Strong Stability of Bounded Evolution Families and Semigroups

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∈X. These results are connected to the asymptotic stability of the well-posed linear nonautonomous Cauchy problemu(t)=A(t)u(t),t⩾s⩾0,u(s)=x,x∈X. 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.