Methods of Hankel and Toeplitz Operators in Noncommutative Function Theory

建议：DMS-9970561首席研究员：Vladimir V.Peller摘要：V.V.Peller将继续研究Hankel和Toeplitz算子及其在不同数学领域的应用：控制理论、预测理论、逼近理论。显然，研究带矩阵值符号的Hankel和Toeplitz算子在应用中尤为重要。具有矩阵值符号的Hankel和Toeplitz算子理论的发展依赖于非交换函数理论(矩阵值或算子值函数理论)的进一步发展。另一方面，非对易函数论中的许多问题可以借助于具有矩阵值符号的Hankel和Toeplitz算子来解决。特别是，V.V.Peller将研究以下主题：完全正则向量平稳过程的谱特征；消失平均振荡(和其他类)的酉值函数的Wiener-Hopf分解刻画；连续矩阵函数以及其他函数类中矩阵函数的酉插值；Wiener-Hopf分解与主题分解之间的关系；与酉算子的相似性；预解式的估计；证明了具有矩阵值符号的Toeplitz算子的非平凡不变子空间的存在性。Hankel算子和Toeplitz算子是一类特殊的算子，它们在许多数学领域以及在控制理论和电气工程中的应用都被证明是非常有用的。V·V·佩勒将继续研究这些有趣的算子在函数值为矩阵而不是实数或复数的函数论中的应用。这类函数的理论比经典函数理论要微妙得多，因为这些函数的值不必交换；即，如果改变乘积中各因子的顺序，就会有改变乘积价值的风险。另一方面，这类矩阵值函数在应用中非常重要。例如，当一个人必须处理多个输入和多个输出时，它们在系统论中以一种自然的方式出现。V.V.Peller将研究涉及矩阵值函数的几个重要问题(有技术名称，如因式分解问题、逼近问题、内插问题等)。这些问题的解决将反过来导致关于Hankel算子和Toeplitz算子的新结果以及许多新的具体应用。

Proposal: DMS-9970561Principal Investigator: Vladimir V. PellerAbstract: V.V. Peller is going to continue to work on Hankel and Toeplitz operators and their applications in different fields of mathematics: control theory, prediction theory, approximation theory. It has become clear that it is especially important in applications to study Hankel and Toeplitz operators with matrix-valued symbols. The development of the theory of Hankel and Toeplitz operators with matrix-valued symbols depends on further development of noncommutative function theory (the theory of matrix-valued or operator-valued functions). On the other hand, many problems arising in noncommutative function theory can be solved with the help of Hankel and Toeplitz operators having matrix-valued symbols. In particular, V.V. Peller is going to study the following topics: spectral characterizations of completely regular vectorial stationary processes; characterization of unitary-valued functions of vanishing mean oscillation (and other classes) in terms of Wiener-Hopf factorizations; unitary interpolants of continuous matrix functions as well as of matrix functions in other function classes; the relationship between Wiener-Hopf factorizations and thematic factorizations; similarity to a unitary operator; estimates of resolvents; and the existence of nontrivial invariant subspaces for Toeplitz operators with matrix-valued symbols.Hankel and Toeplitz operators are certain special classes of operators which have proved to be very helpful in many fields of mathematics and for applications in control theory and electrical engineering. V.V. Peller is going to continue his research on applications of these interesting operators to the theory of functions whose values are matrices rather than real or complex numbers. The theory of such functions is considerably more delicate than classical function theory because the values of such functions do not have to commute; i.e., the order of the factors in a product cannot be changed without risk of changing the value of the product. On the other hand, such matrix-valued functions are extremely important in applications. For example, they appear in a natural way in systems theory when one must deal with multiple inputs and multiple outputs. V.V. Peller is going to study several important problems that involve matrix-valued functions (with technical names such as factorization problems, approximation problems, interpolation problems, etc.). The solution of these problems would lead in turn to new results on Hankel and Toeplitz operators as well as to numerous new concrete applications.