On Some Aspects of Pseudonorm Compact and Montel Operators on Locally Solid Vector Lattices

TL;DR

This paper explores the properties of various pseudonorms on vector lattices and investigates the compactness of different classes of operators, including Montel and pseudonorm compact operators, within locally solid topologies.

Contribution

It introduces new classes of operators like pseudonorm compact and pseudo-AM-compact, extending the understanding of operator compactness in locally solid vector lattices.

Findings

Characterization of pseudonorms on vector lattices.

Introduction of pseudonorm compact and pseudo-AM-compact operators.

Analysis of compactness properties in non-Hausdorff topologies.

Abstract

Unbounded convergences have been applied successfully to locally solid topologies on vector lattices. In the present paper, we first expose several properties of various classes of Riesz pseudonorms on vector lattices. We accomplish this by abstracting some generalities of the norm of an AM-space with strong norm unit to locally solid topologies induced by a pseudonorm. By using these classes of pseudonorms, we study compactness properties of operators (not necessarily linear) between locally solid (not necessarily Hausdorff) topologies. We study new classes of operators such as pseudonorm compact, pseudo-semicompact and pseudo-AM-compact operators as well as the classical Montel operators.