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On normal solvability of the Riemann problem with singular coefficient


Authors: M. Rakowski and I. Spitkovsky
Journal: Proc. Amer. Math. Soc. 125 (1997), 815-826
MSC (1991): Primary 45E05, 45F15, 47A68
DOI: https://doi.org/10.1090/S0002-9939-97-03631-9
MathSciNet review: 1353395
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Abstract: Suppose $G$ is a singular matrix function on a simple, closed, rectifiable contour $\Gamma $. We present a necessary and sufficient condition for normal solvability of the Riemann problem with coefficient $G$ in the case where $G$ admits a spectral (or generalized Wiener-Hopf) factorization $G_{+} \Lambda G_{-}$ with $G_{-}^{\pm 1}$ essentially bounded. The boundedness of $G_{-}^{\pm 1}$ is not required when $G$ takes injective values a.e. on $\Gamma $.


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  • 1. J. A. Ball and J. W. Helton, Beurling-Lax Representations Using Classical Lie Groups with Many Applications II: $GL(n,{\mathbf C} )$ and Wiener-Hopf Factorization, Int. Equ. and Op. Th. 7 (1984), 291-309. MR 86e:47020
  • 2. K. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, OT 3, Birkhäuser Verlag, Basel/Boston/Stuttgart, 1981. MR 84a:47016
  • 3. K. Clancey and M. Rakowski, Factorization of Rectangular Matrix Functions Relative to a Contour, manuscript, 1990.
  • 4. G. David, Opérateurs intégraux singuliers sur certains courbes du plan complexe, Ann. Scient. Éc. Norm. Sup. 17 (1984), 157-189. MR 85k:42026
  • 5. P. Duren, Theory of $H^{p}$ spaces, Academic Press, New York, 1970. MR 42:3552
  • 6. G. S. Litvinchuk and I. M. Spitkovsky, Factorization of Measurable Matrix Functions, OT 25, Birkhäuser Verlag, Basel/Boston, 1987. MR 90g:47030
  • 7. M. Rakowski, Spectral Factorization of Rectangular Rational Matrix Functions with Application to Discrete Wiener-Hopf Equations, J. Functional Analysis 10 (1992), 410-433. MR 93j:47031
  • 8. M. Rakowski and I. Spitkovsky, Spectral Factorization of Measurable Rectangular Matrix Functions and the Vector Valued Riemann Problem, Revista Matemática Iberoamericana, to appear.
  • 9. I. Spitkovsky, Factorization of Measurable Matrix-Value Functions and its Relation to the Theory of Singular Integral Equations and the Vector Riemann Boundary-Value Problem, I, English translation:, Differential Equations 17 (1981), 477-485.

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Additional Information

M. Rakowski
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email: rakowski@math.ohio-state.edu

I. Spitkovsky
Affiliation: Department of Mathematics, The College of William and Mary, Williamsburg, Virginia 23187-8795
Email: ilya@cs.wm.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03631-9
Received by editor(s): September 8, 1995
Additional Notes: This research was partially supported by the NSF Grants DMS-9302706 and DMS-9401848.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society

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