  We generalize, on one hand, some results known for composition operators on
Hardy spaces to the case of Hardy-Orlicz spaces $H^\Psi$: construction of a
"slow" Blaschke product giving a non-compact composition operator on $H^\Psi$;
construction of a surjective symbol whose composition operator is compact on
$H^\Psi$ and, moreover, is in all the Schatten classes $S_p (H^2)$, $p > 0$. On
the other hand, we revisit the classical case of composition operators on
$H^2$, giving first a new, and simplier, characterization of closed range
composition operators, and then showing directly the equivalence of the two
characterizations of membership in the Schatten classes of Luecking and
Luecking and Zhu.
