Separated Pairs of Submodules in Hilbert $C^*$-modules
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arXiv
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| Format: | Preprint |
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2024
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| author | Eskandari, R. Luo, W. Moslehian, M. S. Xu, Q. Zhang, H. |
| author_facet | Eskandari, R. Luo, W. Moslehian, M. S. Xu, Q. Zhang, H. |
| contents | We introduce the notion of the separated pair of closed submodules in the setting of Hilbert $C^*$-modules. We demonstrate that even in the case of Hilbert spaces this concept has several nice characterizations enriching the theory of separated pairs of subspaces in Hilbert spaces. Let $\mathscr H$ and $\mathscr K$ be orthogonally complemented closed submodules of a Hilbert $C^*$-module $\mathscr E$. We establish that $ (\mathscr H,\mathscr K)$ is a separated pair in $\mathscr{E}$ if and only if there are idempotents $Π_1$ and $Π_2$ such that $Π_1Π_2=Π_2Π_1=0$ and $\mathscr R(Π_1)=\mathscr H$ and $\mathscr R(Π_2)=\mathscr K$. We show that $\mathscr R(Π_1+λΠ_2)$ is closed for each $λ\in \mathbb{C}$ if and only if $\mathscr R(Π_1+Π_2)$ is closed.
We use the localization of Hilbert $C^*$-modules to define the angle between closed submodules. We prove that if $(\mathscr H^\perp,\mathscr K^\perp)$ is concordant, then $(\mathscr H^{\perp\perp},\mathscr K^{\perp\perp})$ is a separated pair if the cosine of this angle is less than one. We also present some surprising examples to illustrate our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04852 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Separated Pairs of Submodules in Hilbert $C^*$-modules Eskandari, R. Luo, W. Moslehian, M. S. Xu, Q. Zhang, H. Operator Algebras Functional Analysis 46L08, 46C05, 47A05, 47A30 We introduce the notion of the separated pair of closed submodules in the setting of Hilbert $C^*$-modules. We demonstrate that even in the case of Hilbert spaces this concept has several nice characterizations enriching the theory of separated pairs of subspaces in Hilbert spaces. Let $\mathscr H$ and $\mathscr K$ be orthogonally complemented closed submodules of a Hilbert $C^*$-module $\mathscr E$. We establish that $ (\mathscr H,\mathscr K)$ is a separated pair in $\mathscr{E}$ if and only if there are idempotents $Π_1$ and $Π_2$ such that $Π_1Π_2=Π_2Π_1=0$ and $\mathscr R(Π_1)=\mathscr H$ and $\mathscr R(Π_2)=\mathscr K$. We show that $\mathscr R(Π_1+λΠ_2)$ is closed for each $λ\in \mathbb{C}$ if and only if $\mathscr R(Π_1+Π_2)$ is closed. We use the localization of Hilbert $C^*$-modules to define the angle between closed submodules. We prove that if $(\mathscr H^\perp,\mathscr K^\perp)$ is concordant, then $(\mathscr H^{\perp\perp},\mathscr K^{\perp\perp})$ is a separated pair if the cosine of this angle is less than one. We also present some surprising examples to illustrate our results. |
| title | Separated Pairs of Submodules in Hilbert $C^*$-modules |
| topic | Operator Algebras Functional Analysis 46L08, 46C05, 47A05, 47A30 |
| url | https://arxiv.org/abs/2405.04852 |