Banach Spaces: Theory and Application

作者将致力于框架理论中的问题和几何泛函分析中的几个基本问题。框架代表了信号处理的理论基础。在A/D转换的背景下，它是至关重要的信号，或者，抽象地说，在帧扩展的系数。利用无限维Banach空间理论，作者打算制定和证明的结果，提供了这样的量化的理论背景。他将建立在已经证明的基础扩展的结果。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有界线性算子，它不是单位的倍数的紧扰动？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。其次，作者提出了研究Banach空间中与坐标系有关的几个结构理论问题。这些问题可以归纳为以下一般问题：给定一个可分的Banach空间，如何接近我们可以找到一个其他空间包含第一个有一个有限维分解（甚至一个基础）？

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)?