Invariant projections, hyperinvariant subspaces, and transitive algebras

What OpenAI claims

Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II1 factor. (Lean)

Papers (2)

1. Invariant-projection counterexamples for every irrational rotation

   27 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract For every irrational angle, we construct a continuous nonnegative circle weight with exactly one zero and logarithmic integral -∞ whose weighted rotation has no nontrivial invariant projection in the associated hyperfinite type II1 factor. This answers negatively the question of Zhu, Fang, and Shi, with the angle prescribed in advance. The resulting operator is nonzero and norm-quasinilpotent.

2. Backward intertwiners and a transitive commutant

   27 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We give a negative answer to the hyperinvariant-subspace problem by constructing, on every infinite-dimensional separable complex Hilbert space, a nonzero bounded norm-quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. Its commutant is a proper strongly closed unital transitive complex operator algebra.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.