Spaceability of sets of p-compact maps

TL;DR

This paper investigates conditions under which sets of linear and non-linear operators, including holomorphic mappings, that are $q$-compact but not $p$-compact are spaceable in Banach spaces, revealing structural and geometric properties.

Contribution

It establishes sufficient conditions for the spaceability of $q$-compact but not $p$-compact operators and extends these results to non-linear and holomorphic mappings.

Findings

Set of $q$-compact but not $p$-compact operators is spaceable under certain conditions.

Contains a copy of $ ext{ell}_s$ in the set when $F = ext{ell}_s$.

Applications to non-linear mappings like polynomial and Lipschitz functions.

Abstract

We provide quite sufficient conditions on the Banach spaces $E$ and $F$ in order to obtain the spaceability of the set of all linear operators from $E$ into $F$ which are $q$-compact but not $p$-compact. Also, under similar conditions over $E$, we prove that this set contains (up to the null operator) a copy of $\ell_s$ whenever $F = \ell_s$. Finally, we give some applications of our previous results to show the spaceability of some sets formed by non-linear mappings (polynomial and Lipschitz) which are $q$-compact but not $p$-compact. The spaceability in the space of holomorphic mappings determined by $p$-compact sets is also considered.