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 , 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...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2013 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/46350 |
| Acceso en línea: | http://hdl.handle.net/11441/46350 https://doi.org/10.1016/j.jfa.2012.12.004 |
| Access Level: | acceso abierto |
| Palabra clave: | Bergman space Bergman-Orlicz space Composition operator Dirichlet space Hardy space Hardy-Orlicz space Logarithmic capacity Schatten classes |
| Sumario: | 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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