Mathematical Sciences: Unconditional Structure in Banach Spaces

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Kalton 9500125 The principal investigator plans to study a number of fundamental questions in the theory of unconditional structure in Banach spaces. Among the questions to be considered is the longstanding problem of whether a complemented subspace of a Banach space with an unconditional basis also has an unconditional basis, and a similar problem for complemented subspaces of Banach lattices. The resolution of these problems could have a profound effect on the theory of Banach spaces. Another question to be studied is whether a Banach space in which every closed subpsace has an unconditional basis (or just local unconditional structure) is necessarily a Hilbert space. A number of other problems concerning unconditional structure are also to be investigated. Unconditional bases are a natural framework for the study of the decomposition of functions (or signals) into basis elements (or simpler pure components) as in the modern theory of wavelets. This proposal is concerned with studying situations where such decompositions exist and where they fail to exist. By studying the possible unconditional bases of a certain space or class of functions one is able to obtain important information concerning the space. ***