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Dunford–Pettis Properties and Spaces of Operators

Published online by Cambridge University Press:  20 November 2018

Ioana Ghenciu  and
Paul Lewis
Ioana Ghenciu
Affiliation:
Mathematics Department, University of Wisconsin-River Falls, River Falls, WI 54022-5001, USA e-mail: [email protected]
Paul Lewis
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA e-mail: [email protected]
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Abstract

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J. Elton used an application of Ramsey theory to show that if $X$ is an infinite dimensional Banach space, then ${{c}_{0}}$ embeds in $X$, ${{\ell }_{1}}$ embeds in $X$, or there is a subspace of $X$ that fails to have the Dunford–Pettis property. Bessaga and Pelczynski showed that if ${{c}_{0}}$ embeds in ${{X}^{*}}$ , then ${{\ell }_{\infty }}$ embeds in ${{X}^{*}}.$ Emmanuele and John showed that if ${{c}_{0}}$ embeds in $K\left( X,\,Y \right)$, then $K\left( X,\,Y \right)$ is not complemented in $L\left( X,\,Y \right)$. Classical results from Schauder basis theory are used in a study of Dunford–Pettis sets and strong Dunford–Pettis sets to extend each of the preceding theorems. The space ${{L}_{{{w}^{*}}}}\left( {{X}^{*}},\,Y \right)$ of ${{w}^{*}}\,-\,w$ continuous operators is also studied.

Keywords

46B2046B28Dunford–Pettis propertyDunford–Pettis setbasic sequencecomplemented spaces of operators
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 2 , 01 June 2009 , pp. 213 - 223
Copyright
Copyright © Canadian Mathematical Society 2009

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