Mathematical Sciences: Extending Hilbert Space Operators

Agler将继续发展Hilbert空间上算子的一个非常普遍的扩展理论，该理论将适用于许多类算子，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运算符相对应的扩展类，为该扩展找到一个好的模型理论，最后应用该构造来证明新的定理。这些问题的进展将引起许多分析部门的兴趣。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。

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.