Halmos's Third Problem on Inverses of Intransitive Operators

Determine whether the inverse of every invertible intransitive operator on a separable, infinite-dimensional complex Hilbert space is also intransitive, equivalently, whether having a nontrivial closed invariant subspace is preserved under taking inverses.

Background

Halmos's third problem asks whether an invertible operator with a nontrivial invariant subspace must have an inverse with a nontrivial invariant subspace. The paper reduces this question to invertible operators with conjugate bi-geometric form and proves affirmative results for important subclasses, including operators whose off-diagonal block has rank one.

The reduction theorem establishes equivalence between the general problem and its restriction to conjugate bi-geometric operators, but the quoted passage records the unresolved general question. The paper also notes subsequent progress for narrower classes.