Mathematical Sciences: The Nevanlinna Counting Function, Linear-Fractional Models, and Orbits of Operators

数学科学：Nevanlinna 计数函数、线性分数模型和算子轨道

• 批准号: 9706614
• 负责人: Paul Bourdon
• 金额: $ 7.25万
• 依托单位: Washington and Lee University
• 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: 1997-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706614&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nevanlinna; Counting; Function

ABSTRACT Bourdon The project entails an investigation of function-theoretic issues associated with the following question posed by Walter Rudin (in 1988): when will the set of nonnegative powers of a bounded, holomorphic function on the open unit disk constitute an orthogonal subset of the classical Hardy-two space? Previous work by the project's principal investigator suggests that Rudin's question may be answered through an analysis of properties of the Nevanlinna counting function and radial subharmonic functions. The project also involves a continuation of the principal investigator's study of the behavior of functions and operators under iteration: chief objectives here are to characterize the cyclic behavior of a certain class of composition operators acting on the classical Hardy space and to resolve the invariant-subspace problem for hyponormal operators on Hilbert space. Methods to be employed include computer studies of the Nevanlinna counting function (carried out by undergraduates) and an analysis of the extent to which information about orbits of a linear operator can contribute to the establishment of the existence of a nontrivial, invariant subspace for that operator. The project focuses on problems in the areas of function theory and operator theory. The goal of work in function theory is to provide information and tools that enable scientists to understand/predict how functions of certain types will behave in various situations. For example, the study of the behavior of functions under iteration (which forms a component of the project) connects function theory to the analysis of dynamical systems, including systems that behave chaotically. The goal of much current research in operator theory (including the operator-theoretic component of the project) is to determine when a given linear operator preserves a small part of the whole space on which it acts. When smaller parts are preserved, one hopes the whole operator may be understood in terms of its simpler parts. Linear operations include simple processes such as rotation in space and complex processes such as the application of Schroedinger's equation of quantum mechanics to certain function spaces. The project will also contribute to the development of the nation's human-resources: three undergraduate students will receive significant training and research experiences through their participation in the project.

该项目涉及与Walter Rudin（1988）提出的以下问题相关的泛函理论问题的研究：开放单位盘上有界全纯函数的非负幂的集合何时构成经典Hardy-two空间的正交子集？该项目的首席研究员之前的工作表明，Rudin的问题可以通过分析内万林纳计数函数和径向次谐波函数的性质来回答。该项目还涉及到主要研究者对迭代下函数和算子行为的研究的延续：这里的主要目标是表征作用于经典Hardy空间的某一类复合算子的循环行为，并解决希尔伯特空间上次正规算子的不变子空间问题。所采用的方法包括对奈万林纳计数函数的计算机研究（由本科生进行）和对线性算子的轨道信息在多大程度上有助于建立该算子的非平凡不变子空间的存在性的分析。该项目侧重于函数理论和算子理论领域的问题。功能理论工作的目标是提供信息和工具，使科学家能够理解/预测某些类型的功能在各种情况下的行为。例如，对迭代下函数行为的研究（这是项目的一个组成部分）将函数理论与动态系统的分析联系起来，包括行为混乱的系统。当前许多算子理论研究（包括该项目的算子理论部分）的目标是确定给定的线性算子何时保留其作用的整个空间的一小部分。当较小的部分被保留下来时，人们希望整个操作符可以根据其更简单的部分来理解。线性操作包括空间旋转等简单过程和量子力学薛定谔方程在某些函数空间中的应用等复杂过程。该项目还将有助于国家人力资源的发展：三名本科生将通过参与该项目获得重要的培训和研究经验。