The Shift Operator on Spaces of Analytic and Harmonic Functions

解析函数和调和函数空间上的移位算子

• 批准号: 9732649
• 负责人: William Ross
• 金额: $ 5.98万
• 依托单位: University of Richmond
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9732649&HistoricalAwards=false
• 关键词: Shift; Operator; Spaces; Analytic; Harmonic

Proposal: DMS-9732649Principal Investigator: William T. RossAbstract: In this project, Ross plans to investigate the shift operator (i.e., multiplication by the independent variable) on various spaces of analytic and harmonic functions. Specifically, he plans to look at the lattice of invariant subspaces of the shift operator on the Hardy space of a "slit domain" in the complex plane. Invariant subspaces of Hardy spaces on other types of domains such as the disk, the annulus, and "crescent" domains were examined by Beurling, Sarason, Hitt, Aleman, Richter, Olin, and Yakubovich. Ross also plans to look at the 1 and 2 invariant subspaces of the (formal) shift operator on spaces of harmonic functions on the disk (e.g., the Bergman and Dirichlet spaces) and, in particular, to examine (via duality) how complications in one space are realized in the other. This work combines two fields of mathematics, harmonic function theory and operator theory, both of which stem from classical problems in physics (such as heat diffusion and electrostatics) which are still active areas of research today. It is the hope that a better mathematical understanding of the functions and operators involved will shed new light on the physical processes themselves. Investigations of the mathematics generated by the above physics problems goes back to 18th and 19th century mathematicians such as Fourier, Dirichlet, Kelvin, Weierstrass, and von Neumann and continues today with ground breaking work of mathematicians from all over the world.

提案：DMS-9732649原理研究人员：William T. Rossabstract：在该项目中，Ross计划在分析和谐波功能的各个空间上调查转移操作员（即通过自变量乘法）。具体而言，他计划在复杂平面的“缝隙域”的耐寒空间上查看移位操作员不变子空间的晶格。在其他类型的域上，磁盘，环和“新月”域等其他类型的域上不变的子空间，通过Beurling，Sarason，Hitt，Aleman，Richter，Richter，Olin和Yakubovich检查了。罗斯还计划查看（正式）换档操作员的1和2不变子空间，磁盘上的谐波功能空间（例如，伯格曼和迪里奇的空间），尤其是（通过二元性）如何检查一个空间在另一个空间中实现。这项工作结合了两个数学领域，谐波功能理论和操作者理论，这两者都源于物理学（例如热扩散和静电学）的经典问题，这些问题仍然是当今的活跃研究领域。 希望对所涉及的功能和运营商有更好的数学理解能够为物理过程本身提供新的启示。对上述物理问题产生的数学的调查可以追溯到18世纪和19世纪的数学家，例如Fourier，Dirichlet，Kelvin，Weierstrass和Von Neumann，并继续进行，并继续随着来自世界各地的数学家的艰巨工作。