Basic Classes of Linear Operators

Front cover......Page 1
Title page......Page 3
Date-line......Page 4
Dedication......Page 5
Table of Contents......Page 7
Preface......Page 13
Introduction......Page 15
1.1 Complex $n$-Space......Page 19
1.2 The Hilbert Space $\\mathcal{l}_2$......Page 21
1.3 Definition of Hilbert Space and its Elementary Properties......Page 23
1.4 Distance from a Point to a Finite Dimensional Space......Page 26
1.5 The Gram Determinant......Page 28
1.6 Incompatible Systems of Equations......Page 31
1.7 Least Square Fit......Page 33
1.8 Distance to a Convex Set and Projections onto Subspaces......Page 34
1.9 Orthonormal Systems......Page 36
1.10 Szego Polynomials......Page 37
1.11 Legendre Polynomials......Page 42
1.12 Orthonormal Bases......Page 44
1.13 Fourier Series......Page 47
1.14 Completeness of the Legendre Polynomials......Page 49
1.15 Bases for the Hilbert Space of Functions on a Square......Page 50
1.16 Stability of Orthonormal Bases......Page 52
1.17 Separable Spaces......Page 53
1.18 Isometry of Hilbert Spaces......Page 54
Exercises......Page 56
2.1 Properties of Bounded Linear Operators......Page 69
2.2 Examples of Bounded Linear Operators with Estimates of Norms......Page 70
2.3 Continuity of a Linear Operator......Page 74
2.4 Matrix Representations of Bounded Linear Operators......Page 75
2.5 Bounded Linear Functionals......Page 78
2.6 Operators of Finite Rank......Page 81
2.7 Invertible Operators......Page 82
2.8 Inversion of Operators by the Iterative Method......Page 87
2.9 Infinite Systems of Linear Equations......Page 89
2.10 Integral Equations of the Second Kind......Page 91
2.11 Adjoint Operators......Page 94
2.12 Self Adjoint Operators......Page 98
2.13 Orthogonal Projections......Page 99
2.14 Two Fundamental Theorems......Page 100
2.15 Projections and One-Sided Invertibility of Operators......Page 102
2.16 Compact Operators......Page 109
2.17 The Projection Method for Inversion of Linear Operators......Page 114
2.18 The Modified Projection Method......Page 123
2.19 Invariant Subspaces......Page 126
2.20 The Spectrum of an Operator......Page 127
Exercises......Page 136
3.1 Laurent Operators......Page 153
3.2 Toeplitz Operators......Page 159
3.3 Band Toeplitz operators......Page 161
3.4 Toeplitz Operators with Continuous Symbols......Page 170
3.5 Finite Section Method......Page 177
3.6 The Finite Section Method for Laurent Operators......Page 181
Exercises......Page 184
4.1 Example of an Infinite Dimensional Generalization......Page 189
4.2 The Problem of Existence of Eigenvalues and Eigenvectors......Page 190
4.3 Eigenvalues and Eigenvectors of Operators of Finite Rank......Page 192
4.4 Existence of Eigenvalues......Page 193
4.5 Spectral Theorem......Page 196
4.6 Basic Systems of Eigenvalues and Eigenvectors......Page 198
4.7 Second Form of the Spectral Theorem......Page 200
4.8 Formula for the Inverse Operator......Page 201
4.9 Minimum-Maximum Properties of Eigenvalues......Page 203
Exercises......Page 206
5.1 Hilbert-Schmidt Theorem......Page 211
5.2 Preliminaries for Mercer\'s Theorem......Page 214
5.3 Mercer\'s Theorem......Page 215
Exercises......Page 218
6.1 Closed Operators and First Examples......Page 221
6.2 The Second Derivative as an Operator......Page 222
6.3 The Graph Norm......Page 224
6.4 Adjoint Operators......Page 226
6.5 Sturm-Liouville Operators......Page 229
6.6 Self Adjoint Operators with Compact Inverse......Page 232
Exercises......Page 233
7.1 The Displacement Function......Page 237
7.2 Basic Harmonic Oscillations......Page 238
7.3 Harmonic Oscillations with an External Force......Page 240
8.1 Functions of a Compact Self Adjoint Operator......Page 243
8.2 Differential Equations in Hilbert Space......Page 248
8.3 Infinite Systems of Differential Equations......Page 250
8.4 Integro-Differential Equations......Page 251
Exercises......Page 252
9.1 The Main Theorem......Page 255
9.2 Preliminaries for the Proof......Page 256
9.3 Proof of the Main Theorem......Page 258
9.4 Application to Integral Equations......Page 260
10.1 Simultaneous Diagonalization......Page 261
10.2 Compact Normal Operators......Page 262
10.3 Unitary Operators......Page 264
10.4 Singular Values......Page 266
10.5 Trace Class and Hilbert Schmidt Operators......Page 271
Exercises......Page 272
11.1 Definitions and Examples......Page 277
11.2 Finite Dimensional Normed Linear Spaces......Page 280
11.3 Separable Banach Spaces and Schauder Bases......Page 282
11.4 Conjugate Spaces......Page 283
11.5 Hahn-Banach Theorem......Page 285
Exercises......Page 290
12.1 Description of Bounded Operators......Page 295
12.2 Closed Linear Operators......Page 297
12.3 Closed Graph Theorem......Page 299
12.4 Applications of the Closed Graph Theorem......Page 301
12.5 Complemented Subspaces and Projections......Page 304
12.6 One-Sided Invertibility Revisited......Page 306
12.7 The Projection Method Revisited......Page 307
12.8 The Spectrum of an Operator......Page 308
12.9 Volterra Integral Operator......Page 311
12.10 Analytic Operator Valued Functions......Page 313
Exercises......Page 314
13.1 Examples of Compact Operators......Page 317
13.2 Decomposition of Operators of Finite Rank......Page 320
13.3 Approximation by Operators of Finite Rank......Page 321
13.4 First Results in Fredholm Theory......Page 323
13.5 Conjugate Operators on a Banach Space......Page 324
13.6 Spectrum of a Compact Operator......Page 328
13.7 Applications......Page 331
Exercises......Page 332
14.1 Determinant and Trace......Page 335
14.2 Finite Rank Operators, Determinants and Traces......Page 339
14.3 Theorems about the Poincare Determinant......Page 345
14.4 Determinants and Inversion of Operators......Page 348
14.5 Trace and Determinant Formulas for Poincare Operators......Page 354
Exercises......Page 358
15.2 First Properties......Page 365
15.3 Perturbations Small in Norm......Page 370
15.4 Compact Perturbations......Page 373
15.5 Unbounded Fredholm Operators......Page 374
Exercises......Page 377
16.1 Laurent Operators on $\\mathcal{l}_p(\\mathbb{Z})$......Page 379
16.2 Toeplitz Operators on $\\mathcal{l}_p$......Page 382
16.3 An Illustrative Example......Page 390
16.4 Applications to Pair Operators......Page 395
16.5 The Finite Section Method Revisited......Page 402
16.6 Singular Integral Operators on the Unit Circle......Page 408
Exercises......Page 413
17.1 Fixed Point Theorems......Page 419
17.2 Applications of the Contraction Mapping Theorem......Page 420
17.3 Generalizations......Page 423
Appendix 1: Countable sets and Separable Hilbert Spaces......Page 427
Appendix 2: The Lebesgue integral and $L_p$ Spaces......Page 429
Suggested Reading......Page 433
References......Page 435
List of Symbols......Page 437
Index......Page 439
Back cover......Page 442