| import Mathlib |
| import OAI.Analysis.Commutator.DecomposableAlgebra |
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| namespace OAI |
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| noncomputable section |
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| section |
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| noncomputable section |
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| open MeasureTheory Filter |
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| namespace Kadison.Decomposable |
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| variable {Ω H : Type*} [MeasurableSpace Ω] (μ : Measure Ω) |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| omit [CompleteSpace H] in |
| lemma add_field_bound (T S : Ω → H →L[ℂ] H) {b c : ℝ} |
| (hb : ∀ᵐ a ∂μ,‖T a‖≤b) (hc : ∀ᵐ a ∂μ,‖S a‖≤c) : |
| ∀ᵐ a ∂μ,‖T a+S a‖≤b+c := by |
| filter_upwards [hb,hc] with a ha hc |
| exact (norm_add_le _ _).trans (add_le_add ha hc) |
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| omit [CompleteSpace H] in |
| lemma operator_add (T S : Ω → H →L[ℂ] H) |
| (hT : ∀ ξ,AEStronglyMeasurable (fun a => T a ξ) μ) |
| (hS : ∀ ξ,AEStronglyMeasurable (fun a => S a ξ) μ) |
| {b c : ℝ} (hb : ∀ᵐ a ∂μ,‖T a‖≤b) (hc : ∀ᵐ a ∂μ,‖S a‖≤c) : |
| operator μ (fun a => T a+S a) (add_measurable μ T S hT hS) |
| (add_field_bound μ T S hb hc)=operator μ T hT hb+operator μ S hS hc := by |
| apply ContinuousLinearMap.ext |
| intro f |
| apply Lp.ext |
| filter_upwards [operator_ae μ (fun a => T a+S a) (add_measurable μ T S hT hS) |
| (add_field_bound μ T S hb hc) f,operator_ae μ T hT hb f,operator_ae μ S hS hc f, |
| Lp.coeFn_add (operator μ T hT hb f) (operator μ S hS hc f)] with a h1 h2 h3 h4 |
| change operator μ (fun a => T a+S a) (add_measurable μ T S hT hS) |
| (add_field_bound μ T S hb hc) f a=(operator μ T hT hb f+operator μ S hS hc f) a |
| rw [h1,h4] |
| change T a (f a)+S a (f a)=operator μ T hT hb f a+operator μ S hS hc f a |
| rw [h2,h3] |
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| end Kadison.Decomposable |
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| end |
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| end |
| end |
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| end OAI |
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