# Revista Matemática Iberoamericana

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**Volume 12, Issue 3, 1996, pp. 669–696**

**DOI: 10.4171/RMI/211**

Published online: 1996-12-31

Spectral factorization of measurable rectangular matrix functions and the vector-valued Riemann problem

Marek Rakowski^{[1]}and Ilya Spitkovsky

^{[2]}(1) Ohio State University, Columbus, USA

(2) The College of William and Mary, Williamsburg, USA

We define spectral factorization in $L_p$ or a generalized Wiener–Hopf factorization of a measurable singular matrix function on a simple closed rectifiable contour $\Gamma$. 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 $\Gamma$. 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 $L_p$.

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Rakowski Marek, Spitkovsky Ilya: Spectral factorization of measurable rectangular matrix functions and the vector-valued Riemann problem. *Rev. Mat. Iberoam.* 12 (1996), 669-696. doi: 10.4171/RMI/211