lean/OAI/Analysis/BackwardIntertwiners/HilbertClassification.lean

import OAI.Analysis.BackwardIntertwiners.ModelProperties

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 MainClaim : Prop :=
  ∃ T : H →L[ℂ] H, T ≠ 0 ∧
    Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) ∧
    TransitiveCommutant T


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 Hyperinvariant233
open Submodule TopologicalSpace
universe u v
variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
  [CompleteSpace H] [SeparableSpace H]


theorem orthonormal_subtype_countable (w : Set H)
    (hw : Orthonormal ℂ (Subtype.val : w → H)) : Countable w := by
  classical
  let : DiscreteTopology w := discreteTopology_iff_isOpen_singleton.mpr (by
    intro i
    have ho : IsOpen {j : w | (1 / 2 : ℝ) < ‖inner ℂ (i : H) (j : H)‖} :=
      isOpen_lt continuous_const (continuous_const.inner continuous_subtype_val).norm
    have he : {j : w | (1 / 2 : ℝ) < ‖inner ℂ (i : H) (j : H)‖} = {i} := by
      ext j
      rw [Set.mem_ofPred_eq, orthonormal_iff_ite.mp hw i j]
      by_cases hij : i = j <;> norm_num [hij, eq_comm]
    rwa [he] at ho)
  exact separableSpace_iff_countable.mp inferInstance



theorem exists_nat_hilbertBasis (hInf : ¬FiniteDimensional ℂ H) :
    Nonempty (HilbertBasis ℕ ℂ H) := by
  classical
  obtain ⟨w, b, hb⟩ := exists_hilbertBasis ℂ H
  have hw : Orthonormal ℂ (Subtype.val : w → H) := by
    simpa only [hb] using b.orthonormal
  let : Countable w := orthonormal_subtype_countable w hw
  let : Infinite w := by
    refine not_finite_iff_infinite.mp ?_
    intro hfin
    let : Finite w := hfin
    let : Fintype w := Fintype.ofFinite w
    exact hInf (Module.Finite.of_basis b.toOrthonormalBasis.toBasis)
  let e : ℕ ≃ w := Classical.choice (inferInstance : Nonempty (ℕ ≃ w))
  have ho : Orthonormal ℂ (b ∘ e) := b.orthonormal.comp e e.injective
  have hr : Set.range (b ∘ e) = Set.range b := by
    ext x
    constructor
    · rintro ⟨n, rfl⟩
      exact ⟨e n, rfl⟩
    · rintro ⟨i, rfl⟩
      exact ⟨e.symm i, by simp⟩
  exact ⟨HilbertBasis.mk ho (by rw [hr, b.dense_span])⟩


theorem nonempty_linearIsometryEquiv
    {K : Type v} [NormedAddCommGroup K] [InnerProductSpace ℂ K]
    [CompleteSpace K] [SeparableSpace K]
    (hH : ¬FiniteDimensional ℂ H) (hK : ¬FiniteDimensional ℂ K) :
    Nonempty (H ≃ₗᵢ[ℂ] K) := by
  obtain ⟨bH⟩ := exists_nat_hilbertBasis hH
  obtain ⟨bK⟩ := exists_nat_hilbertBasis hK
  exact ⟨bH.repr.trans bK.repr.symm⟩

end Hyperinvariant233

end OAI