Extending Hilbert Space Operators

艾格勒。摘要：在谱定理之后，很难想到还有什么定理比Sz.-Nagy膨胀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更深远的影响。将指定的算子类中的一般算子表示为该类中良好算子的一部分的想法已经取得了许多成功，我们试图发展这一观点，主要集中在矩阵理论、算子理论以及一元多复变量函数理论中尚未解决的问题上。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。另一组涉及在比经典Hardy空间更一般的空间上推导Adamyan、Arov和Krein定理的类比。我们将从事的算子理论的内在研究包括平面内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。算符理论是我们提议研究的特殊类型的数学，它对人类工作的许多领域都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。线性规划是线性规划的一种扩展，它使大规模的资源分配和经济预测成为可能，目前流行于多个工程领域。最后，我们提出要研究的函数理论构成了最近发展起来的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的聚变反应控制系统和航天飞机的反馈稳定系统。

Agler. Abs Proposal: DMS-9801461 Principal Investigator: Jim Agler 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 myriad applications to mathematics and science than the Sz.-Nagy Dilation Theorem. The idea of representing a general operator in a specified 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 unsolved problems in matrix theory, operator theory, and the theories of functions in one and several complex variables. A particular group of problems that we propose to attack involve 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 con- nected 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 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. LMIs, which currently are all the rage in several areas of engineering, are an extension of linear programming, which has made large scale resource allocation and economic prediction possible. Finally, the particular brand of functi on 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.