The hyperinvariant subspace problem
OpenAI's negative answer (2026) to the hyperinvariant subspace problem: on every infinite-dimensional separable complex Hilbert space there is a nonzero operator S with ‖S^n‖^{1/n} → 0 whose commutant is a proper, weakly closed, unital, transitive algebra, so S has no hyperinvariant subspace other than 0 and the whole space. The course proves Domar's theorem that a bilateral weighted shift with regular, quadratically decaying logarithmic weights has only the tails as invariant subspaces; builds a measurable field of such shifts over the 2-adic integers with the odometer and proves the estimate ‖S^n‖ ≤ exp(−10⌊(n+1)²/4⌋); shows that operators commuting with the scalar multipliers are fields of operators; and constructs a contraction commuting with S that moves every tail one step backward, so that a hyperinvariant subspace would carry a strictly decreasing label along the odometer, which a probability space does not allow.
Three lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Weighted shifts whose invariant subspaces are tails
- Measurable fields of weighted shifts over the 2-adic integers
- A backward intertwiner and a transitive commutant
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Hilbert spaces and compact operators · Foundations of von Neumann algebras
- The double commutant theorem · Foundations of von Neumann algebras
- The spectral theorem for bounded self-adjoint operators · Foundations of von Neumann algebras
- The Stone–Weierstrass theorem for functions vanishing at infinity · Open Mathematics Courses (Function algebras and approximation)
- Measure and Hilbert space tools for Haar integration (Harmonic analysis on locally compact groups)
- The projection and Riesz–Fréchet theorems, orthonormal bases, compact and multiplication operators, the double commutant theorem, products of probability spaces and the uniqueness of measures, and the Borel functional calculus (Foundations of von Neumann algebras); monotone and dominated convergence and completeness of L² (course Harmonic analysis on locally compact groups, measure tools); the Weierstrass approximation theorem (course Function algebras and approximation).