The Riemann-Hilbert problem for singular positive loops (Q1890787)

The problem here is to factor a positive matrix-valued function \(g\) on the unit circle into the product of two such functions, one analytic inside the circle and the other outside the circle. The easier case when \(g\in L^\infty\) is positive and invertible and \(g^{-1}\in L^\infty\) is settled first. The main result, based on the calculus of operators, is that a positive continuous (noninvertible) \(g\) admits such a factorization provided \(\log g\) is integrable on the circle. The problem arises in the study of the scattering matrix, in prediction theory and in amalgamation procedures in several complex variables, cf. \textit{R. C. Gunning} and \textit{H. Rossi}, ``Analytic functions of several complex variables'' (1965; Zbl 0141.08601).