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

TL;DR

This paper proves that unconditional bases in finite direct sums of _p spaces decompose into bases of each summand, resolving a long-standing question and establishing uniqueness of bases in certain combined spaces.

Contribution

It demonstrates that unconditional bases in finite direct sums of _p spaces split into bases of each component, solving a 40-year-old open problem.

Findings

Unconditional bases in finite _p sums decompose into bases of each summand.

Spaces formed by finite _p sums have unique unconditional bases up to permutation.

Resolved a long-standing question from 1981 about bases in _p _q sums.

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.