Extending into Isometries of K(X,Y)

In this paper we generalize a result of Hopenwasser and Plastiras (1997) that gives a geometric condition under which into isometries from$\mathcal{L}(\ell^{2})$to$\mathcal{L}(\ell^{2})$have a unique extension to an isometry in$\mathcal{L}(\mathcal{L}(\ell^{2}))$. We show that when X and Y are separable reflexive Banach spaces having the metric approximation property with X strictly convex and Y smooth and such that$\mathcal{K}(X, Y)$is a Hahn-Banach smooth subspace of$\mathcal{L}(X,Y)$, any nice into isometry$\Psi_{0} : \mathcal{K}(X,Y) \rightarrow \mathcal{L}(X, Y)$has a unique extension to an isometry in$\mathcal{L}(\mathcal{L}(X,Y))$.