Invariant Subspaces in Bergman and Dirichlet Spaces

[摘要]Richter Richter和Sundberg研究了有界解析函数的不变子空间格作为解析函数的Banach或Hilbert空间上的乘法算子的结构。关于单侧移位的不变子空间的知识（即在单位盘的Hardy空间上乘以z）通过膨胀和模型理论来增强我们对Hilbert空间上的收缩算子的知识。所考虑的空间包括单位圆盘的狄利克雷空间和伯格曼空间。由此产生的算子理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。研究作用于希尔伯特空间的线性算子可以看作是线性代数在无限维空间中的扩展。它在偏微分方程、流体力学和量子力学等领域有历史渊源和应用。Richter和Sundberg的工作关注的是这些操作者的某些自然模型。期望对这些模型的研究将对希尔伯特空间算子的一般结构性质产生深入的了解。

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.