On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm

Mohit Bansil; Ilya Kachkovskiy

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arXiv · 2608.31000

On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm

Mohit Bansil·Ilya Kachkovskiy

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Abstract

In this paper, we consider the Rosenthal -- Halmos problem of almost commuting matrices with respect to the normalized Hilbert -- Schmidt norm $\|\cdot\|_{2,d}=d^{-1/2}\|\cdot\|_2$. We show that if $X$ and $Y$ are self-adjoint matrices with $\|X\|\le 1$, $\|Y\|\le 1$, then there exist commuting self-adjoint matrices $X',Y'$ such that $\|X-X'\|_{2,d}+\|Y-Y'\|_{2,d}\le 5\|[X,Y]\|_{2,d}^{1/3}$, and $[X,X']=0$. Within these constraints, the exponent $1/3$ cannot be improved.

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BibTeX RIS

Mohit Bansil, Ilya Kachkovskiy. 2026-08-31. On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm. https://arxiv.org/abs/2608.31000

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