Completely Continuous operators

• Autorzy: Ioana Ghenciu , Paul Lewis
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 46B25: Classical Banach spaces in the general theory
• 46B28: Spaces of operators; tensor products; approximation properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2012
• Tom: 126
• Numer: 2
• Strony: 231-256

Daty

wydano

2012

Twórcy

autor

Ioana Ghenciu

• Mathematics Department, University of Wisconsin-River Falls, River Falls, WI 54022, U.S.A.

autor

Paul Lewis

• Department of Mathematics, University of North Texas, Box 311430, Denton, TX 76203-1430, U.S.A.

Identyfikatory

DOI

10.4064/cm126-2-7