Slowly changing vectors and the asymptotic finite-dimensionality of an operator semigroup

Let $X$ be a Banach space and let $T\colon X\to X$ be a linear power bounded operator. Put $X_0=\{x\in X\mid T^nx\to0\}$. We prove that if $X_0\ne X$ then there exists $\lambda\in\mathrm{Sp}(T)$ such that, for every $\varepsilon>0$, there is $x$ such that $\|Tx-\lambda x\|<\varepsilon$ but $\|T^nx\|>1-\varepsilon$ for all $n$. The technique we develop enables us to establish that if $X$ is reflexive and there exists a compactum $K\subset X$ such that $\lim\inf_{n\to\infty}\rho\{T^nx,K\}<\alpha(T)<1$ for every norm-one $x\in X$ then $\operatorname{codim}X_0<\infty$. The results hold also for a one-parameter semigroup.