On subordination of holomorphic semigroups

We prove that for any Bernstein function ψ   the operator −ψ(A)−ψ(A) generates a bounded holomorphic C0C0-semigroup (e−tψ(A))t≥0(e−tψ(A))t≥0 on a Banach space, whenever −A   does. This answers a question posed by Kishimoto and Robinson. Moreover, giving a positive answer to a question by Berg, Boyadzhiev and de Laubenfels, we show that (e−tψ(A))t≥0(e−tψ(A))t≥0 is holomorphic in the holomorphy sector of (e−tA)t≥0(e−tA)t≥0, and if (e−tA)t≥0(e−tA)t≥0 is sectorially bounded in this sector then (e−tψ(A))t≥0(e−tψ(A))t≥0 has the same property. We also obtain new sufficient conditions on ψ in order that, for every Banach space X  , the semigroup (e−tψ(A))t≥0(e−tψ(A))t≥0 on X   is holomorphic whenever (e−tA)t≥0(e−tA)t≥0 is a bounded C0C0-semigroup on X. These conditions improve and generalize well-known results by Carasso–Kato and Fujita.