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.
The principal investigator (PI) continued his earlier work on the behavior of functions of operators under perturbations. In a joint paper of the PI with Aleksandrov, sharp estimates were obtained for functions with a given modulus of continuity. This extends earlier results for Hölder functions. Jointly with Aleksandrov (upper and lower) sharp estimates were found for Lipschitz functions of perturbed self-adjoint operators. It was shown that for piecewise convex-concave functions, perturbations of functions of operators admit much better estimates that for general functions. In particular, famous Kato's inequality for the differnce of the moduli of operators was significantly improved and the estimate obtained is proved to be sharp. Sharp estimates were obtained for Hölder functions under perturbations of Schatten--von Neumann classes. This improves considerably earlier results of Naboko. In a joint paper of the PI with Nazarov sharp estimates were obtained for trace class perturbations and Lipschitz functions. In another joint paper with Aleksandrov nontrivial results were obtained for functions of perturbed maximal dissipative operators. In a joint paper of the PI with Aleksandrov, Potapov, and Sukochev the problem of the behavior of functions of normal operators under perturbation was studied. It turned out that the case of functions of normal operators is considerable more complicated than the case of functions of self-adjoint operators. The authors found new methods that allowed them to obtain highly nontrivial results for functions of normal operators. The PI has been working jointly with Aleksandrov on the book "Operator smooth functions in peturbation theory". Intellectual merit: the outcomes of the award are very important to better understand various aspects of perturbation theory related to the behavior of functions of operators under perturbations, discover new connections with Schur multipliers and Hankel operators, 1find new applications in mathematical physics, control theory, prediction theory and approximation theory. Broader impacts resulting from the award outcomes: the results obtained will find applications in mathematical physics, control theory and systems theory. This will result in a deeper collaboration between pure mathematicians, physicists, and engineers. The results of the proposed activity are broadly disseminated via internet, journals, books, lectures at various conferences. This will lead to teaching new advanced graduate courses and it will raise the number of researchers in this field and in applications, and it will broaden the participation of different ethnic groups. The PI has been teaching graduate courses based on his book on Hankel operators and their applications. The PI conducted mathematical seminars for 5-9 graders and organized annual Mid-Michigan mathematical olympiads for 5-12 grades. The PI published in Notices of the American Mathematical Society an article, in which he dicussed effects of computer presentations at mathematical conferences on the quality of talks.
