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2fca478 | 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 | import Mathlib
namespace OAI
/-! Compact Alternative. -/
section
noncomputable section
namespace GlobalElliptic
variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℝ H] [CompleteSpace H]
lemma compact_one_sub_surjective (T : H →L[ℝ] H) (hT : IsCompactOperator T)
(hker : ∀ u : H, T u = u → u = 0) :
Function.Surjective (1 - T : H →L[ℝ] H) := by
have hn : ¬ Module.End.HasEigenvalue (T : Module.End ℝ H) 1 := by
intro hev
obtain ⟨u, hu⟩ := hev.exists_hasEigenvector
apply hu.2
apply hker u
have heq : T u = (1 : ℝ) • u := Module.End.HasEigenvector.apply_eq_smul hu
exact heq.trans (one_smul ℝ u)
have hr := (hT.hasEigenvalue_or_mem_resolventSet (by norm_num : (1 : ℝ) ≠ 0)).resolve_left hn
rw [spectrum.mem_resolventSet_iff] at hr
have hu : IsUnit (1 - T : H →L[ℝ] H) := by simpa only [map_one] using hr
exact (ContinuousLinearMap.isUnit_iff_bijective.mp hu).2
end GlobalElliptic
end
end
end OAI
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