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

Jayanta Manna; Somdatta Barik; Kallol Paul; Debmalya Sain

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arXiv · 2507.18871

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

Jayanta Manna·Somdatta Barik·Kallol Paul·Debmalya Sain

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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.

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Jayanta Manna, Somdatta Barik, Kallol Paul, Debmalya Sain. 2025-07-25. On $A$-orthogonality preservation and Blanco-Koldobsky-Turn\v{s}ek theorem in semi-Hilbert spaces. https://arxiv.org/abs/2507.18871

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