(加权)复合算子的谱与循环性的研究

This project involves in fields such as several complex variables, operator theory, as well as linear dynamics systems. The aim of this project is to investigate the spectra, cyclicity, and complex symmetric of composition operators and weighted composition operators on several different function spaces. The composition operators and weighted composition operators are critical issues in the study of several complex variables and operator theory. Moreover, the topics being considered in this project are being highly discussed and active researched internationally. The scheme will be proceeded in three fields. First of all, we will characterize the spectra of weighted composition operators on the Hardy space and weighted Bergman spaces in unit ball. Then we will investigate the dynamic properties of weighted composition operators acting on holmorphic and smooth function spaces on pseudo-convex domains. These properties including the supercyclicity, disk-cyclicity, and the disjoint-hypercyclicity. Finally, we will consider the complex symmetric of composition operators and weighted composition operators on the classical Hardy space. So far, there are still several gaps between people and the truth in these fields. The accomplishment of this project can fill these gaps, which may have a considerable influence on both internal development of these fields and their diffusion within the mathematical community.

本项目是一项涉及到多复变函数论、算子理论以及线性动力系统等领域的综合性数学研究课题，主要致力于研究各类函数空间上复合算子及加权复合算子的谱、循环性与复对称性。复合算子与加权复合算子是多复变函数论与算子理论领域极为重要的研究对象，而本项目所涉及到的具体研究问题则更是国际上关于（加权）复合算子研究领域的热点及前沿课题。项目计划首先探讨在多复变量单位球上的Hardy空间和加权Bergman空间上各类加权复合算子的谱及其谱的结构。在此基础上，进一步研究拟凸域上全纯函数空间与光滑函数空间上加权复合算子的亚超循环性，圆盘循环性以及有限个复合算子的不相交超循环性等动力学性质。最后，项目还将研究在经典Hardy空间上复合算子以及加权复合算子的复对称性。国际上对于这些关键目标问题的研究目前在理论、方法及已知的结果上还有很大的空白。因此，本项目的研究可以极大地推动相关领域的发展，在学术上具有重要的理论意义。

该项目主要研究了各类函数空间上复合算子及加权复合算子的谱、循环性与复对称性。复合算子与加权复合算子是多复变函数论与算子理论领域极为重要的研究对象，而本项目所涉及到的具体研究问题则更是国际上关于（加权）复合算子研究领域的热点及前沿课题。项目探讨了单位圆盘上Hardy空间与加权Bergman空间上加权复合算子的谱及其谱的结构；加权位移算子与（加权）复合算子的超循环性与不交超循环性研究的动力学性质的研究；微分与复合算子的积与差分研究；单位圆盘上全纯函数空间上的复合算子的拓扑结构与紧缠绕；单位多圆柱以及弱拟凸域上两个单型Toeplitz算子的交换子与半交换子；单位圆盘上与单位球上加权Hardy空间上复合算子与加权复合算子的复对称性研究；单位球与单位多圆柱上复对称的Toeplitz算子及其性质等.