File size: 3,295 Bytes
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 73 74 75 76 77 78 79 80 81 | import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.InnerProductSpace.Projection.Submodule
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Algebra.Ring.GeomSum
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Abel
namespace OAI
noncomputable section
open Filter
open scoped Topology BigOperators
namespace Nonhyperlinear.Diffuseness
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
/-- Cesàro means of the powers of a bounded operator. -/
def cesaro (S : H →L[ℂ] H) (n : ℕ) : H →L[ℂ] H :=
((n+1 : ℕ) : ℂ)⁻¹ • ∑ k ∈ Finset.range (n+1), S^k
omit [CompleteSpace H] in
lemma pow_norm_le_one [CompleteSpace H] (S : H →L[ℂ] H) (hS : ‖S‖≤1) (k : ℕ) : ‖S^k‖≤1 :=
by
induction k with
| zero =>
rw [pow_zero]
apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
intro x
simp
| succ k ih =>
rw [pow_succ]
exact (norm_mul_le _ _).trans (by nlinarith [norm_nonneg (S^k),norm_nonneg S])
lemma cesaro_norm_le_one (S : H →L[ℂ] H) (hS : ‖S‖≤1) (n : ℕ) : ‖cesaro S n‖≤1 := by
have hn : (0:ℝ)<(n+1:ℕ) := Nat.cast_pos.mpr (Nat.succ_pos n)
have hs : ‖∑ k ∈ Finset.range (n+1), S^k‖≤(n+1:ℕ) := by
calc
_ ≤ ∑ k ∈ Finset.range (n+1), ‖S^k‖ := norm_sum_le _ _
_ ≤ ∑ k ∈ Finset.range (n+1), (1:ℝ) := Finset.sum_le_sum (fun k _ => pow_norm_le_one S hS k)
_ = _ := by simp
rw [cesaro,norm_smul,norm_inv]
simp only [Complex.norm_natCast]
exact (mul_le_mul_of_nonneg_left hs (inv_nonneg.mpr hn.le)).trans_eq (inv_mul_cancel₀ hn.ne')
lemma cesaro_apply_norm_le (S : H →L[ℂ] H) (hS : ‖S‖≤1) (n : ℕ) (x : H) :
‖cesaro S n x‖≤‖x‖ :=
((cesaro S n).le_opNorm x).trans (by nlinarith [cesaro_norm_le_one S hS n,norm_nonneg x])
omit [CompleteSpace H] in
lemma cesaro_mul_sub [CompleteSpace H] (S : H →L[ℂ] H) (n : ℕ) :
cesaro S n*(1-S)=((n+1:ℕ):ℂ)⁻¹ • (1-S^(n+1)) := by
rw [cesaro,smul_mul_assoc,geom_sum_mul_neg]
lemma cesaro_range_bound (S : H →L[ℂ] H) (hS : ‖S‖≤1) (n : ℕ) (x : H) :
‖cesaro S n ((1-S) x)‖≤((n+1:ℕ):ℝ)⁻¹*(2*‖x‖) := by
have hn : (0:ℝ)<(n+1:ℕ) := Nat.cast_pos.mpr (Nat.succ_pos n)
have hp : ‖(S^(n+1)) x‖≤‖x‖ :=
((S^(n+1)).le_opNorm x).trans (by nlinarith [pow_norm_le_one S hS (n+1),norm_nonneg x])
have he := congrArg (fun T:H→L[ℂ]H => T x) (cesaro_mul_sub S n)
simp only [mul_apply_eq_comp,smul_apply] at he
rw [he,norm_smul,norm_inv]
simp only [Complex.norm_natCast]
apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hn.le)
change ‖x-(S^(n+1)) x‖≤_
exact (norm_sub_le _ _).trans (by linarith)
lemma cesaro_on_range (S : H →L[ℂ] H) (hS : ‖S‖≤1) (x : H) :
Tendsto (fun n => cesaro S n ((1-S) x)) atTop (𝓝 0) := by
apply tendsto_zero_iff_norm_tendsto_zero.mpr
have ht := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜:=ℝ)).mul_const (2*‖x‖)
simp only [zero_mul,one_div] at ht
apply squeeze_zero (fun n=>norm_nonneg _) (cesaro_range_bound S hS · x)
simpa only [Nat.cast_add,Nat.cast_one] using ht
end Nonhyperlinear.Diffuseness
end
end OAI
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