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 K ( ℓ 2 ) {\mathcal K}(\ell ^2) to L ( ℓ 2 ) {\mathcal L}(\ell ^2) have a unique extension to an isometry in L ( L ( ℓ 2 ) ) {\mathcal L}({\mathcal L}(\ell ^2)) . We show that when X X and Y Y are separable reflexive Banach spaces having the metric approximation property with X X strictly convex and Y Y smooth and such that K ( X , Y ) {\mathcal K}(X,Y) is a Hahn-Banach smooth subspace of L ( X , Y ) {\mathcal L}(X,Y) , any nice into isometry Ψ 0 : K ( X , Y ) → L ( X , Y ) \Psi _0 :{\mathcal K}(X,Y)\rightarrow {\mathcal L}(X,Y) has a unique extension to an isometry in L ( L ( X , Y ) ) {\mathcal L}({\mathcal L}(X,Y)) .