Normal approximations of commuting square-summable matrix families

Abstract

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 $\ell^2$ 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 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.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Remark 1.1
• Theorem 1.2
• Corollary 1.3
• Proof 1
• Remark 1.4
• Theorem 1.6
• Remark 1.7