Vol. 19, No. 2, 1966

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Vol. 286: 1  2
Vol. 285: 1  2
Vol. 284: 1  2
Vol. 283: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Subscriptions
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Fractional powers of operators

Hikosaburo Komatsu

Vol. 19 (1966), No. 2, 285–346
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 < ∞} is bounded; fundamental properties such as additivity (AαAβ = Aα+β), coincidence with the iterations Aα = An for integers α = n, and analytic dependence on α are discussed. Since the fractional powers Aα are generally unbounded in both of the cases Re α > 0 and Re α < 0, attention is paid to the domains D(Aα), which are related to the spaces Dσ and Rτ 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
Authors
Hikosaburo Komatsu