Elements of operator theory / Carlos S. Kubrusly.

Format:

Book

Author/Creator:

Kubrusly, Carlos S., 1947-

Language:

English

Subjects (All):

Operator theory.

Physical Description:

xiii, 527 pages : illustrations; 24 cm

Place of Publication:

Boston : Birkhäuser, [2001]

Summary:

Elements of Operator Theory is aimed at graduate students as well as a new generation of mathematicians and scientists who need to apply operator theory to their field. Written in a user-friendly, motivating style, fundamental topics are presented in a systematic fashion, i.e., set theory, algebraic structures, topological structures, Banach spaces, Hilbert spaces, culminating with the Spectral Theorem, one of the landmarks in the theory of operators on Hilbert spaces. The exposition is concept-driven and as much as possible avoids the formula-computational approach.

This work is an excellent text for the classroom as well as a self-study resource for researchers. Prerequisites include an introduction to analysis and to functions of a complex variable, which most first-year graduate students in mathematics, engineering, or another formal science have already acquired. Measure theory and integration theory are required only for the last section of the final chapter.

Contents:

1 Set-Theoretic Structures 1

1.2 Sets and Relations 3

1.3 Functions 5

1.4 Equivalence Relations 7

1.5 Ordering 8

1.6 Lattices 10

1.7 Indexing 12

1.8 Cardinality 14

2 Algebraic Structures 37

2.1 Linear Spaces 37

2.2 Linear Manifolds 43

2.3 Linear Independence 46

2.4 Hamel Basis 48

2.5 Linear Transformations 55

2.6 Isomorphisms 58

2.7 Isomorphic Equivalence 64

2.8 Direct Sum 66

2.9 Projections 70

3 Topological Structures 85

3.1 Metric Spaces 85

3.2 Convergence and Continuity 93

3.3 Open Sets and Topology 100

3.4 Equivalent Metrics and Homeomorphisms 106

3.5 Closed Sets and Closure 113

3.6 Dense Sets and Separable Spaces 119

3.7 Complete Spaces 127

3.8 Continuous Extension and Completion 134

3.9 The Baire Category Theorem 142

3.10 Compact Sets 148

3.11 Sequential Compactness 155

4 Banach Spaces 197

4.1 Normed Spaces 197

4.3 Subspaces and Quotient Spaces 209

4.4 Bounded Linear Transformations 215

4.5 The Open Mapping Theorem and Continuous Inverses 223

4.6 Equivalence and Finite-Dimensional Spaces 230

4.7 Continuous Linear Extension and Completion 237

4.8 The Banach-Steinhaus Theorem and Operator Convergence 242

4.9 Compact Operators 250

4.10 The Hahn-Banach Theorem and Dual Spaces 258

5 Hilbert Spaces 311

5.1 Inner Product Spaces 311

5.3 Orthogonality 323

5.4 Orthogonal Complement 328

5.5 Orthogonal Structure 335

5.6 Unitary Equivalence 339

5.7 Summability 343

5.8 Orthonormal Basis 352

5.9 The Fourier Series Theorem 359

5.10 Orthogonal Projection 367

5.11 The Riesz Representation Theorem and Weak Convergence 376

5.12 The Adjoint Operator 387

5.13 Self-Adjoint Operators 396

5.14 Square Root and Polar Decomposition 401

6 The Spectral Theorem 441

6.1 Normal Operators 441

6.2 The Spectrum of an Operator 449

6.3 Spectral Radius 457

6.4 Numerical Radius 462

6.5 Examples of Spectra 466

6.6 The Spectrum of a Compact Operator 474

6.7 The Spectral Theorem for Compact Normal Operators 480

6.8 A Glimpse at the Spectral Theorem for Normal Operators 489.

Notes:

Includes bibliographical references (pages [509]-515) and index.

ISBN:

0817641742

OCLC:

45743352