Weak Precompactness And Related Operators On Banach Lattices

The literature related to the theory of weakly precompact sets and weakly precompact operators on Banach spaces is by now very large,but there are few related results in the context of Banach lattices.Based on a new characterization of weakly precompact sets in Banach spaces,we study weakly precompact sets and weakly precompact operators on Banach lattices and their properties,and also consider disjointly weakly compact sets and disjointly weakly compact operators on Banach lattices and their properties.Firstly,we consider solid hulls of weakly precompact sets in Banach lattices,and then apply the related results to the study on relevant properties of weakly precompact operators,such as domination properties,etc.Moreover,the class of order weakly precompact operators on Banach lattices are introduced,and their relations with order weakly compact operators and weakly precompact operators are investigated.Based on the research of weak precompactness of closed unit balls and order intervals in Banach lattices,we give some characterizations of discrete Banach lattices with order continuous norms.Secondly,we consider disjointly weakly compact sets in Banach lattices and introduce disjointly weakly compact operators on Banach lattices.We give some equivalent characterizations of disjointly weakly compact sets in Banach lattices,and also some Banach lattices with special properties are characterized in terms of disjointly weakly compact sets.Moreover,some properties of disjointly weakly compact operators and the relations between the class of disjointly weakly compact operators and other classes of operators are also given.