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

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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. I do realize that this second result is somewhat incompatible with the flavor of the rest of the book (namely, size estimates for various operators). Finally, the original presentation con­ cerning Toeplitz operators in the Schatten classes Sp was only for the case p > 1, but the new edition now covers the full range 0 < p < oo. In fact, I believe the precise range of p in part (d) of Theorem 7.18 was obtained only recently by the author in [439]. Another significant result that appeared after the publication of the first edition of the book was Luecking's approach to Hankel operators on the Bergman space [264]. This has been added in the new edition as Section 8.6. It is still not clear if this approach can be made to work for all weighted Bergman spaces Ll(dAa). As a result of the improvements made in Chapter 7, composition opera­ tors on the Hardy space that belong to the Schatten classes Sp are studied in Chapter 11 for the full range 0 < p < oo, while the old text only covered the case p > 1. Section 11.3 is completely rewritten; it has a new proof for the characterization of compact composition operators on Bergman spaces, and PREFACE TO THE SECOND EDITION xm a new result concerning Schatten class composition operators on Bergman spaces has also been added in this section. In addition to the various changes in content, the style of the book has changed as well. This is mandated by the new publisher, the American Mathematical Society. The new book series has a different format. There­ fore, the second edition is set in AMS-MgX, while the first edition was set in plain TfnX. In particular, results and equations are numbered and refer­ enced differently now. The new text consists of chapters and sections, while the old one was further divided into subsections. Additional comments and references are added to accompany the new material introduced. However, because the number of papers dealing with Toeplitz operators, Hankel operators, composition operators, and functions spaces that have appeared since the appearance of the first edition is prob­ ably in the hundreds, so the updated bibliography is by no means exhaus­ tive. When updating the bibliography, I paid more attention to the area of Toeplitz and Hankel operators on Bergman spaces, as this book is the only one in the market that covers this area of analysis. In the area of composition operators, several monographs have since appeared, most notably the books by Cowen-MacCluer [119] and Shapiro [351]. The recent book of Peller [293] gives a detailed account of the theory of Hankel operators on the Hardy space as well as the various applications of Hankel operators. The function theory of Bergman spaces has experienced several break­ throughs during the last fifteen years or so. This includes Seip's description of interpolating and sampling sequences for Bergman spaces [342], Heden- malm's study of contractive zero divisors [184] in the Bergman space, and the work on invariant subspaces of the Bergman space by Aleman, Richter, and Sundberg [8]. For readers interested in this area of analysis, several re­ lated books have appeared, including Duren-Schuster [134], Hedenmalm- Korenblum-Zhu [187], and Zhu [438]. There are some obvious overlaps between the present book and the books mentioned above, but this book presents a unique perspective and emphasizes several useful techniques about size estimates of Hankel, Toeplitz, and composition operators that cannot be found in other books. In particular, none of the other books cov­ ers Toeplitz and Hankel operators on Bergman spaces. I am grateful to Dr. Ina Lindemann, senior editor in the publications department of the American Mathematical Society, for her enthusiasm and encouragement to publish this second edition with the AMS. I wish to thank my former student Eric Grossmann, who studied the first edition of the book very carefully and pointed out several mistakes to me. I am also grateful to Miroslav Englis, who read an early version of this second edition and spotted numerous misprints. XIV PREFACE TO THE SECOND EDITION

As always, I have been able to count on my wife Peijia and our sons, Peter and Michael, for their constant and continuous support during the preparation of the new manuscript. Thank you for bearing with me!

