Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let 9 be a Hilbert space. We say that x = (xi)i[epsilon]q, xi [epsilon] 9, is a bounded operator-forming vector ([epsilon]9Fq) if the Gram matrix <x, x> = [(xi, xj)]i[epsilon]q,j[epsilon]q is the matrix of a bounded (necessarily >= 0) operator on 6, the Hilbert space of square-summable complex-valued functions on q. Let A be p - q, i.e., let A be a linear operator from 6 to 6. Then exists a linear operator A from (the Banach space) 9Fq to 9Fp on 5(A) = {x:x [epsilon] 9Fq, A<x, x>1/2 is p - q bounded on 6} such that Y = Ax satisfies yj [epsilon] [sigma](x) = {space spanned by the xi}, <y, x> = A<x, x> and <y, y> = A<x, x>1/2(A<x, x>1/2)*. This is a generalization of our earlier [J. Multivariate Anal. 4 (1974), 166-209; 6 (1976), 538-571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.