无穷维Hamilton算子的局部谱性质

The infinite dimensional Hamiltonian operator is the nonselfadjoint operator matrix derived from a linear infinite dimensional Hamiltonian system and it has deep mechanics background. Many scholars studied the Hamiltonian operator and some interesting results are obtained frequently. But there is no any report on the result related to local spectral properties. The contents of the present project are as follows. (1) We study the SVEP of block operator matrices and further discuss the SVEP and Weyl's theorem of Hamiltonian operators; (2) The decomposability, (β) property and (C) property of block operator matrices are studied and, combining with the structural feature, we investigate the local spectral properties of Hamiltonian operators; (3) We study the symplectic selfadjoint problem of Hamiltonian operators and, based on this, discuss the connections between the local spectrum and the spectrum; (4) The invariant subspace problem of Hamiltonian operators is studied by means of spectral subspaces, which is a good preparation for solving the open problem on the existence of invariant subspaces. Furthermore, we introduce the above results into elasticity and related fields, which would be a good mathematical foundation for some applications.

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非自共轭算子矩阵。许多学者研究过无穷维Hamilton算子，并得到了一些有趣的结论，但至今未见有关局部谱性质的研究成果。本项目主要研究如下问题：(1)研究分块算子矩阵的SVEP，并进一步研究无穷维Hamilton算子的SVEP和Weyl型定理；(2)考察分块算子矩阵的可分性、(β)性质和(C)性质，再结合算子本身的结构特性研究无穷维Hamilton算子的局部谱性质；(3)深入研究无穷维Hamilton算子辛自共轭性问题，并在此基础上考察Hamilton算子的谱与局部谱的内在联系；(4)用谱子空间等方法研究Hamilton算子的不变子空间问题，从而为解决关于不变子空间的公开问题做准备。最后，将上述结果应用于弹性力学及其相关领域，为有关应用提供数学依据。

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非自共轭算子矩阵。许多学者研究过无穷维Hamilton算子，并得到了一些有趣的结论，但有关局部谱性质的研究成果较少。本项目围绕无穷维Hamilton算子的SVEP、Weyl型定理、局部谱性质、辛自共轭性、不变子空间等展开研究，得到了一系列重要结论。(1)刻画了分块算子矩阵的SVEP，进而得到了无穷维Hamilton算子SVEP和Weyl型定理；(2)给出了分块算子矩阵的可分性、(β)性质、(C)性质、(δ)性质等以及这些性质相互之间的关系，进而给出了无穷维Hamilton算子和Hamilton型算子的对应性质；(3)得到了无穷维Hamilton算子辛自共轭的条件，在此基础上给出了无穷维Hamilton算子的谱和局部谱的关系；(4)刻画了无穷维Hamilton算子的不变子空间代数核和解析核的性质及其关系。本项目初步建立了无穷维Hamilton 算子局部谱框架，为弹性力学及其相关领域提供数学依据。