A remark on contraction semigroups on Banach spaces

Abstract

Let $X$ be a complex Banach space and let $J:X\to X^{\ast }$ be a duality section on $X$ (i.e. $⟨x,J(x)⟩=\Vert J(x)\Vert \Vert x\Vert =\Vert J(x){\Vert }^2=\Vert x{\Vert }^2$). For any unit vector $x$ and any ($C_0$) contraction semigroup $T=\{e^{tA}:t\ge 0\}$, Goldstein proved that if $X$ is a Hilbert space and if $|⟨T(t)x,J(x)⟩|\to 1$ 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.