Characterization of Exponential Stability of a Semigroup of Operators in Terms of Its Action by Convolution on Vector-Valued Function Spaces overR

It is proved that aC0-semigroupT={T(t)}t 0of linear operators on a Banach spaceXis uniformly exponentially stable if and only if it acts boundedly on one of the spacesLp( +, X) orC0( +, X) by convolution. As an application, it is shown thatTis uniformly exponentially stable if and only if[formula]whereAP( +, X) is the space ofX-valued almost periodic functions on +.