File size: 1,912 Bytes
9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | 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
|