关于不可分Calkin代数的研究

经典Calkin代数理论涉及诸多数学领域，引起了国内外众多专家的关注。本项目基于经典Calkin 代数和Fredholm理论，应用算子矩阵、谱理论、基数等理论与方法，深入研究不可分Calkin代数的性质，探索与紧算子、Fredholm理论以及BDF理论相关的问题，具体内容有：广义算子拓扑问题、广义Weyl-von Neumann-Berg定理、无限重数的算子与逼近问题及广义本性数值域等。旨在建立相应的BDF定理、Weyl-von Neumann-Berg定理等，解决一些与无限重数算子有关的逼近问题。特色与创新之处在于: 打破常规关于Hilbert空间可分性的假设，提出并研究不可分Calkin代数，首次关注不可分Calkin代数与基数之间、算子逼近论与算子无限重数之间的内在联系。研究成果必将丰富与发展经典Calkin代数与Fredholm理论，也可为算子理论与算子代数提出新的研究问题与方向。

The theory of classical Calkin algebras has been an important part of many mathematical fields, it has attracted much attention of many domestic and foreign experts. In this project, based on the theory of classical Calkin algebra and Fredholm theory, starting with the methods from operator matrix, spectral theory and set theory, we will deeply study the properties of nonseparable Calkin algebras, and problems related to compact operator theory, Fredholm theory and BDF theory. The main research contents are listed as follows: generalized operator topology problem, generalized Weyl-von Neumann-Berg theorem, the approximation problem concerning operators of infinite multiplicity and the generalized essential numerical range problem. We aim at establishing the corresponding BDF theorem and the corresponding Weyl-von Neumann-Berg theorem, and solving the approximation problem related to operators of infinite multiplicity. Innovations: without assuming the separability of a Hilbert space, we study the nonseparable Calkin algebras, and focus on the relations between the nonseparable Calkin algebras and cardinal numbers, operator approximation theory and operators of infinite multiplicity. In this process, the theory of classic Calkin algebra will be enriched, new problems and new directions in the study of operator theory and operator algebra will be explored.