调和Bergman空间上Toeplitz算子和Hankel算子的性质研究

发布时间：2018-02-08 10:21

本文关键词： Bergman空间 Toeplitz算子小Hankel算子乘积交换性拟齐次　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是线性算子理论中十分活跃并引起广泛关注的分支之一,这是因为算子理论中许多深层次的问题都可以模型化为具体的函数空间上的、由具有某些特殊性质的函数所诱导出的算子的相应问题.人们通过对这些“具体”算子的研究来揭示“抽象”算子的内在性质.Bergman空间上的Toeplitz算子和Hankel算子作为算子理论中的一个重要分支,近半个世纪以来,得到了广大学者的关注.一方面,它们与函数论和算子理论中的诸多经典问题密切相关,如不变子空间问题.另一方面,它们在量子力学、控制理论、小波分析等学科中有着十分重要的应用.关于这两种算子的研究对探索算子理论乃至线性算子的结构及其应用将会产生积极的作用,同时也将会促进算子理论与代数、几何、拓扑等领域的融合.本文主要研究了单位圆盘调和Bergman空间上以拟齐次函数为符号Toeplitz算子和小Hankel算子的交换性、乘积问题,以及单位球多重调和Bergman空间上以拟齐次和分别拟齐次函数为符号的Toeplitz算子的交换性、乘积等问题.第一章主要介绍Hardy空间、Bergman空间、调和Bergman空间上Toeplitz算子和Hankel算子等相关背景知识,以及关于Toeplitz算子和Hankel算子的有界性、紧性、有限秩、交换性、乘积等方面的研究历史和研究现状.第二章主要在单位圆盘调和Bergman空间上,利用Mellin变换,研究了以径向和拟齐次函数为符号的Toeplitz算子和小Hankel算子的代数性质,解决了拟齐次Toeplitz算子和拟齐次小Hankel算子的乘积问题.同时,给出了拟齐次Toeplitz算子与小Hankel算子可交换的条件.第三章主要在单位球多重调和Bergman空间上,研究了以拟齐次函数和分别拟齐次函数为符号的Toeplitz算子的一些代数性质.首先给出了两个以特殊分别拟齐次函数为符号的Toeplitz算子乘积为Toeplitz算子的条件.其次,讨论了其一为分别拟齐次Toeplitz算子,其它为拟齐次Toeplitz算子的多个Toeplitz算子的零积问题,并且证明了与一个分别拟齐次Toeplitz算子乘积为零的Toeplitz算子只有平凡的形式.最后,得到了特殊拟齐次和分别拟齐次Toeplitz算子的交换性.
[Abstract]:The operator theory on the function space is one of the most active branches of the linear operator theory. This is because many deep problems in the operator theory can be modeled into the specific function space. The corresponding problems of operators induced by functions with some special properties are studied to reveal the intrinsic properties of "abstract" operators. The Toeplitz operators and Hankel operators on the Bergman space are regarded as computations. An important branch of subtheory, In the past half century, many scholars have paid close attention to them. On the one hand, they are closely related to many classical problems in function theory and operator theory, such as invariant subspace problems. On the other hand, they are used in quantum mechanics and control theory. Wavelet analysis and other disciplines have very important applications. The study of these two operators will play a positive role in exploring the structure and application of operator theory and linear operator, and will also promote the theory of operator, algebra, geometry, and so on. In this paper, we study the commutativity and product of Toeplitz operators and small Hankel operators with quasi homogeneous functions as sign on the unit disk harmonic Bergman space. The commutativity and product of Toeplitz operators with the sign of quasi homogeneous and quasi homogeneous functions on the unit sphere multiharmonic Bergman spaces are also discussed in the first chapter. Toeplitz operator and Hankel operator on harmonic Bergman space, and the boundedness, compactness, finite rank, commutativity of Toeplitz operator and Hankel operator. In chapter 2, the algebraic properties of Toeplitz operators and small Hankel operators with radial and quasi-homogeneous functions are studied by using Mellin transform on the unit disk harmonic Bergman space. The product problem of quasi homogeneous Toeplitz operators and quasi homogeneous small Hankel operators is solved. At the same time, the conditions under which quasi homogeneous Toeplitz operators and small Hankel operators can be exchanged are given. In this paper, we study some algebraic properties of Toeplitz operators with the signs of quasi homogeneous functions and quasi homogeneous functions respectively. Firstly, we give the conditions under which the product of two Toeplitz operators signed by special respectively quasi homogeneous functions are Toeplitz operators. In this paper, we discuss the zero product of several Toeplitz operators with quasi homogeneous Toeplitz operators and quasi homogeneous Toeplitz operators respectively, and prove that there is only a trivial form of Toeplitz operators with a zero product of a quasi homogeneous Toeplitz operator. The commutativity of special quasi homogeneous and respectively quasi homogeneous Toeplitz operators is obtained.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177

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