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

Abstract

P. Masani and the author have previously answered the question, “When is an operator on a Hilbert space H 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 H q to H 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 H q is the “inflation” of a spectral measure for H . In the course of this paper, we make available tools for handling the spectral analysis of q-variate weakly stationary processes, 1 ≤ q ≤ ∞, which should enable researchers to deal in the future with the case q = ∞. We show as one application of our theory that if U = ∫( in0, 2 π] e − iθ E( dθ) is a unitary operator on H and if T is a bounded linear operator from H q to H q (1 ≤ q ≤ ∞) which is a prediction operator for each stationary process ( U nx ) −∞ ∞ ⊆ H q (for each x = (x i) i j ∈ H q , U n x = ( U n x i ) i=1 q ), then T is a spectral integral, ∫( 0,2 π )] Φ( θ) E( dθ), and the Banach norm of T, | T| B = ess sup | Φ( θ)| B .