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 $H_1,H_2$, $\Vert H_1\Vert \le 1$, $\Vert H_2\Vert \le 1$, 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 $\Vert [H_1,H_2]\Vert =\delta$, then we can choose $A_i$ such that $\Vert H_i-A_i\Vert \le C(\delta )$, $i=1,2$, 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 Rørdam. 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 $δ=\|[H_1,H_2]\|_{\tr}$, 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.

LaTeX, 5 pages. Important references added, and the case of $n$ operators is considered

References in corpus (3)

• Almost commuting matrices with respect to normalized Hilbert-Schmidt norm
• A Note on Approximate Liftings
• On the relation between an operator and its self-commutator