Complex Analysis and Spectral Theory

复分析与谱理论

• 批准号: RGPIN-2020-04263
• 负责人: Ransford, Thomas
• 金额: $ 2.7万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759468
• 关键词: Complex; Analysis; Spectral; Theory

The long-term objective of my research is to gain a better understanding of the interactions between complex analysis and operator theory. In recent years my work has focused on holomorphic function spaces. These are closely connected to operators. Indeed, the functions that act on a given operator typically emerge from a suitably chosen holomorphic function space. In the other direction, holomorphic function spaces are often best understood in terms of concrete operators that act on them. The proposed research program pursues both lines of enquiry. The first part of the proposal is to study Hilbert-space operators via their numerical range. It comprises two interlinked projects, one concerning mapping theorems for numerical range, and the other to prove the so-called Crouzeix conjecture. A proof of this conjecture would very like lead to an improved structure theorem for operators on a Hilbert space. The second part of the proposal consists of three projects concerning function spaces. The first project is to establish an automatic continuity theorem, and thereby extend the classic Gleason-Kahane-Zelazko theorem to a wide variety of function spaces. The second is to find constructive methods for polynomial approximation in certain function spaces where these methods are currently lacking. And the third project is to investigate a curious, recently-discovered phenomenon arising in the Nyman-Beurling approach to the Riemann hypothesis. For no apparent reason, at least the first billion entries of a certain infinite triangular matrix associated to the Riemann zeta function are all positive. Is this is true of all the entries? If so, what are the implications, if any, for the Riemann hypothesis?

我研究的长期目标是更好地理解复分析和算子理论之间的相互作用。近年来我的工作重点是全纯函数空间。这些都与运营商密切相关。事实上，作用于给定算子的函数通常来自适当选择的全纯函数空间。另一方面，全纯函数空间通常可以通过作用于它们的具体算子来最好地理解。拟议的研究计划同时进行这两种调查。该提案的第一部分是通过希尔伯特空间算子的数值范围来研究它们。它包括两个相互关联的项目，一个涉及数值范围的映射定理，另一个是证明所谓的克鲁泽猜想。这个猜想的证明很可能会导致希尔伯特空间上算子的结构定理的改进。该提案的第二部分由三个有关功能空间的项目组成。第一个项目是建立自动连续性定理，从而将经典的 Gleason-Kahane-Zelazko 定理扩展到各种函数空间。第二是寻找目前缺乏的某些函数空间中多项式逼近的构造方法。第三个项目是研究黎曼假设的尼曼-伯林方法中出现的一个奇怪的、最近发现的现象。没有明显的原因，与黎曼 zeta 函数相关的某个无限三角矩阵的至少前十亿项都是正值。所有条目都是如此吗？如果是这样，那么对黎曼假设有何影响（如果有的话）？