| import Mathlib | |
| namespace OAI | |
| /-! Numerical ranges, completed tensor amplification, and holomorphic and rational operator evaluation. -/ | |
| noncomputable section | |
| open scoped TensorProduct Matrix.Norms.L2Operator InnerProductSpace | |
| open Set | |
| namespace CrouzeixHilbert | |
| universe u | |
| abbrev Operator (H : Type u) [NormedAddCommGroup H] [NormedSpace ℂ H] := H →L[ℂ] H | |
| abbrev Coeff (m : ℕ) := Matrix (Fin m) (Fin m) ℂ | |
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| def numericalRange (A : Operator H) : Set ℂ := | |
| {z | ∃ x : H, ‖x‖ = 1 ∧ ⟪x, A x⟫_ℂ = z} | |
| def numericalClosure (A : Operator H) : Set ℂ := closure (numericalRange A) | |
| def supNorm {E : Type*} [NormedAddCommGroup E] (S : Set ℂ) (F : ℂ → E) : ℝ := | |
| sSup (insert 0 ((fun z => ‖F z‖) '' S)) | |
| abbrev Amplification (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| (m : ℕ) := UniformSpace.Completion (H ⊗[ℂ] EuclideanSpace ℂ (Fin m)) | |
| def tensorOperator (A : Operator H) {m : ℕ} (B : Coeff m) : | |
| Operator (Amplification H m) := | |
| (TensorProduct.mapL A (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin m) B)).completion | |
| def matrixPolynomial {m d : ℕ} (B : Fin (d + 1) → Coeff m) (z : ℂ) : Coeff m := | |
| ∑ k : Fin (d + 1), z ^ (k : ℕ) • B k | |
| def polynomialEval (A : Operator H) {m d : ℕ} (B : Fin (d + 1) → Coeff m) : | |
| Operator (Amplification H m) := | |
| ∑ k : Fin (d + 1), tensorOperator (A ^ (k : ℕ)) (B k) | |
| structure SmoothContour where | |
| path : ℝ → ℂ | |
| smooth : ContDiff ℝ 1 path | |
| closed : path 0 = path 1 | |
| def SmoothContour.index (Γ : SmoothContour) (z : ℂ) : ℂ := | |
| (2 * (Real.pi : ℂ) * Complex.I)⁻¹ * | |
| ∫ t in (0 : ℝ)..1, deriv Γ.path t / (Γ.path t - z) | |
| structure CalculusContour (K U : Set ℂ) extends SmoothContour where | |
| avoids : ∀ t ∈ Icc (0 : ℝ) 1, path t ∈ U \ K | |
| index_inside : ∀ z ∈ K, toSmoothContour.index z = 1 | |
| index_outside : ∀ z ∉ U, toSmoothContour.index z = 0 | |
| def contourEval (A : Operator H) (Γ : SmoothContour) (f : ℂ → ℂ) : Operator H := | |
| (2 * (Real.pi : ℂ) * Complex.I)⁻¹ • | |
| ∫ t in (0 : ℝ)..1, (deriv Γ.path t * f (Γ.path t)) • | |
| Ring.inverse (algebraMap ℂ (Operator H) (Γ.path t) - A) | |
| def holomorphicEval (A : Operator H) (U : Set ℂ) (f : ℂ → ℂ) : Operator H := by | |
| classical | |
| exact if h : Nonempty (CalculusContour (numericalClosure A) U) then | |
| contourEval A (Classical.choice h).toSmoothContour f | |
| else 0 | |
| def matrixHolomorphicEval (A : Operator H) {m : ℕ} (U : Set ℂ) | |
| (F : ℂ → Coeff m) : Operator (Amplification H m) := | |
| ∑ i, ∑ j, tensorOperator (holomorphicEval A U (fun z => F z i j)) | |
| (Matrix.single i j 1) | |
| def EntrywiseHolomorphic {m : ℕ} (U : Set ℂ) (F : ℂ → Coeff m) : Prop := | |
| ∀ i j, DifferentiableOn ℂ (fun z => F z i j) U | |
| abbrev RationalMatrix (m : ℕ) := Matrix (Fin m) (Fin m) (RatFunc ℂ) | |
| def PolesOutside {m : ℕ} (K : Set ℂ) (R : RationalMatrix m) : Prop := | |
| ∀ i j z, z ∈ K → (R i j).denom.eval z ≠ 0 | |
