Linear structure in certain subsets of quasi-Banach sequence spaces

Abstract

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 $1\le s\le r<\infty$. This extends a result proved by Maddox in 1987. Moreover, the result is sharp in the sense that it is not valid for $p\ge 1.$