Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

Abstract

<abstract><p>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 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}) $ for $ 1/2 < \beta < 1 $. 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 range of $ C_{\varphi } $ and the cyclic vector of the multiplier $ M_{\varphi}^{\beta}. $</p></abstract>