Mathematical Sciences: Subnormal Operators and Related Topics

数学科学：次正规运算符及相关主题

• 批准号: 8700835
• 负责人: John Conway
• 金额: $ 6.07万
• 依托单位: Indiana University
• 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: 1987-06-15 至 1989-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8700835&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Subnormal; Operators; Related

Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. Thus, 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. So it has been, that from the fertile minds of Hilbert, von Neumann, and other giants that the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. At the heart of this methodology is the deep investigation of the spectrum of an operator and the concommitant study of its invariant subspaces. For the so-called self-adjoint operators, this theory is now a standard technique throughout analysis, and the spectral theorem provides the necessary building blocks for all such operators. Although not quite as complete, spectral theory is significantly well understood for an extensive generalization of the self-adjoint case; viz., the "normal" operators. The current frontier, therefore, in the study of this structure theory rests in the non-normal theory. Professors Conway has long been a leader in the study of subnormal operators, a broad class of operators that arise in applications, and for which there is enough residual normal information to attempt a deeper understanding of their inherent structure. Recent results, suchas the fact that subnormal operators are reflexive and have invariant subspaces, have given an added impetus to the study of their spectral properties. Since many explicit operators that arise in complex function theory, differential geometry, and approximation theory are subnormal, there is a broad range of developing interaction. Professor Conway will study subnormal operators that arise naturally in actions on Hardy spaces, thereby exploring the interaction with function theory. He will also investigate the internal structure theory of subnormal operators, with particular consideration of unitary equivalence, bundle shifts, and the relationship with general operator theory.

操作者理论是现代分析中的中心学科。 它的起源在于20世纪初期对数学物理学和部分微分方程的研究。 因此，可以看到，可以通过模拟现象的积分方程来有效地研究平衡，振动，量子理论等的许多物理问题。 因此，从希尔伯特（Hilbert），冯·诺伊曼（von Neumann）和其他巨人的肥沃思想中，操作者理论的主题在此类调查和核心数学方面也发展到了中心地位。 这种方法的核心是对操作员光谱的深入研究以及对其不变子空间的同一研究。 对于所谓的自动接合运算符，该理论现在是整个分析过程中的标准技术，光谱定理为所有此类操作员提供了必要的构建块。 尽管不太完整，但是对于自我偶相病例的广泛概括，光谱理论已得到充分理解。即“正常”操作员。 因此，在对该结构理论的研究中，当前的边界取决于非正常理论。 Conway教授长期以来一直是亚正态操作员研究的领导者，正常运营商，在应用中出现的广泛运营商，并且有足够的残留正常信息来尝试更深入地了解其固有结构。 最近的结果是，亚正态算子是反身且具有不变子空间的事实，它为其光谱特性的研究增加了动力。 由于许多在复杂函数理论中出现的显式运算符，差异几何学和近似理论是亚正式的，因此存在广泛的发展相互作用。 Conway教授将研究自然出现在Hardy空间的行动中的亚正态操作员，从而探索与功能理论的相互作用。 他还将研究亚正态操作员的内部结构理论，并特别考虑了单一等效性，捆绑转移以及与普通运营商理论的关系。