Extending Hilbert Space Operators

我们的研究方案涉及算子论和函数论在一个和多个变量中的相互作用。我们提出要解决的一组特殊问题涉及到单位圆盘上一些经典的矩和内插问题的几个复变量的推广，如Nevanlinna和Pick的插值定理，Caratheodory和Hergltz的矩定理。其他小组涉及几何函数论到解析簇的推广，以及到对称双圆盘的极值映射的理解。我们将承担的算子理论的内在研究包括涉及平面上非单连通区域上的一个变量的模型理论以及除双圆盘以外的区域上的多个变量的模型理论。算符理论函数论是我们提议研究的一种特殊类型的数学，它对人类努力的许多领域具有直接和具体的好处。例如，我们建议的模型理论方面都涉及到对交换提升结构的推广，这导致了一种从地球表面的声学数据中发现石油的有效算法。其他方面将会增加线性矩阵不等式的理论。线性矩阵-S的数学理论是线性规划的推广，它使大规模的资源分配和经济预测成为可能，目前在许多工程领域具有重要意义。最后，我们打算研究的函数理论的特定品牌，构成了H-in；&&64257；度控制理论的数学核心，该理论已被用于设计托卡马克内部聚变反应的控制系统和航天飞机的反馈稳定系统。

Our research proposal concerns the interplay between Operator Theory and Function Theory in one and several variables. A particular group of problems that we propose to attack involves the generalizations to several complex variables of some of the classical moment and interpolation problems on the unit disc such as the interpolation theorem of Nevanlinna and Pick, and the moment theorems of Caratheodory and Herglotz. Other groups involve the generalization of geometric function theory to analytic varieties and the understanding of extremal mappings into the symmetrized bidisc. Research intrinsic to operator theory that we will undertake includes issues involving model theory in one variable on nonsimply connected domains in the plane and in several variables on domains other than the bidisc. Operator Theoretic Function Theory, the particular type of mathematics that we are proposing to investigate, has direct and concrete benefits for a number of areas of human endeavor. For example, the model theory aspects of our proposal all involve the generalization of the Commutant Lifting Structure which leads to an efficient algorithm for the discovery of oil from acoustical data taken on the surface of the earth. Other aspects would add to the theory of Linear Matrix Inequalities. The mathematical theory of LMI?s, which currently is of great importance in several areas of engineering, is an extension of linear programming, which has made large scale resource allocation and economic prediction possible. Finally, the particular brand of function theory we propose to study, forms the mathematical core of H-infinity control theory, which has been used to design control systems for fusion reactions inside Tokamaks and feedback stabilization systems for the space shuttle.