双圆盘商模Nψ上Toeplitz算子的向量丛模型

A Vector Bundle Model of Toeplitz Operators on the Quotient Module Nψ in the Bidisc

设D为复平面上的开单位圆盘,H 2(D 2)为双圆盘D 2上的Hardy模,ψ(z 2)为D上的有限Blaschke乘积.首先定义了H 2(D 2)的Nψ-商模,利用Blaschke积的性质给出了商模的等价刻画,然后根据等价刻画构造出商模的一组正交正规基,并给出Nψ-商模的具体刻画.最后研究了Nψ-商模上以有限Blaschke乘积B(z 1)为符号的解析Toeplitz算子T B(z 1),通过分析B(z 1)的全体逆的结构,建立了T B(z 1)的Bergman向量丛模型,并根据该模型给出了该Toeplitz算子的一些性质的几何刻画.

Let D be an open unit disc in the complex plane,H 2(D 2)the Hardy module on the bidisc D 2,andψ(z 2)a finite Blaschke product.Firstly,the Nψ-quotient module of H 2(D 2)is defined,and an equivalent characterization of the quotient module Nψis given by the properties of a finite Blaschke product.Secondly,an orthonormal basis is constructed according to this equivalent characterization and a more concrete characterization of Nψis given.Finally,the analytic Toeplitz operators on Nψ-quotient module with the finite Blaschke product B(z 1)as symbols are studied,and a Bergman bundle shift model is constructed by investigation of the set of inverses of B(z 1).Furthermore,the geographical characterization of some properties of the Toeplitz operator is given using this model.