Extending into isometries of $\mathcal {K}(X,Y)$

Abstract:

In this paper we generalize a result of Hopenwasser and Plastiras (1997) that gives a geometric condition under which into isometries from ${\mathcal K}(\ell ^2)$ 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 ${\mathcal L}({\mathcal L}(X,Y))$.