Operator Theory in the Hardy Space Over the Bidisk

提案：DMS-9970932主要研究者：道格拉斯N.克拉克，杨荣伟摘要：这个项目的目的是开发一个具体的设置，其中从单算子理论过渡到多变量理论变得非常自然。在单算子理论中，向量值哈代空间被用来提供压缩的正则模型。在不失一般性的情况下，我们可以将向量空间与单位圆盘上的哈代空间的另一个副本标识，并且向量值哈代空间于是成为双圆盘上的哈代空间。通过在这种新的背景下陈述和修正模型论中的一些经典结果，我们可以发现双圆盘上的哈代空间中的一些算子理论上的重要内容，并为进一步的研究提供一些一般性的技巧。研究的重点是两个移位算子的压缩，即乘两个坐标函数，商哈代modules.Single算子理论，起源于线性方程组的研究，有着广泛的应用，从图像处理，机器人设计，金融市场预测。“算子对”提供了一个研究不同系统之间相互作用的模型。两个简单的例子是不同电力网络之间的相互作用和一个金融市场对其他市场的影响。这个项目的目的是发展一个具体的设置为研究运营商对。这种设置有两个主要优点。首先，它列举了许多可计算的例子，使研究更容易应用。其次，它以非常自然的方式结合了单算子理论。就一个实际例子而言，这一特点意味着它将使人们能够以统一的方式监测单一市场和多市场体系。

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.