Stabilizability in asymptotically finite-dimensional semigroups

We study a semigroup $\varphi$ of linear operators on a Banach space $X$ which satisfies the condition $\operatorname{codim}X_0<\infty$, where $X_0=\{x\in X|\varphi_t(x)\xrightarrow[t\to\infty]{}0\}$. We show that $X_0$ is closed and establish some properties of the asymptotic behavior of the subspaces complementing $X_0$ to $X$.