A Lumer-Phillips type generation theorem for bi-continuous semigroups (Q2107768)

Summary: 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 \(\lambda > 0\). 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.