算子空间上与谱，局部谱以及零斜Lie积相关的完全保持问题研究

项目名称： 算子空间上与谱，局部谱以及零斜Lie积相关的完全保持问题研究

项目编号： No.11501401

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 黄丽

作者单位： 太原科技大学

项目金额： 16万元

中文摘要： 谱和局部谱是具有重要理论价值和应用价值的算子代数的特征，算子空间上与谱相关的完全保持问题与著名的Kaplansky's问题密切相关。本申请项目拟在算子空间框架下，利用全纯函数理论和二阶算子矩阵的性质，研究算子代数上压缩乘积谱及谱函数的映射，进一步刻画算子代数上完全压缩谱及谱函数的映射的结构，研究算子代数上完全保持局部谱，局部可逆性以及零斜Lie积的映射，并将上述完全保持问题推广到其它代数上，所研究的映射一般为非线性映射。通过本项目的研究，发掘二阶算子矩阵和算子之间相关性质的内在联系，寻找同构的完全不变量，对于认清算子代数和算子空间的结构性质有重要意义，为完全保持问题研究的全面开展和Kaplansky's问题的解决提供一些理论依据和应用基础。

中文关键词： 谱理论；局部谱；算子矩阵；不变量；保持问题

英文摘要： The spectrum and the local spetrum, as the characteristics of operator algebras, have important theoretical values and application values. Complete preserver problems about the spectrum are closely related to the well-known Kaplansky 's problem. Under the framework of operator spaces, we make use of holomorphic function theory and properties of second-moment operator matrix, and research the maps compressing the spectrum or spectral functions of operator products on operator algebras, and further give the characterization of maps completely compressing the spectrum and spectral functions on operator algebras. Then we go on researching the characterizations of maps completely preserving the local spectrum, the local invertibility, and zero skew Lie products, etc., and generalize the relevant complete preserver problems to other algebras. Note that the above maps are generally nonlinear maps. Through the research of this project, we try to discover the internal relation of properties of the second-moment operator matrices and operator, and to find out the complete invariants of isomorphisms. And this project is very important to recognizing the structure of operator algebras and operator spaces, and provides some theoretical basis and the application foundation for the total development of complete preserver problems and the solutions of the Kaplansky 's problem.

英文关键词： the spectral theory;the local spetrum;operator matrices;invariants;preserver problems