Domination properties and extension of positive compact operators on pre-Riesz spaces

TL;DR

This paper investigates the positive domination property of compact operators on pre-Riesz spaces, extending order continuous norms to Riesz completions and establishing compactness of the third power of positive operators in spaces with an order unit.

Contribution

It introduces a method to embed pre-Riesz spaces into Riesz completions and extends order continuous norms, advancing the understanding of positive compact operators in these spaces.

Findings

Extended order continuous norms to Riesz completions.

Established compactness of the third power of positive operators.

Provided new insights into domination properties in pre-Riesz spaces.

Abstract

This paper concerns positive domination property of compact operators on pre-Riesz spaces. The method is embedding the pre-Riesz space to the Riesz completion. It extends the order continuous norms in pre-Riesz spaces to Riesz completions. The compactness of third power of a positive operator is obtained in a pre-Riesz space which has an order unit.