Normal approximations of commuting square-summable matrix families

For any square-summable commuting family (A_i)_i∈I of complex n×n matrices there is a normal commuting family (B_i)_i no farther from it, in squared normalized ℓ² distance, than the diameter of the numerical range of ∑_iA_i^⁎A_i. Specializing in one direction (limiting case of the inequality for finite I) this recovers a result of M. Fraas: if ∑_i=1^ℓA_i^⁎A_i is a multiple of the identity for commuting A_i∈M_n(C) then the A_i are normal; specializing in another (singleton I) retrieves the well-known fact that close-to-isometric matrices are close to isometries.