Almost Limited Sets in Banach Lattices

TL;DR

This paper introduces almost limited sets in Banach lattices, characterizes when they coincide with L-weakly compact sets via order continuity, and explores operator implications in $\sigma$-Dedekind complete lattices.

Contribution

It defines almost limited sets, establishes their equivalence with L-weakly compact sets under order continuity, and characterizes operators related to these sets in $\sigma$-Dedekind Banach lattices.

Findings

Almost limited sets are characterized in Banach lattices.

Order continuity of the norm is equivalent to the coincidence of almost limited and L-weakly compact sets.

Relatively weakly compact sets are almost limited if and only if all operators into c0 are almost Dunford-Pettis.

Abstract

We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak$^{*}$ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and $L$-weakly compact sets coincide. In particular, in terms of almost Dunford-Pettis operators into $c_{0}$, we give an operator characterization of those $\sigma$-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a $\sigma$-Dedekind Banach lattice $E$, every relatively weakly compact set in $E$ is almost limited if and only if every continuous linear operator $T:E\rightarrow c_{0}$ is an almost Dunford-Pettis operator.