【摘要】：本文主要研究Hilbert空間中的無界算子矩陣的譜性質(zhì)和補(bǔ)問題.考慮無界上三角算子矩陣的一些譜由其對角元算子的此類譜刻畫的性質(zhì),給出某些Hamilton算子矩陣的點(diǎn)譜的漸近估計(jì),采用空間分解法研究無界上三角缺項(xiàng)算子矩陣的補(bǔ)問題.具體如下：首先,為了研究無界上三角算子矩陣的譜性質(zhì),先考慮有界情形,即研究有界算子矩陣給出MC的本質(zhì)譜、Weyl譜、Browder譜、本質(zhì)近似點(diǎn)譜和Browder本質(zhì)近似點(diǎn)譜等于對角元算子A和B的對應(yīng)譜的并集的充要條件,并由子塊算子A和B的性質(zhì)刻畫出Mc滿足幾個(gè)Weyl型定理的等價(jià)性的充分條件.其次,考慮對角定義的無界上三角算子矩陣的譜性質(zhì),得到TB的本質(zhì)譜、Weyl譜、Browder譜、近似點(diǎn)譜和虧譜等于對角元算子A和D的相應(yīng)譜的并集的充要條件.作為應(yīng)用,給出上三角Hamilton算子矩陣的這些譜的相應(yīng)性質(zhì).然后,討論某些Hamilton算子矩陣的點(diǎn)譜性質(zhì).利用最小值最大值原理確定一類斜對角Hamilton算子矩陣的點(diǎn)譜的上下界,估計(jì)出一類對角定義的Hamilton算子矩陣的點(diǎn)譜上界或下界,并將此結(jié)論運(yùn)用于數(shù)學(xué)物理方程中.最后,研究無界上三角缺項(xiàng)算子矩陣的補(bǔ)問題.對給定的稠定閉算子A,D,得到存在可閉算子B使得算子矩陣TB為半Weyl和半Fredholm算子的充要條件,并且刻畫出其所有補(bǔ)的剩余譜(連續(xù)譜、閉值域譜)交集和閉值域譜并集.特別地,當(dāng)A是有界線性算子時(shí),給出了它的所有補(bǔ)的點(diǎn)譜交集,以及它的剩余譜和連續(xù)譜并集.
[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.
【學(xué)位授予單位】：內(nèi)蒙古大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O151.21;O177

【參考文獻(xiàn)】

相關(guān)期刊論文 前10條

1 Ya-ru QI;Jun-jie HUANG;ALATANCANG;;Left Invertible Completions of Upper Triangular Operator Matrices with Unbounded Entries[J];Acta Mathematicae Applicatae Sinica;2015年02期

2 海國君;阿拉坦倉;;上三角算子矩陣的(α,β)-本質(zhì)譜[J];數(shù)學(xué)學(xué)報(bào);2014年03期

3 額布日力吐;阿拉坦倉;;一類板彎曲方程的辛本征函數(shù)展開方法(英文)[J];應(yīng)用數(shù)學(xué);2013年01期

4 王華;阿拉坦倉;黃俊杰;;Completeness of system of root vectors of upper triangular infinitedimensional Hamiltonian operators appearing in elasticity theory[J];Applied Mathematics and Mechanics(English Edition);2012年03期

5 黃俊杰;阿拉坦倉;王華;;Completeness of the system of eigenvectors of off-diagonal operator matrices and its applications in elasticity theory[J];Chinese Physics B;2010年12期

6 侯國林;阿拉坦倉;;Symplectic eigenfunction expansion theorem for elasticity of rectangular planes with two simply-supported opposite sides[J];Applied Mathematics and Mechanics(English Edition);2010年10期

7 海國君;阿拉坦倉;;2×2階上三角型算子矩陣的Moore-Penrose譜[J];系統(tǒng)科學(xué)與數(shù)學(xué);2009年07期

8 Alatancang;;Spectra of Off-diagonal Infinite-Dimensional Hamiltonian Operators and Their Applications to Plane Elasticity Problems[J];Communications in Theoretical Physics;2009年02期

9 ;A Note on the Left Essential Spectra of Operator Matrices[J];Acta Mathematica Sinica(English Series);2007年12期

10 曹小紅;郭懋正;孟彬;;Drazin譜和算子矩陣的Weyl定理(英文)[J];數(shù)學(xué)研究與評論;2006年03期

相關(guān)博士學(xué)位論文 前2條

1 范小英;上三角型無窮維Hamilton算子的譜及其應(yīng)用[D];內(nèi)蒙古大學(xué);2009年

2 吳德玉;無窮維Hamilton算子的譜與特征函數(shù)系的完備性[D];內(nèi)蒙古大學(xué);2008年

，

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