Duality Property for Positive Weak Dunford-Pettis Operators

We prove that an operator is weak Dunford-Pettis if its adjoint is one but the converse is false in general, and we give some necessary and sufficient conditions under which each positive weak Dunford-Pettis operator has an adjoint which is weak Dunford-Pettis. 1. Introduction and Notation Let us recall that an operator from a Banach space into another is called Dunford-Pettis if it carries weakly compact subsets of onto compact subsets of . The operator is said to be weak Dunford-Pettis if converges to 0 whenever converges weakly to 0 in and converges weakly to 0 in . The class of weak Dunford-Pettis operators was used by Aliprantis and Burkinshaw [1] and Kalton and Saab [2] when they studied the domination property of Dunford-Pettis operators. As this latter class [3], weak Dunford-Pettis operators do not satisfy the duality property. In fact, there exist weak Dunford-Pettis operators whose adjoints are not weak Dunford-Pettis. For example, as the Banach space has the Schur property, its identity operator is Dunford-Pettis and then weak Dunford-Pettis, but its adjoint , which is the identity operator of the Banach space , is not weak Dunford-Pettis (because the Banach space does not have the Dunford-Pettis property (see [4], page 22)). However, each operator is weak Dunford-Pettis if its adjoint is. On the other hand, if and are two Banach spaces such that is reflexive, then the class of weak Dunford-Pettis operators from into coincides with that of Dunford-Pettis operators from into , and therefore some results of [5] can be applied here to give some answers to our duality problem. Morever, if and are both reflexive, then the class of weak Dunford-Pettis operators from into coincides with that of compact operators from into , and hence if is an operator such that is weak Dunford-Pettis, then its adjoint is weak Dunford-Pettis. Also, if and are two Banach spaces such that or has the Dunford-Pettis property, then each operator from into is weak Dunford-Pettis, and hence each weak Dunford-Pettis has an adjoint which is one. As we have already done for Dunford-Pettis operators [3] and almost Dunford-Pettis operators [6], one of the aims of this paper is to characterize Banach lattices for which each weak Dunford-Pettis operator has an adjoint which is weak Dunford-Pettis. We refer the reader to [5] for unexplained terminologies on Banach lattice theory and positive operators. 2. Some Preliminaries Let us recall that an operator from a Banach lattice into a Banach space is said to be AM-compact if it carries each order-bounded subset of onto a relatively