$DW$-DP operators and $DW$-limited operators on Banach lattices

This paper is devoted to the study of two classes of operators related to disjointly weakly compact sets, which we call $DW$-DP operators and $DW$-limited operators, respectively. They carry disjointly weakly compact subsets of a Banach lattice onto Dunford-Pettis sets and limited sets, respectively. We show that $DW$-DP (resp. $DW$-limited) operators are precisely the operators which are both weak Dunford-Pettis and order Dunford-Pettis (resp. weak${}^{\ast }$ Dunford-Pettis and order limited) operators. Furthermore, the approximation properties of positive $DW$-DP and positive $DW$-limited operators are given.