Operator Theory in the Hardy Space Over the Bidisk

建议：DMS-9970932主要研究人员：道格拉斯·N·克拉克，杨荣伟摘要：这个项目旨在开发一个具体的背景，在这个背景下，从单算子理论到多变量理论的转变变得非常自然。在单算子理论中，向量值Hardy空间被用来提供压缩的正则模型。在不丧失一般性的情况下，我们可以将单位圆盘上的向量空间与Hardy空间的另一个副本相一致，然后向量值的Hardy空间就成为双圆盘上的Hardy空间。在这种新的背景下，通过陈述和反驳模型理论中的一些经典结果，我们可以在双盘上的Hardy空间中发现一些理论上重要的东西，并为进一步的研究发展一些通用的方法。研究的重点是两个移位算子的压缩，即两个坐标函数的乘法到商Hardy模的压缩。单算子理论起源于对线性方程组的研究，有着广泛的应用，从图像处理到机器人设计，再到金融市场预测。“算符对”为研究不同系统之间的相互作用提供了一个模型。两个简单的例子是不同电力网络之间的相互作用，以及一个金融市场对其他市场的影响。这个项目的目的是为研究算子对建立一个具体的环境。这种设置有两个主要优势。首先，它提供了许多可计算的例子，使研究更容易应用。其次，它以一种非常自然的方式融入了单算子理论。就一个实际例子而言，这一功能意味着它将允许人们以统一的方式监控单一市场和多市场系统。

Proposal: DMS-9970932Principal Investigators: Douglas N. Clark, Rongwei YangAbstract: This project aims to develop a concrete setting in which the transition from single operator theory to a multivariable theory becomes very natural. In single operator theory, the vector valued Hardy space is used to provide canonical models for contractions. Without loss of generality, we can identify the vector space with another copy of the Hardy space over the unit disk, and the vector valued Hardy space then becomes the Hardy space over the bidisk. By stating and reproving some of the classical results in model theory in this new context, we can identify some operator theoretically important things in the Hardy space over the bidisk and develop some general techniques for further research. The research focuses on the compressions of two shift operators, namely multiplications by the two coordinate functions, to quotient Hardy modules.Single operator theory, which originated in the study of systems of linear equations, has a wide range of applications, from image processing, to robotic design, to financial market predictions. "Operator pairs" provide a model for studying interactions between different systems. Two simple examples are interactions between different electric networks and the effects of one financial market on others. This project aims to develop a concrete setting for the study of operator pairs. There are two main advantages to this setting. First, it furnishes many computable examples which make the study easier to apply. Second, it incorporates single operator theory in a very natural way. This feature, in terms of a practical example, means that it will allow one to monitor single markets and multi-market systems in a unified way.