We define spectral factorization in or a generalized Wiener–Hopf factorization of a measurable singular matrix function on a simple closed rectifiable contour . Such factorization has the same uniqueness properties as in the nonsingular case. We discuss basic properties of the vector valued Riemann problem whose coefficient takes singular values almost everywhere on . In particular, we introduce defect numbers for this problem which agree with the usual defect numbers in the case of a nonsingular coefficient. Based on the Riemann problem, we obtain a necessary and sufficient condition for existence of a spectral factorization in .
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Marek Rakowski, Ilya Spitkovsky, Spectral factorization of measurable rectangular matrix functions and the vector-valued Riemann problem. Rev. Mat. Iberoam. 12 (1996), no. 3, pp. 669–696DOI 10.4171/RMI/211