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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 | import OAI.Geometry.NodalSets.Spectral.CompactPositiveMaxEigenvector
namespace OAI
namespace Yau.Analysis
open Set
noncomputable section
theorem compact_positive_invariant_max {E : Type*} [NormedAddCommGroup E]
[InnerProductSpace ℝ E] [CompleteSpace E]
(T : E →L[ℝ] E) (hc : IsCompactOperator T) (hs : T.IsSymmetric)
(hp : ∀ x : E, x ≠ 0 → 0 < inner ℝ (T x) x)
(V : Submodule ℝ E) (hV : IsClosed (V : Set E))
(hInv : ∀ x ∈ V, T x ∈ V) (x : E) (hxV : x ∈ V) (hx : x ≠ 0) :
∃ mu > 0, ∃ v : E, v ∈ V ∧ ‖v‖=1 ∧ T v=mu • v ∧
∀ y ∈ V, inner ℝ (T y) y ≤ mu*‖y‖^2 := by
let : CompleteSpace V := hV.completeSpace_coe
let : Nontrivial V := ⟨⟨⟨x,hxV⟩,0,fun h ↦ hx (congrArg Subtype.val h)⟩⟩
let S : V →L[ℝ] V := T.restrict hInv
have hS : S ≠ 0 := by
intro hz
have hh : 0 < inner ℝ (S ⟨x,hxV⟩) (⟨x,hxV⟩ : V) := hp x hx
rw [hz,zero_apply,inner_zero_left] at hh
exact (lt_irrefl 0) hh
have hSp : ∀ v : V, 0 ≤ inner ℝ (S v) v := by
intro v
by_cases hz : (v : E)=0
· have hv : v=0 := Subtype.ext hz
simp only [hv,map_zero,inner_zero_left,le_refl]
· exact (hp v hz).le
obtain ⟨v,hvn,hve,hvi,hmax⟩ := compact_positive_max_eigenvector S
(hc.restrict' hInv) (hs.restrict_invariant hInv) hSp hS
have hnorm : (0 : ℝ) < ‖S‖ := lt_of_le_of_ne (norm_nonneg S)
(fun hz ↦ hS ((ContinuousLinearMap.opNorm_zero_iff S).mp hz.symm))
refine ⟨‖S‖,hnorm,(v : E),v.property,hvn,?_,?_⟩
· exact congrArg Subtype.val hve
· intro y hy
exact hmax ⟨y,hy⟩
end
end Yau.Analysis
end OAI
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