| import Mathlib.Analysis.VonNeumannAlgebra.Basic | |
| import Mathlib.Analysis.SpecialFunctions.Pow.Real | |
| import Mathlib.Analysis.Normed.Module.FiniteDimension | |
| namespace OAI | |
| namespace Hyperinvariant | |
| open Filter | |
| open scoped Topology | |
| universe u | |
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| /-- The ordinary commutant in all bounded operators, not only in a chosen factor. -/ | |
| def commutant (T : H →L[ℂ] H) : Subalgebra ℂ (H →L[ℂ] H) := | |
| Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H)) | |
| /-- Closed common invariant subspaces, including all closed complex subspaces. -/ | |
| 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 = ⊤ | |
| /-- Pointwise norm convergence restricted to bounded linear operators. -/ | |
| @[instance_reducible] def strongOperatorTopology : TopologicalSpace (H →L[ℂ] H) := | |
| TopologicalSpace.induced (fun A : H →L[ℂ] H => (fun x : H => A x)) inferInstance | |
| /-- Nonzero, norm-quasinilpotent, and with no nontrivial closed hyperinvariant subspace. -/ | |
| def MainClaim : Prop := | |
| ∃ T : H →L[ℂ] H, T ≠ 0 ∧ | |
| Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) ∧ | |
| TransitiveCommutant T | |
| /-- The same witness has a proper strongly closed unital complex commutant. -/ | |
| 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 Hyperinvariant | |
| end OAI | |