Fractional powers of generators of equicontinuous semigroups and fractional derivatives

Journal of the Australian Mathematical Society

Oscar E. Lanford¹ 

[1] Derek W. Robinson

关键词: 47 A 99;   

DOI  :  10.1017/S1446788700030950

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

We analyze fractional powers Hα, α > 0, of the generators H of uniformly bounded locally equicontinuous semigroups S. The Hα are defined as the αth derivative δα of the Dirac measure δ evaluated on S. We demonstrate that the Hα are closed operators with the natural properties of fractional powers, for example, HαHβ = Hα+β for α, β > 0, and (Hα)β = Hαβ for 1 > α > 0 and β > 0. We establish that Hα can be evaluated by the Balakrishnan-Lions-Peetre algorithmwhere m is an integer larger than α, Cα, m is a suitable constant, and the limit exists in the appropriate topology if, and only if, x ∈ D(Hα). Finally we prove that H∈ is the fractional derivation of S in the sensewhere the limit again exists if, and only if, x ∈ D(Hα).

【 授权许可】

Unknown