The Factorization of Operator Valued Functions

Abstract

In the theory of multivariate prediction one of the basic problems is to factor a non-negative matrix valued function into certain canonical components. This problem was solved by Wiener and Masani [14] using ideas from the theory of stationary stochastic processes. The problem was solved in a different way by Helson and Lowdenslager [5] using their ideas on the solution of a minimum problem of Szeg6 [12]. The interesting methods used by the second named authors will work for the situation where the domain of the function is a compact abelian group with an ordered dual. The main object of this paper is to extend these results to the situation where the range of functions involved are bounded linear operators acting on a separable Hilbert space. The proofs given in [5] and [14] depend in an essential way on the facts that finite matrices have traces and determinants. Since this is in general not true for operators on a Hilbert space, we have had to use somewhat different ideas to attack the problem. A kernel of the ideas we have used is already to be found, in part, in a paper of M. Krein [8] and to some extent in a paper of A. Beurling [3]. The fact that a bounded operator on a Hilbert space has neither a trace nor a determinant, happens not to be an essential difficulty. In the infinite dimensional case we have found that a more essential difficulty stems from the fact that if A and B are bounded operators and AB = I (identity operator), then it is not necessarily true that A has an inverse. It is precisely because of this fact that we have been unable to overcome all of the difficulties involved when the domain of the operator valued function is a compact abelian group with an ordered dual (see [5, p. 198]), but have had to restrict our attention to the circle group. However, most of the things we say are valid for the more general situation where the operator valued function is defined on a compact abelian group, and in particular our method provides another proof for the factorization theorem for the case of matrix valued functions in the general situation. Let us now state two theorems of Szeg6 [12], [13] which provide the