关于算子谱结构和正交投影对的研究
发布时间：2018-08-22 08:19
【摘要】：算子理论是泛函分析重要的研究领域之一,它对于微分方程,调和分析及理论物理等学科都有着深刻应用.其中谱结构,谱保持问题以及正交投影对一直是众多学者研究的热点问题.对于谱结构,Weyl型定理能很好的反映算子谱的分布特点,因此对Weyl型定理及其变形推广的研究是许多学者一直关注的问题.同时,谱结构和部分谱子集作为代数的同构不变量研究也引起了学者们的广泛关注,即谱保持问题.另一方面,基于Halmos正交投影对分解,许多学者运用谱理论和Fredholm理论来研究正交投影对,研究与正交投影对有关的范数,谱及正交投影对的差积等.这些结果对算子谱理论有着非常重要的影响,同时仍有一些问题引起学者们的关注.本文对Weyl型定理及其变形在紧摄动下的稳定性问题,保持谱子集的可加映射以及具有固定差的正交投影对问题进行了更进一步的研究.具体研究内容有三方面.在谱结构方面,根据算子semi-Fredholm域的特点讨论了算子Weyl定理在紧摄动下有稳定性的特征,其次探究了算子T在紧摄动下有Weyl定理稳定性和T2在紧摄动下有Weyl定理稳定性的关系,之后研究了 Weyl定理的一种变形(ω)性质在紧摄动下有稳定性的等价条件,最后根据2×2上三角算子矩阵的特点,利用对角线上元素的性质来刻画它的单值延拓性质.对于谱保持问题中,首先由正规特征值定义了 m-正规特征值,然后讨论m-正规特征值和m+1-正规特征值作为B(H)上的一个同构或者反同构不变量,之后刻画了 B(X)上保持算子谱中semi-Fredholm域的可加映射的结构.最后,也讨论了保持拓扑一致降标集和保持单值延拓性质稳定性的线性映射.同时,在正交投影对方面,研究了能表示成两个正交投影差的自伴算子.首先讨论自伴算子A是纯的情形,先给出自伴算子A标准型的定义,探讨一个自伴算子A表示成两个正交投影差的充要条件,然后在此基础上,给出满足差为A的所有的正交投影对的一般表示,之后再把得到的结论推广到一般自伴算子的情形,最后,从代数角度考虑,探讨了差为A的所有正交投影生成的von Neumann代数及其换位的形式与结构.
[Abstract]:Operator theory is one of the important research fields of functional analysis. It has profound applications in the fields of differential equation harmonic analysis and theoretical physics. Among them, spectral structure, spectral preservation and orthogonal projection pairs have been the hot topics of many scholars. Weyl type theorem of spectral structure can well reflect the distribution characteristics of operator spectrum, so the study of Weyl type theorem and its extension is a problem that many scholars have been paying close attention to. At the same time, the study of spectral structure and partial spectral subsets as isomorphic invariants of algebras has also attracted extensive attention of scholars, that is, the problem of spectral preservation. On the other hand, based on the decomposition of Halmos orthogonal projection pairs, many scholars use spectral theory and Fredholm theory to study orthogonal projection pairs, and study the norm related to orthogonal projection pairs, the difference product of spectrum and orthogonal projection pairs, etc. These results have a very important influence on the theory of operator spectrum, and there are still some problems that attract the attention of scholars. In this paper, the problem of the stability of Weyl type theorem and its deformation under compact perturbation, the additive mapping of preserving spectral subsets and the orthogonal projection pair problem with fixed difference are studied further. The concrete research content has three aspects. In terms of spectral structure, according to the characteristics of operator semi-Fredholm domain, the stability of operator Weyl theorem under compact perturbation is discussed. Secondly, the relation between Weyl theorem stability of operator T under compact perturbation and Weyl theorem stability under compact perturbation is discussed. Then we study the equivalent condition that a kind of deformation (蠅) property of Weyl theorem has stability under compact perturbation. Finally, according to the characteristics of 2 脳 2 upper triangular operator matrix, we use the properties of elements on diagonal line to characterize its single-valued continuation property. For spectral preserving problem, m- normal eigenvalues are defined by normal eigenvalues, and then m- normal eigenvalues and m1-normal eigenvalues are discussed as an isomorphism or anti-isomorphism invariant on B (H). Then we characterize the structure of additive mappings in the semi-Fredholm domain in the spectrum of preserving operators on B (X). Finally, we also discuss the linear mapping that preserves the topological uniform scale-reducing set and the stability of the single-valued continuation property. At the same time, the self-adjoint operators which can be expressed as two orthogonal projection difference are studied on the orthogonal projection surface. In this paper, the definition of the canonical form of self-adjoint operator A is given, and the necessary and sufficient conditions for a self-adjoint operator A to be expressed as two orthogonal projection differences are discussed. In this paper, we give the general representation of all orthogonal projection pairs satisfying difference A, then generalize the result to the case of general self-adjoint operator. Finally, from the algebraic point of view, The form and structure of von Neumann algebras generated by all orthogonal projections with difference A and their commutations are discussed.
