A remark on contraction semigroups on Banach spaces

Abstract

Let $X$ be a complex Banach space and let $J:X \to X^*$ be a duality section on $X$ (i.e. $\langle x,J(x)\rangle=\|J(x)\|\|x\|=\|J(x)\|^2=\|x\|^2$). For any unit vector $x$ and any ($C_0$) contraction semigroup $T=\{e^{tA}:t \geq 0\}$, Goldstein proved that if $X$ is a Hilbert space and if $|\langle T(t) x,J(x)\rangle| \to 1 $ as $t \to \infty$, then $x$ is an eigenvector of $A$ corresponding to a purely imaginary eigenvalue. In this article, we prove the similar result holds if $X$ is a strictly convex complex Banach space.