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
  USA
  kaminska@memphis.edu
and
Mieczyslaw Mastylo
  Faculty of Mathematics and
  Computer Science
  A. Mickiewicz University
  and
  Institute of Mathematics
  Poznan Branch
  Polish Academy of Sciences
  Matejki 48/49
  60-769 Poznan
  Poland
  mastylo@amu.edu.pl


Abstract
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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