Operator Theory in Function Spaces

Second Edition

http://dx.doi.org/10.1090/surv/138

Mathematical I Surveys I

and I Monographs I

Volume 138 I

Operator Theory in Function Spaces

I Second Edition

I Kehe Zhu

E D I T O R I A L C O M M I T T E E Jer ry L. Bona Ra lph L. Cohen Michael G. Eas twood Michael P. Loss

J. T . Stafford, Chair

2000 Mathematics Subject Classification. P r i m a r y 47-02, 30-02, 46-02, 32-02.

For addi t ional information and upda t e s on this book, visit w w w . a m s . o r g / b o o k p a g e s / s u r v - 1 3 8

Library of C o n g r e s s Cataloging-in-Publicat ion D a t a 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 515/724dc22 2007060704

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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

Preface to the First Edition

Chapter 1. Bounded Linear Operators 1.1. Bounded Operators on Banach Spaces 1.2. Bounded Operators on Hilbert Spaces 1.3. Compact Operators on Hilbert Spaces 1.4. Schatten Class Operators 1.5. Notes 1.6. Exercises

Chapter 2. Interpolation of Banach Spaces 2.1. Interpolation Spaces 2.2. Complex Interpolation 2.3. W Spaces and Schatten Classes 2.4. The Marcinkiewicz Interpolation Theorem 2.5. Notes 2.6. Exercises

Chapter 3. Integral Operators on IP Spaces 3.1. Holder's Inequalities 3.2. Hilbert-Schmidt Integral Operators 3.3. Schur's Theorem 3.4. Integral Operators on the Unit Disk 3.5. Notes 3.6. Exercises

Chapter 4. Bergman Spaces 4.1. The Mobius Group 4.2. The Bergman Metric 4.3. Bergman Spaces 4.4. Kernel Functions and Related Projections 4.5. Atomic Decomposition 4.6. Notes 4.7. Exercises

xi

XV

1 1 5

10 18 30 30

33 33 36 39 41 45 46

47 47 49 52 54 61 62

65 65 66 72 76 87 94 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 10.1. Toeplitz Operators on H2

285 285

CONTENTS IX

10.2. 10.3. 10.4. 10.5. 10.6.

Chapter 11.1. 11.2. 11.3. 11.4. 11.5. 11.6.

Bounded Hankel Operators on H2 Compact Hankel Operators on H2 Schatten Class Hankel Operators on H2 Notes Exercises

11. Composition Operators Littlewood's Subordination Principle The Nevanlinna Counting Function Composition Operators on the Bergman Space Composition Operators on the Hardy Space Notes Exercises

Bibliography

287 289 291 299 301

303 303 305 308 317 323 324

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 in LP(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

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Index

C*-algebra, 13,251

346 INDEX

conjugate operator, 257 conjugation preserving, 142 continuous inclusions, 131 contravariant symbols, 159 coset, 4 counting function, 305 covariant symbols, 159

decent functional, 127 Denjoy-Wolff point, 323 direct sum, 158 Dirichlet integral, 129, 295 Dirichlet space, 115 discrete spectrum, 202 distribution function, 42 distribution of zeros, 126 dominated convergence theorem, 112 dual space, 2 duality of Lp spaces, 2

eigenspace, 149 eigenvalue, 15 eigenvector, 15 embedding, 113 equivalence relation, 4 essential norm, 32 essential spectrum, 32, 252 Euclidean center, 70 Euclidean circle, 70 Euclidean closure, 204 Euclidean disk, 69 Euclidean radius, 70 extreme points, 126

Fefferman's theorem, 272 finite Blaschke product, 126 finite rank operator, 17 finite rank Toeplitz operators, 177 fixed points, 141,323 Fock space, 203 Forelli-Rudin estimates, 234 Fourier coefficients, 46, 255 Fourier expansion, 260 Fredholm operator, 32

gamma function, 55 generalized half-planes, 126 generalized subharmonic property, 204 geodesic, 210, 228 Gleason part, 125

Green's function, 260 Green's theorem, 260

Hadamard gap series, 128 Hadamard three lines theorem, 36 Hahn-Banach extension theorem, 4 half-order differential operator, 302 Hankel matrix, 289 Hankel operator, 287 Hankel operators, 207 Hankel operators on the Bergman space,

