Analysis of Toeplitz Operators (eBook)

Preface to the Second Edition 5
Preface to the First Edition 7
Contents 11
1 Auxiliary Material 14
1.1 Operator Ideals 14
1.2 Operator Determinants 17
1.3 Fredholm Operators 22
1.4 Operator Matrices and Their Determinants 24
1.5 Banach Algebras 25
1.6 C*- Algebras 30
1.7 Local Principles 34
1.8 Lp and Hp 40
1.9 BMO and VMO 49
1.10 Smoothness Classes 51
1.11 Notes and Comments 54
2 Basic Theory 57
2.1 Multiplication Operators 57
2.2 Toeplitz Operators 61
2.3 Hankel Operators 64
2.4 Invertibility of Toeplitz Operators on H2 69
2.5 Spectral Inclusion Theorems 76
2.6 The Connection Between Fredholmness and Invertibility 83
2.7 Compactness of Hankel Operators and C + H8 Symbols 89
2.8 Local Methods for Scalar Toeplitz Operators 97
2.9 Matrix Symbols 111
2.10 Notes and Comments 115
3 Symbol Analysis 122
3.1 Local Sectoriality 122
3.2 Asymptotic Multiplicativity 130
3.3 Piecewise Quasicontinuous Functions 139
3.4 Harmonic Approximation: Algebraization 151
3.5 Harmonic Approximation: Essentialization 160
3.6 Harmonic Approximation: Localization 163
3.7 Harmonic Approximation: Local Spectra 168
3.8 Local Sectoriality Continued 177
3.9 Notes and Comments 180
4 Toeplitz Operators on H2 182
4.1 Fredholmness 182
4.2 Stable Convergence 189
4.3 Index Computation 198
4.4 Transfinite Localization 202
4.5 Local Toeplitz Operators 222
4.6 Symbols with Specific Local Range 228
4.7 Toeplitz Algebras 237
4.8 The Role of the Harmonic Extension 251
4.9 Notes and Comments 255
5 Toeplitz Operators on Hp 259
5.1 General Theorems 259
5.2 Khvedelidze Weights 263
5.3 Locally p, -Sectorial Symbols 264
5.4 Localization 274
5.5 PC Symbols 277
5.6 P2C Symbols 287
5.7 Fisher-Hartwig Symbols 291
5.8 Notes and Comments 293
6 Toeplitz Operators on 297
6.1 Multipliers on Weighted 297
Spaces 297
6.2 Continuous Symbols 303
6.3 Piecewise Continuous Symbols 307
6.4 Analytic Symbols 318
6.5 Notes and Comments 330
7 Finite Section Method 333
7.1 Basic Facts 333
7.2 C + H8 Symbols 345
7.3 Locally Sectorial Symbols 354
7.4 PC Symbols: 357
Theory 357
7.5 PC Symbols: Hp Theory 363
7.6 Operators from alg 367
T (PC) 367
7.7 Fisher-Hartwig Symbols: H2( ) Theory 384
7.8 Fisher-Hartwig Symbols: 388
Theory 388
7.9 Invertibility Versus Finite 399
Section 399
Method 399
7.10 Pseudospectra 404
7.11 Notes and Comments 410
8 Toeplitz Operators over the Quarter-Plane 418
8.1 Function Classes on the Torus 418
8.2 Elementary Properties of Quarter-Plane Operators 426
8.3 Continuous Symbols 430
8.4 The Invertibility Problem 437
8.5 Bilocal Fredholm Theory 447
8.6 PQC . PQC Symbols 457
8.7 Finite Section Method: Kozak’s Theory 463
8.8 Finite Section Method: Bilocal Theory 470
8.9 Higher Dimensions 479
8.10 Notes and Comments 484
9 Wiener-Hopf Integral Operators 489
9.1 Basic Properties 489
9.2 Continuous Symbols 494
9.3 Piecewise Continuous Symbols 496
9.4 AP and SAP Symbols 500
9.5 Some Phenomena Caused by SAP Symbols 505
9.6 Other Oscillating Symbols 509
9.7 Finite Section Method 512
9.8 Operators over the Quarter Plane 521
9.9 Notes and Comments 528
10 Toeplitz Determinants 533
10.1 The First Szeg. o Limit Theorem 533
10.2 Krein Algebras 537
10.3 Wiener-Hopf Factorization 541
10.4 The Strong Szeg. o Limit Theorem 545
10.5 Higher Order Asymptotics 551
10.6 Semirational Symbols 562
10.7 Nonvanishing Index 566
10.8 Selfadjoint Symbols 574
10.9 The Pure Fisher-Hartwig Singularity 578
10.10 Separation Theorems 580
10.11 Fisher-Hartwig Symbols 590
10.12 More on Unbounded Symbols 600
10.13 Wiener-Hopf Determinants 603
10.14 Eigenvalues 613
10.15 Notes and Comments 619
References 628
A 628
B 629
C 634
D 635
E 637
F 638
G 638
H 641
I 643
J 643
K 643
L 645
M 646
N 647
O 647
P 647
R 649
S 651
T 654
U 655
V 655
W 656
X 658
Y 658
Z 658
Notation 659
Index 666