A note on the relationship between three classes of operators on Riesz spaces

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.

Keywords

• counterexamples
• locally solid riesz spaces
• order bounded operators
• regular operators
• topologically bounded operators
• topologically continuous operators