  Let $\psi$ be a Bernstein function in one variable. 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 and deduced necessary
conditions as well. We generalize their results to the multidimensional case
and also give sufficient conditions for the property mentioned above.
