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

TL;DR

This paper extends the classical Lumer-Phillips generation theorem to bi-continuous semigroups, providing new criteria for their generation and applying these results to transport equations and flows on infinite networks.

Contribution

It introduces a Lumer-Phillips type theorem for bi-continuous semigroups, broadening the scope of semigroup generation theory.

Findings

Established a generation theorem for bi-continuous semigroups.

Applied the theorem to transport equations.

Analyzed flows on infinite networks.

Abstract

The famous 1960s Lumer-Phillips Theorem states that a closed and densely defined operator $A\colon D(A)\subseteq X\rightarrow X$ 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.