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dae740b | 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 44 45 | 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
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