无穷维Hamilton算子单值扩张性研究

Infinite dimensional Hamiltonian operator is derived from linear infinite dimensional Hamiltonian system and it has deep mechanic background. Its spectrum and local spectrum are the key problems of new systematic methodolody for theory of elasticity. The single-valued extension property(SVEP) of an operator determines the solution of its spectral and local spectral problem. We will use Schur complement theory, block diagonalization technique and perturbation theory to study the SVEP of Hamiltonian operator:.1.To investigate the Fredholmness of Hamiltonian operator, according to the close link between the SVEP and the Fredholmness, this project characterizes the SVEP of Hamiltonian operator;.2.The symplectic self-adjointness of Hamiltonian operator ensures that T and T* have the SVEP at the same time, and the study on the symplectic self-adjointness of Hamiltonian operator helps to solve its SVEP;.3.To investigate comprehensively the invertibility of Hamiltonian operator, according to the relation between the SVEP and the invertibility, this project characterizes the SVEP of Hamiltonian operator;.4.By studying the quadratic numerical range of Hamiltonian operator, and with the help of the decomposablity of the quadratic numerical range that makes Hamiltonian operator block diagonalization, to study the SVEP of Hamiltonian operator..The successful implementation of this project can provide a preliminary local spectral theory framework of Hamiltonian operator, and can also provide a theoretical guarantee for new systematic methodology for theory of elasticity.

无穷维哈密顿算子是由无穷维哈密顿系统导出的具有深刻力学背景的线性算子，它的谱和局部谱问题是弹性力学求解新体系面临的关键问题，而算子单值扩张性(SVEP)对谱和局部谱问题的解决起决定性作用。本项目拟运用Schur补理论、分块对角化技术及扰动理论研究哈密顿算子的SVEP：1.研究哈密顿算子的Fredholm性，根据Fredholm性与SVEP之间的紧密联系，刻画哈密顿算子的SVEP；2.由于哈密顿算子T辛自伴性保证T与T*同时有SVEP，研究哈密顿算子的辛自伴性，有助于解决其SVEP问题；3.深入研究哈密顿算子的可逆性，根据可逆性与SVEP之间的关系，刻画哈密顿算子的SVEP；4.研究哈密顿算子的二次数值域，利用二次数值域的可分性，将哈密顿算子分块对角化后研究其SVEP。本项目的成功实施可为哈密顿算子局部谱理论提供初步框架，同时对弹性力学求解新体系中提出的重要问题提供理论保障。

无穷维哈密顿算子是由无穷维哈密顿系统导出的具有深刻力学背景的线性算子，它的谱和局部谱问题是弹性力学求解新体系面临的关键问题，而算子单值扩张性(SVEP)对谱和局部谱问题的解决起决定性作用。本项目主要研究了哈密顿算子的SVEP。具体内容包括: (1)研究了哈密顿算子的Fredholm性质，得到了Hamilton算子的Weyl型定理；(2)研究了哈密顿算子的SVEP、(β)性质和可分性；(3)研究了哈密顿算子的次标量性质，解决了Hamilton算子的不变子空间问题；(4)研究了Weyl型定理及其他类算子的局部谱性质。本项目建立了哈密顿算子局部谱理论的初步框架，为弹性力学求解新体系中提出的重要问题提供了理论保障。