The Dunford-Pettis Property for Symmetric Spaces

Kamińska, Anna ; Mastyło, Mieczysław

Canadian journal of mathematics, Tome 52 (2000) no. 4, pp. 789-803

Résumé

A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that ${{\ell }^{1}},{{c}_{0}}$ and ${{\ell }^{\infty }}$ are the only symmetric sequence spaces with the Dunford- Pettis property, and that in the class of symmetric spaces on $(0,\,\alpha ),\,0\,<\,\alpha \,\le \,\infty$ , the only spaces with the Dunford-Pettis property are ${{\text{L}}^{1}},{{\text{L}}^{\infty }},{{\text{L}}^{1}}\cap {{\text{L}}^{\infty }},{{\text{L}}^{1}}+{{\text{L}}^{\infty }},{{({{\text{L}}^{\infty }})}^{\text{o}}}$ and ${{({{\text{L}}^{1}}+{{\text{L}}^{\infty }})}^{\text{o}}}$ , where ${{\text{X}}^{\text{o}}}$ denotes the norm closure of ${{\text{L}}^{1}}\cap {{\text{L}}^{\infty }}$ in $X$ . It is also proved that all Banach dual spaces of ${{\text{L}}^{1}}\cap {{\text{L}}^{\infty }}$ and ${{\text{L}}^{1}}+{{\text{L}}^{\infty }}$ have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces ${{({{\text{L}}^{1}}+{{\text{L}}^{\infty }})}^{\text{o}}}$ and ${{({{\text{L}}^{\infty }})}^{\text{o}}}$ have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some Köthe-Bochner spaces.

Détail

DOI : 10.4153/CJM-2000-033-9

Mots-clés : 46E30, 46B42

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