Introduction to linear operator theory

Cover......Page 1
Half Title......Page 2
Series Page......Page 3
Title Page......Page 8
Copyright Page......Page 9
Dedication......Page 10
PREFACE......Page 12
CONTENTS......Page 14
 1.1 The Algebra of Sets......Page 22
 1.2 Partially Ordered Sets......Page 25
 1.3 Topology and Topological Spaces......Page 27
 1.4 Baire's Theorem......Page 37
 2.1 Linear Spaces......Page 40
 2.2 Linear Independence......Page 42
 2.3 Sets in Linear Spaces......Page 43
 2.4 Classes of Spaces: Isomorphic Spaces, Quotient Spaces, and Complementary Spaces......Page 46
 2.5 Seminorms and Norms on Linear Spaces......Page 48
 2.6 Linear Topological Spaces......Page 50
 2.7 Banach Spaces......Page 51
 2.8 Linear Operators on Banach Spaces......Page 53
 2.9 Uniformly Convex and Rotund Banach Spaces: Some Generalizations......Page 56
 2.10 The Hahn-Banach Extension Theorem......Page 60
 2.11 Extension Theorems for Complex Banach Spaces......Page 64
 2.12 Three Basic Theorems of Linear Analysis......Page 66
 2.13 Convergence in Banach Spaces......Page 77
 2.14 The Adjoint of an Operator......Page 84
 2.15 The Spectrum of an Operator......Page 86
 2.16 The Local Spectrum of an Operator......Page 91
 2.17 Analytic Representation of the Dual of Some Banach Spaces......Page 97
 2.18 Measures of Noncompactness and Classes of Mappings on Banach Spaces......Page 105
 3.1 Inner Products on Linear Spaces......Page 130
 3.2 Orthonormal Bases and the Bessel Inequality......Page 136
 3.3 Separable Hilbert Spaces: Gram-Schmidt Orthogonalization Method......Page 140
 3.4 Orthogonal Subspaces of a Hilbert Space......Page 146
 3.5 The Dual of a Hilbert Space......Page 147
 3.6 Classes of Bounded Linear Operators on Hilbert Spaces......Page 150
 4.1 Definitions and Some Examples......Page 163
 4.2 The Spectrum of an Element in a Banach Algebra with Unit......Page 168
 4.3 Representation Theorems for Commutative Banach Algebras......Page 172
 4.4 Structure Theorems for Commutative Banach Algebras......Page 175
 4.5 Representation Theorems for Noncommutative Banach Algebras......Page 188
 5.1 Semispectral and Spectral Families of Radon Measures......Page 195
 5.2 Measurability and Integrability with Respect to Spectral Families......Page 200
 5.3 A Representation Theorem for L∞......Page 202
 5.4 Spectral Decomposition of Some Classes of Operators......Page 209
 5.5 Some Remarks on the Spectral Mapping Theorem for Hermitian and Normal Operators......Page 212
 6.1 The Numerical Range for Bounded Linear Operators on Hilbert Spaces......Page 221
 6.2 The Numerical Range and the Spectrum......Page 227
 6.3 The Numerical Range and Its Closure......Page 235
 6.4 The Essential Numerical Range for Bounded Linear Operators on Hilbert Spaces......Page 238
 6.5 The Maximal Numerical Range of a Bounded Operator on a Hilbert Space......Page 240
 6.6 The Extreme Points of the Numerical Range for Hyponormal Operators and (WN) Operators......Page 242
 6.7 The Numerical Range and Some Classes of Operators......Page 244
 6.8 The Numerical Range and Tensor Products......Page 245
 6.9 The Numerical Range for Bounded Linear Operators on Banach Spaces......Page 247
 6.10 The Exponential Function on the Set of All Bounded Linear Operators on a Banach Space......Page 254
 6.11 The Numerical Radius, the Spectral Radius, and the Norm of a Bounded Linear Operator on a Banach Space......Page 257
 6.12 Hermitian and Normal Operators on Banach Spaces......Page 259
 6.13 Normal Operators on Banach Spaces......Page 265
 6.14 Classes of Elements in Banach Algebras with Unit: The Vidav-Palmer Theorem......Page 267
 6.15 Some Properties of Hermitian and Normal Elements of a Banach Algebra......Page 276
 6.16 The Numerical Radius and the Iterates of an Element......Page 281
 6.17 The Numerical Range of Elements of Locallym-Convex Algebras......Page 283
7 NONNORMAL CLASSES OF OPERATORS......Page 286
 7.1 Classes of Nonnormal Operators......Page 287
 7.2 Spectral Sets and Dilations of Operators......Page 296
 7.3 Operators with G1 Property and Some Generalizations......Page 307
 7.4 Operators with Property Re σ(T) = σ(Re T)......Page 318
 7.5 The Class R1......Page 323
 7.6 Other Classes of Bounded Operators......Page 328
 8.1 Conditions Implying Hermitianity......Page 331
 8.2 Conditions Implying Unitarity......Page 338
 8.3 Conditions Implying Normality......Page 342
 9.1 Symmetrizable Operators on Hilbert Spaces......Page 365
 9.2 Symmetrizable Elements in Banach Algebras......Page 370
 9.3 Inner Products on Banach Spaces: Symmetrizable Operators and Some Generalizations......Page 373
 9.4 Some Applications of Symmetrizable Operators and Quasi-Normalizable Operators......Page 390
 9.5 Further Results on Symmetrizable Operators on Hilbert Spaces......Page 395
 10.1 Invariant Subspaces: Some Existence Theorems......Page 401
 10.2 Reducing Invariant Subspaces......Page 419
 10.3 Some Structure Theorems......Page 429
 11.1 Preliminaries and Some General Results......Page 433
 11.2 Weyl's Theorem......Page 441
 11.3 Weyl's Theorem for Some Classes of Operators......Page 447
 11.4 The Weyl Spectrum of an Element in a von Neumann Algebra......Page 454
 11.5 The von Neumann Theorem......Page 460
 12.0 Introduction......Page 465
 12.1 Self-Adjoint Operators......Page 467
 12.2 Classes of Vectors for an Operator......Page 473
 12.3 Analytic and Quasi-Analytic Vectors and Essentially Self-Adjoint Operators......Page 476
 12.4 Quasi-Analytic Vectors and Semigroups of Operators......Page 484
 12.5 Analytic and Quasi-Analytic Elements in Commutative Banach Algebras......Page 487
13 SCHWARZ NORMS......Page 489
 13.1 Schwarz Norms......Page 490
 13.2 A New Class of Schwarz Norms......Page 497
 13.3 Schwarz Norms on Banach Spaces......Page 499
 14.1 Holomorphic Functions......Page 504
 14.2 Subharmonic Functions......Page 506
 14.3 Maximum Theorems for the Norm......Page 519
 14.4 Maximum Theorems for the Spectral Radius and for the Essential Spectral Radius......Page 529
 14.5 Maximum Theorems for Other Operator-Valued Holomorphic Functions......Page 549
 15.1 Classes of Operators......Page 558
 15.2 Applications to Markov Processes......Page 566
APPENDIX. Cp CLASSES......Page 570
REFERENCES......Page 574
SYMBOL INDEX......Page 594
SUBJECT INDEX......Page 596