Vol. 61, No. 2, 1975

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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