Hyponormal Toeplitz operators with non-harmonic algebraic symbol


Given a bounded function \(\varphi \) on the unit disk in the complex plane, we consider the operator \(T_{\varphi }\), defined on the Bergman space of the disk and given by \(T_{\varphi }(f)=P(\varphi f)\), where P denotes the orthogonal projection to the Bergman space in \(L^2({\mathbb {D}},dA)\). For algebraic symbols \(\varphi \), we provide new necessary conditions on \(\varphi \) for \(T_{\varphi }\) to be hyponormal, extending recent results of Fleeman and Liaw. Our approach is perturbative and aims to understand how small changes to a symbol preserve or destroy hyponormality of the corresponding operator. We consider both additive and multiplicative perturbations of a variety of algebraic symbols. One of our main results provides a necessary condition on the complex constant C for the operator \(T_{z^n+C|z|^s}\) to be hyponormal. This condition is also sufficient if \(s\ge 2n\).

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  1. 1.

    Ahern, P., Cuckovic, Z.: A mean value inequality with applications to Bergman space operators. Pac. J. Math. 173(2), 295–305 (1996)

    Google Scholar 

  2. 2.

    Beneteau, C., Khavinson, D., Liaw, C., Seco, D., Simanek, B.: Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems. arXiv:1606.08615 Rev. Mat. Iberoam

  3. 3.

    Cowen, C.: Hyponormal and Subnormal Toeplitz Operators, Surveys of Some Recent Results in Operator Theory I. Pitman Research Notes in Mathematics Series, vol. 171, pp. 155–167. Longman Scientific & Technical, Harlow (1988)

    Google Scholar 

  4. 4.

    Cuckovic, Z., Curto, R.: A new necessary condition for the hyponormality of Toeplitz operators on the Bergman space. J. Oper. Theory 79(2), 287–300 (2018)

    Google Scholar 

  5. 5.

    Duren, P., Schuster, A.: Bergman Spaces. Mathematical Surveys and Monographs, vol. 100. American Mathematical Society, Providence (2004)

    Google Scholar 

  6. 6.

    Fleeman, M., Liaw, C.: Hyponormal Toeplitz operators with non-harmonic symbol acting on the Bergman space. Oper. Math. 13(1), 61–83 (2019)

    Google Scholar 

  7. 7.

    Hwang, I.S.: Hyponormal Toeplitz operators on the Bergman space. J. Korean Math. Soc. 42(2), 387–403 (2005)

    Google Scholar 

  8. 8.

    Hwang, I.S.: Hyponormality of Toeplitz operators on the Bergman space. J. Korean Math. Soc. 45(4), 1027–1041 (2008)

    Google Scholar 

  9. 9.

    Lu, Y., Liu, C.: Commutativity and hyponormality of Toeplitz operators on the weighted Bergman space. J. Korean Math. Soc. 46(3), 621–642 (2009)

    Google Scholar 

  10. 10.

    Lu, Y., Shi, Y.: Hyponormal Toeplitz operators on the weighted Bergman space. Integral Equ. Oper. Theory 65(1), 115–129 (2009)

    Google Scholar 

  11. 11.

    Putnam, C.R.: An inequality for the area of hyponormal spectra. Math. Z. 116, 323–330 (1970)

    Google Scholar 

  12. 12.

    Sadraoui, H.: Hyponormality of Toeplitz operators and composition operators. Ph.D. thesis, Purdue University (1992)

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Correspondence to Brian Simanek.

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Simanek, B. Hyponormal Toeplitz operators with non-harmonic algebraic symbol. Anal.Math.Phys. 9, 1613–1626 (2019). https://doi.org/10.1007/s13324-018-00279-2

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  • Hyponormal operator
  • Toeplitz operator
  • Bergman space
  • Perturbation theory

Mathematics Subject Classification

  • 47B20
  • 47B35