Table Of ContentOperator Theory: Advances and Applications
Vol. 185
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Commutative Algebras
of Toeplitz Operators
on the Bergman Space
Nikolai L. Vasilevski
Birkhäuser
Basel · Boston · Berlin
Author:
Nikolai L. Vasilevski
Departamento de Matemáticas
CINVESTAV del I.P.N.
Apartado Postal 14-740
07360 Mexico, D.F.
Mexico
e-mail: [email protected]
2000 Mathematical Subject Classification: 30C40, 46E22, 47A25, 47B10, 47B35, 47C15,
47L15, 81S10
Library of Congress Control Number: 2008930643
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Highlights of the chapters . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Preliminaries 1
1.1 General local principle for C∗-algebras . . . . . . . . . . . . . . . . 1
1.2 C∗-Algebras generated by orthogonal projections . . . . . . . . . . 14
2 Prologue 33
2.1 On the term “symbol” . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Bergman space and Bergman projection . . . . . . . . . . . . . . . 34
2.3 Representation of the Bergman kernel function . . . . . . . . . . . 38
2.4 Some integral operators and representation of the Bergman
projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5 “Continuous” theory and local properties of the Bergman
projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 Model discontinuous case . . . . . . . . . . . . . . . . . . . . . . . 50
2.7 Symbol algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.8 Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.9 Some further results on compactness . . . . . . . . . . . . . . . . . 61
3 Bergman and Poly-Bergman Spaces 65
3.1 Bergman space and Bergman projection . . . . . . . . . . . . . . . 66
3.2 Connections between Bergman and Hardy spaces . . . . . . . . . . 71
3.3 Poly-Bergman spaces, decomposition of L (Π) . . . . . . . . . . . . 73
2
3.4 Projections onto the poly-Bergman spaces . . . . . . . . . . . . . . 76
3.5 Poly-Bergman spaces and two-dimensional singular integral
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Bergman Type Spaces on the Unit Disk 89
4.1 Bergman space and Bergman projection . . . . . . . . . . . . . . . 89
4.2 Poly-Bergman type spaces, decomposition of L (D) . . . . . . . . . 96
2
vi Contents
5 Toeplitz Operators with Commutative Symbol Algebras 101
5.1 Semi-commutator versus commutator . . . . . . . . . . . . . . . . . 102
5.2 Infinite dimensional representations . . . . . . . . . . . . . . . . . . 105
5.3 Spectra and compactness . . . . . . . . . . . . . . . . . . . . . . . 110
5.4 Finite dimensional representations . . . . . . . . . . . . . . . . . . 114
5.5 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Toeplitz Operators on the Unit Disk with Radial Symbols 121
6.1 Toeplitz operators with radial symbols . . . . . . . . . . . . . . . . 122
6.2 Algebras of Toeplitz operators . . . . . . . . . . . . . . . . . . . . . 132
7 Toeplitz Operators on the Upper Half Plane with Homogeneous
Symbols 135
7.1 Representation of the Bergman space . . . . . . . . . . . . . . . . . 135
7.2 Toeplitz operators with homogeneous symbols . . . . . . . . . . . . 138
7.3 Bergman projection and homogeneous functions . . . . . . . . . . . 146
7.4 Algebra generated by the Bergman projection and discontinuous
coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.5 Some particular cases . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.6 Toeplitz operator algebra. A first look . . . . . . . . . . . . . . . . 162
7.7 Toeplitz operator algebra. Some more analysis . . . . . . . . . . . . 165
8 Anatomy of the Algebra Generated by Toeplitz Operators with
Piece-wise Continuous Symbols 175
8.1 Symbol class and operators . . . . . . . . . . . . . . . . . . . . . . 177
8.2 Algebra T (PC(D,T)) . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.3 Operators of the algebra T (PC(D,T)) . . . . . . . . . . . . . . . . 180
8.4 Toeplitz operators of the algebra T (PC(D,T)) . . . . . . . . . . . 183
8.5 More Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . . 187
8.6 Semi-commutators involving unbounded symbols . . . . . . . . . . 198
8.7 Toeplitz or not Toeplitz . . . . . . . . . . . . . . . . . . . . . . . . 206
8.8 Technical statements . . . . . . . . . . . . . . . . . . . . . . . . . . 209
9 Commuting Toeplitz Operators and Hyperbolic Geometry 215
9.1 Bergman metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
9.2 Basic properties of M¨o¨bius transformations . . . . . . . . . . . . . 217
9.3 Fixed points and commuting Mo¨¨bius transformations . . . . . . . . 220
9.4 Elements of hyperbolic geometry . . . . . . . . . . . . . . . . . . . 221
9.5 Action of Mo¨¨bius transformations . . . . . . . . . . . . . . . . . . . 224
9.6 Classification theorem . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.7 Proof of the classification theorem . . . . . . . . . . . . . . . . . . 228
Contents vii
10 Weighted Bergman Spaces 233
10.1 Unit disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
10.2 Upper half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
10.3 Representations of the weighted Bergman space . . . . . . . . . . . 240
10.4 Model classes of Toeplitz operators . . . . . . . . . . . . . . . . . . 250
10.5 Boundedness, spectra, and invariant subspaces . . . . . . . . . . . 260
11 Commutative Algebras of Toeplitz Operators 263
11.1 On symbol classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
11.2 Commutativity on a single Bergman space . . . . . . . . . . . . . . 267
11.3 Commutativity on each weighted Bergman space . . . . . . . . . . 270
11.4 First term: common gradient and level lines . . . . . . . . . . . . . 272
