Topological algebras of bounded operators with locally solid Riesz spaces

TL;DR

This paper investigates the algebraic properties of $ob$-bounded operators between vector lattices and locally solid Riesz spaces, focusing on their behavior under uniform and equicontinuous convergence.

Contribution

It introduces and analyzes the concept of $ob$-bounded operators, exploring their algebraic structure in the context of locally solid Riesz spaces.

Findings

$ob$-bounded operators form an algebra under certain conditions

Characterization of convergence types for these operators

Identification of algebraic properties related to boundedness and continuity

Abstract

Let $X$ be a vector lattice and $(E,\tau)$ be a locally solid vector lattice. An operator $T:X\to E$ is said to be $ob$-bounded if, for each order bounded set $B$ in $X$, $T(B)$ is topologically bounded in $E$. In this paper, we study on algebraic properties of $ob$-bounded operators with respect to the topology of uniform convergence and equicontinuous convergence.