STABILITY OF INDIVIDUAL ELEMENTS UNDER ONE-PARAMETER SEMIGROUPS

Let {T(t) : t greater-than-or-equal-to 0} be a C0-semigroup on a Banach space X with generator A, and let x is-a-member-of X. If sigma-(A) union iR is empty and t-->T(t)x is uniformly continuous, then \\T(t)x\\-->0 as t-->infinity. If the semigroup is sun-reflexive, sigma-(A) union iR is countable, Psigma-(A) union iR is empty, and t-->T(t)x is uniformly weakly continous, then T(t)x-->0 weakly as t-->infinity. Questions of almost periodicity and of stabilization of contraction semigroups on Hilbert space are also discussed.