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Completely Continuous operators

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A Banach space X has the Dunford-Pettis property (DPP) provided that every weakly compact operator T from X to any Banach space Y is completely continuous (or a Dunford-Pettis operator). It is known that X has the DPP if and only if every weakly null sequence in X is a Dunford-Pettis subset of X. In this paper we give equivalent characterizations of Banach spaces X such that every weakly Cauchy sequence in X is a limited subset of X. We prove that every operator T: X → c₀ is completely continuous if and only if every bounded weakly precompact subset of X is a limited set.
We show that in some cases, the projective and the injective tensor products of two spaces contain weakly precompact sets which are not limited. As a consequence, we deduce that for any infinite compact Hausdorff spaces K₁ and K₂, $C(K₁) ⊗_{π} C(K₂)$ and $C(K₁) ⊗_{ϵ} C(K₂)$ contain weakly precompact sets which are not limited.
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  • Mathematics Department, University of Wisconsin-River Falls, River Falls, WI 54022, U.S.A.
autor
  • Department of Mathematics, University of North Texas, Box 311430, Denton, TX 76203-1430, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm126-2-7
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