线性算子动力系统的研究进展  

Progress in research on dynamics of linear operators

作　　者：张亮[1] 周泽华[2] 

机构地区：[1]天津大学海洋科学与技术学院,天津300072 [2]天津大学数学系,天津300072

出　　处：《中国科学：数学》2015年第11期1811-1832,共22页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11371276;11301373和11401426)资助项目

摘　　要：线性算子动力系统主要研究线性算子的超循环性、混沌性、混合性等动力学性质,它与复分析、算子理论、拓扑理论、微分几何等学科有着重要的联系,有广泛的应用范围.作用在无穷维空间上的某些线性算子有着有趣的动力学性质.特别地,超循环性是无穷维空间情形下的性质,即算子迭代形成的轨道能形成稠密的子空间.一个局部凸的完备度量空间存在超循环算子的充分必要条件是空间可分且是无穷维的.近几十年来,线性算子动力系统的研究成为非常活跃的领域,并有了许多精彩的研究成果.本文将对线性算子动力系统的研究内容进行系统的梳理,并对近年来关于线性算子动力性质方面的精彩研究成果作简要的回顾和总结,其中也包括本课题组近年来关于此方向的研究结论.The research of dynamics of linear operators mainly involves hypercyclic, chaotic, mixing properties and so on. It has close links with complex analysis, theory of operator and differential geometry, with a wide range of applications. Some linear operators on infinite dimensional spaces can display interesting dynamical properties. In particular, hypercyclicity is an essentially infinite dimensional property, when iterations of the operator generate a dense subspace. A local convex complete metric space admits a hypercyclic operator if and only if it is separable and infinite dimensional. Over more than two decades, the study of dynamics of linear operators has turned into a very active research area and many fascinating research results have been given. In this paper, we will systematically summarize the contents of dynamics of linear operators and will give a brief review of the recent research results about wonderful dynamical properties of linear operators, among which related conclusions of our research group are involved.

关 键 词：完备度量空间 线性算子 超循环性 

分 类 号：O177[理学—数学]