| import Mathlib |
|
|
| namespace OAI |
|
|
| open MeasureTheory Filter |
| open scoped InnerProductSpace |
|
|
| namespace Invariant233.FieldTraceCyclic |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] |
|
|
| theorem hasSum_matrix_product (b : HilbertBasis ℤ ℂ E) |
| (A B : E →L[ℂ] E) (k : ℤ) : |
| HasSum (fun r : ℤ => inner ℂ (b k) (A (b r)) * inner ℂ (b r) (B (b k))) |
| (inner ℂ (b k) (A (B (b k)))) := by |
| simpa only [ContinuousLinearMap.adjoint_inner_left] using |
| b.hasSum_inner_mul_inner ((ContinuousLinearMap.adjoint A) (b k)) (B (b k)) |
|
|
| theorem summable_norm_matrix_product (b : HilbertBasis ℤ ℂ E) |
| (A B : E →L[ℂ] E) (k : ℤ) : |
| Summable (fun r : ℤ => |
| ‖inner ℂ (b k) (A (b r)) * inner ℂ (b r) (B (b k))‖) := |
| summable_norm_iff.mpr (hasSum_matrix_product b A B k).summable |
|
|
| theorem tsum_norm_matrix_product_le (b : HilbertBasis ℤ ℂ E) |
| (A B : E →L[ℂ] E) (k : ℤ) : |
| (∑' r : ℤ, ‖inner ℂ (b k) (A (b r)) * inner ℂ (b r) (B (b k))‖) ≤ |
| ‖A‖ * ‖B‖ := by |
| have hc := lp.tsum_mul_le_mul_norm' (p := 2) (q := 2) |
| (by simpa using Real.HolderConjugate.two_two) |
| (b.repr ((ContinuousLinearMap.adjoint A) (b k))) (b.repr (B (b k))) |
| simp only [HilbertBasis.repr_apply_apply, LinearIsometryEquiv.norm_map] at hc |
| calc |
| (∑' r : ℤ, ‖inner ℂ (b k) (A (b r)) * inner ℂ (b r) (B (b k))‖) = |
| ∑' r : ℤ, ‖inner ℂ (b r) ((ContinuousLinearMap.adjoint A) (b k))‖ * |
| ‖inner ℂ (b r) (B (b k))‖ := by |
| apply tsum_congr |
| intro r |
| rw [norm_mul, ← ContinuousLinearMap.adjoint_inner_left, norm_inner_symm] |
| _ ≤ ‖(ContinuousLinearMap.adjoint A) (b k)‖ * ‖B (b k)‖ := hc |
| _ ≤ (‖ContinuousLinearMap.adjoint A‖ * ‖b k‖) * (‖B‖ * ‖b k‖) := |
| mul_le_mul ((ContinuousLinearMap.adjoint A).le_opNorm (b k)) |
| (B.le_opNorm (b k)) (norm_nonneg _) (by positivity) |
| _ = ‖A‖ * ‖B‖ := by |
| rw [ContinuousLinearMap.adjoint.norm_map, b.orthonormal.norm_eq_one] |
| ring |
|
|
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] |
|
|
| theorem hasSum_integral_matrix_product (b : HilbertBasis ℤ ℂ E) |
| (A B : Ω → E →L[ℂ] E) |
| (hA : ∀ r : ℤ, AEStronglyMeasurable (fun x => A x (b r)) μ) |
| (hB : ∀ r : ℤ, AEStronglyMeasurable (fun x => B x (b r)) μ) |
| (CA CB : ℝ) (hCA : 0 ≤ CA) |
| (hAbound : ∀ᵐ x ∂μ, ‖A x‖ ≤ CA) (hBbound : ∀ᵐ x ∂μ, ‖B x‖ ≤ CB) |
| (k : ℤ) : |
| HasSum (fun r : ℤ => ∫ x, |
| inner ℂ (b k) (A x (b r)) * inner ℂ (b r) (B x (b k)) ∂μ) |
| (∫ x, inner ℂ (b k) (A x (B x (b k))) ∂μ) := by |
| let F : ℤ → Ω → ℂ := fun r x => |
| inner ℂ (b k) (A x (b r)) * inner ℂ (b r) (B x (b k)) |
| have hFm : ∀ r, AEStronglyMeasurable (F r) μ := by |
| intro r |
| exact (aestronglyMeasurable_const.inner (hA r)).mul |
| (aestronglyMeasurable_const.inner (hB k)) |
| have hsumInt : Integrable (fun x => ∑' r, ‖F r x‖) μ := by |
| apply (integrable_const (CA * CB)).mono' (AEStronglyMeasurable.tsum (fun r => (hFm r).norm)) |
