Banach Spaces and Operators on them

本文作者将致力于几何泛函分析中的几个公开问题。第一个问题是问是否每个无限维Banach空间都有一个非平凡算子，即一个不是恒等式倍数的紧扰动的算子。首席研究员建议探索这个主题的几个有趣的变体，每个变体都有明显的结构理论味道。上述问题的一个反例将给出无限维Banach空间的第一个例子，对于该空间，已知其上的每个算子都有一个不变子空间。其次，他给出了不变子空间问题最一般的表述方法，即：每个伴随算子都有一个非平凡不变子空间吗？该提案的第三部分涉及准贪婪基，这是一种有助于为图像压缩和重建提供理论基础的数学思想之一，并要求在每个Banach空间中存在拟贪婪基。Banach空间及其几何和拓扑结构以及它们上的算子为研究动力系统、微分方程、物理和多分辨分析提供了一个自然的框架。例如，提案中指出了Banach空间理论思想与图像压缩和重建理论的关系。

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.