On fractional powers of generators of fractional resolvent families

Abstract

We show that if − A generates a bounded α-times resolvent family for some α ∈ ( 0 , 2 ] , then − A β generates an analytic γ-times resolvent family for β ∈ ( 0 , 2 π − π γ 2 π − π α ) and 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 ] , then − A 1 / α generates an analytic C 0 -semigroup. Such relations are applied to study the solutions of Cauchy problems of fractional order and first order.