Admissibilityand general dichotomies for evolution families

For an arbitrary noninvertible evolution family on the half-line and for ρ : [0, ∞) → [0, ∞) in a large class of rate functions, we consider the notion of a ρdichotomy with respect to a family of norms and characterize it in terms of two admissibility conditions. In particular, our results are applicable to exponential as well as polynomial dichotomies with respect to a family of norms. As a nontrivial application of our work, we establish the robustness of general nonuniform dichotomies.