According to Wikipedia, an operator is compact if it can be written in the form
$T(u)=\sum_{n=1}^\infty \lambda_n<f_n, u> g_n$, where $\{f_n\}$ and $\{g_n\}$ are orthonormal sets and $\lambda_n$ is a positive sequence with limit zero.
I understand how this is a necessary condition, but wouldn't we also need $\sum \lambda_n <\infty$ to guarantee that $T$ is compact? I am thinking of using $T_n$ defined by the parital sums is compact and using the fact that $\sum \lambda_n<\infty$ to show that $T_n\to T$. Otherwise, if $\lambda_n\to 0$ but $\sum \lambda_n$ diverges, it does not seem that this even has to be bounded.