Operator Theory in the Hardy Space Over the Bidisk

Bidisk上Hardy空间的算子理论

• 批准号: 9970932
• 负责人: Edward Azoff
• 金额: $ 4.53万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2001-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970932&HistoricalAwards=false
• 关键词: Operator; Theory; Hardy; Space; Over

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.

提案：DMS-9970932 主要研究者：Douglas N. Clark、Rongwei Yang 摘要：该项目旨在开发一个具体的环境，使从单算子理论到多变量理论的过渡变得非常自然。在单算子理论中，向量值 Hardy 空间用于提供收缩的规范模型。不失一般性，我们可以将向量空间与单位圆盘上的 Hardy 空间的另一个副本等同起来，并且向量值 Hardy 空间就成为 Bidisk 上的 Hardy 空间。通过在这个新的背景下陈述和反驳模型论中的一些经典结果，我们可以识别出 Bidisk 上的 Hardy 空间中一些算子理论上重要的东西，并开发一些用于进一步研究的通用技术。研究重点是两个移位算子的压缩，即两个坐标函数的乘法，对商Hardy模的压缩。单算子理论起源于线性方程组的研究，具有广泛的应用，从图像处理、到机器人设计，到金融市场预测。 “算子对”提供了研究不同系统之间交互的模型。两个简单的例子是不同电网之间的相互作用以及一个金融市场对其他市场的影响。该项目旨在为算子对的研究开发一个具体的环境。这种设置有两个主要优点。首先，它提供了许多可计算的例子，使研究更容易应用。其次，它以非常自然的方式结合了单算子理论。从实际例子来看，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。