Vol. 61, No. 2, 1975

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Vol. 320: 1  2
Vol. 319: 1  2
Vol. 318: 1  2
Vol. 317: 1  2
Vol. 316: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
An operator version of a theorem of Kolmogorov

Graham Donald Allen, Francis Joseph Narcowich and James Patrick Williams

Vol. 61 (1975), No. 2, 305–312

Let 𝒢 be a (separable) Hausdorff space and let K be a continuous nonnegative-definite kernel (covariance) from 𝒢×𝒢 to C. The well known theorem of Kolmogorov states that in the case 𝒢 is the set of integers there is a continuous mapping (stochastic process) x() from 𝒢 into a (separable) Hilbert space 𝒦 such that K(s,t) = (x(s),x(t)). The theorem is also known for any separable Hausdorff space. The purpose of this paper is to replace the complex numbers C by the algebra B(,) of bounded linear operators from a Hilbert space into itself. The factorization is then K(t,s) = X(t)X(s) with X a continuous map from 𝒢 to B(,𝒦) for a suitable Hilbert space 𝒦. If 𝒢 is separable we may take 𝒦 = .

Mathematical Subject Classification 2000
Primary: 47B99
Secondary: 60G05
Received: 25 March 1975
Revised: 4 November 1975
Published: 1 December 1975
Graham Donald Allen
Francis Joseph Narcowich
James Patrick Williams