本文選題：模型空間 + 截?cái)郥oeplitz算子��； 參考：《重慶大學(xué)》2016年博士論文

【摘要】：算子理論是泛函分析的重要組成部分.作為是算子理論的重要分支,解析函數(shù)空間上的算子理論,一直得到國(guó)內(nèi)外學(xué)者的持續(xù)關(guān)注.因?yàn)門oeplitz算子理論是解析函數(shù)空間上的一類非常重要的算子,所以Toeplitz算子的研究雖然已經(jīng)超過半個(gè)世紀(jì),取得了大量的成果,但直到今天Toeplitz算子的相關(guān)研究依然很活躍.主要的原因包括下面兩個(gè)方面:一方面,Toeplitz算子與von Neumann代數(shù)、非交換幾何、隨機(jī)矩陣、量子信息、工程控制理論等有密切的關(guān)系;另外一方面,研究函數(shù)空間上的Toeplitz算子和Toeplitz代數(shù)無論是對(duì)數(shù)學(xué)科學(xué)本身,還是對(duì)物理學(xué)以及工程技術(shù)的發(fā)展都會(huì)起著緊要的作用.本文主要研究模型空間(Model space)上的截?cái)郥oeplitz算子和Bergman空間上的乘法算子.首先,我們研究了模型空間上的截?cái)郥oeplitz算子的緊性.Hardy空間上有界的Toeplitz算子的符號(hào)是唯一的,而模型空間上有界的截?cái)郥oeplitz算子對(duì)應(yīng)的符號(hào)是不唯一的.Baranov,Chalendar,Fricain已經(jīng)構(gòu)造出了沒有有界符號(hào)的有界的截?cái)郥oeplitz算子.所以,本文只考慮具有有界符號(hào)的截?cái)郥oeplitz算子的緊性.我們主要利用Hardy空間上的Hankel算子的乘積和函數(shù)代數(shù)中極大理想空間的相關(guān)技巧,得到了具有有界符號(hào)的截?cái)郥oeplitz算子的緊性的充分必要條件.這樣,Sarason和Bessonov關(guān)于截?cái)郥oeplitz算子緊性的結(jié)果只是這個(gè)充分必要條件的特殊情況.其次,我們研究了Bergman空間上的乘法算子生成的von Neumann代數(shù)的性質(zhì)和可交換性.在高維區(qū)域上的Bergman空間,考慮以全純真映射為符號(hào)的乘法算子生成的von Neumann代數(shù)的性質(zhì)和可交換性.在一些有趣的情形下,這些性質(zhì)依賴于一個(gè)特殊的黎曼流形.算子理論,幾何和復(fù)分析在研究中交互出現(xiàn).最后,我們總結(jié)了本學(xué)位論文研究的主要結(jié)果,并提出本文尚未克服的困難和希望進(jìn)一步考慮的問題。
[Abstract]:Operator theory is an important part of functional analysis. As an important branch of operator theory, the operator theory in analytic function space has been continuously concerned by scholars at home and abroad. Because Toeplitz operator theory is a very important class of operators on analytic function space, the study of Toeplitz operator has been studied for more than half a century, and a lot of achievements have been made, but the research on Toeplitz operator is still very active up to now. The main reasons are as follows: on the one hand, Toeplitz operators are closely related to von Neumann algebra, noncommutative geometry, random matrix, quantum information, engineering control theory, etc. On the other hand, The study of Toeplitz operators and Toeplitz algebras on function spaces will play an important role in the development of mathematical science, physics and engineering technology. In this paper, the truncated Toeplitz operator on the model space and the multiplication operator on the Bergman space are studied. First, we study the compactness of truncated Toeplitz operators on model spaces. The sign of bounded Toeplitz operators on Hardy spaces is unique. The bounded truncation Toeplitz operator corresponding to the sign is not unique. Baranovi Chalendarn Fricain has constructed a bounded truncated Toeplitz operator without bounded sign. Therefore, we only consider the compactness of truncated Toeplitz operators with bounded symbols. By using the product of Hankel operators on Hardy spaces and the relevant techniques of maximal ideal spaces in function algebra, we obtain a sufficient and necessary condition for the compactness of truncated Toeplitz operators with bounded symbols. So Sarason's and Bessonov's results on truncating the compactness of Toeplitz operators are only a special case of this necessary and sufficient condition. Secondly, we study the properties and commutativity of von Neumann algebras generated by multiplication operators on Bergman spaces. The properties and commutativity of von Neumann algebras generated by multiplicative operators with all pure mappings as symbols are considered in Bergman spaces of high dimensional domains. In some interesting cases, these properties depend on a special Riemannian manifold. Operator theory, geometry and complex analysis interact in the study. Finally, we summarize the main results of this dissertation, and put forward the difficulties which have not been overcome in this paper and the problems we hope to consider further.
【學(xué)位授予單位】：重慶大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O177

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本文編號(hào)：1917094