DOI QR코드

DOI QR Code

REDUCING SUBSPACES FOR TOEPLITZ OPERATORS ON THE POLYDISK

  • Shi, Yanyue (College of Mathematical Science Ocean University of China) ;
  • Lu, Yufeng (School of Mathematical Sciences Dalian University of Technology)
  • Received : 2012.01.25
  • Published : 2013.03.31

Abstract

In this note, we completely characterize the reducing subspaces of $T_{{z^N_1}{z^M_2}}$ on $A^2_{\alpha}(D^2)$ where ${\alpha}$ > -1 and N, M are positive integers with $N{\neq}M$, and show that the minimal reducing subspaces of $T_{{z^N_1}{z^M_2}}$ on the unweighted Bergman space and on the weighted Bergman space are different.

Keywords

Toeplitz operator;reducing subspace;Bergman space

References

  1. K. Guo and H. Huang, On Multiplication operators on the Bergman space: similarity, unitary equivalence and reducing subspaces, J. Operator Theory 65 (2011), no. 2, 355-378.
  2. K. Guo, S. Sun, D. Zheng, and C. Zhong, Multiplication operators on the Bergman space via the Hardy space of the bidisk, J. Reine Angew. Math. 628 (2009), 129-168.
  3. Y. Lu and Y. Shi, Hyponormal Toeplitz operators on the weighted Bergman space, Integral Equations Operator Theory 65 (2009), no. 1, 115-129. https://doi.org/10.1007/s00020-009-1712-z
  4. Y. Lu and X. Zhou, Invariant subspaces and reducing subspaces of weighted Bergman space over bidisk, J. Math. Soc. Japan 62 (2010), no. 3, 745-765. https://doi.org/10.2969/jmsj/06230745
  5. S. Shimorin, On Beurling-type theorems in weighted $l^{2}$ and Bergman spaces, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1777-1787. https://doi.org/10.1090/S0002-9939-02-06721-7
  6. L. Trieu, On Toeplitz operators on Bergman spaces of the unit polydisk, Proc. Amer. Math. Soc. 138 (2010), no. 1, 275-285. https://doi.org/10.1090/S0002-9939-09-10060-6
  7. X. Zhou, Y. Shi, and Y. Lu, Invariant subspaces and reducing subspaces of weighted Bergman space over polydisc, Sci. Sin. Math. 41 (2011), no. 5, 427-438. https://doi.org/10.1360/012010-627
  8. K. Zhu, Reducing subspaces for a class of multiplication operators, J. London Math. Soc. 62 (2000), no. 2, 553-568. https://doi.org/10.1112/S0024610700001198
  9. K. Zhu, Operator Theory in Function Spaces, 2nd ed. Providence, R.I.: American Mathematical Society, 2007.

Cited by

  1. Reducing subspaces of tensor products of weighted shifts vol.59, pp.4, 2016, https://doi.org/10.1007/s11425-015-5089-y
  2. A Note on Reducing Subspaces of Toeplitz Operator on the Weighted Analytic Function Spaces of the Bidisk Hw2D2 vol.2017, 2017, https://doi.org/10.1155/2017/5807909
  3. REDUCING SUBSPACES FOR A CLASS OF TOEPLITZ OPERATORS ON THE BERGMAN SPACE OF THE BIDISK vol.52, pp.5, 2015, https://doi.org/10.4134/BKMS.2015.52.5.1649
  4. Multiplication Operators Defined by a Class of Polynomials on $${L_a^2(\mathbb{D}^2)}$$ L a 2 ( D 2 ) vol.80, pp.4, 2014, https://doi.org/10.1007/s00020-014-2176-3
  5. Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk vol.2015, 2015, https://doi.org/10.1155/2015/209307
  6. Reducing subspaces for a class of non-analytic Toeplitz operators on the bidisk vol.445, pp.1, 2017, https://doi.org/10.1016/j.jmaa.2016.08.012
  7. Reducing subspaces of multiplication operators with the symbol αz k + βw l on $$L_a^2 (\mathbb{D}^2 )$$ vol.58, pp.10, 2015, https://doi.org/10.1007/s11425-015-4973-9
  8. Joint Reducing Subspaces of Multiplication Operators and Weight of Multi-variable Bergman Spaces vol.40, pp.2, 2019, https://doi.org/10.1007/s11401-019-0125-9