Maximal Spaceability of the set of non-absolutely summing operators in quasi-Banach spaces

Abstract

For every $0<p<1$, we establish maximal spaceability of the bounded operators on $\ell_p$ which fail to be $(r,s)$-absolutely summing for all $1\le s\le r<\infty$, using the coordinate-disjointization construction and Maddox's theorem. We give detailed isometric estimates and prove a corresponding maximal spaceability theorem inside the compact operators. The compact result is based on an explicit diagonal multiplier and a single weakly $1$-summable sequence witnessing the simultaneous failure of all admissible absolute summability conditions. A diagonal characterization, a logarithmic compact example, and extensions to operators with values in quasi-Banach spaces containing a copy of $\ell_p$ are also obtained. The constructed subspaces are isomorphic to $\ell_p$ and have maximal algebraic dimension whenever the ambient operator spaces have dimension $\mathfrak c$.