Analysis of Toeplitz Operators

von: Albrecht Böttcher, Bernd Silbermann

Springer-Verlag, 2006

ISBN: 9783540324362 , 665 Seiten

2. Auflage

Format: PDF, OL

Kopierschutz: Wasserzeichen

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Mehr zum Inhalt

Analysis of Toeplitz Operators


 

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