An operator version of a theorem of Kolmogorov
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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 (ℋ,ℋ)of bounded linear operators from a space into itself. The factorization is then K(t, s) = X(t)*X(s) with X a continuous map from ℊ to B(ℋ, K) for a suitable Hilbert K. space If ℊ is separable we may take K=ℋ. © 1975 Pacific Journal of Mathematics.
author list (cited authors)
Allen, G., Narcowich, F., & Williams, J.