We establish that the summability of the series $ \sum\varepsilon_n$ is the necessary and sufficient criterion ensuring that every $(1+\varepsilon_n)$-bounded 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 $(1+ \varepsilon_n)$-bounded 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 in some 1-symmetric separable Banach spaces of Auerbach bases, no permutations of which are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis.
(joint work with Michal Wojciechowski and Pavel Zatitskii)
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