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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. Vasilveski最近在C^n中的复单位球上的标准加权Bergman空间上发现了由算子构成的新的交换Toeplitz Banach代数。这些代数只出现在更高的维度设置n>1，他们仍然远远没有被完全理解。在本项目中，我们专注于他们的分类和分析。具体来说，我们的主要目标是：1. C ^n中n维复单位球B的自同构群的极大交换子群所从属的交换Toeplitz Banach代数类型的完全分类。在1中得到的代数的内部结构的描述。特别地，我们打算研究他们的Gelfand理论，确定极大理想空间和根。哪些代数是半单的？3. 作为2. 我们计划决定这些代数的“谱不变性”问题。并且我们相信我们可以对谱理论和它们的元素的Fredholm性质有更深的理解。类似的问题可以在更一般的有界对称域上的Toeplitz算子或单位球上的不同泛函希尔伯特空间（例如调和L^2-函数）的情况下提出。在这些情况下，完全解决方案1。- 3.似乎很有挑战性