Structure of subspaces of the compact operators having the Dunford-Pettis property
type: Item
instance of: scholarly article
title: Structure of subspaces of the compact operators having the Dunford-Pettis property
zbMATH Open document ID: 0944.47044
DOI: 10.1007/PL00004761
author: Eero Saksman
author: Hans-Olav Tylli
published in: Mathematische Zeitschrift
publication date: 2000-09-14T00:00:00Z
review text: A Banach space \(X\) is said to have the Dunford-Pettis property (DPP) if weakly compact operators on \(X\) map weakly compact sets to norm compact sets. It has turned out to be an important notion for both Banach spaces and Banach algebras. \textit{S. W. Brown} [J. Oper. Theory 33, No. 1, 33-42 (1995; Zbl 0838.47032)] and \textit{A. Ülger} [J. Oper. Theory 37, No. 2, 371-378 (1997; Zbl 0894.47033)] have recently characterized those subspaces of \(K(H)\) (the compact operators on a not-necessarily-separable Hilbert space) that have DPP. Here the authors show that the same characterization is valid for subspaces of \(K(\ell_p)\) \(1 < p < \infty\) and for compact operators on finite sums of \(\ell_p\) spaces. They show, further, that these subspaces are precisely those that can be embedded in ({ i.e.} are linearly isomorphic to a subspace of) \(c_0\). The paper also contains a number of interesting corollaries and examples.
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B28
zbMATH DE Number: 1384885
zbMATH Keywords: Dunford-Pettis property
zbMATH Keywords: weakly compact operators
MaRDI profile type: Publication
Wikidata QID: Q109553847
full work available at URL: pl00004761
OpenAlex ID: W2014809951
description: scientific article; zbMATH DE number 1384885