  For an arbitrary evolution family, we consider the notion of a polynomial
dichotomy with respect to a family of norms and characterize it in terms of the
admissibility property, that is, the existence of a unique bounded solution for
each bounded perturbation. In particular, by considering a family of Lyapunov
norms, we recover the notion of a (strong) nonuniform polynomial dichotomy. As
a nontrivial application of the characterization, we establish the robustness
of the notion of a strong nonuniform polynomial dichotomy under sufficiently
small linear perturbations.
