File size: 3,471 Bytes
110f4f3 | 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | 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
|