openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /HyperinvariantProjection.lean
| 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 | |