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舒尔乘子理论的问题，以及矩阵值函数的近似和因子分解和近似问题。主要研究者最近在微扰理论方面取得了重要进展。在他与A.B.亚历山德罗夫它已被证明，一个霍尔德函数的真实的线的秩序小于1也必须经营霍尔德相同的秩序。首席研究员将发展这一理论。特别是，他将工作类似的问题的情况下扰动的无界运营商，扰动耗散运营商。他还将攻击的问题估计职能的扰动正常运营商。这些问题的扰动理论密切相关的问题所产生的研究舒尔乘子。特别是，首席研究员将致力于著名的问题，以确定是否舒尔乘数的Schatten？von Neumann类必须完全有界。主要研究人员将使用Hankel和Toeplitz算子与矩阵值符号来解决非交换分析中的各种问题。特别是，他将致力于问题的分析和亚纯近似矩阵值函数。在他最近与F. Nazarov和L. Baratchart是一种新的现象，它导致了令人尊敬的矩阵函数类和奇怪的矩阵函数类的发现。他将发展这种方法，并扩大结果的情况下，亚纯近似。 微扰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，结果将应用于研究随机薛定谔算子和数学物理的非线性方程。非交换分析中的因子分解和逼近问题在控制论和系统论中有着重要的应用。特别是，这类问题是非常重要的反馈控制器的设计和建模的线性系统的状态空间，其维数控制给定的约束。在应用中考虑涉及矩阵值函数的问题是特别重要的，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.