Mathematical Sciences: Topics in Operator Theory

算符理论起源于量子力学的数学基础。在那里，人们需要无限多的自由度(在无限维希尔伯特空间中实现得最好)和一种用不遵守普通算术定律的物体来表示物理量的方法。希尔伯特空间上的算子很好地服务于后一目的。给定的运算符在底层空间的各个部分(子空间)周围移动。这个主题中的一个基本问题是，仅从算子的不变子空间格(由算子映射到自身的子空间的集合)可以知道多少关于算子的信息？这个项目的首席研究员的重要发现表明，这种关系比迄今所猜测的更加微妙。这项工作将继续跟进。要进行的研究的另一个方面与实现希尔伯特空间和它上的某些算子的特定方式有关。也就是说，空间是由多个复变量的解析函数组成的空间，它们都定义在某个公共区域上，而算子来自于将函数与区域的变换组合在一起。当涉及多个复变量时，域的行为良好的转换可能会导致行为不佳的复合运算符。探索这一神秘现象是该项目议程的一部分。

Operator theory has its origins in the mathematical foundations of quantum mechanics. There, one needs infinitely many degrees of freedom (best realized in infinite-dimensional Hilbert space) and a way of representing physical quantities by objects that do not obey the ordinary laws of arithmetic. Operators on Hilbert space serve the latter purpose quite well. A given operator moves around pieces (subspaces) of the underlying space. One of the fundamental questions in the subject is, how much can you tell about the operator just from its invariant subspace lattice (the collection of subspaces that are mapped into themselves by the operator)? Important discoveries by the principal investigator on this project have shown that the relationship is more subtle than was hitherto conjectured. This work will be followed up. Another aspect of the research to be performed has to do with a particular way of realizing Hilbert space and certain operators on it. Namely, the space comes as a space of analytic functions of several complex variables, all of them defined on some common domain, and the operator comes from composing the functions with a transformation of the domain. When more than one complex variable is involved, well-behaved transformations of the domain can give rise to composition operators that are not well behaved. Exploration of this mysterious phenomenon is part of the agenda of the project.