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


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

Mathematics Subject Classification

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