Complex Analysis and Spectral Theory

我研究的长期目标是更好地理解复杂分析和算子理论之间的相互作用。近年来，我的工作主要集中在全纯函数空间上。这些都与运营商密切相关。实际上，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。在另一个方向上，全纯函数空间通常最好地理解为作用于它们的具体运算符。拟议的研究计划兼顾了这两个方面的问题。建议的第一部分是通过它们的数值范围来研究希尔伯特空间算子。它包括两个相互关联的项目，一个是关于数值值域的映射定理，另一个是证明所谓的Crouzeix猜想。这一猜想的证明很可能导致希尔伯特空间上算子的一个改进的结构定理。提案的第二部分包括三个有关功能空间的项目。第一个项目是建立一个自动连续定理，从而将经典的Gleason-Kahane-Zelazko定理推广到各种函数空间。二是寻找在某些函数空间中多项式逼近的构造性方法，这是目前所缺乏的方法。第三个项目是调查在尼曼-比尔林对黎曼假设的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Riemann Zeta函数相关的某个无限三角矩阵的至少前10亿个条目都是正的。所有条目都是这样吗？如果是这样的话，对黎曼假说的影响是什么？

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?