221 Hardy spaces, 253 harmonic conjugate, 98, 256 harmonic extension, 255 harmonic function, 98 harmonic symbols, 197 Hartman's theorem, 290, 299 Hausdorff space, 13 Hausdorff-Young inequality, 46 Havin's lemma, 234 Hilbert space, 6 Hilbert-Schmidt class, 18 Hilbert-Schmidt operator, 196 holomorphic functional calculus, 148 homogeneous domains, 126 hyperbolic disk, 68 hyperbolic metric, 67

ideal, 11 identity operator, 8 inclusion map, 97, 166 infinite Blaschke product, 126 infinitesimal Bergman metric, 67 inner functions, 126 inner product, 5 inner product space, 5 inner-outer factorization, 262 integeral pairing, 113 integral kernel, 50 integral operator, 5 integral pairing, 3 integral representation, 83, 310 intermediate space, 35 interpolation, 33 interpolation space, 33 invariant mean value property, 198 invariant metric, 66 invariant pairing, 123, 295

INDEX 347

invariant semi-norm, 123 involution, 13 involutions, 65 involutive automorphisms, 65 isometric isomorphisms, 126 isometry, 8, 123 isomorphism, 115

Jacobian determinant, 66 Jensen's formula, 305 John-Nirenberg theorem, 266

kernel, 5 kernel functions, 76 Kronecker's theorem, 300

lacunary series, 100, 128 Laplacian, 201 level curves, 109 linear isometries, 126 Lipschitz, 215 Lipschitz estimate, 133 little Bloch space, 108 little Hankel operators, 207, 244 Littlewood's subordination principle, 303 Littlewood-Paley identity, 260 local homeomorphism, 307 local sub-mean value property, 303 logarithmic differentiation, 154 Lorentz spaces, 41

Mobius group, 65 Mobius invariance, 70, 78 Mobius invariant Banach spaces, 127 Mobius invariant Hilbert space, 97 Mobius invariant inner product, 131 Mobius invariant measure, 114 Mobius map, 65 Marcinkiewicz interpolation theorem, 41 maximal ideal space, 125 maximal ideal space of H, 249 maximal operator, 46, 261 maximum principle, 36, 166, 286 mean oscillation, 208 mean value property, 73 min-max characterization, 27 monic polynomial, 157 monomial, 119 monotone convergence theorem, 305 multiplicity, 30

Nehari's theorem, 289, 299 net, 4 Nevanlinna counting function, 305 nontangential limit, 312 norm, 1 norm topology, 4 normal family, 167 normal operator, 8 normal Toeplitz operators, 201 normalized area measure, 54 normalized reproducing kernel, 134, 285 normed space, 1 null space, 157

open mapping theorem, 3 open unit ball, 96 optimal growth rate, 74 orthogonal complement, 9, 26 orthogonal projection, 8 orthonormal basis, 6 orthonormal set, 8

Parseval's identity, 6 partial counting function, 305 partial isometry, 7 partition of D, 90 partition of unity, 236 Poincare metric, 96 point evaluation, 75 point masses, 47 point spectrum, 158 pointwise multipliers, 104 Poisson extension, 255 Poisson transform, 146, 255 polar coordinates, 54 polar decomposition, 7 polar rectangles, 307 polydisk, 249 positive definite kernels, 61 positive operator, 7 predual, 81 probability measure, 57 projection, 76 pseudo-hyperbolic disk, 68 pseudo-hyperbolic metric, 66 pullback measure, 313 Pythagorean theorem, 6

quasi-continuous functions, 291

348 INDEX

quasi-normed spaces, 41 quotient algebra, 32 quotient norm, 4, 81 quotient space, 5, 81

radial growth, 106 rank-one operator, 29, 137 rank-one projection, 244 rank-two operator, 138 Rayleigh's equation, 26, 31 real interpolation, 41 real method, 41 rectangular coordinates, 54 reflexive Banach space, 17 regular r-lattice, 71 reiteration theorem, 41 reproducing formula, 83 reproducing Hilbert spaces, 94 reproducing kernel, 76, 133 reproducing property, 80, 133 Riemannian metric, 96 Riemman map, 109 Riesz representation, 3, 6 Riesz-Thorin theorem, 39 rotation invariance, 55, 143

Schatten p-ideal, 173 Schatten class, 18 Schlicht disk, 125 Schur's test, 52 Schur's theorem, 52 Schwarz's lemma, 102, 304 Segal-Bargmann space, 203 self-adjoint subalgebra, 251 separable Hilbert space, 6 Siegel domains, 126 singular inner function, 108 singular integral operators, 299 singular value decomposition, 293 singular values, 17 size estimate, 207 spectral decomposition, 9 spectral mapping theorem, 20, 158 spectral theorem, 9 spectral theory, 9, 300 spectrum, 9 square root of an operator, 7 standard weights, 72 Stirling's formula, 56

