File size: 2,775 Bytes
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 | import Mathlib.Analysis.VonNeumannAlgebra.Basic
namespace OAI
namespace Invariant233
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
[CompleteSpace H]
theorem hyperinvariant_projection_mem
(M : VonNeumannAlgebra H) {T p : H →L[ℂ] H}
(hT : T ∈ M) (hp : IsStarProjection p)
(hhyper : ∀ A : H →L[ℂ] H, Commute A T →
p.range ∈ Module.End.invtSubmodule A) :
p ∈ M ∧ (1 - p) * T * p = 0 := by
constructor
· apply (VonNeumannAlgebra.IsStarProjection.mem_iff hp M).2
intro A hA
apply hhyper A
exact (VonNeumannAlgebra.mem_commutant_iff.mp hA T hT).symm
· have h := (ContinuousLinearMap.IsIdempotentElem.range_mem_invtSubmodule_iff
hp.isIdempotentElem).mp
(hhyper T (Commute.refl T))
change p * T * p = T * p at h
rw [sub_mul, one_mul, sub_mul, h, sub_self]
theorem hyperinvariant_subspace_projection_mem
(M : VonNeumannAlgebra H) {T : H →L[ℂ] H}
(hT : T ∈ M) (S : Submodule ℂ H) [S.HasOrthogonalProjection]
(hhyper : ∀ A : H →L[ℂ] H, Commute A T →
S ∈ Module.End.invtSubmodule A) :
S.starProjection ∈ M ∧ (1 - S.starProjection) * T * S.starProjection = 0 := by
apply hyperinvariant_projection_mem M hT isStarProjection_starProjection
simpa only [Submodule.range_starProjection] using hhyper
theorem hyperinvariant_subspace_trivial
(M : VonNeumannAlgebra H) {T : H →L[ℂ] H} (hT : T ∈ M)
(htrivial : ∀ p : H →L[ℂ] H, p ∈ M → IsStarProjection p →
(1 - p) * T * p = 0 → p = 0 ∨ p = 1)
(S : Submodule ℂ H) [S.HasOrthogonalProjection]
(hhyper : ∀ A : H →L[ℂ] H, Commute A T →
S ∈ Module.End.invtSubmodule A) : S = ⊥ ∨ S = ⊤ := by
obtain ⟨hmem, hinv⟩ := hyperinvariant_subspace_projection_mem M hT S hhyper
rcases htrivial S.starProjection hmem isStarProjection_starProjection hinv with h0 | h1
· left
apply le_antisymm ?_ bot_le
intro x hx
change x = 0
have h := (S.starProjection_eq_self_iff).mpr hx
simpa only [h0, zero_apply] using h.symm
· right
apply Submodule.eq_top_iff'.mpr
intro x
have h := Submodule.starProjection_apply_mem S x
simpa only [h1, one_apply_eq_self] using h
theorem closed_hyperinvariant_subspace_trivial
(M : VonNeumannAlgebra H) {T : H →L[ℂ] H} (hT : T ∈ M)
(htrivial : ∀ p : H →L[ℂ] H, p ∈ M → IsStarProjection p →
(1 - p) * T * p = 0 → p = 0 ∨ p = 1)
(S : ClosedSubmodule ℂ H)
(hhyper : ∀ A : H →L[ℂ] H, Commute A T →
S.toSubmodule ∈ Module.End.invtSubmodule A) :
S.toSubmodule = ⊥ ∨ S.toSubmodule = ⊤ := by
exact hyperinvariant_subspace_trivial M hT htrivial S.toSubmodule hhyper
end Invariant233
end OAI
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