On multi-ideals and polynomial ideals of Banach spaces: a new approach to coherence and compatibility

TL;DR

This paper introduces a new framework for extending operator ideals to polynomial and multilinear settings using pairs of polynomial ideals and multi-ideals, addressing coherence issues in real scalar cases.

Contribution

It proposes a novel approach considering pairs of polynomial and multi-ideals, improving the theory of coherence and compatibility for operator ideal extensions.

Findings

The new approach resolves coherence issues for real scalar cases.

It offers a more consistent extension framework for operator ideals.

The framework enhances understanding of polynomial and multilinear ideal structures.

Abstract

What is an adequate extension of an operator ideal I to the polynomial and multilinear settings? This question motivated the appearance of the concepts of coherent sequences of polynomial ideals and compatibility of a polynomial ideal with an operator ideal, introduced by D. Carando el al. We propose a different approach by considering pairs (U_{k},M_{k})_{k=1}^{\infty}, where (U_{k})_{k=1}^{\infty} is a polynomial ideal and (M_{k})_{k=1}^{\infty} is a multi-ideal, instead of considering just polynomial ideals. Our approach ends a discomfort caused by the previous theory: for real scalars the canonical sequence (P_{k})_{k=1}^{\infty} of continuous k-homogeneous polynomials is not coherent according to the definition of Carando et al.