| import OAI.Analysis.BackwardIntertwiners.HilbertClassification | |
| namespace OAI | |
| namespace Hyperinvariant233 | |
| open Filter | |
| open scoped Topology | |
| universe u v | |
| variable {H : Type u} {K : Type v} | |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K] | |
| theorem norm_conjugate_le (e : H ≃ₗᵢ[ℂ] K) (T : H →L[ℂ] H) : | |
| ‖e.conjStarAlgEquiv T‖ ≤ ‖T‖ := by | |
| apply ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) | |
| intro x | |
| change ‖e (T (e.symm x))‖ ≤ ‖T‖ * ‖x‖ | |
| rw [e.norm_map] | |
| simpa only [e.symm.norm_map] using T.le_opNorm (e.symm x) | |
| theorem norm_conjugate (e : H ≃ₗᵢ[ℂ] K) (T : H →L[ℂ] H) : | |
| ‖e.conjStarAlgEquiv T‖ = ‖T‖ := by | |
| apply le_antisymm (norm_conjugate_le e T) | |
| have he : e.symm.conjStarAlgEquiv (e.conjStarAlgEquiv T) = T := by | |
| ext x | |
| simp only [LinearIsometryEquiv.conjStarAlgEquiv_apply_apply, | |
| LinearIsometryEquiv.symm_symm, LinearIsometryEquiv.symm_apply_apply] | |
| have h := norm_conjugate_le e.symm (e.conjStarAlgEquiv T) | |
| rwa [he] at h | |
| theorem transitiveCommutant_conjugate (e : H ≃ₗᵢ[ℂ] K) (T : H →L[ℂ] H) | |
| (hT : TransitiveCommutant T) : TransitiveCommutant (e.conjStarAlgEquiv T) := by | |
| intro W hW hInv | |
| let U : Submodule ℂ H := W.comap e.toLinearMap | |
| have hU : IsClosed (U : Set H) := hW.preimage e.continuous | |
| have hUI : ∀ A : H →L[ℂ] H, A * T = T * A → ∀ x ∈ U, A x ∈ U := by | |
| intro A hA x hx | |
| change e (A x) ∈ W | |
| have hc : e.conjStarAlgEquiv A * e.conjStarAlgEquiv T = | |
| e.conjStarAlgEquiv T * e.conjStarAlgEquiv A := by | |
| simpa only [map_mul] using congrArg e.conjStarAlgEquiv hA | |
| have hm := hInv (e.conjStarAlgEquiv A) hc (e x) hx | |
| simpa using hm | |
| rcases hT U hU hUI with hz | ho | |
| · left | |
| apply le_antisymm _ bot_le | |
| intro y hy | |
| have hx : e.symm y ∈ U := by change e (e.symm y) ∈ W; simpa using hy | |
| rw [hz] at hx | |
| have hzero : e.symm y = 0 := hx | |
| have := congrArg e hzero | |
| simpa using this | |
| · right | |
| apply le_antisymm le_top | |
| intro y _ | |
| have hx : e.symm y ∈ U := by rw [ho]; trivial | |
| change e (e.symm y) ∈ W at hx | |
| simpa using hx | |
| theorem mainClaim_transport (e : H ≃ₗᵢ[ℂ] K) (h : MainClaim (H := H)) : | |
| MainClaim (H := K) := by | |
| obtain ⟨T, hT0, hq, ht⟩ := h | |
| refine ⟨e.conjStarAlgEquiv T, ?_, ?_, transitiveCommutant_conjugate e T ht⟩ | |
| · exact fun hzero => hT0 (e.conjStarAlgEquiv.injective (by simpa using hzero)) | |
| · have he : (fun n : ℕ => ‖(e.conjStarAlgEquiv T) ^ n‖ ^ (1 / (n : ℝ))) = | |
| (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) := by | |
| funext n | |
| rw [← map_pow, norm_conjugate] | |
| rwa [he] | |
| end Hyperinvariant233 | |
| namespace Hyperinvariant233 | |
| open Filter | |
| open scoped Topology | |
| universe u | |
