Spectral integrals of operator-valued functions. II. From the study of stationary processes

This paper is a continuation of the study made in [38]. Using Douglas' operator range theorem and Crimmins' corollary we obtain several new results on the "square-integrability of operator-valued functions with respect to a nonnegative hermitian measure". Using these facts we are able to extend in an important way theorems on the "spectral integral of an operator-valued function" which were obtained in [38], to wit, we are able to drop assumptions that functions are closed operator-valued. We apply these results to Wiener-Masani type infinite-dimensional stationary processes, representing a purely non-deterministic process as a "moving average" and obtaining a "factorization" of its spectral density. Next, anticipating global applications of our tools, we investigate the adjoint and generalized inverse of spectral integrals. Our definition of measurability for closed-operator-valued functions plays a key role here. Finally, we partially prove a conjecture (J. Multivariate Anal. (1974), 166-209) on simpler necessary and sufficient conditions on "when is a closed densely defined operator T from q to p a spectral integral T = f [Phi] dE?": Let q be finite and E be of countable multiplicity for . Then (i) Tx [set membership, variant] xp each x [set membership, variant] T (T is E-subordinate), and (ii) E(B)T [subset, double equals] TE(B) each B [set membership, variant] (T is E-commutative) implies LxpT [subset, double equals] TLxq each x [set membership, variant] q (T commutes with all the cyclic projections), and thus T = f [Phi] dE.