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

解析函数理论是十九世纪和二十世纪数学中最成功和最美丽的分支之一。这些函数足够光滑，在复平面区域的每个点上都有一个梯度。这一理论对于理解物理学和工程学的几个分支以及在纯数学中起到了至关重要的作用。自20世纪20年代左右以来，多变量解析函数理论也有了类似的发展，这对科学和技术也具有重要意义。特别是，一些工程设计问题要求构造单变量函数是有理的(即，可以用一个只涉及加法、乘法和除法的公式表示)，在高维复杂空间的某个指定目标区域取值，并满足某些进一步的规范。对于某些特定的目标区域，已经有了比较成熟的理论，这一理论在控制工程的一个分支“H_无穷控制”中起着重要的作用。本项目将为PI提供机会，主要由Jim Agler教授在加州大学圣地亚哥分校学习新的算子理论方法。他对几个复杂变量和算子论方法的深刻理解对这个项目至关重要。拟议的研究将扩展现有的理论，以包括其他与工程相关的目标区域，建立在许多数学家对这些区域的几何和函数理论的发现的基础上。我们的主要目标之一是将有理函数理论从圆盘或半平面发展到与经典理论平行的区域，如对称双圆盘。进行这样的发展有很大的困难：这里有三个困难。首先，经典的目标区域是均匀的，这意味着任何点都是相似的，而对称双圆盘是不均匀的，因此一些点具有特殊的几何性质。其次，虽然经典区域是凸的，但对称的双圆盘甚至不同构于凸域。此外，对称双圆盘具有尖角，因此主流几个复变量的许多结果不适用于对称双圆盘。然而，过去十年的研究表明，对称双圆盘和一些类似的区域具有丰富的几何和函数理论，显示出经典区域中不存在的迷人的新特征。我们将利用对称双圆盘与两个经典区域(双圆盘和2×2矩阵空间的单位球)之间的密切联系来识别对称双圆盘中具有保范数扩张性质的集合，并得到$-Gamma压缩的新性质。我们打算对其他目标域做同样的事情，我们称之为准Cartan域，以表明它们与经典的‘Cartan域’的密切联系。该项目的结果将对几个复变量和线性算子理论的研究人员具有重要意义；世界上有许多这两类。它们对控制工程师也将具有重要意义，特别是那些使用“单位综合”技术为结构不确定的线性对象设计自动控制器的人。

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.