Exponential behavior in Banach spaces: robustness of trichotomies in discrete time

Abstract

For difference equations in Banach spaces, we consider a generalization of the notion of exponential dichotomy, usually called trichotomy in the literature, for which the behaviors in $$\mathbb {Z}^+$$ and $$\mathbb {Z}^-$$ are still exponential but need not agree at the origin. Our main aim is to show that this exponential behavior is robust, in the sense that it persists under sufficiently small linear perturbations.