Lipschitz Grothendieck-integral operators

Abstract

Let $X$ be a pointed metric space and let $E$ be a Banach space. It is known that the Lipschitz space $\mathrm{Lip}_o(X,E^*)$ is isometrically isomorphic to $(\mathcal{F}(X)\widehat{\otimes}_\pi E)^*$, the dual of the projective tensor product of the Lipschitz-free space $\mathcal{F}(X)$ and $E$. Since the injective norm $\varepsilon$ on $\mathcal{F}(X)\otimes E$ is smaller than the projective norm $\pi$, we study Lipschitz Grothendieck-integral operators which are exactly those elements in $\mathrm{Lip}_o(X,E^*)$ which correspond to the elements of $(\mathcal{F}(X)\widehat{\otimes}_\varepsilon E)^*$, the dual of the injective tensor product of $\mathcal{F}(X)$ and $E$.