Show Summary Details
More options …

# Concrete Operators

Ed. by Ross, William / Mashreghi, Javad

Mathematical Citation Quotient (MCQ) 2017: 0.34

Open Access
Online
ISSN
2299-3282
See all formats and pricing
More options …
Just Accepted

# A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

A. Perälä
/ J. A. Virtanen
/ L. Wolf
Published Online: 2013-09-16 | DOI: https://doi.org/10.2478/conop-2012-0004

## Abstract

We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

MSC: 35Q15; 45E05; 30E25; 47B35

• [1] A. Böttcher and B. Silbermann, Analysis of Toeplitz operators. Second edition, Springer-Verlag, Berlin, 2006. Google Scholar

• [2] J. B. Garnett, Bounded analytic functions. Revised first edition. Graduate Texts in Mathematics, 236. Springer, New York, 2007. Google Scholar

• [3] L. A. Coburn, Weyl’s theorem for nonnormal operators. Michigan Math. J. 13 1966 285–288. Google Scholar

• [4] I. C. Gohberg, On the number of solutions of a homogeneous singular integral equation with continuous coefficients. (Russian) Dokl. Akad. Nauk SSSR 122 1958 327–330. Google Scholar

• [5] I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations, Birkhäuser Verlag, Basel, 1992. Google Scholar

• [6] P. Koosis, Introduction to Hp spaces. Second edition. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. Cambridge Tracts in Mathematics, 115. Cambridge University Press, Cambridge, 1998. Google Scholar

• [7] S. G. Mihlin, Singular integral equations. (Russian) Uspehi Matem. Nauk (N.S.) 3, (1948). no. 3(25), 29–112. Google Scholar

• [8] N. I. Muskhelishvili, Singular integral equations. Second edition. Dover Publications, New York, 1992. Google Scholar

• [9] M. Papadimitrakis and J. A. Virtanen, Hankel and Toeplitz transforms on H1: continuity, compactness and Fredholm properties. Integral Equations Operator Theory 61 (2008), no. 4, 573–591. Google Scholar

• [10] I. B. Simonenko, Riemann’s boundary value problem with a continuous coefficient. (Russian) Dokl. Akad. Nauk SSSR 124 1959 278–281. Google Scholar

• [11] I. B. Simonenko, Some general questions in the theory of the Riemann boundary problem. Math. USSR Izvestiya 2 (1968), 1091–1099. Google Scholar

• [12] E. Shargorodsky, J. F. Toland, A Riemann-Hilbert problem and the Bernoulli boundary condition in the variational theory of Stokes waves. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 1, 37–52. Google Scholar

• [13] E. Shargorodsky, J. F. Toland, Bernoulli free-boundary problems. Mem. Amer. Math. Soc. 196 (2008), no. 914, viii+70 pp. Google Scholar

• [14] E. Shargorodsky, J. A. Virtanen, Uniqueness results for the Riemann-Hilbert problem with a vanishing coefficient. Integral Equations Operator Theory 56 (2006), no. 1, 115-127. Google Scholar

• [15] J. A. Virtanen, A remark on the Riemann-Hilbert problem with a vanishing coefficient. Math. Nachr. 266 (2004), 85–91. Google Scholar

• [16] J. A. Virtanen, Fredholm theory of Toeplitz operators on the Hardy space H1. Bull. London Math. Soc. 38 (2006), no. 1, 143–155. Google Scholar

• [17] D. Vukotic, A note on the range of Toeplitz operators. (English summary) Integral Equations Operator Theory 50 (2004), no. 4, 565–567.Google Scholar

Accepted: 2013-08-06

Published Online: 2013-09-16

Citation Information: Concrete Operators, Volume 1, Pages 28–36, ISSN (Online) 2299-3282,

Export Citation

©2013 Versita Sp. z o.o.. This content is open access.