No greedy bases for matrix spaces with mixed ${\ell }_p$ and ${\ell }_q$ norms

Abstract

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

Cited by in corpus (2)

• Renorming spaces with greedy bases
• Non-existence of greedy bases in direct sums of mixed ${\ell }_p$ spaces