Every Banach ideal of polynomials is compatible with an operator ideal

TL;DR

This paper establishes a unique correspondence between Banach ideals of homogeneous polynomials and Banach operator ideals, and demonstrates the coherence of polynomial ideals across degrees.

Contribution

It proves the existence and uniqueness of a compatible Banach operator ideal for each Banach ideal of homogeneous polynomials, and shows polynomial ideals form coherent sequences.

Findings

Every Banach ideal of homogeneous polynomials has a unique compatible Banach operator ideal.

Polynomial ideals of different degrees form coherent sequences.

The results unify the structure of polynomial and operator ideals.

Abstract

We show that for each Banach ideal of homogeneous polynomials, there exists a (necessarily unique) Banach operator ideal compatible with it. Analogously, we prove that any ideal of $n$-homogeneous polynomials belongs to a coherent sequence of ideals of $k$-homogeneous polynomials.