| import OAI.Analysis.HilbertCrouzeix.Compression |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Complex Set |
| open scoped TensorProduct Matrix.Norms.L2Operator Classical ComplexConjugate |
|
|
| namespace HilbertCrouzeix |
|
|
| universe u |
|
|
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| def sharpOperator : CoeffSpace 2 →L[ℂ] CoeffSpace 2 := |
| (EuclideanSpace.proj (1 : Fin 2)).smulRight |
| ((2 : ℂ) • EuclideanSpace.single (0 : Fin 2) 1) |
|
|
| def sharpPolynomial : Fin (1 + 1) → Coeff 1 := ![0, 1] |
|
|
| theorem sharpPolynomial_apply (z : ℂ) : matrixPolynomial sharpPolynomial z = z • (1 : Coeff 1) := by |
| simp [matrixPolynomial, sharpPolynomial, Fin.sum_univ_two] |
|
|
| theorem sharpPolynomial_norm (z : ℂ) : ‖matrixPolynomial sharpPolynomial z‖ = ‖z‖ := by |
| rw [sharpPolynomial_apply, norm_smul, norm_one, mul_one] |
| |
| theorem sharpOperator_apply (x : CoeffSpace 2) : |
| sharpOperator x = x.ofLp 1 • ((2 : ℂ) • EuclideanSpace.single (0 : Fin 2) 1) := by |
| simp only [sharpOperator, ContinuousLinearMap.smulRight_apply, EuclideanSpace.proj, |
| PiLp.proj_apply] |
|
|
| theorem sharpOperator_numerical_le_one {z : ℂ} (hz : z ∈ numericalRange sharpOperator) : |
| ‖z‖ ≤ 1 := by |
| obtain ⟨x, hx, rfl⟩ := hz |
| have hsq : ‖x.ofLp 0‖ ^ 2 + ‖x.ofLp 1‖ ^ 2 = 1 := by |
| have h := EuclideanSpace.norm_sq_eq x |
| simpa only [hx, one_pow, Fin.sum_univ_two] using h.symm |
| rw [sharpOperator_apply] |
| rw [inner_smul_right, inner_smul_right, EuclideanSpace.inner_single_right] |
| simp only [norm_mul, norm_conj, one_mul] |
| norm_num |
| nlinarith [sq_nonneg (‖x.ofLp 0‖ - ‖x.ofLp 1‖)] |
|
|
| theorem sharpPolynomial_sup_le_one : |
| normSup (numericalRange sharpOperator) (matrixPolynomial sharpPolynomial) ≤ 1 := by |
| rw [normSup, ite_eq_left (numericalRange_nonempty sharpOperator)] |
| apply csSup_le ((numericalRange_nonempty sharpOperator).image _) |
| rintro _ ⟨z, hz, rfl⟩ |
| change ‖matrixPolynomial sharpPolynomial z‖ ≤ 1 |
| rw [sharpPolynomial_norm] |
| exact sharpOperator_numerical_le_one hz |
|
|
| theorem sharpPolynomial_sup_nonneg : |
| 0 ≤ normSup (numericalRange sharpOperator) (matrixPolynomial sharpPolynomial) := by |
| obtain ⟨z, hz⟩ := numericalRange_nonempty sharpOperator |
| have hb : BddAbove ((fun z => ‖matrixPolynomial sharpPolynomial z‖) '' |
| numericalRange sharpOperator) := by |
| refine ⟨1, ?_⟩ |
| rintro _ ⟨w, hw, rfl⟩ |
| change ‖matrixPolynomial sharpPolynomial w‖ ≤ 1 |
| rw [sharpPolynomial_norm] |
| exact sharpOperator_numerical_le_one hw |
| rw [normSup, ite_eq_left (numericalRange_nonempty sharpOperator)] |
| exact (norm_nonneg _).trans (le_csSup hb (mem_image_of_mem _ hz)) |
| |
| theorem sharpPolynomialEval_norm_ge_two : |
| 2 ≤ ‖polynomialEval sharpOperator sharpPolynomial‖ := by |
| let e : CoeffSpace 2 := EuclideanSpace.single (1 : Fin 2) 1 |
| let v : CoeffSpace 1 := EuclideanSpace.single (0 : Fin 1) 1 |
| let ξ : Amplification (CoeffSpace 2) 1 := ↑(e ⊗ₜ[ℂ] v) |
| have hξ : ‖ξ‖ = 1 := by |
| simp only [ξ, UniformSpace.Completion.norm_coe, TensorProduct.norm_tmul, |
| e, v, EuclideanSpace.single, PiLp.norm_single, norm_one, mul_one] |
| have heval : polynomialEval sharpOperator sharpPolynomial ξ = |
| (↑(((2 : ℂ) • EuclideanSpace.single (0 : Fin 2) (1 : ℂ)) ⊗ₜ[ℂ] v) : |
| Amplification (CoeffSpace 2) 1) := by |
| simp [polynomialEval, Fin.sum_univ_two, sharpPolynomial, ξ, tensorOp_tmul, |
| sharpOperator_apply, e] |
| have hvalue : ‖polynomialEval sharpOperator sharpPolynomial ξ‖ = 2 := by |
| rw [heval] |
| simp [UniformSpace.Completion.norm_coe, TensorProduct.norm_tmul, |
| norm_smul, v] |
| simpa only [hvalue, hξ, mul_one] using |
| (polynomialEval sharpOperator sharpPolynomial).le_opNorm ξ |
| |
| theorem uniformlySharp : UniformlySharp := by |
| intro c hc |
| refine ⟨2, 1, 1, by norm_num, by norm_num, sharpOperator, sharpPolynomial, ?_⟩ |
| apply lt_of_lt_of_le _ sharpPolynomialEval_norm_ge_two |
| by_cases h : 0 ≤ c |
| · exact lt_of_le_of_lt |
| (by simpa using mul_le_mul_of_nonneg_left sharpPolynomial_sup_le_one h) hc |
| · exact lt_of_le_of_lt |
| (mul_nonpos_of_nonpos_of_nonneg (le_of_not_ge h) sharpPolynomial_sup_nonneg) |
| (by norm_num) |
| |
| |
| end HilbertCrouzeix |
| |
| end |
| |
| end OAI |
| |
