Admissibility and general dichotomies for evolution families

For an arbitrary noninvertible evolution family on the half-line and for $ρ\colon [0, \infty)\to [0, \infty)$ in a large class of rate functions, we consider the notion of a $ρ$-dichotomy 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.

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