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

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