Characteristic Operators and Spectral Properties of Periodic Evolutionary Systems

TL;DR

This paper introduces characteristic operators for closable linear operators, explores their spectral properties, and applies these concepts to periodic evolution equations, resolving an open problem about Floquet exponents.

Contribution

It develops a new framework for characteristic operators in periodic systems and extends existing spectral theory to a broader class of differential equations.

Findings

Resolved an open problem on Floquet exponents for delay differential equations.

Constructed explicit schemes for characteristic operators in periodic evolution systems.

Demonstrated applications to classical and delay differential equations.

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.