| import OAI.Analysis.BackwardIntertwiners.DiagonalSymbols |
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| namespace OAI |
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| namespace BackwardIntertwiners |
| open Filter |
| open scoped Topology |
| noncomputable section |
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| open Topology MeasureTheory Set |
| open scoped ENNReal |
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| open scoped Classical |
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| lemma L2_norm_eq_sqrt {X E : Type*} [MeasurableSpace X] [NormedAddCommGroup E] |
| {μ : Measure X} (f : Lp E 2 μ) : |
| ‖f‖ = Real.sqrt (∫ x, ‖f x‖^2 ∂μ) := by |
| rw [Lp.norm_def, toReal_eLpNorm, |
| lpNorm_eq_integral_norm_rpow_toReal (by norm_num : (2:ENNReal) ≠ 0) |
| (by norm_num : (2:ENNReal) ≠ ∞) (Lp.aestronglyMeasurable f)] |
| simp [Real.sqrt_eq_rpow] |
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| lemma L2_tendsto_zero_of_dominated {X E ι : Type*} {l : Filter ι} [l.IsCountablyGenerated] [MeasurableSpace X] [NormedAddCommGroup E] |
| {μ : Measure X} (F : ι → Lp E 2 μ) (b : X → ℝ) |
| (hb : Integrable (fun x => b x ^ 2) μ) |
| (hB : ∀ n, ∀ᵐ x ∂μ, ‖F n x‖ ≤ b x) |
| (hlim : ∀ᵐ x ∂μ, Tendsto (fun n => F n x) l (𝓝 0)) : |
| Tendsto F l (𝓝 0) := by |
| apply tendsto_zero_iff_norm_tendsto_zero.mpr |
| have hi : Tendsto (fun n => ∫ x, ‖F n x‖^2 ∂μ) l (𝓝 0) := by |
| simpa using tendsto_integral_filter_of_dominated_convergence (f:=fun _ : X => (0:ℝ)) |
| (fun x => b x^2) |
| (Eventually.of_forall fun n => ((Lp.aestronglyMeasurable (F n)).norm.aemeasurable.pow_const 2).aestronglyMeasurable) |
| (Eventually.of_forall fun n => (hB n).mono fun x hx => by |
| rw [Real.norm_of_nonneg (sq_nonneg _)] |
| exact pow_le_pow_left₀ (norm_nonneg _) hx 2) hb |
| (hlim.mono fun x hx => by simpa using hx.norm.pow 2) |
| simpa only [Function.comp_def, Real.sqrt_zero, L2_norm_eq_sqrt] using Real.continuous_sqrt.continuousAt.tendsto.comp hi |
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| lemma fiber_norm_sq (v : Fiber) : ‖v‖^2 = ∑' j, ‖v j‖^2 := by |
| simpa using lp.norm_rpow_eq_tsum (by norm_num : (0:ℝ) < (2:ENNReal).toReal) v |
| lemma fiber_tendsto_zero_of_dominated {ι : Type*} {l : Filter ι} (v : ι → Fiber) (b : ℤ → ℝ) |
| (hb : Summable (fun j => b j^2)) |
| (hB : ∀ n j, ‖v n j‖ ≤ b j) |
| (hlim : ∀ j, Tendsto (fun n => v n j) l (𝓝 0)) : |
| Tendsto v l (𝓝 0) := by |
| apply tendsto_zero_iff_norm_tendsto_zero.mpr |
| have hi : Tendsto (fun n => ‖v n‖^2) l (𝓝 0) := by |
| simp_rw [fiber_norm_sq] |
| simpa using tendsto_tsum_of_dominated_convergence (g:=fun _ : ℤ => (0:ℝ)) hb |
| (fun j => by simpa using (hlim j).norm.pow 2) |
| (Filter.Eventually.of_forall fun n j => by |
| rw [Real.norm_of_nonneg (sq_nonneg _)] |
| exact pow_le_pow_left₀ (norm_nonneg _) (hB n j) 2) |
