### 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 Γ. 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. 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}.

Original language | English (US) |
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Pages (from-to) | 669-696 |

Number of pages | 28 |

Journal | Revista Matematica Iberoamericana |

Volume | 12 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1996 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Spectral factorization of measurable rectangular matrix functions and the vector-valued Riemann problem.** / Rakowski, Marek; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Revista Matematica Iberoamericana*, vol. 12, no. 3, pp. 669-696. https://doi.org/10.4171/RMI/211

}

TY - JOUR

T1 - Spectral factorization of measurable rectangular matrix functions and the vector-valued Riemann problem

AU - Rakowski, Marek

AU - Spitkovsky, Ilya

PY - 1996/1/1

Y1 - 1996/1/1

N2 - We define spectral factorization in Lp (or a generalized Wiener-Hopf factorization) of a measurable singular matrix function on a simple closed rectifiable contour Γ. 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. 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 Lp.

AB - We define spectral factorization in Lp (or a generalized Wiener-Hopf factorization) of a measurable singular matrix function on a simple closed rectifiable contour Γ. 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. 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 Lp.

UR - http://www.scopus.com/inward/record.url?scp=0030354946&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030354946&partnerID=8YFLogxK

U2 - 10.4171/RMI/211

DO - 10.4171/RMI/211

M3 - Article

AN - SCOPUS:0030354946

VL - 12

SP - 669

EP - 696

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

SN - 0213-2230

IS - 3

ER -