Relative compactness of orbits and geometry of Banach spaces

We investigate for a bounded semigroup of linear operators $S$ on a Banach space $E$ and a vector $x\in E$, when relative compactness of $S(I-T)x$ for every $T\in S$ implies relative compactness of the orbit $Sx$. In particular, we derive characterizations of separable Banach spaces not containing ${\mathrm{c}}_0$ and of reflexivity of Banach spaces with a Schauder basis in terms of such compactness results.