Hyperinvariant subspaces of Hilbert-space operators
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Does every bounded operator on a separable infinite-dimensional complex Hilbert space have a nonzero proper closed hyperinvariant subspace, invariant under its entire commutant? The counterexample target permits a nonzero norm-quasinilpotent operator that still has ordinary invariant subspaces.
Why it matters and what counts as progress
Why it matters: It asks for an invariance property stronger than invariance under the operator alone, separating two different subspace problems.
What counts as progress: Prove a common proper closed invariant subspace exists for the full commutant, or construct an operator with none; ordinary invariant subspaces do not settle this.
How the problem developed
- 1973 · Partial theorem [1]
V. I. Lomonosov Proved that a bounded operator commuting with a nonzero compact operator has a nontrivial hyperinvariant subspace; this covers a restricted class of operators.
- 29 Jul 2025 · Open status [2]
I. Chalendar and J. R. Partington Surveyed the invariant-subspace problem and recent progress, treating the universal Hilbert-space question as unresolved before the later counterexample claim.
- 27 Sept 2026 · Counterexample claim [3]
OpenAI A preprint claims a nonzero quasinilpotent operator on every separable infinite-dimensional complex Hilbert space with no nontrivial closed hyperinvariant subspace.
AI contribution
Backward intertwiners and a transitive commutant
On every separable infinite-dimensional complex Hilbert space, the paper constructs a nonzero bounded norm-quasinilpotent operator S whose commutant has no nonzero proper closed invariant subspace. The operator S itself has ordinary invariant subspaces.
Sources
- V. I. Lomonosov, Invariant subspaces for the family of operators which commute with a completely continuous operator
- I. Chalendar and J. R. Partington, Recent perspectives on the Invariant Subspace Problem
- OpenAI backward-intertwiner paper
- Backward intertwiners and a transitive commutant
- OpenAI mathematics release
- Lean scope for the Hilbert-space construction
- Hyperinvariant-subspace comparator statement
- Comparator solution mapping for hyperinvariant subspaces
Website timeline
Publication dates and editorial changes.
OpenAI reports a Hilbert-space hyperinvariant-subspace counterexample