| import Mathlib | |
| import OAI.Analysis.Crouzeix.MetricBoundary | |
| namespace OAI | |
| /-! Complete Finite. -/ | |
| noncomputable section | |
| open Set Filter Metric Topology Function Complex ComplexConjugate MeasureTheory | |
| open scoped Matrix Matrix.Norms.L2Operator MatrixOrder ComplexOrder | |
| namespace CrouzeixHilbert | |
| open Boundary | |
| lemma entrywiseHolomorphic_matrixPolynomial {m d : ℕ} (B : Fin (d + 1) → Coeff m) | |
| (V : Set ℂ) : EntrywiseHolomorphic V (matrixPolynomial B) := by | |
| intro i j | |
| simp only [matrixPolynomial, Matrix.sum_apply, Matrix.smul_apply, smul_eq_mul] | |
| apply DifferentiableOn.fun_sum | |
| intro k _ | |
| exact (differentiableOn_id.pow (k : ℕ)).mul_const _ | |
| lemma matrixHolomorphicEval_matrixPolynomial | |
| {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] [Nontrivial H] (A : Operator H) | |
| {V : Set ℂ} (hV : IsOpen V) (hKV : numericalClosure A ⊆ V) | |
| {m d : ℕ} (B : Fin (d + 1) → Coeff m) : | |
| matrixHolomorphicEval A V (matrixPolynomial B) = polynomialEval A B := by | |
| obtain ⟨Γ⟩ := exists_calculusContour (isCompact_numericalClosure A) | |
| (convex_numericalClosure A) (numericalClosure_nonempty A) hV hKV | |
| have he : matrixHolomorphicEval A V (matrixPolynomial B) = | |
| matrixContourEval A Γ.toSmoothContour (matrixPolynomial B) := by | |
| simp only [matrixHolomorphicEval, matrixContourEval, | |
| holomorphicEval_eq_contourEval A hV (entrywiseHolomorphic_matrixPolynomial B V _ _) Γ] | |
| rw [he, matrixContourEval_matrixPolynomial A Γ B] | |
| lemma norm_tensorOperator_one_le | |
| {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| (A : Operator H) (m : ℕ) : ‖tensorOperator A (1 : Coeff m)‖ ≤ ‖A‖ := by | |
| have h1 : ‖(1 : Coeff m)‖ ≤ 1 := by | |
| rw [Matrix.cstar_norm_def, map_one] | |
| exact ContinuousLinearMap.norm_id_le | |
| exact (norm_tensorOperator_le A (1 : Coeff m)).trans | |
| ((mul_le_mul_of_nonneg_left h1 (norm_nonneg _)).trans_eq (mul_one _)) | |
| namespace Conformal.ExteriorCollar | |
| variable {U : Set ℂ} (C : ExteriorCollar U) (R : InteriorCollar U) | |
| variable {n : ℕ} (hn : 0 < n) (A : Operator (EuclideanSpace ℂ (Fin n))) | |
| (hKU : numericalClosure A ⊆ U) (hU : IsOpen U) (hc : Convex ℝ U) | |
| (hb : Bornology.IsBounded U) | |
| include C R hn hKU hU hc hb | |
| lemma holomorphic_bound {V : Set ℂ} (hV : IsOpen V) (hUV : closure U ⊆ V) | |
| {m : ℕ} {F : ℂ → Coeff m} (hF : EntrywiseHolomorphic V F) : | |
| ‖matrixHolomorphicEval A V F‖ ≤ 2 * supNorm (closure U) F := by | |
| let : Nonempty (Fin n) := ⟨⟨0,hn⟩⟩ | |
| obtain ⟨S,J,D,hJS,_,hcond,hN,hD,hI⟩ := C.exists_conformal_contraction R hn A hKU hU hc hb | |
| let Λ := C.operatorDensity R D hD | |
| let F₀ := C.matrixAnalyticTrace hUV F hF.continuousOn | |
| let L := ∫ t, tensorOperator (Λ t) (F₀ t) ∂circleMeasure | |
| have hs : tensorOperator S (1 : Coeff m) * matrixHolomorphicEval A V F = | |
| L * tensorOperator S (1 : Coeff m) := | |
| C.physical_intertwines R hU hc A D S hKU hN hD hI hV hUV hF | |
| have hi : tensorOperator J (1 : Coeff m) * tensorOperator S (1 : Coeff m) = 1 := by | |
| rw [← tensorOperator_mul, hJS, one_mul, tensorOperator_one] | |
| have he : matrixHolomorphicEval A V F = | |
| tensorOperator J (1 : Coeff m) * L * tensorOperator S (1 : Coeff m) := by | |
| calc | |
| _ = (tensorOperator J (1 : Coeff m) * tensorOperator S (1 : Coeff m)) * | |
