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# The Dunford-Pettis Property for Symmetric Spaces

Published:2000-08-01
Printed: Aug 2000
• Anna Kamińska
• Mieczysław Mastyło
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## Abstract

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 $(0, \alpha)$, $0 < \alpha \leq \infty$, the only spaces with the Dunford-Pettis property are $L^1$, $L^{\infty}$, $L^1 \cap L^{\infty}$, $L^1 + L^{\infty}$, $(L^{\infty})^\circ$ and $(L^1 + L^{\infty})^\circ$, where $X^\circ$ denotes the norm closure of $L^1 \cap L^{\infty}$ in $X$. It is also proved that all Banach dual spaces of $L^1 \cap L^{\infty}$ and $L^1 + L^{\infty}$ 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 $(L^1 + L^{\infty})^\circ$ and $(L^{\infty})^\circ$ have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some K\"othe-Bochner spaces.
 MSC Classifications: 46E30 - Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B42 - Banach lattices [See also 46A40, 46B40]