On uniform exponential stability and exact admissibility of discrete semigroups
The authors show that a discrete semigroup \((T(n))_{n\in\mathbb Z_+}\) of bounded linear operators acting on a complex Banach space \(X\) is uniformly exponentially stable if and only if for each \(x\in AP_0(\mathbb Z_+,X)\), the sequence \(n\to\sum_{k=0}^nT(n-k)x(k):\mathbb Z_+\to X\) belongs to \(AP_0(\mathbb Z_+,X)\).
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