| filter_upwards [hAbound, hBbound] with x hxA hxB |
| rw [Real.norm_of_nonneg (tsum_nonneg (fun r => norm_nonneg (F r x)))] |
| exact (tsum_norm_matrix_product_le b (A x) (B x) k).trans |
| (mul_le_mul hxA hxB (norm_nonneg _) hCA) |
| exact hasSum_integral_of_dominated_convergence |
| (fun r x => ‖F r x‖) hFm (fun _ => ae_of_all _ (fun _ => le_rfl)) |
| (ae_of_all _ (fun x => summable_norm_matrix_product b (A x) (B x) k)) |
| hsumInt (ae_of_all _ (fun x => hasSum_matrix_product b (A x) (B x) k)) |
|
|
| theorem integral_central_matrix_product_cyclic (b : HilbertBasis ℤ ℂ E) |
| (A B : Ω → E →L[ℂ] E) |
| (hA : ∀ r : ℤ, AEStronglyMeasurable (fun x => A x (b r)) μ) |
| (hB : ∀ r : ℤ, AEStronglyMeasurable (fun x => B x (b r)) μ) |
| (CA CB : ℝ) (hCA : 0 ≤ CA) (hCB : 0 ≤ CB) |
| (hAbound : ∀ᵐ x ∂μ, ‖A x‖ ≤ CA) (hBbound : ∀ᵐ x ∂μ, ‖B x‖ ≤ CB) |
| (σ : ℤ → Ω → Ω) (hσ : ∀ r, MeasurePreserving (σ r) μ μ) |
| (hcovA : ∀ r k l : ℤ, ∀ᵐ x ∂μ, |
| inner ℂ (b k) (A (σ r x) (b l)) = inner ℂ (b (k + r)) (A x (b (l + r)))) |
| (hcovB : ∀ r k l : ℤ, ∀ᵐ x ∂μ, |
| inner ℂ (b k) (B (σ r x) (b l)) = inner ℂ (b (k + r)) (B x (b (l + r)))) : |
| (∫ x, inner ℂ (b 0) (A x (B x (b 0))) ∂μ) = |
| ∫ x, inner ℂ (b 0) (B x (A x (b 0))) ∂μ := by |
| have hterm (r : ℤ) : |
| (∫ x, inner ℂ (b 0) (A x (b r)) * inner ℂ (b r) (B x (b 0)) ∂μ) = |
| ∫ x, inner ℂ (b 0) (B x (b (-r))) * inner ℂ (b (-r)) (A x (b 0)) ∂μ := by |
| let f : Ω → ℂ := fun x => |
| inner ℂ (b 0) (B x (b (-r))) * inner ℂ (b (-r)) (A x (b 0)) |
| have hfm : AEStronglyMeasurable f μ := |
| (aestronglyMeasurable_const.inner (hB (-r))).mul |
| (aestronglyMeasurable_const.inner (hA 0)) |
| have hm : AEStronglyMeasurable f (μ.map (σ r)) := by |
| rw [(hσ r).map_eq] |
| exact hfm |
| calc |
| (∫ x, inner ℂ (b 0) (A x (b r)) * inner ℂ (b r) (B x (b 0)) ∂μ) = |
| ∫ x, f (σ r x) ∂μ := by |
| apply integral_congr_ae |
| filter_upwards [hcovA r (-r) 0, hcovB r 0 (-r)] with x hxA hxB |
| simp only [neg_add_cancel, zero_add] at hxA hxB |
| dsimp [f] |
| rw [hxA, hxB] |
| exact mul_comm _ _ |
| _ = ∫ x, f x ∂μ := by |
| simpa only [(hσ r).map_eq] using |
| (integral_map (hσ r).measurable.aemeasurable hm).symm |
| rw [← (hasSum_integral_matrix_product b A B hA hB CA CB hCA hAbound hBbound 0).tsum_eq, |
| ← (hasSum_integral_matrix_product b B A hB hA CB CA hCB hBbound hAbound 0).tsum_eq] |
| calc |
| (∑' r : ℤ, ∫ x, inner ℂ (b 0) (A x (b r)) * inner ℂ (b r) (B x (b 0)) ∂μ) = |
| ∑' r : ℤ, ∫ x, inner ℂ (b 0) (B x (b (-r))) * inner ℂ (b (-r)) (A x (b 0)) ∂μ := |
| tsum_congr hterm |
| _ = ∑' r : ℤ, ∫ x, inner ℂ (b 0) (B x (b r)) * inner ℂ (b r) (A x (b 0)) ∂μ := |
| by |
| simpa only [Equiv.neg_apply] using |
| (Equiv.neg ℤ).tsum_eq (fun r : ℤ => |
| ∫ x, inner ℂ (b 0) (B x (b r)) * inner ℂ (b r) (A x (b 0)) ∂μ) |
| |
| end Invariant233.FieldTraceCyclic |
| |
| end OAI |
| |
