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 subset of partial derivative D of logarithmic capacity Cap K = 0, there exists a Schur function phi both in the disk algebra A (D) and in D-* such that the composition operator C-phi is in all Schatten classes S-p (D-*), p > 0, and for which K = {e(jt); vertical bar phi(e(it))vertical bar = 1} = {e(it); phi(e(it)) = 1). For every bounded composition operator C-phi on D-* and every xi is an element of partial derivative D, the logarithmic capacity of {e(it); phi(*)(e(it)) = xi} is 0. Every compact composition operator C-phi on D-* is compact on B-2(psi) and on H-2(psi); in particular, C-phi is in every Schatten class S-p, p > 0, both on H-2 and on B-2. There exists a Schur function phi such that C-phi is compact on H-2(psi), but which is not even bounded on D-*. There exists a Schur function phi such that C-phi is compact on D-*, but in no Schatten class S-p (D-*). (C) 2012 Elsevier Inc. All rights reserved.