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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 | 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
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