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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 44 45 46 | import OAI.Analysis.IrrationalRotation.Operators.Conjugation
import Mathlib.Analysis.InnerProductSpace.l2Space
namespace OAI
noncomputable section
namespace Invariant233
open scoped lp
variable {H ι : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
theorem exists_conjugation_fixing_hilbertBasis (b : HilbertBasis ι ℂ H) :
∃ C : HilbertConjugation H, ∀ i, C.val (b i) = b i := by
classical
let C : H → H := fun x => b.repr.symm (star (b.repr x))
have hC : IsConjugation C := by
refine ⟨fun x => ?_, fun x y => ?_, fun c x => ?_, fun x => ?_⟩
· simp [C]
· simp [C]
· simp [C]
· simp [C]
refine ⟨hC.toHilbertConjugation, fun i => ?_⟩
change b.repr.symm (star (b.repr (b i))) = b i
rw [b.repr_self]
have hs : star (lp.single 2 i (1 : ℂ) : ℓ²(ι, ℂ)) = lp.single 2 i (1 : ℂ) := by
ext j
by_cases hij : j = i
· subst j; simp [lp.single_apply]
· simp [lp.single_apply, Pi.single_eq_of_ne hij]
rw [hs, b.repr_symm_single]
theorem nonempty_hilbertConjugation [CompleteSpace H] : Nonempty (HilbertConjugation H) := by
obtain ⟨w, b, _⟩ := exists_hilbertBasis ℂ H
obtain ⟨C, _⟩ := exists_conjugation_fixing_hilbertBasis b
exact ⟨C⟩
theorem exists_isConjugation [CompleteSpace H] : ∃ C : H → H, IsConjugation C := by
obtain ⟨C⟩ := nonempty_hilbertConjugation (H := H)
exact ⟨C.val, C.isConjugation⟩
end Invariant233
end
end OAI
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