Family 293Operator algebrasLeanNot yet digestedfeatured
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 II factor.
Manuscripts in this family · 2
Invariant-projection counterexamples for every irrational rotation
Sep 27, 2026·paper.pdf
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 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
Sep 27, 2026·paper.pdfLean
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.