Exponential stability for abstract linear autonomous functional differential equations with infinite delay

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 equationx˙(t)=L(xt)                              (∗)whereLis a continuous linear operator on some abstract phase spaceBinto a Banach spaceE. We prove that the solution semigroup of(∗)is exponentially stable if and only if the fundamental operator(∗)is exponentially stable and the phase spaceBis an exponentially fading memory space.