English
For a real Banach algebra A, the real spectral radius is in the spectrum or its negative is in the spectrum.
Русский
Для вещественной банаховой алгебры A спектр радиуса either входит в спектр или его отрицание входит в спектр.
LaTeX
$$((spectralRadius Real a).toReal ∈ spectrum Real a) ∨ (-(spectralRadius Real a).toReal ∈ spectrum Real a)$$
Lean4
theorem _root_.NNReal.spectralRadius_mem_spectrum {A : Type*} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A]
{a : A} (ha : (spectrum ℝ a).Nonempty) (ha' : SpectrumRestricts a ContinuousMap.realToNNReal) :
(spectralRadius ℝ a).toNNReal ∈ spectrum ℝ≥0 a :=
by
obtain ⟨x, hx₁, hx₂⟩ := spectrum.exists_nnnorm_eq_spectralRadius_of_nonempty ha
rw [← hx₂, ENNReal.toNNReal_coe, ← spectrum.algebraMap_mem_iff ℝ, NNReal.algebraMap_eq_coe]
have : 0 ≤ x := ha'.rightInvOn hx₁ ▸ NNReal.zero_le_coe
convert hx₁
simpa
