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.
Author(s): Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
Series: Operator Theory: Advances and Applications, 288
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 357
City: Cham
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
