Normal approximations of commuting square-summable matrix families

2024/02/26 by Alexandru Chirvăsitu, Chirvasitu, Alexandru
Mathematics · Computer Science · Economics, Econometrics and Finance · #Approximation Theory and Sequence Spaces #Matrix Theory and Algorithms #Impulse Buying and Technology Impacts

paper · pdf · doi:10.48550/arxiv.2402.16559

Abstract

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 ∑_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 ∑_i=1^ℓ A_i^* A_i is scalar 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.

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