Commutative algebras generated by Toeplitz operators - Gelfand theory and spectral properties

Toeplitz 算子生成的交换代数 - Gelfand 理论和谱特性

基本信息

批准号：
237774273
负责人：
Professor Dr. Wolfram Bauer
金额：
--
依托单位：
Consejo Nacional de Humanidades Ciencias y Tecnologías (CONAHCYT)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2013
资助国家：
德国
起止时间：
2012-12-31 至 2016-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/237774273?language=en
关键词：
Commutative algebras generated Toeplitz operators

项目摘要

In a series of papers by N. Vasilevski and W. Bauer/N. Vasilveski new types of commutative Toeplitz Banach algebras consisting of operators on the standard weighted Bergman spaces over the complex unit ball in C^n have been discovered recently. These algebras only show up in the higher dimensional setting n>1 and they are still far from being completely understood. In the current project we focus on their classification and analysis. More precisely, the following tasks are our main objectives:1. The full classification of the types of commutative Toeplitz Banach algebras that are sub-ordinate to the maximal commutative subgroups of the automorphism group of the n-dimensional complex unit ball B in C^n.2. A description of the internal structure of the algebras obtained in 1. In particular, we intend to study their Gelfand theory, determine the maximal ideal spaces and the radical. Which of these algebras are semi-simple?3. As an application of the results in 2. we plan to decide the question of the "spectral invariance" of these algebras. Moreover we believe that we can reach a deeper understanding of the spectral theory and the Fredholm property of their elements. Similar questions can be asked in the case of Toeplitz operators on more general bounded symmetric domains or for different functional Hilbert spaces (e.g. harmonic L^2-functions) over the unit ball. In these cases a complete solution of 1. - 3. seems quite challenging.

在N. Vasilevski和W. Bauer/N的一系列论文中。 Vasilveski最近发现了C^n中C^n中的标准加权伯格曼空间的操作员组成的新型托管托管代数。这些代数仅在较高的维度设置n> 1中出现，并且它们远非完全理解。在当前的项目中，我们专注于他们的分类和分析。更确切地说，以下任务是我们的主要目标：1。与C^n.2中N维复合物单位球B B的最大交换亚组相对于自动形态组的最大交换亚组的全部分类。尤其是在1中获得的代数的内部结构的描述。我们打算研究其gelfand理论，确定最大的理想空间和自由基。这些代数中的哪个是半简单的？3。作为结果的应用，我们计划决定这些代数的“光谱不变性”问题。此外，我们相信我们可以对光谱理论和弗雷德霍尔姆元素的财产有更深入的了解。如果在更通用的对称域或不同功能性的希尔伯特空间（例如谐波l^2函数）上，可以提出类似的问题。在这些情况下，1。 -3。的完整解决方案似乎很具有挑战性。