Order Dunford-Pettis Operators On Banach Lattices

The order Dunford-Pettis operator is a new type of important operators on Banach lattices. It can be said to the extention of the Dunford-Pettis operator. The order Dunford-Pettis operator is closed to the order weakly compact operator and the AM-compact operator, which has the research value. The paper defines the order Dunford-Pettis operator and discusses its controllability, conjugacy,(order)weakly compactness,(almost)Dunford-Pettis property and AM-compactness.The main results:1. We introduce the research results on some operators on Banach lattices, such as the controllability, conjugacy and the relations of the AM-compact, Dunford-Pettis and compact operators. These are prepare for introducing and further reseaching order Dunford-Pettis operator.2. We study some characterizations of order Dunford-Pettis opertors and establish sufficent conditions under which each operator between Banach latties is order Dunford-Pettis. First, introduce the class of order Dunford-Pettis operators and study its controllability, two sides ideal and the product of order Dunford-Pettis operator with others. Second, establish sufficent space conditions under which each operator between Banach latties is order Dunford-Pettis. Third, discuss its dulity property.3. We study the (order)weakly compactness,(almost)Dunford-Pettis property and AM-compactness of order Dunford-Pettis operators on Banach lattices. We establish a necessary and sufficient confitions on Banach lattices which guarantees that each order Dunford-Pettis operator is weakly compact operator. In addition, We disccuss the relationship between AM-compact operators and Dunford-Pettis operators.