Generation of subordinated holomorphic semigroups via Yosida's theorem

Abstract

Using functional calculi theory, we obtain several estimates for $\Vert \psi (A)g(A)\Vert$, where $\psi$ is a Bernstein function, $g$ is a bounded completely monotone function and $-A$ is the generator of a holomorphic $C_0$-semigroup on a Banach space, bounded on $[0,\infty )$. Such estimates are of value, in particular, in approximation theory of operator semigroups. As a corollary, we obtain a new proof of the fact that $-\psi (A)$ generates a holomorphic semigroup whenever $-A$ does, established recently in [8] by a different approach.