Application of \((\text{L})\) sets to some classes of operators. (Q2828794)

In this paper, the authors present some applications of the notion of \((\text{L})\) sets to several classes of operators on Banach lattices. They establish some characterizations of weak Dunford-Pettis operators and Banach spaces with the Dunford-Pettis property. They introduce another weak version of Dunford-Pettis operators called order \((\text{L})\)-Dunford-Pettis operators and derive some characterizations of this kind of operators. The authors also study the relationship between the class of order \((\text{L})\)-Dunford-Pettis operators and that of \((\text{AM})\)-compact (respectively, order weakly compact, weak Dunford-Pettis) operators. They characterize Banach lattices \(E\) and \(F\) on which each operator from \(E\) into \(F\) which is both order \((\text{L})\)-Dunford-Pettis and weak\(^*\) Dunford-Pettis is Dunford-Pettis.