| import OAI.Analysis.IrrationalRotation.Operators.HyperinvariantProjection |
| import OAI.Analysis.IrrationalRotation.Analysis.UnitaryTransport |
| import OAI.Analysis.IrrationalRotation.Operators.CommutantAlgebra |
| import OAI.Analysis.IrrationalRotation.Spectral.QuasinilpotentNonscalar |
| import OAI.Analysis.IrrationalRotation.Model.Analytic |
| import OAI.Analysis.IrrationalRotation.Operators.HilbertModelSize |
| import Mathlib.MeasureTheory.Function.L2Space |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Invariant233 |
|
|
| open scoped Topology |
| open TopologicalSpace |
|
|
| variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
| [CompleteSpace E] [NormedAddCommGroup F] [InnerProductSpace ℂ F] |
| [CompleteSpace F] |
|
|
| theorem no_hyperinvariant_subspace_of_no_invariant_projection |
| (M : VonNeumannAlgebra E) {T : E →L[ℂ] E} (hT : T ∈ M) |
| (hproj : ∀ p : E →L[ℂ] E, p ∈ M → IsStarProjection p → |
| (1 - p) * T * p = 0 → p = 0 ∨ p = 1) : |
| ∀ S : Submodule ℂ E, IsClosed (S : Set E) → |
| (∀ A : E →L[ℂ] E, Commute A T → ∀ x ∈ S, A x ∈ S) → S = ⊥ ∨ S = ⊤ := by |
| intro S hS hhyper |
| let : CompleteSpace S := hS.completeSpace_coe |
| have hhyper' : ∀ A : E →L[ℂ] E, Commute A T → S ∈ Module.End.invtSubmodule A := |
| hhyper |
| obtain ⟨hmem, hinv⟩ := hyperinvariant_subspace_projection_mem M hT S hhyper' |
| rcases hproj S.starProjection hmem isStarProjection_starProjection hinv with hzero | hone |
| · left |
| apply le_antisymm _ bot_le |
| intro x hx |
| have heq := Submodule.starProjection_eq_self_iff.mpr hx |
| simpa [hzero] using heq.symm |
| · right |
| apply le_antisymm le_top |
| intro x _ |
| apply Submodule.starProjection_eq_self_iff.mp |
| simp [hone] |
|
|
| theorem hilbert_counterexample_of_factor_witness |
| [SeparableSpace E] [SeparableSpace F] |
| (hE : ¬ FiniteDimensional ℂ E) (hF : ¬ FiniteDimensional ℂ F) |
| (M : VonNeumannAlgebra E) {T : E →L[ℂ] E} (hT : T ∈ M) (hne : T ≠ 0) |
| (hquasi : Filter.Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) |
| Filter.atTop (𝓝 (0 : ℝ))) |
| (hproj : ∀ p : E →L[ℂ] E, p ∈ M → IsStarProjection p → |
| (1 - p) * T * p = 0 → p = 0 ∨ p = 1) : |
| ∃ T' : F →L[ℂ] F, T' ≠ 0 ∧ |
| Filter.Tendsto (fun n : ℕ => ‖T' ^ n‖ ^ (1 / (n : ℝ))) |
| Filter.atTop (𝓝 (0 : ℝ)) ∧ |
| (∀ S : Submodule ℂ F, IsClosed (S : Set F) → |
| (∀ A : F →L[ℂ] F, Commute A T' → ∀ x ∈ S, A x ∈ S) → S = ⊥ ∨ S = ⊤) := by |
| obtain ⟨e⟩ := UnitaryTransport.nonempty_linearIsometryEquiv hE hF |
| exact UnitaryTransport.transport_counterexample e T hne hquasi |
| (no_hyperinvariant_subspace_of_no_invariant_projection M hT hproj) |
|
|
| theorem hasNoNontrivialHyperinvariantSubspace_of_no_invariant_projection |
| (M : VonNeumannAlgebra E) {T : E →L[ℂ] E} (hT : T ∈ M) |
| (hproj : ∀ p : E →L[ℂ] E, p ∈ M → IsStarProjection p → |
