Linear structure in certain subsets of quasi-Banach sequence spaces

TL;DR

This paper demonstrates the existence of large linear subspaces within certain quasi-Banach sequence spaces where vectors do not possess absolutely summing properties, extending Maddox's 1987 result and establishing sharpness for p<1.

Contribution

It proves the existence of high-dimensional subspaces with non-absolutely summing vectors in quasi-Banach spaces, extending previous results and showing sharpness for p<1.

Findings

Existence of a continuum-dimensional subspace with non-absolutely summing vectors for 0<p<1.

Extension of Maddox's 1987 result to a broader class of spaces.

Result is sharp, not valid for p≥1.

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 $1\leq s\leq r<\infty$. This extends a result proved by Maddox in 1987. Moreover, the result is sharp in the sense that it is not valid for $p\geq1.$