【学位授予单位】：陕西师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177
本文编号：2196530
[Abstract]:Operator theory is one of the important research fields of functional analysis. It has profound applications in the fields of differential equation harmonic analysis and theoretical physics. Among them, spectral structure, spectral preservation and orthogonal projection pairs have been the hot topics of many scholars. Weyl type theorem of spectral structure can well reflect the distribution characteristics of operator spectrum, so the study of Weyl type theorem and its extension is a problem that many scholars have been paying close attention to. At the same time, the study of spectral structure and partial spectral subsets as isomorphic invariants of algebras has also attracted extensive attention of scholars, that is, the problem of spectral preservation. On the other hand, based on the decomposition of Halmos orthogonal projection pairs, many scholars use spectral theory and Fredholm theory to study orthogonal projection pairs, and study the norm related to orthogonal projection pairs, the difference product of spectrum and orthogonal projection pairs, etc. These results have a very important influence on the theory of operator spectrum, and there are still some problems that attract the attention of scholars. In this paper, the problem of the stability of Weyl type theorem and its deformation under compact perturbation, the additive mapping of preserving spectral subsets and the orthogonal projection pair problem with fixed difference are studied further. The concrete research content has three aspects. In terms of spectral structure, according to the characteristics of operator semi-Fredholm domain, the stability of operator Weyl theorem under compact perturbation is discussed. Secondly, the relation between Weyl theorem stability of operator T under compact perturbation and Weyl theorem stability under compact perturbation is discussed. Then we study the equivalent condition that a kind of deformation (蠅) property of Weyl theorem has stability under compact perturbation. Finally, according to the characteristics of 2 脳 2 upper triangular operator matrix, we use the properties of elements on diagonal line to characterize its single-valued continuation property. For spectral preserving problem, m- normal eigenvalues are defined by normal eigenvalues, and then m- normal eigenvalues and m1-normal eigenvalues are discussed as an isomorphism or anti-isomorphism invariant on B (H). Then we characterize the structure of additive mappings in the semi-Fredholm domain in the spectrum of preserving operators on B (X). Finally, we also discuss the linear mapping that preserves the topological uniform scale-reducing set and the stability of the single-valued continuation property. At the same time, the self-adjoint operators which can be expressed as two orthogonal projection difference are studied on the orthogonal projection surface. In this paper, the definition of the canonical form of self-adjoint operator A is given, and the necessary and sufficient conditions for a self-adjoint operator A to be expressed as two orthogonal projection differences are discussed. In this paper, we give the general representation of all orthogonal projection pairs satisfying difference A, then generalize the result to the case of general self-adjoint operator. Finally, from the algebraic point of view, The form and structure of von Neumann algebras generated by all orthogonal projections with difference A and their commutations are discussed.
【学位授予单位】：陕西师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177
【参考文献】
相关期刊论文 前2条
1 M.BERKANI;;On the Equivalence of Weyl Theorem and Generalized Weyl Theorem[J];Acta Mathematica Sinica(English Series);2007年01期
2 ;SPECTRUM-PRESERVING ELEMENTARY OPERATORS ON B(X)[J];Chinese Annals of Mathematics;1998年04期
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