This is an announcement for the paper “Strategically reproducible bases and the factorization property” by Richard Lechnerhttps://arxiv.org/search/math?searchtype=author&query=Lechner%2C+R, Pavlos Motakishttps://arxiv.org/search/math?searchtype=author&query=Motakis%2C+P, Paul F. X. Müllerhttps://arxiv.org/search/math?searchtype=author&query=M%C3%BCller%2C+P+F+X, Thomas Schlumprechthttps://arxiv.org/search/math?searchtype=author&query=Schlumprecht%2C+T.
Abstract: We introduce the concept of strategically reproducible bases in Banach spaces and show that operators which have large diagonal with respect to strategically reproducible bases are factors of the identity. We give several examples of classical Banach spaces in which the Haar system is strategically reproducible: multi-parameter Lebesgue spaces, mixed-norm Hardy spaces and most significantly the space $L^1$. Moreover, we show the strategical reproducibility is inherited by unconditional sums.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.09423