Mathematical Sciences: Extending Hilbert Space Operators

[501397]阿格勒继谱定理之后，很难想象还有哪个定理对算符理论的发展及其在数学和科学上的无数应用具有比谱定理更深远的影响。——纳吉膨胀定理。将特定算子类中的一般算子表示为该类中一个好的算子的一部分的想法已经取得了许多成功。本项目进一步发展了这一观点，主要集中在算子理论和一个和几个复杂变量的理论中未解决的问题。希尔伯特空间算子是矩阵的无限推广。向量的无限泛化通常是一个函数，因此希尔伯特空间算子通常被建模为函数空间上的乘法算子。这个项目包括为运营商寻找模型。一旦获得了这样的模型，关于这些算子结构的许多基本问题就变得更加自然了。＊＊＊

9501397 Agler After the spectral theorem it is difficult to think of a theorem that has had a more profound effect on the development of operator theory and its myriad of applications to mathematics and science than the Sz.-Nagy Dilation Theorem. The idea of representing a general operator in a specified class of operators as part of a nice operator in the class has had many successes. This project further develops this point of view with a primary focus on unsolved problems in operator theory and the theories of one and several complex variables. Hilbert space operators are infinite generalizations of matrices. The infinite generalization of a vector is frequently a function and for this reason Hilbert space operators are frequently modeled as the operator of multiplication on a space of functions. This project involves finding models for operators. Once such models are obtained many basic questions about the structure of these operators become more natural. ***