This is an announcement for the paper "Non-existence of greedy bases in direct sums of mixed $\ell_{p}$ spaces" by Fernando Albiac and Jose L. Ansorena.

Abstract: The fact that finite direct sums of two or more mutually different spaces from the family ${\ell_{p} : 1\le p<\infty}\cup c_{0}$ fail to have greedy bases is stated in [Dilworth et al., Greedy bases for Besov spaces, Constr. Approx. 34 (2011), no. 2, 281-296]. However, the concise proof that the authors give of this fundamental result in greedy approximation relies on a fallacious argument, namely the alleged uniqueness of unconditional basis up to permutation of the spaces involved. The main goal of this note is to settle the problem by providing a correct proof. For that we first show that all greedy bases in an $\ell_{p}$ space have fundamental functions of the same order. As a by-product of our work we obtain that {\it every} almost greedy basis of a Banach space with unconditional basis and nontrivial type contains a greedy subbasis.

Archive classification: math.FA

Mathematics Subject Classification: 41A35, 46B15 46B45, 46T99

Submitted from: joseluis.ansorena@unirioja.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1401.0693

or

http://arXiv.org/abs/1401.0693