Notes on the numerical radius for adjointable operators on Hilbert $C^*$-modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909515140562944 |
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| author | Li, J. Wu, K. Xu, Q. |
| author_facet | Li, J. Wu, K. Xu, Q. |
| contents | Given a Hilbert module $H$ over a $C^*$-algebra, let $\mathcal{L}(H)$ be the set of all adjointable operators on $H$. For each $T\in\mathcal{L}(H)$, its numerical radius is defined by $w(T)=\sup\big\{\|\langle Tx, x \rangle\|: x\in H, \|x\|=1\big\}$. It is proved that $w(T)=\|T\|$ whenever $T$ is normal. Examples are constructed to show that there exist Hilbert module $H$ over certain $C^*$-algebra and $T_1,T_2\in \mathcal{L}(H)$ with $T_1^2=0$ such that $w(T_1)\ne \frac12 \|T_1\|$ and $\sup\limits_{θ\in [0,2π]}\|\mbox{Re}(e^{iθ}T_2)\|<w(T_2)$. In addition, a new characterization of the spatial numerical radius is given, and it is proved that $w\big(π(T)\big)\le w(T)$ for every faithful representation $(π, X)$ of $\mathcal{L}(H)$ and every $T\in\mathcal{L}(H)$. Some inequalities are derived based on the newly obtained results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20259 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Notes on the numerical radius for adjointable operators on Hilbert $C^*$-modules Li, J. Wu, K. Xu, Q. Functional Analysis Given a Hilbert module $H$ over a $C^*$-algebra, let $\mathcal{L}(H)$ be the set of all adjointable operators on $H$. For each $T\in\mathcal{L}(H)$, its numerical radius is defined by $w(T)=\sup\big\{\|\langle Tx, x \rangle\|: x\in H, \|x\|=1\big\}$. It is proved that $w(T)=\|T\|$ whenever $T$ is normal. Examples are constructed to show that there exist Hilbert module $H$ over certain $C^*$-algebra and $T_1,T_2\in \mathcal{L}(H)$ with $T_1^2=0$ such that $w(T_1)\ne \frac12 \|T_1\|$ and $\sup\limits_{θ\in [0,2π]}\|\mbox{Re}(e^{iθ}T_2)\|<w(T_2)$. In addition, a new characterization of the spatial numerical radius is given, and it is proved that $w\big(π(T)\big)\le w(T)$ for every faithful representation $(π, X)$ of $\mathcal{L}(H)$ and every $T\in\mathcal{L}(H)$. Some inequalities are derived based on the newly obtained results. |
| title | Notes on the numerical radius for adjointable operators on Hilbert $C^*$-modules |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2502.20259 |