Fractional powers of generators of equicontinuous semigroups and fractional derivatives
Fractional powers of generators of equicontinuous semigroups and fractional derivatives
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DOI：
10.1017/s1446788700030950
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发表时间：
1989-06
期刊：
影响因子：
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通讯作者：
O. Lanford;D. W. Robinson
中科院分区：
文献类型：
--
作者：
O. Lanford;D. W. Robinson
本文分析了一致有界局部等度连续半群S的生成元H的分数幂Hα，α > 0。Hα定义为S上Dirac测度δ的α阶导数δα。我们证明了Hα是具有分数幂性质的闭算子，例如，对α，β > 0，HαHβ = Hα+β，对1 > α > 0，β > 0，（Hα）β = Hαβ.我们证明了Hα可以用Balakrishnan-Lions-Peetre算法计算，其中m是大于α的整数，Cα，m是适当的常数，极限存在于适当的拓扑中当且仅当x ∈ D（Hα）.最后证明了H∈是S的分数阶导子，在此意义下极限存在当且仅当x ∈ D（Hα）.
Abstract 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 algorithm where 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 sense where the limit again exists if, and only if, x ∈ D(Hα).