多复变函数空间上的算子理论

This project is mainly devoted to explore some issues in the operator theory on spaces of analytic fuctions in several complex variables. The spaces we focus are Hardy spaces, Bergman spaces and Dirichlet spaces over the unit polydisc and the unit ball in higher dimensional complex Ecleaden space. Firstly, we consider some algebraic properties of Toeplitz operators, such as commutativity, semicommutativity, commutativity model finite rank operators, and the structure of the commutator ideal of the Toeplitz algebra. Secondly, the structure of the invariant subspaces under certain Toeplitz operators is taken into account. In this line, we consider whether or not such invariant subspace can be generated by its wandering subspace, and how to represent it as an invariant subspace of a space of vector-valued analytic functions such that the dimension of the wandering subspace equals to its fiber dimension. Based on this representation, we will study the structure of invariant subspaces by using some algebraic invariant as fiber dimension and Samuel multiplicity. The unitary equivalence of reductive subspaces is an important issue in operator theory. Motivated by the pre-existing results, we will study the corresponding problem under the multiplication of rational functions. Lastly, some basic properties of the truncated Toeplitz are discussed. Meanwhile, we consider the relationship between the isomorphism of the truncated Toeplitz algebras and the unitary equivalence of the corresponding invariant subspaces.

本项目主要研究多复变解析函数空间上算子理论若干问题.我们主要关注高维复欧式空间中多圆盘和单位球上的Hardy空间、Bergman空间和Dirichlet空间上的问题.首先，研究Toeplitz算子的代数性质，如交换性、半交换性、模有限秩算子交换和换位代数，Toeplitz代数的换位子理想的结构等.其次，研究某些Toeplitz算子的不变子空间的结构，考虑其是否可由游荡子空间生成，如何将其表示为向量值解析函数空间的不变子空间，满足游荡子空间的维数等于纤维维数，并通过这种表示使用纤维维数、Samuel重数等代数方法讨论不变子空间的结构.约化子空间的酉等价是算子论的一个研究重点，我们将在前人的基础上研究有理函数符号乘法算子的约化子空间的酉等价.最后，我们讨论截断Toeplitz算子的一些基本性质，同时讨论截断Toeplitz代数的同构与对应不变子空间酉等价的关系.

本项目主要研究多变量解析函数空间上Toeplitz算子、对偶Toeplitz算子、截断Toeplitz算子和复合算子的交换性、约化性等代数性质，以及有界性和紧性等分析性质；研究Samuel重数、纤维维数等不变量及解析函数空间的不变子空间结构等问题。.(1)研究了双圆盘加权Bergman空间上以测度为符号的Toeplitz算子的有界性和紧性, 使用对数型Bloch空间概念给出一个充要条件。.(2)讨论了Dirichlet空间上有限Blaschke乘子的约化性及其与多重加权移位的酉等价, 给出了判别条件, 并给出在二重Blaschke乘子的应用。.(3)研究Hilbert空间上Fredholm算子的Samuel重数, 用算子论语言描述了Samuel重数, 讨论了Fredholm算子的结构。.(4) 将Cantor极小系统的弱逼近共轭概念推广到原代数为AF-代数的C*-代数动力系统上来, 讨论了原代数上具有Rokhlin性质的*-自同构之间弱逼近共轭与相应交叉积上*-自同构之间的关系。.(5)研究了多圆盘Hardy空间上Toeplitz 算子的代数性质和单位圆盘和单位球调和Dirichlet空间的对偶Toeplitz算子的代数性质和谱理论. .(6)研究了向量值解析函数空间的不变子空间的纤维维数,研究满足纤维维数等于余维数的一类不变子空间的结构, 得到一个纤维维数和Samuel重数的加法公式。.(7)研究了 Bloch型空间和加权Bergman空间之间加权复合算子的本性范数，单位圆盘上Bloch型空间之间加权复合算子与高阶微分算子的乘积C_φ D_m的本性范数。