  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.
