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| Asıl Yazarlar: | , , , |
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| Materyal Türü: | Preprint |
| Baskı/Yayın Bilgisi: |
2020
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| Konular: | |
| Online Erişim: | https://arxiv.org/abs/2004.01444 |
| Etiketler: |
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| _version_ | 1866916628136984576 |
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| author | Eskandari, Rasoul Frank, Michael Manuilov, Vladimir Moslehian, Mohammad Sal |
| author_facet | Eskandari, Rasoul Frank, Michael Manuilov, Vladimir Moslehian, Mohammad Sal |
| contents | We introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued sesquilinear form on a Hilbert $C^*$-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert $C^*$-module is self-dual. Extending a bounded $C^*$-valued sesquilinear form on a Hilbert $C^*$-module to a sesquilinear form on its second dual, we then provide some necessary and sufficient conditions for the $B$-spline interpolation problem to have a solution. Passing to the setting of Hilbert $W^*$-modules, we present our main result by characterizing when the spline interpolation problem for the extended $C^*$-valued sesquilinear to the dual $\mathscr{X}'$ of the Hilbert $W^*$-module $\mathscr{X}$ has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert $W^*$-module $\mathscr{X}$ is orthogonally complemented with respect to another $C^*$-inner product on $\mathscr{X}$. Finally, solutions of the $B$-spline interpolation problem for Hilbert $C^*$-modules over $C^*$-ideals of $W^*$-algebras are extensively discussed. Several examples are provided to illustrate the existence or lack of a solution for the problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_01444 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | B-spline interpolation problem in Hilbert C*-modules Eskandari, Rasoul Frank, Michael Manuilov, Vladimir Moslehian, Mohammad Sal Operator Algebras Functional Analysis 46L08, 46L05, 47A62 We introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued sesquilinear form on a Hilbert $C^*$-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert $C^*$-module is self-dual. Extending a bounded $C^*$-valued sesquilinear form on a Hilbert $C^*$-module to a sesquilinear form on its second dual, we then provide some necessary and sufficient conditions for the $B$-spline interpolation problem to have a solution. Passing to the setting of Hilbert $W^*$-modules, we present our main result by characterizing when the spline interpolation problem for the extended $C^*$-valued sesquilinear to the dual $\mathscr{X}'$ of the Hilbert $W^*$-module $\mathscr{X}$ has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert $W^*$-module $\mathscr{X}$ is orthogonally complemented with respect to another $C^*$-inner product on $\mathscr{X}$. Finally, solutions of the $B$-spline interpolation problem for Hilbert $C^*$-modules over $C^*$-ideals of $W^*$-algebras are extensively discussed. Several examples are provided to illustrate the existence or lack of a solution for the problem. |
| title | B-spline interpolation problem in Hilbert C*-modules |
| topic | Operator Algebras Functional Analysis 46L08, 46L05, 47A62 |
| url | https://arxiv.org/abs/2004.01444 |