Banach Spaces: Theory and Application

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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)?