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

Abstract

We study a semigroup $\varphi$ of linear operators acting on a Banach space $X$ which satisfies the condition $\codim X_0<\infty$, where $X_0=\{x\in X\mid {\varphi }_t(x)\underset{t\to \infty }{⟶}0\}.$ 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 $X$.

10 pages