On $A$-orthogonality preservation and Blanco-Koldobsky-Turnšek theorem in semi-Hilbert spaces
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909705078571008 |
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| author | Manna, Jayanta Barik, Somdatta Paul, Kallol Sain, Debmalya |
| author_facet | Manna, Jayanta Barik, Somdatta Paul, Kallol Sain, Debmalya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18871 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $A$-orthogonality preservation and Blanco-Koldobsky-Turnšek theorem in semi-Hilbert spaces Manna, Jayanta Barik, Somdatta Paul, Kallol Sain, Debmalya Functional Analysis Primary 46B20, 47L05, Secondary 46C50 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. |
| title | On $A$-orthogonality preservation and Blanco-Koldobsky-Turnšek theorem in semi-Hilbert spaces |
| topic | Functional Analysis Primary 46B20, 47L05, Secondary 46C50 |
| url | https://arxiv.org/abs/2507.18871 |