多复变全纯函数空间上复合算子的谱理论研究

In this project, we would like to investigate the spectra of composition operators and related problems. From the composition operators on the unit disk, we are trying to discuss some topics of (weighted) composition operators on some holomorphic functions, such as the norms, essential norms, spectral radii, essential spectral radii, adjoints, etc. of these operators. We would like to focus on the spectra of composition operators on different holomorphic function spaces of the unit ball, especially the linear fractional composition operators on some Hilbert spaces of holomorphic functions.The main tool is the linear fractional model of the unit ball. Using this model, we want to study the eigenvectors of composition operators on function spaces, and characterize the spectra of different classes of linear fractional composition operators. Furthermore, we would like to get the complete characterization of spectra of all kinds of composition operators. More than above, we would like to explore the normality, cyclicity, invariant subspaces of composition operators, since these characterization are closely related to the spectral characterization.

本课题拟研究全纯函数空间上复合算子谱理论及相关问题.拟从单位圆盘上复合算子出发,首先拟讨论一些全纯函数空间上复合算子与加权复合算子的范数,本性范数,谱半径,本性谱半径,伴随算子等相关问题.着重考虑单位球上复合算子在不同的全纯函数空间上谱,特别是分式线性复合算子在一些Hilbert函数空间上的谱.主要通过单位球上分式线性模型的研究和对函数空间中复合算子的特征向量的讨论,拟给出一些全纯函数空间上几种不同类的分式线性复合算子的谱的刻画,进一步得到完整的不同类型复合算子的谱.并借助复合算子谱的性质,给出其相关的正规性,循环性,不变子空间等方面的刻画.

本项目属于多复变函数论与算子理论领域。主要研究了单位圆和球上一些全纯函数空间和一些相关的算子理论，包括：Dirichlet-型空间上复合算子的紧性；Bloch型函数到全纯 F(p,q,s) 空间的距离；单位球上一类保持 s-Carleson 测度的积分算子等。共完成论文4篇，其中一篇被 Acta Mathematica Sinica, English Series 接收，另三篇均投稿于SCI检索的杂志。