Mathematical Sciences: Extending Hilbert Space Operators

Agler将继续发展他关于Hilbert空间上的算子的非常一般的扩张理论，该理论将适用于许多类型的算子，包括次正规算子、n-对称算子和m-等距算子。这将需要首先确定与给定算子相匹配的扩展类，为扩展找到一个好的模型理论，最后应用构造来证明新的定理。这些问题上的进展将是许多分析部门感兴趣的。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。

Agler will continue his development of a very general extension theory for operators on Hilbert spaces that would apply to many classes of operators including subnormal operators, n-symmetric operators, and m-isometries. This will require first identifying the class of extensions that would go with a given operator, finding a good model theory for the extension, and finally applying the construction to prove new theorems. Progress on these problems will be of interest to many parts of analysis. 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.