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

Abstract

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⁎).