For any real $\beta$ let $H^2_\beta$ be the Hardy-Sobolev space on the unit disc $\mathbb{ D}$. $H^2_\beta$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $\beta>1/2$. In this paper, we characterize that for a non-constant analytic function $\varphi:\mathbb{ D}\to\mathbb{ D}$, when the composition operator $C_{ \varphi }$ on $H^{ 2}_{ \beta }$ is Fredholm. For $1/2<\beta<1$, we also prove that $C_{ \varphi }$ has dense range in $H_{ \beta }^{ 2}$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $\varphi(\mathbb{ D})$. It follows that if the range of $C_{ \varphi }$ is dense in $H_{ \beta }^{ 2}$, then $\varphi $ is a weak-star generator of $H^{ \infty}$, although the conclusion is false for the classical Dirichlet space $\mathfrak{ D}$. Moreover, we study the relation between the density of the rang of $C_{ \varphi }$ and the cyclic vector of the multiplier $M_{ \varphi}^{ \beta}.$
Categories: math.CV math.FA