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

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.

AbstractSchlumprecht作者将致力于解决几何泛函分析中的几个开放问题。第一个问题询问是否每个无限维 Banach 空间都承认一个非平凡算子，即不是多个恒等式的紧扰动的算子。首席研究员建议探索这个主题的几个有趣的变体，每个变体都具有明显的结构理论风格。上述问题的反例将呈现无限维 Banach 空间的第一个示例，已知该空间上的每个算子都具有不变子空间。其次，他提出了一种解决不变子空间问题最一般形式的方法，即：是否每个伴随算子都有一个非平凡的不变子空间？该提案的第三部分涉及拟贪婪基，这是有助于为图像压缩和重建提供理论基础的数学思想之一，并要求每个 Banach 空间中都存在拟贪婪基。Banach 空间，它们的几何和拓扑结构及其算子为研究动力系统、微分方程、物理和多分辨率分析提供了一个自然的框架。 例如，提案中指出Banach空间理论思想如何与图像压缩和重建理论联系起来。