Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach spaces

TL;DR

This paper constructs explicit Markushevich bases in Hilbert and Banach spaces with specific boundedness properties, demonstrating sharp conditions under which these bases are not hereditary complete or Schauder after permutations.

Contribution

It provides the first explicit construction of nearly Auerbach, Markushevich bases with sharp bounds in Hilbert spaces and extends these results to Banach spaces using advanced geometric theorems.

Findings

Constructed explicit $(1+ ext{varepsilon}_n)$-bounded Markushevich bases in Hilbert spaces.

Showed these bases are not hereditary complete under certain conditions on $( ext{varepsilon}_n)$.

Proved the existence of non-Schauder Markushevich bases in Banach spaces with specific bounds.

Abstract

For any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varepsilon_n = \infty$ we provide an explicit simple construction of $(1+\varepsilon_n)$-bounded Markushevich basis in a separable Hilbert space which is not strong, or, in other terminology, is not hereditary complete; this condition on the sequence $(\varepsilon_n)_{n=1}^\infty$ is sharp. Using a finite-dimensional version of this construction, Dvoretzky's theorem and a construction of Vershynin, we conclude that in any Banach space for any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varepsilon_n^2 = \infty$ there exists a $(1+\varepsilon_n)$-bounded Markushevich basis which is not a Schauder basis after any permutation of its elements.