关于紧算子的奇异值不等式和可测算子的范数不等式的研究

发布时间：2018-10-19 20:12

【摘要】：算子理论在数学和其他科学中都占有重要地位,具有广泛的应用。Hilbert空间和Banach空间上的有界线性算子理论是算子理论和算子代数的基础。紧算子和可测算子是两类重要的算子。紧算子的奇异值是算子理论的研究热点之一。非交换Lp空间是泛函分析的重要组成部分,它的性质是当前泛函分析的研究热点。可测算子及其范数不等式的研究是算子理论重要的研究领域之一。近几年来,众多学者对紧算子和可测算子的奇异值和范数等相关问题进行了广泛的研究,并取得了大量的成果。本文运用算子理论和算子代数的相关知识和技巧,进一步研究紧算子的奇异值不等式和可测算子的范数不等式。本文研究了三类问题。一是利用算子分块矩阵的技巧,研究了Hilbert空间上紧算子的奇异值不等式。二是利用von Neumann代数的性质,研究了非交换L~p空间的一些性质。三是利用von Neumann代数M上的可测算子的性质,研究了M上的可测算子的范数不等式。本文的主要内容分为四部分。第一部分首先介绍了泛函分析以及算子理论和算子代数的起源和发展,其次介绍了紧算子和可测算子的国内外研究现状,最后介绍了本论文研究的内容、目的及相关的预备知识。第二部分主要研究了Hilbert空间上紧算子的奇异值不等式。首先介绍了一些相关概念及性质,其次利用算子分块矩阵的技巧和紧算子的性质,得到了紧算子的一些奇异值不等式。第三部分利用von Neumann代数的性质,研究了非交换L~p空间的一些性质。第四部分首先介绍了可测算子的范数和它的性质,其次研究了一系列的范数不等式,最后证明了几个可测算子的奇异值不等式的等价性。
[Abstract]:Operator theory plays an important role in mathematics and other sciences and has wide applications. The bounded linear operator theory on Hilbert space and Banach space is the basis of operator theory and operator algebra. Compact operators and measurable operators are two kinds of important operators. The singular value of compact operator is one of the hotspots in operator theory. Non-commutative Lp space is an important part of functional analysis. The study of estimators and their norm inequalities is one of the important research fields of operator theory. In recent years, many scholars have made extensive research on the singular values and norms of compact operators and estimators, and have made a lot of achievements. In this paper, the singular value inequalities of compact operators and the norm inequalities of estimators are further studied by means of operator theory and related knowledge and techniques of operator algebra. In this paper, three kinds of problems are studied. One is to study singular value inequalities of compact operators on Hilbert spaces by using the technique of partition matrix of operators. Second, by using the properties of von Neumann algebra, we study some properties of noncommutative LNP spaces. Thirdly, by using the properties of measurable subunits on von Neumann algebra M, the norm inequality of measurable subunits on M is studied. The main content of this paper is divided into four parts. The first part introduces the functional analysis, the origin and development of operator theory and operator algebra, then introduces the research status of compact operators and measurable operators at home and abroad, and finally introduces the contents of this paper. Objective and related preparatory knowledge. In the second part, the singular value inequalities of compact operators on Hilbert spaces are studied. Firstly, some related concepts and properties are introduced, then some singular value inequalities of compact operators are obtained by using the technique of partition matrix of operators and the properties of compact operators. In the third part, by using the properties of von Neumann algebra, we study some properties of noncommutative LNP spaces. In the fourth part, we first introduce the norm of the estimator and its properties, then we study a series of norm inequalities, and finally prove the equivalence of the singular value inequalities of several estimators.
【学位授予单位】：西安建筑科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177

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