JournalsrmiVol. 12, No. 3pp. 669–696

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

Abstract

We define spectral factorization in LpL_p or a generalized Wiener–Hopf factorization of a measurable singular matrix function on a simple closed rectifiable contour Γ\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 Γ\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 LpL_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