Invariant Subspaces in Spaces of Analytic Functions

Richter和Sundberg将继续研究解析函数及其自然关联算子的空间。在这个领域中，最好的和最完全理解的例子是单边移位，它简单地取Hilbert空间中由非负整数索引的正交基中的每个元素，并将其“移位”到具有一个更高索引的元素。该算子是具有真正无限维性质的非正常算子的最简单的例子，它的研究在算子理论中具有重要的意义。单侧位移是用复平面上单位圆盘上复解析函数在Hardy空间上的坐标函数z的乘法运算来表示的。正是这种模型一直是研究单边位移的核心，它使我们对它有了非常透彻的理解，并在一般收缩算子理论中得到了重要的结果。自20世纪80年代以来，已经有许多研究表明，与单边位移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。这些空间的重要例子包括狄利克雷空间、伯格曼空间和加权伯格曼空间。Richter和Sundberg将继续研究这些空间和其他空间，特别是为了更好地理解它们的子空间的格在乘以z的操作下不变，以及有关零集、非切极限行为和多项式近似的相关问题。提议的工作涉及纯数学和应用数学的几个领域。算符理论作为泛函分析的一个分支，在19世纪80年代兴起于对物理学和工程学中出现的偏微分方程的研究，并在20世纪随着量子力学的出现而变得越来越重要。复分析是一门有着悠久而杰出的历史和广泛适用性的学科——事实上，它在数学的几乎每个领域以及物理学的许多领域都有重要的应用。特别是，复分析从一开始就在算子理论中占有重要地位，这些领域之间的研究和联系仍然是一个非常富有成果的研究领域。联系的一个重要来源是通过解析函数空间上的自然运算对算子进行建模。这种算子在Hardy空间上的研究在纯数学和应用数学中都具有重要的意义。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的重要领域。自20世纪80年代以来，许多研究人员（包括本作者）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。

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.