Kernels of a Class of Toeplitz Plus Hankel Operators with Piecewise Continuous Generating Functions

  • Victor D. Didenko
  • Bernd Silbermann


Toeplitz T(a) and Toeplitz plus Hankel operators T(a) + H(b) acting on sequence space lp, 1 < p < , are considered. If a ∈ PCp is a piecewise continuous lp-multiplier, a complete description of the kernel of the Fredholm operator T(a) is derived. Moreover, the kernels of Fredholm Toeplitz plus Hankel operators T(a) + H(b) the generating functions a and b of which belong to PCp and satisfy the condition a(t)a(1∕t) = b(t)b(1∕t), \(t\in {\mathbb {T}}\), are also determined.


  1. 1.
    Böttcher, A., Silbermann, B.: Analysis of Toeplitz operators. Springer Monographs in Mathematics. Springer, Berlin (2006)Google Scholar
  2. 2.
    Didenko, V.D., Silbermann, B.: Index calculation for Toeplitz plus Hankel operators with piecewise quasi-continuous generating functions. Bull. Lond. Math. Soc. 45(3), 633–650 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Didenko, V.D., Silbermann, B.: Some results on the invertibility of Toeplitz plus Hankel operators. Ann. Acad. Sci. Fenn. Math. 39(1), 443–461 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Didenko, V.D., Silbermann, B.: Structure of kernels and cokernels of Toeplitz plus Hankel operators. Integr. Equ. Oper. Theory 80(1), 1–31 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Didenko, V.D., Silbermann, B.: Generalized inverses and solution of equations with Toeplitz plus Hankel operators. Bol. Soc. Mat. Mex. 22(2), 645–667 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Didenko, V.D., Silbermann, B.: Invertibility and inverses of Toeplitz plus Hankel operators. J. Operator Theory 72(2), 293–307 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Duduchava, R.: On discrete Wiener-Hopf equations in l p spaces with weight. Soobsh. Akad. Nauk Gruz. SSR 67(1), 17–20 (1972)MathSciNetGoogle Scholar
  8. 8.
    Duduchava, R.: The discrete Wiener-Hopf equations. Proc. A. Razmadze Math. Inst. 50, 42–59 (1975)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Edwards, R.: Fourier Series. A Modern Introduction. Vol. 2. Graduate Texts in Mathematics, vol. 85. Springer, New York (1982)Google Scholar
  10. 10.
    Gohberg, I.C., Feldman, I.A.: Convolution Equations and Projection Methods for Their Solution. American Mathematical Society, Providence (1974)Google Scholar
  11. 11.
    Hagen, R., Roch, S., Silbermann, B.: Spectral theory of approximation methods for convolution equations, Operator Theory: Advances and Applications, vol. 74. Birkhäuser Verlag, Basel (1995)CrossRefGoogle Scholar
  12. 12.
    Roch, S., Santos, P.A., Silbermann, B.: Non-commutative Gelfand Theories. A Tool-kit for Operator Theorists and Numerical Analysts. Universitext. Springer-Verlag London Ltd., London (2011)CrossRefGoogle Scholar
  13. 13.
    Roch, S., Silbermann, B.: Algebras of convolution operators and their image in the Calkin algebra. Report MATH, vol. 90. Akademie der Wissenschaften der DDR Karl-Weierstrass-Institut für Mathematik, Berlin (1990)Google Scholar
  14. 14.
    Roch, S., Silbermann, B.: A handy formula for the Fredholm index of Toeplitz plus Hankel operators. Indag. Math. 23(4), 663–689 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Silbermann, B.: The C -algebra generated by Toeplitz and Hankel operators with piecewise quasicontinuous symbols. Integr. Equ. Oper. Theory 10(5), 730–738 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSouthern University of Science and TechnologyShenzhenChina
  2. 2.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany

Personalised recommendations