A note on the factorization of operator-valued functions

Abstract

1. Wiener (cf. (3, p. 122]) gave the factorization theorem of nonsingular positive matrix valued functions into analytic components. The theorem has the complement for singular positive matrix valued functions proved by Helson and Lowdenslager (cf. [3, p. 122]). Devinatz [2] extended this theorem to the situation where the ranges of functions are bounded linear operators acting on a separable Hilbert space. Let SC be a separable Hilbert space and let L2(9C) denote the Hilbert space of functions defined on the unit circle with values in SC that are weakly measurable and have square summable norm with respect to normalized Lebesgue measure dO. The L2(9C)-inner product of two functions f = f(ei") and g = g(eie) is given by 0'(f(e' ), g(eiG))X dM. The subspace H2(9C) is the subset of L2(9C) consisting of the functions f = f(ei9) such that J2ff(f(e' ), x)xe "9 dO = 0 for every x in 9C when n >0. Let fi3 (9C) be the algebra of bounded operators on 9C and let L'(63 (9C)) denote the algebra of functions defined on the unit circle with values in @ (SC) that are weakly measurable and have essentially bounded norm IIW v1 = ess supj IW(e'0)IIf(x. The subalgebra H'(93C)) is the subset of L'(6 (9C)) consisting of the functions A = A(eiG) such that Ax is in H2(9C) for every x in 9C. When W is a function in L?( (JC)) whose values are nonnegative almost everywhere (a.e.), W is said to be factorable if W(eie) = A(eO)*A(ei9) a.e. 0 for some A in H (@(9C)). Let us now state the factorization theorem, associated with Wiener and Devinatz (cf. [3, pp. 119-1231).