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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 | 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
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