Operator Theory and Complex Analysis

提案：DMS-9970376主要研究者：Nathan S. Feldman摘要：Feldman将研究算子理论及其与复分析的相互作用。主要重点将放在研究特殊类的线性算子，是接近正常的运营商，如subnormal，hyponnormal，基本上subnormal运营商。关于余次正规和余亚正规算子的各种形式的循环性的问题将被研究，以及循环性和交织映射的存在之间的关系。其他要探讨的主题包括：次正规算子的超不变子空间的存在性;次正规算子的伴随的“广义特征向量”;次正规算子的主函数，以及它与Toeplitz算子、Hankel算子和自相关算子等算子类的关系。算子理论是矩阵和线性代数思想到无限维的自然延伸。简单来说，运算符只是一个无限数组或数字矩阵，用于涉及无限多未知变量的线性问题。虽然许多棘手的真实的世界问题不是线性的，人们可以经常“线性化”一个给定的问题，以产生一个新的问题，这是线性和可解的，并有一个解决方案，密切接近-因此提供了有价值的信息-原来的物理问题的解决方案。也许这方面最简单的例子是微积分中导数的概念，它用于用直线（线性的东西）近似曲线。在多变量设置中，导数的精确模拟是线性算子。认识到如何有用和强大的工具，即使是一个单一的变量函数的衍生物已成为在所有领域的科学，人们可以很容易地认识到它是多么重要，以充分了解结构的线性算子。虽然有些算子在性质上可能相当“病态”，但作为本项目研究对象的次正常算子具有大量的结构：它们本身非常自然，并且以自然的方式出现。算子理论和线性代数对数学、物理学和工程学的所有分支都产生了深远的影响。人们希望，费尔德曼的研究同样将产生影响，不仅对纯数学，但对应用科学以及。

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.