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Characteristic Operators and Spectral Properties of Periodic Evolutionary Systems

Bram Lentjes, Babette A. J. de Wolff

Abstract

In this paper, we introduce the notion of a characteristic operator for closable linear operators and explore their connected spectral properties via equivalence. Additionally, we develop an explicit scheme for constructing characteristic operators for a broad class of closable linear operators which are commonly encountered in periodic evolution equations. Our findings are illustrated through examples involving classical delay differential equations, delay differential equations with infinite delay and mixed functional differential equations. Notably, we resolve an open problem concerning the discrete spectral structure of the Floquet exponents for this latter class of differential equations. This work can be regarded as a natural and significant extension of the powerful framework developed by Kaashoek and Verduyn Lunel [40] on characteristic matrices and spectral properties induced by autonomous evolution equations.

Characteristic Operators and Spectral Properties of Periodic Evolutionary Systems

Abstract

In this paper, we introduce the notion of a characteristic operator for closable linear operators and explore their connected spectral properties via equivalence. Additionally, we develop an explicit scheme for constructing characteristic operators for a broad class of closable linear operators which are commonly encountered in periodic evolution equations. Our findings are illustrated through examples involving classical delay differential equations, delay differential equations with infinite delay and mixed functional differential equations. Notably, we resolve an open problem concerning the discrete spectral structure of the Floquet exponents for this latter class of differential equations. This work can be regarded as a natural and significant extension of the powerful framework developed by Kaashoek and Verduyn Lunel [40] on characteristic matrices and spectral properties induced by autonomous evolution equations.
Paper Structure (17 sections, 55 theorems, 196 equations, 1 figure)

This paper contains 17 sections, 55 theorems, 196 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Challenges and overview
  3. Characteristic operators for closable linear operators
  4. Spectral properties of closable linear operators
  5. Equivalence and Jordan chains
  6. Characteristic operators and spectral properties
  7. General scheme for constructing characteristic operators
  8. Construction of $\Delta$ in terms of $\mathcal{A}$
  9. Construction of explicit equivalences and spectral properties
  10. Periodic spectral properties and similarity
  11. Applications to classes of periodic evolutionary systems
  12. Classical delay differential equations
  13. Delay differential equations with infinite delay
  14. Mixed functional differential equations
  15. Conclusion and outlook
  16. ...and 2 more sections

Key Result

Lemma 3

The following inclusions hold:

Figures (1)

  • Figure 1: Relations between the spaces and operators involved in \ref{['def:equi']}.

Theorems & Definitions (122)

  • Remark 1
  • Definition 2
  • Lemma 3
  • proof
  • Definition 4
  • Remark 5
  • Proposition 6
  • Lemma 7
  • proof
  • Definition 8
  • ...and 112 more