On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups

Abstract

We prove that a discrete semigroup $𝕋=\{T(n):n\in {\mathbb{Z}}_{+}\}$ of bounded linear operators acting on a complex Banach space $X$ is uniformly exponentially stable if and only if, for each $x\in A{P}_0({\mathbb{Z}}_{+},X)$, the sequence $n\mapsto {\sum }_{k=0}^n\mathrm{‍}T(n-k)x(k):{\mathbb{Z}}_{+}\to X$ belongs to $A{P}_0({\mathbb{Z}}_{+},X)$. Similar results for periodic discrete evolution families are also stated.