Commutative algebras of Toeplitz operators on the Bergman space

Contents......Page 6
Preface......Page 10
Introduction......Page 12
1. Preliminaries......Page 15
1.1 General local principle for C*-algebras......Page 30
1.2 C*-Algebras generated by orthogonal projections......Page 43
2.1 On the term \"symbol\"......Page 62
2.2 Bergman space and Bergman projection......Page 63
2.3 Representation of the Bergman kernel function......Page 67
2.4 Some integral operators and representation of the Bergman projection......Page 71
2.5 \"Continuous\" theory and local properties of the Bergman projection......Page 74
2.6 Model discontinuous case......Page 79
2.7 Symbol algebra......Page 82
2.8 Toeplitz operators......Page 86
2.9 Some further results on compactness......Page 90
3. Bergman and Poly-Bergman Spaces......Page 93
3.1 Bergman space and Bergman projection......Page 94
3.2 Connections between Bergman and Hardy spaces......Page 99
3.3 Poly-Bergman spaces, decomposition of L[sup(2)](II)......Page 101
3.4 Projections onto the poly-Bergman spaces......Page 104
3.5 Poly-Bergman spaces and two-dimensional singular integral operators......Page 110
4.1 Bergman space and Bergman projection......Page 117
4.2 Poly-Bergman type spaces, decomposition of L[sup(2)](D)......Page 124
5. Toeplitz Operators with Commutative Symbol Algebras......Page 129
5.1 Semi-commutator versus commutator......Page 130
5.2 Infinite dimensional representations......Page 133
5.3 Spectra and compactness......Page 138
5.4 Finite dimensional representations......Page 142
5.5 General case......Page 144
6. Toeplitz Operators on the Unit Disk with Radial Symbols......Page 148
6.1 Toeplitz operators with radial symbols......Page 149
6.2 Algebras of Toeplitz operators......Page 159
7.1 Representation of the Bergman space......Page 162
7.2 Toeplitz operators with homogeneous symbols......Page 165
7.3 Bergman projection and homogeneous functions......Page 173
7.4 Algebra generated by the Bergman projection and discontinuous coefficients......Page 178
7.5 Some particular cases......Page 185
7.6 Toeplitz operator algebra. A first look......Page 189
7.7 Toeplitz operator algebra. Some more analysis......Page 192
8. Anatomyof the Algebra Generated by Toeplitz Operators with Piece-wise Continuous Symbols......Page 201
8.1 Symbol class and operators......Page 203
8.2 Algebra T (PC(D, T))......Page 204
8.3 Operators of the algebra T (PC(D, T))......Page 206
8.4 Toeplitz operators of the algebra T (PC(D, T))......Page 209
8.5 More Toeplitz operators......Page 213
8.6 Semi-commutators involving unbounded symbols......Page 224
8.7 Toeplitz or not Toeplitz......Page 232
8.8 Technical statements......Page 235
9. Commuting Toeplitz Operators and Hyperbolic Geometry......Page 241
9.1 Bergmanmetric......Page 242
9.2 Basic properties of Möbius transformations......Page 243
9.3 Fixed points and commuting Möbius transformations......Page 246
9.4 Elements of hyperbolic geometry......Page 247
9.5 Action of Möbius transformations......Page 250
9.6 Classification theorem......Page 252
9.7 Proof of the classification theorem......Page 254
10.1 Unit disk......Page 258
10.2 Upper half-plane......Page 262
10.3 Representations of the weighted Bergman space......Page 265
10.4 Model classes of Toeplitz operators......Page 275
10.5 Boundedness, spectra, and invariant subspaces......Page 285
11. Commutative Algebras of Toeplitz Operators......Page 287
11.1 On symbol classes......Page 288
11.2 Commutativity on a single Bergman space......Page 291
11.3 Commutativity on each weighted Bergman space......Page 294
11.4 First term: common gradient and level lines......Page 296
11.5 Second term: gradient lines are geodesics......Page 299
11.6 Curves with constant geodesic curvature......Page 302
11.7 Third term: level lines are cycles......Page 309
11.8 Commutative Toeplitz operator algebras and pencils of geodesics......Page 314
12. Dynamics of Properties of Toeplitz Operators with Radial Symbols......Page 316
12.1 Boundedness and compactness properties......Page 317
12.2 Schatten classes......Page 328
12.3 Spectra of Toeplitz operators, continuous symbols......Page 337
12.4 Spectra of Toeplitz operators, piece-wise continuous symbols......Page 341
12.5 Spectra of Toeplitz operators, unbounded symbols......Page 347
13.1 Boundedness of Toeplitz operators with symbols depending on y = Im z......Page 351
13.2 Continuous symbols......Page 361
13.3 Piece-wise continuous symbols......Page 363
13.4 Oscillating symbols......Page 365
13.5 Unbounded symbols......Page 367
14.1 Boundedness of Toeplitz operators with symbols depending on θ = arg z......Page 370
14.2 Continuous symbols......Page 374
14.3 Piece-wise continuous symbols......Page 376
14.4 Unbounded symbols......Page 379
A.1 General approach to coherent states......Page 382
A.2 Numerical range and spectra......Page 386
A.3 Coherent states in the Bergman space......Page 388
A.4 Berezin transform......Page 389
B.1 Definition of the quantization......Page 393
B.2 Quantization on the unit disk......Page 395
B.3 Two first terms of asymptotic of the Wick symbol......Page 396
B.4 Three first terms of asymptotic in a commutator......Page 400
Bibliographical Remarks......Page 411
Bibliography......Page 416
List of Figures......Page 432
F......Page 434
P......Page 435
W......Page 436