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

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.

- 1. Preliminaries. - 2. Completeness Theorems for Compact Hilbert Space Operators. - 3. Compact Hilbert Space Operators of Order One. - 4. Completeness for a Class of Banach Space Operators. - 5. Characteristic Matrix Functions for a Class of Operators. - 6. Finite Rank Perturbations of Volterra Operators. - 7. Finite Rank Perturbations of Operators of Integration. - 8. Discrete Case: Infinite Leslie Operators. - 9. Semi-Separable Operators and Completeness. - 10. Periodic Delay Equations. - 11. Completeness Theorems for Period Maps. - 12. Completeness for Perturbations of Unbounded Operators. - 13. Applications to Dynamical Systems. - 14. Results from the Theory of Entire Functions. - Epilogue.

Marinus A. Kaashoek is a Dutch mathematician, and Emeritus Professor Analysis and Operator Theory at the Vrije Universiteit in Amsterdam. Kaashoek’s research interests are in the field of Analysis and Operator Theory, and various connections between Operator Theory, Matrix Theory and Mathematical Systems Theory. In particular, Wiener–Hopf integral equations and Toeplitz operators, their nonstationary variants, and other structured operators, such as continuous operator analogs of Bezout and resultant matrices. State space methods for problems in analysis are shown to be useful. Also metric constrained interpolation problems and completion problems for partially given operators, including relaxed commutant lifting problems, are proved to be solvable.

Sjoerd M. Verduyn Lunel is Professor of Applied Analysis at Utrecht University. He held positions at Brown University, Georgia Institute of Technology, University of Amsterdam, Vrije Universiteit Amsterdam, and Leiden University. His research interests are at the interface of Analysis and infinite dimensional Dynamical Systems Theory with focus on the theory of Functional Differential Equations. He was co-Editor-in-Chief of Integral Equations and Operator Theory (2000-2009) and is currently associate editor of SIAM Journal on Mathematical Analysis and of Integral Equations and Operator Theory. In 2012 he was elected member of the Royal Holland Society of Sciences and Humanities and in 2014 he was appointed honorary member of the Indonesian Mathematical Society.