Normal approximations of commuting square-summable matrix families

For any square-summable commuting family $(A_i{)}_{i\in I}$ of complex $n\times n$ matrices there is a normal commuting family $(B_i{)}_i$ no farther from it, in squared normalized ${\mathrm{\ell }}^2$ distance, than the diameter of the numerical range of ${\sum }_i{A}_i^{\ast }{A}_i$. Specializing in one direction (limiting case of the inequality for finite $I$) this recovers a result of M. Fraas: if ${\sum }_{i=1}^{\mathrm{\ell }}{A}_i^{\ast }{A}_i$ is scalar for commuting $A_i\in M_n(\mathbb{C})$ 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.