上三角型无穷维Hamilton算子的谱及其应用
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摘要
无穷维Hamilton算子来源于线性无穷维Hamilton系统,具有深刻的力学背景。本文主要研究了无穷维Hamilton算子的谱刻画、可逆性及其应用。
一直以来,非自伴算子的谱理论没形成完善的理论框架,一些有重要力学背景的非自伴算子:无穷维Hamilton算子、J-自伴算子、迁移算子的谱研究没有统一的处理方法。本文首先定义了两类算子:同点谱算子、反点谱算子;自伴算子、J-自伴算子、迁移算子均为同点谱算子;反自伴算子、无穷维Hamilton算子均为反点谱算子;对Hilbert空间中的稠定闭线性算子的剩余谱利用其点谱进行了完全的刻画,利用这个刻画给出了其剩余谱为空的充要条件;由此给出了同点谱算子及反点谱算子的谱结构;由此得到上述非自伴算子的谱结构。
为了解决应用的问题,研究了上三角型无穷维Hamilton算子的谱刻画。无穷维Hamilton算子是一类特殊的2×2算子矩阵,利用其谱结构及结构特性给出对角定义的上三角型无穷维Hamilton算子的谱能由其第一个对角元的谱等价刻画的充要条件;由此建立了不计体力的平面弹性力学问题的谱理论模型。并且给出上行占优的上三角型无穷维Hamilton算子的谱、连续谱的刻画及可逆性。
本文还得到了判定一类无穷维Hamilton算子纯虚本征值的代数指标是1的充分条件。
非负Hamilton算子在线性二次最优控制问题中有着重要的应用,本文给出了一类非负Hamilton算子的点谱分布及其可逆性。
Infinite Hamiltonian operators are originated from infinite-dimensional Hamilto-nian system, which is engaged in profound mechanical background. This dissertation mainly investigates the spectrum and invertibility of the upper triangle infinite dimensional Hamiltonian operators and applications.
All along, spectral theory of non-selfadjiont operators did not develop a satisfactory theoretical framework. Non-selfadjiont operators which is engaged in profound mechanical background: infinite dimensional Hamiltonian operators, J-selfadjiont operators, transport operators, have not uniform way about their spectral investigation. In this dissertation , firstly the definitions of operator of same point spectrum and operator of opposite point spectrum are given. J-selfadjiont operators and transport operators are all operators of same point spectrum, infinite dimensional Hamiltonian operators are all operators of opposite point spectrum. The description of residual spectrum of the closed densely defined operators in Hilbert space is obtained, by the above description, a necessary and sufficient condition of the residual spectrum of the closed densely defined operators in Hilbert space being empty is given; by the result, the spectral structures of operator of same point spectrum and operator of opposite point spectrum are given; by the results, the spectral structures of the above non-selfadjiont operators are obtained.
In order to pratical applications, the spectral description of upper triangle infinite dimensional Hamiltonian operators is studied. Infinite dimensional Hamiltonian operators are a class of special 2×2 operator matrices, by its spectral structure and structure character, we get the necessary and sufficient condition of the spectrum of upper triangle infinite dimensional Hamiltonian operators with diagonal domain being equivalently described by the spectrum of the first diagonal element. by the result, the spectral theoretical mode of plane elasticity problems without the body force is established. And we obtain the invertibility and description of spectrum and continuous spectrum of upper triangle infinite dimensional Hamiltonian operators with upper dominant.
In addition, we obtain the sufficient conditions of algebric index of the pure imaginary piont spectrum of a class of infinite dimensional Hamiltonian operators being one.
Nonnegative Hamiltonian operators have important application in linear quadratic optimal control problem. The inverbility and distribution of point spectrum of a class of nonnegative Hamiltonian operators are obtained.
一直以来,非自伴算子的谱理论没形成完善的理论框架,一些有重要力学背景的非自伴算子:无穷维Hamilton算子、J-自伴算子、迁移算子的谱研究没有统一的处理方法。本文首先定义了两类算子:同点谱算子、反点谱算子;自伴算子、J-自伴算子、迁移算子均为同点谱算子;反自伴算子、无穷维Hamilton算子均为反点谱算子;对Hilbert空间中的稠定闭线性算子的剩余谱利用其点谱进行了完全的刻画,利用这个刻画给出了其剩余谱为空的充要条件;由此给出了同点谱算子及反点谱算子的谱结构;由此得到上述非自伴算子的谱结构。
为了解决应用的问题,研究了上三角型无穷维Hamilton算子的谱刻画。无穷维Hamilton算子是一类特殊的2×2算子矩阵,利用其谱结构及结构特性给出对角定义的上三角型无穷维Hamilton算子的谱能由其第一个对角元的谱等价刻画的充要条件;由此建立了不计体力的平面弹性力学问题的谱理论模型。并且给出上行占优的上三角型无穷维Hamilton算子的谱、连续谱的刻画及可逆性。
本文还得到了判定一类无穷维Hamilton算子纯虚本征值的代数指标是1的充分条件。
非负Hamilton算子在线性二次最优控制问题中有着重要的应用,本文给出了一类非负Hamilton算子的点谱分布及其可逆性。
Infinite Hamiltonian operators are originated from infinite-dimensional Hamilto-nian system, which is engaged in profound mechanical background. This dissertation mainly investigates the spectrum and invertibility of the upper triangle infinite dimensional Hamiltonian operators and applications.
All along, spectral theory of non-selfadjiont operators did not develop a satisfactory theoretical framework. Non-selfadjiont operators which is engaged in profound mechanical background: infinite dimensional Hamiltonian operators, J-selfadjiont operators, transport operators, have not uniform way about their spectral investigation. In this dissertation , firstly the definitions of operator of same point spectrum and operator of opposite point spectrum are given. J-selfadjiont operators and transport operators are all operators of same point spectrum, infinite dimensional Hamiltonian operators are all operators of opposite point spectrum. The description of residual spectrum of the closed densely defined operators in Hilbert space is obtained, by the above description, a necessary and sufficient condition of the residual spectrum of the closed densely defined operators in Hilbert space being empty is given; by the result, the spectral structures of operator of same point spectrum and operator of opposite point spectrum are given; by the results, the spectral structures of the above non-selfadjiont operators are obtained.
In order to pratical applications, the spectral description of upper triangle infinite dimensional Hamiltonian operators is studied. Infinite dimensional Hamiltonian operators are a class of special 2×2 operator matrices, by its spectral structure and structure character, we get the necessary and sufficient condition of the spectrum of upper triangle infinite dimensional Hamiltonian operators with diagonal domain being equivalently described by the spectrum of the first diagonal element. by the result, the spectral theoretical mode of plane elasticity problems without the body force is established. And we obtain the invertibility and description of spectrum and continuous spectrum of upper triangle infinite dimensional Hamiltonian operators with upper dominant.
In addition, we obtain the sufficient conditions of algebric index of the pure imaginary piont spectrum of a class of infinite dimensional Hamiltonian operators being one.
Nonnegative Hamiltonian operators have important application in linear quadratic optimal control problem. The inverbility and distribution of point spectrum of a class of nonnegative Hamiltonian operators are obtained.
引文
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