On $A$-orthogonality preservation and Blanco-Koldobsky-Turn\v{s}ek theorem in semi-Hilbert spaces

TL;DR

This paper characterizes how $A$-orthogonality is locally preserved by $A$-bounded operators in semi-Hilbert spaces, exploring properties of $A$-norm attainment sets and characterizing $A$-isometries.

Contribution

It provides complete characterizations of $A$-orthogonality preservation, properties of $A$-norm attainment sets, and characterizations of $A$-isometries in semi-Hilbert spaces.

Findings

Complete characterization of local $A$-orthogonality preservation.

Properties of $A$-norm attainment and minimum $A$-norm attainment sets.

Characterization of $A$-isometries as $A$-norm one operators preserving $A$-orthogonality.

Abstract

We investigate the local preservation of $A$-orthogonality at a point by $A$-bounded operators within the semi-Hilbertian framework induced by a positive operator $A$ on a Hilbert space $\mathbb{H}.$ We provide complete characterizations of such preservation. Additionally, we explore properties of the $A$-norm attainment set of an $A$-bounded operator in light of $A$-orthogonality preservation. We also study analogous properties for the minimum $A$-norm attainment set of an $A$-bounded operator. We then characterize the $A$-isometries as the $A$-norm one operators preserving $A$-orthogonality. Finally, we characterize those subsets of Hilbert spaces for which such preservation by an $A$-norm one operator implies that the operator is an $A$-isometry.