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Chinese Annals of Mathematics, Series B

, Volume 40, Issue 2, pp 187–198 | Cite as

Joint Reducing Subspaces of Multiplication Operators and Weight of Multi-variable Bergman Spaces

  • Hansong HuangEmail author
  • Peng Ling
Article
  • 6 Downloads

Abstract

This paper mainly concerns a tuple of multiplication operators defined on the weighted and unweighted multi-variable Bergman spaces, their joint reducing subspaces and the von Neumann algebra generated by the orthogonal projections onto these subspaces. It is found that the weights play an important role in the structures of lattices of joint reducing subspaces and of associated von Neumann algebras. Also, a class of special weights is taken into account. Under a mild condition it is proved that if those multiplication operators are defined by the same symbols, then the corresponding von Neumann algebras are *-isomorphic to the one defined on the unweighted Bergman space.

Keywords

Joint reducing subspaces Von Neumann algebras Weighted Bergman spaces 

2000 MR Subject Classification

47A13 47B35 

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References

  1. [1]
    Arveson, W., An Invitation to C*-Algebras, GTM 39, Springer-Verlag, New York, 1998.zbMATHGoogle Scholar
  2. [2]
    Arveson, W., A Short Course on Spectral Theory, GTM 209, Springer-Verlag, New York, 2001.zbMATHGoogle Scholar
  3. [3]
    Bell, S., The Bergman kernel function and proper holomorphic mappings, Trans. Amer. Math. Soc., 270, 1982, 685–691.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Bjorn, A., Removable singularities for weighted Bergman spaces, Czechoslovak Mathematical Journal, 56, 2006, 179–227.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Bochner, S., Weak solutions of linear partial differential equations, J. Math. Pure Appl., 35, 1956, 193–202.MathSciNetzbMATHGoogle Scholar
  6. [6]
    Carleson, L., Selected Problems on Exceptional Sets, Van Nostrand, Princeton, NJ, 1967.zbMATHGoogle Scholar
  7. [7]
    Cowen, C., The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc., 239, 1978, 1–31.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Cowen, C., The commutant of an analytic Toeplitz operator, II, Indiana Univ. Math. J., 29, 1980, 1–12.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Cowen, C., An analytic Toeplitz operator that commutes with a compact operator and a related class of Toeplitz operators, J. Funct. Anal., 36, 1980, 169–184.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Cowen, C. and Wahl, R., Commutants of finite Blaschke product multiplication operators, preprint.Google Scholar
  11. [11]
    Dan, H. and Huang, H., Multiplication operators defined by a class of polynomials on \(L_a^2(\mathbb{D}^2)\), Integr. Equ. Oper. Theory, 80, 2014, 581–601.CrossRefzbMATHGoogle Scholar
  12. [12]
    Douglas, R., Putinar, M. and Wang, K., Reducing subspaces for analytic multipliers of the Bergman space, J. Funct. Anal., 263, 2012, 1744–1765.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Douglas, R., Sun, S. and Zheng, D., Multiplication operators on the Bergman space via analytic continuation, Adv. Math., 226, 2011, 541–583.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Guo, K. and Huang, H., On multiplication operators of the Bergman space: Similarity, unitary equivalence and reducing subspaces, J. Operator Theory, 65, 2011, 355–378.MathSciNetzbMATHGoogle Scholar
  15. [15]
    Guo, K. and Huang, H., Multiplication operators defined by covering maps on the Bergman space: The connection between operator theory and von Neumann algebras, J. Funct. Anal., 260, 2011, 1219–1255.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Guo, K. and Huang, H., Geometric constructions of thin Blaschke products and reducing subspace problem, Proc. London Math. Soc., 109, 2014, 1050–1091.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Guo, K. and Huang, H., Multiplication Operators on the Bergman Space, Lecture Notes in Mathematics, 2145, Springer-Verlag, Heidelberg, 2015.CrossRefGoogle Scholar
  18. [18]
    Guo, K., Sun, S., Zheng, D. and Zhong, C., Multiplication operators on the Bergman space via the Hardy space of the bidisk, J. Reine Angew. Math., 629, 2009, 129–168.MathSciNetzbMATHGoogle Scholar
  19. [19]
    Guo, K. and Wang, X., Reducing subspaces of tensor products of weighted shifts, Sci. China Ser. A, 59, 2016, 715–730.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Huang, H. and Zheng, D., Multiplication operators on the Bergman space of bounded domains in ℂd, 2015, arXiv: math.OA//1511.01678v1.Google Scholar
  21. [21]
    Lu, Y. and Zhou, X., Invariant subspaces and reducing subspaces of weighted Bergman space over bidisk, J. Math. Soc. Japan, 62(3), 2010, 745–765.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Rudin, W., Real and Complex Analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987.zbMATHGoogle Scholar
  23. [23]
    Shi, Y. and Lu, Y., Reducing subspaces for Toeplitz operators on the polydisk, Bull. Korean Math. Soc., 50, 2013, 687–696.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Sun, S., Zheng, D. and Zhong, C., Classification of reducing subspaces of a class of multiplication operators via the Hardy space of the bidisk, Canad. J. Math., 62, 2010, 415–438.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Thomson, J., The commutant of a class of analytic Toeplitz operators, II, Indiana Univ. Math. J., 25, 1976, 793–800.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Thomson, J., The commutant of a class of analytic Toeplitz operators, Amer. J. Math., 99, 1977, 522–529.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Tikaradze, A., Multiplication operators on the Bergman spaces of pseudoconvex domains, New York J. Math., 21, 2015, 1327–1345.MathSciNetzbMATHGoogle Scholar
  28. [28]
    Wang, X., Dan, H. and Huang, H., Reducing subspaces of multiplication operators with the symbol αz k + βw l on \(L_a^2(\mathbb{D}^2)\), Sci. China Ser. A, 58, 2015, 1–14.CrossRefGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsEast China University of Science and TechnologyShanghaiChina
  2. 2.School of MathematicsFudan UniversityShanghaiChina

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