CHARACTERIZATIONS OF STABILITY FOR STRONGLY CONTINUOUS SEMIGROUPS BY BOUNDEDNESS OF ITS CONVOLUTIONS WITH ALMOST PERIODIC FUNCTIONS

Abstract

Let ${\bf T}=\{T(t)\}_{t\ge 0}$ be a strongly continuous semigroup  of bounded linear operators acting on a Banach space $X.$ We prove that if the convolution ${\bf T}*(e^{-i\mu \cdot}f)$ is bounded for every continuous and 1-periodic function which is null in $t=0$ and some $\mu\in{\bf R}$, then $T(1)$ is power bounded and $e^{i\mu}\in\rho(T(1)).$ Applications to questions of exponential stability are also presented.