| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Complex Set |
| open scoped TensorProduct Matrix.Norms.L2Operator Classical ComplexConjugate |
|
|
| namespace HilbertCrouzeix |
|
|
| universe u |
|
|
| abbrev Coeff (m : ℕ) := Matrix (Fin m) (Fin m) ℂ |
| abbrev CoeffSpace (m : ℕ) := EuclideanSpace ℂ (Fin m) |
|
|
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] |
|
|
| def numericalRange (A : H →L[ℂ] H) : Set ℂ := |
| {z | ∃ x : H, ‖x‖ = 1 ∧ inner ℂ x (A x) = z} |
|
|
| def numericalClosure (A : H →L[ℂ] H) : Set ℂ := closure (numericalRange A) |
|
|
| abbrev Amplification (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H] |
| (m : ℕ) := UniformSpace.Completion (H ⊗[ℂ] CoeffSpace m) |
|
|
| def tensorOp {m : ℕ} (A : H →L[ℂ] H) (B : Coeff m) : |
| Amplification H m →L[ℂ] Amplification H m := |
| (TensorProduct.mapL A (Matrix.toEuclideanCLM (n := Fin m) (𝕜 := ℂ) B)).completion |
|
|
| def matrixPolynomial {m d : ℕ} (B : Fin (d + 1) → Coeff m) (z : ℂ) : Coeff m := |
| ∑ k : Fin (d + 1), z ^ k.val • B k |
|
|
| def polynomialEval {m d : ℕ} (A : H →L[ℂ] H) (B : Fin (d + 1) → Coeff m) : |
| Amplification H m →L[ℂ] Amplification H m := |
| ∑ k : Fin (d + 1), tensorOp (A ^ k.val) (B k) |
|
|
| def normSup {E : Type*} [Norm E] (S : Set ℂ) (f : ℂ → E) : ℝ := |
| if S.Nonempty then sSup ((fun z => ‖f z‖) '' S) else 0 |
|
|
| def contourIntegral {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] |
| (γ : ℝ → ℂ) (f : ℂ → E) : E := |
| ∫ t : ℝ in (0 : ℝ)..1, deriv γ t • f (γ t) |
|
|
| def contourIndex (γ : ℝ → ℂ) (z : ℂ) : ℂ := |
| (2 * (Real.pi : ℂ) * Complex.I)⁻¹ * contourIntegral γ (fun w => (w - z)⁻¹) |
|
|
| def IsCauchyContour (K U : Set ℂ) (γ : ℝ → ℂ) : Prop := |
| ContDiff ℝ 1 γ ∧ γ 0 = γ 1 ∧ |
| (∀ t ∈ Set.Icc (0 : ℝ) 1, γ t ∈ U \ K) ∧ |
| (∀ z ∈ K, contourIndex γ z = 1) ∧ |
| (∀ z ∉ U, contourIndex γ z = 0) |
|
|
| def chosenContour (K U : Set ℂ) : ℝ → ℂ := |
| Classical.epsilon (IsCauchyContour K U) |
|
|
| variable [CompleteSpace H] |
|
|
| def dunfordIntegral (A : H →L[ℂ] H) (γ : ℝ → ℂ) (f : ℂ → ℂ) : H →L[ℂ] H := |
| (2 * (Real.pi : ℂ) * Complex.I)⁻¹ • |
| contourIntegral γ (fun z => f z • resolvent A z) |
|
|
| def holomorphicCalculus (A : H →L[ℂ] H) (U : Set ℂ) (f : ℂ → ℂ) : H →L[ℂ] H := |
| dunfordIntegral A (chosenContour (numericalClosure A) U) f |
|
|
| def matrixUnit {m : ℕ} (i j : Fin m) : Coeff m := Matrix.single i j 1 |
|
|
| def holomorphicEval {m : ℕ} (A : H →L[ℂ] H) (U : Set ℂ) (F : ℂ → Coeff m) : |
| Amplification H m →L[ℂ] Amplification H m := |
| ∑ i : Fin m, ∑ j : Fin m, |
| tensorOp (holomorphicCalculus A U (fun z => F z i j)) (matrixUnit i j) |
|
|
| def contourEval {m : ℕ} (A : H →L[ℂ] H) (γ : ℝ → ℂ) (F : ℂ → Coeff m) : |
| Amplification H m →L[ℂ] Amplification H m := |
| ∑ i : Fin m, ∑ j : Fin m, |
| tensorOp (dunfordIntegral A γ (fun z => F z i j)) (matrixUnit i j) |
|
|
| def HolomorphicOn {m : ℕ} (F : ℂ → Coeff m) (U : Set ℂ) : Prop := |
| ∀ i j, DifferentiableOn ℂ (fun z => F z i j) U |
|
|
| abbrev RationalMatrix (m : ℕ) := Matrix (Fin m) (Fin m) (RatFunc ℂ) |
|
|
| def poleFreeSet {m : ℕ} (R : RationalMatrix m) : Set ℂ := |
| {z | ∀ i j, (R i j).denom.eval z ≠ 0} |
|
|
