On individual stability of $C_0-$semigroups

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

(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