Compact composition operators on the Dirichlet space and capacity of sets of contact points

We prove several results about composition operators on the Dirichlet space D∗. For every compact set K ⊆ ∂D of logarithmic capacity CapK = 0, there exists a Schur function ϕ both in the disk algebra A(D) and in D∗ such that the composition operator Cϕ is in all Schatten classes Sp(D∗), p >0, and for which K = {eit; |ϕ(eit )| = 1} = {eit ; ϕ(eit ) = 1}. For every bounded composition operator Cϕ on D∗ and every ξ ∈ ∂D, the logarithmic capacity of {eit ; ϕ ∗ (eit ) = ξ } is 0. Every compact composition operator Cϕ on D∗ is compact on BΨ2 and on HΨ2 ; in particular, Cϕ is in every Schatten class Sp, p > 0, both on H2 and on B2. There exists a Schur function ϕ such that Cϕ is compact on HΨ2 , but which is not even bounded on D∗. There exists a Schur function ϕ such that Cϕ is compact on D∗, but in no Schatten class Sp(D∗). © 2012 Elsevier Inc. All rights reserved