Weak forms of unconditionality of bases in greedy approximation

From the abstract perspective of Banach spaces, the theory of (nonlinear) greedy approximation using bases sprang from the seminal characterization of greedy bases by Konyagin and Temlyakov in 1999 as those bases that are simultaneously unconditional and democratic [16]. These two properties are, a priori, independent of each other and we find examples of unconditional bases which are not democratic and the other way around already in the very early stages of the theory (see, e.g., [7, Example 10.4.4]). However, the geometry of some spaces X can make these properties intertwine, to the extent that the unconditional semi-normalized bases in X end up being democratic (hence greedy). This is the case of unconditional bases in Hilbert spaces, and also in the spaces l1 and c0 for instance (see [12, Theorem 4.1], [21, Theorem 3] and [10, Corollary 8.6]).