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ddc84c5 | 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 | import OAI.Analysis.FiniteFactor.Diffuse.OperatorCesaro
namespace OAI
noncomputable section
open Filter
open scoped Topology BigOperators
namespace Nonhyperlinear.Diffuseness
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
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)])
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]
/-- 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]
end Nonhyperlinear.Diffuseness
end
end OAI
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