Hermitian and Positive <i>C</i>-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 Lp(μ) (1 < p < ∞) 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 Lp(μ) and _C_0(Ω).