Operators on Banach Spaces

数学家Stefan Banach定义了一个数学对象，我们现在称之为Banach空间。Banach空间理论推广了Volterra和Hilbert等人的某些工作。Banach空间上的算子构成了物理和工程中许多对象的数学模型。Banach空间算子理论在数学中有很长的历史，并与数学的其他领域相联系，包括偏微分方程组和代数拓扑学。这个项目的重点是Banach空间上的算子理论的某些方面，这些方面与物理、工程和数学的各个领域有关。这个项目的主要目的是将关于Hilbert空间上的算子的有用结果推广到Banach空间的设置上。例如，关于C*-代数扩张的结果在多大程度上适用于Banach空间上的算子代数。由于许多关于Hilbert空间上的算子的证明不能直接推广到Banach空间上的算子，因此推广这些结果需要不同的方法。来自Banach空间理论的结果，特别是概率技术的使用，预计将是有用的。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The mathematician Stefan Banach defined a mathematical object that we now call a Banach space. The theory of Banach spaces extended certain work of Volterra and Hilbert, among others. Operators on Banach spaces form the mathematical models for many objects in physics and engineering. Banach space operator theory has a long history within mathematics itself and is connected to other areas of mathematics, including partial differential equations and algebraic topology. The focus of this project is on certain aspects of the theory of operators on Banach spaces that have relevance for physics, engineering, and various areas of mathematics. The main goal of this project is to extend useful results concerning operators on Hilbert spaces to the setting of Banach spaces. For instance, the extent to which results about extensions of C*-algebras hold for algebras of operators on Banach spaces is considered. Since many proofs about operators on Hilbert spaces do not directly generalize to operators on Banach spaces, it is expected that extending these results requires different approaches. Results from Banach space theory, especially the use of probabilistic techniques, are expected to be useful.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.