Operators Affiliated to Banach Lattice Properties and the Enveloping Norms

TL;DR

This paper explores operators related to Banach lattice properties, showing they form Banach spaces and examining their domination, thereby advancing understanding of operator structures in Banach lattices.

Contribution

It demonstrates that operators affiliated with key Banach lattice properties form Banach spaces and investigates the domination problem for these operators.

Findings

Operators affiliated with Banach lattice properties form Banach spaces.

Many operators related to these properties can be characterized via enveloping norms.

The domination problem for these operators is addressed.

Abstract

Several recent papers were devoted to various modifications of limited, Grothendieck, and Dunford--Pettis operators, etc., through involving the Banach lattice structure. In the present paper, it is shown that many of these operators appear as operators affiliated to well known properties of Banach lattices, like the disjoint (dual) Schur property, the disjoint Grothendieck property, the property (d), and the sequential w*-continuity of the lattice operations. It is proved that the spaces consisting of regularly versions of the above operators are all Banach spaces. The domination problem for such operators is investigated.