openAI-math-dataset / lean /OAI /Analysis /JiangSu /DimensionFunctions /FiniteSpectralIntegralRootBounds.lean
| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.DimensionFunctions.FiniteSpectralIntegralBounds | |
| import OAI.Analysis.JiangSu.CStarAlgebras.FiniteDimensionPowers | |
| namespace OAI | |
| noncomputable section | |
| namespace Pureness | |
| open TraceCone MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology | |
| variable {D : Type*} [NonUnitalCStarAlgebra D] [PartialOrder D] [StarOrderedRing D] | |
| theorem positive_root_cut_monotone (M : Positive D) (hM : ‖M.1‖ ≤ 1) (ε : ℝ≥0) : | |
| Monotone (fun k => Positive.cut ε (Positive.power (Positive.rootExponent k) M)) := by | |
| intro j k hjk | |
| change cfcₙ (fun t : ℝ => max (t-ε) 0) (M.1 ^ Positive.rootExponent j) ≤ | |
| cfcₙ (fun t : ℝ => max (t-ε) 0) (M.1 ^ Positive.rootExponent k) | |
| rw [CFC.nnrpow_eq_cfcₙ_real M.1 _ M.2,CFC.nnrpow_eq_cfcₙ_real M.1 _ M.2, | |
| ← cfcₙ_comp' (fun t : ℝ => max (t-ε) 0) (fun t : ℝ => t ^ (Positive.rootExponent j : ℝ)) M.1 | |
| (hf0 := Real.zero_rpow (show (Positive.rootExponent j : ℝ) ≠ 0 by exact_mod_cast (Positive.rootExponent_pos j).ne')), | |
| ← cfcₙ_comp' (fun t : ℝ => max (t-ε) 0) (fun t : ℝ => t ^ (Positive.rootExponent k : ℝ)) M.1 | |
| (hf0 := Real.zero_rpow (show (Positive.rootExponent k : ℝ) ≠ 0 by exact_mod_cast (Positive.rootExponent_pos k).ne'))] | |
| apply (cfcₙ_le_iff _ _ M.1 | |
| (hf0 := by simp [Real.zero_rpow (show (Positive.rootExponent j : ℝ) ≠ 0 by exact_mod_cast (Positive.rootExponent_pos j).ne')]) | |
| (hg0 := by simp [Real.zero_rpow (show (Positive.rootExponent k : ℝ) ≠ 0 by exact_mod_cast (Positive.rootExponent_pos k).ne')]) | |
| (ha := IsSelfAdjoint.of_nonneg M.2)).mpr | |
| intro t ht | |
| have ht0 := quasispectrum_nonneg_of_nonneg M.1 M.2 t ht | |
| have ht1 : t ≤ 1 := (le_abs_self t).trans | |
| ((NonUnitalIsometricContinuousFunctionalCalculus.norm_quasispectrum_le M.1 ht | |
| (IsSelfAdjoint.of_nonneg M.2)).trans hM) | |
| exact max_le_max (sub_le_sub_right (Real.rpow_le_rpow_of_exponent_ge' ht0 ht1 | |
| (Positive.rootExponent k).2 (Positive.rootExponent_antitone hjk)) ε) le_rfl | |
| theorem positive_cut_le_smul_root_cut (M : Positive D) (hM : ‖M.1‖ ≤ 1) | |
| (δ r α : ℝ≥0) (hr : 0 < r) (hα : (α : ℝ) < (δ : ℝ) ^ (r : ℝ)) : | |
| (Positive.cut δ M).1 ≤ (1/((δ : ℝ)^(r : ℝ)-α)) • | |
| (Positive.cut α (Positive.power r M)).1 := by | |
| change cfcₙ (fun t : ℝ => max (t-δ) 0) M.1 ≤ | |
| (1/((δ : ℝ)^(r : ℝ)-α)) • cfcₙ (fun t : ℝ => max (t-α) 0) (M.1 ^ r) | |
