Weak compactness and strongly summing multilinear operators

TL;DR

This paper investigates the properties of strongly summing multilinear operators and polynomials, revealing that certain subclasses exhibit weak compactness and favorable factorization properties, unlike the general case.

Contribution

It identifies a subclass of strongly summing multilinear operators and polynomials that possess weak compactness and enhanced properties, extending classical theorems.

Findings

A subclass shares properties like Grothendieck's and Pietsch's theorems.

This subclass exhibits weak compactness unlike the general case.

A natural factorization theorem is established for this subclass.

Abstract

Every absolutely summing linear operator is weakly compact. However, for strongly summing multilinear operators and polynomials - one of the most natural extensions of the linear case to the non linear framework - weak compactness does not hold in general. We show that a subclass of the class of strongly summing multilinear operators/polynomials, sharing its main properties such as Grothendieck's Theorem, Pietsch Domination Theorem and Dvoretzky-Rogers Theorem, has even better properties like weak compactness and a natural factorization theorem.