Spectral Theory of Bounded Linear Operators

Description

This textbook introduces spectral theory for bounded linear operators by focusing on (i) the spectral theory and functional calculus for normal operators acting on Hilbert spaces; (ii) the Riesz-Dunford functional calculus for Banach-space operators; and (iii) the Fredholm theory in both Banach and Hilbert spaces. Detailed proofs of all theorems are included and presented with precision and clarity, especially for the spectral theorems, allowing students to thoroughly familiarize themselves with all the important concepts. Covering both basic and more advanced material, the five chapters and two appendices of this volume provide a modern treatment on spectral theory. Topics range from spectral results on the Banach algebra of bounded linear operators acting on Banach spaces to functional calculus for Hilbert and Banach-space operators, including Fredholm and multiplicity theories. Supplementary propositions and further notes are included as well, ensuring a wide range of topics in spectral theory are covered. Spectral Theory of Bounded Linear Operators is ideal for graduate students in mathematics, and will also appeal to a wider audience of statisticians, engineers, and physicists. Though it is mostly self-contained, a familiarity with functional analysis, especially operator theory, will be helpful. Carlos Kubrusly is Emeritus Professor at Catholic University of Rio de Janeiro. His research focuses on operator theory, particularly on weak and strong dynamics of Hilbert-space operators and their connection with the Invariant Subspace Problem. He has published over 100 scientific articles in international journals, five books, and served as the editor-in-chief of the journal Computational and Applied Mathematics. Preface......Page 7 Contents......Page 10 1.1 Background: Notation and Terminology......Page 12 1.2 Inverse Theorems on Banach Spaces......Page 14 1.3 Orthogonal Structure in Hilbert Spaces......Page 16 1.4 Orthogonal Projections on Inner Product Spaces......Page 18 1.5 The Adjoint Operator on Hilbert Space......Page 20 1.6 Normal and Hyponormal Operators......Page 23 1.7 Orthogonal Reducing Eigenspaces......Page 27 1.8 Compact Operators and the Fredholm Alternative......Page 28 1.9 Supplementary Propositions......Page 32 Suggested Readings......Page 37 2.1 Basic Spectral Properties......Page 38 2.2 A Classical Partition of the Spectrum......Page 41 2.3 Spectral Mapping Theorems......Page 46 2.4 Spectral Radius and Normaloid Operators......Page 49 2.5 Numerical Radius and Spectraloid Operators......Page 52 2.6 Spectrum of a Compact Operator......Page 55 2.7 Supplementary Propositions......Page 58 Suggested Readings......Page 64 3.1 Spectral Theorem for Compact Operators......Page 65 3.2 Diagonalizable Operators......Page 69 3.3 Spectral Measure......Page 73 3.4 Spectral Theorem: The General Case......Page 78 3.5 Fuglede Theorems and Reducing Subspaces......Page 89 3.6 Supplementary Propositions......Page 94 Suggested Readings......Page 100 4.1 Rudiments of Banach and C*-Algebras......Page 101 4.2 Functional Calculus for Normal Operators......Page 105 4.3 Analytic Functional Calculus: Riesz Functional Calculus......Page 112 4.4 Riesz Idempotents and the Riesz Decomposition Theorem......Page 125 4.5 Supplementary Propositions......Page 136 Suggested Readings......Page 140 5.1 Fredholm and Weyl Operators — Fredholm Index......Page 141 5.2 Essential (Fredholm) Spectrum and Spectral Picture......Page 151 5.3 Riesz Points and Weyl Spectrum......Page 163 5.4 Browder Operators and Browder Spectrum......Page 172 5.5 Remarks on Browder and Weyl Theorems......Page 188 5.6 Supplementary Propositions......Page 194 Suggested Readings......Page 196 Appendices......Page 197 A.1 Quotient Space......Page 199 A.2 Complementation......Page 206 A.3 Range-Kernel Complementation......Page 210 A.4 Upper-Lower and Left-Right Approaches......Page 214 A.5 Essential Spectrum and Spectral Picture Again......Page 219 Suggested Readings......Page 227 B.1 Meanings of Multiplicity......Page 228 B.2 Complete Set of Unitary Invariants......Page 229 B.3 Multiplicity Function for Compact Operators......Page 230 B.4 Multiplicity-Free Normal Operators......Page 233 B.5 Multiplicity Function for Normal Operators......Page 236 Suggested Readings......Page 240 References......Page 241 List of Symbols......Page 247 Index......Page 251