Resolvent representations for functions of sectorial operators

TL;DR

This paper develops integral resolvent representations for functions of sectorial operators, showing how certain functions preserve sectorial properties and semigroup generation, with implications for operator calculus and semigroup theory.

Contribution

It introduces new integral representations for resolvents of functions of sectorial operators and establishes sectoriality preservation under a broad class of functions, including Bernstein functions.

Findings

Operator $- ext{psi}(A)$ generates a sectorially bounded holomorphic semigroup.

Preservation of sectorial angle for functions $ ext{psi}$.

New results on functions mapping generators of bounded semigroups.

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.