Extending Hilbert Space Operators

【摘要】在谱定理之后，很难想到一个定理能比谱定理对算子理论的发展及其在数学和科学上的无数应用产生更深远的影响。——纳吉膨胀定理。在一个特殊的算子类中，将一个一般算子表示为该类中一个好的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，主要关注在一个和多个复变量函数理论中的应用问题。我们提出要解决的一组特殊问题涉及到对单位圆盘上的一些经典矩和插值问题的几个复变量的推广，例如Nevanlinna和Pick的插值定理以及Caratheodory和Herglotz的矩定理。我们项目的另一个重点将是使用算子理论方法来研究解析变量的函数理论。我们将进行算子理论固有的研究，包括平面上非单连通域上的一个变量模型理论和双曲面以外的域上的几个变量模型理论。这将包括我们在对称双曲面上的工作的继续，特别是在复杂几何和mu合成问题上的应用。算子理论，这个我们打算发展的特殊类型的数学，对人类努力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换向举升结构的泛化，这导致了从地球表面的声学数据中发现石油的有效算法。其他方面将丰富线性矩阵不等式的理论。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，而且使对经济市场的准确预测成为可能。最后，我们提出要研究的函数理论的特定分支，形成了h -∞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。

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.