Slowly Changing Vectors and the Asymptotic Finite-Dimensionality of an Operator Semigroup

Let $T:X\to X$ be a linear power bounded operator on Banach space. Let $X_0$ is a subspace of vectors tending to zero under iterating of $T$. We prove that if $X_0$ is not equal to $X$ then there exists $\lambda$ in Sp(T) such that, for every $\epsilon>0$, there is $x$ such that $|Tx-\lambda x|<\epsilon$ but $|T^nx|>1-\epsilon$ for all $n$. The technique we develop enables us to establish that if $X$ is reflexive and there exists a compactum $K$ in $X$ such that for every norm-one $x\in X$ $\rho\{T^nx, K\}<\alpha (T)<1$ for some $n=n_1, n_2,...$ then $codim(X_0)<\infty$. The results hold also for a one-parameter semigroup.