Normal approximations of commuting square-summable matrix families

For any square-summable commuting family (Aᵢ)i∈ I 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 ∑ᵢ₌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.