The paper is devoted to the following question: consider two self-adjoint$n\times n$-matrices $H_1,H_2$, $\|H_1\|\le 1$, $\|H_2\|\le 1$, such that theircommutator $[H_1,H_2]$ is small in some sence. Do there exist such self-adjointcommuting matrices $A_1,A_2$, such that $A_i$ is close to $H_i$, $i=1,2$? Theanswer to this question is positive if the smallness is considered with respectto the operator norm. The following result was established by Huaxin Lin: if$\|[H_1,H_2]\|=\delta$, then we can choose $A_i$ such that $\|H_i-A_i\|\leC(\delta)$, $i=1,2$, where $C(\delta)\to 0$ as $\delta\to 0$. Notice that$C(\delta)$ does not depend on $n$. The proof was simplified by Friis andR{\o}rdam. A quantitative version of the result with$C(\delta)=E(1/\delta)\delta^{1/5}$, where $E(x)$ grows slower than any powerof $x$, was recently established by Hastings. We are interested in the samequestion, but with respect to the normalized Hilbert-Schmidt norm. An analog ofLin's theorem for this norm was established by Hadwin and independently byFilonov and Safarov. A quantitative version with $C(\delta)=12\delta^{1/6}$,where $\delta=\|[H_1,H_2]\|_{\tr}$, was recently obtained by Glebsky. In thepresent paper, we use the same ideas to prove a similar result with$C(\delta)=2\delta^{1/4}$. We also refine Glebsky's theorem concerning the caseof $n$ operators.
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