林华新定理的Hilbert-Schmidt类比
A Hilbert-Schmidt analog of Huaxin Lin's Theorem
摘要：
本文研究了两个自伴$n\times n$ -矩阵$H_1,H_2$，$\|H_1\|\le 1$，$\|H_2\|\le 1$，使得它们的交换子$[H_1,H_2]$在某种意义下是小的.是否存在这样的自伴交换矩阵$A_1,A_2$，使得$A_i$接近于$H_i$，$i=1,2$？这个问题的答案是肯定的，如果考虑到小的运营商规范。Huaxin Lin建立了以下结果：如果是$\|[H_1,H_2]\|=δ$，那么我们可以选择$A_i$，使得$\|H_i-A_i\|\le C(δ)$，$i=1,2$，其中$C(δ)\to 0$为$δ\to 0$。请注意，$C(δ)$不依赖于$n$。这个证明被Friis和Rørdam简化。Hastings最近建立了$C(δ)=E(1/δ)δ^{1/5}$结果的定量版本，其中$E(x)$的生长速度比$x$的任何幂次都慢。我们感兴趣的是同一个问题，但关于规范化的希尔伯特-施密特范数。一个类似的林的定理，这个规范是由Hadwin和独立的Filonov和Safarov。Glebsky最近获得了$C(δ)=12δ^{1/6}$的定量版本，其中$δ=\|[H_1,H_2]\|_{\tr}$。在本文中，我们使用相同的思想，证明了类似的结果与$C(δ)=2δ^{1/4}$。我们还完善了Glebsky定理的$n$运营商的情况下。
英文摘要：
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 $\|[H_1,H_2]\|=δ$, then we can choose $A_i$ such that $\|H_i-A_i\|\le C(δ)$, $i=1,2$, where $C(δ)\to 0$ as $δ\to 0$. Notice that $C(δ)$ does not depend on $n$. The proof was simplified by Friis and Rørdam. A quantitative version of the result with $C(δ)=E(1/δ)δ^{1/5}$, 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 $C(δ)=12δ^{1/6}$, 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 $C(δ)=2δ^{1/4}$. We also refine Glebsky's theorem concerning the case of $n$ operators.
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