On individual stability of $C_0-$semigroups

Abstract:

Let $\{T(t)\}_{t\ge 0}$ be a $C_0$-semigroup with generator $A$ on a Banach space $X$. Let $x_0\in X$ be a fixed element. We prove the following individual stability results.

(i) Suppose $X$ is an ordered Banach space with weakly normal closed cone $C$ and assume there exists $t_0\ge 0$ such that $T(t)x_0\in C$ for all $t\ge t_0$. If the local resolvent $\lambda \mapsto ( \lambda -A)^{-1} x_0$ admits a bounded analytic extension to the right half-plane $\{\operatorname {Re}\lambda >0\}$, then for all $\mu \in \varrho (A)$ and $x^\ast \in X^\ast$ we have \[ \lim _{t\to \infty } \bigl \langle T(t)(\mu -A)^{-1} x_0, x^\ast \bigr \rangle = 0.\]

(ii) Suppose $E$ is a rearrangement invariant Banach function space over $[0,\infty )$ with order continuous norm. If $x_0^\ast \in X^\ast$ is an element such that $t\mapsto \langle T(t)x_0, x_0^\ast \rangle$ defines an element of $E$, then for all $\mu \in \varrho (A)$ and $\beta \ge 1$ we have \[ \lim _{t\to \infty } \bigl \langle T(t)(\mu -A)^{-\beta } x_0, x_0^\ast \bigr \rangle = 0.\]