Stability of Individual Elements Under One-Parameter Semigroups

Abstract

Let {r(Z):Z>0} be a C0-semigroup on a Banach space X with generator A , and let x X.If a (A) n ;'R is empty and t -> T(t)x is uniformly continuous, then ||7"(Z)jc|| -> 0 as t - oo .If the semigroup is sun-reflexive, o(A)CiiR is countable, Pa(A)DiS.is empty, and 1 >-> T(t)x is uniformly weakly continuous, then T(t)x - 0 weakly as t - oo .Questions of almost periodicity and of stabilization of contraction semigroups on Hubert space are also discussed.