J. Math. Anal. Appl. 354 (2009) 295–300 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Some properties of the class of positive Dunford–Pettis operators Belmesnaoui Aqzzouz a,∗ , Aziz Elbour b , Jawad Hmichane b a Université Mohammed V-Souissi, Faculté des Sciences Economiques, Juridiques et Sociales, Département d’Economie, B.P. 5295, SalaEljadida, Morocco b Université Ibn Tofail, Faculté des Sciences, Département de Mathématiques, B.P. 133, Kénitra, Morocco article info abstract Article history: Received 31 March 2008 Available online 8 January 2009 Submitted by R. Curto Keywords: M-weakly compact operator L-weakly compact operator Dunford–Pettis operator Order continuous norm We characterize Banach lattices for which each positive Dunford–Pettis operator is M-weakly compact (resp. L-weakly compact) and we give some consequences.  2008 Elsevier Inc. All rights reserved. 1. Introduction and notation In [9], Meyer-Nieberg gave an interesting study of the class of M-weakly compact operators and the class of L-weakly compact operators. The introduction of these two classes of operators was justified by the difficulties meeting in the study of the class of weakly compact operators on Banach lattices. Recall that an operator T from a Banach lattice E into a Banach space F is said to be M-weakly compact if for each disjoint bounded sequence (x n ) of E , we have lim n ‖T (x n )‖= 0. And an operator T from a Banach space E into a Banach lattice F is called L-weakly compact if for each disjoint bounded sequence ( y n ), in the solid hull of T ( B E ), we have lim n ‖ y n ‖= 0 where B E is the closed unit ball of E . Note that by Proposition 3.6.11 of [9], an operator between two Banach lattices is M-weakly compact (resp. L-weakly compact) if and only if its adjoint is L-weakly compact (resp. M-weakly compact). On the other hand, an operator T from a Banach space E into another F is said to be Dunford–Pettis if it carries weakly compact subsets of E onto compact subsets of F . Note that a M-weakly compact (resp. L-weakly compact) operator is not necessary Dunford–Pettis. The converse is not always true. More than, the class of Dunford–Pettis operators and the class of M-weakly compact (resp. L-weakly compact) operators do not coincide even when the Banach lattice E is reflexive. However, Meyer-Nieberg [9, Theorem 3.7.10] established that each Dunford–Pettis operator from a Banach lattice E into a Banach space F is M-weakly compact if and only if the norm of the topological dual of E is order continuous. The objective of this paper is to establish necessary and sufficient conditions for which each positive Dunford–Pettis operator is M-weakly compact (resp. L-weakly compact). We will first prove that each Dunford–Pettis (resp. compact) oper- ator T , from a Banach lattice E into a Banach space F , is M-weakly compact if and only if the norm of E ′ is order continuous or F ={0}. As a consequence, we will give a generalization of Theorem 2.26 of [5] about the weak compactness of Dunford– Pettis operators. Next, we will establish an analogue to Riesz Theorem, by proving that the closed unit ball B E , of a Banach lattice E , is L-weakly compact if and only if E is finite-dimensional. Finally, we will use the last result to show that each positive Dunford–Pettis operator T : E → F , between Banach lattices, is L-weakly compact if and only if E ={0} or F is finite-dimensional or the norms of E ′ and F are order continuous. Also, we will deduce some interesting consequences. * Corresponding author. E-mail address: baqzzouz@hotmail.com (B. Aqzzouz). 0022-247X/$ – see front matter  2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2008.12.063 brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector