Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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REDUCING SUBSPACES FOR A CLASS OF TOEPLITZ OPERATORS ON THE BERGMAN SPACE OF THE BIDISK

  • ALBASEER, MOHAMMED (SCHOOL OF MATHEMATICAL SCIENCES DALIAN UNIVERSITY OF TECHNOLOGY) ;
  • LU, YUFENG (SCHOOL OF MATHEMATICAL SCIENCES DALIAN UNIVERSITY OF TECHNOLOGY) ;
  • SHI, YANYUE (SCHOOL OF MATHEMATICAL SCIENCES OCEAN UNIVERSITY OF CHINA)
  • Received : 2014.11.03
  • Published : 2015.09.30
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Abstract

In this paper, we completely characterize the nontrivial reducing subspaces of the Toeplitz operator $T{_{z{^N_1{\bar{z}}^M_2}}$ on the Bergman space $A^2(\mathbb{D}^2)$, where N and M are positive integers.

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References

  1. H. Dan, and H. Huang, Multiplication operators defined by a class of polynomials on $L^2_{\alpha}({\mathbb{D}}^2)$, Integral Equations Operator Theory 80 (2014), no. 4, 581-601. https://doi.org/10.1007/s00020-014-2176-3
  2. R. G. Douglas, M. Putinar, and K. Wang, Reducing subspaces for analytic multipliers of the Bergman space, J. Funct. Anal. 263 (2012), no. 6, 1744-1765. https://doi.org/10.1016/j.jfa.2012.06.008
  3. R. G. Douglas, S. Sun, and D. Zheng, Multiplication operators on the Bergman space via analytic continuation, Adv. Math. 226 (2011), no. 1, 541-583. https://doi.org/10.1016/j.aim.2010.07.001
  4. 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.
  5. K. Guo and H. Huang, Multiplication operators defined by covering maps on the Bergman space: the connection between operator theory and von Neumann algebras, J. Funct. Anal. 260 (2011), no. 4, 1219-1255. https://doi.org/10.1016/j.jfa.2010.11.002
  6. K. Guo and H. Huang, Geometric constructions of thin Blaschke products and reducing subspace problem, Proc. Lond. Math. Soc. 109 (2014), no. 4, 1050-1091. https://doi.org/10.1112/plms/pdu027
  7. K. Guo and H. Huang, Multiplication Operators on the Bergman Space, Lecture Notes in Mathematics 2145, Springer-Verlag Berlin Heidelberg 2015.
  8. 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.
  9. J. Hu, S. Sun, X. Xu, and D. Yu, Reducing subspace of analytic Toeplitz operators on the Bergman space, Integral Equations Operator Theory 49 (2004), no. 3, 387-395. https://doi.org/10.1007/s00020-002-1207-7
  10. 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
  11. Y. Shi and Y. Lu, Reducing subspaces for Toeplitz operators on the polydisk, Bull. Korean Math. Soc. 50 (2013), no. 2, 687-696. https://doi.org/10.4134/BKMS.2013.50.2.687
  12. M. Stessin and K. Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2631-2639. https://doi.org/10.1090/S0002-9939-02-06382-7
  13. S. L. Sun and Y. Wang, Reducing subspaces of certain analytic Toeplitz operators on the Bergman space, Northeast. Math. J. 14 (1998), no. 2, 147-158.
  14. S. Sun, D. Zheng, and C. Zhong, Classification of reducing subspaces of a class of multiplication operators on the Bergman space via the Hardy space of the bidisk, Canad. J. Math. 62 (2010), no. 2, 415-438. https://doi.org/10.4153/CJM-2010-026-4
  15. X. Wang, H. Dan, and H. Huang, Reducing subspaces of multiplication operators with the symbol ${\alpha}z^k$ +${\beta}w^l$ on $L^2_{\alpha}({\mathbb{D}}^2)$, Sci. China Math. 58 (2015), doi:10.1007/s11425-015-4973-9.
  16. K. Zhu, Reducing subspaces for a class of multiplication operators, J. Lond. Math. Soc. 62 (2000), no. 2, 553-568. https://doi.org/10.1112/S0024610700001198

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