On the Exponential Stability of Discrete Semigroups

Abstract

Let $q$ be a positive integer and let $X$ be a complex Banach space. We denote by $\mathbb {Z}_+$ the set of all nonnegative integers. Let $P_q (\mathbb {Z}_+,X)$ is the set of all $X$-valued, q-periodic sequences. Then $P_1 (\mathbb {Z}_+,X)$ is the set of all $X$-valued constant sequences. When $q\ge 2$, we denote by $P^0_q (\mathbb {Z}_+,X)$, the subspace of $P_q (\mathbb {Z}_+,X)$ consisting of all sequences $z(.)$ with $z(0) = 0$. Let $T$ be a bounded linear operator acting on $X$. It is known, that the discrete semigroup generated (from the algebraic point of view) of $T$, i.e. the operator valued sequence $T= (T^n)$, is uniformly exponentially stable (i.e. $\lim _{n\rightarrow \infty } \frac{\ln \Vert T^n\Vert }{n} <0$), if and only if for each real number $\mu$ and each sequences $z(.)$ in $P_1 (\mathbb {Z}_+,X)$ the sequences $(y_n)$ given by

$$\begin{aligned} \left\{ \begin{aligned} y_{n+1}&= T(1)y_{n}+e^{i\mu (n+1)}z(n+1), \\ y(0)&= 0 \end{aligned}\right. \end{aligned}$$

is bounded. In this paper we prove a complementary result taking $P^0_q (\mathbb {Z}_+,X)$ with some integer $q\ge 2$ instead of $P_1 (\mathbb {Z}_+,X)$.