On the relations between Auerbach or almost Auberbach Markushevich systems and Schauder bases
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914830328266752 |
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| author | Randrianantoanina, Beata Wojciechowski, Michał Zatitskii, Pavel |
| author_facet | Randrianantoanina, Beata Wojciechowski, Michał Zatitskii, Pavel |
| contents | We establish that the summability of the series $\sum\varepsilon_n$ is the necessary and sufficient criterion ensuring that every $(1+\varepsilon_n)$ Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if $n\varepsilon_n\to \infty$, then in $\ell_2$ there exists a $(1+\varepsilon_n)$ Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00612 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the relations between Auerbach or almost Auberbach Markushevich systems and Schauder bases Randrianantoanina, Beata Wojciechowski, Michał Zatitskii, Pavel Functional Analysis Primary: 46B15, Secondary: 46C15, 46B20, 46B03, 46B04, 46B45 We establish that the summability of the series $\sum\varepsilon_n$ is the necessary and sufficient criterion ensuring that every $(1+\varepsilon_n)$ Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if $n\varepsilon_n\to \infty$, then in $\ell_2$ there exists a $(1+\varepsilon_n)$ Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis. |
| title | On the relations between Auerbach or almost Auberbach Markushevich systems and Schauder bases |
| topic | Functional Analysis Primary: 46B15, Secondary: 46C15, 46B20, 46B03, 46B04, 46B45 |
| url | https://arxiv.org/abs/2401.00612 |