| rw [CFC.nnrpow_eq_cfcₙ_real M.1 r M.2, | |
| ← cfcₙ_comp' (fun t : ℝ => max (t-α) 0) (fun t : ℝ => t^(r : ℝ)) M.1 | |
| (hf0 := Real.zero_rpow (show (r : ℝ) ≠ 0 by exact_mod_cast hr.ne')), | |
| ← cfcₙ_smul (1/((δ : ℝ)^(r : ℝ)-α)) (fun t : ℝ => max (t^(r : ℝ)-α) 0) M.1 | |
| (h0 := by simp [Real.zero_rpow (show (r : ℝ) ≠ 0 by exact_mod_cast hr.ne')])] | |
| apply (cfcₙ_le_iff _ _ M.1 (hf0 := by simp) | |
| (hg0 := by simp [Real.zero_rpow (show (r : ℝ) ≠ 0 by exact_mod_cast hr.ne')]) | |
| (ha := IsSelfAdjoint.of_nonneg M.2)).mpr | |
| intro t ht | |
| change max (t-δ) 0 ≤ (1/((δ : ℝ)^(r : ℝ)-α))*max (t^(r : ℝ)-α) 0 | |
| by_cases htδ : t ≤ δ | |
| · rw [max_eq_right (sub_nonpos.mpr htδ)] | |
| exact mul_nonneg (one_div_nonneg.mpr (sub_pos.mpr hα).le) (le_max_right _ _) | |
| · have hpow : (δ : ℝ)^(r : ℝ) ≤ t^(r : ℝ) := | |
| Real.rpow_le_rpow δ.2 (le_of_not_ge htδ) r.2 | |
| have ht1 : t ≤ 1 := (le_abs_self t).trans | |
| ((NonUnitalIsometricContinuousFunctionalCalculus.norm_quasispectrum_le M.1 ht | |
| (IsSelfAdjoint.of_nonneg M.2)).trans hM) | |
| calc | |
| max (t-δ) 0 ≤ 1 := max_le ((sub_le_self t δ.2).trans ht1) zero_le_one | |
| _ = (1/((δ : ℝ)^(r : ℝ)-α))*((δ : ℝ)^(r : ℝ)-α) := by field_simp [(sub_pos.mpr hα).ne'] | |
| _ ≤ _ := mul_le_mul_of_nonneg_left | |
| ((sub_le_sub_right hpow α).trans (le_max_left _ _)) | |
| (one_div_nonneg.mpr (sub_pos.mpr hα).le) | |
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| lemma finiteSpectralIntegral_roots_monotone | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) (hM : ‖M.1‖ ≤ 1) : | |
| Monotone (fun k => finiteSpectralIntegral d n (Positive.power (Positive.rootExponent k) M)) := by | |
| intro j k hjk | |
| apply lintegral_mono | |
| intro t | |
| apply hd | |
| apply (finitePositiveClass_le_iff n _ _).mpr | |
| exact (cuntzSubequiv_iff_shared _ _).mpr | |
| (CuntzBelow.of_le _ _ (positive_root_cut_monotone M hM t.toNNReal hjk)) | |
| lemma finiteSpectralIntegral_root_le | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) (hM : ‖M.1‖ ≤ 1) (k : ℕ) : | |
| finiteSpectralIntegral d n (Positive.power (Positive.rootExponent k) M) ≤ | |
| d (finitePositiveClass n M) := by | |
| have h := finiteSpectralIntegral_le d hd n (Positive.power (Positive.rootExponent k) M) | |
| rw [finitePositiveClass_power n M _ (Positive.rootExponent_pos k)] at h | |
| exact h.trans (mul_le_of_le_one_right bot_le (by | |
| simpa using ENNReal.ofReal_le_ofReal (Positive.power_norm_le_one _ (Positive.rootExponent_pos k) M hM))) | |
| end Pureness | |
| end | |
| end OAI | |