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

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This project is devoted to the study of various problems in noncommutative analysis. In particular, it is concentrated on problems in perturbation theory. One of the most important problems of the proposal is the problem to estimate functions of perturbed noncommuting pairs of self-adjoint operators. This problem is very important in many applications, in particular, in quantum mechanics. Other important problems in perturbation theory deal with trace formulas for functions of perturbed operators, functional calculus for almost commuting self-adjoint operators, and estimates for operator and commutator moduli of continuity. The project hopes to reveal new connections with Schur multipliers, Hankel operators, and Hankel tensors; comprehend the role of Hankel and Toeplitz operators in noncommutative analysis; develop techniques of Schur multipliers in operator theory and harmonic analysis; find new applications in mathematical physics, control theory, prediction theory, and approximation theory. The problems of perturbation theory that are the focus of this project are important in applications in quantum mechanics, mathematical physics, electrical engineering, and control theory, and in other fields of science. The importance of perturbation theory in applications stems from the fact that if one is faced with an equation that describes a given situation (say, a physical one), the equation is often very complicated. One can replace it with a simpler (perturbed) equation that is reasonably close to the original equation and solve the new equation. Then it is time to invoke the latest results of perturbation theory to find out whether the solutions of the perturbed equation are reasonably close to the solutions of the initial equation that is needed for applications. This project considers important problems in perturbation theory that can help in such commonly occurring situations. Other problems described in the proposal are also important in applications such as prediction theory, control theory, and approximation theory.