File size: 3,181 Bytes
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 | import OAI.Analysis.IrrationalRotation.Operators.FieldOperator
import OAI.Analysis.IrrationalRotation.Analysis.StrongFieldMeasurability
namespace OAI
noncomputable section
open MeasureTheory Filter TopologicalSpace
open scoped ENNReal Topology
namespace Invariant233.DirectIntegral
variable {X E : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ]
[NormedAddCommGroup E] [NormedSpace ℂ E] [SeparableSpace E]
{A : X → E →L[ℂ] E} {C : ℝ} {hA : ∀ x, ‖A x‖ ≤ C}
{hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ}
theorem ae_isIdempotentElem_of_fieldOperator (hC : 0 ≤ C)
(hP : IsIdempotentElem (fieldOperator A C hA hm)) :
∀ᵐ x ∂μ, IsIdempotentElem (A x) := by
have hAA : ∀ x, ‖A x * A x‖ ≤ C * C := fun x =>
(norm_mul_le _ _).trans (mul_le_mul (hA x) (hA x) (norm_nonneg _) hC)
have hmAA : ∀ v, AEStronglyMeasurable (fun x => (A x * A x) v) μ := fun v =>
apply_aestronglyMeasurable A hm (hm v)
have hmul : fieldOperator (fun x => A x * A x) (C * C) hAA hmAA =
fieldOperator A C hA hm * fieldOperator A C hA hm :=
fieldOperator_comp_of_ae_eq (Filter.Eventually.of_forall fun x => rfl)
have hfield := fieldOperator_ae_eq_of_eq (hmul.trans hP.eq)
exact hfield
end Invariant233.DirectIntegral
namespace Invariant233.DirectIntegral
variable {X E ι : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ]
[NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]
[SeparableSpace E] [Countable ι]
{A : X → E →L[ℂ] E} {C : ℝ} {hA : ∀ x, ‖A x‖ ≤ C}
theorem ae_isSelfAdjoint_of_fieldOperator (b : HilbertBasis ι ℂ E)
(hm : ∀ v, StronglyMeasurable (fun x => A x v))
(hP : IsSelfAdjoint (fieldOperator (μ := μ) A C hA (fun v => (hm v).aestronglyMeasurable))) :
∀ᵐ x ∂μ, IsSelfAdjoint (A x) := by
have hB : ∀ x, ‖star (A x)‖ ≤ C := by
intro x
rw [ContinuousLinearMap.star_eq_adjoint, LinearIsometryEquiv.norm_map]
exact hA x
have hn : ∀ v, AEStronglyMeasurable (fun x => star (A x) v) μ := by
intro v
simpa only [ContinuousLinearMap.star_eq_adjoint] using
(StrongField.stronglyMeasurable_adjoint_apply_of_columns b A
(fun i => hm (b i)) v).aestronglyMeasurable
have hglobal : fieldOperator (fun x => star (A x)) C hB hn =
star (fieldOperator (μ := μ) A C hA (fun v => (hm v).aestronglyMeasurable)) := by
rw [ContinuousLinearMap.star_eq_adjoint]
exact fieldOperator_adjoint_of_ae_eq (Filter.Eventually.of_forall fun x =>
ContinuousLinearMap.star_eq_adjoint (A x))
exact fieldOperator_ae_eq_of_eq (hglobal.trans hP.star_eq)
theorem ae_isStarProjection_of_fieldOperator (b : HilbertBasis ι ℂ E) (hC : 0 ≤ C)
(hm : ∀ v, StronglyMeasurable (fun x => A x v))
(hP : IsStarProjection (fieldOperator (μ := μ) A C hA (fun v => (hm v).aestronglyMeasurable))) :
∀ᵐ x ∂μ, IsStarProjection (A x) := by
filter_upwards [ae_isIdempotentElem_of_fieldOperator hC hP.isIdempotentElem,
ae_isSelfAdjoint_of_fieldOperator b hm hP.isSelfAdjoint] with x hid hself
exact ⟨hid, hself⟩
end Invariant233.DirectIntegral
end
end OAI
|