| import Mathlib |
| import OAI.Analysis.Crouzeix.HolomorphicSpectrum |
|
|
| namespace OAI |
|
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| /-! Functional Composition. -/ |
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| noncomputable section |
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| open Set Filter Metric Topology Complex MeasureTheory |
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| open scoped InnerProductSpace Matrix.Norms.L2Operator |
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| namespace CrouzeixHilbert |
|
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| lemma intertwines_aeval {B : Type*} [Ring B] [Algebra ℂ B] |
| (S T D : B) (h : S * T = D * S) (p : Polynomial ℂ) : |
| S * Polynomial.aeval T p = Polynomial.aeval D p * S := by |
| have hp (n : ℕ) : S * T^n = D^n * S := by |
| induction n with |
| | zero => simp |
| | succ n ih => rw [pow_succ, ← mul_assoc, ih, mul_assoc, h, ← mul_assoc, ← pow_succ] |
| induction p using Polynomial.induction_on' with |
| | add p q hq hr => simp only [map_add, mul_add, add_mul, hq, hr] |
| | monomial n c => |
| simp only [Polynomial.aeval_monomial] |
| rw [← mul_assoc, (Algebra.commutes c S).symm, mul_assoc, hp, ← mul_assoc] |
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| universe u |
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| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] |
| [CompleteSpace H] [Nontrivial H] |
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| lemma holomorphicEval_comp_intertwines (A D S : Operator H) (hN : ‖D‖ ≤ 1) |
| {U V : Set ℂ} (hU : IsOpen U) (hKU : numericalClosure A ⊆ U) |
| (hV : IsOpen V) (hKV : closedBall (0 : ℂ) 1 ⊆ V) |
| {f g : ℂ → ℂ} (hf : DifferentiableOn ℂ f U) (hg : DifferentiableOn ℂ g V) |
| (hfU : MapsTo f U V) (hfK : MapsTo f (numericalClosure A) (closedBall 0 1)) |
| (hI : S * holomorphicEval A U f = D * S) : |
| S * holomorphicEval A U (g ∘ f) = holomorphicEval D V g * S := by |
| let K : Set ℂ := closedBall 0 1 |
| have hK : IsCompact K := isCompact_closedBall _ _ |
| have hKc : Convex ℝ K := convex_closedBall _ _ |
| have hDK : numericalClosure D ⊆ K := |
| (numericalClosure_subset_closedBall D).trans (closedBall_subset_closedBall hN) |
| obtain ⟨ε,hε,hεV⟩ := hK.exists_cthickening_subset_open hV hKV |
| let L := cthickening ε K |
| have hKL : K ⊆ interior L := |
| (self_subset_thickening hε _).trans (thickening_subset_interior_cthickening _ _) |
| let W := U ∩ f ⁻¹' interior L |
| have hW : IsOpen W := hf.continuousOn.isOpen_inter_preimage hU isOpen_interior |
| have hAW : numericalClosure A ⊆ W := fun z hz => ⟨hKU hz,hKL (hfK hz)⟩ |
| have hfW : DifferentiableOn ℂ f W := hf.mono inter_subset_left |
| have hAWL : MapsTo f W L := fun z hz => interior_subset hz.2 |
| have hWL : W ⊆ U := inter_subset_left |
| have hgf := hg.comp hf hfU |
| obtain ⟨Γ⟩ := exists_calculusContour (isCompact_numericalClosure A) |
| (convex_numericalClosure A) (numericalClosure_nonempty A) hW hAW |
| obtain ⟨Δ₀⟩ := exists_calculusContour (isCompact_numericalClosure D) |
| (convex_numericalClosure D) (numericalClosure_nonempty D) isOpen_interior (hDK.trans hKL) |
| let Δ : CalculusContour (numericalClosure D) V := { |
| toSmoothContour := Δ₀.toSmoothContour |
| avoids := fun t ht => ⟨hεV (interior_subset (Δ₀.avoids t ht).1), (Δ₀.avoids t ht).2⟩ |
| index_inside := Δ₀.index_inside |
| index_outside := fun z hz => Δ₀.index_outside z (fun he => hz (hεV (interior_subset he))) } |
| have hΔL : Δ.toSmoothContour.trace ⊆ L := by |
| rintro z ⟨t,ht,rfl⟩ |
| exact interior_subset (Δ₀.avoids t ht).1 |
| have hΓW : Γ.toSmoothContour.trace ⊆ W := by |
| rintro z ⟨t,ht,rfl⟩ |
| exact (Γ.avoids t ht).1 |
| obtain ⟨p,hp⟩ := exists_polynomial_tendstoUniformlyOn hK.cthickening (hKc.cthickening ε) hV hεV hg |
| have hpd := tendsto_contourEval D Δ (fun n => (p n).continuous.continuousOn) (hp.mono hΔL) |
| simp_rw [contourEval_polynomial, ← holomorphicEval_eq_contourEval D hV hg Δ] at hpd |
| have hpa : Tendsto (fun n => Polynomial.aeval (holomorphicEval A U f) (p n)) atTop |
| (𝓝 (holomorphicEval A U (g ∘ f))) := by |
| have hp' : TendstoUniformlyOn (fun n z => (p n).eval (f z)) (g ∘ f) atTop Γ.toSmoothContour.trace := by |
| apply Metric.tendstoUniformlyOn_iff.mpr |
| intro δ hδ |
| filter_upwards [Metric.tendstoUniformlyOn_iff.mp hp δ hδ] with n hn z hz |
| exact hn (f z) (hAWL (hΓW hz)) |
| have hh := tendsto_contourEval A Γ |
| (fun n => (p n).continuous.comp_continuousOn (hf.continuousOn.mono (hΓW.trans hWL))) hp' |
| have he (n : ℕ) : contourEval A Γ.toSmoothContour (fun z => (p n).eval (f z)) = |
| Polynomial.aeval (holomorphicEval A U f) (p n) := by |
| calc |
| _ = holomorphicEval A W (fun z => (p n).eval (f z)) := |
| (holomorphicEval_eq_contourEval A hW ((p n).differentiable.comp_differentiableOn hfW) Γ).symm |
| _ = Polynomial.aeval (holomorphicEval A W f) (p n) := |
| holomorphicEval_polynomial_comp A hW hAW hfW (p n) |
| _ = _ := by rw [holomorphicEval_restrict A hW hU hAW hWL hf] |
| change Tendsto (fun n => contourEval A Γ.toSmoothContour (fun z => (p n).eval (f z))) atTop |
| (𝓝 (contourEval A Γ.toSmoothContour (g ∘ f))) at hh |
| simp_rw [he, ← holomorphicEval_eq_contourEval A hW (hgf.mono hWL) Γ, |
| holomorphicEval_restrict A hW hU hAW hWL hgf] at hh |
| exact hh |
| have hh := (tendsto_const_nhds (x := S)).mul hpa |
| have he (n : ℕ) : S * Polynomial.aeval (holomorphicEval A U f) (p n) = |
| Polynomial.aeval D (p n) * S := intertwines_aeval S _ D hI (p n) |
| simp_rw [he] at hh |
| exact (tendsto_nhds_unique hh (hpd.mul tendsto_const_nhds)) |
|
|
| end CrouzeixHilbert |
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| end |
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| end OAI |
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