Extending Hilbert Space Operators

摘要 在谱定理之后，很难想象有一个定理对算子理论的发展及其在数学和科学中的无数应用产生了比Sz定理更深刻的影响。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在理论中的问题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。另一组涉及推导类似的定理阿达米扬，Arov，和克莱因的空间更一般比经典的哈代空间。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。算子理论是我们打算研究的一种特殊类型的数学，它对人类奋进的许多领域都有直接和具体的好处。例如，我们的建议的模型理论方面都涉及到推广的换位提升结构，导致一个有效的算法，从地球表面上的声学数据发现石油。其他方面将添加到理论的线性矩阵不等式。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理论的特定分支，形成了最近发展的H-无穷控制理论的数学核心，该理论已被用于设计托卡马克内聚变反应的控制系统和航天飞机的反馈稳定系统。

ABSTRACT After 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 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 group involves deriving analogs of the theorem of Adamyan, Arov, and Krein on spaces more general than the classical Hardy space. 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 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 effcient 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. 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 recently developed 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.