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On fractional powers of generators of fractional resolvent families. (English) Zbl 1203.47021
The authors 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 $\gamma \in (0,2)$. They also derive a generalized subordination principle. 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. These results allow to the authors to give a unitary perspective to the variegate phenomena related to fractional operators and to establish a connection between solutions of fractional Cauchy problems and Cauchy problems of first order.

47D06One-parameter semigroups and linear evolution equations
35K90Abstract parabolic equations
47D60$C$-semigroups, regularized semigroups
Full Text: DOI arXiv
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