Strong stability of bounded evolution families and semigroups

Abstract

We prove several characterizations of strong stability of uniformly bounded evolution families (U(t, s))t≥s≥0 of bounded operators on a Banach space X, i.e. we characterize the property limt→∞U(t, s)x = 0 for all s ≥ 0 and all x ε X. These results are connected to the asymptotic stability of the well-posed linear nonautonomous Cauchy problem In the autonomous case, i.e. when U(t, s) = T(t - s) for some C0-semigroup (T(t))t≥0, we present, in addition, a range condition on the generator A of (T(t))t≥0 which is sufficient for strong stability. This condition is more general than the condition in the ABLV-Theorem involving countability of the imaginary part of the spectrum of A. © 2002 Elsevier Science (USA).