Normal approximations of commuting square-summable matrix families

Abstract

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