Admissibility and Polynomial Dichotomy of Discrete Nonautonomous Systems

We give new admissibility criteria for dichotomic behaviours of discrete nonautonomous systems, in infinite dimensional spaces. First, we present admissibility conditions for uniform and exponential dichotomy. Next, our study is focused on polynomial dichotomy, providing new characterizations for this notion by means of some double admissibilities. We obtain two categories of criteria for polynomial dichotomy, based on input-output conditions imposed to some suitable systems such that, for each one, the input sequences belong to certain -spaces and the outputs are bounded. We point out the importance of the assumptions regarding the complementarity of the stable subspaces at the initial time and we also discuss the relevance of the concept of solvability (unique or not) in the admissibility criteria for polynomial dichotomies on the half-line. All the results are obtained in the general case, without any additional hypotheses on the systems coefficients and without assuming any growth type properties for the associated propagators. Furthermore, as an application of the admissibility results we establish a robustness property of the polynomial dichotomy under small perturbations.