Weak limited sets and operators on Banach lattices

Abstract

In this paper, we prove that an operator T : E → F, between two Banach lattices, maps order intervals onto weak limited sets if and only if the modulus jSTj exists and is Dunford-Pettis for every Dunford-Pettis operator S : F → c₀. Next, we establish that a Banach lattice E does not contain any isomorphic copy of ℓ¹ if and only if the order intervals of E are weak limited and the norm of E′ is order continuous. We also investigate the domination problem of the class of weak limited operators.