Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4590908 | Journal of Functional Analysis | 2013 | 25 Pages |
Abstract
We prove several results about composition operators on the Dirichlet space D⁎. For every compact set K⊆∂D of logarithmic capacity , 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 . For every bounded composition operator Cφ on D⁎ and every ξ∈∂D, the logarithmic capacity of 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⁎).
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