If [e~' ] is a uniformly bounded C0 semigroup on a complex Banach space X, then -A" , 0 < a < 1, generates a holomorphic semigroup on X , and [e ] is subordinated to [e ] through the Levy stable density function. This was proved by Yosida in 1960, by suitably deforming the contour in an inverse Laplace transform representation. Using other methods, we exhibit a large class of probability measures such that the subordinated semigroups are always holomorphic, and obtain a necessary condition on

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... e measure's Laplace transform for that to be the case. We then construct probability measures that do not have this property. Jo We express this symbolically by U(t) -(pa(t), T), where, for fixed t, pa(t) is the probability distribution with density p°(t). We write T(t) = e~tA , where -A is the infinitesimal generator of [T(t)]. Whenever multivalued functions \p(z) appear, the particular branch where Re y/(z) > 0 for Re z > 0, is understood. The above is an example of a subordinated semigroup: [U(t)] is said to be subordinated to [7X0] through the directing process [pa(t)]. See e.g. Feller,