Asymptotical behavior of subspaces under action of asymptotically   finite-dimensional semigroups of operators
  We study a semigroup $\phi$ of linear operators acting on a Banach space $X$ which satisfies the condition $\codim X_0<\infty$, where $X_0=\{x\in X \mid \phi_t(x)\underset{t\to\infty}\longrightarrow 0\}.$ We show that $X_0$ is closed under these conditions. We establish some properties concerning the asymptotic behavior of subspaces which complement $X_0$ in $X$. 
K.Storozhuk