Reciprocal Dunford--Pettis sets and V$^*$-sets in Banach lattices

Jin Xi Chen; Xi Li

Reciprocal Dunford--Pettis sets and V$^*$-sets in Banach lattices

Functional Analysis 2024-06-17 v1

Authors: Jin Xi Chen , Xi Li

Abstract

In this short note, we show that one cannot differentiate between reciprocal Dunford--Pettis sets and V $^*$ -sets in a Banach lattice. That is, for a bounded subset $K$ of a Banach lattice $E$ , $K$ is a V $^*$ -set if and only if $K$ is a reciprocal Dunford--Pettis set, or equivalently, if and only if every disjoint sequence in the solid hull of $K$ is weakly null (i.e. $K$ is disjointly weakly compact).

Keywords

Dunford-Pettis sets V*-sets Banach lattices disjoint sequences weak compactness

Cite

@article{arxiv.2406.09853,
  title  = {Reciprocal Dunford--Pettis sets and V$^*$-sets in Banach lattices},
  author = {Jin Xi Chen and Xi Li},
  journal= {arXiv preprint arXiv:2406.09853},
  year   = {2024}
}

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4 pages

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