| import OAI.Analysis.Hyperinvariant.Witness | |
| import OAI.Analysis.Hyperinvariant.HilbertClassification | |
| import OAI.Analysis.Hyperinvariant.Transport | |
| import OAI.Analysis.Hyperinvariant.CommutantTopology | |
| namespace OAI | |
| namespace Hyperinvariant | |
| universe u | |
| variable (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] [TopologicalSpace.SeparableSpace H] | |
| /-- Every infinite-dimensional separable complex Hilbert space admits a nonzero | |
| norm-quasinilpotent operator with a proper strongly closed transitive commutant. -/ | |
| theorem main_theorem (hInf : ¬FiniteDimensional ℂ H) : FullClaim (H := H) := by | |
| let : Nontrivial H := not_subsingleton_iff_nontrivial.mp (by | |
| intro hs | |
| let : Subsingleton H := hs | |
| exact hInf inferInstance) | |
| obtain ⟨e⟩ := nonempty_linearIsometryEquiv | |
| (H := FiniteFactor.SourceHilbert FiniteFactor.SourceCoordinate) | |
| (K := H) source_infiniteDimensional hInf | |
| exact fullClaim_of_mainClaim (mainClaim_transport e source_mainClaim) | |
| end Hyperinvariant | |
| end OAI | |