| import Mathlib.Analysis.InnerProductSpace.Spectrum |
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| namespace OAI |
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| namespace Yau.Analysis |
| open Filter Set Module.End |
| noncomputable section |
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| theorem compact_positive_max_eigenvector {E : Type*} [NormedAddCommGroup E] |
| [InnerProductSpace ℝ E] [CompleteSpace E] [Nontrivial E] |
| (T : E →L[ℝ] E) (hc : IsCompactOperator T) (hs : T.IsSymmetric) |
| (hp : ∀ x, 0 ≤ inner ℝ (T x) x) (hT : T ≠ 0) : |
| ∃ v : E, ‖v‖=1 ∧ T v=‖T‖ • v ∧ |
| inner ℝ (T v) v=‖T‖ ∧ ∀ x : E, inner ℝ (T x) x ≤ ‖T‖*‖x‖^2 := by |
| have hn : 0 < ‖T‖ := norm_pos_iff.mpr hT |
| have hspec : ‖T‖ ∈ spectrum ℝ T ∨ -‖T‖ ∈ spectrum ℝ T := by |
| by_contra! h |
| have h1 : ‖T‖ ∈ resolventSet ℝ T := by simpa only [spectrum,mem_compl_iff,not_not] using h.1 |
| have h2 : -‖T‖ ∈ resolventSet ℝ T := by simpa only [spectrum,mem_compl_iff,not_not] using h.2 |
| obtain ⟨c,hc0,hc⟩ := T.abs_rayleighQuotient_le_of_norm_mem_resolventSet h1 h2 |
| have hb : (⨆ x, |T.rayleighQuotient x|) ≤ ‖T‖-c := ciSup_le hc |
| rw [← T.norm_eq_iSup_rayleighQuotient hs] at hb |
| linarith |
| have he : HasEigenvalue T.toLinearMap ‖T‖ := by |
| rcases hspec with h | h |
| · exact (hc.hasEigenvalue_iff_mem_spectrum hn.ne').mpr h |
| · obtain ⟨v,hv⟩ := ((hc.hasEigenvalue_iff_mem_spectrum (neg_ne_zero.mpr hn.ne')).mpr h).exists_hasEigenvector |
| have heq : T v = -‖T‖ • v := hv.apply_eq_smul |
| have hh := hp v |
| rw [heq,inner_smul_left_eq_smul,smul_eq_mul,real_inner_self_eq_norm_sq] at hh |
| have hvpos : 0 < ‖v‖^2 := sq_pos_of_pos (norm_pos_iff.mpr hv.2) |
| nlinarith |
| obtain ⟨v,hv⟩ := he.exists_hasEigenvector |
| have hvnorm : ‖v‖ ≠ 0 := norm_ne_zero_iff.mpr hv.2 |
| let u := ‖v‖⁻¹ • v |
| have hun : ‖u‖=1 := by |
| simp only [u,norm_smul,Real.norm_eq_abs,abs_inv,abs_norm,inv_mul_cancel₀ hvnorm] |
| have heq : T v=‖T‖ • v := hv.apply_eq_smul |
| have hue : T u=‖T‖ • u := by |
| dsimp [u] |
| rw [map_smul,heq,smul_comm] |
| refine ⟨u,hun,hue,?_,?_⟩ |
| · rw [hue,inner_smul_left_eq_smul,smul_eq_mul,real_inner_self_eq_norm_sq,hun] |
| ring |
| · intro x |
| calc |
| inner ℝ (T x) x ≤ ‖T x‖*‖x‖ := real_inner_le_norm _ _ |
| _ ≤ (‖T‖*‖x‖)*‖x‖ := mul_le_mul_of_nonneg_right (T.le_opNorm x) (norm_nonneg x) |
| _ = ‖T‖*‖x‖^2 := by ring |
|
|
| end |
| end Yau.Analysis |
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| end OAI |
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