  We analyze the ergodic properties of power-bounded operators on a reflexive
Banach space of the form "scalar plus compact-power", and show that they are
almost periodic (all the orbits are conditionally compact). If such an operator
is weakly mixing, then it is stable (its powers converge in the strong operator
topology).Let X-ISP be the separable reflexive indecomposable Banach space
constructed by Argyros and Motakis, in which every operator has an invariant
subspace. We conclude that every power-bounded operator on a closed subspace of
X-ISP is almost periodic.
