On approximate preservation of orthogonality and its application to isometries
Kalidas Mandal, Jayanta Manna, Kallol Paul, Debmalya Sain
TL;DR
This work investigates when approximate preservation of Birkhoff–James orthogonality by linear operators implies isometric structure, introducing $\epsilon$-orthogonality and the Property P. It develops a detailed polyhedral space framework in which preserving $\epsilon$-orthogonality yields a correspondence between extreme supporting functionals and a facet-to-facet structure, often forcing the operator to be a scalar multiple of an isometry and refining the Blanco-Koldobsky-Turnšek characterization for specific spaces. The main results show that, in particular, the sets $\mathcal{A}_f$ are well behaved (nonempty, open, convex), and preservation induces a unique $g$ for each $f$, leading to a global structural constraint when the counts of extreme points match. Notably, Property P holds for two-dimensional spaces with unit spheres as regular $2n$-gons and for $\ell_\infty^n$, while it fails for $\ell_p$ spaces with $1\le p<\infty$, highlighting which geometries yield isometric rigidity under approximate orthogonality preservation. These findings sharpen isometry characterizations in certain Banach spaces and delineate when approximate orthogonality preservation enforces strong geometric constraints.
Abstract
Motivated by the famous Blanco-Koldobsky-Turnšek characterization of isometries, we study the \textit{approximate preservation of Birkhoff-James orthogonality by a linear operator between Banach spaces}. In particular, we investigate various geometric and analytic properties related to such preservation on finite-dimensional polyhedral Banach spaces. As an application of the results obtained here, we present refinements of the Blanco-Koldobsky-Turnšek characterization of isometries on certain Banach spaces.