Taiwanese Journal of Mathematics

Algebraic Properties of Cauchy Singular Integral Operators on the Unit Circle

Caixing Gu

Full-text: Open access


In this paper we study algebraic properties of singular integral operators with Cauchy kernel on the $L^{2}$ space of the unit circle. We give an operator equation characterization for this class of Cauchy singular integral operators. This characterization provides a direct connection between the singular integral operators and multiplication operators. We then use this characterization to study when two Cauchy singular integral operators commute. Our approach also leads to generalizations of several results on normal Cauchy singular integral operators obtained recently by Nakazi and Yamamoto.

Article information

Taiwanese J. Math., Volume 20, Number 1 (2016), 161-189.

First available in Project Euclid: 1 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) [See also 47B35] 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 47L05: Linear spaces of operators [See also 46A32 and 46B28] 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.)

singular integral operator Cauchy kernel Toeplitz operator Hankel operator normal operator


Gu, Caixing. Algebraic Properties of Cauchy Singular Integral Operators on the Unit Circle. Taiwanese J. Math. 20 (2016), no. 1, 161--189. doi:10.11650/tjm.20.2016.6188. https://projecteuclid.org/euclid.twjm/1498874427

Export citation


  • A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Second edition, Springer-Verlag, Berlin, 2006.
  • A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963), 89–102.
  • J. B. Garnett, Bounded Analytic Functions, Graduate Texts in Mathematics 236, Springer, New Yourk, 2007.
  • I. Gohberg and N. Krupnik, One-dimensional Linear Singular Integral Equations, Vol. I, Introduction; Operator Theory: Advances and Applications, 53, Birkhäuser Verlag, Basel, 1992.
  • ––––, One-dimensional Linear Singular Integral Equations, Vol. II, General theory and applications; Operator Theory: Advances and Applications 54, Birkhäuser Verlag, Basel, 1993.
  • C. Gu, Products of several Toeplitz Operators, J. Funct. Anal. 171 (2000), no. 2, 483–527.
  • ––––, When is the product of Hankel operators also a Hankel operator, J. Operator Theory 49 (2003), 347–362.
  • N. Krupnik, The conditions of selfadjointness of the operator of singular integration, Integral Equations Operator Theory 14 (1991), no. 5, 760–763.
  • ––––, Survey on the best constants in the theory of one-dimensional singular integral operators, Oper. Theory Adv. Appl. 202, 2010, 365–393.
  • B. N. Mandal and A. Chakrabarti, Applied Singular Integral Equations, CRC press, Boca Raton, FL, 2011.
  • T. Nakazi and T. Yamamoto, Norms and essential norms of the singular integral operator with Cauchy kernel on weighted Lebesque spaces, Integral Equations Operator Theory 68 (2010), no. 1, 101–113.
  • ––––, Normal singular integral operators with Cauchy kernel on $L^{2}$, Integral Equations Operator Theory 78 (2014), no. 2, 233–248.