Completeness Theorems and Characteristic Matrix Functions: Applications to Integral and Differential Operators

Description

This monograph presents necessary and sufficient conditions for completeness of the linear span of eigenvectors and generalized eigenvectors of operators that admit a characteristic matrix function in a Banach space setting. Classical conditions for completeness based on the theory of entire functions are further developed for this specific class of operators. The classes of bounded operators that are investigated include trace class and Hilbert-Schmidt operators, finite rank perturbations of Volterra operators, infinite Leslie operators, discrete semi-separable operators, integral operators with semi-separable kernels, and period maps corresponding to delay differential equations. The classes of unbounded operators that are investigated appear in a natural way in the study of infinite dimensional dynamical systems such as mixed type functional differential equations, age-dependent population dynamics, and in the analysis of the Markov semigroup connected to the recently introduced zig-zag process. Preface Review of Contents Acknowledgements Contents List of Symbols 1 Preliminaries 1.1 Basic Elements of Operator Theory and Definition of Completeness 1.1.1 Elements of Banach Space Operator Theory 1.1.2 Definition of Completeness 1.1.3 Examples Illustrating Proposition 1.1.1 and Schmidt Representations of Compact Operators 1.2 Spectral Preliminaries I 1.3 Compact Hilbert Space Operator of Finite Order 1.3.1 The Operator Tg Revisited 2 Completeness Theorems for Compact Hilbert Space Operators 2.1 First Hilbert Space Completeness Theorem 2.2 Two Additional Completeness Theorems 2.3 A First Application of Theorem 2.2.2 2.3.1 Three Special Cases 2.4 Classical Completeness Theorems Revisited 2.5 The Dense Range Property 3 Compact Hilbert Space Operators of Order One 3.1 Some Remarks About Trace Class Operators 3.2 Preliminaries About Hilbert-Schmidt Operators 3.3 Resolvent Estimates for Compact Operators of Order One 3.4 A Completeness Theorem 3.5 Supplementary Remarks 4 Completeness for a Class of Banach Space Operators 4.1 A Special Class of Operators 4.2 Spectral Preliminaries II 4.3 Theorem 4.1.3 Reduced to the Case When z0 Is Zero 4.4 Proof of Theorem 4.1.3 4.5 An Additional Example 4.6 Some Additional Remarks 4.7 Theorem 3.4.1 Revisited 5 Characteristic Matrix Functions for a Class of Operators 5.1 Equivalence and Jordan Chains 5.1.1 Entire Matrix Functions 5.2 The Characteristic Matrix Function 6 Finite Rank Perturbations of Volterra Operators 6.1 The Characteristic Matrix Function 6.2 A Completeness Theorem 6.3 The Volterra Operator Replaced by a Quasi-Nilpotent Operator 6.4 Examples of Non-compact Quasi-Nilpotent Operators 7 Finite Rank Perturbations of Operators of Integration 7.1 Preliminaries 7.2 Rank One Perturbations of the Operator of Integration on C[0,1], Part 1 7.3 Rank One Perturbations of the Operator of Integration on C[0,1], Part 2 7.4 Rank One Perturbations of the Operator of Integration on L2[0,1] 8 Discrete Case: Infinite Leslie Operators 8.1 Definition of a Leslie Operator 8.2 Associated Boundary Value Systems 8.3 The Characteristic Function and Related Properties 8.4 Completeness for a Concrete Class of Leslie Operators 8.5 A Generalised Leslie Operator 9 Semi-Separable Operators and Completeness 9.1 Discrete Semi-Separable Operators 9.1.1 A Completeness Theorem (A Scalar Case) 9.2 Integral Operators with Semi-Separable Kernels 9.2.1 A Completeness Result for Semi-Separable Integral Operators 9.3 Intermezzo: Fundamental Solutions of ODE and Volterra Operators 9.3.1 A Related Volterra Operator 10 Periodic Delay Equations 10.1 Time Dependent Delay Equations 10.2 A Family of Time Dependent Delay Equations 10.3 A Two-Parameter Family of Solution Operators 10.4 Solution Operators for Periodic Delay Equations 11 Completeness Theorems for Period Maps 11.1 The Period Map and Its Generalisations 11.2 Spectral Properties of the Period Map 11.3 Completeness of the Period Map in Case the Period Is Equal to the Delay 11.4 Scalar Periodic Delay Equations and Completeness (One Periodic) 11.5 Scalar Periodic Delay Equations and Completeness (Two Periodic) 12 Completeness for Perturbations of Unbounded Operators 12.1 The Associated Characteristic Matrix Function 12.2 A Completeness Theorem for a Class of Unbounded Operators 13 Applications to Dynamical Systems 13.1 Mixed Type Functional Differential Equations 13.1.1 Three Examples 13.2 Age-Dependent Population Dynamics 13.3 The Zig-Zag Semigroup 14 Results from the Theory of Entire Functions 14.1 Basic Definitions 14.2 Applications of the Phragmén-Lindelöf Theorem 14.3 Applications of the Paley-Wiener Theorem 14.4 The Phragmén-Lindelöf Indicator Function 14.5 Properties of the Indicator Function 14.6 Entire Functions of Completely Regular Growth 14.7 The Dominating Property 14.8 Distribution of Zeros of Entire Functions and Related Properties 14.8.1 Distribution of Zeros and Completely Regular Growth 14.8.2 Genus, Convergence Exponent, and Order of Entire Functions 14.9 Vector-Valued and Operator-Valued Entire Functions Epilogue Bibliography Subject Index