Mathematical Sciences: Lifting Properties of Invariant Forms and Applications

数学科学：提升不变形式的性质及其应用

• 批准号: 8911717
• 负责人: Cora Sadosky
• 金额: $ 3.8万
• 依托单位: Howard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-01-15 至 1992-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8911717&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Lifting; Properties; Invariant

The primary theme of this mathematical research is that of analyzing integral transforms of classical type in several new settings. They include transforms generated by toeplitz kernels and their liftings to abstract domains. A central feature of this analysis is the so-called generalized Bochner theorem which can be viewed as a Fourier representation of generalized invariant kernels. The integral representation feature is linked with the general theory of Krein and Gelfand on eigenfunction expansions, while the lifting property is related to the Nagy- Foias dilation theory. While the Nagy-Foias theory provides a powerful approach to the spectral theory of dissipative operators, clarifying the notions of scattering and characteristic functions, and includes interpolation theory, it is only remotely connected to the theories of Bochner and Krein. The purpose of this work is to explore the deeper connections between the different aspects of the Bochner theorem to understand the relations between the above-mentioned theories. In particular, work will be done in determining whether the Nagy-Foias commutant lifting is related to bound mean oscillation and prediction theory. In addition to direct application to complex interpolation and moment theory, this work is expected to have applications to scattering structures and control.

这项数学研究的主要主题是在几种新的环境下分析经典类型的积分变换。它们包括由Toeplitz核生成的变换及其到抽象域的提升。这种分析的一个中心特征是所谓的广义Bochner定理，它可以被视为广义不变核的傅里叶表示。积分表示性质与Krein和Gelfand关于本征函数展开的一般理论相联系，而提升性质与Nagy-FOIas展开理论有关。虽然Nagy-FOIAS理论为耗散算符的谱理论提供了一种强有力的方法，澄清了散射和特征函数的概念，并包括了内插理论，但它只与Bochner和Krein的理论有很小的联系。这项工作的目的是探索Bochner定理不同方面之间的更深层次联系，以理解上述理论之间的关系。特别是确定Nagy-FOIAs对位提升是否与边界平均振荡和预测理论有关。除了直接应用于复数插值理论和矩理论之外，这项工作还有望应用于散射结构和控制。