Mathematical Sciences: Hankel Operators and Their Applications

Treil和Peller将继续研究Hankel算子及其应用。特别是，他们将研究自伴随Hankel算子的谱性质及其与连续时间和离散时间平衡实现的关系；完全正则矢量平稳随机过程的谱表征矩阵值函数的超优逼近；以及由有限秩Hankel算子的特殊逼近，该算子重复初始算子的第一个奇异值。算符理论是研究矩阵的无限维推广的数学部分。特别地，当限制在有限维子空间中时，算子具有通常的线性性质，因此可以用矩阵表示。算子理论的中心问题是对满足附加条件的算子进行分类，这些附加条件是根据相关算子（如伴随算子）或根据基础空间给出的。算子理论是许多数学的基础，也是数学在其他科学中的许多应用的基础。

Treil and Peller will continue their work on Hankel operators and their applications. In particular, they will investigate spectral properties of self-adjoint Hankel operators and their relations with balanced realizations with continuous time and with discrete time; spectral characterizations of the completely regular vectorial stationary random processes; superoptimal approximations of matrix-valued functions; and special approximation by finite rank Hankel operators that repeat first singular values of the initial operator. Operator theory is that part of mathematics that studies the infinite dimensional generalizations of matrices. In particular, when restricted to finite dimensional subspaces, an operator has the usual linear properties, and thus can be represented by a matrix. The central problem in operator theory is to classify operators satisfying additional conditions given in terms of associated operators (e.g. the adjoint) or in terms of the underlying space. Operator theory underlies much of mathematics, and many of the applications of mathematics to other sciences.