Journal of the Australian Mathematical Society

Research Article

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

Anna Kamińskaa1 and Mieczysław Mastyłoa2

a1 Department of Mathematical Sciences, The University of Memphis, Memphis TN 38152, USA e-mail: kaminska@memphis.edu

a2 Faculty of Mathematics, and Computer Science, A. Mickiewicz University, and Institute of Mathematics, Poznań Branch, Polish Academy of Sciences, Matejki 48/49, 60–769 Poznań, Poland e-mail: mastylo@amu.edu.pl

Abstract

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 MusielakOrlicz 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 xs2227. 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.

(Received November 13 2000)

2000 Mathematics subject classification

  • primary 46B20;
  • 46E30;
  • 46B42;
  • 46B45