Sequence of operators that commute imply the limit commutes?

Question

Given a sequence of compact operators $A_n\to A$ as $n\ \to \infty$ and $B$ (which has finite rank).

$\varphi \in L^2([a,b])$

If $A_nB\varphi = BA_n \varphi$

Am I able to say anything about $A$, i.e. does it also commute?

Answer 1

Yes. Simply note $$ \|AB \phi - BA \phi \| \leq \|AB \phi - A_n B \phi\| + \|A_n B\phi - BA\phi\| = \|AB \phi- A_n B \phi\| + \|B A_n \phi - BA\phi\|\leq \|B\phi\|\cdot \|A-A_n\| + \|B\| \|A_n \phi - A\phi\| $$ as $n\to \infty$.

Here, I assumed $A_n \to A$ in operator norm. But actually, an easy modification of the argument shows that it suffices to have $A_n \to A$ strongly.