Banach Spaces and Operators on them

Banach 空间及其上的算子

• 批准号: 0300058
• 负责人: Thomas Schlumprecht
• 金额: $ 12万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-15 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300058&HistoricalAwards=false
• 关键词: Banach; Spaces; Operators; them; Banach; Spaces; Operators; them

AbstractSchlumprechtThe author will work on several open problems in Geometrical Functional Analysis. The first problem asks whether or not every infinite dimensional Banach space admits a non-trivial operator, i.e. an operator that is not a compact perturbation of a multiple of the identity. The principal investigator proposes to explore several interesting variations on this theme, each with a distinctly structure-theoretical flavor. A counter example to aforementioned question would present the first example of an infinite dimensional Banach space for which it is known that every operator on it has an invariant subspace. Secondly, he presents an approach to the most general formulation of the invariant subspace problem, namely: does every adjoint operator have a nontrivial invariant subspace? The third part of the proposal deals with quasi-greedy bases, one of the mathematical ideas that help provide a theoretical groundwork for image compression and reconstruction and asks for the existence of quasi-greedy bases in each Banach space.Banach spaces, their geometric and topological structure, as well as operators on them, provide a natural framework for studying dynamical systems, differential equations, physics and multiresolution analysis. For example, it is pointed out in the proposal how Banach space theoretical ideas relate to the theory of image compression and reconstruction.

摘要Schlumprecht作者将在几何功能分析中处理一些开放问题。第一个问题询问是否每个无限的尺寸Banach空间都承认一个非平凡的操作员，即不是对身份倍数的紧凑扰动的操作员。主要研究者建议探索有关此主题的几种有趣的变体，每个变体都有明显的结构理论风味。上述问题的一个反面示例将介绍一个无限维度Banach空间的第一个例子，众所周知，它的每个操作员都有一个不变的子空间。其次，他为不变子空间问题的最通用表述提供了一种方法，即：每个伴随操作员是否都有非平凡的不变子空间？该提案的第三部分涉及准绿色基础，这是一个数学思想，有助于为图像压缩和重建提供理论基础，并要求在每个Banach空间中存在准绿色的基础。BanachSpace.Banach空间，几何学和拓扑结构，以及在它们上的自然框架，以及对整个方程式进行了分析，并提供了整个方程式。 例如，在提案中指出，巴纳克空间理论思想如何与图像压缩和重建理论相关。