Order Dunford-Pettis Operators On Banach Lattices

Operator theory on Banach lattices, as an important part of Banach lattice theory, mainly investigate the properties of special classes operators and relationships between them, such as, compact operators, weakly compact operators and Dunford-Pettis operators etc. Based on the consideration of Dunford-Pettis operators, the concept of order Dunford-Pettis operator has been given in this paper. We also investigated the basic properties of order Dunford-Pettis operators and relationships between order Dunford-Pettis operators and other operators on Banach lattices. The main contents of this paper consists of two parts as follow.The first part of this paper deals with the basic properties of order Dunford-Pettis operators. First, in terms of weak Cauchy sequences the Dunford-Pettis operators are characterized as follow: The linear operator T is a Dunford-Pettis operator if and only if T maps weak Cauchy sequences to norm convergent sequences. Next, we study the domination property of order Dunford-Pettis operators, conclude that if the domain of T has weakly sequentially continuous lattice operations or the range of T has order continuous norm, then the positive operator T dominated by a order Dunford-Pettis operator is likewise order Dunford-Pettis. At last, two conclusions about lattice property of Dunfor-Pettis operators has been given.In the second part of this paper, we discussed the relationships between order Dunford-Pettis operators and other operators on Banach lattices. Particulur emphasis is given to relationships between order Dunford-Pettis operators and order weakly compact operators, as well as AM-compact operators. First, every order Dunford-Pettis operator is order weakly compact, conversly, if the domain of operator T has an unit, then each order weakly operator is necessarily an order Dunford-Pettis operator. Next, on the relationship between oder Dunford-Pettis operators and AM-compact operators, we claim that if the domain of oder Dunford-Pettis operator T has order continuous norm, then T is AM-compact. conversely, if the domain of AM-compact operator T has order continuous norm, then T is oder Dunford-Pettis. During the discussion, we also obtained some related results.