Extensions of the Hilbert-multi-norm in Hilbert $C^*$-modules
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| Formato: | Preprint |
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2024
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| _version_ | 1866914789514543104 |
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| author | Abedi, Sajjad Moslehian, Mohammad Sal |
| author_facet | Abedi, Sajjad Moslehian, Mohammad Sal |
| contents | Dales and Polyakov introduced a multi-norm $\left( \left\|\cdot\right\|_n^{(2,2)}:n\in\mathbb{N}\right)$ based on a Banach space $\mathscr{X}$ and showed that it is equal with the Hilbert-multi-norm $\left( \left\|\cdot\right\|_n^{\mathscr{H}}:n\in\mathbb{N}\right)$ based on an infinite-dimensional Hilbert space $\mathscr{H}$. We enrich the theory and present three extensions of the Hilbert-multi-norm for a Hilbert $C^*$-module $\mathscr{X}$. We denote these multi-norms by $\left( \left\|\cdot\right\|_n^{\mathscr{X}}:n\in\mathbb{N}\right)$, $\left( \left\|\cdot\right\|_n^{*}:n\in\mathbb{N}\right)$, and $\left( \left\|\cdot\right\|_n^{\mathcal{P}\left(\mathfrak{A} \right) }:n\in\mathbb{N}\right)$. We show that $\left\|x\right\|_n^{\mathcal{P}\left(\mathfrak{A} \right) }\geq\left\|x\right\|_n^{\mathscr{X}}\leq \left\|x\right\|_n^{*}$ for each $x\in\mathscr{X}^n$. In the case when $\mathscr{X}$ is a Hilbert $\mathbb{K}\left(\mathscr{H}\right)$-module, for each $x\in\mathscr{X}^n$, we observe that $\left\|\cdot\right\|_n^{\mathcal{P}\left(\mathfrak{A} \right)}=\left\|\cdot\right\|_n^{\mathscr{X}}$. Furthermore, if $\mathscr{H}$ is separable and $\mathscr{X}$ is infinite-dimensional, we prove that $\left\|x\right\|_n^{\mathscr{X}}=\left\|x\right\|_n^{*}$. Among other things, we show that small and orthogonal decompositions with respect to these multi-norms are equivalent. Several examples are given to support the new findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05291 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extensions of the Hilbert-multi-norm in Hilbert $C^*$-modules Abedi, Sajjad Moslehian, Mohammad Sal Functional Analysis Operator Algebras 46L05, 46B15, 47L10 Dales and Polyakov introduced a multi-norm $\left( \left\|\cdot\right\|_n^{(2,2)}:n\in\mathbb{N}\right)$ based on a Banach space $\mathscr{X}$ and showed that it is equal with the Hilbert-multi-norm $\left( \left\|\cdot\right\|_n^{\mathscr{H}}:n\in\mathbb{N}\right)$ based on an infinite-dimensional Hilbert space $\mathscr{H}$. We enrich the theory and present three extensions of the Hilbert-multi-norm for a Hilbert $C^*$-module $\mathscr{X}$. We denote these multi-norms by $\left( \left\|\cdot\right\|_n^{\mathscr{X}}:n\in\mathbb{N}\right)$, $\left( \left\|\cdot\right\|_n^{*}:n\in\mathbb{N}\right)$, and $\left( \left\|\cdot\right\|_n^{\mathcal{P}\left(\mathfrak{A} \right) }:n\in\mathbb{N}\right)$. We show that $\left\|x\right\|_n^{\mathcal{P}\left(\mathfrak{A} \right) }\geq\left\|x\right\|_n^{\mathscr{X}}\leq \left\|x\right\|_n^{*}$ for each $x\in\mathscr{X}^n$. In the case when $\mathscr{X}$ is a Hilbert $\mathbb{K}\left(\mathscr{H}\right)$-module, for each $x\in\mathscr{X}^n$, we observe that $\left\|\cdot\right\|_n^{\mathcal{P}\left(\mathfrak{A} \right)}=\left\|\cdot\right\|_n^{\mathscr{X}}$. Furthermore, if $\mathscr{H}$ is separable and $\mathscr{X}$ is infinite-dimensional, we prove that $\left\|x\right\|_n^{\mathscr{X}}=\left\|x\right\|_n^{*}$. Among other things, we show that small and orthogonal decompositions with respect to these multi-norms are equivalent. Several examples are given to support the new findings. |
| title | Extensions of the Hilbert-multi-norm in Hilbert $C^*$-modules |
| topic | Functional Analysis Operator Algebras 46L05, 46B15, 47L10 |
| url | https://arxiv.org/abs/2405.05291 |