An improvement of the Blanco-Koldobsky-Turnšek characterization of isometries
Jayanta Manna, Kalidas Mandal, Kallol Paul, Debmalya Sain
TL;DR
The paper enhances the Blanco-Koldobsky-Turnšek framework by characterizing level vectors of bounded linear operators via directional preservation of Birkhoff-James orthogonality and linking these vectors to the operator’s adjoint. It proves that every nonzero element being a level vector is equivalent to the operator being a scalar multiple of an isometry, and it provides structural and geometric consequences, including relationships with faces of the unit ball and Krein-Milman type refinements. The results generalize previous smooth-case findings to arbitrary normed spaces and offer a sharp refinement of the original BK-T characterization with additional refinements under Krein-Milman property. These insights deepen the understanding of when norm-attainment and orthogonality-preservation properties force linear operators to be essentially isometric up to scalar factors, with potential implications in operator theory on normed spaces.
Abstract
We present an improvement of the Blanco-Koldobsky-Turnšek characterization of isometries in normed linear spaces by using the concept of level vectors of an operator. In this context, we characterize level vectors entirely through directional preservation of Birkhoff-James orthogonality and analyze the associated geometric and structural phenomena that they induce. Furthermore, in spaces whose unit balls possess the \textit{Krein-Milman property}, we derive an additional refinement of the Blanco-Koldobsky-Turnšek characterization of isometries.