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, and contain weakly precompact sets which are not limited.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm126-2-7,
author = {Ioana Ghenciu and Paul Lewis},
title = {Completely Continuous operators},
journal = {Colloquium Mathematicae},
volume = {126},
year = {2012},
pages = {231-256},
zbl = {1256.46009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm126-2-7}
}
Ioana Ghenciu; Paul Lewis. Completely Continuous operators. Colloquium Mathematicae, Tome 126 (2012) pp. 231-256. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm126-2-7/