Methods of Hankel and Toeplitz Operators in Noncommutative Function Theory

摘要：V.V. Peller将继续研究Hankel和Toeplitz算子及其在控制理论、预测理论、逼近理论等不同数学领域的应用。研究矩阵值符号的Hankel算子和Toeplitz算子在实际应用中具有重要意义。具有矩阵值符号的Hankel和Toeplitz算子理论的发展取决于非交换函数理论（即矩阵值或算子值函数理论）的进一步发展。另一方面，在非交换函数理论中出现的许多问题可以借助具有矩阵值符号的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表征；用Wiener-Hopf分解表征消失平均振荡（及其他类）的一值函数连续矩阵函数及其它函数类中矩阵函数的幺正插值Wiener-Hopf分解与主题分解的关系；与酉算子相似；解决方案的估计；以及矩阵值符号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，已被证明在数学的许多领域以及控制理论和电气工程中的应用中非常有用。V.V. Peller将继续研究这些有趣的算子在函数理论中的应用，这些函数的值是矩阵而不是实数或复数。这些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；也就是说，在不改变产品价值的情况下，改变产品中各因素的顺序是不可能的。另一方面，这种矩阵值函数在应用中是极其重要的。例如，当一个人必须处理多个输入和多个输出时，它们以一种自然的方式出现在系统理论中。V.V. Peller将研究几个涉及矩阵值函数的重要问题（技术名称如因式分解问题，近似问题，插值问题等）。这些问题的解决将反过来导致关于Hankel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.