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

在国家科学基金会的支持下，提议者将继续他一直在进行的研究。 具体的研究内容包括：（1）模不同模理想的自伴算子交换元组的同时对角化。 这个问题已经解决（与NSF支持）的情况下，规范理想是Schatten p-类时，p严格大于1。 提议者接下来将考虑p为1的情况，即，其中范数理想是迹类。 这伊萨一个困难的问题，但也是一个重要的问题，因为它的潜在应用。提议者也将考虑一类与Schatten类相关的理想。 (2)单位圆上的全Toeplitz代数的由圆的同胚诱导的自同构的完全确定。 这是在以前的NSF支持下完成的，在这种情况下，同胚的问题是双Lipschitz。 最终的目标是消除双Lipschitz条件。 这涉及到Toeplitz代数的范数的一些仔细估计和某些奇异积分算子的使用。 (3)与极小流相关的Toeplitz代数。 这里的最终目标是使用K-理论来描述与这种流相关联的Toeplitz算子系统的可逆性。 （四） 再生核Hilbert空间上的Hankel算子 这里的主要问题是这些算子的Schatten类成员。 再生核将涉及到某些定量的估计.所提出的问题是相当有代表性的当前研究兴趣的算子理论和算子代数，这是一个研究，除其他事项外，各种线性算子的谱性质。 这项研究在一定程度上受到了世纪早期量子理论发展的启发和要求，由伟大的数学家如H. Weyl和J. von Neumann。因为可加性（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到最优控制的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起源于这些应用领域。例如，为了理论上的原因和实际应用，人们常常必须处理或引入在某种程度上是“小”的扰动，问题（1）就是关于这种扰动的。 这个问题的根源可以追溯到Weyl在1909年发表的一篇论文，其中断言，连续谱可以通过紧扰动（这是一种“小”的度量）变成离散谱。 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，边界或增长率）。 一般来说，估计越精确，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.