| simpa only [Function.comp_def, Real.sqrt_zero, Real.sqrt_sq_eq_abs,abs_of_nonneg (norm_nonneg _)] |
| using Real.continuous_sqrt.continuousAt.tendsto.comp hi |
| lemma diagonalFiber_bound (a : DiagonalSymbol) {C : ℝ} (hC : 0 ≤ C) |
| (ha : ∀ x j, ‖a x j‖ ≤ C) (x : Base) (v : Fiber) : |
| ‖diagonalFiber a x v‖ ≤ C*‖v‖ := by |
| calc |
| _ ≤ ‖(C:ℂ) • v‖ := by |
| apply lp.norm_mono (by norm_num : (2:ENNReal) ≠ 0) |
| intro j |
| simp only [diagonalFiber_apply,lp.coeFn_smul,Pi.smul_apply,norm_smul,norm_mul, |
| Complex.norm_real,Real.norm_of_nonneg hC] |
| exact mul_le_mul_of_nonneg_right (ha x j) (norm_nonneg _) |
| _ = C*‖v‖ := by rw [norm_smul,Complex.norm_real,Real.norm_of_nonneg hC] |
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| lemma diagonalFiber_tendsto_zero {ι : Type*} {l : Filter ι} (a : ι → DiagonalSymbol) {C : ℝ} |
| (ha : ∀ n x j, ‖a n x j‖ ≤ C) (x : Base) (v : Fiber) |
| (hlim : ∀ j, Tendsto (fun n => a n x j) l (𝓝 0)) : |
| Tendsto (fun n => diagonalFiber (a n) x v) l (𝓝 0) := by |
| apply fiber_tendsto_zero_of_dominated _ (fun j => C*‖v j‖) |
| · have hv : Summable (fun j => ‖v j‖^2) := by |
| simpa using (lp.memℓp v).summable (by norm_num : (0:ℝ) < (2:ENNReal).toReal) |
| simpa only [mul_pow] using hv.mul_left (C^2) |
| · intro n j |
| rw [diagonalFiber_apply,norm_mul] |
| exact mul_le_mul_of_nonneg_right (ha n x j) (norm_nonneg _) |
| · intro j |
| simpa using (hlim j).mul_const (v j) |
| lemma L2_integrable_bound_sq (f : GlobalHilbert) (C : ℝ) : |
| Integrable (fun x => (C*‖f x‖)^2) baseMeasure := by |
| have hf : Integrable (fun x => ‖f x‖^2) baseMeasure := |
| (Lp.memLp f).integrable_norm_pow (by norm_num : (2:ℕ) ≠ 0) |
| simpa only [mul_pow] using hf.const_mul (C^2) |
| lemma diagonalOperator_tendsto_zero {ι : Type*} [Countable ι] {l : Filter ι} [l.IsCountablyGenerated] (a : ι → DiagonalSymbol) {C : ℝ} |
| (hC : 0 ≤ C) (ha : ∀ n x j, ‖a n x j‖ ≤ C) |
| (hlim : ∀ᵐ x ∂baseMeasure, ∀ j, Tendsto (fun n => a n x j) l (𝓝 0)) |
| (f : GlobalHilbert) : Tendsto (fun n => diagonalOperator (a n) f) l (𝓝 0) := by |
| apply L2_tendsto_zero_of_dominated _ (fun x => C*‖f x‖) (L2_integrable_bound_sq f C) |
| · intro n |
| filter_upwards [diagonalOperator_ae (a n) f] with x hx |
| rw [hx] |
| exact diagonalFiber_bound _ hC (ha n) _ _ |
| · filter_upwards [hlim,ae_all_iff.mpr (fun n => diagonalOperator_ae (a n) f)] with x hx hn |
| have he : (fun n => diagonalOperator (a n) f x) = (fun n => diagonalFiber (a n) x (f x)) := funext hn |
| rw [he] |
| exact diagonalFiber_tendsto_zero a ha x (f x) hx |
| lemma diagonalOperator_tendsto {ι : Type*} [Countable ι] {l : Filter ι} [l.IsCountablyGenerated] (a : ι → DiagonalSymbol) (b : DiagonalSymbol) |
| {C : ℝ} (hC : 0 ≤ C) (ha : ∀ n x j, ‖a n x j‖ ≤ C) |
| (hb : ∀ x j, ‖b x j‖ ≤ C) |
| (hlim : ∀ᵐ x ∂baseMeasure, ∀ j, Tendsto (fun n => a n x j) l (𝓝 (b x j))) |
