解析函数空间上复合算子的缠绕关系和紧差分

项目名称： 解析函数空间上复合算子的缠绕关系和紧差分

项目编号： No.11301132

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 仝策中

作者单位： 河北工业大学

项目金额： 22万元

中文摘要： 本项目致力于各种解析函数空间上的复合算子及相关算子之间的缠绕性质与紧差分的研究，属于多复变函数论及算子理论中的前沿热点课题。我们将研究单复变量和多复变量Hardy空间、Bergman空间、Bloch空间等函数空间上的复合算子、Toeplitz算子、Volterra算子等相关算子之间的缠绕关系与差分紧性，并且研究它们的广义特征值和谱性质。我们还将研究在紧扰动下上述算子之间的缠绕关系，刻画这些算子之间的本性可交换性，而后将重点刻画其本性换位代数中包含全体有界复合算子的相关算子类，即找出与任意有界复合算子均本性可交换的Toeplitz算子、Volterra算子等其他相关算子。我们还将刻画其本性换位代数中包含全体有界Toeplitz算子或Volterra算子等的复合算子类。本项目将通过研究这些常见的算子之间的缠绕关系和紧差分进一步揭示单复变和多复变的差异与联系，以及揭示各种函数空间之间的联系。

中文关键词： 解析函数空间；复合算子；缠绕关系；拓扑结构；Carleson测度

英文摘要： This project is dedicated to the research of intertwining relations and compact differences for composition operators and related operators, which is a hot topic in several complex variables and operator theory. In one complex variable and several complex variables, the intertwining relations and compact differences of composition operators, Toeplitz operators, Volterra operators and other related operators on Hardy spaces, Bergman spaces, Bloch spaces etc. will be studied first, and their extended eigenvalues and spectrum will be studied. We will also investigate the intertwining relations for those operators under a compact perturbation, and characterize the essential commutativity for those operators. We will focus on determining the class of those related operators, whose essential commutant contains all the bounded composition operators, that is, to determine the class of Toeplitz operators, Volterra operators and other related operators which can commute all the bounded composition operators essentially. We will also determine the class of composition operators, whose essential commutant contains all the Toeplitz operators or Volterra operators or other related operators. Through the research of intertwining relations and compact differences between those operators, this project aims to further reveal the

英文关键词： analytic function space；composition operator；intertwining relation；topological structure；Carleson measure