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

Let $\mathcal{F}$ be a hereditary collection of finite subsets of $\mathbb{N}$. In this paper, we introduce and characterize $\mathcal{F}$-(almost) greedy bases. Given such a family $\mathcal{F}$, a basis $(e_n)_n$ for a Banach space $X$ is called $\mathcal{F}$-greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$, $m \in \mathbb{N}$, and $G_m(x)$, we have $$\|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}.$$ Here $G_m(x)$ is a greedy sum of $x$ of order $m$, and $\mathbb{K}$ is the scalar field. From the definition, any $\mathcal{F}$-greedy basis is quasi-greedy and so, the notion of being $\mathcal{F}$-greedy lies between being greedy and being quasi-greedy. We characterize $\mathcal{F}$-greedy bases as being $\mathcal{F}$-unconditional, $\mathcal{F}$-disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for $\mathcal{F}$-almost greedy bases. Furthermore, we provide several examples of bases that are nontrivially $\mathcal{F}$-greedy. For a countable ordinal $\alpha$, we consider the case $\mathcal{F}=\mathcal{S}_\alpha$, where $\mathcal{S}_\alpha$ is the Schreier family of order $\alpha$. We show that for each $\alpha$, there is a basis that is $\mathcal{S}_{\alpha}$-greedy but is not $\mathcal{S}_{\alpha+1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta$, $$\mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_\alpha\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_\beta\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}.$$