局部α次积分C半群的谱映射定理  

Spectral Mapping Theorems for Local Times Integrated Semigroups

作　　者：秦喜梅[1] Qin Ximei(School of Mathematics and Big Data,Chaohu University,Hefei,Anhui,238024)

机构地区：[1]巢湖学院数学与大数据学院,安徽合肥238024

出　　处：《绥化学院学报》2023年第8期148-152,共5页Journal of Suihua University

基　　金：安徽省高校自然科学研究重点项目“双连续α次积分C半群的概率逼近和抽象柯西问题研究”(KJ2021A1032);巢湖学院校级教学研究重点项目“OBE理念下《实变函数》课程教学模式探索与实践”(ch21jxyj02)

摘　　要：线性算子的谱问题是算子半群理论中的一个重要组成部分,它被广泛应用于数学和物理学的许多分支,如矩阵理论、微分方程、积分方程、控制论和量子力学等。而如何利用生成元的特性来研究生成元的谱与算子的谱之间的关系,从而更好地理解该半群的整体特征和结构:压缩性、稳定性、渐近行为等,这些都是算子半群理论讨论的经典话题。因此对每一个半群,它的谱映射定理都是算子半群理论中研究的重要内容,对分析与理解算子半群性质起着重要作用。受强连续半群谱理论的启发,文章给出了局部α次积分C半群生成元的谱与半群的谱之间的关系,详细讨论了生成元的点谱、连续谱和剩余谱与局部α次积分C半群的点谱、连续谱和剩余谱之间的包含关系,丰富了算子半群的谱理论。The spectral problem of linear operators is an important part of the theory of operator semigroups,which is widely used in many branches of mathematics and physics,such as matrix theory,differential equations,integral equations,cybernetics and quantum mechanics.And how to use the characteristics of generator to study the relationship between the spectrum of generator and the spectrum of operators,so as to better understand the overall characteristics and structure of this semigroup:compressibility,stability,asymptotic behavior,etc.These are classic topics discussed in the theory of operator semigroups.Therefore,for each semigroup,its spectral mapping theorem is an important element in the study of operator semigroup theory and plays an important role in the analysis and understanding of the properties of operator semigroups.Inspired by the spectral theory of strongly continuous semigroups,this paper first gives the relationship between the spectrum of the generator of a locally integrated semigroup and the spectrum of the semigroup,and then discusses in detail the inclusion relations between the point spectrum,continuous spectrum and residual spectrum of the generator and the point spectrum,continuous spectrum and residual spectrum of the locally integrating semigroup,which enrich the spectral theory of operator semigroups.

关 键 词：局部α次积分C半群 谱映射定理 生成元 

分 类 号：O177.7[理学—数学]