adjoint_apply_eq_zero_iff — Mathlib

English

For a star-normal operator T, the condition adjoint T x = 0 is equivalent to T x = 0 for any x.

Русский

Для нормального по звёздочке оператора T условие adjoint T x = 0 эквивалентно T x = 0 для любого x.

LaTeX

$$$\\text{adjoint } T\\, x = 0 \\;\\iff\\; T x = 0$$$

Lean4

theorem adjoint_apply_eq_zero_iff (hT : IsStarNormal T) (x : E) : adjoint T x = 0 ↔ T x = 0 := by
  simp_rw [← norm_eq_zero (E := E), ← isStarNormal_iff_norm_eq_adjoint.mp hT]

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