On Some Functions That Send Each Generator of a C0-Semigroup to the Generator of a Holomorphic Semigroup

We give some conditions on functions of the Schoenberg class T for them to send the generators of uniformly bounded semigroup of class C ₀ to the generators of holomorphic semigroups. This generalizes Yosida, Balakrishnan, and Kato's result relating to fractional powers of operators. The functional calculus of generators of C ₀-semigroups which uses the class T was constructed in the preceding articles of the author.