J Austral Math Soc Ser A 46 pp473--504, 1989.

Fractional Powers of Generators of Equicontinuous Semigroups and Fractional Derivatives

Oscar E. Lanford III and Derek W. Robinson

(Received 21 September 1987)

Abstract

We analyze fractional powers H^a, a > 0, of the generators H of uniformly bounded locally equicontinuous semigroup S. The H^a are defined as the ath derivative of the Dirac measure d evaluated on S. We demonstrate that the H^a are closed operators with the natural properties of fractional powers, for example, H^a H^b = H^a+b for a, b > 0, and (H^a)^b = H^ab for 1 > a > 0 and b > 0. We establish that Ha can be evaluated by the Balakrishnan-Lions-Peetre algorithm

H^a x = lim_{eŽ 0}c_a,m^-1 ó
ő Ľ

e
dt
t
(I - S_t)^m
t^a
x

where m is an integer larger than a, c_a,m is a suitable constant, and the limit exists in the appropriate topology if, and only if, x Î D(H^a). Finally we prove that H^a is the fractional derivative of S in the sense

H^a x = lim_{tŽ 0+}((I - S_t)/t)^a x

where the limit again exists if, and only if, x Î D(H^a).

1980 AMS Subject Classification (1985 Revision): 47A99

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Authors

Oscar E. Lanford III: IHES, 91440 Bures-sur-Yvette, France.
Derek W. Robinson: Mathematics Department, Institute of Advanced Studies, Australian National University, Canberra, Australia.

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Last Modified: Wed Feb 19 10:27:49 2003