The Schur and (weak) Dunford-Pettis properties in Banach lattices


We study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that ℓ1, c0 and ℓ∞ are the only Banach symmetric sequence spaces with the weak Dunford-Pettis property. We also characterize a large class of Banach lattices without the (weak) Dunford-Pettis property. In Musielak-Orlicz sequence spaces we give some necessary and sufficient conditions for the Schur property, extending the Yamamuro result. We also present a number of results on the Schur property in weighted Orlicz sequence spaces, and, in particular, we find a complete characterization of this property for weights belonging to class A. We also present examples of weighted Orlicz spaces with the Schur property which are not ℒ1-spaces. Finally, as an application of the results in sequence spaces, we provide a description of the weak Dunford-Pettis and the positive Schur properties in Orlicz spaces over an infinite non-atomic measure space.

Publication Title

Journal of the Australian Mathematical Society