No greedy bases for matrix spaces with mixed $\ell_p$ and $\ell_q$ norms

TL;DR

This paper proves that certain mixed $igoplus olimits ext{l}_p$ and $ ext{l}_q$ matrix spaces do not admit greedy bases, resolving an open problem and extending the result to related function spaces.

Contribution

It establishes the non-existence of greedy bases in a broad class of mixed $ ext{l}_p$ and $ ext{l}_q$ matrix spaces, solving a previously open problem.

Findings

Spaces $(igoplus_{n=1}^inite ext{l}_p)_{ ext{l}_q}$ lack greedy bases.

Spaces $(igoplus_{n=1}^inite ext{l}_p)_{c_0}$ and $(igoplus_{n=1}^inite c_0)_{ ext{l}_q}$ lack greedy bases.

Certain Besov spaces on $ ext{R}^n$ also do not have greedy bases.

Abstract

We show that non of the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{\ell_q}$, $1\le p\not= q<\infty$, have a greedy basis. This solves a problem raised by Dilworth, Freeman, Odell and Schlumprect. Similarly, the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{c_0}$, $1\le p<\infty$, and $(\bigoplus_{n=1}^\infty c_o)_{\ell_q}$, $1\le q<\infty$, do not have greedy bases. It follows from that and known results that a class of Besov spaces on $\R^n$ lack greedy bases as well.