Hilbert spaces of analytic functions and their applications

在这个项目中的大多数拟议的研究活动可以被视为基本上是压缩算子半群的扰动问题。对这些问题的拟议处理办法可简述如下。一个表示希尔伯特空间上的压缩半群作为一个链的子空间，使每个子空间上的半群几乎作为一组酉。相应的谱图产生了一系列解析函数的空间。然后，我们的目标是通过研究这些空间以及它们与由某个模型半群生成的空间的关系来理解半群的性质。后者往往减少到一个问题的可逆性某些Toeplitz运营商的性质。从本质上讲，研究者提出的处理这些问题的方法可以被看作是经典论证原则的一种微妙形式。它的基础是N. Makarov和A. Poltoratski关于Toeplitz算子的内射性问题。正如调和分析中经常出现的情况一样，希尔伯特变换的微妙性质再次发挥了核心作用。本项目中提出的大多数问题都属于调和分析的范围。谐波分析的核心思想是将复杂信号表示为简单信号的组合的可能性-原子在某种意义上是手头问题的规范。由于测量的不精确性，有时需要使用稍微不同的原子系统，其可能具有或不具有正则系统的理想特性。需要仔细分析以确定理想测量下存在的属性在多大程度上继续适用于真实的世界情况。该项目的很大一部分是致力于进一步开发这种分析所需的数学工具。在信号处理和控制理论领域有潜在的应用。此外，该项目的另一个目标是通过软化一些现有的深层技术来普及调和和复分析的经典领域，从而使未来几代数学家以及其他科学家更容易获得它们。作为一所主要科技大学的成员，研究人员还将纳入其中一些新的想法，为新一代STEM专业的本科生和研究生提供最新的优质教育。

Most of the proposed research activities in this project can be viewed essentially as perturbation problems for contractive operator semigroups. The proposed treatment of these questions can be briefly described as follows. One represents the Hilbert space on which the contractive semigroup acts as a chain of subspaces so that on each of the subspaces the semigroup acts almost as a group of unitaries. The corresponding spectral picture produces a chain of spaces of analytic functions. The goal is then to understand the properties of the semigroup by investigating these spaces and their relationship to the spaces generated by a certain model semigroup. The latter often reduces to a problem about invertibility properties of certain Toeplitz operators. In its essence, the method that the investigator suggests to treat these problems can be viewed as a delicate form of the classical argument principle. It has in its basis the powerful method of N. Makarov and A. Poltoratski regarding the injectivity problem for Toeplitz operators. As is often the case in harmonic analysis, the delicate properties of the Hilbert transform again should play the central role.The majority of the problems proposed in this project lie within the area of harmonic analysis. The core idea of harmonic analysis is the possibility of representing complicated signals as a combination of simpler signals - atoms which are in a sense canonical for the problem at hand. Due to the imprecision of measurements one is sometimes required to use a slightly different system of atoms, which may or may not possess the ideal properties of the canonical system. Careful analysis is required to determine to which extent the properties present under ideal measurements continue to hold in a real world situation. A large part of this project is devoted to the further development of the mathematical tools necessary for such an analysis. Potential applications are possible in the areas of signal processing and control theory. In addition, another goal of this project is to popularize the classical areas of harmonic and complex analysis by softening some existing deep techniques, thus making them more accessible to the future generations of mathematicians as well as to other scientists. As a member of a major science-technology university, the investigator will also incorporate some of these new ideas to offer an up-to-date, quality education of the new generations of STEM majors, at both the undergraduate and graduate level.