Large structures within the class of summing operators

TL;DR

This paper explores the large linear structures within classes of multilinear summing operators on quasi-Banach spaces, introducing new geometric notions and demonstrating significant spaceability results.

Contribution

It introduces pointwise-lineability and $(eta,eta)$-lineability in the context of multilinear summing operators, proving large spaceability results for specific operator classes.

Findings

The class of absolutely but not multiple summing multilinear operators on $\, ext{ell}_q$ is pointwise $ ext{c}$-spaceable.

Spaceability is established for Dunford-Pettis operators.

Large linear structures are found in multilinear operators on summable scalar families.

Abstract

We investigate lineability/spaceability problems within the setting of multilinear summing operators on quasi-Banach sequence spaces. Furthermore, we deal with the contemporary geometric notions of pointwise-lineability and $(\alpha,\beta)$-lineability. Among other results, we prove that the class of multilinear operators taking values on $\ell_q \, (0<q \leq \infty)$ that are absolutely but not multiple summing is pointwise $\mathfrak{c}$-spaceable when non-empty. Spaceability in other classes, such as Dunford-Pettis operators and multilinear operators on general summable scalar families, is also studied.