On some functions that map each generator of a $C_0$-semigroup into 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.