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