File size: 2,110 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 | import OAI.Analysis.FiniteFactor.Diffuse.OperatorSupport
namespace OAI
noncomputable section
open Filter
open scoped Topology BigOperators
namespace Nonhyperlinear.Diffuseness
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
lemma one_sub_star_mul_contraction (T:H→L[ℂ]H) (hT:‖T‖≤1) :
‖1-star T*T‖≤1 := by
apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
intro x
change ‖x-(star T) (T x)‖≤1*‖x‖
rw [one_mul]
have ht : ‖star T‖≤1 := by simpa [ContinuousLinearMap.star_eq_adjoint] using hT
have hb : ‖(star T) (T x)‖≤‖T x‖ := by
calc
_ ≤ ‖star T‖*‖T x‖ := (star T).le_opNorm _
_ ≤ 1*‖T x‖ := mul_le_mul_of_nonneg_right ht (norm_nonneg _)
_ = _ := one_mul _
have hi : RCLike.re (inner ℂ x ((star T) (T x)))=‖T x‖^2 := by
rw [ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_inner_right]
exact inner_self_eq_norm_sq (𝕜:=ℂ) _
have he := norm_sub_sq (𝕜:=ℂ) x ((star T) (T x))
rw [hi] at he
nlinarith [norm_nonneg (x-(star T) (T x)),norm_nonneg x,norm_nonneg (T x),
norm_nonneg ((star T) (T x))]
lemma rangeSupport_eq_one_sub_kernel (T:H→L[ℂ]H) :
rangeSupport T=1-(star T).ker.starProjection := by
have h := ContinuousLinearMap.orthogonal_ker (star T)
rw [←ContinuousLinearMap.star_eq_adjoint,star_star] at h
change T.range.topologicalClosure.starProjection = ContinuousLinearMap.id ℂ H - (star T).ker.starProjection
simpa only [h] using (Submodule.starProjection_orthogonal (star T).ker)
lemma cesaro_star_mul_tendsto (T:H→L[ℂ]H) (hT:‖T‖≤1) (x:H) :
Tendsto (fun n=>cesaro (1-star T*T) n x) atTop
(𝓝 (T.ker.starProjection x)) := by
have hstar : star (1-star T*T)=1-star T*T := by simp
have hker : (1-(1-star T*T)).ker=T.ker := by
rw [sub_sub_cancel]
exact ContinuousLinearMap.ker_adjoint_comp_self T
simpa only [hker] using cesaro_tendsto_fixedProjection (1-star T*T)
(one_sub_star_mul_contraction T hT) hstar x
end Nonhyperlinear.Diffuseness
end
end OAI
|