函数空间上Toeplitz算子和Hankel算子的乘积及交换性

发布时间：2018-04-08 11:51

本文选题：Hardy空间　切入点：Bergman空间　出处：《大连理工大学》2015年博士论文

【摘要】：函数空间上的算子理论是算子理论的一个重要组成部分,如何用符号函数的性质来刻画这些算子的性质是函数空间上算子理论的核心问题.本文研究若干函数空间上的Toeplitz算子,Hankel算子以及对偶Toeplitz算子的乘积和交换性.本文结构如下：第一章简要介绍了函数空间上算子理论的研究背景,发展历程和现状.第二章在单位圆盘Hardy空间上研究了Hankel算子和Toeplitz算子的乘积的有限和何时为零,然后描述了Toeplitz算子和Hankel算子的乘积何时是一个Hankel算子或Toeplitz算子的紧扰动,最后刻画了该乘积何时为一个Toeplitz算子的有限秩扰动.第三章研究了多圆盘Hardy空间上Toeplitz算子的交换性问题,并通过对区域的维数进行数学归纳法,得到结论：两个Toeplitz算子可交换当且仅当它们的换位子的Berezin变换是n-调和函数.第四章考虑单位球加权多重调和Bergman空间上的Toeplitz算子.当TCoeplitz算子的符号是某种分别径向函数或者全纯单项式时,我们描述了与之可交换的Toeplitz算子,然后我们给出Toeplitz算子有限秩乘积问题的一个部分解.第五章研究对偶Toeplitz算子的乘积问题.首先在单位球Bergman空间的直交补空间上建立对偶Toeplitz代数的结构定理,并用它给出了对偶Toeplitz算子的有限乘积的有限和是单个对偶Toeplitz算子的紧扰动的一个等价条件.然后描述了对偶Toeplitz算子的乘积的有限和何时等于一个对偶Toeplitz算子.最后在单位球Hardy空间的直交补空间上得到了类似的结论.
[Abstract]:Operator theory on function space is an important part of operator theory. How to characterize the properties of these operators by the properties of symbolic functions is the core problem of operator theory in function space.In this paper, we study the product and commutativity of Toeplitz operators and dual Toeplitz operators on some function spaces.The structure of this paper is as follows: in chapter 1, the background, development and present situation of operator theory in function space are briefly introduced.In chapter 2, we study the finite and zero product of Hankel operator and Toeplitz operator on the unit disk Hardy space. Then we describe when the product of Toeplitz operator and Hankel operator is a compact perturbation of Hankel operator or Toeplitz operator.Finally, the finite rank perturbation of the product is described when the product is a Toeplitz operator.In chapter 3, we study the commutativity of Toeplitz operators on multi-disk Hardy spaces, and by means of mathematical induction of the dimension of regions, we conclude that two Toeplitz operators are commutative if and only if the Berezin transformation of their commutators is n-harmonic function.In chapter 4, we consider the Toeplitz operator on the unit ball weighted polyharmonic Bergman space.When the symbol of TCoeplitz operator is some separate radial function or Holomorphic monomial, we describe the commutative Toeplitz operator, and then we give a partial solution of the finite rank product problem of Toeplitz operator.In chapter 5, we study the product of dual Toeplitz operators.In this paper, the structure theorem of dual Toeplitz algebras is established on the orthogonal complementary spaces of unit sphere Bergman spaces, and an equivalent condition is given that the finite sum of the finite product of dual Toeplitz operators is a compact perturbation of a single dual Toeplitz operator.Then we describe when the product of dual Toeplitz operator is equal to a dual Toeplitz operator.Finally, a similar conclusion is obtained on the orthogonal complement space of unit ball Hardy space.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O177

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