Collective order convergence and collectively qualified sets of operators

TL;DR

This paper investigates collective order convergence and qualified operator sets in vector lattices, proving boundedness properties of these sets in terms of order and norm convergence.

Contribution

It introduces and analyzes collective order convergence concepts and establishes boundedness results for collectively qualified operator sets in vector lattices.

Findings

Collectively order to norm bounded sets are norm bounded.

Collectively order continuous sets are collectively order bounded.

The paper develops new properties of operator sets in vector lattices.

Abstract

Collective versions of order convergences and corresponding types of collectively qualified sets of operators in vector lattices are investigated. It is proved that collectively order to norm bounded sets are bounded in the operator norm and collectively order continuous sets are collectively order bounded.