Continuation of an Investigation of Certain Operators and Operator Algebras

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.