The study of model spaces, the closed invariant subspaces of the backward shift operator, is a vast area of research with connections to complex analysis, operator theory and functional analysis. This self-contained text is the ideal introduction for newcomers to the field. It sets out the basic ideas and quickly takes the reader through the history of the subject before ending up at the frontier of mathematical analysis. Open questions point to potential areas of future research, offering plenty of inspiration to graduate students wishing to advance further.
Author(s): Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
Series: Cambridge Studies in Advanced Mathematics 148
Publisher: Cambridge University Press
Year: 2016
Language: English
Pages: 339
Contents......Page 8
Preface......Page 12
Notation......Page 15
1.1 Measure and integral......Page 18
1.2 Poisson integrals......Page 25
1.3 Hilbert spaces and their operators......Page 39
1.4 Notes......Page 48
2.1 Disk automorphisms......Page 49
2.2 Bounded analytic functions......Page 50
2.3 Inner functions......Page 53
2.4 Unimodular boundary limits......Page 59
2.5 Angular derivatives......Page 63
2.6 Frostman’s Theorem......Page 69
2.7 Notes......Page 72
3.1 Three approaches to the Hardy space......Page 75
3.2 The Riesz projection......Page 83
3.3 Factorization......Page 84
3.4 A growth estimate......Page 90
3.5 Associated classes of functions......Page 91
3.6 Notes......Page 95
3.7 For further exploration......Page 98
4.1 The shift operator......Page 100
4.2 Toeplitz operators......Page 107
4.3 A characterization of Toeplitz operators......Page 110
4.4 The commutant of the shift......Page 113
4.5 The backward shift......Page 116
4.6 Difference quotient operator......Page 117
4.8 For further exploration......Page 119
5.1 Model spaces as invariant subspaces......Page 121
5.2 Stability under conjugate analytic Toeplitz operators......Page 123
5.3 Containment and lattice operations......Page 125
5.4 A decomposition for Ku......Page 126
5.5 Reproducing kernels......Page 128
5.6 The projection Pu......Page 129
5.7 Finite-dimensional model spaces......Page 132
5.8 Density results......Page 135
5.9 Takenaka–Malmquist–Walsh bases......Page 137
5.10 Notes......Page 138
5.11 For further exploration......Page 141
6.1 Littlewood Subordination Principle......Page 143
6.2 Composition operators on model spaces......Page 146
6.3 Unitary maps between model spaces......Page 151
6.4 Multipliers of Ku......Page 154
6.5 Multipliers between two model spaces......Page 156
6.6 Notes......Page 158
6.7 For further exploration......Page 159
7.1 Pseudocontinuation......Page 161
7.2 Cyclicity via pseudocontinuation......Page 168
7.3 Analytic continuation......Page 169
7.4 Boundary limits......Page 175
7.5 Notes......Page 184
8.1 Abstract conjugations......Page 187
8.2 Conjugation on Ku......Page 190
8.3 Inner functions in Ku......Page 194
8.4 Generators of Ku......Page 195
8.5 Cartesian decomposition......Page 197
8.6 2 × 2 inner functions......Page 199
8.7 Notes......Page 202
9.1 What is a compression?......Page 204
9.2 The compressed shift......Page 206
9.3 Invariant subspaces and cyclic vectors......Page 210
9.4 The Sz.-Nagy–Foias¸ model......Page 212
9.5 Functional calculus for Su......Page 214
9.6 The spectrum of Su......Page 218
9.7 The C*-algebra generated by Su......Page 223
9.8 Notes......Page 229
9.9 For further exploration......Page 230
10 The commutant lifting theorem......Page 232
10.1 Minimal isometric dilations......Page 233
10.2 Existence and uniqueness......Page 234
10.3 Strong convergence......Page 239
10.4 An associated partial isometry......Page 240
10.5 The commutant lifting theorem......Page 241
10.6 The characterization of {Su}′......Page 246
10.7 Notes......Page 247
11.1 The family of Clark measures......Page 248
11.2 The Clark unitary operators......Page 252
11.3 Spectral representation of the Clark operator......Page 256
11.4 The Aleksandrov disintegration theorem......Page 262
11.5 A connection to composition operators......Page 264
11.6 Carleson measures......Page 267
11.7 Isometric embeddings......Page 268
11.8 Notes......Page 273
11.9 For further exploration......Page 275
12.1 Minimal sequences......Page 277
12.2 Uniformly minimal sequences......Page 280
12.3 Uniformly separated sequences......Page 282
12.4 The mappings Λ, V, and Γ......Page 285
12.5 Abstract Riesz sequences......Page 288
12.6 Riesz sequences in KB......Page 293
12.7 Completeness problems......Page 294
12.8 Notes......Page 295
13.1 The basics......Page 299
13.2 A characterization......Page 304
13.3 C-symmetric operators......Page 308
13.4 The spectrum of Auϕ......Page 309
13.5 An operator disintegration formula......Page 316
13.6 Norm of a truncated Toeplitz operator......Page 317
13.7 Notes......Page 318
13.8 For further exploration......Page 322
References......Page 324
Index......Page 335
