Mathematical Sciences: Krein Space Operators and Topics in Analysis

本项目旨在构建Krein空间上的收缩算子理论，该理论在膨胀和扩展性质等方面与Hilbert空间情况相似。最近的研究表明，膨胀和延伸是存在的，并且是可以表征的。将调查新的标签问题。例子包括由单位圆盘子区域的归一化黎曼映射引起的替换算子。一个长期目标是将该理论应用于黎曼映射的系数插值问题。这个项目属于现代分析的一般领域，涉及对空间和这些空间上的变换或算子的研究。希尔伯特空间可能是物理学、分析学和几何学中研究最多、最有价值的空间之一。在这个项目中研究的Krein空间是一个标量积的空间，它可以分解为Hilbert空间和Hilbert空间的反空间的正交直和。

This project is aimed at constructing a theory of contraction operators on Krein spaces which, in areas such as dilation and extension properties, parallels the Hilbert space case. Recent work has shown that dilations and extensions exist and can be characterized. New problems of labeling will be investigated. Examples include substitution operators induced by normalized Riemann mappings for subregions of the unit disk. A long-range goal is to apply the theory to a coefficient interpolation problem for Riemann mappings. This project is in the general area of modern analysis and involves the study of spaces and the transformations or operators on these spaces. Probably one of the most studied and valuable spaces in physics, analysis, and geometry is the Hilbert space. The Krein space examined in this project is a space with a scalar product which is decomposable as the orthogonal direct sum of a Hilbert space and the anti-space of a Hilbert space.