Angle Preserving Mappings
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2015
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| _version_ | 1866913808489906176 |
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| author | Moslehian, Mohammad Sal Zamani, Ali Frank, Michael |
| author_facet | Moslehian, Mohammad Sal Zamani, Ali Frank, Michael |
| contents | In this paper, we give some characterizations of orthogonality preserving mappings between inner product spaces. Furthermore, we study the linear mappings that preserve some angles. One of our main results states that if $\mathcal{X}, \mathcal{Y}$ are real inner product spaces and $θ\in(0, π)$, then an injective nonzero linear mapping $T:\mathcal{X}\longrightarrow \mathcal{Y}$ is a similarity whenever (i) $x\undersetθ{\angle} y\, \Leftrightarrow \,Tx\undersetθ{\angle} Ty$ for all $x, y\in \mathcal{X}$; (ii) for all $x, y\in \mathcal{X}$, $\|x\|=\|y\|$ and $x\undersetθ{\angle} y$ ensure that $\|Tx\|=\|Ty\|$. We also investigate orthogonality preserving mappings in the setting of inner product $C^{*}$-modules. Another result shows that if $\mathbb{K}(\mathscr{H})\subseteq\mathscr{A}\subseteq\mathbb{B}(\mathscr{H})$ is a $C^{*}$-algebra and $T\,:\mathscr{E}\longrightarrow \mathscr{F}$ is an $\mathscr{A}$-linear mapping between inner product $\mathscr{A}$-modules, then $T$ is orthogonality preserving if and only if $|x|\leq|y|\, \Rightarrow \,|Tx|\leq|Ty|$ for all $x, y\in \mathscr{E}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1504_06293 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Angle Preserving Mappings Moslehian, Mohammad Sal Zamani, Ali Frank, Michael Functional Analysis Operator Algebras Primary 46L08, Secondary 46C05, 46B20 In this paper, we give some characterizations of orthogonality preserving mappings between inner product spaces. Furthermore, we study the linear mappings that preserve some angles. One of our main results states that if $\mathcal{X}, \mathcal{Y}$ are real inner product spaces and $θ\in(0, π)$, then an injective nonzero linear mapping $T:\mathcal{X}\longrightarrow \mathcal{Y}$ is a similarity whenever (i) $x\undersetθ{\angle} y\, \Leftrightarrow \,Tx\undersetθ{\angle} Ty$ for all $x, y\in \mathcal{X}$; (ii) for all $x, y\in \mathcal{X}$, $\|x\|=\|y\|$ and $x\undersetθ{\angle} y$ ensure that $\|Tx\|=\|Ty\|$. We also investigate orthogonality preserving mappings in the setting of inner product $C^{*}$-modules. Another result shows that if $\mathbb{K}(\mathscr{H})\subseteq\mathscr{A}\subseteq\mathbb{B}(\mathscr{H})$ is a $C^{*}$-algebra and $T\,:\mathscr{E}\longrightarrow \mathscr{F}$ is an $\mathscr{A}$-linear mapping between inner product $\mathscr{A}$-modules, then $T$ is orthogonality preserving if and only if $|x|\leq|y|\, \Rightarrow \,|Tx|\leq|Ty|$ for all $x, y\in \mathscr{E}$. |
| title | Angle Preserving Mappings |
| topic | Functional Analysis Operator Algebras Primary 46L08, Secondary 46C05, 46B20 |
| url | https://arxiv.org/abs/1504.06293 |