The Lipschitz injective hull of Lipschitz operator ideals and applications (Q778780)

In this paper, the authors extend the injective hull procedure to the Lipschitz environment, namely, they establish the injective hull of a Lipschitz operator ideal in pointed metric spaces. The definition of this procedure is: For pointed metric spaces \(X\) and \(Y\), a~Lipschitz operator \(T \in\mathrm{Lip}_0(X, Y)\) belongs to the Lipschitz injective hull of \(\mathcal{I}_{\mathrm{Lip}}\) if there exists a pointed metric space \(Z\) and a Lipschitz operator \(S \in\mathcal{I}_{\mathrm{Lip}}(X, Z)\) such that \(d(Tx, Tx') \le d(Sx, Sx')\), for all \(x, x' \in X\). The authors prove a number of properties/characterizations of ideals generated by their procedure and these results leads them to define new classes of operators such as the class of quasi \(p\)-nuclear Lipschitz operators. Some related issues are also investigated: the relationship between their new procedure and the injective hull procedure of Banach Lipschitz operator ideals, maximal hull and minimal kernel of the involved (new and known) classes, and its connections with the method of composition ideals.