  Let $\psi$ be a Bernstein function. A.~Carasso and T.~Kato obtained necessary
and sufficient conditions for $\psi$ to have a property that $\psi(A)$
generates a quasibounded holomorphic semigroup for every generator $A$ of a
bounded $C_0$-semigroup in a Banach space, in terms of some convolution
semigroup of measures associated with $\psi$. We give an alternative to
Carasso-Kato's criterium, and derive several sufficient conditions for $\psi$
to have the above-mentioned property.
