Hermitian and Positive $C$-Semigroups on Banach Spacest

Two classes of operator families, namely «-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 w-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 Csemigroup on L''(/^)(l ) 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(in) and C()(Q.).