# Completely Continuous operators

Colloquium Mathematicae (2012)

- Volume: 126, Issue: 2, page 231-256
- ISSN: 0010-1354

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topIoana 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 -

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