Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes

P. Masani and the author have previously answered the question, "When is an operator on a Hilbert space the integral of a complex-valued function with respect to a given spectral (projection-valued) measure?" In this paper answers are given to the question, "When is a linear operator from q to p the integral of a spectral measure?"; here the values of the integrand are linear operators from the square-summable q-tuples of complex numbers to the square-summable p-tuples of complex numbers, and our spectral measure for q is the "inflation" of a spectral measure for . In the course of this paper, we make available tools for handling the spectral analysis of q-variate weakly stationary processes, 1 <= q <= [infinity], which should enable researchers to deal in the future with the case q = [infinity]. We show as one application of our theory that if U = [integral operator](in0, 2[pi]]e-i[theta]E(d[theta]) is a unitary operator on and if T is a bounded linear operator from q to q (1 <= q <= [infinity]) which is a prediction operator for each stationary process (Unx)-[infinity][infinity] [subset, double equals]q (for each x = (xi)ij [set membership, variant] q, Unx = (Unxi)i=1q), then T is a spectral integral, [integral operator](0,2[pi])][Phi]([theta]) E(d[theta]), and the Banach norm of T, TB = ess sup [Phi]([theta])B.