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

The proposer 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) 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 proposer 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 proposer will also consider a class of ideals which are related to the Schatten class. (2) The complete determination of automorphisms of the full Toeplitz algebra on the unit circle which are induced by homeomorphisms of the circle. This has been accomplished under previous NSF support in the case where the homeomorphism in question is bi-Lipschitz. The final goal is to remove the bi-Lipschitz condition. This involves some careful estimates of norms in the Toeplitz algebra and the use of certain singular integral operators. (3) Toeplitz algebras associated with minimal flows. The ultimate goal here is to use K-theory to characterize the invertibility of systems of Toeplitz operators associated with such flows. (4) Hankel operators on certain reproducing-kernel Hilbert spaces. Here the main question is the Schatten-class membership of these operator. The reproducing kernel will be involved in certain quantitative estimates. 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 20th 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. 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 (1) 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 a measure of "smallness") perturbation. Problem (2) requires both modern techniques and classical-style mathematical analysis. A theme which underlies all these problems is the establishment of various estimates (i.e., bounds or growth rates). In general, the sharper the estimates, the better theorems one obtains.