Bergman型空间上的Toeplitz算子和Hankel算子

Toeplitz operators and Hankel operators are very important content in operator theory and operator algebra, they are not only tightly connected with many branches of mathematics,such as function theory and differential equations, but also broadly applied in control theory, quantum mechanics and electronic information, etc. So the research of Toeplitz and Hankel operators has an important value for theory and a great potential for application. Nowadays, the mainly techniques in studying the properties of Toeplitz and Hankel operators on Bergman-type spaces are from the function theory. It follows that the essential difference between the functions in different spaces may cover the possible similarity between the operators structures. In this project, we will study the properties of the Toeplitz operators and the Hankel operators on the Bergman-type space with symmetric measures. Combining the matrix theory, we will use some related sequences to characterize when the operators are bounded or compact, and characterize some algebra properties. The Hardy space and Bergman spaces on many areas are all Bergman-type space with some special symmetric measure. Then we can systematically analyze the properties of the operators on different space with the similar symbols, and find the real reasons for the changes of operators properties. By this way, we can know more about the structures of the Toeplitz and Hankel operators. On the basis of above research, we also will explore the new applications of Toeplitz and Hankel operators in other disciplines.

Toeplitz算子和Hankel算子是算子理论和算子代数中非常重要的内容,与函数论、微分方程等数学分支联系紧密，其若干性质在控制论、量子力学、电子信息等领域也具有许多应用，因此研究该理论具有重要的理论价值与应用前景.目前，Bergman型空间上对这两种算子的研究主要是利用函数论的方法,但在有些研究过程中,不同空间上函数的本质差别会掩盖算子结构可能存在的相同性.本项目中,我们将在对称测度诱导的Bergman型空间上,结合矩阵理论,利用数列刻画Toeplitz算子和Hankel算子的有界性、紧性及代数等方面的性质.这样可以将Hardy空间、多种域Bergman空间上的算子理论放到统一框架下,系统分析同一函数在不同空间上诱导算子的性质,找到空间变化影响算子性质的本质原因,由此加深对Toeplitz和Hankel算子结构的认识.在此基础上,我们也将探索这两种算子在其他领域中的应用.

解析函数空间上的Toeplitz算子和Hankel算子是算子理论中重要的组成部分，其研究工具具有多样性，其性质具有广泛的代表性和应用性. 近些年，为满足理论发展和实际应用的需求，定义在高维空间有界区域或者无界区域上的Toeplitz算子的性质受到越来越多的关注. . 本项目主要研究多圆盘加权Bergman空间、单位球Bergman空间和Fock空间等Bergman型空间上Toeplitz算子、块Toeplitz算子的性质和Hankel算子的性质. 主要方法是将底空间维数和测度等方面的信息转化为相关数列的性质，再结合复变函数、代数和算子理论等方面的技巧进行研究，进而给出Toeplitz算子相关性质的刻画. 特别地，我们给出了在Toeplitz算子亚正规性和约化子空间等代数方面性质的刻画. 该方法还适用于对多种域Bergman空间上以拟其次多项式函数为符号的Toeplitz算子的相关性质的研究，从一个侧面揭示出底空间的维数及测度对算子相关性质的影响. 通过本项目的研究， 我们首次在多圆盘加权Bergman空间上刻画了一些非解析函数诱导的Toeplitz算子的约化子空间方面的性质.. 此外，应用方面，我们在鲁棒控制器的设计问题中，突破了应用系统本身素分解的限制，只考虑一个系统的素分解，就能得到一类系统的控制器的参数化.