Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 1353395
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $.

References [Enhancements On Off] (What's this?)

  • 1. Joseph A. Ball and J. William Helton, Beurling-Lax representations using classical Lie groups with many applications. II. 𝐺𝐿(𝑛,𝐶) and Wiener-Hopf factorization, Integral Equations Operator Theory 7 (1984), no. 3, 291–309. MR 756761,
  • 2. Kevin F. Clancey and Israel Gohberg, Factorization of matrix functions and singular integral operators, Operator Theory: Advances and Applications, vol. 3, Birkhäuser Verlag, Basel-Boston, Mass., 1981. MR 657762
  • 3. K. Clancey and M. Rakowski, Factorization of Rectangular Matrix Functions Relative to a Contour, manuscript, 1990.
  • 4. Guy David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 1, 157–189 (French). MR 744071
  • 5. Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • 6. Georgii S. Litvinchuk and Ilia M. Spitkovskii, Factorization of measurable matrix functions, Operator Theory: Advances and Applications, vol. 25, Birkhäuser Verlag, Basel, 1987. Translated from the Russian by Bernd Luderer; With a foreword by Bernd Silbermann. MR 1015716
  • 7. Marek Rakowski, Spectral factorization of rectangular rational matrix functions with application to discrete Wiener-Hopf equations, J. Funct. Anal. 110 (1992), no. 2, 410–433. MR 1194992,
  • 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.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 45E05, 45F15, 47A68

Retrieve articles in all journals with MSC (1991): 45E05, 45F15, 47A68

Additional Information

M. Rakowski
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210

I. Spitkovsky
Affiliation: Department of Mathematics, The College of William and Mary, Williamsburg, Virginia 23187-8795

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