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 < infinity) 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(1)(mu) and C-0(Omega).
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