Let $A,B,\tilde A,\tilde B$ be positive operators on a complex separable Hilbert space, and suppose \begin{gather} \|A\| = 1 \hspace{1cm} \|A-\tilde A\|\ll 1 \\ \|B\| = 1 \hspace{1cm} \|B-\tilde B\|\ll 1 \\ [\tilde A,\tilde B]=0. \end{gather} Does this imply $\big\|[A,B]\big\| \ll 1$? Surely the answer is yes, but how can we prove/quantify this?
2026-03-30 07:08:52.1774854532
If $A$ and $B$ approximate a pair of commuting operators, then do $A$ and $B$ approximately commute?
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Yes, it does if the norm is sub-multiplicative, because then
$$ \begin{align} \|[A, B]\| &= \|AB-BA\|\\ &= \|AB-A\tilde{B}+A\tilde{B}-BA\|\\ &\le\|A\|\|B-\tilde{B}\|+\|A\tilde{B}-BA\|\\ &= \|A\|\|B-\tilde{B}\|+\|A\tilde{B}-\tilde{A}\tilde{B}+\tilde{A}\tilde{B}-BA\|\\ &\le \|A\|\|B-\tilde{B}\|+\|A-\tilde{A}\|\|\tilde{B}\|+\|\tilde{A}\tilde{B}-BA\|\\ &= \|A\|\|B-\tilde{B}\|+\|A-\tilde{A}\|\|\tilde{B}\|+\|\tilde{B}\tilde{A}-BA\|\\ &= \|A\|\|B-\tilde{B}\|+\|A-\tilde{A}\|\|\tilde{B}\|+\|\tilde{B}\tilde{A}-\tilde{B}A+\tilde{B}A-BA\|\\ &\le \|A\|\|B-\tilde{B}\|+\|A-\tilde{A}\|\|\tilde{B}\|+\|\tilde{B}\|\|\tilde{A}-A\|+\|\tilde{B}-B\|\|A\|\\ &=2\|A\|\|B-\tilde{B}\|+2\|A-\tilde{A}\|\|\tilde{B}\|. \end{align} $$