# Lattice copies and applications for weak L- and M-weakly compact operators on Banach lattices

## TL;DR

This paper introduces weak L- and M-weakly compact operators on Banach lattices, explores their properties and relationships with existing classes, and investigates their compactness characteristics using lattice copies and unbounded convergence.

## Contribution

It extends the theory of compact operators on Banach lattices by defining and analyzing weak L- and M-weakly compact operators, utilizing lattice copies and unbounded convergence.

## Key findings

- Solved the RV and LA problem related to James distortion theorem.
- Established relationships between weak L- and M-weakly compact operators and classical compact operators.
- Analyzed the compactness properties of these new operator classes.

## Abstract

Several recent papers investigated lattice copies and unbounded convergences in Banach lattices. In this paper, we first solve the problem of RV and LA which is an extension of the well-known James distortion theorem. Using lattice copies and unbounded convergence, then we introduce weak L- and M-weakly compact operators on Banach lattices and research the relationship between these operators and L- and M-weakly compact operators. Finally, we study the compactness of weak L-weakly compact and weak M-weakly compact operators.