Negative answer to the hyperinvariant subspace problem on Hilbert space

An OpenAI manuscript constructs, on every infinite-dimensional separable complex Hilbert space, a nonzero quasinilpotent operator with no nontrivial closed subspace invariant under every operator commuting with it, answering the hyperinvariant subspace problem in the negative, with a Lean formalization of the statement attached.

What was found

An operator has no nontrivial hyperinvariant subspace exactly when its commutant is a transitive algebra. The construction takes bilateral weighted shifts on $\ell^2(\mathbb{Z})$ fibered over the 2-adic integers, whose invariant subspaces in each fiber are exactly the coordinate tails, and adds a commuting contraction that moves every tail one step backward while the base is shifted by the odometer; a subspace invariant under the whole commutant would then need a tail index that strictly decreases along an ergodic, measure-preserving map, which is impossible. The operator satisfies $\|S^n\| \le \exp(-10\lfloor (n+1)^2/4 \rfloor)$. Since its commutant is a proper, strongly closed, transitive unital algebra, the same example answers negatively the transitive algebra question of whether every such algebra is strongly dense. A companion manuscript in the family gives operators in the hyperfinite type $\mathrm{II}_1$ factor with no nontrivial invariant projection, answering a question of Zhu, Fang and Shi.

Novelty check

Read the introduction at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb on 2026-10-12. It situates the result against Douglas and Pearcy's 1972 work, Arveson's 1967 density theorem (whose maximal abelian hypothesis the construction avoids), and Atzmon and Sodin's 1999 shifts, whose hyperinvariant subspaces were left open, and notes that a Kérchy and Pearcy preprint claiming the invariant and hyperinvariant problems are equivalent was withdrawn. No earlier negative answer was found in the registry.

Also recorded at

History of this entry

1. 2026-10-16CorrectedRecorded the vibemathed record for this result as a registration. It cites the same manuscript and labels it Lean-checked, which is the attached formalization rather than an outside review. No grade moved.
2. 2026-10-12AddedEntered the registry graded Claimed and AI-led.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Community discussion

Graded claimed for verification and ai-led for autonomy. What these mean.

Cite this entry

whataifound.org. (2026). Negative answer to the hyperinvariant subspace problem on Hilbert space. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-06-hyperinvariant-subspace-problem

@misc{whataifound-openai-2026-problem,
  title        = {Negative answer to the hyperinvariant subspace problem on Hilbert space},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by OpenAI. Verification: Claimed. Autonomy: AI-led.},
  url          = {https://whataifound.org/finding/2026-10-06-hyperinvariant-subspace-problem}
}

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