| def matrixRationalFunction {m : ℕ} (R : RationalMatrix m) (z : ℂ) : Coeff m := | |
| fun i j => RatFunc.eval (RingHom.id ℂ) z (R i j) | |
| def rationalEval (A : Operator H) (r : RatFunc ℂ) : Operator H := | |
| (Polynomial.aeval A r.num) * Ring.inverse (Polynomial.aeval A r.denom) | |
| def matrixRationalEval (A : Operator H) {m : ℕ} (R : RationalMatrix m) : | |
| Operator (Amplification H m) := | |
| ∑ i, ∑ j, tensorOperator (rationalEval A (R i j)) (Matrix.single i j 1) | |
| def NonzeroConclusion (A : Operator H) : Prop := | |
| IsCompact (numericalClosure A) ∧ | |
| Convex ℝ (numericalClosure A) ∧ | |
| spectrum ℂ A ⊆ numericalClosure A ∧ | |
| (∀ (m : ℕ), 0 < m → ∀ (d : ℕ) (B : Fin (d + 1) → Coeff m), | |
| ‖polynomialEval A B‖ ≤ 2 * supNorm (numericalRange A) (matrixPolynomial B) ∧ | |
| supNorm (numericalRange A) (matrixPolynomial B) = | |
| supNorm (numericalClosure A) (matrixPolynomial B) ∧ | |
| ∃ z ∈ numericalClosure A, | |
| supNorm (numericalClosure A) (matrixPolynomial B) = ‖matrixPolynomial B z‖) ∧ | |
| (∀ (m : ℕ), 0 < m → ∀ (U : Set ℂ) (F : ℂ → Coeff m), | |
| IsOpen U → numericalClosure A ⊆ U → EntrywiseHolomorphic U F → | |
| Nonempty (CalculusContour (numericalClosure A) U) ∧ | |
| (∀ Γ : CalculusContour (numericalClosure A) U, ∀ i j, | |
| holomorphicEval A U (fun z => F z i j) = | |
| contourEval A Γ.toSmoothContour (fun z => F z i j)) ∧ | |
| ‖matrixHolomorphicEval A U F‖ ≤ 2 * supNorm (numericalClosure A) F ∧ | |
| ∃ z ∈ numericalClosure A, supNorm (numericalClosure A) F = ‖F z‖) ∧ | |
| (∀ (m : ℕ), 0 < m → ∀ (R : RationalMatrix m), | |
| PolesOutside (numericalClosure A) R → | |
| (∀ i j, IsUnit (Polynomial.aeval A (R i j).denom)) ∧ | |
| ‖matrixRationalEval A R‖ ≤ | |
| 2 * supNorm (numericalClosure A) (matrixRationalFunction R) ∧ | |
| (∃ z ∈ numericalClosure A, | |
| supNorm (numericalClosure A) (matrixRationalFunction R) = | |
| ‖matrixRationalFunction R z‖) ∧ | |
| ∃ U : Set ℂ, IsOpen U ∧ numericalClosure A ⊆ U ∧ | |
| EntrywiseHolomorphic U (matrixRationalFunction R) ∧ | |
| matrixRationalEval A R = matrixHolomorphicEval A U (matrixRationalFunction R)) | |
| def ZeroConclusion (A : Operator H) : Prop := | |
| numericalRange A = ∅ ∧ | |
| (∀ (m d : ℕ) (B : Fin (d + 1) → Coeff m), | |
| polynomialEval A B = 0 ∧ | |
| supNorm (numericalRange A) (matrixPolynomial B) = 0 ∧ | |
| ‖polynomialEval A B‖ ≤ 2 * supNorm (numericalRange A) (matrixPolynomial B)) ∧ | |
| (∀ (m : ℕ) (U : Set ℂ) (F : ℂ → Coeff m), | |
| matrixHolomorphicEval A U F = 0 ∧ | |
| supNorm (numericalRange A) F = 0 ∧ | |
| ‖matrixHolomorphicEval A U F‖ ≤ 2 * supNorm (numericalRange A) F) ∧ | |
| (∀ (m : ℕ) (R : RationalMatrix m), | |
| matrixRationalEval A R = 0 ∧ | |
| supNorm (numericalRange A) (matrixRationalFunction R) = 0 ∧ | |
| ‖matrixRationalEval A R‖ ≤ | |
| 2 * supNorm (numericalRange A) (matrixRationalFunction R)) | |
| def SharpConstant : Prop := | |
| ∃ (A : Operator (EuclideanSpace ℂ (Fin 2))) (B : Fin 2 → Coeff 1), | |
| supNorm (numericalRange A) (matrixPolynomial (d := 1) B) = 1 ∧ | |
| ‖polynomialEval A (d := 1) B‖ = 2 | |
| end CrouzeixHilbert | |
| end | |
| end OAI | |