Mathematical Sciences: Lifting Properties of Invariant Forms and Applications

本数学研究的主要主题是分析几种新情况下经典类型的积分变换。它们包括由toeplitz内核生成的转换及其对抽象域的提升。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。积分表示特性与Krein和Gelfand关于特征函数展开的一般理论有关，而提升特性与Nagy- Foias膨胀理论有关。虽然Nagy-Foias理论为耗散算符的光谱理论提供了一个强有力的方法，澄清了散射和特征函数的概念，并包括插值理论，但它与Bochner和Krein的理论只有很小的联系。本研究的目的在于探究博奇纳定理不同方面之间的深层联系，从而理解上述理论之间的关系。特别是，工作将在确定Nagy-Foias换向提升是否与界平均振荡和预测理论有关。除了直接应用于复杂插值和矩理论之外，这项工作还有望应用于散射结构和控制。

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.