Extending Hilbert Space Operators

在谱定理之后，很难再想到一个对算子理论的发展及其在数学和科学中的无数应用有着比Sz. Nagy膨胀定理的想法表示一个一般的运营商在一个专门的一类运营商的一部分，一个很好的运营商在类已经取得了许多成功，我们寻求发展这一观点的主要重点是应用问题的理论中的函数在一个和几个复杂的变量。一组特殊的问题，我们建议攻击涉及的推广到几个复杂的变量的一些经典的时刻和插值问题的单位光盘，如插值定理的Nevanlinna和挑选和矩定理的Caratheodory和Herglotz。我们的项目的另一个重点将是使用算子理论的方法来研究函数理论的分析品种。研究固有的算子理论，我们将进行包括模型理论的问题，在一个变量上的非单连通域的平面和几个变量上的域以外的bidisc。这将包括继续我们的工作对称bidisc与一个特别注重应用复杂的几何形状和μ-synthesis problem.Operator Theory，特定类型的数学，我们建议发展，有直接和具体的好处，为一些领域的人类奋进。例如，我们的建议的模型理论方面都涉及到推广的换向提升结构，这导致了一个有效的算法，发现石油从声学数据在地球表面上。其他方面也将丰富线性矩阵不等式理论。线性矩阵不等式（LMI）是线性规划的一种扩展，它不仅使大规模资源配置的优化成为可能，而且还使经济市场的准确预测成为可能。最后，我们建议研究的函数论的特定分支，形成了H ∞控制理论的数学核心，它已被用于设计托卡马克内的聚变反应控制系统和航天飞机的反馈稳定系统。

ABSTRACTAglerAfter the spectral theorem it is difficult to think of a theorem that has had a more profound effect on the development of operator theory and its myriads of applications to mathematics and science than the Sz.-Nagy Dilation Theorem. The idea of representing a general operator in a specialized class of operators as a part of a nice operator in the class has had many successes and we seek to develop this point of view with a primary focus on applications to problems in the theory of functions in one and several complex 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. Another focus of our project will be the use of operator-theoretic methods to study function theory on analytic varieties. 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. This will include the continuation of our work on the symmetrized bidisc with a particular focus on applications to complex geometry and the mu-synthesis problem.Operator Theory, the particular type of mathematics that we are proposing to develop, 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 enrich the theory of Linear Matrix Inequalities. LMI 's, which currently are all the rage in several areas of engineering, are an extension of linear programming, a mathematics which has made possible not only the optimization of large scale resource allocation but the accurate prediction of economic markets as well. Finally, the particular branch of function theory we propose to study, forms the mathematical core of the 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.