On fractional powers of generators of fractional resolvent families

We show that if −A generates a bounded α-times resolvent family for some α∈(0,2]α∈(0,2], then −Aβ−Aβ generates an analytic γ-times resolvent family for View the MathML sourceβ∈(0,2π−πγ2π−πα) and γ∈(0,2)γ∈(0,2). And a generalized subordination principle is derived. In particular, if −A generates a bounded α-times resolvent family for some α∈(1,2]α∈(1,2], then −A1/α−A1/α generates an analytic C0C0-semigroup. Such relations are applied to study the solutions of Cauchy problems of fractional order and first order.