openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Rotation /ChoKoLeeCircleReduction.lean
| import OAI.Analysis.IrrationalRotation.Model.OperatorProperties | |
| import OAI.Analysis.IrrationalRotation.Analysis.ChoKoLeeComparison | |
| import OAI.Analysis.IrrationalRotation.Operators.HilbertConsequence | |
| import OAI.Analysis.IrrationalRotation.Rotation.RotationAlgebra | |
| import OAI.Analysis.IrrationalRotation.Spectral.RotationQuasinilpotent | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233 | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] [TopologicalSpace.SeparableSpace H] | |
| theorem cho_ko_lee_comparison_of_continuous_circle_obstruction | |
| (hH : ¬ FiniteDimensional ℂ H) {θ : ℝ} (hθ : Irrational θ) | |
| (f : C(Circle, ℝ)) (hf : IsContinuousCircleWeight f) | |
| (hinv : HasOnlyTrivialInvariantProjections | |
| (Rotation.algebra θ) (Rotation.weightedReal θ f)) : | |
| ∃ Q : H →L[ℂ] H, Q ≠ 0 ∧ NormQuasinilpotent Q ∧ | |
| HasNoNontrivialHyperinvariantSubspace Q ∧ | |
| operatorCommutant Q ≠ ⊤ ∧ IsStronglyClosed (operatorCommutant Q) ∧ | |
| IsTransitive (operatorCommutant Q) ∧ | |
| operatorCommutant (1 + Q) = operatorCommutant Q ∧ | |
| ∀ C : H → H, ∀ hC : IsConjugation C, | |
| IsInfiniteConjugationIsometry hC.toHilbertConjugation (1 + Q) ∧ | |
| HasNoNontrivialHyperinvariantSubspace (1 + Q) := by | |
| obtain ⟨Q, hne, hQ, hhyper, hproper, hclosed, htrans⟩ := | |
| transitive_commutant_of_circle_witness (F := H) hH (Rotation.algebra θ) | |
| (Rotation.weightedReal_mem_algebra θ f) | |
| (Rotation.weightedReal_ne_zero_of_isContinuousCircleWeight θ f hf) | |
| (Rotation.weightedReal_normQuasinilpotent hθ f hf) hinv | |
| refine ⟨Q, hne, hQ, hhyper, hproper, hclosed, htrans, | |
| operatorCommutant_one_add Q, fun C hC => ?_⟩ | |
| exact ⟨isInfiniteConjugationIsometry_one_add_of_isConjugation hC hQ, | |
| (hasNoNontrivialHyperinvariantSubspace_one_add_iff Q).2 hhyper⟩ | |
| end Invariant233 | |
| end | |
| end OAI | |