  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.
