On the duality of Dunford-Pettis operators on Banach lattices

We establish sufficient conditions for the duality of regular Dunford-Pettis operators on Banach lattices and necessary conditions for the duality condition “if the adjoint of a (positive) operator is Dunford-Pettis, then the operator itself is”. In particular, we show that if each operator T: E → F from a Banach lattice E with an order continuous norm to another Banach lattice F is Dunford-Pettis whenever its adjoint T′: F′ → E′ is Dunford-Pettis, then E has the Schur property or F is a KB-space. As consequences, we deduce a characterization of the Schur property (and KB-spaces).