| import Mathlib | |
| namespace OAI | |
| namespace Hyperinvariant233 | |
| open Filter | |
| open scoped Topology | |
| universe u | |
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| def commutant (T : H →L[ℂ] H) : Subalgebra ℂ (H →L[ℂ] H) := | |
| Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H)) | |
| def TransitiveCommutant (T : H →L[ℂ] H) : Prop := | |
| ∀ K : Submodule ℂ H, IsClosed (K : Set H) → | |
| (∀ A : H →L[ℂ] H, A * T = T * A → ∀ x ∈ K, A x ∈ K) → | |
| K = ⊥ ∨ K = ⊤ | |
| @[instance_reducible] | |
| def strongOperatorTopology : TopologicalSpace (H →L[ℂ] H) := | |
| TopologicalSpace.induced (fun A : H →L[ℂ] H => (fun x : H => A x)) inferInstance | |
| def FullClaim : Prop := | |
| ∃ T : H →L[ℂ] H, T ≠ 0 ∧ | |
| Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) ∧ | |
| TransitiveCommutant T ∧ commutant T ≠ ⊤ ∧ | |
| @IsClosed (H →L[ℂ] H) strongOperatorTopology (commutant T : Set (H →L[ℂ] H)) | |
| end Hyperinvariant233 | |
| namespace BackwardIntertwiners | |
| open Filter Topology MeasureTheory Set | |
| noncomputable section | |
| open scoped ENNReal | |
| theorem direct_algebra_corollary (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] [TopologicalSpace.SeparableSpace H] (hInf : ¬FiniteDimensional ℂ H) : | |
| Hyperinvariant233.FullClaim (H:=H) := by | |
| sorry | |
| end | |
| end BackwardIntertwiners | |
| end OAI | |