### Abstract

The solvability of the Riemann-Hilbert boundary value problem on the real line is described in the case when its matrix coefficient admits a Wiener-Hopf-type factorization with bounded outer factors, but rather general diagonal elements of its middle factor. This covers, in particular, the almost periodic setting, when the factorization multiples belong to the algebra generated by the functions eλ(x) := eiλx, λ ε R. Connections with the corona problem are discussed. Based on those, constructive factorization criteria are derived for several types of triangular 2 × 2 matrices with diagonal entries e±λ and non-zero off diagonal entry of the form a?e?β + a+e? with ?, β ≥? +β >nd a± analytic and bounded in the upper/lower half plane.

Original language | English (US) |
---|---|

Pages (from-to) | 852-878 |

Number of pages | 27 |

Journal | Journal of the London Mathematical Society |

Volume | 86 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2012 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of the London Mathematical Society*,

*86*(3), 852-878. https://doi.org/10.1112/jlms/jds033

**Factorizations, Riemann-Hilbert problems and the corona theorem.** / Câmara, M. C.; Diogo, C.; Karlovich, Yu I.; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Journal of the London Mathematical Society*, vol. 86, no. 3, pp. 852-878. https://doi.org/10.1112/jlms/jds033

}

TY - JOUR

T1 - Factorizations, Riemann-Hilbert problems and the corona theorem

AU - Câmara, M. C.

AU - Diogo, C.

AU - Karlovich, Yu I.

AU - Spitkovsky, Ilya

PY - 2012/1/1

Y1 - 2012/1/1

N2 - The solvability of the Riemann-Hilbert boundary value problem on the real line is described in the case when its matrix coefficient admits a Wiener-Hopf-type factorization with bounded outer factors, but rather general diagonal elements of its middle factor. This covers, in particular, the almost periodic setting, when the factorization multiples belong to the algebra generated by the functions eλ(x) := eiλx, λ ε R. Connections with the corona problem are discussed. Based on those, constructive factorization criteria are derived for several types of triangular 2 × 2 matrices with diagonal entries e±λ and non-zero off diagonal entry of the form a?e?β + a+e? with ?, β ≥? +β >nd a± analytic and bounded in the upper/lower half plane.

AB - The solvability of the Riemann-Hilbert boundary value problem on the real line is described in the case when its matrix coefficient admits a Wiener-Hopf-type factorization with bounded outer factors, but rather general diagonal elements of its middle factor. This covers, in particular, the almost periodic setting, when the factorization multiples belong to the algebra generated by the functions eλ(x) := eiλx, λ ε R. Connections with the corona problem are discussed. Based on those, constructive factorization criteria are derived for several types of triangular 2 × 2 matrices with diagonal entries e±λ and non-zero off diagonal entry of the form a?e?β + a+e? with ?, β ≥? +β >nd a± analytic and bounded in the upper/lower half plane.

UR - http://www.scopus.com/inward/record.url?scp=84870033996&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870033996&partnerID=8YFLogxK

U2 - 10.1112/jlms/jds033

DO - 10.1112/jlms/jds033

M3 - Article

VL - 86

SP - 852

EP - 878

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 3

ER -