Unconditional bases and unconditional finite-dimensional decompositions\n in Banach spaces

1994/08/24 by Peter G. Casazza, Casazza, Peter G., N. J. Kalton +1
Mathematics · #Holomorphic and Operator Theory #Advanced Operator Algebra Research #Advanced Banach Space Theory

paper · pdf · doi:10.48550/arxiv.math/9408207

Abstract

Let X be a Banach space with an unconditional finite-dimensional Schauder\ndecomposition (E_n). We consider the general problem of characterizing\nconditions under which one can construct an unconditional basis for X by\nforming an unconditional basis for each E_n. For example, we show that if\n\sup \dim E_n<\∞ and X has Gordon-Lewis local unconditional structure\nthen X has an unconditional basis of this type. We also give an example of a\nnon-Hilbertian space X with the property that whenever Y is a closed\nsubspace of X with a UFDD (E_n) such that \sup\dim E_n<\∞ then Y\nhas an unconditional basis, showing that a recent result of Komorowski and\nTomczak-Jaegermann cannot be improved.\n

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