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8fadf68 | 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 | import Mathlib.Analysis.InnerProductSpace.Spectrum
namespace OAI
namespace Yau.Analysis
open Filter Set Module.End
noncomputable section
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
end OAI
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