Normal approximations of commuting square-summable matrix families

For any square-summable commuting family (Aᵢ) ₈ ₈ of complex n n matrices there is a normal commuting family (Bᵢ) ᵢ no farther from it, in squared normalized ² distance, than the diameter of the numerical range of ᵢ Aᵢ^* Aᵢ. Specializing in one direction (limiting case of the inequality for finite I) this recovers a result of M. Fraas: if ₈=₁^ Aᵢ^* Aᵢ is scalar for commuting Aᵢ Mₙ (C) then the Aᵢ are normal; specializing in another (singleton I) retrieves the well-known fact that close-to-isometric matrices are close to isometries.