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 |ST| 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.

Mathematics Subject Classification (2020):

• Primary: 47B60
• Secondary: 47B65
• 46B42

Key words:

• Weak limited set
• weakly precompact set
• weak limited operator
• Dunford-Pettis operator
• Banach lattice
• order continuous norm