Operators on Banach Spaces

数学家斯蒂芬·巴拿赫定义了一个数学对象，我们现在称之为巴拿赫空间。巴拿赫空间理论扩展了沃尔泰拉和希尔伯特等人的某些工作。巴拿赫空间上的算子构成了物理和工程中许多对象的数学模型。巴拿赫空间算子理论在数学本身有着悠久的历史，并与其他数学领域有关，包括偏微分方程和代数拓扑。这个项目的重点是巴拿赫空间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。本课题的主要目的是将Hilbert空间上算子的有用结果推广到Banach空间的集。例如，考虑了C*-代数的扩展结果在Banach空间上对算子代数成立的程度。由于希尔伯特空间上算子的许多证明不能直接推广到巴拿赫空间上的算子，因此我们期望扩展这些结果需要不同的方法。巴拿赫空间理论的结果，特别是对概率技术的使用，被认为是有用的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.