Continuous Operators with Convergence in Lattice-Normed Locally Solid Riesz Spaces

TL;DR

This paper introduces and studies continuous and bounded linear operators in lattice-normed locally solid Riesz spaces, generalizing many existing classes of operators and exploring their properties.

Contribution

It defines new classes of $p_ au$-continuous and $up_ au$-continuous operators, extending the framework of operator theory in lattice-normed locally solid Riesz spaces.

Findings

$p_ au$-continuous operators generalize known classes.

Characterization of $up_ au$-continuous operators.

Relationships between different operator classes.

Abstract

A linear operator $T$ between two lattice-normed locally solid Riesz spaces is said to be $p_\tau$-continuous if, for any $p_\tau$-null net $(x_\alpha)$, the net $(Tx_\alpha)$ is $p_\tau$-null, and $T$ is also said to be $p_\tau$-bounded operator if it sends $p_\tau$-bounded subsets to $p_\tau$-bounded subsets. They are generalize several known classes of operators such as continuous, order continuous, $p$-continuous, order bounded, $p$-bounded operators, etc. We also study $up_\tau$-continuous operators between lattice-normed locally solids Riesz spaces.