Characterizations of stability for discrete semigroups of bounded linear operators

Let $mathbb{T}={T(n)}_{ngeq 0}$ be a discrete semigroup of bounded linear operators acting on a Banach space $X$. We prove that if for each $mu in mathbb{R}$ and every $q$-periodic sequence $f$ with $f(0)=0$, the sequence $n ightarrow sum_{k=0}^{n}e^{imu k}T(n-k) f(k)$ is bounded, then the semigroup $mathbb{T}$ is uniformly exponentially stable.