For $0<p<1,$ we prove that there is a $\mathfrak{ c}$-dimensional subspace of $\mathcal{ L}\left( \ell_{ p},\ell_{ p}\right) $ such that, except for the null vector, all of its vectors fail to be absolutely $(r,s)$-summing regardless of the real numbers $r,s$, with $1\leq s\leq r<\infty$. This extends a result proved by Maddox in 1987. Moreover, the result is sharp in the sense that it is not valid for $p\geq1.$
Categories: math.FA