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

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 $L_p$.