Perron conditions for exponential expansiveness of one-parameter semigroup

Abstract

We present a new approach for the theorems of Perron type for exponential expansiveness of one-parameter semigroups in terms of l^p(N, X) spaces. We prove that an exponentially bounded semigroup is exponentially expansive if and only if the pair (l^p (N, X), l^q(N, X)) is completely admissible relative to a discrete equation associated to the semigroup, where p, q ∈ [1, ∞), p ≥ q. We apply our results in order to obtain very general characterizations for exponential expansiveness of C_0-semigroups in terms of the complete admissibility of the pair (L^ p (R_+ , X), L^ q (R_+ , X)) and for exponential dichotomy, respectively, in terms of the admissibility of the pair (L^p(R_+,X), L^q(R_+,X)).