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 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.