Schreier families and \(\mathcal{F}\)-(almost) greedy bases (Q6622790)

This paper concerns the characterization of \(\mathcal{F}\)-greedy bases which interpolate between greedy bases and quasi-greedy bases. It is shown that a basis \((e_n)\) in a Banach space is \(\mathcal{F}\)-greedy if and only if it is quasi-greedy, \(\mathcal{F}\)-unconditional, and \(\mathcal{F}\)-disjoint democratic. Moreover, among others, the authors prove that a basis \((e_n)\) is \(\mathcal{F}\)-almost greedy if and only if it is quasi-greedy and \(\mathcal{F}\)-disjoint democratic. Examples of bases being nontrivially \(\mathcal{F}\)-greedy are also given.