广义(复)Volterra型算子动力学性质的研究

This project is dedicated to the research on the dynamic properties of generalized (complex) Volterra type operators on different function spaces, which is a hot topic in several complex variables, operator theory and dynamics of linear operators. More specifically, firstly, we will study disk-cyclicity and disk-cyclicity subspaces of generalized Volterra type operators which act on different Frechet spaces. Secondly, we will discuss supercyclicity and hypercyclicity of generalized complex Volterra type operators which act on one complex variable classical Hardy space, and give sufficient and necessary conditions. Furthermore, we will give the characterizations of disjoint disk-cyclicity and disjoint hypercyclicity of generalized (complex) Volterra type operators. Finally, by using the results obtained above, we will characterize dynamical properties of generalized complex Volterra type integral operators on several complex variables Hardy space. The results of this project could offer theoretical foundations for discovering dynamical properties of other integral type operators (Cesaro type operator), and also reveal the dynamical tendency and evolution laws about dynamics of linear operators on one or several complex variables holomorphic function spaces.

本项目致力于研究不同函数空间上广义(复)Volterra型算子的动力学性质，属于多复变函数论与算子理论以及线性算子动力系统的前沿热点课题。具体来讲，首先我们将研究不同Fréchet空间上的广义Volterra型算子的圆盘循环性和圆盘循环子空间等动力学性质；然后研究作用在单复变量经典Hardy空间上广义复Volterra型积分算子的亚超循环性和超循环性等性质，并给出等价条件；进一步探讨有限个广义(复)Volterra型积分算子不交圆盘循环性和不交的超循环性等性质；最后，在此研究的基础上，刻画作用在多复变量单位球Hardy空间上广义复Volterra型算子的超循环性等动力学性质。该项目结果可为进一步探索其他积分型算子(如Cesaro算子)的动力学性质提供理论依据，同时对进一步揭示单复变量和多复变量全纯函数空间上线性算子动力系统运动趋势及变化规律，具有重要的意义。

该项目主要研究了各类函数空间上广义Volterra型算子和加权位移算子的动力学性质。积分型算子与加权位移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研究问题更是国际上关于线性算子动力系统研究领域的热点及前沿课题。具体来讲，项目探讨了不同F空间上的广义Volterra型算子的圆盘循环性等动力学性质；然后研究有限个加权位移算子超循环性和不交的超循环性等性质；进一步探讨有限个加权复合算子超循环性和不交的超循环性等动力学性质；该项目结果可为进一步探索其他积分型算子(如Cesaro算子)的动力学性质提供理论依据，同时也揭示了与海洋动力系统中集合滤波数据同化自适应协方差膨胀方法的联系，具有重要的意义。