unitarySelfAddISMul — Mathlib

English

The nn-norm is preserved by the inverse of a star-equivalence as well, giving equality of nnnorms for φ and φ⁻¹.

Русский

Обратный к звездному эквиваленту сохраняет nnnorm, следовательно, nnnorm(φ(a)) = nnnorm(a) и nnnorm(φ⁻¹(b)) = nnnorm(b).

LaTeX

$$$\mathrm{nnnorm}(\varphi(a)) = \mathrm{nnnorm}(a)$ and $\mathrm{nnnorm}(\varphi^{-1}(b)) = \mathrm{nnnorm}(b)$$$

Lean4

/-- For `a` selfadjoint with `‖a‖ ≤ 1`, this is the unitary `a + I • √(1 - a ^ 2)`. -/
@[simps]
noncomputable def unitarySelfAddISMul (a : selfAdjoint A) (ha_norm : ‖a‖ ≤ 1) : unitary A :=
  ⟨(a : A) + I • CFC.sqrt (1 - a ^ 2 : A), a.2.self_add_I_smul_cfcSqrt_sub_sq_mem_unitary _ ha_norm⟩

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