Fractional powers of generators of equicontinuous semigroups and fractional derivatives

Abstract

AbstractWe analyze fractional powersHα, α > 0, of the generatorsHof uniformly bounded locally equicontinuous semigroupsS. TheHαare defined as the αth derivative δαof the Dirac measure δ evaluated onS. We demonstrate that theHαare closed operators with the natural properties of fractional powers, for example,HαHβ=Hα+βfor α, β > 0, and (Hα)β=Hαβfor 1 > α > 0 and β > 0. We establish thatHα can be evaluated by the Balakrishnan-Lions-Peetre algorithmwheremis an integer larger than α,Cα, mis a suitable constant, and the limit exists in the appropriate topology if, and only if,x ∈ D(Hα). Finally we prove thatH∈is the fractional derivation ofSin the sensewhere the limit again exists if, and only if,x∈ D(Hα).