| 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 |
|
|
