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Generalizing the approximation lemma beyond monotone bases

Generalize Lemma 2.1 (the approximation/density lemma requiring a monotone basis to produce SOT and SOT* density results for classes of positive contractions) to Banach spaces that merely admit a basis not necessarily monotone.

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Background

Lemma 2.1 (Lemma \ref{lemapprox}) is a key tool used to obtain density results for classes of positive contractions in SOT and SOT*, but it relies on the basis being monotone to control norms of truncations.

Extending this lemma to non-monotone bases would broaden the scope of the paper’s main techniques and results, including typical failure of the AAB criterion.

References

Lemma \ref{lemapprox} requires the basis $(e_n)_{n \geq 0}$ to be monotone. This lemma was useful to prove Theorem \ref{thaabreflexivesec4}. We thus have the following open question. Can Lemma \ref{lemapprox} be generalized to Banach spaces admitting a basis which is not necessarily monotone?

— Typical properties of positive contractions and the invariant subspace problem (2409.14481 - Gillet, 22 Sep 2024) in Section 5 (Further remarks and questions)