Two characterizations of power compact operators

Author:
D. G. Tacon

Journal:
Proc. Amer. Math. Soc. **73** (1979), 356-360

MSC:
Primary 47B05; Secondary 03H05

DOI:
https://doi.org/10.1090/S0002-9939-1979-0518519-0

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 there is an integer *k* such that has a convergent subsequence then *T* is power compact. The equivalent nonstandard characterization is that for each finite point *p* in an -saturated enlargement of the space there is an integer *k* such that is near-standard. Similar results are shown to hold for countable families of operators and for operators possessing weakly compact powers.

**[1]**N. Dunford and J. T. Schwartz,*Linear operators*, Part I, Interscience, New York, 1958.**[2]**W. A. J. Luxemburg,*A general theory of monads*, Applications of Model Theory to Algebra, Analysis, and Probability (Inte rnat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR**0244931****[3]**Catherine L. Olsen,*A structure theorem for polynomially compact operators*, Amer. J. Math.**93**(1971), 686–698. MR**0405152**, https://doi.org/10.2307/2373464**[4]**Abraham Robinson,*Non-standard analysis*, North-Holland Publishing Co., Amsterdam, 1966. MR**0205854****[5]**D. G. Tacon,*Weak compactness in normed linear spaces*, J. Austral. Math. Soc.**14**(1972), 9–16. MR**0315405**

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DOI:
https://doi.org/10.1090/S0002-9939-1979-0518519-0

Keywords:
Banach space,
operator,
compact,
weakly compact,
-saturated enlargement,
nonstandard analysis

Article copyright:
© Copyright 1979
American Mathematical Society