Almost order-weakly compact operators on Banach lattices

TL;DR

This paper introduces and characterizes almost order-weakly compact operators between Banach lattices, establishing conditions under which positive operators exhibit this property and exploring their relationships with other operator classes.

Contribution

It provides a new characterization of almost order-weakly compact operators on Banach lattices, especially for positive operators into Dedekind complete lattices.

Findings

Characterization of almost order-weakly compact operators via disjoint sequences.

Equivalence of almost order-weakly compactness and norm convergence for positive operators.

Analysis of properties and relationships with other operator classes.

Abstract

A continuous operator $T$ between two Banach lattices $E$ and $F$ is called almost order-weakly compact, whenever for each almost order bounded subset $A$ of $E$, $T(A)$ is a relatively weakly compact subset of $F$. In Theorem 4, we show that the positive operator $T$ from $E$ into Dedekind complete $F$ is almost order-weakly compact if and only if $T(x_n) \xrightarrow{\|.\|}0$ in $F$ for each disjoint almost order bounded sequence $\{x_n\}$ in $E$. In this manuscript, we study some properties of this class of operators and its relationships with others known operators.