| 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 | |