Compact-Like Operators in Vector Lattices Normed by Locally Solid Lattices

TL;DR

This paper introduces and studies a new class of operators called $p_ au$-compact operators in vector lattices, generalizing many existing operator classes within locally solid Riesz space frameworks.

Contribution

It defines $p_ au$-compact operators and explores their properties, extending the understanding of operator classes in vector lattice theory.

Findings

$p_ au$-compact operators generalize known classes like compact and order continuous operators.

Properties of $p_ au$-compact operators are systematically analyzed.

The framework unifies various operator concepts in locally solid Riesz spaces.

Abstract

A linear operator $T$ between two vector lattices normed by 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 said to be $p_\tau$-bounded operator if it sends $p_\tau$-bounded subsets to $p_\tau$-bounded subsets. Also, $T$ is called $p_\tau$-compact if, for any $p_\tau$-bounded net $(x_\alpha)$, the net $(Tx_\alpha)$ has a $p_\tau$-convergent subnet. They generalize several known classes of operators such as norm continuous, order continuous, $p$-continuous, order bounded, $p$-bounded, compact and AM-compact operators. We study the general properties of these operators.