Invariant projections, hyperinvariant subspaces, and transitive algebras

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)

Manuscripts (2)

Invariant-projection counterexamples for every irrational rotation

September 27, 2026 · source · cite as \cite{OAI:Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026}

For every irrational angle, we construct a continuous nonnegative circle weight with exactly one zero and logarithmic integral -\infty 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.

Backward intertwiners and a transitive commutant

September 27, 2026 · source · Lean formalization available · cite as \cite{OAI:Backward-intertwiners-and-a-transitive-commutant-September-27-2026}

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.

Result 293 · Operator algebras. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.