解析函数空间上复合算子的缠绕关系和紧差分

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 differences and contact between one complex variable and several complex variables, together with the contact between function spaces.

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。

本项目致力于各种解析函数空间上的复合算子与积分算子之间的缠绕性质与拓扑结构的研究，属于多复变函数论及算子理论中的前沿热点课题。我们研究了单位圆盘上Bergman空间、Bloch空间、有界全纯函数空间、F(p,q,s)空间上的复合算子与Volterra算子之间的缠绕关系和紧缠绕关系，Carleson测度等问题。我们还研究了在单位球上F(p,q,s)空间、Hibert-Hardy空间、Hilbert-Bergman空间等全纯函数空间上的积分算子、Carleson测度及加权复合算子的拓扑连通性等问题。我们得到了如下4个方面主要结果：1. 完整刻画单位圆盘上Bergman空间、Bloch空间、有界全纯函数空间之间的复合算子、Volterra算子的紧缠绕关系，得到了本性换位代数中包含全体有界复合算子的Volterra算子的符号函数集合其实就是某类加权的小Bloch空间。我们还给出了本性换位代数中包含全体有界Volterra算子的复合算子的一些充分性和必要性的刻画。2. 在单位圆盘上的F(p,q,s)空间上推广了Peter-Jones的距离公式。3. 研究了单位球上某些Hilbert解析函数空间上的复合函数的拓扑结构，并进一步得到该拓扑空间为单连通的。4. 在单位球上的F(p,q,s)空间上研究了Carleson测度，并推广了Peter-Jones的距离公式。本项目进一步从算子理论方面揭示了单复变与多复变之间的差异与联系，以及揭示了各种函数空间的联系。