| 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 | |