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| Main Authors: | , |
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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2111.12605 |
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| _version_ | 1866911871759548416 |
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| author | Abedi, Sajjad Moslehian, Mohammad Sal |
| author_facet | Abedi, Sajjad Moslehian, Mohammad Sal |
| contents | Suppose that $\mathscr{E}$ and $\mathscr{F}$ are Hilbert $C^*$-modules. We present a power-norm $\left(\left\|\cdot\right\|^{\mathscr{E}}_n:n\in\mathbb{N}\right)$ based on $\mathscr{E}$ and obtain some of its fundamental properties. We introduce a new definition of the absolutely $(2,2)$-summing operators from $\mathscr{E}$ to $\mathscr{F}$, and denote the set of such operators by $\tildeΠ_2(\mathscr{E},\mathscr{F})$ with the convention $\tildeΠ_2(\mathscr{E})=\tildeΠ_2(\mathscr{E},\mathscr{E})$. It is known that the class of all Hilbert--Schmidt operators on a Hilbert space $\mathscr{H}$ is the same as the space $\tildeΠ_2(\mathscr{H})$. We show that the class of Hilbert--Schmidt operators introduced by Frank and Larson coincides with the space $\tildeΠ_2(\mathscr{E})$ for a countably generated Hilbert $C^*$-module $\mathscr{E}$ over a unital commutative $C^*$-algebra. These results motivate us to investigate the properties of the space $\tildeΠ_2(\mathscr{E},\mathscr{F})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_12605 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Power-norms based on Hilbert $C^*$-modules Abedi, Sajjad Moslehian, Mohammad Sal Operator Algebras Functional Analysis 46L08, 46L05, 46B15, 47L10 Suppose that $\mathscr{E}$ and $\mathscr{F}$ are Hilbert $C^*$-modules. We present a power-norm $\left(\left\|\cdot\right\|^{\mathscr{E}}_n:n\in\mathbb{N}\right)$ based on $\mathscr{E}$ and obtain some of its fundamental properties. We introduce a new definition of the absolutely $(2,2)$-summing operators from $\mathscr{E}$ to $\mathscr{F}$, and denote the set of such operators by $\tildeΠ_2(\mathscr{E},\mathscr{F})$ with the convention $\tildeΠ_2(\mathscr{E})=\tildeΠ_2(\mathscr{E},\mathscr{E})$. It is known that the class of all Hilbert--Schmidt operators on a Hilbert space $\mathscr{H}$ is the same as the space $\tildeΠ_2(\mathscr{H})$. We show that the class of Hilbert--Schmidt operators introduced by Frank and Larson coincides with the space $\tildeΠ_2(\mathscr{E})$ for a countably generated Hilbert $C^*$-module $\mathscr{E}$ over a unital commutative $C^*$-algebra. These results motivate us to investigate the properties of the space $\tildeΠ_2(\mathscr{E},\mathscr{F})$. |
| title | Power-norms based on Hilbert $C^*$-modules |
| topic | Operator Algebras Functional Analysis 46L08, 46L05, 46B15, 47L10 |
| url | https://arxiv.org/abs/2111.12605 |