11.5 Second term: gradient lines are geodesics . . . . . . . . . . . . . . . 275
11.6 Curves with constant geodesic curvature . . . . . . . . . . . . . . . 278
11.7 Third term: level lines are cycles . . . . . . . . . . . . . . . . . . . 285
11.8 Commutative Toeplitz operator algebras and pencils of geodesics . 290
12 Dynamics of Properties of Toeplitz Operators with Radial Symbols 293
12.1 Boundedness and compactness properties . . . . . . . . . . . . . . 294
12.2 Schatten classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
12.3 Spectra of Toeplitz operators, continuous symbols . . . . . . . . . . 314
12.4 Spectra of Toeplitz operators, piece-wise continuous symbols . . . 318
12.5 Spectra of Toeplitz operators, unbounded symbols . . . . . . . . . 324
13 Dynamics of Properties of Toeplitz operators
on the Upper Half Plane: Parabolic case 329
13.1 Boundedness of Toeplitz operators with symbols depending on
y =Imz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
13.2 Continuous symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 339
13.3 Piece-wise continuous symbols . . . . . . . . . . . . . . . . . . . . . 341
13.4 Oscillating symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
13.5 Unbounded symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 345
14 Dynamics of Properties of Toeplitz operators
on the Upper Half Plane: Hyperbolic case 349
14.1 Boundedness of Toeplitz operators with symbols depending on
θ =argz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
14.2 Continuous symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 353
14.3 Piece-wise continuous symbols . . . . . . . . . . . . . . . . . . . . . 355
14.4 Unbounded symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 358
viii Contents
Appendices
A Coherent states and Berezin transform 361
A.1 General approach to coherent states . . . . . . . . . . . . . . . . . 361
A.2 Numerical range and spectra . . . . . . . . . . . . . . . . . . . . . 365
A.3 Coherent states in the Bergman space . . . . . . . . . . . . . . . . 367
A.4 Berezin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
B Berezin Quantization on the Unit Disk 373
B.1 Definition of the quantization . . . . . . . . . . . . . . . . . . . . . 373
B.2 Quantization on the unit disk . . . . . . . . . . . . . . . . . . . . . 375
B.3 Two first terms of asymptotic of the Wick symbol . . . . . . . . . 376
B.4 Three first terms of asymptotic in a commutator . . . . . . . . . . 380
Bibliographical Remarks 391
Bibliography 397
List of Figures 413
Index 415
Preface
The book is devoted to the spectral theory of commutative C∗-algebrasof Toeplitz
operators on Bergman spaces, and its applications. For each such commutative
algebra we construct a unitary operator which reduces each Toeplitz operator from
this algebra to a certain multiplication operator, thus also providing its spectral
type representation.This givesus a powerful researchtool allowing direct access to
the majority of the important properties of the Toeplitz operators studied herein.
The presence and exploitation of these spectral type representations forms the
basis for an essential part of the results presented in this book.
We give a criterion of when the algebras are commutative on each commonly
considered weighted Bergman space. For Toeplitz operators generating such com-
mutative algebras we describe their boundedness, compactness, and spectral prop-
erties. Furthermore, the above commutative algebras serve as model or local cases
for a number of problems treated in the book, thus making their solutions possible.
We note that in the Bergman space case considered in the book the un-
derlying manifold (the unit disk equipped with the hyperbolic metric) possesses
a richer geometric structure in comparison with the Hardy space case (the unit
circle). This fact has an important reflection in the presented theory.
We mention as well that from the general operator point of view the Toeplitz
operators, both on Hardy space and on Bergman space, are compressions of multi-
plication operators onto certain subspaces, and thus they represent two interesting
different models of operators having similar structure. At the same time the pre-
sented results show clearly essential differences between the theories for these two
species of operators.
The book is addressed to a wide audience of mathematicians, from graduate
students to researchers, whose primary interests lie in complex analysis and op-
erator theory. The prerequisites for reading this book include a basic knowledge
in one-dimensional complex analysis, functional analysis, and operator theory. An
acquaintance with some facts of the theory of Banach and C∗-algebras will be use-
ful as well. Among various excellent sources which may serve for the preliminary
reading we mention, for example, the books by I. Gohberg, S. Goldberg, and M.
Kaashoek [81], and R. Douglas [58].
Theauthorisgreatlyindebtedto hiscolleaguesSergueiGrudsky,hiscoauthor
in many papers, and Michael Porter who read the manuscript and made many
important suggestions essentially improving the book. The author would like to
address special words of gratitude to Olga Grudskaia who tragically died in a car
accident in February 2004. She generously assisted in the preparation of this book,
beautifully elaborating all the figures of the last three chapters.
The author is sincerelygrateful to IsraelGohberg,the editor of the “Operator
Theory: Advances and Applications” book series, for his invitation to publish the
book in this series and whose friendly remarks and advice greatly assisted in the
final stage of its preparation.
