Dirichlet空间上Toeplitz算子和对偶Toeplitz算子的若干性质

发布时间：2018-02-05 00:35

  本文关键词： (对偶)Toeplitz算子 Dirichlet空间 紧性 交换性 乘积问题　出处：《大连理工大学》2016年博士论文　论文类型：学位论文

【摘要】：函数空间上的算子理论的核心问题是用算子符号的分析,几何等性质去描述算子的性质,由此搭建了复分析与算子理论之间的桥梁,是泛函分析中的活跃领域.由于Toeplitz算子,Hankel算子在控制论,信息学,概率论及其它数学领域有广泛应用,因此有重要的实际应用与理论价值.本文主要研究Dirichlet空间上的Toeplitz算子和对偶Toeplitz算子的交换性,紧性和乘积问题.第一章,介绍了函数空间算子与之相关的基本概念以及Toeplitz算子和对偶Toeplitz算子的乘积问题,紧性和交换性的发展现状与历史.第二章,利用Sobolev空间分解和拟齐次分解,研究了调和Dirichlet空间的直交补空间上两个对偶Toeplitz算子乘积的交换性和半交换性,并给出符号满足的充分必要条件.第三章,利用Riesz函数的性质,给出了加权Dirichlet空间上的紧Toeplitz算子的充分必要条件.第四章,通过建立单位球Dirichlet空间上多重调和函数为符号的Toeplitz算子和单位球Hardy空间上多重调和函数为符号的Toeplitz算子的联系,利用己知的单位球Hardy空间上的Toeplitz算子的代数性质,描述了单位球Dirichlet空间上多重调和函数为符号Toeplitz算子的有限乘积有限和何时为有限秩算子,进而解决了两个Toeplitz算子交换性问题和乘积问题.第五章,对于单位球上的解析函数f1,…fN和g1,…,gN,通过刻画f1g1+…+fNgN何时是多重调和函数问题,给出了单位球上的多重调和Dirichlet直交补空间上两个以多重调和函数为符号的对偶Toeplitz算子的交换性和半交换性的充分必要条件.
[Abstract]:The core problem of operator theory in function space is to describe the properties of operator by means of operator symbol analysis, geometry and other properties, thus building a bridge between complex analysis and operator theory. Toeplitz operator is widely used in cybernetics, informatics, probability theory and other mathematical fields. This paper mainly studies the commutativity of Toeplitz operator and dual Toeplitz operator on Dirichlet space. In chapter 1, we introduce the basic concepts of function space operator and the product of Toeplitz operator and dual Toeplitz operator. The present situation and history of compactness and commutativity. Chapter 2, using Sobolev space decomposition and quasi homogeneous decomposition. In this paper, the commutativity and semi-commutativity of the product of two dual Toeplitz operators on the direct complement space of harmonic Dirichlet space are studied, and the sufficient and necessary conditions for the symbol to satisfy are given. Chapter 3. By using the properties of Riesz functions, the sufficient and necessary conditions for compact Toeplitz operators on weighted Dirichlet spaces are given. Chapter 4th. By establishing the Toeplitz operator with polyharmonic function as symbol on unit sphere Dirichlet space and Toeplitz calculation with multiple harmonic function as symbol on unit ball Hardy space. Son's connection. The algebraic properties of Toeplitz operators on unit sphere Hardy spaces are used. In this paper, we describe the finite product and when finite rank operator of polyharmonic function is signed Toeplitz operator on unit sphere Dirichlet space. Then, the commutativity problem and product problem of two Toeplitz operators are solved. Chapter 5th, for the analytic functions f _ 1, 鈥，

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