Mathematical Sciences: Operator Theory and Systems Engineering

Helton将致力于用几种不同的方式扩展经典的矩阵值函数内插理论和称为交换提升的算子理论的分支：扩展基本的线性理论，为乘法算子和算子模型的非线性推广发展这样的理论，以及发展函数理论扩展和它们的数值。另一个主要的努力将是将这种类型的数学扩展到非线性算子、控制系统和电路。算子论是研究矩阵的无限维广义的数学部分。特别地，当被限制在有限维子空间时，算子具有通常的线性性质，因此可以用矩阵来表示。算子理论中的中心问题是根据伴随算子(例如伴随算子)或基础空间对满足附加条件的算子进行分类。算符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。

Helton will work to extend the classical theory of interpolation of matrix-valued functions, and the branch of operator theory called the commutant lifting, in several different ways: to extend the basic linear theory, to develop such a theory for nonlinear generalizations of multiplication operators and operator models, and to develop function theoretic extensions and their numerics. Another major effort will be to extend this type of mathematics to nonlinear operators, control systems, and circuits. 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.