Commutative Algebraic Methods in Multivariable Operator Theory and Applications

多变量算子理论中的交换代数方法及其应用

• 批准号: 0801174
• 负责人: Xiang Fang
• 金额: $ 10.66万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801174&HistoricalAwards=false
• 关键词: Commutative; Algebraic; Methods; Multivariable; Operator

The objective of this proposal is develop a systematic theory of commutative algebraic invariants in operator theory, centering around the dimension invariant and the Samuel multiplicity of a tuple of commuting operators acting on a Hilbert space. These algebraic invariants have found applications to purely analytic problems in operator theory including the study of Hilbert spaces of analytic functions, and Fredholm theory. Further connection with Nevanllina-Pick kernels will be explored in the proposed research.Hilbert space operators can be viewed as quantization of classical objects such as holomorphic functions. Operator theory provides an infinite dimensional approach to problems in a finite dimensional space. The subject originates from providing a mathematical foundation to Quantum Mechanics, and has found applications in quality control in Electrical Engineering. The proposed research will introduce new techniques from commutative algebra, another area of mathematics, into operator theory, hence not only enriching the subject, but also enhancing the ties between two subjects in mathematics.

本文的目的是围绕作用于Hilbert空间上的交换算子组的维不变性和Samuel多重性，发展算子理论中交换代数不变量的系统理论。这些代数不变量已被应用于算子理论中的纯解析问题，包括研究解析函数的希尔伯特空间和Fredholm理论。希尔伯特空间算子可以看作是经典对象的量子化，例如全纯函数。算子理论为有限维空间中的问题提供了一种无限维的方法。该学科起源于为量子力学提供数学基础，并已在电气工程的质量控制中得到应用。这项研究将把交换代数的新技术引入到算子论中，从而不仅丰富了这一学科，而且加强了数学中两门学科之间的联系。