调和函数空间上Toeplitz算子的相关问题

在Hardy空间和Bergman空间上的Toeplitz算子已经被广泛研究的基础上，由于理论和实际问题的需要，调和Bergman空间、调和Dirichlet空间、Fock空间以及调和Fock空间等更多更复杂的函数空间的Toeplitz算子的研究成为重要课题。由于对偶Toeplitz算子与Toeplitz算子的天然密切联系，研究Toeplitz算子的同时必然要研究对偶Toeplitz算子。本项目主要研究三方面的内容: 1.拟齐次函数为符号的Toeplitz算子的有限秩、交换性问题。2.调和Bergman(Dirichlet)空间上对偶Toeplitz算子的本质交换性等代数性质。3.调和Fock空间及其上Toeplitz算子的交换性、紧性。

Based on the extensive research of Toeplitz operators on the Hardy space and Bergman space, the study of Toeplitz operators on the harmonic Bergman space, harmonic Dirichlet space, Fock space , harmonic Fock space and other more complicated function spaces have become an important project for the theoretical and practical need. Due to the closed relationship betsween Toeplitz operators and dual Toeplitz operators, we must study dual Toeplitz operators when we characterize Toeplitz operators. The contents of this project are as follows: 1. Finite rank and commutativity for quasihomogeneous Toeplitz operators. 2. Essential commutativity and other properties of dual Toeplitz operators on the harmonic Bergman(Dirichlet) space. 3. Commutativity, Compactness of Toeplitz operators on harmonic Fock space.