About positive weak Dunford–Pettis operators on Banach lattices

We characterize Banach lattices for which each positive weak Dunford–Pettis operator from a Banach lattice into another dual Banach lattice is almost Dunford–Pettis. Also, we give some sufficient and necessary conditions for which the class of positive weak Dunford–Pettis operators coincides with that of positive Dunford–Pettis operators, and we derive some consequences.