Continuation of an Investigation of Certain Operators and Operator Algebras

PI将继续他在国家科学基金的支持下一直从事的研究。具体的研究内容包括：(1)某些再生核Hilbert空间上的Hankel算子。这里的主要问题是这些算子在各种范数理想中的成员资格。再生核将涉及到许多定量估计。(2)单位球面上么模函数的因式分解。这里的目的是确定单模函数在单变量情况下的某些众所周知的因式分解是否可以推广到多个变量的情况。(3)模不同范数理想的自伴算子交换元组的同时对角化。这个问题已经在范数理想是Schatten p-类的情况下得到了解决(在NSF的支持下)，当p严格大于1时。PI接下来将考虑p是1的情况，即范数理想是迹类的情况。这是一个难题，但由于其潜在的应用前景，这也是一个重要的问题。PI还将考虑与Schatten类相关的一类理想。(4)关于单位球面上柯西投影的某些范数估计。如果它们能够成立，这些估计将为柯西投影提供新的见解。拟议的研究的智力价值在于它突破了一些现有技术的极限，有助于我们更好地理解上述空间、算子和算子理想的结构。所提出的问题很好地代表了当前算子理论和算子代数的研究兴趣，其中包括研究各种线性算子的谱性质。这项研究部分是受到二十世纪早期量子理论发展的启发和要求，由H·韦尔和J·冯·诺伊曼等伟大的数学家发起。由于加性(即线性)出现在自然界的许多基本方面，算符理论为从原子物理到最优控制的科学领域提供了正确的数学工具。算子论和算子代数中的许多抽象问题都源于这些应用领域。在这个意义上，研究算子理论和算子代数的更广泛的影响是它对我们理解物理世界的贡献。例如，无论是出于理论上的原因，还是为了实际应用，人们经常必须处理或引入这样或那样的“小”扰动。问题(3)是关于这种扰动的。这个问题的根源可以追溯到Weyl在1909年发表的一篇论文，该论文断言连续谱可以通过紧致扰动(这是一种“小”的度量)变成离散谱。这里提出的问题既需要现代技术，也需要古典风格的数学分析。所有这些问题背后的一个主题是建立各种估计(即界限)。一般来说，估计得越准确，结果就越好。

The PI will continue the research that he has been pursuing under the support of the National Science Foundation. The specific areas of the proposed research include the following: (1) Hankel operators on certain reproducing-kernel Hilbert spaces. Here the main question is the membership of these operators in various norm ideals. The reproducing kernel will be involved in many quantitative estimates. (2) Factorization of unimodular functions on the unit sphere. The goal here is to determine whether or not certain well-known factorization for unimodular functions in the one-variable case can be generalized to the case of several variables. (3) Simultaneous diagonalization of commuting tuples of self-adjoint operators modulo various norm ideals. This problem has been solved (with NSF support) in the case where the norm ideal is the Schatten p-class when p is strictly greater than 1. The PI will next consider the case where p is 1, i.e., where the norm ideal is the trace class. This is a difficult problem, but this is also an important problem because of its potential applications. The PI will also consider a class of ideals which are related to the Schatten class. (4) Certain norm estimates related to the Cauchy projection on the unit sphere. If they can be established, these estimates will provide new insight on the Cauchy projection.The intellectual merit of the proposed research is that it pushes the limit of some of the existing techniques and it helps us better understand the structure of the spaces, operators and operator ideals mentioned above. The proposed problems are fairly representative of the current research interests in operator theory and operator algebras, which is a study of, among other things, the spectral properties of various linear operators. In part inspired and demanded by the development of the quantum theory in the early part of the twentieth century, this study was initiated by great mathematicians such as H. Weyl and J. von Neumann. Because additivity (i.e., linearity) appears in many fundamental aspects of nature, operator theory provides the right mathematical tools for scientific fields ranging from atomic physics to optimal control. Many abstract problems in operator theory and operator algebras owe their origin to these fields of applications. In this sense the broader impact of research in operator theory and operator algebras is its contribution to our understanding of the physical world. For example, both for theoretical reasons and for practical applications, quite often one must deal with, or introduce, perturbations which are "small" by some measure or other. Problem (3) is about such perturbations. The root of this problem can be traced back to a paper of Weyl published in 1909, which asserts that a continuous spectrum can be turned into a discrete one by a compact (which is a measure of "smallness") perturbation. The problems proposed here require both modern techniques and classical-style mathematical analysis. A theme which underlies all these problems is the establishment of various estimates (i.e., bounds). In general, the sharper the estimates, the better results one obtains.