  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.
