Smooth robustness of parameterized perturbations of exponential dichotomies

We establish the robustness of nonuniform exponential dichotomies in Banach spaces, under sufficiently small C1-parameterized perturbations. Moreover, we show that the stable and unstable subspaces of the exponential dichotomies obtained from the perturbation are also of class C1 on the parameter, thus yielding an optimal smoothness.