  We prove that a separable Banach space $E$ does not contain a copy of the
space $\co$ of null-sequences if and only if for every doubly power-bounded
operator $T$ on $E$ and for every vector $x\in E$ the relative compactness of
the sets $\{T^{n+m}x-T^nx: n\in \NN\}$ (for some/all $m\in\NN$, $m\geq 1$) and
$\{T^nx:n\in \NN\}$ are equivalent. With the help of the Jacobs--de
Leeuw--Glicksberg decomposition of strongly compact semigroups the case of (not
necessarily invertible) power-bounded operators is also handled.
