On directional preservation of orthogonality and its application to isometries
Jayanta Manna, Kalidas Mandal, Kallol Paul, Debmalya Sain
TL;DR
This work analyzes the local preservation of Birkhoff-James orthogonality under bounded linear operators between normed spaces and develops a complete framework to understand when such preservation holds locally. By introducing associated cones $\mathcal{V}(x)$ built from the extreme support functionals $Ext\,J(x)$, it provides a precise local-preservation criterion: $T$ preserves BJ-orthogonality at $x$ if and only if $T(V)\subset S$ for some convex $S\subset (Tx)^{\perp_B}$ for every $V\in \mathcal{V}(x)$, with dual-space versions using $Ext\,J(x)$ and $V\in\mathcal{V}(x^*)$. A directional preservation condition is established: for $y\in x^{\perp_B}\setminus\{0\}$, preservation holds iff there exist $u,v\in x^{\perp_B}\setminus\{0\}$ with $Tu\in (Tx)^+$ and $Tv\in (Tx)^-$, and the paper derives kernel-based results and order-of-smoothness transfer in finite dimensions. These results yield refinements of the Blanco-Koldobsky-Turnšek Theorem for polyhedral spaces, including $\ell_{\infty}^{n}$ and $\ell_{1}^{n}$, and provide isometry-type criteria based on the images of extreme points, enhancing geometric understanding of BJ-orthogonality and its preservation in key normed spaces.
Abstract
We study the local preservation of Birkhoff-James orthogonality by linear operators between normed linear spaces, at a point and in a particular direction. We obtain a complete characterization of the same, which allows us to present refinements of the local preservation of orthogonality explored earlier. We also study the directional preservation of orthogonality with respect to certain special subspaces of the domain space, and apply the results towards identifying the isometries on a polyhedral normed linear space. In particular, we obtain refinements of the Blanco-Koldobsky-Turnšek Theorem for polyhedral normed linear spaces, including $ \ell_{\infty}^{n}, \ell_{1}^{n}. $