  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.
