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

Bourdon该项目需要研究与Walter Rudin(1988)提出的下列问题有关的函数论问题：开单位圆盘上有界全纯函数的非负幂的集合何时会构成经典Hardy-2空间的正交子集？该项目的主要研究人员之前的工作表明，鲁丁的问题可以通过分析内瓦林纳计数函数和径向亚调和函数的性质来回答。该项目还包括主要研究者对函数和算子在迭代下的行为的研究的继续：这里的主要目标是刻画作用于经典Hardy空间上的一类复合算子的循环行为，并解决Hilbert空间上亚正规算子的不变子空间问题。所采用的方法包括(由本科生完成的)奈万林纳计数函数的计算机研究，以及分析关于线性算子的轨道的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。该项目专注于函数论和算子理论领域的问题。函数论工作的目标是提供信息和工具，使科学家能够了解/预测某些类型的函数在不同情况下的表现。例如，研究函数在迭代下的行为(形成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。目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持其作用的整个空间的一小部分。当较小的部分被保存下来时，人们希望可以用更简单的部分来理解整个操作员。线性运算包括简单的过程，如空间中的旋转，以及复杂的过程，如将量子力学的薛定谔方程应用于某些函数空间。该项目还将有助于国家人力资源的开发：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。

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.