# A Lumer–Phillips type generation theorem for bi-continuous semigroups

### Christian Budde

University of the Free State, Bloemfontein, South Africa### Sven-Ake Wegner

Universität Hamburg, Germany

## Abstract

The famous 1960s Lumer–Phillips theorem states that a closed and densely defined operator $A\colon \operatorname{D}(A)\subseteq X \to X$ on a Banach space $X$ generates a strongly continuous contraction semigroup if and only if $(A,\operatorname{D}(A))$ is dissipative and the range of $\lambda-A$ is surjective in $X$ for some $\lambda>0$. In this paper, we establish a version of this result for bi-continuous semigroups and apply the latter amongst other examples to the transport equation as well as to flows on infinite networks.

## Cite this article

Christian Budde, Sven-Ake Wegner, A Lumer–Phillips type generation theorem for bi-continuous semigroups. Z. Anal. Anwend. 41 (2022), no. 1/2, pp. 65–80

DOI 10.4171/ZAA/1695