Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

Abstract

For any real β let H 2 β be the Hardy-Sobolev space on the unit disk D. H 2 β is a reproducing kernel Hilbert space and its reproducing kernel is bounded when β > 1/2. In this paper, we study composition operators C ϕ on H 2 β for 1/2 < β < 1. Our main result is that, for a non-constant analytic function ϕ : D → D, the operator C ϕ has dense range in H 2 β if and only if the polynomials are dense in a certain Dirichlet space of the domain ϕ(D). It follows that if the range of C ϕ is dense in H 2 β , then ϕ is a weak-star generator of H ∞ . Note that this conclusion is false for the classical Dirichlet space D. We also characterize Fredholm composition operators on H 2 β . 2010 Mathematics Subject Classification. 47B33, 47A53.