Completely Continuous operators

Ioana Ghenciu ¹   ; Paul Lewis ²

¹ Mathematics Department University of Wisconsin-River Falls River Falls, WI 54022, U.S.A.
² Department of Mathematics University of North Texas Box 311430 Denton, TX 76203-1430, U.S.A.

Colloquium Mathematicum, Tome 126 (2012) no. 2, pp. 231-256

Résumé

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 $T:X\to c_0$ 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 $K_1$ and $K_2$, $C(K_1)\otimes _\pi C(K_2)$ and $C(K_1)\otimes _\epsilon C(K_2)$ contain weakly precompact sets which are not limited.

Détail

DOI : 10.4064/cm126-2-7

Keywords: banach space has dunford pettis property dpp provided every weakly compact operator banach space completely continuous dunford pettis operator known has dpp only every weakly null sequence dunford pettis subset nbsp paper equivalent characterizations banach spaces every weakly cauchy sequence limited subset prove every operator completely continuous only every bounded weakly precompact subset limited set cases projective injective tensor products nbsp spaces contain weakly precompact sets which limited consequence deduce infinite compact hausdorff spaces nbsp otimes otimes epsilon contain weakly precompact sets which limited