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We prove that a separable Banach space $E$ does not contain a copy of thespace $\co$ of null-sequences if and only if for every doubly power-boundedoperator $T$ on $E$ and for every vector $x\in E$ the relative compactness ofthe 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--deLeeuw--Glicksberg decomposition of strongly compact semigroups the case of (notnecessarily invertible) power-bounded operators is also handled.

A Bohl-Bohr-Kadets type theorem characterizing Banach spaces not containing c0