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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 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 | import OAI.Analysis.IrrationalRotation.Model.ConjugationDefect
import Mathlib.Analysis.Normed.Operator.Mul
import Mathlib.Topology.Order.LiminfLimsup
namespace OAI
noncomputable section
namespace Invariant233
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
@[simp] theorem IsConjugation.toHilbertConjugation_apply {C : H → H}
(hC : IsConjugation C) (x : H) : hC.toHilbertConjugation.val x = C x := rfl
theorem HilbertConjugation.isConjugation (C : HilbertConjugation H) :
IsConjugation C.val := by
exact ⟨C.property, C.val.map_add, C.val.map_smul', C.val.norm_map⟩
namespace HilbertConjugation
@[simp] theorem conjugate_apply (C : HilbertConjugation H) (T : H →L[ℂ] H) (x : H) :
C.conjugate T x = C.val (T (C.val x)) := rfl
@[simp] theorem apply_apply (C : HilbertConjugation H) (x : H) :
C.val (C.val x) = x := C.property x
@[simp] theorem conjugate_one (C : HilbertConjugation H) :
C.conjugate (1 : H →L[ℂ] H) = 1 := by
ext x
simp
@[simp] theorem conjugate_add (C : HilbertConjugation H) (A B : H →L[ℂ] H) :
C.conjugate (A + B) = C.conjugate A + C.conjugate B := by
ext x
simp
@[simp] theorem conjugate_mul (C : HilbertConjugation H) (A B : H →L[ℂ] H) :
C.conjugate (A * B) = C.conjugate A * C.conjugate B := by
ext x
simp
@[simp] theorem conjugate_pow (C : HilbertConjugation H) (A : H →L[ℂ] H) (n : ℕ) :
C.conjugate (A ^ n) = (C.conjugate A) ^ n := by
induction n with
| zero => simp
| succ n ih => simp [pow_succ, ih]
@[simp] theorem conjugate_conjugate (C : HilbertConjugation H) (A : H →L[ℂ] H) :
C.conjugate (C.conjugate A) = A := by
ext x
simp
theorem norm_conjugate_le (C : HilbertConjugation H) (A : H →L[ℂ] H) :
‖C.conjugate A‖ ≤ ‖A‖ := by
apply ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _)
intro x
simpa only [conjugate_apply, LinearIsometryEquiv.norm_map] using A.le_opNorm (C.val x)
@[simp] theorem norm_conjugate (C : HilbertConjugation H) (A : H →L[ℂ] H) :
‖C.conjugate A‖ = ‖A‖ := by
apply le_antisymm (C.norm_conjugate_le A)
simpa only [conjugate_conjugate] using C.norm_conjugate_le (C.conjugate A)
theorem normQuasinilpotent_conjugate (C : HilbertConjugation H) {A : H →L[ℂ] H}
(hA : NormQuasinilpotent A) : NormQuasinilpotent (C.conjugate A) := by
simpa only [NormQuasinilpotent, ← conjugate_pow, norm_conjugate] using hA
end HilbertConjugation
theorem isInfiniteConjugationIsometry_iff_limsup [CompleteSpace H]
(C : HilbertConjugation H) (T : H →L[ℂ] H) :
IsInfiniteConjugationIsometry C T ↔
Filter.limsup (fun m : ℕ => ENNReal.ofReal
(‖conjugationDefect C T m‖ ^ (1 / (m : ℝ)))) Filter.atTop = 0 := by
constructor
· intro h
have h' := ENNReal.continuous_ofReal.continuousAt.tendsto.comp h
simpa only [Function.comp_def, ENNReal.ofReal_zero] using h'.limsup_eq
· intro h
have h' : Filter.Tendsto (fun m : ℕ => ENNReal.ofReal
(‖conjugationDefect C T m‖ ^ (1 / (m : ℝ)))) Filter.atTop (nhds 0) :=
tendsto_of_le_liminf_of_limsup_le bot_le h.le
have hreal := (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp h'
simpa only [IsInfiniteConjugationIsometry, Function.comp_def, ENNReal.toReal_zero,
ENNReal.toReal_ofReal (Real.rpow_nonneg (norm_nonneg _) _)] using hreal
end Invariant233
end
end OAI
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