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Geodesics, extension of holomorphic functions and the spectral theory of multioperators

测地线、全纯函数的扩展和多算子谱理论

基本信息

批准号：
EP/N03242X/1
负责人：
Zinaida Lykova
金额：
$ 4.71万
依托单位：
Newcastle University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN03242X%2F1
关键词：
Geodesics extension holomorphic functions spectral

项目摘要

One of the most successful and beautiful branches of mathematics in the nineteenth and twentieth centuries was the theory of analytic functions. These are functions which are smooth enough to have a gradient at every point of a region of the complex plane. The theory has had enormous importance for the understanding of several branches of physics and engineering, as well as playing an essential role in pure mathematics. Since the 1920s or thereabouts there has been an analogous development of a theory of analytic functions of several variables, which is also significant for science and technology. In particular, some engineering design problems require the construction of functions of a single variable which are rational (that is, expressible by a formula involving only addition, multiplication and division), take their values in some prescribed target region of higher-dimensional complex space and meet some further specifications. For certain special target regions there is a well-developed theory already; this theory plays a significant role in `H infinity control', a branch of control engineering. The present project will provide the opportunity for the PI to study new operator-theoretic methods at the University of California at San Diego, mainly, from Professor Jim Agler. His deep understanding of several complex variables and operator-theoretic methods are vital for the project. The proposed research will extend existing theory to include other target regions of engineering relevance, building on discoveries about the geometry and function theory of such regions by many mathematicians. One of our principal aims is to develop a theory of rational functions from a disc or half-plane to regions such as the symmetrised bidisc which parallels the classical theory. There are substantial difficulties in carrying out such a development: here are three of them. Firstly, whereas the classical target regions are homogeneous, meaning that any point is like any other, the symmetrised bidisc is inhomogeneous, so that some points have special geometric properties. Secondly, whereas classical domains are convex, the symmetrised bidisc is not even isomorphic to a convex domain. Furthermore, the symmetrised bidisc has sharp corners, for which reason many of the results of mainstream several complex variables do not apply to it. Nevertheless, research over the past decade has shown that the symmetrised bidisc and some similar domains have a rich geometry and function theory, exhibiting fascinating new features that do not appear in classical domains. We shall exploit the close connection between the symmetrised bidisc and two classical domains (the bidisc and the unit ball of the space of 2 x 2 matrices) to identify sets in the symmetrized bidisc with the norm-preserving extension property and to get new properties of $\Gamma$-contractions. We intend to do likewise for other target domains, which we call quasi-Cartan domains, to indicate their close connection with the classical `Cartan domains'.The results of the project will be significant for researchers in several complex variables and in the theory of linear operators; there are many of both categories worldwide. They will also be significant for control engineers, particularly those who use the technique of `mu-synthesis' for the design of automatic controllers for linear plants subject to structured uncertainty.

十九世纪和二十世纪最成功、最美丽的数学分支之一是解析函数理论。这些函数足够平滑，可以在复平面区域的每个点处具有梯度。该理论对于理解物理和工程学的几个分支具有巨大的重要性，并且在纯数学中发挥着重要作用。自 20 年代左右以来，多变量解析函数理论也出现了类似的发展，这对于科学和技术也具有重要意义。特别是，一些工程设计问题需要构造有理的单个变量的函数（即，可以用仅涉及加法、乘法和除法的公式来表达），在高维复杂空间的某个规定的目标区域中取它们的值，并满足一些进一步的规范。对于某些特殊目标区域，已经有成熟的理论；该理论在控制工程的一个分支“H无穷控制”中发挥着重要作用。目前的项目将为 PI 提供在加州大学圣地亚哥分校主要由 Jim Agler 教授学习新算子理论方法的机会。他对几个复杂变量和算子理论方法的深刻理解对于该项目至关重要。拟议的研究将扩展现有理论，以包括工程相关的其他目标区域，以许多数学家关于这些区域的几何和函数理论的发现为基础。我们的主要目标之一是发展一种有理函数理论，从圆盘或半平面到类似于经典理论的对称双圆盘等区域。进行这样的开发存在很大的困难：以下是其中的三个。首先，经典的目标区域是均匀的，这意味着任何点都与其他点相似，而对称的 bidisc 是不均匀的，因此某些点具有特殊的几何属性。其次，虽然经典域是凸域，但对称 Bidisc 甚至不与凸域同构。此外，对称bidisc具有尖角，因此主流的几个复杂变量的许多结果不适用于它。然而，过去十年的研究表明，对称bidisc和一些类似的域具有丰富的几何和函数理论，展现出经典域中没有出现的令人着迷的新特征。我们将利用对称 bidisc 和两个经典域（bidisc 和 2 x 2 矩阵空间的单位球）之间的紧密联系来识别对称 bidisc 中具有范数保持扩展属性的集合，并获得 $\Gamma$ 收缩的新属性。我们打算对其他目标域（我们称之为准嘉当域）做同样的事情，以表明它们与经典“嘉当域”的密切联系。该项目的结果对于几个复杂变量和线性算子理论的研究人员来说具有重要意义；世界范围内这两个类别都有很多。它们对于控制工程师也很重要，特别是那些使用“mu-综合”技术来设计受结构不确定性影响的线性设备自动控制器的工程师。