Analysis on Hardy and Bergman Modules

Suny At Albany, Albany NY

Investigators

Abstract

Abstract: This project studies several important problems concerning shift operators on invariant subspaces of the Bergman space or the Hardy space over the unit polydisk. Since the spaces are Hilbert modules over the polynomial ring, invariant subspaces are submodules. A key ingredient in this study is the newly discovered core operator for a submodule, which is an integral operator with kernel as a quotient of two reproducing kernels relative to the submodule. The core operator is an essential associate of a submodule and,in some sense, can be considered as a partial replacement for the inner functions which play pivatal roles in the classical one-variable Hardy space theory. Many properties of the shift operators, for instance the spectra or joint commutators, are deeply related to the core operator. Boundary values of the involved reproducing kernels are also closely connected with the core operator. The study in this project will be a clean display of some multivariable concepts and techniques, and will, in return, create many new ideas for multivariable operator theory in general. In many areas of science, interactions among a group of similar objects is an important subject of study. When these objects can be modeled by linear operators, multivariable operator theory provides very useful ideas and techniques for the study, especially in cases when noncommutativity is involved. However, multivariable operator theory has not yet sit in standard graduate curriculum. This reality seems to be a result of two drawbacks in current stage of the theory. One is the unfortunate complication caused by the generality the theory has to seek. This complication obscures the nature of the subject and makes it hard for beginners to see a broad picture. The other drawback is a lack of concrete setting where many of these ideas and techniques are organically integrated in answering natural questions. This drawback makes it hard for beginners to see the values of these ideas or techniques, let alone to appreciate them. A large part of this project, in hope to provide a remedy, aims to build a systematic two variable operator theory in possibly the most manageable setting, namely the Hardy space over the bidisk. When successfully developed, the theory will serve as a ground in multivariable operator theory where more effective education and research could take place. High educational value is a notable feature of this project. Apart from its mathematical and educational values, some aspects of this study see possible applications in Control theory and Signal Processing.