Selected topics in perturbation theory, Schur multipliers, and Hankel and Toeplitz Operators in Noncommutative Analysis

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The project is concentrated on problems of perturbation theory and the theory of Schur multipliers as well as approximation and factorization and approximation problems for matrix-valued functions. The principal investigator has achieved recently important progress in perturbation theory. In his joint work with A.B. Aleksandrov it has been shown that a Hölder function on the real line of order less than 1 must also be operator Hölder of the same order. The principal investigator is going to develop this theory. In particular, he is going to work on similar problems in the case of perturbations by unbounded operators, on perturbations of dissipative operators. He is also going to attack the problem of estimating functions of perturbed normal operators. Such problems of perturbation theory are closely related to problems arising in studying Schur multipliers. In particular, the principal investigator is going to work on the famous problem to determine whether a Schur multiplier of a Schatten ? von Neumann class must be completely bounded. The principal investigator is going to use Hankel and Toeplitz operators with matrix-valued symbols to work on various problems in noncommutative analysis. In particular, he is going to work on problems of analytic and meromorphic approximation of matrix-valued functions. In his recent results with F. Nazarov and L. Baratchart a new phenomenon has been found that has resulted in discovering the class of respectable matrix functions and the class of weird matrix functions. He is going to develop this approach and extend the results to the case of meromorphic approximation. The research in perturbation theory will have an impact on several areas of mathematics and applications such as mathematical physics, quantum mechanics, and physics. In particular, the results will be applied in studying random Schrödinger operators and nonlinear equations of mathematical physics. The factorization and approximation problems in noncommutative analysis are very important in applications in control theory and systems theory. In particular, such problems are extremely important in designing feedback controllers and modeling linear systems with state spaces whose dimension is controlled by given restrains. It is especially important in applications to consider problems that involve matrix-valued functions, because this corresponds to the case of multiple input ? multiple output linear systems.