On the class of order L-weakly and order M-weakly compact operators

TL;DR

This paper introduces new types of compact operators in Banach lattices, characterizes their properties, and explores their relationships with dual operators and lattice norms.

Contribution

It defines order L-weakly and order M-weakly compact operators and establishes their key properties and duality relations in Banach lattices.

Findings

T is order M-weakly compact iff T' is order L-weakly compact.

If T' is order M-weakly compact, then T is order L-weakly compact.

Characterizations of Banach lattices with order continuous norms.

Abstract

In this paper, we introduce and study new concepts of order L-weakly and order M-weakly compact operators. As consequences, we obtain some characterizations of Banach lattices with order continuous norms or whose topological duals have order continuous norms. It is proved that if $T : E \longrightarrow F$ is an operator between two Banach lattices, then T is order M-weakly compact if and only if its adjoint T' is order L-weakly compact. Also, we show that if its adjoint T' is order M-weakly compact, then T is order L-weakly compact. Some related results are also obtained.