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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 47 48 49 50 | 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
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