The Shift Operator on Spaces of Analytic and Harmonic Functions

提案：DMS-9732649主要研究者：William T.罗斯摘要：在这个项目中，罗斯计划研究移位运算符（即，乘法的独立变量）的各种空间的解析和调和函数。具体来说，他计划看看格不变子空间的移位算子的哈代空间的“缝域”在复杂的平面。不变子空间的哈代空间的其他类型的域，如磁盘，环，和“新月”域进行了检查Beurling，Sarason，希特，阿莱曼，里希特，奥林，和Yakubovich。罗斯还计划研究圆盘上调和函数空间上（形式）移位算子的1和2不变子空间（例如，伯格曼和狄利克雷空间），特别是，研究（通过对偶）如何在一个空间的复杂性实现在其他。这项工作结合了两个领域的数学，调和函数理论和算子理论，这两个都源于经典问题的物理学（如热扩散和静电），这仍然是活跃的研究领域今天。 人们希望，对所涉及的函数和算子的更好的数学理解将为物理过程本身带来新的启示。调查的数学所产生的上述物理问题可以追溯到18日和19日世纪的数学家，如傅立叶，狄利克雷，开尔文，维尔斯特拉斯和冯诺依曼，并继续今天的开创性工作的数学家来自世界各地。

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.