分块算子矩阵的二次数值域及其谱的研究

Block operator matrices have been widely considered in many areas such as spectral theory, essential spectrum, quadratic numerical range and numerical range in recent years, and the research of block operator matrices has been made important progress. The quadratic numerical range was introduced by H. Langer, C. Tretter in 1998, which is a newer concept with respect to traditional numerical range, and is a subset of the numerical range. The quadratic numerical range has important applications in the investigation of spectral theory of linear operator, variational principle, block diagonalization and invariant subspaces. In this project, we will study the elementary properties of the quadratic numerical range, for example, the topological properties of quadratic numerical range in complex plane, the detailed inclusion relationship between the quadratic numerical range and the spectrum. And then we will use the quadratic numerical range and its corresponding knowledge to consider the spectral localization and spectral properties of unbounded block operator matrices. Based on these, we will continue to study some important unbounded operator matrices having mechanics background, such as infinite dimensional Hamiltonian operators and Dirac operators. For the operators, we give the properties of the quadratic numerical range, the spectral distribution and the spectral properties. And we provide a theoretical basis for the relevant applications.

近年来，对算子矩阵的研究日益拓广和深化，在诸多方面取得了重要进展，如谱、本质谱、二次数值域、数值域等。二次数值域是1998年由 H. Langer 和 C. Tretter 等学者提出的较新的概念，是传统数值域的一个子集。它在线性算子谱理论、变分原理、算子矩阵分块对角化和不变子空间的研究当中具有重要的应用。 本项目将要研究线性算子矩阵的二次数值域的基本性质，例如二次数值域的拓扑性质、二次数值域和谱之间的包含关系，以及利用二次数值域和相关知识讨论无界分块算子矩阵的谱分布和谱性质。 基此给出某些具有力学背景的重要无界算子矩阵的二次数值域的性质、谱分布和谱性质，如无穷维 Hamilton 算子或 Dirac 算子等，为相关应用提供理论依据。

分块算子矩阵的二次数值域、谱、广义逆及特征值问题在弹性力学、流体力学和量子力学等领域中都有重要的应用。本项目以分块算子矩阵为主线，主要研究了其二次数值域、谱和相关问题。关于二次数值域的基本性质，我们在前人的工作基础上进一步刻画了二次数值域和数值域与谱之间的关系，得到了一些新的结论，并提出了进一步的猜想。在分块算子矩阵的谱估计方面，我们从两个角度进行了研究并得到了成果：1.我们把分块矩阵的Ostrowski定理推广到分块算子矩阵的情形，进一步又得到了近似点谱的广义Brauer-Ostrowski定理，这些结论改进了分块算子矩阵的Gershgorin定理；2.我们还利用二次数值域刻画了两种具有力学背景的反三角分块算子的谱分布，给出了较为详细的谱估计范围。此外，我们研究了分块算子矩阵的本质谱与Fredholm性、Drazin可逆性等性质，得到了利用内部元素的性质刻画分块算子矩阵以上性质的条件；分析了一类反三角分块算子矩阵的特征值问题，刻画了其特征函数系的完备性。以上研究为分块算子矩阵的二次数值域、谱、广义逆及特征值问题在数学物理的实际问题中的应用提供了理论基础。