A characterization of F-spaces containing an isomorph of $\ell_0$ - Fotiy, O.; Popov, M.; Ukrainets, O. (Carpathian Mathematical Publications, 2025)
We prove that an F-space $X$ contains an isomorph of $\ell_0$ if and only if there exists a continuous at zero function $T \colon L_0 → X$ with $T0 = 0$ possessing the following two properties. (1) $(\forall ε > 0)(\exists δ > 0)(\forall x \in L_0) (\forall a > 0) (\exists b > 0) \,\bigl( \|a Tx \| \ge ε \, \Rightarrow \, \|T(bx) \| \ge δ \bigr)$, which is a weak version of the positive homogeneity. (2) For every $x,y,z \in L_0$ with $Tx eq 0$, if $x$ is a disjoint sum of $y$ and $z$, then either $Ty eq 0$ or $Tz eq 0$. We also provide examples showing that all assumptions on $T$ are essential.