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Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

For any real $β$ let $H^2_β$ be the Hardy-Sobolev space on the unit disc $\mathbb{D}$. $H^2_β$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $β>1/2$. In this paper, we characterize that for a non-constant analytic function $φ:\mathbb{D}\to\mathbb{D}$, when the composition operator $C_{φ}$ on $H^{2}_{β}$ is Fredholm. For $1/2<β<1$, we also prove that $C_{φ}$ has dense range in $H_{β}^{2}$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $φ(\mathbb{D})$. It follows that if the range of $C_{φ}$ is dense in $H_{β}^{2}$, then $φ$ 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 $C_{φ}$ and the cyclic vector of the multiplier $M_φ^β.$