| import OAI.Geometry.NodalSets.Spectral.CompactPositiveMaxEigenvector |
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| namespace OAI |
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| namespace Yau.Analysis |
| open Set |
| noncomputable section |
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| 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⟩ |
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|
| end |
| end Yau.Analysis |
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| end OAI |
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