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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
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