On the extension of one-parameter operator semigroups to completions of Archimedean vector lattices

TL;DR

This paper investigates how one-parameter operator semigroups can be extended to completions of Archimedean vector lattices, establishing existence, uniqueness, and new extension theorems for positive semigroups.

Contribution

It provides new results on extending positive semigroups to their completions, relaxing the need for ru-completeness in certain cases.

Findings

Existence and uniqueness of extensions to ru-completions for positive semigroups.

An extension theorem for ru-continuous semigroups on vector lattices with property (R).

Relaxation of ru-completeness requirements in extension results.

Abstract

Extensions of one-parameter operator semigroups on Archimedean vector lattices to their order/ru-completions are studied. Existence and uniqueness of the extension to the ru-completion is established in the class of positive semigroups. An extension theorem for positive ru-continuos semigroups on vector lattices with property (R) is proved. This theorem allows one to abandon the requirement of ru-completeness in various results concerning positive ru-continuous semigroups.