Operator Theory in the Hardy Space Over the Bidisk

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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.