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arXiv · 1909.06829

Projections and unconditional bases in direct sums of $\ell_p$ spaces, $0<p\le \infty$

Fernando Albiac·Jose L. Ansorena
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Abstract

We show that every unconditional basis in a finite direct sum $\bigoplus_{p\in A} \ell_p$, with $A\subset (0,\infty]$, splits into unconditional bases of each summand. This settles a 40 year old question raised in [A. Orty\'nski, Unconditional bases in $\ell_{p}\oplus\ell_{q},$ $0<p<q<1$, Math. Nachr. 103 (1981), 109-116]. As an application we obtain that for any $A\subset (0,1]$ finite, the spaces $Z=\bigoplus_{p\in A} \ell_p$, $Z\oplus \ell_{2}$, and $Z\oplus c_{0}$ have a unique unconditional basis up to permutation.

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Fernando Albiac, Jose L. Ansorena. 2019-09-15. Projections and unconditional bases in direct sums of $\ell_p$ spaces, $0<p\le \infty$. https://arxiv.org/abs/1909.06829

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