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).

我们首先将Izuchi和Yang引入到三圆盘的\（{\ mathcal {N}} _ {\ psi} \）商模块，并给出\（{\ mathcal {N}} _ {\ psi}的表征\）商数模块，然后构造通过道格拉斯，Keshari首先介绍一个加权Bergman束移模型，作者为Toeplitz算\（T_ {B} \）具有有限布拉施克产品符号乙上\（{\ mathcal {N}} _ {\ psi} \）型多盘商模块。最后，是\（T_ {B} \）的可交换代数 以束的形式给出，并且证明了Toeplitz算子类似于加权Bergman移位的副本，该副本回答了道格拉斯的一个广义问题并推广了Jiang和Zheng的结果（J Funct Anal 258（9）：2961– 2982，2010）。