Hankel and Toeplitz Operators in Noncommutative Analysis, Schur Multipliers, and Perturbation Theory

首席研究员将继续他在非对易分析方面的研究。Hankel和Toeplitz算子在本研究中起着重要的作用，它们的符号为矩阵值或算子值。数学分析及其应用的最新发展表明，研究作用于向量空间和算子函数空间上的算子变得非常重要。特别是，Hankel算子将被用来研究关于有理矩阵函数的超优逼近度这一非常重要的问题。这一程度与最小实现空间的维度相吻合，在控制理论的应用中具有重要意义。主要研究人员将继续他在微扰理论方面的工作，并使用他的方法来研究基于积分射影张量积和Schur乘子的多重算子积分。他还将研究VMO(平均振荡消失函数)类酉值矩阵函数的Wiener-Hopf分解，以及带有矩阵值符号的Toeplitz算子预解的估计。该项目还将应用Toeplitz和Hankel算子来刻画满足各种正则性条件的平稳向量高斯过程。首席研究人员目前正在写一本关于微扰理论的书。该项目的预期结果将在控制理论、系统理论、统计学和应用数学中的应用中具有非常重要的意义。这位首席研究员已经成功地将他的结果应用于控制理论和统计学中的非对易分析。他还成功地应用系统论的方法解决了纯数学中的重要问题。拟议的活动将导致纯数学家、应用数学家、统计学家和工程师之间更深层次的合作。拟议活动的结果将通过互联网、期刊、讲座和在各种会议、研讨会等上的演讲广泛传播。这也将导致教授新的高级研究生和本科课程，并招收优秀的研究生，并将扩大包括少数民族在内的不同族裔群体对这项研究的参与。

The principal investigator is going to continue his research in noncommutative analysis. An important role in this research is played by Hankel and Toeplitz operators with matrix-valued or operator-valued symbols. Recent developments of mathematical analysis and its applications show that it becomes very important to study operators acting on spaces of vector and operator functions. In particular, Hankel operators will be used to study the very important problem on the degree of superoptimal approximation of rational matrix functions. This degree coincides with the dimension of the space of minimal realization and is very important in applications in control theory. The principal investigator is going to continue his work in perturbation theory and use his approach to multiple operator integrals based on integral projective tensor products and Schur multipliers. He is also going to work on Wiener-Hopf factorizations of unitary-valued matrix functions of class VMO (functions of vanishing mean oscillation) and on estimates of the resolvents of Toeplitz operators with matrix-valued symbols. The project is also going to apply Toeplitz and Hankel operators to characterize vectorial stationary Gaussian processes satisfying various regularity conditions. The principal investigator is currently at work on a book on perturbation theory.The anticipated results of the project will be very important in applications in control theory, systems theory, statistics, and applied mathematics. The principal investigator has already successfully applied his results in noncommutative analysis to problems in control theory and statistics. He has also successfully applied methods of systems theory to solve important problems in pure mathematics. The proposed activity will result in a deeper collaboration between pure mathematicians, applied mathematicians, statisticians, and engineers. The results of the proposed activity will be broadly disseminated via the internet, journals, lectures and talks at various conferences, seminars, etc. This will also lead to teaching new advanced graduate and undergraduate courses and recruiting strong graduate students and will broaden the participation in this research of different ethnic groups, including ethnic minorities.