On subordination of holomorphic semigroups

We prove that for any Bernstein function $\psi$ the operator $-\psi(A)$ generates a holomorphic $C_0$-semigroup $(e^{-t\psi(A)})_{t \ge 0}$ 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 $(e^{-t\psi(A)})_{t \ge 0}$ is holomorphic in the holomorphy sector of $(e^{-tA})_{t \ge 0},$ and if $(e^{-tA})_{t \ge 0}$ is sectorially bounded in this sector then $(e^{-t\psi(A)})_{t \ge 0}$ has the same property. We also obtain new sufficient conditions on $\psi$ in order that, for every Banach space $X$, the semigroup $(e^{-t\psi(A)})_{t\ge 0}$ on $X$ is holomorphic whenever $(e^{-tA})_{t\ge 0}$ is a bounded $C_0$-semigroup on $X$. These conditions improve and generalize well-known results by Carasso-Kato and Fujita.