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

基本信息

  • 批准号:
    9706614
  • 负责人:
  • 金额:
    $ 7.25万
  • 依托单位:
  • 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

项目摘要

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.
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.

项目成果

期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)

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Paul Bourdon其他文献

Reproducing Kernel Hilbert Spaces Supporting Nontrivial Hermitian Weighted Composition Operators
  • DOI:
    10.1007/s11785-011-0212-8
  • 发表时间:
    2011-12-21
  • 期刊:
  • 影响因子:
    0.800
  • 作者:
    Paul Bourdon;Wenling Shang
  • 通讯作者:
    Wenling Shang

Paul Bourdon的其他文献

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{{ truncateString('Paul Bourdon', 18)}}的其他基金

Collaborative Research in Operator Theory on Holomorphic Function Spaces
全纯函数空间算子理论的协同研究
  • 批准号:
    0100290
  • 财政年份:
    2001
  • 资助金额:
    $ 7.25万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: RUI: Linear- Fractional Models and Their Applications
数学科学:RUI:线性-分数阶模型及其应用
  • 批准号:
    9401206
  • 财政年份:
    1994
  • 资助金额:
    $ 7.25万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Linear Fractional Models and Composition Operators
数学科学:线性分数模型和复合算子
  • 批准号:
    9023427
  • 财政年份:
    1991
  • 资助金额:
    $ 7.25万
  • 项目类别:
    Standard Grant

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