A functional calculus for analytic generators ofC 0-groups

In 9] and 3] anF(S )-functional calculus for sectorial operators is constructed via the Dunford-Riesz integral. This calculus implicitely defines the well-known complex powers of such operators. Sectorial operators with bounded imaginary powers turn out to be of particular interest due to the remarkable Dore-Venni theorem. In 12] this theorem is proved via the theory of analytic generators ofC ₀-groups. These results suggest the existence ofF(S )-functional, calculi forC ₀-groups and their analytic generators. In this paper we show that such functional calculi indeed exsist, however the approach via the Dunford-Riesz integral is no longer viable. Therefore a different approach via an approximation argument is introduced. Existence and uniqueness theorems are given and we show how the functional calculi relate to known results. Examples illustrate the theory.