The Shift Operator on Spaces of Analytic and Harmonic Functions

摘要：在本项目中，Ross计划研究解析函数和调和函数的各种空间上的移位算子（即与自变量相乘）。具体来说，他计划在复平面的“狭缝域”的Hardy空间上观察移位算子的不变子空间的格。Beurling、Sarason、Hitt、Aleman、Richter、Olin和Yakubovich研究了Hardy空间在盘形、环形和新月形等其他类型域上的不变子空间。Ross还计划研究盘上调和函数空间（例如Bergman和Dirichlet空间）上（形式）移位算子的1和2不变子空间（例如，Bergman和Dirichlet空间），特别是（通过对偶性）检查一个空间中的复杂性如何在另一个空间中实现。这项工作结合了数学的两个领域，调和函数理论和算子理论，两者都源于物理学中的经典问题（如热扩散和静电），这些问题在今天仍然是活跃的研究领域。人们希望对所涉及的函数和运算符有更好的数学理解，从而为物理过程本身带来新的曙光。对上述物理问题产生的数学的研究可以追溯到18世纪和19世纪的数学家，如傅里叶、狄利克雷、开尔文、魏尔斯特拉斯和冯·诺伊曼，并在今天继续着来自世界各地数学家的开创性工作。

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.