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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 29 30 31 32 33 34 | import OAI.Analysis.IrrationalRotation.Operators.Commutant
import OAI.Analysis.IrrationalRotation.Operators.CommutantProper
namespace OAI
namespace Invariant233
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
theorem commutant_algebra_properties (T : H →L[ℂ] H)
(hT : ¬ ∃ c : ℂ, T = c • (1 : H →L[ℂ] H)) :
operatorCommutant T ≠ ⊤ ∧ IsStronglyClosed (operatorCommutant T) ∧
(IsTransitive (operatorCommutant T) ↔ HasNoNontrivialHyperinvariantSubspace T) := by
exact ⟨centralizer_singleton_ne_top_of_nonscalar T hT,
operatorCommutant_stronglyClosed T, operatorCommutant_transitive_iff T⟩
theorem operatorCommutant_one_add (T : H →L[ℂ] H) :
operatorCommutant (1 + T) = operatorCommutant T :=
centralizer_singleton_one_add T
theorem isHyperinvariant_one_add_iff (T : H →L[ℂ] H) (L : Submodule ℂ H) :
IsHyperinvariant (1 + T) L ↔ IsHyperinvariant T L := by
unfold IsHyperinvariant
rw [operatorCommutant_one_add]
theorem hasNoNontrivialHyperinvariantSubspace_one_add_iff (T : H →L[ℂ] H) :
HasNoNontrivialHyperinvariantSubspace (1 + T) ↔
HasNoNontrivialHyperinvariantSubspace T := by
simp only [HasNoNontrivialHyperinvariantSubspace, isHyperinvariant_one_add_iff]
end Invariant233
end OAI
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