ON SUBORDINATED HOLOMORPHIC SEMIGROUPS

If [e-tA] is a uniformly bounded C0 semigroup on a complex Banach space X , then -A-alpha, 0 < alpha < 1 generates a holomorphic semigroup on X, and [e-tA-alpha] is subordinated to [e-tA] 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 the measure's Laplace transform for that to be the case. We then construct probability measures that do not have this property.