| matrixHolomorphicEval A V F := by rw [hi, one_mul] | |
| _ = _ := by rw [mul_assoc, hs, ← mul_assoc] | |
| have hsup := supNorm_bddAbove hb.isCompact_closure (hF.continuousOn.mono hUV) | |
| have hc0 := supNorm_nonneg hsup | |
| have hnL : ‖L‖ ≤ supNorm (closure U) F := | |
| norm_integral_tensor_density_le Λ (C.operatorDensity_nonneg R D hD hN) | |
| (C.integral_operatorDensity R D hD hU hc) F₀ _ hc0 | |
| (fun t => norm_le_supNorm hsup (C.boundaryMap_mem_frontier t).1) | |
| rw [he] | |
| calc | |
| ‖tensorOperator J (1 : Coeff m) * L * tensorOperator S (1 : Coeff m)‖ ≤ | |
| (‖tensorOperator J (1 : Coeff m)‖ * ‖L‖) * ‖tensorOperator S (1 : Coeff m)‖ := | |
| (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _)) | |
| _ ≤ (‖J‖ * supNorm (closure U) F) * ‖S‖ := | |
| mul_le_mul (mul_le_mul (norm_tensorOperator_one_le J m) hnL (norm_nonneg _) (norm_nonneg _)) | |
| (norm_tensorOperator_one_le S m) (norm_nonneg _) (mul_nonneg (norm_nonneg _) hc0) | |
| _ = (‖S‖ * ‖J‖) * supNorm (closure U) F := by ring | |
| _ ≤ 2 * supNorm (closure U) F := mul_le_mul_of_nonneg_right hcond hc0 | |
| lemma polynomial_bound {m d : ℕ} (B : Fin (d + 1) → Coeff m) : | |
| ‖polynomialEval A B‖ ≤ 2 * supNorm (closure U) (matrixPolynomial B) := by | |
| let : Nonempty (Fin n) := ⟨⟨0,hn⟩⟩ | |
| rw [← matrixHolomorphicEval_matrixPolynomial A isOpen_univ (subset_univ _) B] | |
| exact C.holomorphic_bound R hn A hKU hU hc hb isOpen_univ (subset_univ _) | |
| (entrywiseHolomorphic_matrixPolynomial B univ) | |
| end Conformal.ExteriorCollar | |
| theorem finite_matrix_polynomial_bound : FiniteMatrixPolynomialBound := by | |
| intro n hn M m _ d B | |
| let : Nonempty (Fin n) := ⟨⟨0,hn⟩⟩ | |
| let A := Matrix.toEuclideanCLM (𝕜 := ℂ) M | |
| change ‖polynomialEval A B‖ ≤ 2 * supNorm (numericalRange A) (matrixPolynomial B) | |
| rw [(polynomial_sup_eq_max A B).1] | |
| let K := numericalClosure A | |
| let P := matrixPolynomial B | |
| have hK := isCompact_numericalClosure A | |
| have hc := convex_numericalClosure A | |
| have hne : K.Nonempty := numericalClosure_nonempty A | |
| have hsup := supNorm_bddAbove hK (continuous_matrixPolynomial B).continuousOn | |
| have hs0 := supNorm_nonneg hsup | |
| apply le_of_forall_pos_le_add | |
| intro ε hε | |
| let V := {z : ℂ | ‖P z‖ < supNorm K P + ε / 2} | |
| have hV : IsOpen V := isOpen_lt (continuous_matrixPolynomial B).norm continuous_const | |
| have hKV : K ⊆ V := fun z hz => (norm_le_supNorm hsup hz).trans_lt (by dsimp only [K,P]; linarith) | |
| obtain ⟨U,hU,hb,hcU,hKU,hUV,hchart⟩ := Conformal.exists_convex_analytic_neighborhood | |
| hK hc hne hV hKV | |
| obtain ⟨a,ha⟩ := hne | |
| obtain ⟨R⟩ := Conformal.exists_interiorCollar hU hb | |
| (Conformal.isSimplyConnected_of_convex hcU ⟨a,hKU ha⟩) ⟨a,hKU ha⟩ hchart | |
| obtain ⟨C⟩ := Conformal.exists_exteriorCollar hU hb hcU (hKU ha) hchart | |
| have hg := C.polynomial_bound R hn A hKU hU hcU hb B | |
| have hs : supNorm (closure U) P ≤ supNorm K P + ε / 2 := | |
| supNorm_le (by dsimp only [K,P]; linarith) (fun z hz => (hUV hz).le) | |
| calc | |
| ‖polynomialEval A B‖ ≤ 2 * supNorm (closure U) P := hg | |
| _ ≤ 2 * (supNorm K P + ε / 2) := mul_le_mul_of_nonneg_left hs (by norm_num) | |
| _ = 2 * supNorm (numericalClosure A) (matrixPolynomial B) + ε := by dsimp only [K,P]; ring | |
| end CrouzeixHilbert | |
| end | |
| end OAI | |