Completely Continuous operators

Ioana Ghenciu, and Paul Lewis. "Completely Continuous operators." Colloquium Mathematicae 126.2 (2012): 231-256. <http://eudml.org/doc/286199>.

@article{IoanaGhenciu2012,
abstract = {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.},
author = {Ioana Ghenciu, Paul Lewis},
journal = {Colloquium Mathematicae},
keywords = {completely continuous operators; limited sets; Dunford-Pettis property; Dunford-Pettis set; Grothendieck space},
language = {eng},
number = {2},
pages = {231-256},
title = {Completely Continuous operators},
url = {http://eudml.org/doc/286199},
volume = {126},
year = {2012},
}

TY - JOUR
AU - Ioana Ghenciu
AU - Paul Lewis
TI - Completely Continuous operators
JO - Colloquium Mathematicae
PY - 2012
VL - 126
IS - 2
SP - 231
EP - 256
AB - 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.
LA - eng
KW - completely continuous operators; limited sets; Dunford-Pettis property; Dunford-Pettis set; Grothendieck space
UR - http://eudml.org/doc/286199
ER -