About positive Dunford–Pettis operators on Banach lattices

We give several characterizations of Banach lattices on which each positive Dunford–Pettis operator is compact. As consequences, we obtain new sufficient and necessary conditions under which a norm of a Banach lattice is order continuous, a positive weakly compact operator is compact and the dual operator of a positive Dunford–Pettis operator is Dunford–Pettis.