A note on the Lumer--Phillips theorem for bi-continuous semigroups

TL;DR

This paper extends the Lumer-Phillips theorem to characterize generators of bi-continuous semigroups on Banach spaces with a coarser topology, using dissipativity and range conditions.

Contribution

It provides a new characterization of generators of bi-continuous semigroups in Banach spaces with a coarser topology, generalizing classical results.

Findings

Characterization of generators via dissipativity with respect to seminorms

Range condition for bi-continuous semigroup generators

Extension of Lumer-Phillips theorem to bi-continuous setting

Abstract

Given a Banach space $X$ and an additional coarser Hausdorff locally convex topology $\tau$ on $X$ we characterise the generators of $\tau$-bi-continuous semigroups in the spirit of the Lumer--Phillips theorem, i.e. by means of dissipativity w.r.t.~a directed system of seminorms and a range condition.