Mathematical Sciences: Joint K-spectral Sets and Subnormal Operators

数学科学：联合 K 谱集和次正规算子

• 批准号: 8701498
• 负责人: Vern Paulsen
• 金额: $ 3.28万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701498&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Joint; spectral; Sets

Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial equations in the early twentieth century. At that time, it was seen that numerous physical problems in the theory of equilibria, vibration, quantum theory, etc. could be studied productively via the integral equations that model the phenomena. In the ensuing years, the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. Also central to Modern Analysis is the related discipline of operator algebras in which one studies collections of operators simultaneously. As the mathematical construct which best transfers the concepts of probability, measure theory, topology, and geometry to noncommutative contexts, operator algebras relate to a rich and important array of applications, from within mathematics itself to physics and microbiology. The research of Professor Paulsen bridges both operator theory and operator algebras. A careful analysis of the internal structure theory of operator algebras and the maps between them is an increasingly important subject, and Professor Paulsen is an expert in such matters. In prior research, Professor Paulsen obtained important results concerning completely bounded maps between C*-algebras. His recent book on this subject is expected to open new areas of research for both single operator theorists as well as operator algebraists. In his current program, Professor Paulsen will study joint K-spectral sets, subnormal models, and Hilbert modules over function algebras. The former study should lead to insights into tensor products of nonself-adjoint operator algebras. The second area explores aspects of dilation theory, and the latter project relates to several topics in operator theory.

算子理论是现代分析的核心学科。 它的起源在于二十世纪初对数学物理和偏方程的研究。 当时，人们发现平衡理论、振动理论、量子理论等中的许多物理问题可以通过对现象进行建模的积分方程来有效地研究。 在接下来的几年里，算子理论的主题已经发展到此类研究以及核心数学中的中心地位。 现代分析的另一个核心是算子代数的相关学科，其中同时研究算子的集合。 作为最能将概率、测度论、拓扑和几何概念转移到非交换环境中的数学构造，算子代数涉及丰富而重要的应用，从数学本身到物理学和微生物学。 Paulsen 教授的研究连接了算子理论和算子代数。 仔细分析算子代数的内部结构理论及其之间的映射是一个日益重要的课题，Paulsen教授是这方面的专家。 在之前的研究中，Paulsen教授获得了关于C*代数之间的完全有界映射的重要成果。 他最近关于这个主题的书预计将为单算子理论家和算子代数学家开辟新的研究领域。 在他当前的项目中，保尔森教授将研究函数代数上的联合 K 谱集、次正规模型和希尔伯特模。 前一项研究应该有助于深入了解非自伴算子代数的张量积。 第二个领域探索膨胀理论的各个方面，后一个项目涉及算子理论中的几个主题。