exists_positive_complex_lift

Plain-English statement

Independently exact-science60 verified 0a77416f5ec38702c46ec1358966b9dd4846c8d3. Arbitrary measure real positive bounded D; actual compLpL scalar maps, norm-preserving embedding, full pointwise-conjugation fixed range, actual positive bounded complex lift with literal formula/intertwining/conjugation. Reused by actual PBPS consumer. Complex-root API test only; real-root preservation/descent/Gamma/B11/polar/fullH1/dynamics/main/error/cost/composition remain open.

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Lean statement

theorem exists_positive_complex_lift
    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
    (D : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 μ) (hD : D.IsPositive) :
    let ι : Lp ℝ 2 μ →L[ℝ] Lp ℂ 2 μ := Complex.ofRealCLM.compLpL 2 μ
    let R : Lp ℂ 2 μ →L[ℝ] Lp ℝ 2 μ := Complex.reCLM.compLpL 2 μ
    let Q : Lp ℂ 2 μ →L[ℝ] Lp ℝ 2 μ := Complex.imCLM.compLpL 2 μ
    let C : Lp ℂ 2 μ →L[ℝ] Lp ℂ 2 μ := Complex.conjCLE.toContinuousLinearMap.compLpL 2 μ
    (∀ u : Lp ℝ 2 μ, ‖ι u‖ = ‖u‖) ∧
    (∀ g : Lp ℂ 2 μ, C g = g ↔ ∃ u : Lp ℝ 2 μ, ι u = g) ∧
    ∃ Dc : Lp ℂ 2 μ →L[ℂ] Lp ℂ 2 μ,
      Dc.IsPositive ∧
      (∀ g : Lp ℂ 2 μ, Dc g = ι (D (R g)) + Complex.I • ι (D (Q g))) ∧
      (∀ u : Lp ℝ 2 μ, Dc (ι u) = ι (D u)) ∧
      (∀ g : Lp ℂ 2 μ, C (Dc g) = Dc (C g)) := by
  dsimp only
  refine ⟨norm_embed μ, fixed_range μ, liftComplex μ D, liftComplex_positive μ D hD,
    liftComplex_apply μ D, liftComplex_embed μ D, liftComplex_conjugate μ D⟩

end
end AutoSamplingTheory.TechnicalLemmas.Measure.L2RealComplexOperator

#print axioms AutoSamplingTheory.TechnicalLemmas.Measure.L2RealComplexOperator.exists_positive_complex_lift

Open AutoSamplingTheory/TechnicalLemmas/Measure/L2RealComplexOperator.lean:211published source at ed709a7f320d

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