  Let $T:X\to X$ be a compact linear (or more generally affine) operator from a
Banach space into itself. For each $x\in X$, the sequence of iterates $T^nx,
n=0,1,...$ and its averages $\frac{1}{k}\sum_{k=0}^nT^{k-1}x, n=0,1,...$ are
either bounded or approach infinity.
