openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Spectral /ProductPolarModulusComplex.lean
| import OAI.Analysis.IrrationalRotation.Model.Product | |
| import OAI.Analysis.IrrationalRotation.Model.ConcreteSpectral | |
| import OAI.Analysis.IrrationalRotation.Spectral.FieldDiagonalPolar | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped ENNReal ComplexOrder | |
| namespace Invariant233.ProductModel.GlobalPolarSupport | |
| local instance productPolarModulusComplexRealModule : Module ℝ (Hilbert →L[ℂ] Hilbert) := | |
| Module.complexToReal (Hilbert →L[ℂ] Hilbert) | |
| local instance productPolarModulusComplexFunctionalCalculus : | |
| NonUnitalContinuousFunctionalCalculus ℝ (Hilbert →L[ℂ] Hilbert) IsSelfAdjoint := by | |
| infer_instance | |
| theorem multiplier_nonneg (θ : ℕ → ℝ) (g : Base → ℂ) (C : ℝ) | |
| (hC : 0 ≤ C) (hg : Measurable g) (hb : ∀ x, ‖g x‖ ≤ C) | |
| (hpos : ∀ x, 0 ≤ g x) : 0 ≤ multiplier θ g C hC hg hb := by | |
| apply WeightedShift.fieldDiagonal_nonneg | |
| exact Filter.Eventually.of_forall fun x k => hpos (rotationIterate θ k x) | |
| theorem operatorModulus_shift_mul_multiplier (θ : ℕ → ℝ) (g : Base → ℂ) | |
| (C : ℝ) (hC : 0 ≤ C) (hg : Measurable g) (hb : ∀ x, ‖g x‖ ≤ C) | |
| (hpos : ∀ x, 0 ≤ g x) : | |
| operatorModulus (shiftOperator * multiplier θ g C hC hg hb) = | |
| multiplier θ g C hC hg hb := by | |
| change CFC.abs (shiftOperator * multiplier θ g C hC hg hb) = _ | |
| apply WeightedShift.abs_global_shift_mul_fieldDiagonal | |
| exact Filter.Eventually.of_forall fun x k => hpos (rotationIterate θ k x) | |
| theorem shiftOperator_mem_unitary : shiftOperator ∈ unitary (Hilbert →L[ℂ] Hilbert) := | |
| WeightedShift.global_shift_mem_unitary 1 | |
| theorem weightedOperator_eq_shift_mul_modulus (θ c : ℕ → ℝ) : | |
| weightedOperator θ c = shiftOperator * operatorModulus (weightedOperator θ c) := by | |
| simp only [weightedOperator, ← ContinuousLinearMap.mul_def] | |
| rw [operatorModulus_shift_mul_multiplier] | |
| intro x | |
| exact Complex.nonneg_iff.mpr ⟨ENNReal.toReal_nonneg, rfl⟩ | |
| theorem operatorModulus_weightedOperator (θ c : ℕ → ℝ) | |
| (hg : Measurable fun x => (weight θ c x : ℂ)) | |
| (hb : ∀ x, ‖(weight θ c x : ℂ)‖ ≤ 1) : | |
| operatorModulus (weightedOperator θ c) = | |
| multiplier θ (fun x => (weight θ c x : ℂ)) 1 (by norm_num) hg hb := by | |
| change operatorModulus (shiftOperator * | |
| multiplier θ (fun x => (weight θ c x : ℂ)) 1 (by norm_num) hg hb) = _ | |
| apply operatorModulus_shift_mul_multiplier | |
| intro x | |
| exact Complex.nonneg_iff.mpr ⟨ENNReal.toReal_nonneg, rfl⟩ | |
| end Invariant233.ProductModel.GlobalPolarSupport | |
| end | |
| end OAI | |