Krein Space Operator Theory and Applications

Rovnyak。摘要提案：DMS-9801016首席研究员：James L. Rovnyak算子理论和函数理论的主题将在涉及Krein空间和不定内积（相对于Hilbert空间）的几个方向上进行。该项目将继续之前关于广义舒尔函数和奈万林纳函数理论的工作。继Krein和Langer之后，它们被定义为亚纯函数，其相关的Pick-Nevanlinna核具有有限个负平方。其中一个主题是插值理论。在确定的情况下，Rosenblum和Rovnyak给出了包含Loewner型问题的边界插值理论，现在将研究广义Schur函数的类似问题。一个统一的工具是广义舒尔函数的规范共等距、等距和幺正实现理论。本文还将研究状态空间结构和差商变换的相关问题。该理论以标量值函数为最具体的形式，Sarason和他的同事在特定情况下的工作为发展指明了其他方向。对克莱恩空间算子的相关问题也将进行研究。该项目将研究几个扩展算子理论和函数理论中的经典问题的主题。这项工作是数学的，但主题是在支持数学工程和线性系统理论的领域。其中一个主题是插值理论，当只有一些值（问题的数据）已知时，需要重构具有规定性质的函数。在线性系统理论中，一个类似的问题是重构一个具有规定输入输出特性的系统；一些状态被输入到系统中，系统（本身由一个未知状态描述）对它起作用，一个新的状态被输出。在经典理论中，状态之间的距离是“确定的”，也就是说，在通常意义上的欧几里得空间中是正的。在数学模型中，距离是“不确定的”，也就是说，有时是正的，有时是负的，这种模型已经存在很长时间了，比如在相对论中。事实上，线性系统理论的数学是插值理论的有力工具，并将在项目中使用。该项目使用“不确定”来代替“确定”的距离概念，并试图探索当经典理论不适用时出现的新现象。要研究的问题包括解的存在性、唯一性、显式构造和状态空间的性质。预算包括支持一名本科生每年在REU暑期研究项目。

Rovnyak.Abs Abstract Proposal: DMS-9801016 Principal Investigator: James L. Rovnyak Topics in operator theory and function theory will be pursued in several directions that involve Krein spaces and indefinite inner products (as opposed to Hilbert spaces). The project will continue previous work on the theory of generalized Schur and Nevanlinna functions. Following Krein and Langer, these are defined as meromorphic functions whose associated Pick-Nevanlinna kernels have a finite number of negative squares. One topic is interpolation theory. In the definite case, a theory of boundary interpolation including problems of Loewner type was given by Rosenblum and Rovnyak, and analogous problems will now be studied for generalized Schur functions. A unifying tool is the theory of canonical coisometric, isometric, and unitary realizations for generalized Schur functions. Related problems on the structure of state spaces and the difference-quotient transformation will also be studied. The theory takes its most concrete form for scalar-valued functions, and work of Sarason and his colleagues in the definite case suggest other directions for development. Related problems for Krein space operators will also be investigated. The project will study several topics that extend classical problems in operator theory and function theory. The work is mathematical, but the topics are in areas that support mathematical engineering and the theory of linear systems. One topic is interpolation theory, where it is required to reconstruct a function with prescribed properties when only some of its values, the data of the problem, are known. In linear system theory, a similar problem is to reconstruct a system with prescribed input-output characteristics; some state is input into the system, the system (itself described by an unknown state) acts upon it, and a new state is output. In the classical theory, distances between states are "definite," that is, positive in the usual sense of Euclidean space. Mathem atical models in which distances are "indefinite," that is, sometimes positive and sometimes negative, have been known for a long time and occur, for example, in relativity theory. In fact, the mathematics of linear system theory is a powerful tool for interpolation theory and will be used in the project. The project uses "indefinite" as opposed to "definite" notions of distance, and it seeks to explore new phenomena that arise when the classical theories do not apply. Questions to be investigated include existence of solutions, uniqueness, explicit construction, and properties of the state space. The budget includes support for one undergraduate student in an REU summer research project each year.