| (1 - p) * T * p = 0 → p = 0 ∨ p = 1) : |
| HasNoNontrivialHyperinvariantSubspace T := by |
| intro S hS |
| apply no_hyperinvariant_subspace_of_no_invariant_projection M hT hproj S hS.1 |
| intro A hA |
| apply hS.2 A |
| exact (operatorCommutant_mem_iff T A).2 hA.symm.eq |
|
|
| theorem transitive_commutant_of_factor_witness |
| [SeparableSpace E] [SeparableSpace F] |
| (hE : ¬ FiniteDimensional ℂ E) (hF : ¬ FiniteDimensional ℂ F) |
| (M : VonNeumannAlgebra E) {T : E →L[ℂ] E} (hT : T ∈ M) (hne : T ≠ 0) |
| (hquasi : NormQuasinilpotent T) |
| (hproj : ∀ p : E →L[ℂ] E, p ∈ M → IsStarProjection p → |
| (1 - p) * T * p = 0 → p = 0 ∨ p = 1) : |
| ∃ T' : F →L[ℂ] F, T' ≠ 0 ∧ NormQuasinilpotent T' ∧ |
| HasNoNontrivialHyperinvariantSubspace T' ∧ |
| operatorCommutant T' ≠ ⊤ ∧ IsStronglyClosed (operatorCommutant T') ∧ |
| IsTransitive (operatorCommutant T') := by |
| classical |
| have : Nontrivial F := by |
| cases subsingleton_or_nontrivial F with |
| | inl h => |
| let : Subsingleton F := h |
| exact False.elim (hF inferInstance) |
| | inr h => exact h |
| obtain ⟨T', hne', hquasi', hhyper'⟩ := |
| hilbert_counterexample_of_factor_witness hE hF M hT hne hquasi hproj |
| have hnonscalar : ¬ ∃ c : ℂ, T' = c • (1 : F →L[ℂ] F) := by |
| rintro ⟨c, hc⟩ |
| exact InvariantProjection.continuousLinearMap_not_scalar_of_nonzero_norm_root_tendsto |
| hne' hquasi' c hc |
| have hhyper : HasNoNontrivialHyperinvariantSubspace T' := by |
| intro S hS |
| apply hhyper' S hS.1 |
| intro A hA |
| exact hS.2 A ((operatorCommutant_mem_iff T' A).2 hA.symm.eq) |
| obtain ⟨hproper, hclosed, htrans⟩ := commutant_algebra_properties T' hnonscalar |
| exact ⟨T', hne', hquasi', hhyper, hproper, hclosed, htrans.2 hhyper⟩ |
|
|
| theorem transitive_commutant_of_circle_witness |
| [SeparableSpace F] (hF : ¬ FiniteDimensional ℂ F) |
| (M : VonNeumannAlgebra (MeasureTheory.Lp (lp (fun _ : ℤ => ℂ) 2) 2 circleMeasure)) |
| {T : (MeasureTheory.Lp (lp (fun _ : ℤ => ℂ) 2) 2 circleMeasure) →L[ℂ] |
| (MeasureTheory.Lp (lp (fun _ : ℤ => ℂ) 2) 2 circleMeasure)} |
| (hT : T ∈ M) (hne : T ≠ 0) (hquasi : NormQuasinilpotent T) |
| (hproj : ∀ p : (MeasureTheory.Lp (lp (fun _ : ℤ => ℂ) 2) 2 circleMeasure) →L[ℂ] |
| (MeasureTheory.Lp (lp (fun _ : ℤ => ℂ) 2) 2 circleMeasure), |
| p ∈ M → IsStarProjection p → (1 - p) * T * p = 0 → p = 0 ∨ p = 1) : |
| ∃ T' : F →L[ℂ] F, T' ≠ 0 ∧ NormQuasinilpotent T' ∧ |
| HasNoNontrivialHyperinvariantSubspace T' ∧ |
| operatorCommutant T' ≠ ⊤ ∧ IsStronglyClosed (operatorCommutant T') ∧ |
| IsTransitive (operatorCommutant T') := by |
| let := HilbertModelSize.circle_representation_separable |
| exact transitive_commutant_of_factor_witness |
| HilbertModelSize.circle_representation_not_finiteDimensional hF M hT hne hquasi hproj |
|
|
| end Invariant233 |
|
|
| end |
|
|
| end OAI |
|
|
