无界算子矩阵的谱和补问题

发布时间：2018-12-12 13:56

【摘要】：本文主要研究Hilbert空间中的无界算子矩阵的谱性质和补问题.考虑无界上三角算子矩阵的一些谱由其对角元算子的此类谱刻画的性质,给出某些Hamilton算子矩阵的点谱的渐近估计,采用空间分解法研究无界上三角缺项算子矩阵的补问题.具体如下：首先,为了研究无界上三角算子矩阵的谱性质,先考虑有界情形,即研究有界算子矩阵给出MC的本质谱、Weyl谱、Browder谱、本质近似点谱和Browder本质近似点谱等于对角元算子A和B的对应谱的并集的充要条件,并由子块算子A和B的性质刻画出Mc满足几个Weyl型定理的等价性的充分条件.其次,考虑对角定义的无界上三角算子矩阵的谱性质,得到TB的本质谱、Weyl谱、Browder谱、近似点谱和亏谱等于对角元算子A和D的相应谱的并集的充要条件.作为应用,给出上三角Hamilton算子矩阵的这些谱的相应性质.然后,讨论某些Hamilton算子矩阵的点谱性质.利用最小值最大值原理确定一类斜对角Hamilton算子矩阵的点谱的上下界,估计出一类对角定义的Hamilton算子矩阵的点谱上界或下界,并将此结论运用于数学物理方程中.最后,研究无界上三角缺项算子矩阵的补问题.对给定的稠定闭算子A,D,得到存在可闭算子B使得算子矩阵TB为半Weyl和半Fredholm算子的充要条件,并且刻画出其所有补的剩余谱(连续谱、闭值域谱)交集和闭值域谱并集.特别地,当A是有界线性算子时,给出了它的所有补的点谱交集,以及它的剩余谱和连续谱并集.
[Abstract]:In this paper, we study the spectral properties and complementarity of unbounded operator matrices in Hilbert spaces. Considering the properties of some spectra of unbounded upper triangular operator matrices characterized by this kind of spectrum of diagonal element operators, the asymptotic estimates of the point spectra of some Hamilton operator matrices are given, and the complementarity problem of unbounded upper triangular operator matrices is studied by space decomposition method. Firstly, in order to study the spectral properties of unbounded upper triangular operator matrices, the bounded case is considered first, that is, the essential spectrum, Weyl spectrum, Browder spectrum of bounded operator matrix are given. The essential approximate point spectrum and the Browder essential approximate point spectrum are the necessary and sufficient conditions for the union of the corresponding spectra of the diagonal operators A and B. the sufficient conditions for Mc to satisfy the equivalence of several Weyl type theorems are described by the properties of the subblock operators A and B. Secondly, we consider the spectral properties of unbounded upper triangular operator matrices defined diagonally, and obtain a sufficient and necessary condition that the essential spectrum, Weyl spectrum, Browder spectrum, approximate point spectrum and deficient spectrum of TB are equal to the corresponding spectra of diagonal operators A and D. As an application, the properties of these spectra of the upper triangular Hamilton operator matrix are given. Then, the point spectral properties of some Hamilton operator matrices are discussed. The upper and lower bounds of the point spectrum of a class of diagonal Hamilton operator matrices are determined by using the principle of minimum maximum. The upper and lower bounds of the point spectrum of a class of diagonally defined Hamilton operator matrices are estimated, and the results are applied to the mathematical and physical equations. Finally, the problem of complements of unbounded upper triangular operator matrices is studied. For a given dense closed operator, we obtain a necessary and sufficient condition for the existence of a closed operator B such that the operator matrix TB is a semi-Weyl and a semi-Fredholm operator, and characterize all complementary residual spectra (continuous spectrum, closed range spectrum) intersection and closed range spectral union. In particular, when A is a bounded linear operator, the intersection of all complementary point spectra and its residual and continuous spectral combinations are given.
【学位授予单位】：内蒙古大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O151.21;O177

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