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

Abstract

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