| import OAI.Analysis.IrrationalRotation.Model.Commutant | |
| namespace OAI | |
| namespace Invariant233.Commutant | |
| variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℂ H] | |
| theorem mem_iff (T A : H →L[ℂ] H) : | |
| A ∈ Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H)) ↔ | |
| T * A = A * T := by | |
| simp [Subalgebra.mem_centralizer_iff] | |
| theorem one_mem (T : H →L[ℂ] H) : | |
| (1 : H →L[ℂ] H) ∈ Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H)) := | |
| (Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H))).one_mem | |
| theorem isClosed_strong_commutant (T : H →L[ℂ] H) : | |
| IsClosed {A : H →Lₚₜ[ℂ] H | ∀ x : H, T (A x) = A (T x)} := by | |
| have h : ∀ x : H, | |
| IsClosed {A : H →Lₚₜ[ℂ] H | T (A x) = A (T x)} := by | |
| intro x | |
| exact isClosed_eq (T.continuous.comp (continuous_eval_const x)) | |
| (continuous_eval_const (T x)) | |
| simpa only [Set.iInter_ofPred] using isClosed_iInter h | |
| theorem commute_of_strong_limit {ι : Type*} {l : Filter ι} [l.NeBot] | |
| (T A : H →L[ℂ] H) (B : ι → H →L[ℂ] H) | |
| (hB : ∀ᶠ i in l, T * B i = B i * T) | |
| (hlim : ∀ x : H, Filter.Tendsto (fun i => B i x) l (nhds (A x))) : | |
| T * A = A * T := by | |
| ext x | |
| have h₁ := T.continuous.continuousAt.tendsto.comp (hlim x) | |
| have h₂ := hlim (T x) | |
| apply tendsto_nhds_unique_of_eventuallyEq h₁ h₂ | |
| filter_upwards [hB] with i hi | |
| exact congrArg (fun S : H →L[ℂ] H => S x) hi | |
| end Invariant233.Commutant | |
| namespace Invariant233 | |
| variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℂ H] | |
| theorem operatorCommutant_mem_iff (T A : H →L[ℂ] H) : | |
| A ∈ operatorCommutant T ↔ T * A = A * T := | |
| Commutant.mem_iff T A | |
| theorem operatorCommutant_stronglyClosed (T : H →L[ℂ] H) : | |
| IsStronglyClosed (operatorCommutant T) := by | |
| unfold IsStronglyClosed | |
| have heq : {A : H →Lₚₜ[ℂ] H | strongOperatorUnderlying A ∈ operatorCommutant T} = | |
| {A : H →Lₚₜ[ℂ] H | ∀ x : H, T (A x) = A (T x)} := by | |
| ext A | |
| change strongOperatorUnderlying A ∈ operatorCommutant T ↔ _ | |
| exact (operatorCommutant_mem_iff T (strongOperatorUnderlying A)).trans | |
| ContinuousLinearMap.ext_iff | |
| rw [heq] | |
| exact Commutant.isClosed_strong_commutant T | |
| theorem operatorCommutant_transitive_iff (T : H →L[ℂ] H) : | |
| IsTransitive (operatorCommutant T) ↔ HasNoNontrivialHyperinvariantSubspace T := by | |
| constructor | |
| · intro h L hL | |
| exact h L hL.1 hL.2 | |
| · intro h L hclosed hinvariant | |
| exact h L ⟨hclosed, hinvariant⟩ | |
| theorem IsHyperinvariant.isInvariant {T : H →L[ℂ] H} {L : Submodule ℂ H} | |
| (h : IsHyperinvariant T L) : IsInvariant T L := by | |
| apply h.2 T | |
| rw [operatorCommutant_mem_iff] | |
| end Invariant233 | |
| end OAI | |