Localised frames for tensor product spaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912272157245440 |
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| author | Bytchenkoff, Dimitri Speckbacher, Michael Balazs, Peter |
| author_facet | Bytchenkoff, Dimitri Speckbacher, Michael Balazs, Peter |
| contents | In this paper, we investigate whether the tensor product of two frames, each individually localised with respect to a spectral matrix algebra, is also localised with respect to a suitably chosen tensor product algebra. We provide a partial answer by constructing an involutive Banach algebra of rank-four tensors that is built from two solid spectral matrix algebras. We show that this algebra is inverse-closed, given that the original algebras satisfy a specific property related to operator-valued versions of these algebras. This condition is satisfied by all commonly used solid spectral matrix algebras. We then prove that the tensor product of two self-localised frames remains self-localised with respect to our newly constructed tensor algebra. Additionally, we discuss generalisations to localised frames of Hilbert-Schmidt operators, which may not necessarily consist of rank-one operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09764 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Localised frames for tensor product spaces Bytchenkoff, Dimitri Speckbacher, Michael Balazs, Peter Functional Analysis 46B28, 42B35, 42C15 In this paper, we investigate whether the tensor product of two frames, each individually localised with respect to a spectral matrix algebra, is also localised with respect to a suitably chosen tensor product algebra. We provide a partial answer by constructing an involutive Banach algebra of rank-four tensors that is built from two solid spectral matrix algebras. We show that this algebra is inverse-closed, given that the original algebras satisfy a specific property related to operator-valued versions of these algebras. This condition is satisfied by all commonly used solid spectral matrix algebras. We then prove that the tensor product of two self-localised frames remains self-localised with respect to our newly constructed tensor algebra. Additionally, we discuss generalisations to localised frames of Hilbert-Schmidt operators, which may not necessarily consist of rank-one operators. |
| title | Localised frames for tensor product spaces |
| topic | Functional Analysis 46B28, 42B35, 42C15 |
| url | https://arxiv.org/abs/2503.09764 |