Operator algebras and operator theory

算子理论是研究无限维欧氏空间上的线性变换（算子）的理论。运算符和它们生成的代数可以用来模拟各种各样的现象，包括动力学系统，量子力学和数据压缩。我们希望我们的研究将在该领域提供重要的新见解。 我对多变量算子理论很感兴趣。这涉及到从几何和代数中产生的不变量。由我们的设置强加的分析增加了额外的结构，允许几个不同的攻击方向。我们已经研究了代数，它是具有关系的可交换压缩的通用模型。从代数几何和解析函数的几个变量的想法提供了一个模板和灵感，以找到适当的类似物在运营商设置。 膨胀理论试图将一般结构理解为规范模型的一部分。这是一个强大的工具，我们已经帮助开发，并寻求进一步推动。特别地，半交叉积是编码半群作用的算子代数。膨胀理论揭示了它们的结构，提供了区分不同系统的不变量。我们最近的工作提供了一般性质的算子代数，使他们的semicrossed产品有一个很好的理解。我们寻求进一步发展这一点。第二个项目认为Arveson的概念的边界表示和超刚性。我们试图扩大这些概念的存在和适用性。 在算子理论中，一个重要的结构特征是不变子空间。虽然它是未知的，是否每个运营商有一个，最近已被证明，每个运营商有一个子空间是不变的1维扰动。已知每个算子都有一个小的紧扰动，该扰动具有一个约化子空间。我们试图确定是否有限秩扰动将足够。

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.