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

这个项目致力于研究非交换分析中的各种问题。特别地，它集中于摄动理论中的问题。该方法的一个重要问题是自伴随算子的摄动非交换对的函数估计问题。这个问题在很多应用中都很重要，特别是在量子力学中。摄动理论中的其他重要问题涉及摄动算子的函数的迹公式，几乎可交换自伴随算子的泛函演算，以及连续性算子和对易子模的估计。该项目希望揭示与舒尔乘子、汉克尔算子和汉克尔张量之间的新联系；理解Hankel算子和Toeplitz算子在非交换分析中的作用；发展了算符理论和谐波分析中的舒尔乘法器技术；在数学物理、控制理论、预测理论和近似理论中寻找新的应用。本课题重点研究的微扰理论问题在量子力学、数学物理、电气工程、控制理论以及其他科学领域的应用中具有重要意义。微扰理论在应用中的重要性源于这样一个事实：如果一个人面对一个描述给定情况（比如物理情况）的方程，这个方程通常是非常复杂的。人们可以用一个更简单的（扰动）方程来代替它，这个方程与原方程相当接近，并求解新方程。然后是时候引用摄动理论的最新结果来确定摄动方程的解是否合理地接近于应用所需的初始方程的解。这个项目考虑了摄动理论中的重要问题，这些问题可以帮助解决这种常见的情况。在建议中描述的其他问题在预测理论，控制理论和近似理论等应用中也很重要。

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.