eq_zero_of_neg — Mathlib

English

The spectrum-restrictive property is preserved under the sum of self-adjoint elements.

Русский

Сохранение ограниченности спектра сохраняется при сумме самопротиворечивых элементов.

LaTeX

$$SpectrumRestricts (a+b) ContinuousMap.realToNNReal$$

Lean4

theorem eq_zero_of_neg {a : A} (ha : IsSelfAdjoint a) (ha₁ : SpectrumRestricts a ContinuousMap.realToNNReal)
    (ha₂ : SpectrumRestricts (-a) ContinuousMap.realToNNReal) : a = 0 :=
  by
  nontriviality A
  rw [SpectrumRestricts.nnreal_iff] at ha₁ ha₂
  apply CFC.eq_zero_of_spectrum_subset_zero (R := ℝ) a
  rw [Set.subset_singleton_iff]
  simp only [← spectrum.neg_eq, Set.mem_neg] at ha₂
  peel ha₁ with x hx _
  linarith [ha₂ (-x) ((neg_neg x).symm ▸ hx)]

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