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

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.