Robustness of dichotomies and trichotomies for difference equations

Abstract

We give a short proof of the robustness of the notion of a nonuniform exponential dichotomy for a sequence of linear operators acting on a Banach space. This means that any sufficiently small linear perturbation of a nonuniform exponential dichotomy exhibits the same type of exponential behavior. The method of proof is based on the notion of admissibility introduced by Perron in the special case of a uniform exponential behavior. In strong contrast to former proofs, we do not need to construct explicitly projections on the stable and unstable directions. As an application, we also give a short proof of the robustness of the notion of a nonuniform exponential trichotomy.