleson Measures 262 9.3. Functions of Bounded Mean Oscillation 266 9.4. Functions of Vanishing Mean Oscillation 275 9.5. Notes 279 9.6. Exercises 280 Chapter 10. Hankel Operators on the Hardy Space 285 10.1. Toeplitz Operators on H2 285 CONTENTS IX 10.2. Bounded Hankel Operators on H2 287 10.3. Compact Hankel Operators on H2 289 10.4. Schatten Class Hankel Operators on H2 291 10.5. Notes 299 10.6. Exercises 301 Chapter 11. Composition Operators 303 11.1. Littlewood's Subordination Principle 303 11.2. The Nevanlinna Counting Function 305 11.3. Composition Operators on the Bergman Space 308 11.4. Composition Operators on the Hardy Space 317 11.5. Notes 323 11.6. Exercises 324 Bibliography 325 Index Preface to the Second Edition The study of Toeplitz operators, Hankel operators, and composition op­ erators has witnessed several major advances since the first edition of the book was published over fifteen years ago. So I decided to undertake a substantial revision when I was offered the opportunity to publish a second edition with the AMS. I rewrote several existing sections completely and added several sec­ tions to reflect new developements in the field. Also, the present Chapter 6 is new. It consists of several results from the old Chapter 6 and several new results that have appeared since the publication of the first editon of the book. As a result, the old Chapters 6, 7, 8, 9, and 10 became the new Chapters 7, 8, 9, 10, and 11, respectively. Except for a few minor corrections and improvements, the contents of Chapters 1, 2, 5, 9, and 10 remain mostly unchanged. Chapter 3 has pretty much been rewritten. I added a section to cover several useful versions of Holder's inequality that are needed elsewhere in the book. I also added a section about integral operators on the unit disk whose kernels are built from the Bergman kernel. Such integral operators have proven useful in a number of problems studied in the book as well as in other related problems in the literature. The section on Schur's theorem has been rewritten, with the theorems here more general and more applicable. Chapter 4 has also been revised substantially. The original text only covered Bergman spaces whose integral exponent p is greater than or equal to 1. Since all earlier results remain true when 0 < p < 1,1 decided to make a little extra effort to cover all integral exponents p as well as all weight parameters a, so the results are now more general and more complete. The atomic decomposition for functions in Bergman spaces is more general now and has a new proof. The proof given in the first edition was based on duality arguments, so it did not work for Bergman spaces whose integral exponent p is less than 1. On the other hand, a complete proof that covers all exponents p and all weight parameters a requires a very elaborate, two- step partition of the unit disk into hyperbolically small pieces. I decided to strike a balance and present a new proof here that only requires a one-step partition of the disk but works only for exponents p > 2/3. Readers who XI Xll PREFACE TO THE SECOND EDITION are really interested in atomic decomposition can find a complete proof in my book [438] or in Coifman and Rochberg's original paper [107]. As far as Bergman spaces are concerned, all results in the first edition were stated and proved for the unweighted case. As a result of the addi­ tional effort made in Chapter 4, these results have all been generalized to the weighted case. It turned out that the size estimates for Toeplitz operators Tf on the Bergman space L%(dAa) with nonnegative symbol / are actually independent of the weight parameter. The same is true for the simultaneous size estimates of the Hankel operators Hf and Hj. As was mentioned earlier, Chapter 6 of this edition is new. In addition to the material about the Berezin transform from various sections (mostly Sec­ tion 2) of the old Chapter 6, it includes an elegant recent result of Coburn's which states that the Berezin transform of any bounded linear operator sat­ isfies a sharp Lipschitz condition involving the pseudo-hyperbolic metric. It also contains a description of the fixed-point set of the a-Berezin transform P in L (D, dAa) for 1 < p < oo. In particular, a function / e Z/°°(D) is fixed by the a-Berezin tranform if and only if it is harmonic. Chapter 7 is the result of a substantial revision of the old Chapter 6. In particular, I added two new results here. First, I added the characterization of compactness for Toeplitz operators on the Bergman space in terms of the Berezin transform, a significant result due to Axler and Zheng [42] that was not yet available when the book was first published. Second, I chose to include the very elegant description of commuting Toeplitz operators with harmonic symbols by Axler and Cuckovic [37] because this result has gen­ erated a flurry of activity in the area.