Fractional powers of operators

Hikosaburo Komatsu

Vol. 19 (1966), No. 2, 285–346

DOI: 10.2140/pjm.1966.19.285

Abstract

A definition of fractional (or complex) powers A^α, 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 < ∞} is bounded; fundamental properties such as additivity (A^αA^β = A^α+β), coincidence with the iterations A^α = Aⁿ for integers α = n, and analytic dependence on α are discussed. Since the fractional powers A^α are generally unbounded in both of the cases Re α > 0 and Re α < 0, attention is paid to the domains D(A^α), which are related to the spaces D^σ and R^τ 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.

Mathematical Subject Classification

Primary: 47.50

Milestones

Received: 17 March 1965

Revised: 21 July 1965

Published: 1 November 1966