Banach space theory and its applications

Banach空间理论及其应用

• 批准号: 0555670
• 负责人: Nigel Kalton
• 金额: $ 26.85万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-04-01 至 2011-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555670&HistoricalAwards=false
• 关键词: Banach; space; theory; its; applications

0555670 KaltonAbstractThis project deals with a number of problems concerning Banach space theory and applications to other areas of mathematics. Part of the project deals with the theory of extensions of Banach spaces and related problems about the existence of extensions of linear operators between Banach spaces. For example, one central problem concerns identifying those subspaces of the classical Lebesgue space of integrable functions that have the property that every operator into a Hilbert space can be extended to the whole space. This amounts to finding subspaces that verify Grothendieck's theorem and is equivalent to the problem of finding spaces with the property that every extension by a Hilbert space is trivial. Similar problems for spaces of continuous functions in place of Hilbert space are also of interest. A second part of the project deals with the nonlinear theory of Banach spaces; a typical open problem here is whether two Lipschitz isomorphic separable Banach spaces are always linearly isomorphic. A third part deals with the study of sectorial operators and possible applications of that study to partial differential equations.Extensions of Banach spaces have a geometrical interpretation in finite dimensions. If, given a centrally symmetric convex set, one only has information about a slice in some directions and the shadow cast in the perpendicular directions, one cannot reconstruct the origin set. However, the hope is to reconstruct the body as accurately as possible, even though this information is incomplete. This is a concrete way of looking at many questions in analysis that arise in quite different forms. The nonlinear theory of Banach spaces and the related Lipschitz structure of metric spaces has potential applications in theoretical computer science in the handling of large data sets. Sectorial operators and semigroups are a basic tool in the study of partial differential equations of evolution type.

0555670 Kalton摘要这个项目涉及到一些问题有关的Banach空间理论和应用到其他领域的数学。该项目的一部分涉及的理论扩展的Banach空间和相关问题的存在扩展的线性算子之间的Banach空间。 例如，一个中心问题涉及确定经典的可积函数的勒贝格空间的那些子空间，这些子空间具有这样的性质，即每个进入希尔伯特空间的算子都可以扩展到整个空间。这相当于找到验证格罗滕迪克定理的子空间，并且等价于找到具有希尔伯特空间的每个扩展都是平凡的性质的空间的问题。 类似的问题空间的连续函数的地方希尔伯特空间也感兴趣。 该项目的第二部分涉及非线性理论的Banach空间;一个典型的开放问题是两个Lipschitz同构可分Banach空间是否总是线性同构。第三部分研究扇形算子及其在偏微分方程中的可能应用。Banach空间的扩展在有限维中有一个几何解释。如果给定一个中心对称的凸集，人们只知道某些方向上的切片信息和垂直方向上的投影信息，就不能重建原点集。 然而，希望是尽可能准确地重建身体，即使这些信息是不完整的。这是一种具体的方法，用来观察精神分析学中以不同形式出现的许多问题。 Banach空间的非线性理论和度量空间的相关Lipschitz结构在处理大数据集的理论计算机科学中有潜在的应用。扇形算子和半群是研究发展型偏微分方程的基本工具。