| import OAI.Analysis.IrrationalRotation.Spectral.ProductFinitePolar |
| import OAI.Analysis.IrrationalRotation.Spectral.PolarFactor |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter |
|
|
| namespace Invariant233.ProductModel |
|
|
| theorem realMultiplier_nonneg |
| (θ : ℕ → ℝ) (f : Base → ℝ) (C : ℝ) (hC : 0 ≤ C) |
| (hf : Measurable f) (hb : ∀ x, ‖(f x : ℂ)‖ ≤ C) |
| (hpos : ∀ x, 0 ≤ f x) : |
| 0 ≤ multiplier θ (fun x => (f x : ℂ)) C hC hf.complex_ofReal hb := by |
| have hsqrtb : ∀ x, ‖(Real.sqrt (f x) : ℂ)‖ ≤ Real.sqrt C := by |
| intro x |
| rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (Real.sqrt_nonneg _)] |
| apply Real.sqrt_le_sqrt |
| simpa only [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (hpos x)] using hb x |
| let R := multiplier θ (fun x => (Real.sqrt (f x) : ℂ)) (Real.sqrt C) |
| (Real.sqrt_nonneg C) hf.sqrt.complex_ofReal hsqrtb |
| let B := multiplier θ (fun x => (f x : ℂ)) C hC hf.complex_ofReal hb |
| have hR : IsSelfAdjoint R := by |
| apply isSelfAdjoint_of_real_sequence_diagonal |
| (fun k x => Real.sqrt (f (rotationIterate θ k x))) R |
| intro v k |
| exact (multiplier_coordinate_ae θ _ _ _ _ hsqrtb v).mono (fun x hx => hx k) |
| have hRR : R * R = B := by |
| apply ContinuousLinearMap.ext |
| intro v |
| apply Lp.ext |
| filter_upwards [multiplier_coordinate_ae θ _ _ _ _ hsqrtb (R v), |
| multiplier_coordinate_ae θ _ _ _ _ hsqrtb v, |
| multiplier_coordinate_ae θ _ C hC hf.complex_ofReal hb v] |
| with x houter hinner hB |
| apply lp.ext |
| funext k |
| change R (R v) x k = B v x k |
| rw [houter, hinner, hB, ← mul_assoc, ← Complex.ofReal_mul, |
| Real.mul_self_sqrt (hpos _)] |
| change 0 ≤ B |
| rw [← hRR] |
| simpa only [hR.star_eq] using star_mul_self_nonneg R |
|
|
| theorem operatorModulus_shift_mul_realMultiplier |
| (θ : ℕ → ℝ) (f : Base → ℝ) (C : ℝ) (hC : 0 ≤ C) |
| (hf : Measurable f) (hb : ∀ x, ‖(f x : ℂ)‖ ≤ C) |
| (hpos : ∀ x, 0 ≤ f x) : |
| operatorModulus |
| (shiftOperator * multiplier θ (fun x => (f x : ℂ)) C hC hf.complex_ofReal hb) = |
| multiplier θ (fun x => (f x : ℂ)) C hC hf.complex_ofReal hb := by |
| let U : unitary (Hilbert →L[ℂ] Hilbert) := |
| ⟨shiftOperator, WeightedShift.fieldShift_mem_unitary (μ := baseMeasure) 1⟩ |
| exact PolarFactor.abs_unitary_mul_of_nonneg U _ (realMultiplier_nonneg θ f C hC hf hb hpos) |
|
|
| end Invariant233.ProductModel |
|
|
| end |
|
|
| end OAI |
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|
