复合算子的拓扑连通性

This project is dedicated to the topological connection of space of composition operator or other similar operators (for example，weighted composition operator and integral type operator). The issue what we will deal with belongs to the field of several complex variables and operator theory, and it is a hot topic which is studied by domestic and international mathematics researchers. First, we research the boundedness and compactness of difference of operators, which defined on different holomorphic function spaces in the unit disk, unit ball or unit polydisk, and give the the estimates of operator norm and essential morm. Second, according to these properties, we characterize the topological structure of space of bounded operators, including the problems of connection and isolated point.

本项目主要致力于研究定义在各种解析函数空间上复合算子及其相关算子（如加权复合算子、积分型复合算子）所构成的算子空间的拓扑连通性，属于多复变函数论与算子理论领域，是目前国内外数学工作者研究的前沿热点课题。我们将首先探讨单位圆盘、单位球、单位多元柱上各种不同经典函数空间上的算子差分的有界性与紧致性问题，并给出算子差分范数与本性范数的上下界估计；其次利用算子差分的性质对算子空间本身的拓扑结构进行研究，包括连通性、孤立点等问题。