Mathematical Sciences: Topics in Operator Theory

数学科学：算子理论主题

• 批准号: 8800130
• 负责人: Warren Wogen
• 金额: $ 4.48万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-05-15 至 1990-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8800130&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Operator; Theory

Operator theory has its origins in the mathematical foundations of quantum mechanics. There, one needs infinitely many degrees of freedom (best realized in infinite-dimensional Hilbert space) and a way of representing physical quantities by objects that do not obey the ordinary laws of arithmetic. Operators on Hilbert space serve the latter purpose quite well. A given operator moves around pieces (subspaces) of the underlying space. One of the fundamental questions in the subject is, how much can you tell about the operator just from its invariant subspace lattice (the collection of subspaces that are mapped into themselves by the operator)? Important discoveries by the principal investigator on this project have shown that the relationship is more subtle than was hitherto conjectured. This work will be followed up. Another aspect of the research to be performed has to do with a particular way of realizing Hilbert space and certain operators on it. Namely, the space comes as a space of analytic functions of several complex variables, all of them defined on some common domain, and the operator comes from composing the functions with a transformation of the domain. When more than one complex variable is involved, well-behaved transformations of the domain can give rise to composition operators that are not well behaved. Exploration of this mysterious phenomenon is part of the agenda of the project.

算子理论起源于数学 量子力学的基础 在那里，一个人需要无限 许多自由度（最好在无限维中实现） 希尔伯特空间）和一种用 不服从算术的普通法则的对象。 希尔伯特空间上的算子很好地服务于后一个目的。 给定的操作符在 底层空间这门学科的基本问题之一 你能从操作员的 不变子空间格（ 由操作员映射到自身）？ 重要发现 这个项目的主要研究者已经表明， 这种关系比迄今为止所揭示的更为微妙。 这 工作将继续进行。 研究的另一个方面是 用一种特殊的方法来实现希尔伯特空间， 也就是说，这个空间是一个解析的空间， 多个复变量的函数，所有这些函数都定义在 一些公共域，运算符来自于组合 函数与域的变换。当多于一个 复杂的变量，行为良好的转换， 域可以产生不太好的组合运算符， 表现良好探索这一神秘现象是 项目的议程。