Grothendieck-type subsets of Banach lattices

TL;DR

This paper explores new classes of sets in Banach lattices related to Grothendieck spaces, analyzing their properties and the associated linear operators to deepen understanding of these structures.

Contribution

It introduces localized classes of sets in Banach lattices and compares them with existing classes, along with the study of related linear operators.

Findings

New classes of sets in Banach lattices are defined and analyzed.

Relationships between these classes and existing Grothendieck space notions are established.

Comparisons of the properties of associated linear operators are provided.

Abstract

In the setting of Banach lattices the weak (resp. positive) Grothendieck spaces have been defined. We localize such notions by defining new classes of sets that we study and compare with some quite related different classes. This allows us to introduce and compare the corresponding linear operators.