Type and cotype of multilinear operators

TL;DR

This paper introduces a new framework for understanding the type and cotype of multilinear operators, extending classical linear operator ideals to the multilinear setting and exploring their properties.

Contribution

It proposes definitions for multilinear type and cotype, studies their properties, and relates them to existing linear ideals, establishing maximality and stability results.

Findings

Multilinear type and cotype classes are defined and analyzed.

The classes are shown to be maximal and Aron-Berner stable.

Connections to linear operator ideals are established.

Abstract

The aim of this paper is to start the study of multilinear generalizations of the classical ideals of linear operators of type $p$ and cotype $q$. As a first step in a theory we believe will be long and fruitful, we propose a notion of type and cotype of multilinear operators and the resulting classes of such mappings are studied in the setting of the theory of Banach/quasi-Banach ideals of multilinear operators. Distinctions between the linear and the multilinear theories are pointed out, typical multilinear features of the theory are emphasized and many illustrative examples are provided. The classes we introduce are related to the multi-ideals generated by the linear ideals of operators of some type/cotype and are proved to be maximal and Aron-Berner stable.