Skip to main content
SpringerLink
Log in

Hyponormal Toeplitz Operators with Matrix-Valued Circulant Symbols

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

In this paper we are concerned with the hyponormality of Toeplitz operators with matrix-valued circulant symbols. We establish a necessary and sufficient condition for Toeplitz operators with matrix-valued partially circulant symbols to be hyponormal and also provide a rank formula for the self-commutator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abrahamse M.B.: Sunormal Toeplitz operators and functions of bounded type. Duke Math. J. 43, 597–604 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brown, A., Halmos, P.R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213, 89–102 (1963/1964)

    Google Scholar 

  3. Cowen, C.: Hyponormal and subnromal Toeplitz operators, Survey of Some Recent Results in Operator Theory, I. In: Conway, J.B., Morrel, B.B. (eds.) Pitman Research Notes in Mathematics, vol. 171. Longman, London, pp. 155–167 (1988)

  4. Cowen C.: Hyponormality of Toeplitz operators. Proc. Am. Math. Soc. 103, 809–812 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Curto, R.E., Hwang, I.S., Lee, W.Y.: Which subnormal Toeplitz operators are either normal or analytic? (preprint, 2010)

  6. Curto, R.E., Lee, W.Y.: Joint hyponormality of Toeplitz pairs. In: Mem. Amer. Math. Soc., vol. 712. Am. Math. Soc., Providence (2001)

  7. Derkach V., Dym H.: On linear fractional transformations associated with generalized J-inner matrix functions. Integral Equ. Oper. Theory 65, 1–50 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Douglas R.G.: Banach algenra techniques in operator theory. Academic Press, New York (1972)

    Google Scholar 

  9. Douglas, R.G.: Banach algenra techniques in the theory of Toeplitz operators. CBMS 15. Amer. Math. Soc., Providence (1973)

  10. Farenick D.R., Krupnik K., Krupnik N., Lee W.Y.: Normal Toeplitz matrices. SIAM J. Matrix Anal. Appl. 17, 1037–1043 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Farenick D.R., Lee W.Y.: Hyponormality and spectra of Toeplitz operators. Trans. Am. Math. Soc. 348, 4153–4174 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Farenick D.R., Lee W.Y.: On hyponormal Toeplitz operators with polynomial and circulant-type symbols. Integral Equ. Oper. Theory 29, 202–210 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Foias, C., Frazo, A.: The commutant lifting approach to interpolation problems. In: Operator Theory: Adv. Appl., vol. 44. Birkhäuser, Boston (1993)

  14. Fuhrmann P.A.: On the corona theorem and its applications to spectral problems in Hilbert space. Trans. Am. Math. Soc. 132, 55–66 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gohberg I., Goldberg S., Kaashoek M.A.: Classes of linear operators, vol. II. Birkhauser, Basel (1993)

    Book  MATH  Google Scholar 

  16. Gu C.: A generalization of Cowen’s characterization of hyponormal Toeplitz operators. J. Funct. Anal. 124, 135–148 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gu C., Hendricks J., Rutherford D.: Hyponormality of block Toeplitz operators. Pac. J. Math. 223, 95–111 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gu C., Shapiro J.E.: Kernels of Hankel operators and hyponormality of Toeplitz operators. Math. Ann. 319, 553–572 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hwang I.S., Kim I.H.: Hyponormality of Toeplitz operators with genralized circulant symbols. J. Math. Anal. Appl. 349(1), 264–271 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hwang I.S., Kim I.H., Lee W.Y.: Hyponormality of Toeplitz operators with polynomial symbols. Math. Ann. 313(2), 247–261 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hwang I.S., Kim I.H., Lee W.Y.: Hyponormality of Toeplitz operators with polynomial symbols: an extremal case. Math. Nachr. 231, 25–38 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hwang I.S., Lee W.Y.: Hyponormality of trigonometric Toeplitz operators. Trans. Am. Math. Soc. 354, 2461–2474 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hwang I.S., Lee W.Y.: Hyponormality of Toeplitz operators with rational symbols. Math. Ann. 335, 405–414 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hwang I.S., Lee W.Y.: Hyponormal Toeplitz operators with rational symbols. J. Oper. Theory 56, 47–58 (2006)

    MathSciNet  MATH  Google Scholar 

  25. Hwang I.S., Lee W.Y.: Block Toeplitz operators with rational symbols. J. Phys. A Math. Theor. 41(18), 185207 (2008)

    Article  MathSciNet  Google Scholar 

  26. Hwang I.S., Lee W.Y.: Block Toeplitz operators with rational symbols (II). J. Phys. A Math. Theor. 41(38), 385205 (2008)

    Article  MathSciNet  Google Scholar 

  27. Ikramov Kh.D., Chugunov V.N.: Normality conditions for a complex Toeplitz matrix. Zh. Vychisl. Mat. i Mat. Fiz. 36, 3–10 (1996)

    MathSciNet  Google Scholar 

  28. Ito T.: Every normal Toeplitz matrix is either of type (I) or type (II). SIAM J. Matrix Anal. Appl. 17, 998–1006 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lee W.Y.: Cowen sets for Toeplitz operators with finite rank selfcommutators. J. Oper. Theory 54(2), 301–307 (2005)

    Google Scholar 

  30. Nakazi T., Takahashi K.: Hyponormal Toeplitz operators and extremal problems of Hardy spaces. Trans. Am. Math. Soc. 338, 753–769 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nikolskii N.K.: Treatise on the shift operator. Springer, New York (1986)

    Book  Google Scholar 

  32. Peller V.V.: Hankel operators and their applications. Springer, New York (2003)

    Book  MATH  Google Scholar 

  33. Zhu K.: Hyponormal Toeplitz operators with polynomial symbols. Integral Equ. Oper. Theory 21, 376–381 (1996)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Korea

    In Sung Hwang

  2. Department of Mathematics, Seoul National University, Seoul, 151-742, Korea

    Dong-O. Kang & Woo Young Lee

Authors
  1. In Sung Hwang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dong-O. Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Woo Young Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Woo Young Lee.

Additional information

Communicated by Harry Dym.

The work of the first author was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2010-0016369). The work of the third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2011-0001250).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hwang, I.S., Kang, DO. & Lee, W.Y. Hyponormal Toeplitz Operators with Matrix-Valued Circulant Symbols. Complex Anal. Oper. Theory 7, 843–861 (2013). https://doi.org/10.1007/s11785-011-0184-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-011-0184-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation