Refinements of the Blanco-Koldobsky-Turn\v{s}ek Theorem

Abstract

We refine the well-known Blanco-Koldobsky-Turn\v{s}ek Theorem which states that a norm one linear operator defined on a Banach space is an isometry if and only if it preserves orthogonality at every element of the space. We improve the result for Banach spaces in which the set of all smooth points forms a dense $G_{\delta}$-set by proving that a norm one linear operator that preserves orthogonality on a dense subset of the space is an isometry. We further demonstrate that if such an operator preserves orthogonality on a hyperplane not passing through the origin then it is an isometry. In the context of finite-dimensional Banach spaces, we prove that preserving orthogonality on the set all extreme points of the unit ball forces the operator to be an isometry, which substantially refines Blanco-Koldobsky-Turn\v{s}ek theorem. Finally, for finite-dimensional polyhedral spaces, we establish the significance of the set of all $k$-smooth points for any possible $k,$ in the study of isometric theory.