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Two characterizations of power compact operators

Author: D. G. Tacon
Journal: Proc. Amer. Math. Soc. 73 (1979), 356-360
MSC: Primary 47B05; Secondary 03H05
MathSciNet review: 518519
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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.

References [Enhancements On Off] (What's this?)

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Keywords: Banach space, operator, compact, weakly compact, $ {\aleph _1}$-saturated enlargement, nonstandard analysis
Article copyright: © Copyright 1979 American Mathematical Society

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