On subordinated holomorphic semigroups

Abstract:

If $[{e^{ - tA}}]$ is a uniformly bounded ${C_0}$ semigroup on a complex Banach space $X$, then $- {A^\alpha },$, $0 < \alpha < 1$, generates a holomorphic semigroup on $X$, and $[{e^{ - t{A^\alpha }}}]$ is subordinated to $[{e^{ - tA}}]$ through the Lévy 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.