Banach Spaces and Operators on them

Schlumprecht作者将研究几何泛函分析中的几个公开问题。第一个问题是问是否每一个无限维的Banach空间承认一个非平凡的运营商，即运营商是不是一个紧凑的扰动的多个身份。主要研究者建议探索这个主题的几个有趣的变化，每一个都具有明显的结构理论风味。上述问题的反例将提出无限维Banach空间的第一个例子，已知其上的每个算子都有不变子空间。其次，他提出了一种方法，最一般的制定不变的子空间问题，即：是否每一个伴随算子有一个非平凡不变的子空间？拟贪婪基是一种数学思想，它为图像压缩和重建提供了理论基础，并要求在每个Banach空间中都存在拟贪婪基。Banach空间，其几何和拓扑结构，以及其上的算子，为研究动力系统，微分方程，物理学和多分辨率分析提供了一个自然的框架。 例如，在提案中指出了Banach空间理论思想如何与图像压缩和重建理论相关联。

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.