Hermitian and Positive $C$-Semigroups on Banach Spacest

Abstract

Two classes of operator families, namely $n$-times integrated $C$-semigroups of hermitian and positive operators on Banach spaces, are studied. By using Gelfand transform and a theorem of Sinclair, we prove some interesting special properties of such $C$-semigroups. For instances, every hermitian nondegenerate $n$-times integrated $C$-semigroup on a reflexive space is the $n$-times integral of some hermitian $C$-semigroup with a densely defined generator; an exponentially bounded $C$-semigroup on $L^p(\mu )$ ($1<p<\infty$) dominates $C$ (a positive injective operator) if and only if its generator $A$ is bounded, positive , and commutes with $C$; when $C$ has dense range, the latter assertion is also true on $L^p(\mu )$ and $C_0(\Omega )$.