Hyperinvariant subspaces for quasinilpotent Hilbert-space operators

Hyperinvariant subspaces for quasinilpotent Hilbert-space operators

OpenAI's claim

Each separable infinite-dimensional complex Hilbert space supports, according to the paper, a nonzero norm-quasinilpotent operator with no proper nonzero closed hyperinvariant subspace and a transitive commutant. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.

Let $H$ be any separable complex Hilbert space of infinite dimension. OpenAI claims a nonzero bounded operator $S$ with \[ \lim_{n\to\infty}\|S^n\|^{1/n}=0 \] such that any closed subspace invariant under all operators commuting with $S$ is either zero or the entire space. Equivalently the commutant $\{S\}'$ is a transitive complex operator algebra that is unital, proper and closed in the strong topology. The claim concerns hyperinvariant subspaces; ordinary invariant subspaces of $S$ may exist.

Outline of the argument

Manuscript 1 (family 293, 2026-09-27): Quadratically weighted bilateral shifts have explicitly classified invariant tails, and the construction organizes them over a 2-adic probability base. A bounded commuting transport moves these tails backward, forcing an impossible strict descent for any hyperinvariant projection field. Uniform estimates on powers establish quasinilpotence of the entire operator, not just of individual fibers.

Formal verification

For family 293, the scope page documents the following selections. The rotation selection gives quasinilpotent weighted operators with no proper invariant projection in the associated finite factor. The backward-intertwiner selection gives a proper strong-closed transitive commutant on each separable infinite-dimensional Hilbert space; Brown-measure and product variants are documented separately. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: BackwardIntertwiners.lean, ContinuousCircleWeight.lean, FiniteFactor.lean, HyperinvariantSubspaces.lean, IrrationalRotation.lean, ProductBrown.lean. Declarations: OAI.BackwardIntertwiners.direct_algebra_corollary, OAI.ContinuousCircleWeight.main_theorem, OAI.FiniteFactor.main_theorem, OAI.Hyperinvariant.main_theorem, OAI.Invariant233.Results.prescribed_irrational_counterexample, OAI.Invariant233.Results.product_realization, OAI.FiniteFactor.productBrown.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

• OpenAI, Backward intertwiners and a transitive commutant, 2026-09-27; citation key OAI:Backward-intertwiners-and-a-transitive-commutant-September-27-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: cor:direct-algebra.
• Family 293: Lean scope.
• Comparator BackwardIntertwiners.json.
• Comparator BackwardIntertwiners.lean.
• Comparator ContinuousCircleWeight.json.
• Comparator ContinuousCircleWeight.lean.
• Comparator FiniteFactor.json.
• Comparator FiniteFactor.lean.
• Comparator HyperinvariantSubspaces.json.
• Comparator HyperinvariantSubspaces.lean.
• Comparator IrrationalRotation.json.
• Comparator IrrationalRotation.lean.
• Comparator ProductBrown.json.
• Comparator ProductBrown.lean.
• OpenAI catalogue entry.