Böttcher / Silbermann Analysis of Toeplitz Operators

2. Auflage 2006
ISBN: 978-3-540-32436-2
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 665 Seiten

Since the late 1980s, Toeplitz operators and matrices have remained a ?eld of extensive research and the development during the last nearly twenty years is impressive. One encounters Toeplitz matrices in plenty of applications on the one hand, and Toeplitz operators con?rmed their role as the basic elementary building blocks of more complicated operators on the other. Several monographs on Toeplitz and Hankel operators were written d- ing the last decade. These include Peller’s grandiose book on Hankel ope- tors and their applications and Nikolski’s beautiful easy reading on operators, functions, and systems, with emphasis on topics connected with the names of Hardy, Hankel, and Toeplitz. They also include books by the authors together withHagen,Roch,Yu.Karlovich,Spitkovsky,Grudsky,andRabinovich.Thus, results, techniques, and developments in the ?eld of Toeplitz operators are now well presented in the monographic literature. Despite these competitive works, we felt that large parts of the ?rst edition of the present monograp- whichismeanwhileoutofstock-havenotlosttheirfascinationandrelevance. Moreover, the ?rst edition has received a warm reception by many colleagues and became a standard reference. This encouraged us to venture on thinking about a second edition, and we are grateful to the Springer Publishing House for showing an interest in this.

Autoren/Hrsg.

