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

We establish that the summability of the series $\sum {\epsilon }_n$ is the necessary and sufficient criterion ensuring that every $(1+{\epsilon }_n)$ Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if $n{\epsilon }_n\to \mathrm{\infty }$, then in ${\mathrm{\ell }}_2$ there exists a $(1+{\epsilon }_n)$ 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.