| import OAI.Analysis.FiniteFactor.Diffuse.OperatorCesaro |
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| namespace OAI |
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| noncomputable section |
| open Filter |
| open scoped Topology BigOperators |
| namespace Nonhyperlinear.Diffuseness |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| lemma cesaro_on_closedRange (S : H →L[ℂ] H) (hS : ‖S‖≤1) (x : H) |
| (hx : x∈(1-S).range.topologicalClosure) : |
| Tendsto (fun n=>cesaro S n x) atTop (𝓝 0) := by |
| apply Metric.tendsto_atTop.mpr |
| intro ε hε |
| have hcl : x∈closure ((1-S).range : Set H) := hx |
| obtain ⟨y,hy,hxy⟩ := Metric.mem_closure_iff.mp hcl (ε/2) (by positivity) |
| obtain ⟨z,rfl⟩ := hy |
| obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp (cesaro_on_range S hS z) (ε/2) (by positivity) |
| refine ⟨N,fun n hn=>?_⟩ |
| rw [dist_zero_right] |
| have he : cesaro S n x = cesaro S n (x-(1-S) z)+cesaro S n ((1-S) z) := by |
| rw [map_sub]; abel |
| rw [he] |
| have hh := hN n hn |
| rw [dist_zero_right] at hh |
| rw [dist_eq_norm] at hxy |
| change ‖x-(1-S) z‖<ε/2 at hxy |
| exact (norm_add_le _ _).trans_lt (by linarith [cesaro_apply_norm_le S hS n (x-(1-S) z)]) |
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| omit [CompleteSpace H] in |
| lemma cesaro_fixed [CompleteSpace H] (S : H →L[ℂ] H) (n : ℕ) (x : H) (hx : S x=x) : |
| cesaro S n x=x := by |
| have hp : ∀k:ℕ,(S^k) x=x := by |
| intro k |
| induction k with |
| | zero => simp |
| | succ k ih => rw [pow_succ',mul_apply_eq_comp,ih,hx] |
| simp only [cesaro,smul_apply,sum_apply,hp, |
| Finset.sum_const,Finset.card_range] |
| rw [← Nat.cast_smul_eq_nsmul ℂ,smul_smul,inv_mul_cancel₀ (by exact_mod_cast Nat.succ_ne_zero n),one_smul] |
|
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| /-- The Hilbert mean ergodic theorem for a selfadjoint contraction, proved |
| using telescoping on the dense range and the fixed-space orthogonal projection. -/ |
| theorem cesaro_tendsto_fixedProjection (S : H →L[ℂ] H) |
| (hS : ‖S‖≤1) (hstar : star S=S) (x : H) : |
| Tendsto (fun n=>cesaro S n x) atTop (𝓝 ((1-S).ker.starProjection x)) := by |
| let K := (1-S).ker |
| let : CompleteSpace K := (1-S).isClosed_ker.completeSpace_coe |
| have hP : S (K.starProjection x)=K.starProjection x := by |
| have h := K.starProjection_apply_mem x |
| change (K.starProjection x)-S (K.starProjection x)=0 at h |
| exact (sub_eq_zero.mp h).symm |
| have hk : Kᗮ=(1-S).range.topologicalClosure := by |
| have hh := ContinuousLinearMap.orthogonal_ker (1-S) |
| rw [←ContinuousLinearMap.star_eq_adjoint] at hh |
| simpa only [star_sub,star_one,hstar] using hh |
| have hr : x-K.starProjection x∈(1-S).range.topologicalClosure := by |
| rw [←hk] |
| exact K.sub_starProjection_mem_orthogonal x |
| have ht := (cesaro_on_closedRange S hS (x-K.starProjection x) hr).add_const (K.starProjection x) |
| simp only [zero_add] at ht |
| convert ht using 1 |
| funext n |
| rw [map_sub,cesaro_fixed S n _ hP,sub_add_cancel] |
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| end Nonhyperlinear.Diffuseness |
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| end |
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| end OAI |
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