| (f : GlobalHilbert) : |
| Tendsto (fun n => diagonalOperator (a n) f) l (𝓝 (diagonalOperator b f)) := by |
| have hB : ∀ n x j, ‖(a n-b) x j‖ ≤ 2*C := fun n x j => |
| (norm_sub_le _ _).trans ((add_le_add (ha n x j) (hb x j)).trans_eq (by ring)) |
| have hl : ∀ᵐ x ∂baseMeasure, ∀ j, Tendsto (fun n => (a n-b) x j) l (𝓝 0) := by |
| filter_upwards [hlim] with x hx j |
| simpa only [DiagonalSymbol.sub_apply,sub_self] using (hx j).sub_const (b x j) |
| have hh := diagonalOperator_tendsto_zero (fun n => a n-b) (mul_nonneg (by norm_num) hC) hB hl f |
| have he (n : ι) : diagonalOperator (a n-b) f = diagonalOperator (a n) f - diagonalOperator b f := |
| congrArg (fun T : GlobalHilbert →L[ℂ] GlobalHilbert => T f) (map_sub diagonalRepresentation (a n) b) |
| simp_rw [he] at hh |
| exact tendsto_sub_nhds_zero_iff.mp hh |
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| universe u |
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [completeSpace : CompleteSpace H] |
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| @[instance_reducible] |
| def strongOperatorTopology : TopologicalSpace (H →L[ℂ] H) := |
| TopologicalSpace.induced (fun T : H →L[ℂ] H => (fun v : H => T v)) inferInstance |
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| def StronglyClosed (A : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop := |
| @IsClosed (H →L[ℂ] H) strongOperatorTopology A |
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| omit completeSpace in |
| lemma strong_continuous_apply [CompleteSpace H] (v : H) : |
| @Continuous (H →L[ℂ] H) H strongOperatorTopology inferInstance (fun T => T v) := by |
| let := strongOperatorTopology (H:=H) |
| exact (continuous_apply v).comp continuous_induced_dom |
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| lemma StronglyClosed.mem_of_tendsto {A : StarSubalgebra ℂ (H →L[ℂ] H)} |
| (hA : StronglyClosed A) {ι : Type*} {l : Filter ι} [l.NeBot] |
| {T : ι → H →L[ℂ] H} {S : H →L[ℂ] H} |
| (hT : ∀ᶠ i in l, T i ∈ A) (hlim : ∀ v, Tendsto (fun i => T i v) l (𝓝 (S v))) : |
| S ∈ A := by |
| let := strongOperatorTopology (H:=H) |
| refine IsClosed.mem_of_tendsto hA (f:=T) ?_ hT |
| change Tendsto T l (@nhds (H →L[ℂ] H) (TopologicalSpace.induced _ _) S) |
| rw [nhds_induced,Filter.tendsto_comap_iff] |
| exact tendsto_pi_nhds.mpr hlim |
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| def strongGenerated (s : Set (H →L[ℂ] H)) : StarSubalgebra ℂ (H →L[ℂ] H) := |
| sInf {A | StronglyClosed A ∧ s ⊆ A} |
| lemma mem_strongGenerated {s : Set (H →L[ℂ] H)} {T : H →L[ℂ] H} : |
| T ∈ strongGenerated s ↔ ∀ A : StarSubalgebra ℂ (H →L[ℂ] H), |
| StronglyClosed A → s ⊆ A → T ∈ A := by |
| simp [strongGenerated,StarSubalgebra.mem_sInf] |
| lemma subset_strongGenerated (s : Set (H →L[ℂ] H)) : s ⊆ strongGenerated s := by |
| intro T hT |
| exact mem_strongGenerated.mpr fun _ _ hs => hs hT |
