Uniform exponential stability of discrete semigroup and space of asymptotically almost periodic sequences (Q891804)

Summary: We prove that the discrete semigroup \(\mathbb T=\{\mathcal T(n):n\in\mathbb Z_+\}\) is uniformly exponentially stable if and only if for each \(z(n)\in \mathbb{AAP}_0(\mathbb Z_+,\mathcal X)\) the solution of the Cauchy problem \[ \begin{cases} y_{n+1} & =\mathcal T(1)y_n+z(n+1), \\ y(0) & = 0,\end{cases} \] belongs to \(\mathbb{AAP}_0(\mathbb Z_+,\mathcal X)\). Here, \(\mathcal T(1)\) is the algebraic generator of \(\mathbb T\), \(\mathbb Z_+\) is the set of all non-negative integers, and \(\mathcal X\) is a complex Banach space. Our proof uses the approach of discrete evolution semigroups.