On T-orthogonality in Banach spaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914911007801344 |
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| author | Sain, Debmalya Ghosh, Souvik Paul, Kallol |
| author_facet | Sain, Debmalya Ghosh, Souvik Paul, Kallol |
| contents | Let $\mathbb{X}$ be a Banach space and let $\mathbb{X}^*$ be the dual space of $\mathbb{X}.$ For $x,y \in \mathbb{X},$ $ x$ is said to be $T$-orthogonal to $y$ if $Tx(y) =0,$ where $T$ is a bounded linear operator from $\mathbb{X}$ to $\mathbb{X}^*.$
We study the notion of $T$-orthogonality in a Banach space and investigate its relation with the various geometric properties, like strict convexity, smoothness, reflexivity of the space. We explore the notions of left and right symmetric elements w.r.t. the notion of $T$-orthogonality. We characterize bounded linear operators on $\mathbb{X}$ preserving $T$-orthogonality. Finally we characterize Hilbert spaces among all Banach spaces using $T$-orthogonality. \end{abstract} |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_05541 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On T-orthogonality in Banach spaces Sain, Debmalya Ghosh, Souvik Paul, Kallol Functional Analysis 46B20, 52A21 Let $\mathbb{X}$ be a Banach space and let $\mathbb{X}^*$ be the dual space of $\mathbb{X}.$ For $x,y \in \mathbb{X},$ $ x$ is said to be $T$-orthogonal to $y$ if $Tx(y) =0,$ where $T$ is a bounded linear operator from $\mathbb{X}$ to $\mathbb{X}^*.$ We study the notion of $T$-orthogonality in a Banach space and investigate its relation with the various geometric properties, like strict convexity, smoothness, reflexivity of the space. We explore the notions of left and right symmetric elements w.r.t. the notion of $T$-orthogonality. We characterize bounded linear operators on $\mathbb{X}$ preserving $T$-orthogonality. Finally we characterize Hilbert spaces among all Banach spaces using $T$-orthogonality. \end{abstract} |
| title | On T-orthogonality in Banach spaces |
| topic | Functional Analysis 46B20, 52A21 |
| url | https://arxiv.org/abs/2407.05541 |