  For any real $\beta$ let $H^2_\beta$ be the Hardy-Sobolev space on the unit
disk $\D$. $H^2_\beta$ is a reproducing kernel Hilbert space and its
reproducing kernel is bounded when $\beta>1/2$. In this paper, we study
composition operators $C_\varphi$ on $H^2_\beta$ for $1/2<\beta<1$. Our main
result is that, for a non-constant analytic function $\varphi:\D\to\D$, the
operator $C_{\varphi }$ has dense range in $H_{\beta }^{2}$ if and only if the
polynomials are dense in a certain Dirichlet space of the domain $\varphi(\D)$.
It follows that if the range of $C_{\varphi }$ is dense in $H_{\beta }^{2}$,
then $\varphi $ is a weak-star generator of $H^{\infty}$. Note that this
conclusion is false for the classical Dirichlet space $\mathfrak{D}$. We also
characterize Fredholm composition operators on $H^{2}_{\beta }$.
