Stability of individual elements under one-parameter semigroups

Abstract:

Let $\{ T(t):t \geqslant 0\}$ be a ${C_0}$-semigroup on a Banach space $X$ with generator $A$, and let $x \in X$. If $\sigma (A) \cap i{\mathbf {R}}$ is empty and $t \mapsto T(t)x$ is uniformly continuous, then $||T(t)x|| \to 0$ as $t \to \infty$. If the semigroup is sun-reflexive, $\sigma (A) \cap i{\mathbf {R}}$ is countable, $P\sigma (A) \cap i{\mathbf {R}}$ is empty, and $t \mapsto T(t)x$ is uniformly weakly continuous, then $T(t)x \to 0$ weakly as $t \to \infty$. Questions of almost periodicity and of stabilization of contraction semigroups on Hilbert space are also discussed.