{
  "id": "the-hyperinvariant-subspace-problem",
  "title": "The hyperinvariant subspace problem",
  "description": "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.",
  "units": [
    {
      "id": "weighted-shifts-whose-invariant-subspaces-are-tails",
      "title": "Weighted shifts whose invariant subspaces are tails",
      "reader": "weighted-shifts-whose-invariant-subspaces-are-tails.html",
      "source": "src/weighted-shifts-whose-invariant-subspaces-are-tails.md",
      "source_sha256": "450d99efe078c0e7624157603672597a6f595d11becc3830463714cdf5b98751",
      "status": "author_self_checked"
    },
    {
      "id": "measurable-fields-of-weighted-shifts",
      "title": "Measurable fields of weighted shifts over the 2-adic integers",
      "reader": "measurable-fields-of-weighted-shifts.html",
      "source": "src/measurable-fields-of-weighted-shifts.md",
      "source_sha256": "e44be843b6d91bde3d7d659377a056fd36cdcec162a429c95feaa704b47cb5ac",
      "status": "author_self_checked"
    },
    {
      "id": "a-transitive-commutant",
      "title": "A backward intertwiner and a transitive commutant",
      "reader": "a-transitive-commutant.html",
      "source": "src/a-transitive-commutant.md",
      "source_sha256": "d187b96b0bea79b3544844fc8b68402c08e20d5f30ef51136268a1a52d1b3a96",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "foundations-of-von-neumann-algebras lessons hilbert-spaces-and-compact-operators (Theorems 2.2 and 2.3, projection and Riesz-Frechet theorems; Theorem 4.1, orthonormal bases; Theorem 5.1, compact operators; Theorem 9.1, multiplication operators), the-double-commutant-theorem (Proposition 2.1 and Definition 2.3; Section 9, the uniqueness lemma for measures and the product of probability spaces), the-spectral-theorem-for-bounded-self-adjoint-operators (Lemma 1.1, Theorem 3.1)",
    "harmonic-analysis-on-locally-compact-groups lesson measure-and-hilbert-space-tools (Theorems 2.1 and 2.2, monotone convergence, Fatou and dominated convergence; Theorem 3.2, completeness of L^2)",
    "function-algebras-and-approximation lesson the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity (Section 14, Weierstrass approximation)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}