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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 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import OAI.Analysis.IrrationalRotation.Model.Commutant
namespace OAI
namespace Invariant233.Commutant
variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℂ H]
theorem mem_iff (T A : H →L[ℂ] H) :
A ∈ Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H)) ↔
T * A = A * T := by
simp [Subalgebra.mem_centralizer_iff]
theorem one_mem (T : H →L[ℂ] H) :
(1 : H →L[ℂ] H) ∈ Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H)) :=
(Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H))).one_mem
theorem isClosed_strong_commutant (T : H →L[ℂ] H) :
IsClosed {A : H →Lₚₜ[ℂ] H | ∀ x : H, T (A x) = A (T x)} := by
have h : ∀ x : H,
IsClosed {A : H →Lₚₜ[ℂ] H | T (A x) = A (T x)} := by
intro x
exact isClosed_eq (T.continuous.comp (continuous_eval_const x))
(continuous_eval_const (T x))
simpa only [Set.iInter_ofPred] using isClosed_iInter h
theorem commute_of_strong_limit {ι : Type*} {l : Filter ι} [l.NeBot]
(T A : H →L[ℂ] H) (B : ι → H →L[ℂ] H)
(hB : ∀ᶠ i in l, T * B i = B i * T)
(hlim : ∀ x : H, Filter.Tendsto (fun i => B i x) l (nhds (A x))) :
T * A = A * T := by
ext x
have h₁ := T.continuous.continuousAt.tendsto.comp (hlim x)
have h₂ := hlim (T x)
apply tendsto_nhds_unique_of_eventuallyEq h₁ h₂
filter_upwards [hB] with i hi
exact congrArg (fun S : H →L[ℂ] H => S x) hi
end Invariant233.Commutant
namespace Invariant233
variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℂ H]
theorem operatorCommutant_mem_iff (T A : H →L[ℂ] H) :
A ∈ operatorCommutant T ↔ T * A = A * T :=
Commutant.mem_iff T A
theorem operatorCommutant_stronglyClosed (T : H →L[ℂ] H) :
IsStronglyClosed (operatorCommutant T) := by
unfold IsStronglyClosed
have heq : {A : H →Lₚₜ[ℂ] H | strongOperatorUnderlying A ∈ operatorCommutant T} =
{A : H →Lₚₜ[ℂ] H | ∀ x : H, T (A x) = A (T x)} := by
ext A
change strongOperatorUnderlying A ∈ operatorCommutant T ↔ _
exact (operatorCommutant_mem_iff T (strongOperatorUnderlying A)).trans
ContinuousLinearMap.ext_iff
rw [heq]
exact Commutant.isClosed_strong_commutant T
theorem operatorCommutant_transitive_iff (T : H →L[ℂ] H) :
IsTransitive (operatorCommutant T) ↔ HasNoNontrivialHyperinvariantSubspace T := by
constructor
· intro h L hL
exact h L hL.1 hL.2
· intro h L hclosed hinvariant
exact h L ⟨hclosed, hinvariant⟩
theorem IsHyperinvariant.isInvariant {T : H →L[ℂ] H} {L : Submodule ℂ H}
(h : IsHyperinvariant T L) : IsInvariant T L := by
apply h.2 T
rw [operatorCommutant_mem_iff]
end Invariant233
end OAI
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