On the relations between Auerbach or almost Auberbach Markushevich systems and Schauder bases

Beata Randrianantoanina; Michał Wojciechowski; Pavel Zatitskii

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arXiv · 2401.00612

On the relations between Auerbach or almost Auberbach Markushevich systems and Schauder bases

Beata Randrianantoanina·Michał Wojciechowski·Pavel Zatitskii

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Abstract

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.

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Beata Randrianantoanina, Michał Wojciechowski, Pavel Zatitskii. 2023-12-31. On the relations between Auerbach or almost Auberbach Markushevich systems and Schauder bases. https://arxiv.org/abs/2401.00612

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