On fractional powers of generators of fractional resolvent families

Abstract

We show that if $-A$ generates a bounded $\alpha$-times resolvent family for some $\alpha \in (0,2]$, then $-A^{\beta }$ generates an analytic $\gamma$-times resolvent family for $\beta \in (0,\frac{2\pi -\pi \gamma }{2\pi -\pi \alpha })$ and $\gamma \in (0,2)$. And a generalized subordination principle is derived. In particular, if $-A$ generates a bounded $\alpha$-times resolvent family for some $\alpha \in (1,2]$, then $-A^{1/\alpha }$ generates an analytic $C_0$-semigroup. Such relations are applied to study the solutions of Cauchy problems of fractional order and first order.

arXiv comment20 pages