Commutative algebras of Toeplitz operators on the Bergman space:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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Basel [u.a.]
Birkhäuser
2008
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| Series: | Operator theory
185 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIX, 417 S. graph. Darst. |
| ISBN: | 9783764387259 3764387254 9783764387266 |
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| 100 | 1 | |a Vasilevski, Nikolai L. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Commutative algebras of Toeplitz operators on the Bergman space |c Nikolai L. Vasilevski |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2008 | |
| 300 | |a XXIX, 417 S. |b graph. Darst. | ||
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| 490 | 1 | |a Operator theory |v 185 | |
| 650 | 4 | |a Bergman spaces | |
| 650 | 4 | |a Commutative algebra | |
| 650 | 4 | |a Toeplitz operators | |
| 650 | 0 | 7 | |a Toeplitz-Operator |0 (DE-588)4191521-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Spektraltheorie |0 (DE-588)4116561-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a C-Stern-Algebra |0 (DE-588)4136693-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Bergman-Raum |0 (DE-588)4347980-7 |2 gnd |9 rswk-swf |
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Record in the Search Index
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|---|---|
| adam_text | 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 I^n)............ 73
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 L2(D)......... 96
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(B,T)) ......................... 178
8.3 Operators of the algebra T(PC(B, 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 Möbius transformations............. 217
9.3 Fixed points and commuting Möbius transformations........ 220
9.4 Elements of hyperbolic geometry................... 221
9.5 Action of Möbius 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 geodesies............... 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 geodesies . 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=lmz................................. 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
e=axgz................................. 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.I 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.I 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
|
| adam_txt |
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 I^n). 73
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 L2(D). 96
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(B,T)) . 178
8.3 Operators of the algebra T(PC(B, 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 Möbius transformations. 217
9.3 Fixed points and commuting Möbius transformations. 220
9.4 Elements of hyperbolic geometry. 221
9.5 Action of Möbius 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 geodesies. 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 geodesies . 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=lmz. 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
e=axgz. 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.I 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.I 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 |
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| index_date | 2024-07-02T19:43:51Z |
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| spelling | Vasilevski, Nikolai L. Verfasser aut Commutative algebras of Toeplitz operators on the Bergman space Nikolai L. Vasilevski Basel [u.a.] Birkhäuser 2008 XXIX, 417 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Operator theory 185 Bergman spaces Commutative algebra Toeplitz operators Toeplitz-Operator (DE-588)4191521-5 gnd rswk-swf Spektraltheorie (DE-588)4116561-5 gnd rswk-swf C-Stern-Algebra (DE-588)4136693-1 gnd rswk-swf Bergman-Raum (DE-588)4347980-7 gnd rswk-swf Toeplitz-Operator (DE-588)4191521-5 s Bergman-Raum (DE-588)4347980-7 s C-Stern-Algebra (DE-588)4136693-1 s Spektraltheorie (DE-588)4116561-5 s DE-604 Operator theory 185 (DE-604)BV000000970 185 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016301211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vasilevski, Nikolai L. Commutative algebras of Toeplitz operators on the Bergman space Operator theory Bergman spaces Commutative algebra Toeplitz operators Toeplitz-Operator (DE-588)4191521-5 gnd Spektraltheorie (DE-588)4116561-5 gnd C-Stern-Algebra (DE-588)4136693-1 gnd Bergman-Raum (DE-588)4347980-7 gnd |
| subject_GND | (DE-588)4191521-5 (DE-588)4116561-5 (DE-588)4136693-1 (DE-588)4347980-7 |
| title | Commutative algebras of Toeplitz operators on the Bergman space |
| title_auth | Commutative algebras of Toeplitz operators on the Bergman space |
| title_exact_search | Commutative algebras of Toeplitz operators on the Bergman space |
| title_exact_search_txtP | Commutative algebras of Toeplitz operators on the Bergman space |
| title_full | Commutative algebras of Toeplitz operators on the Bergman space Nikolai L. Vasilevski |
| title_fullStr | Commutative algebras of Toeplitz operators on the Bergman space Nikolai L. Vasilevski |
| title_full_unstemmed | Commutative algebras of Toeplitz operators on the Bergman space Nikolai L. Vasilevski |
| title_short | Commutative algebras of Toeplitz operators on the Bergman space |
| title_sort | commutative algebras of toeplitz operators on the bergman space |
| topic | Bergman spaces Commutative algebra Toeplitz operators Toeplitz-Operator (DE-588)4191521-5 gnd Spektraltheorie (DE-588)4116561-5 gnd C-Stern-Algebra (DE-588)4136693-1 gnd Bergman-Raum (DE-588)4347980-7 gnd |
| topic_facet | Bergman spaces Commutative algebra Toeplitz operators Toeplitz-Operator Spektraltheorie C-Stern-Algebra Bergman-Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016301211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000970 |
| work_keys_str_mv | AT vasilevskinikolail commutativealgebrasoftoeplitzoperatorsonthebergmanspace |