Banach Spaces: Theory and Application

作者将致力于解决框架理论中的问题和几何泛函分析中的几个基本问题。帧代表了信号处理的理论基础。在A/D转换的背景下，量化信号，或者抽象地说，量化帧扩展中的系数是至关重要的。利用无限维Banach空间理论，给出并证明了为这种量子化提供理论背景的结果。他将在已经证明可以扩大基础的结果的基础上再接再厉。作者还打算继续研究几何泛函分析的下列基本问题：在任何无限维Banach空间上，是否存在一个不是恒等式的倍数的紧扰动的有界线性算子？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。其次，作者提出了关于Banach空间中的几个结构理论问题，这些问题涉及到它们的坐标系统。这些问题可以归结为以下一般问题：给定一个可分的Banach空间，我们能找到一个包含第一个空间的有限维分解(甚至是基)的另一个空间有多近？

The author will work on problems in Frame Theory and on several fundamental questions in Geometrical Functional Analysis. Frames represent a theoretical underpinning of Signal processing. In the context of A/D conversion it is crucial to quantize signals, or, abstractly speaking, the coefficients in a frame-expansion. Using infinite dimensional Banach space theory the author intends to formulate and prove results which provide the theoretical background for such quantizations. He will build on results already proven for basis-expansions. The author intends also to continue his research on the following fundamental questions of Geometrical Functional Analysis: Does there exist a bounded linear operator on any infinite dimensional Banach space which 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. Secondly, the author proposes to work on several structure theoretical questions in Banach spaces, which concern their co-ordinate systems. These problems can be summarized under the following general question: Given a separable Banach space, how close can we find an other space containing the first one which has a finite dimensional decomposition (or even a basis)?