No greedy bases for matrix spaces with mixed ℓpℓp and ℓqℓq norms

We show that none of the spaces (⨁n=1∞ℓp)ℓq, 1≤p≠q<∞1≤p≠q<∞ have a greedy basis. This solves a problem raised by Dilworth, Freeman, Odell and Schlumprecht. Similarly, the spaces (⨁n=1∞ℓp)c0, 1≤p<∞1≤p<∞, and (⨁n=1∞co)ℓq, 1≤q<∞1≤q<∞, do not have greedy bases. It follows from that and known results that a class of Besov spaces on RnRn lack greedy bases as well.