Application of $\rm (L)$ sets to some classes of operators

summary:The paper contains some applications of the notion of $Ł$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $\rm (L)$-Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $\rm (L)$ sets. As a sequence characterization of such operators, we see that an operator $T\colon X\rightarrow E$ from a Banach space into a Banach lattice is order $Ł$-Dunford-Pettis, if and only if $|T(x_{n})|\rightarrow 0$ for $\sigma (E,E')$ for every weakly null sequence $(x_{n})\subset X$. We also investigate relationships between order $Ł$-Dunford-Pettis, $\rm AM$-compact, weak* Dunford-Pettis, and Dunford-Pettis operators. In particular, it is established that each operator $T\colon E\rightarrow F$ between Banach lattices is Dunford-Pettis whenever it is both order $\rm (L)$-Dunford-Pettis and weak* Dunford-Pettis, if and only if $E$ has the Schur property or the norm of $F$ is order continuous