Hankel and Toeplitz Operators in Noncommutative Analysis, Schur Multipliers, and Perturbation Theory

Michigan State University, East Lansing MI

Investigators

Abstract

The principal investigator is going to continue his research in noncommutative analysis. An important role in this research is played by Hankel and Toeplitz operators with matrix-valued or operator-valued symbols. Recent developments of mathematical analysis and its applications show that it becomes very important to study operators acting on spaces of vector and operator functions. In particular, Hankel operators will be used to study the very important problem on the degree of superoptimal approximation of rational matrix functions. This degree coincides with the dimension of the space of minimal realization and is very important in applications in control theory. The principal investigator is going to continue his work in perturbation theory and use his approach to multiple operator integrals based on integral projective tensor products and Schur multipliers. He is also going to work on Wiener-Hopf factorizations of unitary-valued matrix functions of class VMO (functions of vanishing mean oscillation) and on estimates of the resolvents of Toeplitz operators with matrix-valued symbols. The project is also going to apply Toeplitz and Hankel operators to characterize vectorial stationary Gaussian processes satisfying various regularity conditions. The principal investigator is currently at work on a book on perturbation theory. The anticipated results of the project will be very important in applications in control theory, systems theory, statistics, and applied mathematics. The principal investigator has already successfully applied his results in noncommutative analysis to problems in control theory and statistics. He has also successfully applied methods of systems theory to solve important problems in pure mathematics. The proposed activity will result in a deeper collaboration between pure mathematicians, applied mathematicians, statisticians, and engineers. The results of the proposed activity will be broadly disseminated via the internet, journals, lectures and talks at various conferences, seminars, etc. This will also lead to teaching new advanced graduate and undergraduate courses and recruiting strong graduate students and will broaden the participation in this research of different ethnic groups, including ethnic minorities.