J. Austral. Math. Soc.  73 (2002), 251-278
The Schur and (weak) Dunford-Pettis properties in Banach lattices

Anna Kaminska
  Department of Mathematical Sciences
  The University of Memphis
  Memphis TN 38152
Mieczyslaw Mastylo
  Faculty of Mathematics and
  Computer Science
  A. Mickiewicz University
  Institute of Mathematics
  Poznan Branch
  Polish Academy of Sciences
  Matejki 48/49
  60-769 Poznan

We study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that $\ell_1$, $c_0$ and $\ell_{\infty}$ 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 $\Lambda$. We also present examples of weighted Orlicz spaces with the Schur property which are not $\mathcal{L}_{1}$-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.
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