Exceptional sets and Hilbert–Schmidt composition operators

It is shown that an analytic map ϕ of the unit disk into itself inducing a Hilbert–Schmidt composition operator on the Dirichlet space has the property that the set Eϕ={eiθ∈∂D : |ϕ(eiθ)|=1} has zero logarithmic capacity. We also show that this is no longer true for compact composition operators on the Dirichlet space. Moreover, such a condition is not even satisfied by Hilbert–Schmidt composition operators on the Hardy space.