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 KaltonAbstract该项目涉及巴拿赫空间理论及其在其他数学领域的应用的许多问题。该项目的一部分涉及 Banach 空间的扩张理论以及 Banach 空间之间线性算子扩张的存在性相关问题。 例如，一个中心问题涉及识别可积函数的经典勒贝格空间的那些子空间，这些子空间具有希尔伯特空间中的每个算子都可以扩展到整个空间的属性。这相当于寻找验证格洛腾迪克定理的子空间，并且相当于寻找具有希尔伯特空间的每个扩展都是平凡属性的空间的问题。 代替希尔伯特空间的连续函数空间的类似问题也令人感兴趣。 该项目的第二部分涉及 Banach 空间的非线性理论；这里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。第三部分涉及扇形算子的研究以及该研究在偏微分方程中的可能应用。Banach 空间的扩展在有限维上具有几何解释。如果给定一个中心对称凸集，人们仅拥有某些方向上的切片和垂直方向上投射的阴影的信息，则无法重建原点集。 然而，希望尽可能准确地重建身体，尽管这些信息并不完整。这是一种看待分析中以不同形式出现的许多问题的具体方法。 Banach 空间的非线性理论和相关的度量空间 Lipschitz 结构在理论计算机科学中处理大数据集方面具有潜在的应用。扇形算子和半群是研究演化型偏微分方程的基本工具。