Typical positive contractions on ℓq (1<q≠2): invariant subspace

Determine whether a typical positive contraction in (P1(ℓq), SOT) or in (P1(ℓq), SOT*) has a non-trivial closed invariant subspace when 1<q≠2.

Background

The paper proves that typical positive contractions on ℓ2 (and on ℓ1) do have non-trivial invariant subspaces, but the intermediate ℓq cases with 1<q≠2 remain unresolved.

Because typicality is studied in the Polish spaces (P1(ℓq), SOT) and (when applicable) (P1(ℓq), SOT*), the question is whether the comeager set consists of operators admitting non-trivial invariant subspaces.

References

The first natural open question is of course the following. If $X = \ell_q$ with $1 < q \ne 2 < \infty$, does a typical $T \in (, SOT)$ or $T \in (,SOT) $ have a non-trivial invariant subspace?

— Typical properties of positive contractions and the invariant subspace problem (2409.14481 - Gillet, 22 Sep 2024) in Section 5 (Further remarks and questions), Question 1