Operator algebras and operator theory

算子代数和算子理论

• 批准号: 3488-2013
• 负责人: Davidson, Kenneth
• 金额: $ 2.77万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635291
• 关键词: Operator; algebras; operator; theory

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.

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。运算符及其生成的代数可用于对各种现象进行建模，包括动力系统、量子力学和数据压缩。我们期待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。这涉及到几何和代数中的不变量。我们设置的分析增加了额外的结构，允许几种不同的攻击方向。我们研究了代数，它们是具有关系的交换压缩的普遍模型。来自代数几何和多变量解析函数的思想为在运算符设置中找到适当的相似项提供了模板和灵感。膨胀理论试图将一般结构理解为规范模型的一部分。这是我们帮助开发的一个强大的工具，并寻求进一步推动。具体地说，半微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算子代数的一般性质，这使得我们可以很好地理解它们的半微观乘积。我们寻求进一步发展这一点。第二个项目考虑了Arveson的边界表示和超刚性的概念。我们试图扩大这些概念的存在和适用范围。在算子理论中，一个重要的结构特征是不变子空间。虽然还不知道是否每个算子都有一个算子，但最近已经证明每个算子都有一个子空间，这个子空间直到一维扰动都是不变的。众所周知，每个算子都有一个小的紧扰动，它有一个约化子空间。我们试图确定有限秩扰动是否就足够了。