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 $A:D(A)\subseteq X\to X$ on a Banach space $X$ generates a strongly continuous contraction semigroup if and only if $(A,D(A))$ 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.