Fock型空间上有界线性算子组不交动力学性态研究

The disjoint dynamical behaviors are mainly used to describe the dense status of generated orbit by self-iterations of bounded linear operator tuple, which especially play important role in revealing topological structure of the orbit. Based on the crossing theories of several complex variables function, linear dynamical system and functional analysis etc., the project is to explore the equivalent conditions for the disjoint hypercyclicity and disjoint supercyclicity of several bounded linear operator tuples acting on Fock-type space (an important Banach space). Furthermore, we will characterize the structural properties of their disjoint subspaces. Firstly, we will give sufficient and necessary descriptions for the disjoint dynamical behaviors of differentiation operator tuple in one and several variable Fock-type space, and illuminate its specific application in quantum theory. Secondly, we are going to characterize the disjoint dynamical properties of weighted shift operator tuple, weighted composition operator tuple and their adjoint tuples in one-variable Fock-type space, respectively. And then we will derive the results in several-variable spaces. Finally, we will investigate the topological structure and connected situation of their disjoint hypercyclic subspaces and disjoint supercyclic subspaces for above several bounded linear operator tuples. The results of this project could offer theoretical foundations for discovering disjoint dynamical properties of Toeplitz operator tuple and Volterra operator tuple, also reveal the dynamical tendency and evolution laws about several linear dynamical systems on generalized Banach spaces.

不交动力学性态主要用于描述有界线性算子组自身迭代生成轨道的稠密状态，尤其在揭示轨道的拓扑结构方面发挥着重要作用。本项目基于多复变函数论、线性动力系统和泛函分析等多学科交叉理论，探究一类重要巴拿赫空间--Fock型空间上几类不交超循环和不交亚超循环算子组的等价表示，并刻画它们不交子空间的结构性质。首先，给出单变量及多变量Fock型空间上求导算子组不交动力学性态的充要描述条件，阐明其在量子理论中的具体应用；其次，分别刻画单变量Fock型空间上加权移位算子组、加权复合算子组及它们对偶算子组的不交动力学性态，进而将结果延拓到多变量空间上；最后，研究上述几类有界线性算子组不交超循环和不交亚超循环子空间的拓扑结构和连通状况。该项目结果可为进一步探索Toeplitz算子组和Volterra算子组的不交动力学性态提供理论依据，同时对揭示广义巴拿赫空间上多个线性动力系统的运动趋势及演变规律具有深刻意义。

本项目基于多复变函数论、线性动力系统和泛函分析等多学科交叉理论，拟探究定义在整个复平面上一类重要巴拿赫空间--Fock型空间上几种经典线性算子组的静态学及动力学性态的等价表示，并描述它们（不交）子空间的结构。.鉴于上述目标，项目组成员开展了如下的一系列工作：(1)利用分析诱导函数的特点，构造特殊检验函数等技术手段，在多个解析函数空间上等价刻画了加权求导复合型算子、积分型算子及叠加算子等的有界性、紧性、本性范数，又借助函数级数展开式和Strling公式等工具，得到了相关线性算子差分性质的等价判别式；(2)从有界线性算子预解集上的非欧度量及能量泛函等理论出发，依次揭示了Hardy空间及L2[0,1]空间上拟幂零的Volterra算子不变子空间结构与Power-set之间的关联性，进而在Hardy空间上证明了由几类重要诱导符号定义的Toeplitz算子有限秩扰动的核空间都是具有有限defect的几乎不变子空间；(3)在赋予紧开拓扑的H(Ω)空间上推导出了加权复合算子组不交亚超循环性的充分描述条件，并在一般可分Hilbert空间上给出了乘法算子共轭组的不交亚超循环性的判别式，同时反证了可分的小Bloch空间和Besov空间上不存在由自同构符号诱导的超循环复合算子；尤其在广义无穷阶数可分Fock空间上阐明了单个求导算子的动力学性态与有限个递增阶数求导算子组成的算子组不交动力学性态的等价关系；(4)在四元数Fock空间上初次定义了四元数值左（右）加权复合算子，并详细描述了它们的几个经典性质，为量子力学、量子理论等学科的研究提供了一定的理论铺垫。随着研究的深入，研究内容在原计划上有所拓展，在国内外数学杂志上共发表了相关的SCI收录论文14篇。本项目的研究结果对于探究更多广义解析函数空间上线性算子的性态具有促进作用，也可用于揭示一般巴拿赫空间上多个线性动力系统的运动趋势及演变规律。