Stone-Cech compactification, 251 Stone-Weierstrass approximation, 226, 246,

290 strictly pseudo-convex domain, 96, 249 sub-mean value property, 72 subharmonic function, 303 subordination, 303 surjective isometry, 8 symbol of a Toeplitz operator, 164 symmetric kernel, 222 symmetry, 108 Szego kernel, 256 Szego projection, 256, 285 theorem of F. and M. Riesz, 278 Toeplitz algebras, 203 Toeplitz matrix, 202, 287 Toeplitz operator, 163, 285 trace class, 18, 252 trace formula, 174, 225 trigonometric polynomial, 277 two-sided ideal, 11, 251

uniform boundedness principle, 4 unit vector, 6 unitary equivalence, 8 unitary operator, 8 unitary transformation, 8 unweighted Bergman space, 79, 233

valence, 314 vanishing Carleson measure, 166, 264 vanishing mean oscillation, 275 VMO, 219, 275 VMO in the Bergman metric, 219 VMOA, 275 Volterra operator, 63

weak convergence, 130 weak topology, 4 weak-star convergence, 130 weak-star topology, 4 weight parameter, 76 weighted area measure, 57 weighted Bergman spaces, 72

Zorn's lemma, 8

Titles in This Series

138 Kehe Zhu, Operator theory in function spaces, 2007 137 Mikhail G. Katz , Systolic geometry and topology, 2007 136 Jean-Michel Coron, Control and nonlinearity, 2007 135 Bennet t Chow, Sun-Chin Chu, David Glickenstein, Christ ine Guenther , James

Isenberg, Tom Ivey, D a n Knopf, Peng Lu, Feng Luo, and Lei Ni , The Ricci flow: Techniques and applications, Part I: Geometric aspects, 2007

134 Dana P. Wil l iams, Crossed products of C*-algebras, 2007 133 Andrew Knightly and Charles Li, Traces of Hecke operators, 2006 132 J . P . May and J . Sigurdsson, Parametrized homotopy theory, 2006 131 Jin Feng and Thomas G. Kurtz , Large deviations for stochastic processes, 2006 130 Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in

Euclidean spaces, 2006 129 Wil l iam M. Singer, Steenrod squares in spectral sequences, 2006 128 Athanassios S. Fokas, Alexander R. Its , Andrei A. Kapaev , and Victor Yu.

Novokshenov, Painleve transcendents, 2006 127 Nikolai Chernov and Rober to Markarian, Chaotic billiards, 2006 126 Sen-Zhong Huang, Gradient inequalities, 2006 125 Joseph A. Cima, Alec L. Matheson , and Wil l iam T. Ross , The Cauchy Transform,

2006 124 Ido Efrat, Editor, Valuations, orderings, and Milnor K-Theory, 2006 123 Barbara Fantechi, Lothar Gottsche , Luc Illusie, Steven L. Kle iman, Ni t in

Nitsure , and Ange lo Vistol i , Fundamental algebraic geometry: Grothendieck's FGA explained, 2005

122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005

121 Anton Zettl , Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian M a and Shouhong Wang, Geometric theory of incompressible flows with

applications to fluid dynamics, 2005 118 Alexandru Buium, Arithmetic differential equations, 2005 117 Volodymyr Nekrashevych, Self-similar groups, 2005 116 Alexander Koldobsky, Fourier analysis in convex geometry, 2005 115 Carlos Julio Moreno, Advanced analytic number theory: L-functions, 2005 114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005 113 Wil l iam G. Dwyer , Phi l ip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith ,

Homotopy limit functors on model categories and homotopical categories, 2004 112 Michael Aschbacher and Stephen D . Smith , The classification of quasithin groups

II. Main theorems: The classification of simple QTKE-groups, 2004 111 Michael Aschbacher and Stephen D . Smith , The classification of quasithin groups I.

Structure of strongly quasithin K-groups, 2004 110 Bennet t Chow and D a n Knopf, The Ricci flow: An introduction, 2004 109 Goro Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups,

2004 108 Michael Farber, Topology of closed one-forms, 2004 107 Jens Carsten Jantzen, Representations of algebraic groups, 2003 106 Hiroyuki Yoshida, Absolute CM-periods, 2003 105 Charalambos D . Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with

applications to economics, second edition, 2003

TITLES IN THIS SERIES

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 Phi l ip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guil lemin, Viktor Ginzburg, and Yael Karshon, Moment maps,

cobordisms, and Hamiltonian group actions, 2002 97 V . A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Mart in Markl, Steve Shnider, and J im Stasheff, Operads in algebra, topology and

physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D . Neuse l 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 Christ ian 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 Wil l iam 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

For a complete list of t i t les in th is series, visit t he AMS Bookstore a t w w w . a m s . o r g / b o o k s t o r e / .