Operator Theory and Complex Analysis

建议：DMS-9970376首席研究员：内森·S·费尔德曼摘要：费尔德曼将进行算子理论及其与复杂分析相互作用的研究。主要的重点将放在研究接近正常算子的特殊类型的线性算子，如次正规、次正规和本质次正规算子。关于余次正规和次正规算子的各种形式的循环性的问题，以及循环性与交织映射的存在之间的关系也将被研究。简单地说，运算符只是一个无限的数组或数字矩阵，用于涉及无限多个未知变量的线性问题。尽管许多棘手的现实世界问题不是线性的，但人们经常可以将给定的问题线性化，以产生一个既是线性的又可解的新问题，并且它的解非常接近--因此提供了关于--原始物理问题的解的有价值的信息。也许最简单的例子是微积分的导数，它被用来用一条直线(线性的东西)来逼近一条曲线。在多变量设置中，导数的精确模拟是线性运算符。认识到即使是单变量函数的导数在所有科学领域中都已成为一种多么有用和强大的工具，人们很容易理解完全理解线性算子的结构是多么重要。虽然有些运算符的性质可能相当“病态”，但作为本项目研究对象的亚正常运算符具有大量的结构：它们本身是非常自然的，它们以自然的方式出现。算子理论和线性代数对数学、物理和工程的所有分支都产生了深远的影响。人们希望费尔德曼的研究不仅会对纯数学产生影响，也会对应用科学产生影响。

Proposal: DMS-9970376Principal Investigator: Nathan S. FeldmanAbstract: Feldman will conduct research in operator theory and its interplay with complex analysis. Primary emphasis will be placed on studying special classes of linear operators that are close to normal operators, such as subnormal, hyponormal, and essentially subnormal operators. Questions regarding various forms of cyclicity for cosubnormal and cohyponormal operators will be investigated, as will the relationship between cyclicity and the existence of intertwining maps. Other topics to be explored include: the existence of hyperinvariant subspaces for subnormal operators; "generalized eigenvectors" for adjoints of subnormal operators; the principal function for a subnormal operator, and its relations with such classes of operators as Toeplitz operators, Hankel operators, and self-commutators.Operator theory is a natural extension of the ideas of matrices and linear algebra to infinite dimensions. In simple terms, an operator is just an infinite array or matrix of numbers and is used in linear problems involving infinitely many unknown variables. Although many intractable real world problems are not linear, one can frequently "linearize" a given problem to produce a new problem that is both linear and solvable and that has a solution which closely approximates -- hence provides valuable information about -- the solution of the original physical problem. Perhaps the simplest example of this is the idea of the derivative from calculus, which is used to approximate a curve by a straight line (something that is linear). In a multivariable setting, the precise analogue of the derivative is a linear operator. Recognizing how useful and powerful a tool the derivative of even a single variable function has become in all areas of science, one can readily appreciate how important it is to understand fully the structure of linear operators. While some operators can be rather "pathological" in character, the subnormal operators that are the objects of study in this project have a great deal of structure: they are very natural themselves, and they arise in natural ways. Operator theory and linear algebra have had a profound impact on all branches of mathematics, physics, and engineering. It is hoped that Feldman's research will likewise exert an influence not only on pure mathematics but on the applied sciences as well.