Criteria for analyticity of subordinate semigroups

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.