Dirichlet空间上一类乘法算子的酉等价类

赵连阔

数学学报 ›› 2011, Vol. 54 ›› Issue (1) : 169-176.

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PDF(390 KB)
数学学报 ›› 2011, Vol. 54 ›› Issue (1) : 169-176. DOI: 10.12386/A2011sxxb0018
论文

Dirichlet空间上一类乘法算子的酉等价类

    赵连阔
作者信息 +

Unitary Equivalence of a Class of Multiplication Operators on the Dirichlet Space

    Lian Kuo ZHAO
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文章历史 +

摘要

本文研究了Dirichlet空间D上由二阶Blaschke积φ定义的乘法算子 Mφ的酉等价类.主要结果表明对于D的一个乘子ψ,乘法算子 Mψ酉等价于Mφ当且仅当存在常数θ, |θ|=1,使得ψ(z)=φ(θz).这一结果是完全不同于Hardy空间与Bergman空间上的相应结果.

 

Abstract

The unitary equivalence of multiplication operator Mφ on the Dirichlet space D is studied, where φ is a Blaschke product of order two. The main result shows that for a multiplier ψ of D, the multiplication operator Mψ is unitarily equivalent to Mφ if and only if ψ(z)=φ(θz) for some constant θ with |θ|=1. This is very different from the corresponding result in the Hardy space and in the Bergman space.

 

关键词

Dirichlet空间 / 乘法算子 / 酉等价

Key words

Dirichlet space / multiplication operator / unitary equivalence

引用本文

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赵连阔. Dirichlet空间上一类乘法算子的酉等价类. 数学学报, 2011, 54(1): 169-176 https://doi.org/10.12386/A2011sxxb0018
Lian Kuo ZHAO. Unitary Equivalence of a Class of Multiplication Operators on the Dirichlet Space. Acta Mathematica Sinica, Chinese Series, 2011, 54(1): 169-176 https://doi.org/10.12386/A2011sxxb0018

参考文献


[1] Ross W., The classical Dirichlet space, Contemp. Math., 2006, 393: 171--197.

[2] Stegenga D., Multipliers of the Dirichlet space, Illinois J. Math., 1980, 24: 113--139.

[3] Zhao L., Reducing subspaces for a class of multiplication operators on the Dirichlet space, Proc. Amer. Math. Soc., 2009, 13(9): 3091--3097.

[4] Cowen C., On equivalence of Toeplitz operators, J. Opera. Theory, 1982, 7: 167--172.

[5] Sun S., On unitary equivalence of multiplication operators on Bergman space, Northeastern Math. J., 1985, 1(2): 213--222.

[6] Carleson L., A representation formula for the Dirichlet integral, Math. Z., 1960, 73: 190--196.

[7] Zhu K., On certain Unitary Operators and composition operators, Proceedings of Symposia in Pure Mathematics, 1990, 51(2): 371--385.

基金

国家自然科学基金资助项目(10971195);数学天元基金资助项目(10926143);山西省青年科技研究基金(2010021002-2);浙江省自然科学基金(Y6090689)

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