Refinements of the Blanco-Koldobsky-Turnšek Theorem
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915782417448960 |
|---|---|
| author | Manna, Jayanta Mandal, Kalidas Paul, Kallol Sain, Debmalya |
| author_facet | Manna, Jayanta Mandal, Kalidas Paul, Kallol Sain, Debmalya |
| contents | We refine the well-known Blanco-Koldobsky-Turnšek Theorem which states that a norm one linear operator defined on a Banach space is an isometry if and only if it preserves orthogonality at every element of the space. We improve the result for Banach spaces in which the set of all smooth points forms a dense $G_δ$-set by proving that a norm one linear operator that preserves orthogonality on a dense subset of the space is an isometry. We further demonstrate that if such an operator preserves orthogonality on a hyperplane not passing through the origin then it is an isometry. In the context of finite-dimensional Banach spaces, we prove that preserving orthogonality on the set all extreme points of the unit ball forces the operator to be an isometry, which substantially refines Blanco-Koldobsky-Turnšek theorem. Finally, for finite-dimensional polyhedral spaces, we establish the significance of the set of all $k$-smooth points for any possible $k,$ in the study of isometric theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11161 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Refinements of the Blanco-Koldobsky-Turnšek Theorem Manna, Jayanta Mandal, Kalidas Paul, Kallol Sain, Debmalya Functional Analysis 46B20, 46B04 We refine the well-known Blanco-Koldobsky-Turnšek Theorem which states that a norm one linear operator defined on a Banach space is an isometry if and only if it preserves orthogonality at every element of the space. We improve the result for Banach spaces in which the set of all smooth points forms a dense $G_δ$-set by proving that a norm one linear operator that preserves orthogonality on a dense subset of the space is an isometry. We further demonstrate that if such an operator preserves orthogonality on a hyperplane not passing through the origin then it is an isometry. In the context of finite-dimensional Banach spaces, we prove that preserving orthogonality on the set all extreme points of the unit ball forces the operator to be an isometry, which substantially refines Blanco-Koldobsky-Turnšek theorem. Finally, for finite-dimensional polyhedral spaces, we establish the significance of the set of all $k$-smooth points for any possible $k,$ in the study of isometric theory. |
| title | Refinements of the Blanco-Koldobsky-Turnšek Theorem |
| topic | Functional Analysis 46B20, 46B04 |
| url | https://arxiv.org/abs/2504.11161 |