On the relationship between three classes of operators on Riesz spaces

TL;DR

This paper investigates the relationships among three classes of operators on Riesz spaces, providing counterexamples and conditions for their equivalence, thereby clarifying existing theoretical results.

Contribution

It offers new conditions under which order bounded, topologically bounded, and topologically continuous operators coincide, and presents counterexamples to previous claims.

Findings

Counterexamples illustrating gaps in existing results

Conditions for equivalence of operator classes

Clarification of relationships among operator types

Abstract

Following several papers in the prior literature, we study the relationship between order bounded operators, topologically bounded operators and topologically continuous operators. Our main contribution is two folded: (i) we provide a set of counterexamples to illustrate several extant results in the literature; (ii) we give conditions for the space of order bounded operators to coincide with the space of topologically bounded operators as well as conditions for these two spaces to coincide with the space of topologically continuous operators.