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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | 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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