On the Lipschitz operator ideal $\text{Lip}_{0}\circ \mathcal A\circ \text{Lip}_{0}$

TL;DR

This paper develops a systematic approach to construct Lipschitz operator ideals from Banach linear operator ideals, analyzing their maximal and minimal hulls, and establishing connections between nonlinear and linear operator factorizations.

Contribution

It introduces a method to generate Lipschitz operator ideals from Banach ideals and characterizes their maximal and minimal hulls using ultraproduct techniques.

Findings

Characterization of Lipschitz maximal hulls and minimal kernels.

Identification of linear operators within Lipschitz operator ideals.

Conditions under which nonlinear factorizations imply linear ones.

Abstract

We study a systematic way to produce a Lipschitz operator ideal from a Banach linear operator ideal $\mathcal A$. For maximal and minimal operator ideals $\mathcal A$, the Lipschitz maximal hull and minimal kernel of the Lipschitz operator ideals $\text{Lip}_0 \circ \mathcal A \circ \text{Lip}_0$ are investigated, respectively. Using ultraproduct techniques, we obtain the Lipschitz version of a result of K\"ursten and Piestch which characterizes the maximal hull of any Lipschitz operator ideal. Among other results, we characterize the linear operators which belong to $\text{Lip}_0\circ \mathcal A\circ \text{Lip}_0$ which, in many cases, they are precisely those which are in $\mathcal A$. In particular, we give some cases in which a nonlinear factorization of linear operators implies a linear one, in terms of a given Banach linear operator ideal $\mathcal A$.