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

### Marek Rakowski

Ohio State University, Columbus, USA### Ilya Spitkovsky

The College of William and Mary, Williamsburg, USA

## Abstract

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 $L_p$.

## Cite this article

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. 669–696

DOI 10.4171/RMI/211