  We characterize the (essentially) decreasing sequences of positive numbers
$\beta$ = ($\beta$ n) for which all composition operators on H 2 ($\beta$) are
bounded, where H 2 ($\beta$) is the space of analytic functions f in the unit
disk such that $\infty$ n=0 |c n | 2 $\beta$ n < $\infty$ if f (z) = $\infty$
n=0 c n z n. We also give conditions for the boundedness when $\beta$ is not
assumed essentially decreasing.
