Resolvent representations for functions of sectorial operators

Abstract

We obtain integral representations for the resolvent of $\psi (A)$, where $\psi$ is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and $A$ is a sectorial operator on a Banach space. As a corollary, for a wide class of functions $\psi$, we show that the operator $-\psi (A)$ generates a sectorially bounded holomorphic $C_0$-semigroup on a Banach space whenever $-A$ does, and the sectorial angle of $A$ is preserved. When $\psi$ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a permanence property for $A$ can be described, at least on Hilbert spaces, in terms of the existence of a bounded $H^{\infty }$-calculus for $A$. As byproducts of our approach, we also obtain new results on functions mapping generators of bounded semigroups into generators of holomorphic semigroups and on subordination for Ritt operators.

The paper has been accepted for publication in Advances in Mathematics. This is the authors' accepted version

References in corpus (1)

• Product formulas in functional calculi for sectorial operators

Cited by in corpus (1)

• On discrete subordination of power bounded and Ritt operators