Hyper-ideals of multilinear operators

TL;DR

This paper introduces hyper-ideals of multilinear operators, a new nonlinear concept that extends traditional ideals by considering composition with multilinear operators, revealing richer structures and properties.

Contribution

It develops the theory of hyper-ideals, a novel class of multilinear operator ideals based on multilinear composition, expanding the understanding beyond classical multi-ideals.

Findings

Hyper-ideals have desirable general properties.

Hyper-ideals provide examples where multi-ideals do not.

The theory enriches the structure of multilinear operator analysis.

Abstract

We introduce and develop the notion of hyper-ideals of multilinear operators between Banach spaces. While the well studied notion of ideals of multilinear operators (multi-ideals) relies on the composition with linear operators, the notion we propose, by considering the composition with multilinear operators, explores more deeply the nonlinear feature of the subject. The results we prove show that, although more restrictive a priori, hyper-ideals enjoy nice general properties and its theory is rich enough to provide distinguished examples in every situation where the corresponding multi-ideal fails to be a hyper-ideal.