Criteria for Analyticity of Multidimensionally Subordinate Semigroups

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.