Resolvent representations for functions of sectorial operators

We obtain integral representations for the resolvent of (A), where 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 , we show that the operator 􀀀 (A) generates a sectorially bounded holomorphic C0-semigroup on a Banach space whenever 􀀀A does, and the sectorial angle of A is preserved. When 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 H1-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.