Robustness of nonuniform exponential dichotomies under a wider class of perturbations

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.