Basic Classes of Linear Operators

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ISBN: 9783764369309 | 3764369302
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Copyright: 1/1/2004

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This book provides an introduction to functional analysis with an emphasis on the theory of linear operators and its application to differential equations, integral equations, infinite systems of linear equations, approximation theory, and numerical analysis. As textbook designed for senior undergraduate and graduate students, it begins with the geometry of Hilbert spaces and proceeds to the theory of linear operators on these spaces including Banach spaces. Presented as a natural continuation of linear algebra, the book provides a firm foundation in operator theory which is an essential part of mathematical training for students of mathematics, engineering, and other technical sciences.

Preface

xiii

Introduction

Hilbert Spaces

(50)

Complex n-Space

(2)

The Hilbert Space l2

(2)

Definition of Hilbert Space and its Elementary Properties

(3)

Distance from a Point to a Finite Dimensional Space

(2)

The Gram Determinant

(3)

Incompatible Systems of Equations

(2)

Least Square Fit

(1)

Distance to a Convex Set and Projections onto Subspaces

(2)

Orthonormal Systems

(1)

Szego Polynomials

(5)

Legendre Polynomials

(2)

Orthonormal Bases

(3)

Fourier Series

(2)

Completeness of the Legendre Polynomials

(1)

Bases for the Hilbert Space of Functions on a Square

(2)

Stability of Orthonormal Bases

(1)

Separable Spaces

(1)

Isometry of Hilbert Spaces

(2)

Example of a Non Separable Space

(13)

Exercises

(13)

Bounded Linear Operators on Hilbert Spaces

(84)

Properties of Bounded Linear Operators

(1)

Examples of Bounded Linear Operators with Estimates of Norms

(4)

Continuity of a Linear Operator

(1)

Matrix Representations of Bounded Linear Operators

(3)

Bounded Linear Functionals

(3)

Operators of Finite Rank

(1)

Invertible Operators

(5)

Inversion of Operators by the Iterative Method

(2)

Infinite Systems of Linear Equations

(2)

Integral Equations of the Second Kind

(3)

Adjoint Operators

(4)

Self Adjoint Operators

(1)

Orthogonal Projections

(1)

Two Fundamental Theorems

(2)

Projections and One-Sided Invertibility of Operators

(7)

Compact Operators

(5)

The Projection Method for Inversion of Linear Operators

(9)

The Modified Projection Method

105

(3)

Invariant Subspaces

108

(1)

The Spectrum of an Operator

109

(26)

Exercises

118

(17)

Laurent and Toeplitz Operators on Hilbert Spaces

135

(36)

Laurent Operators

135

(6)

Toeplitz Operators

141

(2)

Band Toeplitz operators

143

(9)

Toeplitz Operators with Continuous Symbols

152

(7)

Finite Section Method

159

(4)

The Finite Section Method for Laurent Operators

163

(8)

Exercises

166

(5)

Spectral Theory of Compact Self Adjoint Operators

171

(22)

Example of an Infinite Dimensional Generalization

171

(1)

The Problem of Existence of Eigenvalues and Eigenvectors

172

(2)

Eigenvalues and Eigenvectors of Operators of Finite Rank

174

(1)

Existence of Eigenvalues

175

(3)

Spectral Theorem

178

(2)

Basic Systems of Eigenvalues and Eigenvectors

180

(2)

Second Form of the Spectral Theorem

182

(1)

Formula for the Inverse Operator

183

(2)

Minimum-Maximum Properties of Eigenvalues

185

(8)

Exercises

188

(5)

Spectral Theory of Integral Operators

193

(10)

Hilbert-Schmidt Theorem

193

(3)

Preliminaries for Mercer's Theorem

196

(1)

Mercer's Theorem

197

(3)

Trace Formula for Integral Operators

200

(3)

Exercises

200

(3)

Unbounded Operators on Hilbert Space

203

(16)

Closed Operators and First Examples

203

(1)