| lemma strongGenerated_le {s : Set (H →L[ℂ] H)} {A : StarSubalgebra ℂ (H →L[ℂ] H)} |
| (hA : StronglyClosed A) (hs : s ⊆ A) : strongGenerated s ≤ A := |
| fun _ hT => mem_strongGenerated.mp hT A hA hs |
| lemma strongGenerated_closed (s : Set (H →L[ℂ] H)) : StronglyClosed (strongGenerated s) := by |
| let := strongOperatorTopology (H:=H) |
| have he : (strongGenerated s : Set (H →L[ℂ] H)) = |
| ⋂ A : StarSubalgebra ℂ (H →L[ℂ] H), ⋂ (_ : StronglyClosed A ∧ s ⊆ A), (A : Set _) := by |
| ext T |
| simp [mem_strongGenerated] |
| rw [StronglyClosed,he] |
| exact isClosed_iInter fun A => isClosed_iInter fun h => h.1 |
| lemma strongGenerated_mono {s t : Set (H →L[ℂ] H)} (hst : s ⊆ t) : |
| strongGenerated s ≤ strongGenerated t := |
| strongGenerated_le (strongGenerated_closed t) (hst.trans (subset_strongGenerated t)) |
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| lemma strong_centralizer_closed (s : Set (H →L[ℂ] H)) : |
| StronglyClosed (StarSubalgebra.centralizer ℂ s) := by |
| let := strongOperatorTopology (H:=H) |
| have he : (StarSubalgebra.centralizer ℂ s : Set (H →L[ℂ] H)) = |
| ⋂ A ∈ s ∪ star s, ⋂ v : H, {T | A (T v) = T (A v)} := by |
| ext T |
| change (∀ A ∈ s ∪ star s, A*T=T*A) ↔ _ |
| simp only [mem_iInter,mem_ofPred_eq] |
| exact forall₂_congr fun A hA => ⟨fun h v => DFunLike.congr_fun h v, |
| fun h => ContinuousLinearMap.ext h⟩ |
| rw [StronglyClosed,he] |
| exact isClosed_iInter fun A => isClosed_iInter fun _ => isClosed_iInter fun v => |
| isClosed_eq (A.continuous.comp (strong_continuous_apply v)) (strong_continuous_apply (A v)) |
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| lemma strongGenerated_commutative {s : Set (H →L[ℂ] H)} |
| (hs : ∀ A ∈ s, IsSelfAdjoint A) (hc : ∀ A ∈ s, ∀ B ∈ s, A*B=B*A) : |
| ∀ A ∈ strongGenerated s, ∀ B ∈ strongGenerated s, A*B=B*A := by |
| have hle : strongGenerated s ≤ StarSubalgebra.centralizer ℂ s := by |
| apply strongGenerated_le (strong_centralizer_closed s) |
| intro B hB |
| apply (StarSubalgebra.mem_centralizer_iff ℂ).mpr |
| intro A hA |
| exact ⟨hc A hA B hB, by rw [(hs A hA).star_eq]; exact hc A hA B hB⟩ |
| have hle' : strongGenerated s ≤ StarSubalgebra.centralizer ℂ (strongGenerated s : Set _) := by |
| apply strongGenerated_le (strong_centralizer_closed _) |
| intro B hB |
| apply (StarSubalgebra.mem_centralizer_iff ℂ).mpr |
| intro A hA |
| refine ⟨((StarSubalgebra.mem_centralizer_iff ℂ).mp (hle hA) B hB).1.symm, ?_⟩ |
| exact ((StarSubalgebra.mem_centralizer_iff ℂ).mp (hle ((strongGenerated s).star_mem' hA)) B hB).1.symm |
| intro A hA B hB |
| exact ((StarSubalgebra.mem_centralizer_iff ℂ).mp (hle' hB) A hA).1 |
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| end |
| end BackwardIntertwiners |
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| end OAI |
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