A Hilbert-Schmidt analog of Huaxin Lin's Theorem

Abstract: 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 their commutator $[H_1,H_2]$ is small in some sence. Do there exist such self-adjoint commuting matrices $A_1,A_2$, such that $A_i$ is close to $H_i$, $i=1,2$? The answer to this question is positive if the smallness is considered with respect to 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 $|H_i-A_i|\le C(\delta)$, $i=1,2$, where $C(\delta)\to 0$ as $\delta\to 0$. Notice that $C(\delta)$ does not depend on $n$. The proof was simplified by Friis and R{\o}rdam. A quantitative version of the result with $C(\delta)=E(1/\delta)\delta^{1/5}$, where $E(x)$ grows slower than any power of $x$, was recently established by Hastings. We are interested in the same question, but with respect to the normalized Hilbert-Schmidt norm. An analog of Lin's theorem for this norm was established by Hadwin and independently by Filonov 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 the present 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 case of $n$ operators.