Operator Theory on the Bergman Space and the Hardy Space

郑德超将对函数论和算子论相互作用所产生的问题进行研究。重点研究Bergman空间和哈代空间上的Hankel算子和Toeplitz算子。 要考虑的主题包括Hankel算子的紧扰动、圆盘上的函数代数和Toeplitz代数以及Bergman空间上乘法算子的约化子空间。该项目的重点是建立这些运营商的基本属性和他们的符号的分析和几何性质之间的关系的中心问题。运营商理论是数学的一部分，研究矩阵的无限维推广。特别是，当限制在有限维子空间时，算子具有通常的线性性质，因此可以用矩阵表示。算子理论的中心问题是对满足附加条件的算子进行分类，这些附加条件是根据相伴算子或根据底层空间给出的。算子理论是许多数学的基础，也是数学在其他科学中的许多应用的基础。

Dechao Zheng will conduct research on problems arising from the interaction between function theory and operator theory. Primary emphasis will rest on the study of Hankel operators and Toeplitz operators on the Bergman space and the Hardy space. The topics to be considered include compact perturbation of Hankel operators, function algebras and Toeplitz algebras on the disk, and reducing subspaces for multiplication operators on the Bergman space. This project focuses on the central problem of establishing relationship between the fundamental properties of those operators and analytic and geometric properties of their symbols.Operator theory is that part of mathematics that studies the infinite dimensional generalizations of matrices. In particular, when restricted to finite dimensional subspaces, an operator has the usual linear properties, and thus can be represented by a matrix. The central problems in operator theory is to classify operators satisfying additional conditions given in terms of associated operators or in terms of the underlying space. Operator theory underlies much of mathematics, and many of the applications of mathematics to other sciences.