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

We prove that a separable Banach space $E$ does not contain a copy of the space $\text{co}$ 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 $\{T^{n+m}x-T^{n}x:n\in \text{NN}\}$ (for some/all $m\in \text{NN}$, $m\ge 1$) and $\{T^{n}x:n\in \text{NN}\}$ 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.