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。最近发现了 Vasilveski 新型交换 Toeplitz Banach 代数，该代数由 C^n 中复数单位球上的标准加权 Bergman 空间上的算子组成。这些代数仅出现在 n>1 的高维设置中，而且它们还远未被完全理解。在当前的项目中，我们重点关注它们的分类和分析。更准确地说，以下任务是我们的主要目标： 1．从属于 C^n.2 中 n 维复数单位球 B 的自同构群的最大交换子群的交换 Toeplitz Banach 代数类型的完整分类。 1中得到的代数内部结构的描述。特别是，我们打算研究他们的Gelfand理论，确定最大理想空间和根式。这些代数中哪些是半单代数？3．作为2.中结果的应用，我们计划解决这些代数的“谱不变性”问题。此外，我们相信我们可以对光谱理论及其元素的 Fredholm 性质有更深入的了解。对于更一般的有界对称域上的托普利茨算子或单位球上的不同函数希尔伯特空间（例如调和 L^2 函数），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相当具有挑战性。