Pervasiveness of $\mathcal{L}^r(E,F)$ in $\mathcal{L}^r(E,F^{\delta})$

TL;DR

This paper investigates conditions under which the space of regular operators between certain Riesz spaces is pervasive in their order completions, revealing structural properties and formulas for positive parts of operators.

Contribution

It establishes necessary conditions for the pervasiveness of regular operator spaces in order completions and explores their structural implications.

Findings

Riesz completion of regular operators can be embedded as a Riesz subspace.

The regular part of order continuous operators forms a band.

Positive parts of operators are given by the Riesz-Kantorovich formula.

Abstract

Let $E, F$ be Archimedean Riesz spaces, and let $F^{\delta}$ denote an order completion of $F$. In this note, we provide necessary conditions under which the space of regular operators $\mathcal{L}^r(E, F)$ is pervasive in $\mathcal{L}^r(E, F^{\delta})$. Pervasiveness of $\mathcal{L}^r(E, F)$ in $\mathcal{L}^r(E, F^{\delta})$ implies that the Riesz completion of $ \mathcal{L}^r(E, F)$ can be realized as a Riesz subspace of $ \mathcal{L}^r(E, F^{\delta}$. It also ensures that the regular part of the space of order continuous operators $\mathcal{L}^{oc}(E, F)$ forms a band of $\mathcal{L}^r(E, F)$. Furthermore, the positive part $T^+$ of any operator $T \in \mathcal{L}^r(E, F)$, provided it exists, is given by the Riesz-Kantorovich formula. The results apply in particular to cases where $E = \ell_0^{\infty}$, $E = c$, or $F$ is atomic, and they provide solutions to some problems posed in [3] and [16].