A characterization of F-spaces containing an isomorph of $\ell_0$

Abstract

We prove that an F-space $X$ contains an isomorph of $\ell_0$ if and only if there exists a continuous at zero function $T \colon L_0 \to X$ with $T0 = 0$ possessing the following two properties.

(1) $(\forall \varepsilon > 0)(\exists \delta > 0)(\forall x \in L_0) (\forall a > 0) (\exists b > 0) \,\bigl( \|a Tx \| \ge \varepsilon \, \Rightarrow \, \|T(bx) \| \ge \delta \bigr)$, which is a weak version of the positive homogeneity.

(2) For every $x,y,z \in L_0$ with $Tx \neq 0$, if $x$ is a disjoint sum of $y$ and $z$, then either $Ty \neq 0$ or $Tz \neq 0$.

We also provide examples showing that all assumptions on $T$ are essential.