Invariant projections, hyperinvariant subspaces, and transitive algebras

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 II_(1) 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.

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.