The hyperinvariant subspace problem on Hilbert space
A closed subspace is hyperinvariant for a bounded operator on a Hilbert space if it is invariant under every operator commuting with ; equivalently, has no nontrivial hyperinvariant subspace exactly when its commutant is a transitive algebra. Douglas and Pearcy (1972) studied the problem and its link to transitive algebras; it sits beside the invariant subspace problem and is known to be distinct from it (a claimed equivalence by Kerchy-Pearcy was withdrawn). Does every bounded nonscalar operator on an infinite-dimensional separable complex Hilbert space have a nonzero proper closed hyperinvariant subspace?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Operator theory
- Posed by
- Classical operator-theory problem; the manuscript cites R. G. Douglas and C. Pearcy, Hyperinvariant subspaces and transitive algebras, Michigan Math. J. 19 (1972), and names no original poser
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-27
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 52 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1 and Corollary 1.2: on a fibered bilateral weighted shift with quadratically decaying weights over the 2-adic odometer is nonzero with , hence norm-quasinilpotent, and its commutant (containing a backward intertwiner and all scalar multipliers) is transitive. So every infinite-dimensional separable Hilbert space carries a nonzero quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. The operator has many ordinary invariant subspaces (the fiberwise tails), so the invariant subspace problem itself is untouched. The companion reaches the same existence statement by a different construction.
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.
Verification
No independent mathematician has checked this yet. Checked here: the abstracts and introductions of both manuscripts and Theorem 1.1 and Corollary 1.2 of the principal one, read against the problem as stated there. The proofs were not refereed. Lean-checked on the release's Comparator challenge BackwardIntertwiners (declaration OAI.BackwardIntertwiners.direct_algebra_corollary, listed in lean/formalization.yaml). Its statement was read here: for every complete separable infinite-dimensional complex inner product space it asserts an operator T that is nonzero, with norm of T^n to the power 1/n tending to 0, whose commutant leaves no closed subspace other than 0 and H invariant, is not the whole operator algebra, and is closed in the strong operator topology. That is the headline claim. Not rebuilt here. The companion also asserts that a published corollary (Cho-Ko-Lee 2017, Corollary 2.8) is false.
Sources
- PaperInvariant-projection counterexamples for every irrational rotation (companion, independent proof)
- Lean proofLean Comparator challenge BackwardIntertwiners (OpenAI math release)
- CodeOpenAI math release: Backward intertwiners and a transitive commutant
- Problem recordDouglas-Pearcy 1972, Hyperinvariant subspaces and transitive algebras