| import OAI.Analysis.IrrationalRotation.Rotation.FieldDiagonal | |
| import OAI.Analysis.IrrationalRotation.Spectral.WeightedShiftPolar | |
| import OAI.Analysis.IrrationalRotation.Spectral.PolarFactor | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped ENNReal InnerProductSpace ComplexOrder | |
| namespace Invariant233.WeightedShift | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} | |
| theorem fieldDiagonal_nonneg (a : X → Weight) (C : ℝ) (hC : 0 ≤ C) | |
| (ha : ∀ x, ‖a x‖ ≤ C) | |
| (hm : ∀ k, AEStronglyMeasurable (fun x => a x k) μ) | |
| (hpos : ∀ᵐ x ∂μ, ∀ k, 0 ≤ a x k) : | |
| 0 ≤ fieldDiagonal a C hC ha hm := by | |
| apply ContinuousLinearMap.nonneg_iff_isPositive.mpr | |
| apply ContinuousLinearMap.isPositive_def.mpr | |
| constructor | |
| · intro f g | |
| simp only [L2.inner_def] | |
| apply integral_congr_ae | |
| filter_upwards [fieldDiagonal_ae a C hC ha hm f, | |
| fieldDiagonal_ae a C hC ha hm g, hpos] with x hf hg hx | |
| change inner ℂ ((fieldDiagonal a C hC ha hm f) x) (g x) = | |
| inner ℂ (f x) ((fieldDiagonal a C hC ha hm g) x) | |
| rw [hf, hg] | |
| exact ContinuousLinearMap.nonneg_iff_isPositive.mp | |
| (PolarSupport.diagonal_nonneg (a x) hx) |>.isSymmetric (f x) (g x) | |
| · intro f | |
| change 0 ≤ (inner ℂ (fieldDiagonal a C hC ha hm f) f).re | |
| rw [L2.inner_def] | |
| change 0 ≤ RCLike.re (∫ x, inner ℂ ((fieldDiagonal a C hC ha hm f) x) (f x) ∂μ) | |
| rw [← integral_re (L2.integrable_inner (𝕜 := ℂ) (fieldDiagonal a C hC ha hm f) f)] | |
| apply integral_nonneg_of_ae | |
| filter_upwards [fieldDiagonal_ae a C hC ha hm f, hpos] with x hf hx | |
| rw [hf] | |
| exact (ContinuousLinearMap.nonneg_iff_isPositive.mp | |
| (PolarSupport.diagonal_nonneg (a x) hx)).re_inner_nonneg_left (f x) | |
| theorem star_global_shift (n : ℤ) : | |
| star ((shift n).toLinearIsometry.toContinuousLinearMap.compLpL 2 μ) = | |
| (shift (-n)).toLinearIsometry.toContinuousLinearMap.compLpL 2 μ := by | |
| symm | |
| rw [ContinuousLinearMap.star_eq_adjoint] | |
| apply (ContinuousLinearMap.eq_adjoint_iff _ _).2 | |
| intro f g | |
| simp only [L2.inner_def] | |
| apply integral_congr_ae | |
| filter_upwards [fieldShift_ae (-n) f, fieldShift_ae n g] with x hf hg | |
| rw [hf, hg] | |
| have hinv : shift (-n) (f x) = (shift n).symm (f x) := by | |
| apply (shift n).injective | |
| ext k | |
| simp [shift_apply] | |
| rw [hinv] | |
| have h := (shift n).inner_map_map ((shift n).symm (f x)) (g x) | |
| simpa only [(shift n).apply_symm_apply] using h.symm | |
| theorem global_shift_mem_unitary (n : ℤ) : | |
| (shift n).toLinearIsometry.toContinuousLinearMap.compLpL 2 μ ∈ | |
| unitary (Lp Hilbert 2 μ →L[ℂ] Lp Hilbert 2 μ) := by | |
| apply Unitary.mem_iff.mpr | |
| rw [star_global_shift, ← fieldShift_add, ← fieldShift_add] | |
| simpa using And.intro (fieldShift_zero (μ := μ)) (fieldShift_zero (μ := μ)) | |
| local instance fieldDiagonalPolarRealModule : Module ℝ | |
| (Lp Hilbert 2 μ →L[ℂ] Lp Hilbert 2 μ) := | |
| Module.complexToReal (Lp Hilbert 2 μ →L[ℂ] Lp Hilbert 2 μ) | |
| local instance fieldDiagonalPolarFunctionalCalculus : NonUnitalContinuousFunctionalCalculus ℝ | |
| (Lp Hilbert 2 μ →L[ℂ] Lp Hilbert 2 μ) IsSelfAdjoint := by | |
| infer_instance | |
| theorem abs_global_shift_mul_fieldDiagonal (n : ℤ) (a : X → Weight) | |
| (C : ℝ) (hC : 0 ≤ C) (ha : ∀ x, ‖a x‖ ≤ C) | |
| (hm : ∀ k, AEStronglyMeasurable (fun x => a x k) μ) | |
| (hpos : ∀ᵐ x ∂μ, ∀ k, 0 ≤ a x k) : | |
| CFC.abs ((shift n).toLinearIsometry.toContinuousLinearMap.compLpL 2 μ * | |
| fieldDiagonal a C hC ha hm) = fieldDiagonal a C hC ha hm := | |
| PolarFactor.abs_unitary_mul_of_nonneg | |
| ⟨_, global_shift_mem_unitary n⟩ _ (fieldDiagonal_nonneg a C hC ha hm hpos) | |
| end Invariant233.WeightedShift | |
| end | |
| end OAI | |