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