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9eeb9c7 | 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 | 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
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