Basic Classes of Linear Operators

I Hilbert Spaces.- 1.1 Complex n-Space.- 1.2 The Hilbert Space ?2.- 1.3 Definition of Hilbert Space and its Elementary Properties.- 1.4 Distance from a Point to a Finite Dimensional Space.- 1.5 The Gram Determinant.- 1.6 Incompatible Systems of Equations.- 1.7 Least Square Fit.- 1.8 Distance to a Convex Set and Projections onto Subspaces.- 1.9 Orthonormal Systems.- 1.10 Szegö Polynomials.- 1.11 Legendre Polynomials.- 1.12 Orthonormal Bases.- 1.13 Fourier Series.- 1.14 Completeness of the Legendre Polynomials.- 1.15 Bases for the Hilbert Space of Functions on a Square.- 1.16 Stability of Orthonormal Bases.- 1.17 Separable Spaces.- 1.18 Isometry of Hilbert Spaces.- 1.19 Example of a Non Separable Space.- Exercises.- II Bounded Linear Operators on Hilbert Spaces.- 2.1 Properties of Bounded Linear Operators.- 2.2 Examples of Bounded Linear Operators with Estimates of Norms.- 2.3 Continuity of a Linear Operator.- 2.4 Matrix Representations of Bounded Linear Operators.- 2.5 Bounded Linear Functionals.- 2.6 Operators of Finite Rank.- 2.7 Invertible Operators.- 2.8 Inversion of Operators by the Iterative Method.- 2.9 Infinite Systems of Linear Equations.- 2.10 Integral Equations of the Second Kind.- 2.11 Adjoint Operators.- 2.12 Self Adjoint Operators.- 2.13 Orthogonal Projections.- 2.14 Two Fundamental Theorems.- 2.15 Projections and One-Sided Invertibility of Operators.- 2.16 Compact Operators.- 2.17 The Projection Method for Inversion of Linear Operators.- 2.18 The Modified Projection Method.- 2.19 Invariant Subspaces.- 2.20 The Spectrum of an Operator.- Exercises.- III Laurent and Toeplitz Operators on Hilbert Spaces.- 3.1 Laurent Operators.- 3.2 Toeplitz Operators.- 3.3 Band Toeplitz operators.- 3.4 Toeplitz Operators with Continuous Symbols.- 3.5 Finite Section Method.- 3.6 The Finite Section Method for Laurent Operators.- Exercises.- IV Spectral Theory of Compact Self Adjoint Operators.- 4.1 Example of an Infinite Dimensional Generalization.- 4.2 The Problem of Existence of Eigenvalues and Eigenvectors.- 4.3 Eigenvalues and Eigenvectors of Operators of Finite Rank.- 4.4 Existence of Eigenvalues.- 4.5 Spectral Theorem.- 4.6 Basic Systems of Eigenvalues and Eigenvectors.- 4.7 Second Form of the Spectral Theorem.- 4.8 Formula for the Inverse Operator.- 4.9 Minimum-Maximum Properties of Eigenvalues.- Exercises.- V Spectral Theory of Integral Operators.- 5.1 Hilbert-Schmidt Theorem.- 5.2 Preliminaries for Mercer’s Theorem.- 5.3 Mercer’s Theorem.- 5.4 Trace Formula for Integral Operators.- Exercises.- VI Unbounded Operators on Hilbert Space.- 6.1 Closed Operators and First Examples.- 6.2 The Second Derivative as an Operator.- 6.3 The Graph Norm.- 6.4 Adjoint Operators.- 6.5 Sturm-Liouville Operators.- 6.6 Self Adjoint Operators with Compact Inverse.- Exercises.- VII Oscillations of an Elastic String.- 7.1 The Displacement Function.- 7.2 Basic Harmonic Oscillations.- 7.3 Harmonic Oscillations with an External Force.- VIII Operational Calculus with Applications.- 8.1 Functions of a Compact Self Adjoint Operator.- 8.2 Differential Equations in Hilbert Space.- 8.3 Infinite Systems of Differential Equations.- 8.4 Integro-Differential Equations.- Exercises.- IX Solving Linear Equations by Iterative Methods.- 9.1 The Main Theorem.- 9.2 Preliminaries for the Proof.- 9.3 Proof of the Main Theorem.- 9.4 Application to Integral Equations.- X Further Developments of the Spectral Theorem.- 10.1 Simultaneous Diagonalization.- 10.2 Compact Normal Operators.- 10.3 Unitary Operators.- 10.4 Singular Values.- 10.5 Trace Class and Hilbert Schmidt Operators.- Exercises.- XI Banach Spaces.- 11.1 Definitions and Examples.- 11.2 Finite Dimensional Normed Linear Spaces.- 11.3 Separable Banach Spaces and Schauder Bases.- 11.4 Conjugate Spaces.- 11.5 Hahn-Banach Theorem.- Exercises.- XII Linear Operators on a Banach Space.- 12.1 Description of Bounded Operators.- 12.2 Closed Linear Operators.- 12.3 Closed Graph Theorem.- 12.4 Applications of the Closed Graph Theorem.- 12.5 Complemented Subspaces and Projections.- 12.6 One-Sided Invertibility Revisited.- 12.7 The Projection Method Revisited.- 12.8 The Spectrum of an Operator.- 12.9 Volterra Integral Operator.- 12.10 Analytic Operator Valued Functions.- Exercises.- XIII Compact Operators on a Banach Space.- 13.1 Examples of Compact Operators.- 13.2 Decomposition of Operators of Finite Rank.- 13.3 Approximation by Operators of Finite Rank.- 13.4 First Results in Fredholm Theory.- 13.5 Conjugate Operators on a Banach Space.- 13.6 Spectrum of a Compact Operator.- 13.7 Applications.- Exercises.- XIV Poincaré Operators: Determinant and Trace.- 14.1 Determinant and Trace.- 14.2 Finite Rank Operators, Determinants and Traces.- 14.3 Theorems about the Poincaré Determinant.- 14.4 Determinants and Inversion of Operators.- 14.5 Trace and Determinant Formulas for Poincaré Operators.- Exercises.- XV Fredholm Operators.- 15.1 Definition and Examples.- 15.2 First Properties.- 15.3 Perturbations Small in Norm.- 15.4 Compact Perturbations.- 15.5 Unbounded Fredholm Operators.- Exercises.- XVI Toeplitz and Singular Integral Operators.- 16.1 Laurent Operators on ?p(?).- 16.2 Toeplitz Operators on ?p.- 16.3 An Illustrative Example.- 16.4 Applications to Pair Operators.- 16.5 The Finite Section Method Revisited.- 16.6 Singular Integral Operators on the Unit Circle.- Exercises.- XVII Non Linear Operators.- 17.1 Fixed Point Theorems.- 17.2 Applications of the Contraction Mapping Theorem.- 17.3 Generalizations.- Appendix 1: Countable sets and Separable Hilbert Spaces.- Suggested Reading.- References.- List of Symbols.