On fractional powers of generators of fractional resolvent families

Details On fractional powers of generators of fractional resolvent families On fractional powers of generators of fractional resolvent families

2010-05-11

arxiv.org/abs/1005.1725v2

Mathematics

Analysis of PDEs

20 pages

Scientific paper

10.1016/j.jfa.2010.07.07

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.

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