一类算子矩阵的谱理论及其应用

The theory of operator matrices has many applications in operator theory, differential equations and system theory which occur in various areas of applied sciences, such as mathematical physics, mechanics, biological sciences etc. In all these applications, the spectral properties of the corresponding operator matrices play an important role. Recently, the study of operator matrices attracts much attention of many authors, and now it has become one of important research areas in Functional Analysis. Using matrices decomposition, Schur complement, relative boundedness, perturbation theory and linear relation theory and so on, this project mainly considers spectral theory for a class of operator matrices and its applications, including the eigenvalue, essential spectrum, Goldberg spectrum, Weyl’s theorem, generation theorems of semigroup, and its applications in mechanics, and then improve the spectral theory of operator matrices, provide a theoretical basis for the practical problems in quantum mechanics and elasticity.

算子矩阵理论在数学物理、力学和生物科学等应用学科中出现的算子理论、微分方程理和系统理论等数学和应用数学的众多分支中有着重要的应用. 在这些应用中, 对应算子矩阵的谱性质起着极其重要的作用. 最近几年， 算子矩阵的研究吸引了国内外众多学者, 现在算子矩阵理论已经成为泛函分析的主要研究领域之一. 本项目将借助矩阵分解、Schur补、相对有界、扰动理论以及线性关系理论等，主要研究一类算子矩阵的谱理论及其应用，包括特征值、本质谱、Goldberg谱、Weyl型定理、半群生成定理等及其力学中的应用, 进而完善算子矩阵谱理论, 为弹性力学和量子力学中的的实际问题提供理论依据.

算子矩阵在数学物理、力学和生物科学等学科中出现的算子理论、微分方程理论和系统理论等数学和应用数学的众多分支中有着重要的应用。在这些应用中，对应算子矩阵的相关性质起着极其重要的作用。最近几年， 算子矩阵的研究吸引了国内外众多学者。本项目将借助矩阵分解、空间分解、相对有界性等，主要研究算子矩阵的谱理论及其应用，包括算子矩阵的特征值、本质谱、Weyl型定理、局部谱、数值域、半群生成定理等及其力学中的应用, 进而完善算子矩阵谱理论, 为弹性力学的实际问题提供理论依据。