  Let $T$ be a $C_0$--contraction on a separable Hilbert space. We assume that
$I_H-T^*T$ is compact. For a function $f$ holomorphic in the unit disk $\DD$
and continuous on $\bar\DD$, we show that $f(T)$ is compact if and only if $f$
vanishes on $\sigma (T)\cap \TT$, where $\sigma (T)$ is the spectrum of $T$ and
$\TT$ the unit circle. If $f$ is just a bounded holomorphic function on $\DD$
we prove that $f(T)$ is compact if and only if $\lim_{n\to \infty} T^nf(T) =0$.
