Collective order boundedness of sets of operators between ordered vector spaces

TL;DR

This paper establishes conditions under which sets of operators between ordered vector spaces are bounded, with implications for operator semigroups, enhancing understanding of their structural properties.

Contribution

It proves that certain collectively order continuous and order-to-norm bounded operator sets are bounded, providing new insights into their behavior in ordered vector spaces.

Findings

Collectively order continuous sets are collectively order bounded in Archimedean OVSs.

Order to norm bounded sets from Banach spaces are norm bounded.

Applications to commutative operator semigroups are demonstrated.

Abstract

It is proved that: each collectively order continuous set of operators from an Archimedean OVS with a generating cone to an OVS is collectively order bounded; and each collectively order to norm bounded set of operators from an ordered Banach space with a closed generating cone to a normed space is norm bounded. Several applications to commutative operator semigroups on ordered vector spaces are given.