On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups

We prove that a discrete semigroup ={T(n):n∈ℤ+} of bounded linear operators acting on a complex Banach space X is uniformly exponentially stable if and only if, for each x∈AP0(ℤ+,X), the sequence n↦∑k=0nT(n-k)x(k):ℤ+→X belongs to AP0(ℤ+,X). Similar results for periodic discrete evolution families are also stated