Abstract
We establish that the summability of the series ∑εn is the necessary and sufficient criterion ensuring that every (1+εn)-bounded Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if nεn→∞, then in ℓ2 there exists a (1+εn)-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 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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