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。在这项研究中发挥了重要作用的Hankel和Toeplitz算子与矩阵值或算子值的符号。近年来数学分析及其应用的发展表明，研究作用于向量空间和算子函数空间的算子变得十分重要。特别地，Hankel算子将被用于研究有理矩阵函数的超优逼近度这一非常重要的问题。这个度与最小实现空间的维数一致，在控制理论的应用中非常重要。首席研究员将继续他的工作在微扰理论和使用他的方法，以多个算子积分的基础上，积分投影张量积和舒尔乘数。 他还将工作的维纳-霍普夫因式分解的一元值矩阵函数类VMO（职能消失平均振荡）和估计的解决方案的Toeplitz运营商与矩阵值的符号。该项目还将应用Toeplitz和Hankel算子来表征满足各种正则性条件的向量平稳高斯过程。主要研究者目前正在编写一本关于扰动理论的书。该项目的预期结果将在控制理论、系统理论、统计学和应用数学方面的应用中非常重要。主要研究者已经成功地将他在非交换分析中的结果应用于控制理论和统计学问题。他还成功地运用系统论的方法来解决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入的合作。拟议活动的结果将通过互联网、期刊、各种会议、研讨会等的讲座和谈话广泛传播，这也将导致教授新的高级研究生和本科生课程，招收优秀的研究生，并将扩大包括少数民族在内的不同民族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.