Lineability and uniformly dominated sets of summing nonlinear operators

TL;DR

This paper extends the theory of summing nonlinear operators by providing an abstract framework, exploring applications in multilinear and nonlinear contexts, and analyzing the size of non-uniformly dominated operator sets through lineability.

Contribution

It introduces an abstract generalization of a 2002 result on absolutely summing operators and investigates the lineability of non-uniformly dominated operator sets.

Findings

Extended the theory of summing nonlinear operators with an abstract approach.

Applied results to multilinear and nonlinear operator frameworks.

Analyzed the lineability of non-uniformly dominated sets of linear operators.

Abstract

In this note we prove an abstract version of a result from 2002 due to Delgado and Pi\~{n}ero on absolutely summing operators. Several applications are presented; some of them in the multilinear framework and some in a completely nonlinear setting. In a final section we investigate the size of the set of non uniformly dominated sets of linear operators under the point of view of lineability.