We characterize Banach lattices E and F on which the adjoint of each operator from E into F which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if E and F are two Banach lattices then each order DunfordPettis and weak Dunford-Pettis operator T from E into F has an adjoint Dunford-Pettis operator T ′ from F ′ into E ′ if, and only if, the norm of E ′ is order continuous or F ′ has the Schur property. As a consequence we show that, if E and F are two Banach lattices such that E or F has the Dunford-Pettis property, then each order Dunford-Pettis operator T from E into F has an adjoint T ′ : F ′ → E ′ which is Dunford-Pettis if, and only if, the norm of E ′ is order continuous or F ′ has the Schur property.
We establish some sufficient conditions under which the subspaces of Dunford-Pettis operators, of M-weakly compact operators, of L-weakly compact operators, of weakly compact operators, of semi-compact operators and of compact operators coincide and we give some consequences.
We introduce the notion of order weakly sequentially continuous lattice operations of a Banach lattice, use it to generalize a result regarding the characterization of order weakly compact operators, and establish its converse. Also, we derive some interesting consequences.
We establish some properties of the class of order weakly compact operators on Banach lattices. As consequences, we obtain some characterizations of Banach lattices with order continuous norms or whose topological duals have order continuous norms.
We establish necessary and sufficient conditions under which the linear span of positive AM-compact operators (in the sense of Fremlin) from a Banach lattice $E$ into a Banach lattice $F$ is an order $\sigma $-complete vector lattice.