On almost limited p-convergent operators on Banach lattices

The purpose of this article is to introduce and study the class of almost limited p-convergent and weak\(^*\) almost p-convergent operators (\(1 \le p <\infty \)). Some new characterizations of Banach lattices with the strong limited p-Schur property; that is, spaces on which every almost limited weakly p-compact set is relatively compact and the weak DP\(^*\) property of order p are obtained. The behavior of the class of these operators with the weak DP\(^*\) property of order p (with focus on Banach lattices with the strong limited p-Schur property) is investigated. Moreover, Banach lattices with the positive limited p-Schur property are introduced and Banach lattices in which this property is equivalent to some other known properties are discussed. In addition, the domination properties of almost limited p-convergent and weak\(^*\) almost p-convergent operators are considered. As an application, using almost limited p-convergent operators we establish some necessary and sufficient conditions under which some operator spaces have the strong limited p-Schur property.