Kehe Zhu, Albany, 2007 Preface to the First Edition

This book deals with three types of operators: Toeplitz operators, Han- kel operators, and composition operators. We treat these operators on both the Bergman space and the Hardy space of the open unit disk in the complex plane. The main emphasis of the book is on the size estimates of these oper­ ators: boundedness, compactness, and membership in the Schatten classes. Toeplitz and Hankel operators on the Hardy space have been studied intensively for a long time. However, the study of these operators on the Bergman space began only a few years ago. In particular, the Bergman space theory has never appeared in book form before. Also, this book is the first to deal with composition operators on both the Bergman space and the Hardy space. We choose to develop the theory on the open unit disk because maximal clarity can be achieved in this case without losing many techniques of the subject. Almost all of the results in this book about Bergman spaces can be generalized to the open unit ball or even to a bounded symmetric domain in Cn. We will comment on this matter in detail at the end of each chapter (in the section entitled "Notes"). Another reason for choosing the open unit disk is that composition operators in several variables are not yet well understood. Thus, concentrating on the disk will enable us to achieve some uniformity. This book is intended for both research mathematicians and graduate students in complex analysis and operator theory. The prerequisites for the book are kept to a minimum. A graduate course in each of the following subjects should sufficiently prepare the reader for the book: functions of one complex variable, functional analysis, integration and measure theory. Basic facts in general operator theory are collected in the first two sections of the first chapter. Some exercises are provided at the end of each chapter. Most of these problems are workable. When a relatively difficult problem appears in the exercises, appropriate references are given to the reader. These exercises should be suitable for homework assignments if the book is used as a text­ book.

XV XVI PREFACE TO THE FIRST EDITION

The book can be divided into three parts. Part one of the book provides the necessary preliminaries on operator theory. This includes Chapters 1, 2, and 3. Part two is about operators on the Bergman space, including Chapters 4, 5, 6, 7, 8, and Section 3 of Chapter 11. Part three of the book, including Chapters 9, 10, and most of Chapter 11, deals with operators on the Hardy space. I wish to thank Lewis Coburn, Richard Rochberg, and Donald Sarason for encouragement and indispensable help during the writing of the book. Carl Cowen provided the author with most of the references on composition operators. Finally, I am grateful to my wife, Peijia Tan, for her patience and understanding. It was not easy for her to take care of our new-born son alone while I spent long hours at the computer working on the manuscript.

