无穷维Hamilton算子的辛自共轭性研究

Infinite dimensional Hamiltonian operator is derived from linear infinite dimensional Hamiltonian system and it has deep mechanic background. The properties of spectral symmetry, the skew selfajointness in space with indefinite metric, the completeness and symplectic orthogonality of eigenfunctions system of infinite dimensional Hamiltonian operator have important applications in theory of control, systems theory, symplectic geometry and solving problems of algebraic Riccati equations and new systematic methodology for theory of elasticity. However, infinite dimensional Hamiltonian operator is an symplectic symmetry operator in generally and not necessarily has the properties of spectral symmetry and the skew selfadjointness in space with indefinite metric. When the infinite dimensional Hamiltonian operator satisfies symplectic selfadjointness, it has the properties of spectral symmetry and the skew selfadjointness in space with indefinite metric, and the problem of completeness of eigenfunctions system become easier in this moment. Consequently, in this project through studing the invertibility of infinite dimensional Hamiltonian operator, the adjoint operator and perturbation theory, try to solve the problem of the symplectic selfadjointess of infinite dimensional Hamiltonian operator, and furthermore, provide a theoretical guarantee for the solving problem of algebric Riccati equations and new systematic methodology for theory of elasticity.

无穷维Hamilton算子是由线性无穷维Hamilton系统导出的具有深刻力学背景的线性算子,关于它的谱对称性、不定度规空间中的反自共轭性、特征函数系的完备性以及辛正交性等性质在控制论、系统理论、辛几何、代数Riccati方程的求解问题以及弹性力学求解新体系中有重要应用。然而，一般情况下以分块算子矩阵形式给出的无穷维Hamilton算子只是一个辛对称算子，不一定具有谱对称性和不定度规空间中的反自共轭性等特性。但是无穷维Hamilton算子具有辛自共轭性时，谱的对称性以及不定度规空间中的反自共轭性等性质就能成立，此时，解决特征函数系的完备性问题就会变得得心应手。因此，本项目通过研究无穷维Hamilton算子的可逆性、共轭算子以及扰动理论，试图解决无穷维Hamilton算子何时为辛自共轭算子的问题，进而为代数Riccati方程的求解问题和弹性力学求解新体系面临的关键问题提供理论保障。

无穷维Hamilton算子是由线性无穷维Hamilton系统导出的具有深刻力学背景的线性算子, 关于它的谱对称性、不定度规空间中的反自共轭性、特征函数系的完备性以及辛正交性等性质在控制论、系统理论、辛几何、代数Riccati方程的求解问题以及弹性力学求解新体系中有重要应用. 一般情况下, 以分块算子矩阵形式给出的无穷维Hamilton算子只是一个辛对称算子, 不一定具有谱对称性和不定度规空间中的反自共轭性等特性. 但是, 无穷维Hamilton算子具有辛自共轭性时, 谱的对称性以及不定度规空间中的反自共轭性等性质就能成立, 此时, 解决特征函数系的完备性问题就会变得相对容易. 于是, 本项目围绕着无穷维Hamilton算子的可逆性、分块算子矩阵的共轭表示、无穷维Hamilton算子Schur补问题以及无穷维Hamilton算子辛正交系下的谱方法等内容展开研究, 给出了形式无穷维Hamilton算子左可逆性、分块算子矩阵可逆补、分块算子矩阵共轭算子表示以及基于无穷维Hamilton算子谱理论的半群方法等的刻画, 为解决弹性力学求解新体系面临的数学问题提供了理论保障.