Certain operators and operator algebras in perturbation theory and on function spaces

申请人将在国家科学基金的资助下继续他一直在进行的研究。本文提出的具体研究领域包括：(1)对各种范数理想模的自伴随算子交换元的同时对角化。当p严格大于1时，范数理想是Schatten p类时，这个问题已经解决了（在nsf的支持下）。提议者接下来将考虑p为1的情况，即，范数理想是迹类。这是一个困难的问题，但由于其潜在的应用，这也是一个重要的问题。提议者还将考虑与Schatten类相关的一类理想。(2)由圆的同胚导出的单位圆上的全Toeplitz代数的自同构的完全确定。这是在NSF以前的支持下完成的，在这种情况下，所讨论的同胚是双lipschitz。最终目标是消除双lipschitz条件。这涉及到对Toeplitz代数中范数的一些谨慎估计和某些奇异积分算子的使用。(3)最小流相关的Toeplitz代数。这里的最终目标是使用k理论来描述与此类流相关的Toeplitz算子系统的可逆性。(4)复制核Hilbert空间上的Hankel算子。这里的主要问题是这些操作符的chattenclass成员关系。繁殖核将参与某些定量估计。所提出的问题相当代表了当前对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。在20世纪早期量子理论的发展的部分启发和要求下，这项研究是由H. Weyl和J. von Neumann等伟大数学家发起的。由于可加性（即线性）出现在自然界的许多基本方面，算子理论为从原子物理学到最优控制等科学领域提供了正确的数学工具。算子理论和算子代数中的许多抽象问题都源于这些领域的应用。例如，无论是出于理论原因还是为了实际应用，人们常常必须处理或引入某种程度上“小”的扰动。问题(1)是关于这种扰动的。这个问题的根源可以追溯到1909年Weyl发表的一篇论文，该论文断言连续光谱可以通过紧致（一种“小”的度量）扰动变成离散光谱。问题(2)需要现代技术和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（即界限或增长率）。一般来说，估计越清晰，得到的定理就越好。

The proposer will continue the research that he hasbeen pursuing under the support of the National Science Foundation. Thespecific areas of the proposed research include the following: (1)Simultaneous diagonalization of commuting tuples of self-adjoint operatorsmodulo various norm ideals. This problem has been solved (with NSFsupport) in the case where the norm ideal is the Schatten p-class when p is strictly greater than 1. The proposer will next consider the casewhere p is 1, i.e., where the norm ideal is the trace class. This isa difficult problem, but this is also an important problem because of itspotential applications. The proposer will also consider a class ofideals which are related to the Schatten class. (2) The completedetermination of automorphisms of the full Toeplitz algebra on the unitcircle which are induced by homeomorphisms of the circle. This has beenaccomplished under previous NSF support in the case where thehomeomorphism in question is bi-Lipschitz. The final goal is to removethe bi-Lipschitz condition. This involves some careful estimates of normsin the Toeplitz algebra and the use of certain singular integral operators. (3) Toeplitz algebras associated with minimal flows. The ultimate goalhere is to use K-theory to characterize the invertibility of systems of Toeplitz operators associated with such flows. (4) Hankel operators oncertain reproducing-kernel Hilbert spaces. Here the main question is theSchatten-class membership of these operator. The reproducing kernelwill be involved in certain quantitative estimates.The proposed problems are fairly representative of the currentresearch interests in operator theory and operator algebras, which is astudy of, among other things, the spectral properties of various linearoperators. In part inspired and demanded by the development of thequantum theory in the early part of the 20th century, this study wasinitiated 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 forscientific fields ranging from atomic physics to optimal control. Manyabstract problems in operator theory and operator algebras owe theirorigin to these fields of applications. For example, both for theoreticalreasons and for practical applications, quite often one must deal with, or introduce, perturbations which are "small" by some measure or other.Problem (1) is about such perturbations. The root of this problem can betraced back to a paper of Weyl published in 1909, which asserts that acontinuous spectrum can be turned into a discrete one by a compact (which a measure of "smallness") perturbation. Problem (2) requires bothmodern techniques and classical-style mathematical analysis. A themewhich underlies all these problems is the establishment of various estimates (i.e., bounds or growth rates). In general, the sharper theestimates, the better theorems one obtains.