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a97875e | 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 | import Mathlib
import OAI.Analysis.Commutator.DecomposableAlgebra
namespace OAI
noncomputable section
section
noncomputable section
open MeasureTheory Filter
namespace Kadison.Decomposable
variable {Ω H : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
[NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
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)
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]
end Kadison.Decomposable
end
end
end
end OAI
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