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On some properties of a high order fractional differential operator which is not in general selfadjoint. (English) Zbl 1143.26004
The author extends the definition of the fractional differential operator connected with stable laws to higher order. It is proved that the introduced operator, given via Fourier transform on $\Bbb{L}^{2}(\Bbb{R}^{d}),$ is the infinitesimal generator of an analytic semigroup. It generalizes also the fractional Laplacian operator, Riesz-Feller operators, Riemann-Liouville operator and Jacob pseudodifferential operators. Moreover, it shares most of the well-known properties of these operators. Further, several other properties are discussed in the paper.

MSC:
26A33Fractional derivatives and integrals (real functions)
35S05General theory of pseudodifferential operators