Asymptotical behavior of subspaces under action of asymptotically finite-dimensional semigroups of operators

We study a semigroup $\varphi$ of linear operators acting on a Banach space $X$ which satisfies the condition $\text{codim}{X}_0<\mathrm{\infty }$, where $X_0={x\in X \mid \phi_t(x)\underset{t\to\infty}\longrightarrow 0}.$Weshowthat$X_0$ is closed under these conditions. We establish some properties concerning the asymptotic behavior of subspaces which complement $X_0$ in $X$.