On the surjective Dunford-Pettis Property

Abstract

A Banach space E has the Dunford-Pettis property if every operator from E into a reflexive Banach space is a Dunford-Pettis operator. D. Leung [Math. Z. 197, 21-32 (1988] introduced a formally weaker property, the surjective Dunford-Pettis property, by imposing that every operator from E onto a reflexive Banach space is Dunford-Pettis. This property is used by Leung to obtain sustantial extensions of previous results of Lotz on ergodic operators and strongly continuous semigroups of operators. Also he proved that the surjective Dunford- Pettis property is, in fact, genuinely weaker than the Dunford-Pettis property, building a Banach space L with the surjective Dunford-Pettis property that fails the Dunford-Pettis property. In this paper we obtain several results about the surjective Dunford- Pettis property showing some of the analogies and differences with the Dunford-Pettis property. Also, new properties of the interesting Banach space L built by Leung are obtained.