On weak-star p-convergent operators

Abstract

The purpose of this article is to introduce and study the notion of “weak* p-convergent operator”. We discuss the relationship between the weak* p-convergent operators and the p-convergent operators, a class of operators that was introduced in the paper [4] and which plays an important role in the study of the DP*-property of order p (in the paper [14]). Some new characterizations of Banach spaces with the DP*-property of order p are obtained, the p-Gelfand-Phillips property is introduced and the behaviour of weak* p-convergent operators on Banach spaces with this property (with focus on Banach lattices with the p-Gelfand-Phillips property) is investigated. In the last section of this paper, we consider the domination properties of positive p-convergent and positive weak* p-convergent operators on Banach lattices.