Fractional powers of closed operators and the semigroups generated by them

Abstract

Fractional powers of closed linear operators were first constructed by Bochner [2] and subsequently Feller [3], for the Laplacian operator. These constructions depend in an essential way on the fact that the Laplacian generates a semigroup. Phillips [6] in fact showed that these constructions (for positive indices less than one) were part of a more general one based on the Kolmogoroff-Levy representation theorem for infinitely divisible distributions. Finally, the present author constructed an operational calculus [1] for infinitesimal generators affording in particular a systematic study of the representation and properties of these operators. © 1960 by Pacific Journal of Mathematics.