Abstract

Based on our preceding paper, this note is concerned with the exponential stability of the solution semigroup for the abstract linear autonomous functional differential equation x˙(t)=L(xt)                              (∗) where L is a continuous linear operator on some abstract phase space B into a Banach space E. We prove that the solution semigroup of (∗) is exponentially stable if and only if the fundamental operator (∗) is exponentially stable and the phase space B is an exponentially fading memory space.