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

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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.