算子矩阵谱的自伴扰动及其应用

The partial operator matrix is operator matrix the entries of which are unknown on some subsets of its positions, while a completion of a partial operator matrix is the conventional operator matrix resulting from filling in its unknown entries. Usually, one concerns the conditions under which a partial operator matrix has completions with some given properties. The completion problem was shown to be very useful in various pure and applied mathematical areas. It should be mention that the perturbation of spectra is one of the most active topics in the completion problem. In this project, we concern with self-adjoint perturbations of the spectra for partial operator matrices. Firstly, we study self-adjoint perturbations of various spectra for partial upper triangular operator matrices. It is expected that some complete descriptions of self-adjoint perturbations of the left (right) essential spectrum, left (right) Weyl spectrum, approximate point spectrum and defect spectrum are given, based on the perturbation property of semi-Fredholm operators, the characterization of non-compact operators and the space decomposition method. Secondly, we characterize the self-adjoint perturbations of the spectrum, essential spectrum and Weyl spectrum for 2×2 partial operator matrices in which the left lower entry is unknown. Also, we will discuss the relationship between self-adjoint perturbations and the general bounded perturbations of the various spectra for partial operator matrices. As applications, we further extend these two kinds of perturbation results into Hamiltonian cases, which will provide a good theoretical foundation for the corresponding applications in Hamiltonian systems.

缺项算子矩阵是缺少了某些项的算子矩阵，而（缺项）算子矩阵的补就是补全这些缺失的项所得的算子矩阵。算子矩阵的补问题关注的是由缺项算子矩阵中已知元素的性质研究该算子矩阵具有某种性质的补。算子矩阵的补问题在基础数学与应用数学的众多领域中具有重要应用。谱扰动是最活跃的补问题之一。本项目研究算子矩阵各类谱的自伴扰动。首先采用半Fredholm算子的扰动性质、非紧算子的性质和空间分解方法研究上三角缺项算子矩阵的左（右）本质谱、左（右）Weyl谱、近似点谱和亏谱等的自伴扰动。其次，刻画2×2阶左下角元为未知缺项算子矩阵的谱、本质谱和Weyl谱等的自伴扰动。此外，还要讨论缺项算子矩阵的各类谱的自伴扰动与有界扰动的关系。作为应用，给出上三角缺项Hamilton算子矩阵和2×2阶左下角元为未知缺项Hamilton算子矩阵的相应谱的扰动，为Hamilton系统方面的有关应用提供理论依据。

本项目中，我们研究了算子矩阵谱的自伴扰动。基于半Fredholm算子的扰动性质、非紧算子的性质和空间分解方法给出了缺项算子矩阵的左（右）本质谱、左（右）Weyl谱、近似点谱和亏谱等的自伴扰动的完全描述。结果表明，左（右）本质谱、左（右）Weyl谱、近似点谱和亏谱等的自伴扰动包含通常的相应谱的扰动。结合谱性质，我们进一步刻画了Hamilton算子矩阵的左（右）本质谱、左（右）Weyl谱、近似点谱和亏谱等的有界扰动。从学科发展的趋势来看，算子矩阵谱的自伴扰动正在成为基础数学和应用数学的一个很好的结合点。从本项目研究的的内容和结果来看，这些工作丰富了算子矩阵谱理论的内容，为研究算子矩阵的补问题奠定了理论基础和方法基础，对Hamilton算子矩阵谱理论的进一步发展作出了积极的贡献。由于本项目的研究工作与数学的其他分支(如系统、优化、控制等)有着密切的关系，再加上Hamilton系统有着非常深厚的应用背景，因此，本项目的研究工作具有广泛的潜在应用前景。