Mathematical Sciences: Operator Theory and Systems Engineering

数学科学：算子理论与系统工程

• 批准号: 9207740
• 负责人: J. William Helton
• 金额: $ 18万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9207740&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Operator; Theory; Systems

Helton will work to extend the classical theory of interpolation of matrix-valued functions, and the branch of operator theory called the commutant lifting, in several different ways: to extend the basic linear theory, to develop such a theory for nonlinear generalizations of multiplication operators and operator models, and to develop function theoretic extensions and their numerics. Another major effort will be to extend this type of mathematics to nonlinear operators, control systems, and circuits. Operator theory is that part of mathematics that studies the infinite dimensional generalizations of matrices. In particular, when restricted to finite dimensional subspaces, an operator has the usual linear properties, and thus can be represented by a matrix. The central problem in operator theory is to classify operators satisfying additional conditions given in terms of associated operators (e.g. the adjoint) or in terms of the underlying space. Operator theory underlies much of mathematics, and many of the applications of mathematics to other sciences.

Helton将以几种不同的方式扩展经典的矩阵值函数插值理论，以及算子理论的分支——交换子提升：扩展基本线性理论，发展乘法算子和算子模型的非线性推广理论，发展函数理论的扩展及其数值。另一个主要的努力将是将这类数学扩展到非线性算子、控制系统和电路。算符理论是研究矩阵的无限维推广的数学部分。特别地，当限制在有限维子空间中时，算子具有通常的线性性质，因此可以用矩阵表示。算子理论的中心问题是对满足附加条件的算子进行分类，这些附加条件是根据相关算子（如伴随算子）或根据基础空间给出的。算子理论是许多数学的基础，也是数学在其他科学中的许多应用的基础。