On the Lipschitz operator ideal ${\text{Lip}}_0\circ \mathcal{A}\circ {\text{Lip}}_0$

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 Kürsten 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 $\mathcal{A}$.