Kehe Zhu, Albany, 1989 Bibliography

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C*-algebra, 13,251 BMO, 266 <9-Neumann operator, 203 BMO in the Bergman metric, 207 r-lattice, 70 BMOA, 266 Borel measure, 3 adjoint, 5 bounded linear functional, 1 Alaoglu's theorem, 4 bounded linear operator, 3 analytic Besov spaces, 114 bounded mean oscillation, 208, 266 angular derivative, 312 bounded oscillation, 209 angular limit, 312 bounded symmetric domain, 96, 249 area measure, 54 bounded valence, 314 area version of the invariant mean value property, 198 Calkin algebra, 32 atomic decomposition, 87 canonical decomposition, 17, 218 atomic measure, 118 Caratheodory's theorem, 312 automorphism group, 65 Carleson measure, 166, 262 averaging function, 170 Carleson measure for Bergman spaces, 166 backward shift, 8 Carleson rectangles, 262 Banach algebra, 13, 75 Carleson square, 262 Banach space, 1 Carleson type measure, 166 Banach-Steinhaus theorem, 4 Carleson's theorem, 263 Bell's operator, 127 Cauchy's integral formula, 88 Berezin symbol, 133 Cauchy-Schwarz inequality, 5 Berezin transform, 133 characteristic function, 196 Bergman disk, 69 closed graph theorem, 3 Bergman kernel, 76 closed unit ball, 4 Bergman metric, 67 coefficient multipliers, 126 Bergman projection, 79 cokernel, 301 Bergman spaces, 72 commuting Toeplitz operators, 197 Bergman type projections, 85 compact operator, 10 Besov spaces, 114 compatible Banach spaces, 33 Beurling's theorem, 133, 300 compatible pair, 33 Blaschke condition, 305 complex interpolation, 36 Blaschke product, 126 complex method, 35 Bloch constant, 125 composition operator, 308 Bloch function, 106 conformal invariance, 263 Bloch space, 101 conjugate index, 21 346 INDEX conjugate operator, 257 Green's function, 260 conjugation preserving, 142 Green's theorem, 260 continuous inclusions, 131 contravariant symbols, 159 Hadamard gap series, 128 coset, 4 Hadamard three lines theorem, 36 counting function, 305 Hahn-Banach extension theorem, 4 covariant symbols, 159 half-order differential operator, 302 Hankel matrix, 289 decent functional, 127 Hankel operator, 287 Denjoy-Wolff point, 323 Hankel operators, 207 direct sum, 158 Hankel operators on the Bergman space, Dirichlet integral, 129, 295 221 Dirichlet space, 115 Hardy spaces, 253 discrete spectrum, 202 harmonic conjugate, 98, 256 distribution function, 42 harmonic extension, 255 distribution of zeros, 126 harmonic function, 98 dominated convergence theorem, 112 harmonic symbols, 197 dual space, 2 Hartman's theorem, 290, 299 p duality of L spaces, 2 Hausdorff space, 13 Hausdorff-Young inequality, 46 eigenspace, 149 Havin's lemma, 234 eigenvalue, 15 Hilbert space, 6 eigenvector, 15 Hilbert-Schmidt class, 18 embedding, 113 Hilbert-Schmidt operator, 196 equivalence relation, 4 holomorphic functional calculus, 148 essential norm, 32 homogeneous domains, 126 essential spectrum, 32, 252 hyperbolic disk, 68 Euclidean center, 70 hyperbolic metric, 67 Euclidean circle, 70 Euclidean closure, 204 ideal, 11 Euclidean disk, 69 identity operator, 8 Euclidean radius, 70 inclusion map, 97, 166 extreme points, 126 infinite Blaschke product, 126 Fefferman's theorem, 272 infinitesimal Bergman metric, 67 finite Blaschke product, 126 inner functions, 126 finite rank operator, 17 inner product, 5 finite rank Toeplitz operators, 177 inner product space, 5 fixed points, 141,323 inner-outer factorization, 262 Fock space, 203 integeral pairing, 113 Forelli-Rudin estimates, 234 integral kernel, 50 Fourier coefficients, 46, 255 integral operator, 5 Fourier expansion, 260 integral pairing, 3 Fredholm operator, 32 integral representation, 83, 310 intermediate space, 35 gamma function, 55 interpolation, 33 generalized half-planes, 126 interpolation space, 33 generalized subharmonic property, 204 invariant mean value property, 198 geodesic, 210, 228 invariant metric, 66 Gleason part, 125 invariant pairing, 123, 295 INDEX 347 invariant semi-norm, 123 Nehari's theorem, 289, 299 involution, 13 net, 4 involutions, 65 Nevanlinna counting function, 305 involutive automorphisms, 65 nontangential limit, 312 isometric isomorphisms, 126 norm, 1 isometry, 8, 123 norm topology, 4 isomorphism, 115 normal family, 167 normal operator, 8 Jacobian determinant, 66 normal Toeplitz operators, 201 Jensen's formula, 305 normalized area measure, 54 John-Nirenberg theorem, 266 normalized reproducing kernel, 134, 285 kernel, 5 normed space, 1 kernel functions, 76 null space, 157 Kronecker's theorem, 300 open mapping theorem, 3 lacunary series, 100, 128 open unit ball, 96 Laplacian, 201 optimal growth rate, 74 level curves, 109 orthogonal complement, 9, 26 linear isometries, 126 orthogonal projection, 8 Lipschitz, 215 orthonormal basis, 6 Lipschitz estimate, 133 orthonormal set, 8 little Bloch space, 108 little Hankel