Operator Theory on the Bergman Space and the Hardy Space

郑德超将对函数论与算子论相互作用产生的问题进行研究。主要重点在于研究Bergman空间和Hardy空间上的Hankel算子和Toeplitz算子。所要考虑的主题包括圆盘上的Hankel算子、函数代数和Toeplitz代数的紧扰动，以及Bergman空间上乘法算子的约化子空间。这个项目集中于建立这些算子的基本性质与其符号的解析和几何性质之间的关系的中心问题。算子理论是研究矩阵的无限维广义的数学的一部分。特别地，当被限制在有限维子空间时，算子具有通常的线性性质，因此可以用矩阵来表示。算子理论中的核心问题是对满足附加条件的算子按关联算子或基础空间进行分类。算符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。

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.