Hyperinvariant Subspace Problem: OpenAI Claims a Counterexample

About this video

Does every operator on a Hilbert space leave some nontrivial closed subspace untouched by everything that commutes with it? OpenAI claims the answer is no: on every infinite-dimensional separable complex Hilbert space it constructs a nonzero norm-quasinilpotent operator with no nonzero proper closed hyperinvariant subspace, and, for every irrational angle, a weighted rotation with no nontrivial invariant projection in the hyperfinite II₁ factor (answering a question of Zhu, Fang and Shi). These are OpenAI's preprints, not yet peer reviewed. The animation shows a weighted rotation of the kind in the paper: a weight with exactly one zero and logarithmic integral −∞, and the exact average shrink per turn along one orbit, computed up to a million turns.

Papers (OpenAI, catalogue result 293):
- Invariant-projection counterexamples for every irrational rotation: github.com/openai/math/blob/main/preprints/Invariant-projecti…
- Backward intertwiners and a transitive commutant: github.com/openai/math/blob/main/preprints/Backward-intertwin…

The animation illustrates the result; it is not a proof.

Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/293