Criteria for Analyticity of Multidimensionally Subordinated Semigroups

Let ψ be a Bernstein function in one variable. A. Carasso and T. Kato obtained necessary and sufficient conditions for ψ to have a property that ψ(A) generates a quasibounded holomorphic semigroup for every generator A of a bounded C₀-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.