  We give a complete characterization of the sequences $\beta = (\beta_n)$ of
positive numbers for which all composition operators on $H^2 (\beta)$ are
bounded, where $H^2 (\beta)$ is the space of analytic functions $f$ on the unit
disk ${\mathbb D}$ such that $\sum_{n = 0}^\infty |a_n|^2 \beta_n < + \infty$
if $f (z) = \sum_{n = 0}^\infty a_n z^n$. We prove that all composition
operators are bounded on $H^2 (\beta)$ if and only if $\beta$ is essentially
decreasing and slowly oscillating. We also prove that every automorphism of the
unit disk induces a bounded composition operator on $H^2 (\beta)$ if and only
if $\beta$ is slowly oscillating. We give applications of our results.
