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Completeness Theorems and Characteristic Matrix Functions

Applications to Integral and Differential Operators

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  • © 2022

Overview

Authors:
  1. Marinus A. Kaashoek

    1. Department of Mathematics, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

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  2. Sjoerd M. Verduyn Lunel

    1. Mathematical Institute, Utrecht University, Utrecht, The Netherlands

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • A new and self-contained study of linear operators that admit a characteristic matrix function
  • New completeness theorems for classes of Banach space operators motivated by applications
  • Comprehensive treatment of periodic functional differential equations

Part of the book series: Operator Theory: Advances and Applications (OT, volume 288)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Preliminaries

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 1-18

  3. Completeness Theorems for Compact Hilbert Space Operators

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 19-43

  4. Compact Hilbert Space Operators of Order One

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 45-53

  5. Completeness for a Class of Banach Space Operators

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 55-90

  6. Characteristic Matrix Functions for a Class of Operators

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 91-103

  7. Finite Rank Perturbations of Volterra Operators

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 105-120

  8. Finite Rank Perturbations of Operators of Integration

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 121-140

  9. Discrete Case: Infinite Leslie Operators

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 141-160

  10. Semi-Separable Operators and Completeness

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 161-198

  11. Periodic Delay Equations

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 199-223

  12. Completeness Theorems for Period Maps

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 225-257

  13. Completeness for Perturbations of Unbounded Operators

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 259-267

  14. Applications to Dynamical Systems

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 269-286

  15. Results from the Theory of Entire Functions

    • Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel
    Pages 287-339

  16. Back Matter

    Pages 341-350
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    Marinus A. Kaashoek

  • Mathematical Institute, Utrecht University, Utrecht, The Netherlands

    Sjoerd M. Verduyn Lunel

About the authors

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.



Bibliographic Information

  • Book Title: Completeness Theorems and Characteristic Matrix Functions

  • Book Subtitle: Applications to Integral and Differential Operators

  • Authors: Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

  • Series Title: Operator Theory: Advances and Applications

  • DOI: https://doi.org/10.1007/978-3-031-04508-0

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

  • Hardcover ISBN: 978-3-031-04507-3Published: 14 June 2022

  • Softcover ISBN: 978-3-031-04510-3Published: 15 June 2023

  • eBook ISBN: 978-3-031-04508-0Published: 13 June 2022

  • Series ISSN: 0255-0156

  • Series E-ISSN: 2296-4878

  • Edition Number: 1

  • Number of Pages: XVI, 350

  • Number of Illustrations: 2 illustrations in colour

  • Topics: Operator Theory, Analysis

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