Spectral factorization of measurable rectangular matrix functions and the vector-valued Riemann problem
Marek Rakowski, Ilya M. Spitkovsky · Revista Matemática Iberoamericana · 1996
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 .