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

该项目集中在扰动理论和Schur乘子理论以及矩阵值函数的逼近、因式分解和逼近问题上。这位首席研究员最近在微扰理论方面取得了重要进展。在他与A.B.Aleksandrov的合作中，证明了在小于1的实数行上的Hölder函数也一定是同级算子Hölder。首席研究员将发展这一理论。特别是，他将在无界算子扰动的情况下研究类似的问题，关于耗散算子的扰动。他还将讨论摄动正规算子的函数估计问题。这类微扰理论问题与舒尔乘子研究中出现的问题密切相关。特别是，首席研究员将研究这个著名的问题，以确定Schatten？的Schur乘子？冯·诺依曼类一定是完全有界的。主要研究人员将使用带矩阵值符号的Hankel和Toeplitz算子来研究非对易分析中的各种问题。特别是，他将致力于矩阵值函数的解析和亚纯逼近问题。在他最近与F.Nazarov和L.Baratchart的结果中，发现了一种新的现象，它导致了可尊敬的矩阵函数类和奇异矩阵函数类的发现。他将发展这一方法，并将结果推广到亚纯逼近的情况。微扰理论的研究将对数学和应用的几个领域产生影响，如数学物理、量子力学和物理学。特别是，这些结果将被用于研究随机薛定谔算子和数学物理中的非线性方程。非对易分析中的因式分解和逼近问题在控制理论和系统理论中有着非常重要的应用。特别是，这类问题在反馈控制器的设计和具有状态空间的线性系统的建模中是非常重要的，状态空间的维度由给定的约束控制。在应用中考虑涉及矩阵值函数的问题尤其重要，因为这对应于多输入的情况？多输出线性系统。

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.