SAMS Subject Classification Number: 3, 10, 11

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 \(\lambda>0\). 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).