On Levi operators between normed and vector lattices

TL;DR

This paper investigates Levi operators from normed lattices to vector lattices, establishing their properties, examples, and relationships with other operator classes, advancing the understanding of their structure and behavior.

Contribution

It introduces new results on Levi operators, including their continuity, examples of non-closure under addition, and connections with KB-spaces and domination problems.

Findings

Every finite rank operator is a Levi operator.

Sum of a positive rank one and a positive compact Levi operator need not be Levi.

The set of Levi operators on convergent sequences space is not complete.

Abstract

The notion of a Levi operator is an operator abstraction of the Levy property of a norm or, more generally of the Levi topology on a locally solid vector lattice. Various aspects of Levi operators have been studied recently by several authors. The present paper is devoted to Levi operators from a normed lattice to a vector lattice. It is proved that every finite rank operator is a Levi operator. An example is given showing that the sum of a positive rank one operator and a positive compact Levi operator need not to be a Levi operator. We prove that every quasi Levi operator is continuous. It is shown that the set of Levi operators on the space of convergent sequences is not complete in the operator norm. Several results concerning the domination problem for Levi operators and the relations between Levi operators and KB-spaces are established.