分块算子矩阵的扰动理论及其应用

Block operator matrices are matrices the entries of which are linear operators between Hilbert or Banach spaces, and many of the state operators of physical and mechanics time evolution problems are block operator matrices, hence it has important applications in various areas of mathematics, physics and mechanics. In all these applications, the spectral properties of the corresponding block operator matrices are of vital importance as they govern the process of time evelusion and stability of physical systems. However, the characterization of spectrum of blockoperator matrices is more conmplex than general linear operators, need a powerfull tool and the tool is perturbation theory. Perturbation theory arises from works of L.Rayleight and E.Schrödinger and plays an important roll in eigenvalue problems of quantum mechanics. Thereafter, T.Kato established the perturbation theory for linear operators and devoloped the study of linear operators. In this project, we shall introduce perturbation theory of block operator matrices and by using this theory we will study the spectral analysis of block operator matrices including stability of closeness of block operator matrices, stability of spectra, the adjoint operator and perturbation theory for operator semi-group, and which provides mathematical basis for applying the block operator matrices theroy to physical and mechanical systems.

分块算子矩阵是由Hilbert 或Banach 空间中线性算子构成的特殊矩阵，许多随时间变化的物理过程和力学系统的状态算子均为分块算子矩阵，从而分块算子矩阵在数学、物理和力学的多个领域具有重要应用。在诸多应用中，分块算子矩阵谱的刻画尤为重要，因为状态算子的谱控制着系统的演化过程以及稳定性等特性。然而，刻画分块算子矩阵的谱比一般线性算子要复杂，需要引进一个强有力的工具。而扰动理论是研究分块算子矩阵的重要工具。扰动理论源自Rayleigh 和Schrödinger 等人的工作，对于解决量子力学中的特征值问题发挥了重要作用。此后Kato 建立了线性算子的扰动理论，把线性算子理论研究提高到一个新的高度。本项目试图引进分块算子矩阵的扰动理论，并运用该理论研究分块算子矩阵的谱分析，包括分块算子矩阵闭性稳定性、谱的稳定性、共轭算子、算子半群等内容，为分块算子矩阵理论在物理和力学系统中的应用提供数学依据。

分块算子矩阵是由Hilbert空间或Banach空间中线性算子构成的特殊矩阵，许多随时间变化的物理过程和力学系统的状态算子均为分块算子矩阵，从而分块算子矩阵在数学、物理和力学的多个领域具有重要应用。在诸多应用中，分块算子矩阵谱的刻画尤为重要，因为状态算子的谱控制着系统的演化过程以及稳定性等特性。本项目利用分块算子矩阵的扰动理论研究了分块算子矩阵的谱分析，刻画了分块算子矩阵的闭值域、本质谱、数值域以及谱等式等问题，为分块算子矩阵理论在物理和力学系统中的应用提供了数学依据。