Extending into isometries of 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 K( l²) to L(l²) have a unique extension to an isometry in L(l(l²)) . 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 K(X,Y) is a Hahn-Banach smooth subspace of L(X,Y) , any nice into isometry ψ_o : K(X,Y) → L(X,Y) has a unique extension to an isometry in L(L(X,Y)).