Invariant subspace problem for reflexive Banach and Hilbert spaces

Determine whether every bounded linear operator on an infinite-dimensional separable reflexive complex Banach space, in particular on a separable Hilbert space, has a non-trivial closed invariant subspace.

Background

The classical invariant subspace problem asks whether every bounded linear operator on a Banach space admits a non-trivial invariant subspace. While counterexamples are known on some non-reflexive spaces (e.g., Enflo and Read), the situation for reflexive spaces—and notably for Hilbert spaces—has resisted resolution.

This paper studies typical properties of positive contractions but recalls the status of the general invariant subspace problem to contextualize the results.