| 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 | |