| def rationalFunction {m : ℕ} (R : RationalMatrix m) (z : ℂ) : Coeff m := |
| fun i j => RatFunc.eval (RingHom.id ℂ) z (R i j) |
|
|
| def rationalCalculus (A : H →L[ℂ] H) (r : RatFunc ℂ) : H →L[ℂ] H := |
| Polynomial.aeval A r.num * Ring.inverse (Polynomial.aeval A r.denom) |
|
|
| def rationalEval {m : ℕ} (A : H →L[ℂ] H) (R : RationalMatrix m) : |
| Amplification H m →L[ℂ] Amplification H m := |
| ∑ i : Fin m, ∑ j : Fin m, tensorOp (rationalCalculus A (R i j)) (matrixUnit i j) |
|
|
| def PolynomialConclusion (A : H →L[ℂ] H) : Prop := |
| ∀ (m : ℕ), 0 < m → ∀ (d : ℕ) (B : Fin (d + 1) → Coeff m), |
| ‖polynomialEval A B‖ ≤ 2 * normSup (numericalRange A) (matrixPolynomial B) ∧ |
| 2 * normSup (numericalRange A) (matrixPolynomial B) = |
| 2 * normSup (numericalClosure A) (matrixPolynomial B) ∧ |
| IsGreatest ((fun z => ‖matrixPolynomial B z‖) '' numericalClosure A) |
| (normSup (numericalClosure A) (matrixPolynomial B)) |
|
|
| def HolomorphicConclusion (A : H →L[ℂ] H) : Prop := |
| ∀ (m : ℕ), 0 < m → ∀ (U : Set ℂ), IsOpen U → numericalClosure A ⊆ U → |
| ∀ (F : ℂ → Coeff m), HolomorphicOn F U → |
| (∃ γ, IsCauchyContour (numericalClosure A) U γ) ∧ |
| (∀ γ, IsCauchyContour (numericalClosure A) U γ → |
| ∀ i j, dunfordIntegral A γ (fun z => F z i j) = |
| holomorphicCalculus A U (fun z => F z i j)) ∧ |
| IsGreatest ((fun z => ‖F z‖) '' numericalClosure A) |
| (normSup (numericalClosure A) F) ∧ |
| ‖holomorphicEval A U F‖ ≤ 2 * normSup (numericalClosure A) F |
|
|
| def RationalConclusion (A : H →L[ℂ] H) : Prop := |
| ∀ (m : ℕ), 0 < m → ∀ (R : RationalMatrix m), |
| numericalClosure A ⊆ poleFreeSet R → |
| (∀ i j, IsUnit (Polynomial.aeval A (R i j).denom)) ∧ |
| rationalEval A R = holomorphicEval A (poleFreeSet R) (rationalFunction R) ∧ |
| IsGreatest ((fun z => ‖rationalFunction R z‖) '' numericalClosure A) |
| (normSup (numericalClosure A) (rationalFunction R)) ∧ |
| ‖rationalEval A R‖ ≤ 2 * normSup (numericalClosure A) (rationalFunction R) |
|
|
| def NonzeroConclusion (A : H →L[ℂ] H) : Prop := |
| IsCompact (numericalClosure A) ∧ Convex ℝ (numericalClosure A) ∧ |
| spectrum ℂ A ⊆ numericalClosure A ∧ PolynomialConclusion A ∧ |
| HolomorphicConclusion A ∧ RationalConclusion A |
|
|
| def ZeroConclusion (A : H →L[ℂ] H) : Prop := |
| numericalRange A = ∅ ∧ |
| (∀ (m : ℕ) (F : ℂ → Coeff m), normSup (numericalRange A) F = 0) ∧ |
| ∀ (m : ℕ), 0 < m → |
| (∀ T : Amplification H m →L[ℂ] Amplification H m, T = 0) ∧ |
| (∀ (d : ℕ) (B : Fin (d + 1) → Coeff m), polynomialEval A B = 0 ∧ |
| ‖polynomialEval A B‖ ≤ 2 * normSup (numericalRange A) (matrixPolynomial B)) ∧ |
| (∀ (U : Set ℂ) (F : ℂ → Coeff m), holomorphicEval A U F = 0 ∧ |
| ‖holomorphicEval A U F‖ ≤ 2 * normSup (numericalRange A) F) ∧ |
| (∀ (R : RationalMatrix m), rationalEval A R = 0 ∧ |
| ‖rationalEval A R‖ ≤ 2 * normSup (numericalRange A) (rationalFunction R)) |
|
|
| def UniformlySharp : Prop := |
| ∀ c : ℝ, c < 2 → ∃ (n m d : ℕ), 0 < n ∧ 0 < m ∧ |
| ∃ (A : CoeffSpace n →L[ℂ] CoeffSpace n) (B : Fin (d + 1) → Coeff m), |
| c * normSup (numericalRange A) (matrixPolynomial B) < ‖polynomialEval A B‖ |
|
|
|
|
|
|
| end HilbertCrouzeix |
|
|
| end |
|
|
| end OAI |
|
|
