Uniqueness of unconditional basis in Lorentz sequence spaces

Abstract

We show that the Lorentz sequence spaces $d(\omega ,p)$ with $0<p<1$ and $\inf \frac {\omega _1+\cdots +\omega _n}{n^p}>0$ have unique unconditional basis. This completely settles the question of uniqueness of unconditional basis in Lorentz sequence spaces, and solves a problem raised by Popa in 1981 and Nawrocki and Ortyński in 1985.