Two characterizations of power compact operators

Abstract:

It is shown that if T is an operator on a Banach space with the property that for every bounded sequence $\{ {x_n}\}$ there is an integer k such that $\{ {T^k}({x_n})\}$ has a convergent subsequence then T is power compact. The equivalent nonstandard characterization is that for each finite point p in an ${\aleph _1}$-saturated enlargement of the space there is an integer k such that ${T^k}p$ is near-standard. Similar results are shown to hold for countable families of operators and for operators possessing weakly compact powers.