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| theorem mem_commutant_iff (T A : H →L[ℂ] H) : | |
| A ∈ commutant T ↔ A * T = T * A := by | |
| simp only [commutant, Subalgebra.mem_centralizer_iff, Set.mem_singleton_iff, forall_eq] | |
| exact eq_comm | |
| variable [completeSpace : CompleteSpace H] | |
| omit completeSpace in | |
| theorem commutant_isClosed_strong [CompleteSpace H] (T : H →L[ℂ] H) : | |
| @IsClosed (H →L[ℂ] H) strongOperatorTopology (commutant T : Set (H →L[ℂ] H)) := by | |
| let : TopologicalSpace (H →L[ℂ] H) := strongOperatorTopology | |
| have heval : ∀ x : H, Continuous (fun A : H →L[ℂ] H => A x) := by | |
| intro x | |
| exact (continuous_apply x).comp continuous_induced_dom | |
| have hset : (commutant T : Set (H →L[ℂ] H)) = | |
| ⋂ x : H, {A : H →L[ℂ] H | A (T x) = T (A x)} := by | |
| ext A | |
| simp only [Set.mem_iInter, Set.mem_ofPred_eq, SetLike.mem_coe, mem_commutant_iff] | |
| constructor | |
| · intro h x | |
| exact congrArg (fun B : H →L[ℂ] H => B x) h | |
| · intro h | |
| ext x | |
| exact h x | |
| rw [hset] | |
| exact isClosed_iInter (fun x => isClosed_eq (heval (T x)) (T.continuous.comp (heval x))) | |
| omit completeSpace in | |
| theorem scalar_of_commutant_top [CompleteSpace H] (T : H →L[ℂ] H) (h : commutant T = ⊤) : | |
| ∃ c : ℂ, algebraMap ℂ (H →L[ℂ] H) c = T := by | |
| have hc : T ∈ Subalgebra.center ℂ (H →L[ℂ] H) := by | |
| rw [Subalgebra.mem_center_iff] | |
| intro A | |
| apply (mem_commutant_iff T A).mp | |
| rw [h] | |
| trivial | |
| rw [Algebra.IsCentral.center_eq_bot] at hc | |
| exact Algebra.mem_bot.mp hc | |
| theorem commutant_ne_top_of_nonzero_quasinilpotent [Nontrivial H] | |
| (T : H →L[ℂ] H) (hT : T ≠ 0) | |
| (hlim : Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0)) : | |
| commutant T ≠ ⊤ := by | |
| intro htop | |
| obtain ⟨c, hc⟩ := scalar_of_commutant_top T htop | |
| have hroot (n : ℕ) (hn : n ≠ 0) : ‖T ^ n‖ ^ (1 / (n : ℝ)) = ‖c‖ := by | |
| rw [← hc, Algebra.algebraMap_eq_smul_one, smul_pow, one_pow, | |
| norm_smul, norm_pow, norm_one, mul_one, one_div] | |
| exact Real.pow_rpow_inv_natCast (norm_nonneg c) hn | |
| have hevent : (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) =ᶠ[atTop] | |
| (fun _ : ℕ => ‖c‖) := by | |
| filter_upwards [eventually_gt_atTop 0] with n hn | |
| exact hroot n (Nat.ne_of_gt hn) | |
| have hconst : Tendsto (fun _ : ℕ => ‖c‖) atTop (𝓝 0) := hlim.congr' hevent | |
| have hc0 : ‖c‖ = 0 := tendsto_nhds_unique tendsto_const_nhds hconst | |
| apply hT | |
| rw [← hc, norm_eq_zero.mp hc0, map_zero] | |
| theorem fullClaim_of_mainClaim [Nontrivial H] (h : MainClaim (H := H)) : | |
| FullClaim (H := H) := by | |
| obtain ⟨T, hT, hlim, htrans⟩ := h | |
| exact ⟨T, hT, hlim, htrans, | |
| commutant_ne_top_of_nonzero_quasinilpotent T hT hlim, | |
| commutant_isClosed_strong T⟩ | |
| end Hyperinvariant233 | |
| end OAI | |