Complex analysis and spectral theory

复分析和谱理论

• 批准号: RGPIN-2015-04545
• 负责人: Ransford, Thomas
• 金额: $ 1.82万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=663006
• 关键词: Complex; analysis; spectral; theory

The goal of my research is to develop and apply methods in complex analysis and potential theory to solve problems in other areas of mathematics, in particular in operator theory and spectral theory. This proposal outlines three specific projects on which I plan to work in the next few years: (1) analytic capacity, (2) Dirichlet spaces and (3) pseudospectra. Each project contains a component for the training of research students or postdoctoral fellows.***Analytic capacity measures the size of compact plane sets from the point of view of certain aspects of complex analysis. It was first introduced by Ahlfors in the 1940's in connection with the so-called Painlevé problem of characterizing removable singularities of bounded holomorphic functions. Later, in the 1960's, it played a key role in the solution of several fundamental problems in rational approximation. The last twenty years have seen significant advances in the understanding of analytic capacity, one of the most striking breakthroughs being Tolsa's proof of the long-standing conjecture that analytic capacity is semiadditive (the capacity of the union of two sets is bounded by C times the sum of their individual capacities, where C is a universal constant). However, whether analytic capacity is subadditive (can we take C=1?) still remains an open problem. I propose to attack this problem, exploiting recently developed expertise in the rigorous computation of analytic capacity.***The Dirichlet space (together with its close relatives the Hardy and Bergman spaces) is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. Even though it has been intensively studied over the years, a number of fundamental questions remain unresolved, notably the problem of characterizing the closed shift-invariant subspaces and the cyclic functions (the analogues in this setting of closed ideals and invertible elements). The characterization of the cyclic functions is the subject of a celebrated 30-year-old conjecture of Brown and Shields, on which I propose to work. I also plan to study the several-variable analogue of this conjecture in the special case of polynomials.***The theory of pseudospectra is a modern outgrowth of classical eigenvalue analysis. The general idea is to analyse a matrix by studying how fast its resolvent blows up as one approaches the eigenvalues. In the case of non-normal matrices, this yields considerably more information than merely identifying the eigenvalues. Efficient computation methods have also made pseudospectra a practical tool in numerical linear algebra. It is therefore of interest to determine to what extent the pseudospectra of a matrix really determine its behavior. I plan to study this problem, both for pseudospectra and for certain higher-order generalizations.**

我的研究目标是开发和应用复分析和势理论的方法来解决数学其他领域的问题，特别是算子理论和谱理论。这个建议概述了我计划在未来几年内工作的三个具体项目：（1）分析能力，（2）狄利克雷空间和（3）伪谱。每个项目都包含一个培训研究生或博士后研究员的组成部分。解析容量从复分析的某些方面来度量紧平面集的大小。它首先介绍了Ahlfors在20世纪40年代与所谓的Painlevé问题的特点可移动奇点的有界全纯函数。后来，在20世纪60年代，它发挥了关键作用，在理性近似的几个基本问题的解决方案。在过去的20年里，人们对解析容量的理解有了显著的进步，其中最引人注目的突破之一是托尔萨证明了一个长期存在的猜想，即解析容量是半加性的（两个集合的并集的容量是由C乘以它们各自的容量之和所限定的，其中C是一个普适常数）。然而，分析能力是否是次加性的（我们可以取C=1吗？）仍然是一个悬而未决的问题。我建议利用最近发展起来的严格计算分析能力的专业知识来解决这个问题。Dirichlet空间（与它的近亲哈代和Bergman空间）是单位圆盘上全纯函数的三个基本Hilbert空间之一。尽管它已经被深入研究了多年，一些基本问题仍然没有得到解决，特别是封闭的移位不变子空间和循环函数（类似于封闭理想和可逆元的设置）的特征问题。循环函数的特征是一个著名的30岁的布朗和希尔兹猜想的主题，我建议工作。我还计划在多项式的特殊情况下研究这个猜想的多变量模拟。伪谱理论是经典本征值分析的现代产物。一般的想法是通过研究当一个矩阵接近特征值时，它的预解式爆炸的速度来分析矩阵。在非正规矩阵的情况下，这比仅仅识别特征值产生更多的信息。有效的计算方法也使伪谱成为数值线性代数中的一个实用工具。因此，确定矩阵的伪谱在多大程度上真正决定其行为是很有意义的。我计划研究这个问题，包括伪谱和某些高阶推广。