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ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | import Mathlib
import OAI.Analysis.Crouzeix.HolomorphicSpectrum
namespace OAI
/-! Functional Composition. -/
noncomputable section
open Set Filter Metric Topology Complex MeasureTheory
open scoped InnerProductSpace Matrix.Norms.L2Operator
namespace CrouzeixHilbert
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]
universe u
variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
[CompleteSpace H] [Nontrivial H]
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
end
end OAI
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