Bundle shifts and Toeplitz operators on \({\mathcal {N}}_{\psi }\)-type quotient modules of the tridisc

Abstract

We first generalize the \({\mathcal {N}}_{\psi }\) quotient modules introduced by Izuchi and Yang to the tridisc and give a characterization of the \({\mathcal {N}}_{\psi }\) quotient modules, and then construct a weighted Bergman bundle shift model introduced firstly by Douglas, Keshari, and the author for the Toeplitz operator \(T_{B}\) with a finite Blaschke product symbol B on the \({\mathcal {N}}_{\psi }\)-type quotient modules of the polydisc. Finally, the algebra of commutant of \(T_{B}\) is given in terms of the bundle, and it is shown that the Toeplitz operator is similar to copies of weighted Bergman shift which answers a generalized question of Douglas and generalizes a result of Jiang and Zheng (J Funct Anal 258(9): 2961–2982, 2010).

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References

  1. 1.

    Abrahamse, M.B., Douglas, R.G.: A class of subnormal operators related to multiply-connected domains. Adv. Math. 19(1), 106–148 (1976)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Chen, L.: A dual geometric theory for bundle shifts. J. Funct. Anal. 263(4), 846–868 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cassier, G., Chalendar, I.: The group of the invariants of a finite Blaschke product. Complex. Var. Theory Appl. 42(3), 193–206 (2000)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Conway, J.B.: The Theory of Subnormal Operators. American Mathematical Society, Providence (1992)

    Google Scholar 

  5. 5.

    Cowen, C.: On equivalence of Toeplitz operators. J. Oper. Theory 7(1), 167–172 (1982)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Cowen, M.J., Douglas, R.G.: Complex geometry and operator theory. Acta Math. 141(3-4), 187–261 (1978)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Douglas, R.G.: Operator theory and complex geometry. Extr. Math. 24(2), 135–165 (2009)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Douglas, R.G., Keshari, D.K., Xu, A.J.: Generalized bundle shift with application to multiplication operator on the Bergman space. J. Oper. Theory 75(2), 3–19 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Douglas, R.G., Kim, Y.: Reducing subspaces on the annulus. Integ. Equ. Oper. Theory 70(2), 1–15 (2011)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Douglas, R.G., Putinar, M., Wang, K.: Reducing subspaces for analytic multipliers of the Bergman space. J. Funct. Anal. 263(6), 1744–1765 (2012)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Douglas, R.G., Sun, S., Zheng, D.: Multiplication operators on the Bergman space via analytic continuation. Adv. Math. 226(1), 541–583 (2011)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Fisher, S.D.: Function Theory on Planar Domains. A Wiley-Interscience Publication. Wiley, New York (1983)

    Google Scholar 

  13. 13.

    Gamelin, T.: Complex Analysis. Springer, New York (2001)

    Google Scholar 

  14. 14.

    Guo, K., Huang, H.: On multiplication operators on the Bergman space: similarity, unitary equivalence and reducing subspaces. J. Oper. Theory 65(2), 355–378 (2011)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Kunyu, G., Shunhua, S., Dechao, Z., Changyong, Z.: Multiplication operators on the Bergman space via the Hardy space of the bidisk. J. Reine Angew. Math. 628, 129–168 (2009)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Hu, J., Sun, S., Xu, X., Yu, D.: Reducing subspace of analytic Toeplitz operators on the Bergman space. Integ. Equ. Oper. Theory 49(3), 387–395 (2004)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Husemoller, D.: Fibre Bundle. McGraw-Hill, New York (1966)

    Google Scholar 

  18. 18.

    Izuchi, K.H., Izuchi, Y., Izuchi, K.J.: Wandering subspaces and the Beurling type theorem II. New York J. Math. 16(33), 489–505 (2010)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Izuchi, K.J., Yang, R.: \(N_{\varphi }\) -type quotient modules on the torus. New York J. Math. 14(33), 431–457 (2008)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Jiang, C.-l., Li, Y.-C.: The commutant and similarity invariant of analytic Toeplitz operators on Bergman space. Sci. China Ser. A: Math. 50(5), 651–664 (2008)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Jiang, C.-l., Zheng D.: Similarity of analytic Toeplitz operators on the Bergman spaces. J. Funct. Anal. 258(9), 2961–2982 (2010)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kobayashi, S.: Differential Geometry of Complex Vector Bundles, Publications of the Mathematical Society of Japan, 15. Kanô Memorial Lectures, 5. Princeton University Press, Princeton, NJ; Iwanami Shoten, Tokyo (1987)

  23. 23.

    Stephenson, K.: Analytic functions and hypergroups of function pairs. Indiana Univ. Math. J. 31(6), 843–884 (1982)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Sun, S.: On unitary equivalence of multiplication operators on Bergman space. Northeast. Math. J. 1, 213–222 (1985)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Sun, S., Wang, Y.: The commutants of a class of analytic Toeplitz operators on Bergman spaces. (Chinese). Acta Sci. Natur. Univ. Jilin (2), 4–8 (1997)

  26. 26.

    Thomson, J.: The commutant of a class of analytic Toeplitz operators II. Indiana Univ. Math. J. 25(8), 793–800 (1976)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Thomson, J.: The commutant of a class of analytic Toeplitz operators. Am. J. Math. 99, 522–529 (1977)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Tikaradze, A.: Multiplication operators on the Bergman spaces of pseudoconvex domains. New York J. Math. 21, 1327–1345 (2015)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Wu, Y., Xu, X.-Mi.: Reducing subspaces of Toeplitz operators on \(N_{\varphi }\)-type quotient modules on the torus (Chinese). Commun. Math. Res. 25(1), 19–29 (2009)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Xu, A.J., Yan, C.Q.: Reducing subspace of analytic Toeplitz operators on the quotient module \(N_{\varphi }\) (Chinese). Appl. Math. J. Chin. Univ. Ser. A 24(1), 95–101 (2009)

    MathSciNet  Google Scholar 

  31. 31.

    Zhu, K.: Reducing subspaces for a class of multiplication operators. J. Lond. Math. Soc. (2), 62(2), 553–568 (2000)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

Many thanks for the referee’s important suggestions. The author was supported in part by Chongqing Science and Technology Commission (Grant Nos. cstc2018jcyjA2248 and cstc2019jcyj-msxmX0295), NSF of China (11871127), and Youth Project of Science and Technology Research Program of Chongqing Education Commission of China. (No. KJQN201801110).

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Correspondence to Anjian Xu.

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Communicated by Kehe Zhu.

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Xu, A. Bundle shifts and Toeplitz operators on \({\mathcal {N}}_{\psi }\)-type quotient modules of the tridisc. Ann. Funct. Anal. 12, 26 (2021). https://doi.org/10.1007/s43034-021-00114-z

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Keywords

  • Generalized bundle shift
  • Toeplitz operator
  • \({\mathcal {N}}_{\psi }\)-quotient module
  • Polydisc

Mathematics Subject Classification

  • 47B35
  • 46B32
  • 05A38
  • 15A15