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9eeb9c7 | 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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | 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
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