Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

Abstract

For any real $\beta$ let $H_{\beta }^2$ be the Hardy-Sobolev space on the unit disc $\mathbb{D}$. $H_{\beta }^2$ 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 $\phi :\mathbb{D}\to \mathbb{D}$, when the composition operator $C_{\phi }$ on $H_{\beta }^2$ is Fredholm. For $1/2<\beta <1$, we also prove that $C_{\phi }$ has dense range in $H_{\beta }^2$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $\phi (\mathbb{D})$. It follows that if the range of $C_{\phi }$ is dense in $H_{\beta }^2$, then $\phi$ 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 $C_{\phi }$ and the cyclic vector of the multiplier $M_{\phi }^{\beta }.$

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