| Title: | Birkhoff-James classification of norm's properties |
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| Authors: | ID Guterman, Aleksandr Èmilevič (Author)
ID Kuzma, Bojan (Author)
ID Singla, Sushil (Author)
ID Zhilina, Svetlana (Author) |
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| Files: |  PDF - Presentation file, download (799,67 KB)
MD5: 9B0D132C3A8057AF5DB57D7F032706A6
 URL - Source URL, visit https://link.springer.com/article/10.1007/s43036-024-00321-0
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | For an arbitrary normed space ${\mathcal X}$ over a field ${\mathbb F} \in \{ {\mathbb R}, {\mathbb C} \}$, we define the directed graph $\Gamma({\mathcal X})$ induced by Birkhoff-James orthogonality on the projective space ${\mathbb P}({\mathcal X})$, and also its nonprojective counterpart $\Gamma_0({\mathcal X})$. We show that, in finite-dimensional normed spaces, $\Gamma({\mathcal X})$ carries all the information about the dimension, smooth points, and norm's maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian $C^\ast$-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph $\Gamma_0(\mathcal{R})$ of a (real or complex) Radon plane ${\mathcal R}$ is isomorphic to the graph $\Gamma_0({\mathbb F}^2, \|\cdot\|_2)$ of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes. |
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| Keywords: | normed spaces, Birkhoff-James orthogonality, orthodigraph, dimension, smoothness, rotundness, Radon plane |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.07.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | 33 str. |
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| Numbering: | Vol. 9, iss. 3, [article no.] 43 |
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| PID: | 20.500.12556/DiRROS-21749  |
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| UDC: | 517.9:519.17 |
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| ISSN on article: | 2538-225X |
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| DOI: | 10.1007/s43036-024-00321-0 |
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| COBISS.SI-ID: | 229672195 |
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| Publication date in DiRROS: | 20.03.2025 |
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| Views: | 621 |
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| Downloads: | 432 |
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