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

Part of: Normed linear spaces and Banach spaces; Banach lattices Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Anna Kamińska  and
Mieczysław Mastyło
Anna Kamińska
Affiliation:
Department of Mathematical Sciences, The University of Memphis, Memphis TN 38152, USA e-mail: kaminska@memphis.edu
Mieczysław Mastyło
Affiliation:
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
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Abstract

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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 ∧. 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.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B42: Banach lattices 46B45: Banach sequence spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 2 , October 2002 , pp. 251 - 278
Copyright
Copyright © Australian Mathematical Society 2002

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