| import OAI.Analysis.IrrationalRotation.Model.Conjugation | |
| import Mathlib.Analysis.InnerProductSpace.Adjoint | |
| import OAI.Analysis.IrrationalRotation.Model.Analytic | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators | |
| namespace Invariant233 | |
| variable (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| abbrev HilbertConjugation := | |
| {C : H ≃ₗᵢ⋆[ℂ] H // Function.Involutive C} | |
| variable {H} | |
| def IsConjugation.toHilbertConjugation {C : H → H} (hC : IsConjugation C) : | |
| HilbertConjugation H := | |
| ⟨{ toFun := C | |
| invFun := C | |
| left_inv := hC.1 | |
| right_inv := hC.1 | |
| map_add' := hC.2.1 | |
| map_smul' := hC.2.2.1 | |
| norm_map' := hC.2.2.2 }, hC.1⟩ | |
| def HilbertConjugation.conjugate (C : HilbertConjugation H) (T : H →L[ℂ] H) : | |
| H →L[ℂ] H := | |
| (((C.val : H →L⋆[ℂ] H).comp T).comp (C.val : H →L⋆[ℂ] H)) | |
| variable [CompleteSpace H] | |
| def conjugationDefect (C : HilbertConjugation H) (T : H →L[ℂ] H) (m : ℕ) : | |
| H →L[ℂ] H := | |
| ∑ j ∈ Finset.range (m + 1), | |
| ((-1 : ℂ) ^ j * (m.choose j : ℂ)) • | |
| ((star T) ^ (m - j) * C.conjugate (T ^ (m - j))) | |
| def IsInfiniteConjugationIsometry (C : HilbertConjugation H) (T : H →L[ℂ] H) : Prop := | |
| Filter.Tendsto (fun m : ℕ => ‖conjugationDefect C T m‖ ^ (1 / (m : ℝ))) | |
| Filter.atTop (nhds 0) | |
| end Invariant233 | |
| end | |
| end OAI | |