本文關(guān)鍵詞： (對偶)Toeplitz算子 Dirichlet空間 緊性 交換性 乘積問題　出處：《大連理工大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：函數(shù)空間上的算子理論的核心問題是用算子符號的分析,幾何等性質(zhì)去描述算子的性質(zhì),由此搭建了復(fù)分析與算子理論之間的橋梁,是泛函分析中的活躍領(lǐng)域.由于Toeplitz算子,Hankel算子在控制論,信息學(xué),概率論及其它數(shù)學(xué)領(lǐng)域有廣泛應(yīng)用,因此有重要的實際應(yīng)用與理論價值.本文主要研究Dirichlet空間上的Toeplitz算子和對偶Toeplitz算子的交換性,緊性和乘積問題.第一章,介紹了函數(shù)空間算子與之相關(guān)的基本概念以及Toeplitz算子和對偶Toeplitz算子的乘積問題,緊性和交換性的發(fā)展現(xiàn)狀與歷史.第二章,利用Sobolev空間分解和擬齊次分解,研究了調(diào)和Dirichlet空間的直交補空間上兩個對偶Toeplitz算子乘積的交換性和半交換性,并給出符號滿足的充分必要條件.第三章,利用Riesz函數(shù)的性質(zhì),給出了加權(quán)Dirichlet空間上的緊Toeplitz算子的充分必要條件.第四章,通過建立單位球Dirichlet空間上多重調(diào)和函數(shù)為符號的Toeplitz算子和單位球Hardy空間上多重調(diào)和函數(shù)為符號的Toeplitz算子的聯(lián)系,利用己知的單位球Hardy空間上的Toeplitz算子的代數(shù)性質(zhì),描述了單位球Dirichlet空間上多重調(diào)和函數(shù)為符號Toeplitz算子的有限乘積有限和何時為有限秩算子,進而解決了兩個Toeplitz算子交換性問題和乘積問題.第五章,對于單位球上的解析函數(shù)f1,…fN和g1,…,gN,通過刻畫f1g1+…+fNgN何時是多重調(diào)和函數(shù)問題,給出了單位球上的多重調(diào)和Dirichlet直交補空間上兩個以多重調(diào)和函數(shù)為符號的對偶Toeplitz算子的交換性和半交換性的充分必要條件.
[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，

本文編號：1491610