The Second Derivative as an Operator

204

(2)

The Graph Norm

206

(2)

Adjoint Operators

208

(3)

Sturm-Liouville Operators

211

(3)

Self Adjoint Operators with Compact Inverse

214

(5)

Exercises

215

(4)

Oscillations of an Elastic String

219

(6)

The Displacement Function

219

(1)

Basic Harmonic Oscillations

220

(2)

Harmonic Oscillations with an External Force

222

(3)

Operational Calculus with Applications

225

(12)

Functions of a Compact Self Adjoint Operator

225

(5)

Differential Equations in Hilbert Space

230

(2)

Infinite Systems of Differential Equations

232

(1)

Integro-Differential Equations

233

(4)

Exercises

234

(3)

Solving Linear Equations by Iterative Methods

237

(6)

The Main Theorem

237

(1)

Preliminaries for the Proof

238

(2)

Proof of the Main Theorem

240

(2)

Application to Integral Equations

242

(1)

Further Developments of the Spectral Theorem

243

(16)

Simultaneous Diagonalization

243

(1)

Compact Normal Operators

244

(2)

Unitary Operators

246

(2)

Singular Values

248

(5)

Trace Class and Hilbert Schmidt Operators

253

(6)

Exercises

254

(5)

Banach Spaces

259

(18)

Definitions and Examples

259

(3)

Finite Dimensional Normed Linear Spaces

262

(2)

Separable Banach Spaces and Schauder Bases

264

(1)

Conjugate Spaces

265

(2)

Hahn-Banach Theorem

267

(10)

Exercises

272

(5)

Linear Operators on a Banach Space

277

(22)

Description of Bounded Operators

277

(2)

Closed Linear Operators

279

(2)

Closed Graph Theorem

281

(2)

Applications of the Closed Graph Theorem

283

(3)

Complemented Subspaces and Projections

286

(2)

One-Sided Invertibility Revisited

288

(1)

The Projection Method Revisited

289

(1)

The Spectrum of an Operator

290

(3)

Volterra Integral Operator

293

(2)

Analytic Operator Valued Functions

295

(4)

Exercises

296

(3)

Compact Operators on a Banach Space

299

(18)

Examples of Compact Operators

299

(3)

Decomposition of Operators of Finite Rank

302

(1)

Approximation by Operators of Finite Rank

303

(2)

First Results in Fredholm Theory

305

(1)

Conjugate Operators on a Banach Space

306

(4)

Spectrum of a Compact Operator

310

(3)

Applications

313

(4)

Exercises

314

(3)

Poincare Operators: Determinant and Trace

317

(30)

Determinant and Trace

317

(4)

Finite Rank Operators, Determinants and Traces

321

(6)

Theorems about the Poincare Determinant

327

(3)

Determinants and Inversion of Operators

330

(6)

Trace and Determinant Formulas for Poincare Operators

336

(11)

Exercises

340

(7)

Fredholm Operators

347

(14)

Definition and Examples

347

(1)

First Properties

347

(5)

Perturbations Small in Norm

352

(3)

Compact Perturbations

355

(1)

Unbounded Fredholm Operators

356

(5)

Exercises

358

(3)

Toeplitz and Singular Integral Operators

361

(40)

Laurent Operators on lp(Z)

361

(3)

Toeplitz Operators on lp

364

(8)

An Illustrative Example

372

(5)

Applications to Pair Operators

377

(7)

The Finite Section Method Revisited

384

(6)

Singular Integral Operators on the Unit Circle

390

(11)

Exercises

395

(6)

Non Linear Operators

401

(8)

Fixed Point Theorems

401

(1)

Applications of the Contraction Mapping Theorem

402

(3)

Generalizations

405

(4)

Appendix 1: Countable sets and Separable Hilbert Spaces

409

(2)

Appendix 2: The Lebesgue integral and Lp Spaces

411

(4)

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Basic Classes of Linear Operators

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Table of Contents

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