Operator algebras and operator theory

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Operator theory is the study of linear transformations (operators) on infinite dimensional Euclidean space. Operators and the algebras that they generate can be utilized to model a wide variety of phenomena, including dynamical systems, quantum mechanics, and data compression. We expect that our research will provide significant new insights in the field. I am interested in the multivariable operator theory. This involves invariants arising from geometry and algebra. The analysis imposed by our setting adds additional structure that allows several different directions of attack. We have studied algebras which are universal models for commuting contractions with relations. Ideas from algebraic geometry and analytic functions of several variables provide a template and inspiration to find appropriate analogues in the operator setting. Dilation theory attempts to understand general structures as pieces of a canonical model. It is a powerful tool that we have helped develop, and seek to push further. In particular, semicrossed products are operator algebras which encode a semigroup action. Dilation theory sheds considerable light on their structure, providing invariants to distinguish different systems. Our recent work has provided general properties of operator algebras which allow a good understanding of their semicrossed products. We seek to develop this further. A second project considers Arveson's notions of boundary representation and hyper-rigidity. We seek to extend the existence and applicability of these notions. In operator theory, one important structural feature is an invariant subspace. While it is unknown whether every operator has one, recently it has been shown that every operator has a subspace which is invariant up to a 1-dimensional perturbation. It is known that every operator has a small compact perturbation which has a reducing subspace. We seek to determine whether a finite rank perturbation will suffice.