Invariant Subspaces in Spaces of Analytic Functions

解析函数空间中的不变子空间

• 批准号: 0245384
• 负责人: Stefan Richter
• 金额: $ 24.65万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-01 至 2007-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245384&HistoricalAwards=false
• 关键词: Invariant; Subspaces; Spaces; Analytic; Functions

Richter and Sundberg will continue their research on spaces of analytic functions with their naturally associated operators. The best and most completely understood example in this field is the unilateral shift, which simply takes every element in an orthogonal basis of a Hilbert space indexed by the nonnegative integers and "shifts" it to the element with one higher index. This operator is the simplest example of a nonnormal operator with genuinely infinite dimensional properties and its study has been of major importance in Operator Theory. The unilateral shift is modelled by the operation of multiplication by the coordinate function z on the Hardy space of complex analytic functions on the unit disc in the complex plane. It is this modelling that has been at the heart of the study of the unilateral shift and that has led to our remarkably thorough understanding of it, and also to important results in the theory of general contraction operators. There has been much research since the 1980's that has shown that the ideas used in connection with the study of the unilateral shift have interesting and important extensions to the study of operators modelled in function spaces other than the Hardy space. Among the important examples of such spaces are the Dirichlet Space, the Bergman Space, and the weighted Bergman spaces. Richter and Sundberg will continue to study these and other spaces with a view especially to a better understanding of their lattices of subspaces invariant under the operation of multiplication by z, as well as related questions concerning zero sets, nontangential limiting behavior, and polynomial approximations.The proposed work involves several areas of Pure and Applied Mathematics. Operator Theory as a branch of Functional Analysis, arose in the 1880's in the study of Partial Differential Equations arising in Physics and Engineering, and became increasingly important in the twentieth century with the advent of Quantum Mechanics. Complex Analysis is a subject with a long and distinguished history and a wide applicability - it has in fact important applications in almost every area of Mathematics as well as many areas of Physics. In particular, Complex Analysis has been of importance in Operator Theory from its inception and the investigations and connections between these areas continues to be a very fruitful area of research. One important source of connections is the modelling of operators by natural operations on spaces of analytic functions. The study of such operators on a space called the Hardy space has been of great importance in both Pure and Applied Mathematics. It is at the heart of a certain useful approach in Control Theory, and area of importance in electrical Engineering and the design of guidance systems. Work by a number of researchers since the 1980's (including the present authors), has shown the ideas involved in these studies have important applicability to an extensive class of function spaces and operators.

里希特和桑德伯格将继续他们的研究空间的解析函数与他们的自然相关的运营商。这个领域中最好和最完全理解的例子是单边移位，它简单地将希尔伯特空间中由非负整数索引的正交基中的每个元素“移位”到具有一个更高索引的元素。 这个算子是具有真正无限维性质的非正规算子的最简单的例子，它的研究在算子论中具有重要意义。单向移位是通过在复平面上的单位圆盘上的复解析函数的哈代空间上乘以坐标函数z的运算来模拟的。正是这个模型一直是单边移位研究的核心，并导致我们对它有了非常透彻的理解，也导致了一般收缩算子理论的重要结果。自20世纪80年代以来，已经有许多研究表明，与单边移位研究有关的思想对哈代空间以外的函数空间中建模的算子的研究具有有趣和重要的扩展。 这类空间的重要例子有狄利克雷空间、伯格曼空间和加权伯格曼空间。里希特和桑德伯格将继续研究这些和其他空间，以期特别是更好地了解其格的子空间不变下的操作乘法由z，以及有关问题的零集，非切向限制行为，和多项式approximations.The拟议的工作涉及几个领域的纯数学和应用数学。 算子理论作为泛函分析的一个分支，产生于19世纪80年代物理学和工程学中的偏微分方程的研究中，并随着量子力学的出现在20世纪世纪变得越来越重要。复变分析是一个具有悠久而杰出的历史和广泛适用性的主题-它实际上在几乎所有数学领域以及许多物理领域都有重要的应用。特别是，复杂的分析一直是重要的算子理论从一开始，这些领域之间的调查和连接仍然是一个非常富有成果的研究领域。连接的一个重要来源是通过解析函数空间上的自然操作对算子进行建模。在一个称为哈代空间的空间上研究这样的算子在纯粹数学和应用数学中都具有重要意义。它是控制理论中某些有用方法的核心，也是电气工程和制导系统设计中的重要领域。自20世纪80年代以来，许多研究人员（包括本文作者）的工作表明，这些研究中涉及的思想对广泛的函数空间和算子具有重要的适用性。