Relationship between different classes of sets in the dual of Banach lattices

Abstract

This paper focuses on different classes of sets in the dual of a Banach lattice such as $V$-sets, $L$-sets, Right sets, $L$-limited sets and almost $L$-sets and then, based on the connections between them, some operator characterizations are obtained. As a result, we show that each almost $L$-set and Right set is a $V$-set and a $V$-set is itself an $L$-limited set. In particular, it is proved that almost $L$-sets are relatively weakly compact if and only if $E^*$ has the order continuous norm and also they are relatively compact if and only if $E^*$ is discrete with the order continuous norm.