On Symmetry of Birkhoff-James Orthogonality of Linear Operators on Finite-dimensional Real Banach Spaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2016
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| _version_ | 1866929439124750336 |
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| author | Sain, Debmalya Ghosh, Puja Paul, Kallol |
| author_facet | Sain, Debmalya Ghosh, Puja Paul, Kallol |
| contents | We characterize left symmetric linear operators on a finite dimensional strictly convex and smooth real normed linear space $ \mathbb{X},$ which answers a question raised recently by one of the authors in \cite{S} [D. Sain, \textit{Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, Journal of Mathematical Analysis and Applications, accepted, $ 2016 $}]. We prove that $ T\in B(\mathbb{X}) $ is left symmetric if and only if $ T $ is the zero operator. If $ \mathbb{X} $ is two-dimensional then the same characterization can be obtained without the smoothness assumption. We also explore the properties of right symmetric linear operators defined on a finite dimensional real Banach space. In particular, we prove that smooth linear operators on a finite-dimensional strictly convex and smooth real Banach space can not be right symmetric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1611_03663 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | On Symmetry of Birkhoff-James Orthogonality of Linear Operators on Finite-dimensional Real Banach Spaces Sain, Debmalya Ghosh, Puja Paul, Kallol Functional Analysis Primary 47L05, Secondary 46B20 We characterize left symmetric linear operators on a finite dimensional strictly convex and smooth real normed linear space $ \mathbb{X},$ which answers a question raised recently by one of the authors in \cite{S} [D. Sain, \textit{Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, Journal of Mathematical Analysis and Applications, accepted, $ 2016 $}]. We prove that $ T\in B(\mathbb{X}) $ is left symmetric if and only if $ T $ is the zero operator. If $ \mathbb{X} $ is two-dimensional then the same characterization can be obtained without the smoothness assumption. We also explore the properties of right symmetric linear operators defined on a finite dimensional real Banach space. In particular, we prove that smooth linear operators on a finite-dimensional strictly convex and smooth real Banach space can not be right symmetric. |
| title | On Symmetry of Birkhoff-James Orthogonality of Linear Operators on Finite-dimensional Real Banach Spaces |
| topic | Functional Analysis Primary 47L05, Secondary 46B20 |
| url | https://arxiv.org/abs/1611.03663 |