Böttcher, Albrecht

Silbermann, Bernd

Inhaltsverzeichnis

1;Preface to the Second Edition;5
2;Preface to the First Edition;7
3;Contents;11
4;1 Auxiliary Material;14
4.1;1.1 Operator Ideals;14
4.2;1.2 Operator Determinants;17
4.3;1.3 Fredholm Operators;22
4.4;1.4 Operator Matrices and Their Determinants;24
4.5;1.5 Banach Algebras;25
4.6;1.6 C*- Algebras;30
4.7;1.7 Local Principles;34
4.8;1.8 Lp and Hp;40
4.9;1.9 BMO and VMO;49
4.10;1.10 Smoothness Classes;51
4.11;1.11 Notes and Comments;54
5;2 Basic Theory;57
5.1;2.1 Multiplication Operators;57
5.2;2.2 Toeplitz Operators;61
5.3;2.3 Hankel Operators;64
5.4;2.4 Invertibility of Toeplitz Operators on H2;69
5.5;2.5 Spectral Inclusion Theorems;76
5.6;2.6 The Connection Between Fredholmness and Invertibility;83
5.7;2.7 Compactness of Hankel Operators and C + H8 Symbols;89
5.8;2.8 Local Methods for Scalar Toeplitz Operators;97
5.9;2.9 Matrix Symbols;111
5.10;2.10 Notes and Comments;115
6;3 Symbol Analysis;122
6.1;3.1 Local Sectoriality;122
6.2;3.2 Asymptotic Multiplicativity;130
6.3;3.3 Piecewise Quasicontinuous Functions;139
6.4;3.4 Harmonic Approximation: Algebraization;151
6.5;3.5 Harmonic Approximation: Essentialization;160
6.6;3.6 Harmonic Approximation: Localization;163
6.7;3.7 Harmonic Approximation: Local Spectra;168
6.8;3.8 Local Sectoriality Continued;177
6.9;3.9 Notes and Comments;180
7;4 Toeplitz Operators on H2;182
7.1;4.1 Fredholmness;182
7.2;4.2 Stable Convergence;189
7.3;4.3 Index Computation;198
7.4;4.4 Transfinite Localization;202
7.5;4.5 Local Toeplitz Operators;222
7.6;4.6 Symbols with Specific Local Range;228
7.7;4.7 Toeplitz Algebras;237
7.8;4.8 The Role of the Harmonic Extension;251
7.9;4.9 Notes and Comments;255
8;5 Toeplitz Operators on Hp;259
8.1;5.1 General Theorems;259
8.2;5.2 Khvedelidze Weights;263
8.3;5.3 Locally p, -Sectorial Symbols;264
8.4;5.4 Localization;274
8.5;5.5 PC Symbols;277
8.6;5.6 P2C Symbols;287
8.7;5.7 Fisher-Hartwig Symbols;291
8.8;5.8 Notes and Comments;293
9;6 Toeplitz Operators on ;297
9.1;6.1 Multipliers on Weighted ;297
9.2;Spaces;297
9.3;6.2 Continuous Symbols;303
9.4;6.3 Piecewise Continuous Symbols;307
9.5;6.4 Analytic Symbols;318
9.6;6.5 Notes and Comments;330
10;7 Finite Section Method;333
10.1;7.1 Basic Facts;333
10.2;7.2 C + H8 Symbols;345
10.3;7.3 Locally Sectorial Symbols;354
10.4;7.4 PC Symbols: ;357
10.5;Theory;357
10.6;7.5 PC Symbols: Hp Theory;363
10.7;7.6 Operators from alg;367
10.8;T (PC);367
10.9;7.7 Fisher-Hartwig Symbols: H2( ) Theory;384
10.10;7.8 Fisher-Hartwig Symbols: ;388
10.11;Theory;388
10.12;7.9 Invertibility Versus Finite;399
10.13;Section;399
10.14;Method;399
10.15;7.10 Pseudospectra;404
10.16;7.11 Notes and Comments;410
11;8 Toeplitz Operators over the Quarter-Plane;418
11.1;8.1 Function Classes on the Torus;418
11.2;8.2 Elementary Properties of Quarter-Plane Operators;426
11.3;8.3 Continuous Symbols;430
11.4;8.4 The Invertibility Problem;437
11.5;8.5 Bilocal Fredholm Theory;447
11.6;8.6 PQC . PQC Symbols;457
11.7;8.7 Finite Section Method: Kozak’s Theory;463
11.8;8.8 Finite Section Method: Bilocal Theory;470
11.9;8.9 Higher Dimensions;479
11.10;8.10 Notes and Comments;484
12;9 Wiener-Hopf Integral Operators;489
12.1;9.1 Basic Properties;489
12.2;9.2 Continuous Symbols;494
12.3;9.3 Piecewise Continuous Symbols;496
12.4;9.4 AP and SAP Symbols;500
12.5;9.5 Some Phenomena Caused by SAP Symbols;505
12.6;9.6 Other Oscillating Symbols;509
12.7;9.7 Finite Section Method;512
12.8;9.8 Operators over the Quarter Plane;521
12.9;9.9 Notes and Comments;528
13;10 Toeplitz Determinants;533
13.1;10.1 The First Szeg. o Limit Theorem;533
13.2;10.2 Krein Algebras;537
13.3;10.3 Wiener-Hopf Factorization;541
13.4;10.4 The Strong Szeg. o Limit Theorem;545
13.5;10.5 Higher Order Asymptotics;551
13.6;10.6 Semirational Symbols;562
13.7;10.7 Nonvanishing Index;566
13.8;10.8 Selfadjoint Symbols;574
13.9;10.9 The Pure Fisher-Hartwig Singularity;578
13.10;10.10 Separation Theorems;580
13.11;10.11 Fisher-Hartwig Symbols;590
13.12;10.12 More on Unbounded Symbols;600
13.13;10.13 Wiener-Hopf Determinants;603
13.14;10.14 Eigenvalues;613
13.15;10.15 Notes and Comments;619
14;References;628
14.1;A;628
14.2;B;629
14.3;C;634
14.4;D;635
14.5;E;637
14.6;F;638
14.7;G;638
14.8;H;641
14.9;I;643
14.10;J;643
14.11;K;643
14.12;L;645
14.13;M;646
14.14;N;647
14.15;O;647
14.16;P;647
14.17;R;649
14.18;S;651
14.19;T;654
14.20;U;655
14.21;V;655
14.22;W;656
14.23;X;658
14.24;Y;658
14.25;Z;658
15;Notation;659
16;Index;666