Dunford–Pettis Properties and Spaces of Operators

Ghenciu, Ioana ; Lewis, Paul

Canadian mathematical bulletin, Tome 52 (2009) no. 2, pp. 213-223

Résumé

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.

Détail

DOI : 10.4153/CMB-2009-024-5

Mots-clés : 46B20, 46B28, Dunford–Pettis property, Dunford–Pettis set, basic sequence, complemented spaces of operators