解析函数空间上Hankel算子与Toeplitz算子乘积的有界性研究

本项目研究Hankel乘积以及Haplitz乘积在某些解析函数空间上的有界性。针对解析函数空间Fock和Fock-Sobolev空间，首先借助于再生核Hilbert空间中核函数的特殊性质，建立某些符号函数空间下Toeplitz算子交换性的充分必要条件，分析Hankel算子的零乘积性质；然后再生核函数和符号函数表示成级数形式时，建立Hankel乘积作用在正规正交基下的基表示，分析各项系数与符号函数本身的关系，研究如何用符号函数的性质刻画Hankel乘积的有界性；最后讨论加权的Fock-Sobolev空间上Haplitz乘积的有界性，针对一般的解析符号，研究这类Fock型空间的函数论性质，建立函数的逐点估计、空间的包含关系、Hankel算子的范数估计以及辅助函数有界性的充分必要条件。本项目将揭示算子理论与函数空间理论的研究相互交织相互推动，项目成果将为发展泛函分析领域的重要方法提供理论基础。

This proposal investigates the boundedness of Hankel products and Haplitz products on some analytic function spaces. For the Fock and Fock-Sobolev spaces, firstly, sufficient and necessary conditions of the commuting Toeplitz operators under some symbolic function spaces will be established, using the properties of reproducing kernel functions. The zero-product property of Hankel products will be analyzed. Then when the kernel functions and symbol functions are expressed in series form, the basis representation of Hankel products under the normal orthogonal basis will be proposed. Analyzing the relationship between the coefficients and the symbolic functions, we will identify how to characterize the boundedness of Hankel products by means of the symbolic functions. Finally, the boundedness of Haplitz products under general symbol functions on the weighted Fock-Sobolev spaces will be discussed. The function theoretic property for the Fock type spaces will be studied and the results will be established for pointwise estimation of functions, inclusion relationship of the Fock-Sobolev spaces, norm estimation for Hankel operators, necessary and sufficient conditions for the boundedness of auxiliary functions. This proposal will disclose the operator theory and the function space theory mutually interweave and promote each other. The research results of the proposal will provide theoretical foundations for the development of important methods in the field of functional analysis.