Characterization of weighted Hardy spaces on which all composition operators are bounded

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 ${\sum }_{n=0}^{\mathrm{\infty }}\mathrm{\mid }{a}_n{\mathrm{\mid }}^2{\beta }_n<+\mathrm{\infty }$ if $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$. 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.