Banach Spaces and Operators on them

摘要本文将讨论几何泛函分析中的几个开放性问题。第一个问题是，是否每一个无限维的Banach空间都允许一个非平凡算子，即一个非单位元的紧摄动的算子。首席研究员建议在这个主题上探索几个有趣的变化，每个都具有明显的结构理论风味。上述问题的反例是无限维巴拿赫空间的第一个例子，已知其上的每个算子都有一个不变子空间。其次，他提出了一种解决不变子空间问题的最一般公式的方法，即：是否每个伴随算子都有一个非平凡的不变子空间？第三部分讨论拟贪婪基，拟贪婪基是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。巴拿赫空间及其几何和拓扑结构，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，在提案中指出了巴拿赫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.