File size: 2,800 Bytes
f61e547 | 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 | import OAI.Probability.FreeIsing.Trees.WordAction
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.TreeKoopman
def centeredGenerator {d : ℕ} (c : Fin d) :
centeredL2 (labelMeasure d) →L[ℝ] centeredL2 (labelMeasure d) :=
(generatorOperator c).restrict
(koopmanL2_preserves_centered (reflection c) (reflection_measurePreserving c)
(reflection_involutive c))
@[simp] theorem centeredGenerator_apply_coe {d : ℕ} (c : Fin d)
(f : centeredL2 (labelMeasure d)) :
(centeredGenerator c f : Lp ℝ 2 (labelMeasure d)) =
generatorOperator c (f : Lp ℝ 2 (labelMeasure d)) := rfl
@[simp] theorem centeredGenerator_sq {d : ℕ} (c : Fin d) :
centeredGenerator c * centeredGenerator c = 1 := by
apply ContinuousLinearMap.ext
intro f
apply Subtype.ext
change generatorOperator c (generatorOperator c (f : Lp ℝ 2 (labelMeasure d))) = f
have h := congrArg (fun A : Lp ℝ 2 (labelMeasure d) →L[ℝ] Lp ℝ 2 (labelMeasure d) =>
A (f : Lp ℝ 2 (labelMeasure d))) (generatorOperator_sq c)
exact h
theorem centered_listProduct_apply_coe {d : ℕ} (w : List (Fin d))
(f : centeredL2 (labelMeasure d)) :
((w.map centeredGenerator).prod f : Lp ℝ 2 (labelMeasure d)) =
(w.map generatorOperator).prod (f : Lp ℝ 2 (labelMeasure d)) := by
induction w with
| nil => rfl
| cons c w ih =>
simp only [List.map_cons, List.prod_cons, mul_apply_eq_comp]
rw [centeredGenerator_apply_coe, ih]
theorem centered_wordOperator_apply_coe {d : ℕ} (v : TreeVertex d)
(f : centeredL2 (labelMeasure d)) :
(RadialWalk.wordOperator centeredGenerator v f : Lp ℝ 2 (labelMeasure d)) =
RadialWalk.wordOperator generatorOperator v (f : Lp ℝ 2 (labelMeasure d)) :=
centered_listProduct_apply_coe v.1 f
theorem centered_operatorAverage_eq {d : ℕ} :
RadialWalk.operatorAverage (centeredGenerator (d := d)) = centeredAverage d := by
apply ContinuousLinearMap.ext
intro f
apply Subtype.ext
rw [centeredAverage_apply_coe]
simp only [RadialWalk.operatorAverage, smul_apply, sum_apply,
Submodule.coe_smul, Submodule.coe_sum, centeredGenerator_apply_coe]
rw [← operatorAverage_eq, RadialWalk.operatorAverage_apply]
theorem centered_wordOperator_inner {d : ℕ} (v : TreeVertex d)
(f : centeredL2 (labelMeasure d)) :
inner ℝ f (RadialWalk.wordOperator centeredGenerator v f) =
inner ℝ (f : Lp ℝ 2 (labelMeasure d))
(RadialWalk.wordOperator generatorOperator v (f : Lp ℝ 2 (labelMeasure d))) := by
change inner ℝ (f : Lp ℝ 2 (labelMeasure d))
((RadialWalk.wordOperator centeredGenerator v f : centeredL2 (labelMeasure d)) :
Lp ℝ 2 (labelMeasure d)) = _
rw [centered_wordOperator_apply_coe]
end Problem367.TreeKoopman
end
end OAI
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