多复变函数空间上的算子理论

项目名称： 多复变函数空间上的算子理论

项目编号： No.11271332

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 于涛

作者单位： 浙江师范大学

项目金额： 60万元

中文摘要： 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.

中文关键词： 函数空间；Toeplitz算子；对偶Toeplitz算子；不变子空间；约化子空间

英文摘要： 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 truncat

英文关键词： function space；Toeplitz operator；dual Toeplitz operator；invariant subspace；reducing subspace