Invariant Subspaces in Bergman and Dirichlet Spaces

Bergman 和 Dirichlet 空间中的不变子空间

• 批准号: 9706905
• 负责人: Stefan Richter
• 金额: $ 15.9万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-05-15 至 2001-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706905&HistoricalAwards=false
• 关键词: Invariant; Subspaces; Bergman; Dirichlet; Spaces

Abstract 9706905 Richter Richter and Sundberg will investigate the structure of the invariant subspace lattices of bounded analytic functions acting as multiplication operators on Banach or Hilbert spaces of analytic functions. The knowledge about the invariant subspaces of the unilateral shift (i.e. multiplication by z on the Hardy space of the unit disc) has been used via dilation and model theory to enhance our knowledge about contraction operators on Hilbert spaces. The spaces that will be considered include the Dirichlet and Bergman spaces of the unit disc. The operator theoretic questions that arise are naturally related to questions from classical analysis, such as questions about zero sets and sets of uniqueness. The study of linear operators acting on Hilbert spaces can be viewed as an extension of linear algebra to an infinite dimensional setting. It has historical origins and applications in such fields as partial differential equations, fluid dynamics, and quantum mechanics. The work of Richter and Sundberg is concerned with certain natural models for such operators. It is expected that a study of these models will yield insight into general structural properties of Hilbert space operators.

摘要9706905 Richter、Richter和Sundberg将研究在解析函数的Banach或Hilbert空间上充当乘法算子的有界解析函数不变子空间格的结构。关于单边移位不变子空间的知识(即在单位圆盘的Hardy空间上乘以z)，通过伸缩和模型理论来加强我们对Hilbert空间上的压缩算子的了解。将考虑的空间包括单位圆盘的Dirichlet空间和Bergman空间。出现的算子论问题自然与经典分析中的问题有关，例如关于零集和唯一性集的问题。对作用在Hilbert空间上的线性算子的研究可以看作是线性代数在无限维环境下的推广。它在偏微分方程组、流体力学和量子力学等领域有着悠久的起源和应用。里希特和桑德伯格的工作关注的是这类操作员的某些自然模型。对这些模型的研究有望深入了解希尔伯特空间算子的一般结构性质。