Invariant subspace problem for positive operators on Banach spaces with a basis

Determine whether every bounded positive operator on an infinite-dimensional complex Banach space that admits a basis has a non-trivial closed invariant subspace.

Background

Restricting the invariant subspace problem to positive operators is natural when working on Banach spaces with a (monotone or unconditional) basis, as positivity is defined via the coordinate cone. Despite progress for various classes, the general positive-operator version remains unresolved.

The paper highlights typical properties of positive contractions and criteria (e.g., the Abramovich–Aliprantis–Burkinshaw criterion) but notes that the underlying general problem is still open.