SAMS Subject Classification Number: 3, 10, 11

The famous 1960s Lumer–Phillips theorem states that a closed and densely defined operator $A:\mathrm{D}(A)\subseteq X\to X$ on a Banach space $X$ generates a strongly continuous contraction semigroup if and only if $(A,\mathrm{D}(A))$ is dissipative and the range of $\lambda -A$ is surjective in $X$ for some $\lambda >0$. We will investigate 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. This is joint work with S.-A. Wegner (Hamburg, Germany).