Exponential dichotomies with respect to a sequence of norms and admissibility

For a nonautonomous dynamics defined by a sequence of linear operators, we introduce the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in l (p) spaces, both for the space of perturbations and the space of solutions. This allows unifying the notions of uniform and nonuniform exponential behavior. Moreover, we consider the general case of a noninvertible dynamics. As a nontrivial application we show that the conditional stability of a nonuniform exponential dichotomy persists under sufficiently small linear perturbations.