【作者】 王伟；

【导师】 舒永录；

【作者基本信息】 重庆大学 ， 基础数学， 2017， 博士

【摘要】 从纯粹数学的观点来看,线性算子动力学与函数空间算子理论,复分析,算子代数以及矩阵论等领域有着密切的、深刻的联系;从应用数学的角度来看,线性算子动力学的研究成果又广泛的应用到微分方程、动力系统以及矩阵分析中.此外,线性算子动力学的研究成果在实际应用（如,混沌加密）中也有着极其重要的作用.我们将论文分成以下六章:第一章,介绍线性算子动力学的研究背景、研究现状和已经取得的重要成果.第二章,在第一部分主要介绍了超循环算子（组）、弱混合算子（组）、拓扑混合算子（组）、Devaney混沌算子（组）的基本概念以及它们各自的判别准则.第二部分介绍超循环半群的基本概念及相应的判别准则.第三章,在Feldman和Costakis所得到的研究结果的基础上,我们刻画了在有限维空间Cⁿ上的算子组的超循环性质,得到m（m>n）个上三角的Toeplitz矩阵构成的算子组是超循环的充分必要条件.进一步,我们把这一结果推广到更一般的情形.第四章,我们研究Hardy空间H²（D）,序列空间e^p（l≤p<∞）和c₀上的算子组的动力学性质.我们首次利用联合点谱得到了算子组的特征值准则,在此过程中我们发现一类介于弱混合和拓扑混合之间的新算子组,我们称之为S-mixing算子组,并刻画了算子组S-mixing与构成该算子组的若干个算子的复合算子之间的关系.另外,我进一步研究了算子组S-mixing性与混合性、弱混合性、超循环性以及混沌性之间的关系.第五章,我们主要研究双下标序列空间上（加权）后移位算子的动力学性质.该内容是受单个加权移位算子和双下标序列空间研究的启发,我们利用Feldman的思想,用半群作用替代单个算子,我们首次提出可以在双下标序列空间中研究算子组的动力学性质.首先我们把双下标序列按无限维矩阵的形式重排,在此基础上我们定义了向左和向上的两种移位B^u,B^e ,把它们统称为``backward"shift operators。根据超循环算子的定义,我们很容易判断B^u,B^e 都不肯能是超循环的,由此我们进一步考虑加权后移位算子的性质,根据可不可交换我们把这种研究分成两类,可交换情形与不可交换情形.针对可交换的情况,我们给出了算子λ₁B^u,λ₂B（e）是超循环,拓扑混合,Devaney混沌的充分必要条件的刻画.而对于不可交换的情况,算子(B_w^u,B_v^e)的研究比较麻烦,因为大多数情况下(B_w^u和B_v^e是不可交换的,其中w={w_i,j},v={v_i,j}是有界的正权序列.研究这种情况,我们给出连续路径的定义,在指定路径的前提下,我们给出算子(B_w^u,B_v^e)是超循环,拓扑混合的刻画.在此基础上,我们利用拟共轭映射保持算子的性质,把结果推广到加权的双下标序列空间中.第六章,我们总结全文并对本论文中不足进行了分析,进一步给出接下来需要研究的问题.更多还原

【Abstract】 From a purely mathematical point of view,the linear operator dynamics and function space operator theory,complex analysis,operator algebra and matrix theory,and other fields have close and profound link.From the perspective of applied mathematics,the research of linear operator dynamics is widely applied to the differential equation and dynamical system,matrix analysis.In addition,the kinetics of linear operator to promote the development of chaotic cryptography and other fields also played an important role.The dissertation will be divided into six chapters:In chaper 1,we mainly introduce the background,domestic and foreign research present situation,analysis of developing trend,and then statethe content of our dissertation.In chaper 2,in the first part,we introduce the definition of（tuple of）hypercyclic operators,（tuple of）weakly mixing operators,（tuple of）topological mixing operators,（tuple of）Devaney chaotic operators and their criterions,respectively.In the second part,we mainly introduce some basic notions on hypercyclic semigroup and their criterions.In chaper 3,based on the results of Feldman and Costakis,we mainly characterize the hypercyclicity of tuple operators on finite complex spacen Cⁿ,and obtain a sufficient and necessary condition for an m-tuple of commutative complex upper triangular Toeplitz matrices to be hypercyclic.In addition,we extend the result to more general case.In charpter 4,our main consideration is the dynamical properties of tuple of operators on Hardy spaceH²（D）,sequence space e^p（l≤p<∞） and c₀.Using joint point spectrum,we firstly obtain the eigenvalue criterion for tuple of operators.During the process,we find the new kind of operator that between weakly mixing and mixing,we call it S-mixing operator,and depict the relationship between the S-mixing operator and the composition operators of the several operator in the constitute the tuple of operators.In addition,we further study the relationship between the S-mixing properties and topological mixing,weakly mixing,hypercyclicity and chaos.In charpter 5,we mainly consider the dynamic properties of（weighted）backward shift operators on the double sequence spaces.The content is inspired by the research on single weighted shift operators and double sequence space,using the ideas of Feldman,replace the action of single operator by semigroup action,to promote the study of hypercyclicity of tuple of operators on the double sequence spaces.First,we rearrange the double sequence as the form of infinite dimensional matrix,based on this rearrangement,we define two shift operators B^u,B^e,one is the left shift and another up shift,and they are collectively referred to “backward” shift operators.Based on the definition of hypercyclic operator,it is clear that they are not hypercyclic,thus we further consider the nature of the weighted shift operators B_w^u,B_v^e,according to the commutativity,and the research is divided into two cases.One case they can exchange with each other.In the presence of exchangeable,we present the fully characterization on pair of operators(B_w^u,B_v^e)to be hypercyclic,mixing and chaotic.In the case of noncommutative,the study of the operator more troublesome,because most of the time and is noncommutative,w={w_i,j},v={v_i,j} are the right of bounded positive weight sequences.Research this kind of situation,we give the definition of a continuous path,on the premise of specified path,we obtain the characterization on the pair of operators (B_w^u,B_v^e) to be hypercylic and topological mixing.On this basis,using the properties of qusiconjugate mapping which keep the nature of operator,the results above can be generalized to weighted sequence space.In charpter 6,we summarize the full text and analyze the deficiency in this paper,and then give the further research questions.更多还原