函数空间上复合算子、Toeplitz算子与Volterra型算子的研究

Operator theory and operator algebras on function spaces as an important part of modern mathematics, is closed related to many branches of mathematics. Composite operators, Toeplitz operators and Volterra type operators on function spaces not only provide examples for the study of the general operator theory and operator algebras, the more important is that the operators and their generated algebraic structures associated with the corresponding region and space. This project based on the Hardy-Sobolev space and Fock-Sobolev space as the platform, is to study the properties of composite operator, Toeplitz operator and Volterra operator such as boundedness, compactness, algebraic properties, essential spectrum and Fredholm index of operator. Tthis research project will combine the function theory, operator theory, index theory, C*- algebra, topology and dynamical systems ect, and give full play to the application of the function and C*- algebra techniques in operator theory. It has an important scientific significance to promote the study of operator theory on function spaces.

函数空间上的算子理论与算子代数作为现代数学的重要组成部分，与许多数学分支都有着重要联系。函数空间上的复合算子、Toeplitz算子与Volterra型算子不仅为一般算子理论与算子代数的研究提供例子，更为重要的是这些算子及其生成的代数结构与相应的区域和空间有关。本项目以Hardy-Sobolev空间和Fock-Sobolev空间为平台，研究复合算子、Toeplitz算子与Volterra型算子的有界性、紧性、代数性质、本性谱和Fredholm算子指标。本项目的研究将结合函数论、算子理论、C*-代数、指标理论、拓扑学和动力系统等多个方向，充分发挥函数论与C*-代数技巧在算子理论中的应用。这对于推动函数空间上的算子理论的研究具有重要的科学意义。

函数空间上的算子理论与算子代数作为现代数学的重要组成部分，与许多数学分支都有着重要联系。函数空间上的复合算子、Toeplitz算子与Volterra型算子不仅为一般算子理论与算子代数的研究提供例子，更重要的是这些算子及其生成的代数结构与相应的区域和空间有关。本项目主要研究了对Hardy型和Bloch型等解析函数空间上的复合算子和Volterra型算子的有界性、紧性和下有界性，并给出了复合算子和Volterra型算子的有界性和紧性的充要条件。并作为应用，利用线性算子理论探究我国城乡居民代际收入流动之间的影响。