If $C$ is compact operator and $M_n$ converges pointwise, show that $CM_n$ converges uniformly.

Question

Suppose $C$ is a compact operator and $M_n$ is a sequence of bounded linear operators converging pointwise to another bounded linear operator $M$. Show that $||CM_n - CM|| \rightarrow 0$.

I know how to prove $||M_n C - MC|| \rightarrow 0$ (using Ascoli-Arzela theorem) but get stuck after changing the order of $M_n$ and $C$. Can anyone shed some light? All hints will be appreciated!

Answer 1

You cannot prove that from the assumptions, since $(CM_n)$ need not converge in norm.

Consider your favourite $p \in [1, +\infty)$ and let $X = \ell^p(\mathbb{N})$, $C$ the multiplication operator given by

$$(Cf)(k) = 2^{-k}f(k)$$

and $M_n$ the left shift by $n$ places,

$$(M_nf)(k) = f(k+n).$$

Then $\lVert M_n\rVert = 1$ for all $n$, and $M_nf \to 0$ for every $f \in X$. But $\lVert CM_n\rVert = \lVert C\rVert$ for all $n$, so $\lVert CM_n\rVert \not\to 0$.