Normal approximations of commuting square-summable matrix families

Alexandru Chirvasitu

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arXiv · 2402.16559

Normal approximations of commuting square-summable matrix families

Alexandru Chirvasitu

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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.

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BibTeX RIS

Alexandru Chirvasitu. 2024-02-26. Normal approximations of commuting square-summable matrix families. https://arxiv.org/abs/2402.16559

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