The paper contains some applications of the notion of (L) sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order (L)-Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an (L) sets. As a sequence characterization of such operators, we see that an operator T : X → E from a Banach space into a Banach lattice is order (L)-Dunford-Pettis, if and only if |T (xn)| → 0 for σ(E, E′ ) for every weakly null sequence (xn) ⊂ X. We also investigate relationships between order (L)-DunfordPettis, AM-compact, weak* Dunford-Pettis, and Dunford-Pettis operators. In particular, it is established that each operator T : E → F between Banach lattices is Dunford-Pettis whenever it is both order (L)-Dunford-Pettis and weak* Dunford-Pettis, if and only if E has the Schur property or the norm of F is order continuous.
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.