The Dunford-Pettis Property for Symmetric Spaces

Abstract

AbstractA complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that ℓ1, c0 and ℓ∞ are the only symmetric sequence spaces with the Dunford- Pettis property, and that in the class of symmetric spaces on (0, α), 0 < α ≤ ∞, the only spaces with the Dunford-Pettis property are L1, L∞, L1 ∩ L∞, L1 + L∞, (L∞)◦ and (L1 + L∞)◦, where X◦ denotes the norm closure of L1 ∩ L∞ in X. It is also proved that all Banach dual spaces of L1 ∩ L∞ and L1 + L∞ 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 (L1 + L∞)◦ and (L∞)◦ have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some Köthe-Bochner spaces.