Exponential dichotomies of evolution operators in Banach spaces

TL;DR

This paper explores various types of exponential dichotomies in non-invertible evolution operators within Banach spaces, establishing connections, providing conditions for strong exponential dichotomy, and illustrating the limitations of certain implications with examples.

Contribution

It introduces a comprehensive analysis of exponential dichotomies in Banach spaces, including necessary and sufficient conditions for strong exponential dichotomy and counterexamples for implication limitations.

Findings

Connections between different dichotomy concepts are established.

Necessary and sufficient conditions for strong exponential dichotomy are provided.

Counterexamples show some implications do not hold in general.

Abstract

This paper considers three dichotomy concepts (exponential dichotomy, uniform exponential dichotomy and strong exponential dichotomy) in the general context of non-invertible evolution operators in Banach spaces. Connections between these concepts are illustrated. Using the notion of Green function, we give necessary conditions and sufficient ones for strong exponential dichotomy. Some illustrative examples are presented to prove that the converse of some implication type theorems are not valid.