Fractional powers of operators

Abstract

A definition of fractional (or complex) powers Aα>, α ∈ C, is given for closed linear operators A in a Banach space X with the resolvent set containing the negative real ray (−∞, 0) and such that {λ(λ + A)−1; 0 < λ 0 and Re α < 0, attention is paid to the domains D{Aα), which are related to the spaces Dσ and R┌ of x∈X defined by the regularity of (λ + A)−1x at ∞ and 0. When −A generates a bounded continuous semi-group or a bounded analytic semi-group, more detailed results are obtained. © 1966 by Pacific Journal of Mathematics.