Disjointness preserving $\mathrm{C}_0$-semigroups and local operators on ordered Banach spaces

TL;DR

This paper extends the theory of $ ext{C}_0$-semigroups to ordered Banach spaces, showing that disjointness preserving semigroup generators are local and exploring properties of local operators and their generated semigroups.

Contribution

It generalizes results from Banach lattices to ordered Banach spaces, establishing the locality of generators and analyzing local operators using Taylor series and Yosida approximations.

Findings

Generators of disjointness preserving $ ext{C}_0$-semigroups are local.

Bands are closed under regular norms in these spaces.

Local operators can generate local $ ext{C}_0$-semigroups.

Abstract

We generalize results concerning $\mathrm{C}_0$-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving $\mathrm{C}_0$-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local $\mathrm{C}_0$-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extensions of regular norms.