Invariant projections, hyperinvariant subspaces, and transitive algebras

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

• Invariant-projection counterexamples for every irrational rotation
• Backward intertwiners and a transitive commutant

Scope

For every irrational rotation angle $\theta\in(0,1)$, the formalization constructs a continuous nonnegative circle weight with a single zero and logarithmic integral $-\infty$ whose weighted rotation is nonzero and quasinilpotent. In its associated hyperfinite type $\mathrm{II}_1$ factor, the operator has no nontrivial invariant projection: $(1-p)Tp=0$ forces $p=0$ or $p=1$. The linked statements also include an earlier existence construction, a product realization, and the selected product model's Brown measure $\delta_0$.

The formalization for the companion paper Backward intertwiners and a transitive commutant gives, on every separable infinite-dimensional complex Hilbert space, a nonzero quasinilpotent operator whose commutant is transitive, proper, and closed in the strong operator topology. Thus no nontrivial closed subspace is invariant under every operator in the commutant.

The hyperinvariant-subspace problem asks whether every bounded operator on a complex Hilbert space has a nontrivial closed subspace invariant under every operator that commutes with it. The formalization gives a negative answer on every separable infinite-dimensional complex Hilbert space: there is a nonzero quasinilpotent operator with a transitive commutant. Its commutant is a proper unital operator algebra closed in the strong operator topology. The construction therefore has no nonzero proper closed hyperinvariant subspace.

Comparator links