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2013 | 1 | 28-36
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A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

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Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
1
Strony
28-36
Opis fizyczny
Daty
otrzymano
2013-05-28
zaakceptowano
2013-08-06
online
2013-09-16
Twórcy
autor
autor
Bibliografia
  • [1] A. Böttcher and B. Silbermann, Analysis of Toeplitz operators. Second edition, Springer-Verlag, Berlin, 2006.
  • [2] J. B. Garnett, Bounded analytic functions. Revised first edition. Graduate Texts in Mathematics, 236. Springer, NewYork, 2007.
  • [3] L. A. Coburn, Weyl’s theorem for nonnormal operators. Michigan Math. J. 13 1966 285–288.
  • [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.
  • [5] I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations, Birkhäuser Verlag, Basel, 1992.
  • [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.
  • [7] S. G. Mihlin, Singular integral equations. (Russian) Uspehi Matem. Nauk (N.S.) 3, (1948). no. 3(25), 29–112.
  • [8] N. I. Muskhelishvili, Singular integral equations. Second edition. Dover Publications, New York, 1992.
  • [9] M. Papadimitrakis and J. A. Virtanen, Hankel and Toeplitz transforms on H1: continuity, compactness and Fredholmproperties. Integral Equations Operator Theory 61 (2008), no. 4, 573–591.
  • [10] I. B. Simonenko, Riemann’s boundary value problem with a continuous coefficient. (Russian) Dokl. Akad. Nauk SSSR124 1959 278–281.
  • [11] I. B. Simonenko, Some general questions in the theory of the Riemann boundary problem. Math. USSR Izvestiya 2(1968), 1091–1099.
  • [12] E. Shargorodsky, J. F. Toland, A Riemann-Hilbert problem and the Bernoulli boundary condition in the variationaltheory of Stokes waves. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 1, 37–52.
  • [13] E. Shargorodsky, J. F. Toland, Bernoulli free-boundary problems. Mem. Amer. Math. Soc. 196 (2008), no. 914,viii+70 pp.
  • [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.
  • [15] J. A. Virtanen, A remark on the Riemann-Hilbert problem with a vanishing coefficient. Math. Nachr. 266 (2004),85–91.
  • [16] J. A. Virtanen, Fredholm theory of Toeplitz operators on the Hardy space H1. Bull. London Math. Soc. 38 (2006),no. 1, 143–155.
  • [17] D. Vukotic, A note on the range of Toeplitz operators. (English summary) Integral Equations Operator Theory 50(2004), no. 4, 565–567.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_conop-2012-0004
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