Robustness of nonuniform exponential dichotomies under a wider class of perturbations

TL;DR

This paper extends the class of perturbations under which nonuniform exponential dichotomies remain stable, broadening the understanding of their robustness in general evolution families beyond differential equations.

Contribution

It introduces a wider class of perturbations ensuring the persistence of nonuniform exponential dichotomies in general evolution families, including noninvertible operators.

Findings

Extended the class of perturbations preserving exponential dichotomies

Applied results to noninvertible evolution families

Generalized robustness beyond differential equations

Abstract

The robustness property of exponential dichotomies refers to the stability of this notion under small linear perturbations. In recent work~\cite{PPX}, the authors have identified a new class of perturbations under which the notion of a nonuniform exponential dichotomy persists. In the present paper, we show that it is possible to extend this class. Moreover, unlike~\cite{PPX} where the results are restricted to the case of ordinary differential equations, in the present paper we deal with arbitrary evolution families consisting of possibly noninvertible linear operators.