instNonnegSpectrumClassComplexUnital

English

The cfcHom equals the elemental-subtype composition with continuousFunctionalCalculus.

Русский

cfcHom равен композиции элементарного подткана и непрерывного функционального калькулятора.

LaTeX

$$cfcHom ha = (StarAlgebra.elemental ℂ a).subtype.comp (continuousFunctionalCalculus a)$$

Lean4

instance instNonnegSpectrumClassComplexUnital : NonnegSpectrumClass ℂ A where
  quasispectrum_nonneg_of_nonneg a ha
    x := by
    rw [mem_quasispectrum_iff]
    refine (Or.elim · ge_of_eq fun hx ↦ ?_)
    obtain ⟨y, hy, rfl⟩ := (IsSelfAdjoint.of_nonneg ha).spectrumRestricts.algebraMap_image ▸ hx
    simpa using spectrum_nonneg_of_nonneg ha hy

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