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

 

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Author: Lefèvre, Pascal
Li, Daniel
Queffélec, Hervé
Rodríguez Piazza, Luis
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2013-02-15
Published in: Journal of Functional Analysis, 264 (4), 895-919.
Document type: Article
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⁎).
Cite: Lefèvre, P., Li, D., Queffélec, H. y Rodríguez Piazza, L. (2013). Compact composition operators on the Dirichlet space and capacity of sets of contact points. Journal of Functional Analysis, 264 (4), 895-919.
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URI: http://hdl.handle.net/11441/46350

DOI: 10.1016/j.jfa.2012.12.004

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