operators, 207, 244 Parseval's identity, 6 Littlewood's subordination principle, 303 partial counting function, 305 Littlewood-Paley identity, 260 partial isometry, 7 local homeomorphism, 307 partition of D, 90 local sub-mean value property, 303 partition of unity, 236 logarithmic differentiation, 154 Poincare metric, 96 Lorentz spaces, 41 point evaluation, 75 point masses, 47 Mobius group, 65 point spectrum, 158 Mobius invariance, 70, 78 pointwise multipliers, 104 Mobius invariant Banach spaces, 127 Poisson extension, 255 Mobius invariant Hilbert space, 97 Poisson transform, 146, 255 Mobius invariant inner product, 131 polar coordinates, 54 Mobius invariant measure, 114 polar decomposition, 7 Mobius map, 65 polar rectangles, 307 Marcinkiewicz interpolation theorem, 41 polydisk, 249 maximal ideal space, 125 positive definite kernels, 61 maximal ideal space of H°°, 249 positive operator, 7 maximal operator, 46, 261 predual, 81 maximum principle, 36, 166, 286 probability measure, 57 mean oscillation, 208 projection, 76 mean value property, 73 pseudo-hyperbolic disk, 68 min-max characterization, 27 pseudo-hyperbolic metric, 66 monic polynomial, 157 pullback measure, 313 monomial, 119 Pythagorean theorem, 6 monotone convergence theorem, 305 multiplicity, 30 quasi-continuous functions, 291 348 INDEX quasi-normed spaces, 41 Stone-Cech compactification, 251 quotient algebra, 32 Stone-Weierstrass approximation, 226, 246, quotient norm, 4, 81 290 quotient space, 5, 81 strictly pseudo-convex domain, 96, 249 sub-mean value property, 72 radial growth, 106 subharmonic function, 303 rank-one operator, 29, 137 subordination, 303 rank-one projection, 244 surjective isometry, 8 rank-two operator, 138 symbol of a Toeplitz operator, 164 Rayleigh's equation, 26, 31 symmetric kernel, 222 real interpolation, 41 symmetry, 108 real method, 41 Szego kernel, 256 rectangular coordinates, 54 Szego projection, 256, 285 reflexive Banach space, 17 regular r-lattice, 71 theorem of F. and M. Riesz, 278 reiteration theorem, 41 Toeplitz algebras, 203 reproducing formula, 83 Toeplitz matrix, 202, 287 reproducing Hilbert spaces, 94 Toeplitz operator, 163, 285 reproducing kernel, 76, 133 trace class, 18, 252 reproducing property, 80, 133 trace formula, 174, 225 Riemannian metric, 96 trigonometric polynomial, 277 Riemman map, 109 two-sided ideal, 11, 251 Riesz representation, 3, 6 uniform boundedness principle, 4 Riesz-Thorin theorem, 39 unit vector, 6 rotation invariance, 55, 143 unitary equivalence, 8 Schatten p-ideal, 173 unitary operator, 8 Schatten class, 18 unitary transformation, 8 Schlicht disk, 125 unweighted Bergman space, 79, 233 Schur's test, 52 valence, 314 Schur's theorem, 52 vanishing Carleson measure, 166, 264 Schwarz's lemma, 102, 304 vanishing mean oscillation, 275 Segal-Bargmann space, 203 VMO, 219, 275 self-adjoint subalgebra, 251 VMO in the Bergman metric, 219 separable Hilbert space, 6 VMOA, 275 Siegel domains, 126 Volterra operator, 63 singular inner function, 108 singular integral operators, 299 weak convergence, 130 singular value decomposition, 293 weak topology, 4 singular values, 17 weak-star convergence, 130 size estimate, 207 weak-star topology, 4 spectral decomposition, 9 weight parameter, 76 spectral mapping theorem, 20, 158 weighted area measure, 57 spectral theorem, 9 weighted Bergman spaces, 72 spectral theory, 9, 300 spectrum, 9 Zorn's lemma, 8 square root of an operator, 7 standard weights, 72 Stirling's formula, 56 Titles in This Series

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104 Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, 2003 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanre, Lusternik-Schnirelmann category, 2003 102 Linda Rass and John Radcliffe, Spatial deterministic epidemics, 2003 101 Eli Glasner, Ergodic theory via joinings, 2003 100 Peter Duren and Alexander Schuster, Bergman spaces, 2004 99 Philip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, 2002 97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D. Neusel and Larry Smith, Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2: Model operators and systems, 2002 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1: Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery, A tour of subriemannian geometries, their geodesies and applications, 2002 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, second edition, 2004 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, 2001 84 Laszlo Puchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, 2000 80 Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, 2000 79 Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz, The semicircle law, free random variables and entropy, 2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000

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