Let $f:\mathbb{N} \to \mathbb{C}$ be a bounded function and let $$ M_f : l^2(\mathbb{N}) \to l^2(\mathbb{N}) \hspace{0.2in} (M_fu)(n):=f(n)u(n) $$ be the correspoding multiplication operator. Show that $M_f$ is compact iff $f$ is in $c_0(\mathbb{N}) = \{(a_n) \in l^{\infty}(\mathbb{Z})|a_n \to 0\}$
Another from my analaysis qual exam review sheet! Help is appreciated!!
First of all, note that $M_f$ is an infinite diagonal matrix.
If $f\in c_0(\mathbb N)$, show that $M_f$ can be approximated by finite-rank operators. Hence, it is compact. Conversely, if $M_f$ is compact, it is firstly clear that $f\in\ell^\infty(\mathbb N)$ since $M_f$ is bounded. Now, the spectrum of $M_f$ consists of the $f(n)$ (only). These are the eigenvalues of $M_f$. Now, apply what you know about the spectrum of compact operators (I hope you do).