Generation theorems for fractional resolvent families

Abstract

In this paper, we give some characterizations of generators of fractional resolvent families in terms of the resolvents of the generators and their derivatives. We first give a condition under which a densely defined operator on a Banach space can generate a bounded fractional resolvent family, which generalizes the result of Gomilko for $C_0$-semigroups. And this condition is also necessary in a Hilbert space. Another type of generation theorem for exponentially bounded fractional resolvent families is also presented.