Abstract
Boundedness of weighted composition operators \(W_{u,\varphi }\) acting on the classical Dirichlet space \(\mathcal {D}\) as \(W_{u,\varphi }f= u\, (f\circ \varphi )\) is studied in terms of the multiplier space associated to the symbol \(\varphi \), i.e., \(\mathcal {M}(\varphi )=\{ u \in \mathcal D: W_{u,\varphi } \hbox { is bounded on } \mathcal D\}\). A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of \(W_{u,\varphi }\) in \(\mathcal {D}\) whenever \(\varphi \) is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen et al. (J. Funct. Anal. 265:1749–1777, 2013) to the context of the Dirichlet space.
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Acknowledgments
This work was initiated during a research visit of the first and the third authors to the Departamento de Análisis Matemático at Universidad Complutense de Madrid. They are grateful for the hospitality and the support of research grant MTM2013-42105-P. I. Chalendar also acknowledges support from the London Mathematical Society (under Scheme 2). In the first version of this manuscript, we proved Proposition 3.1 ourselves: we thank the referee for pointing out references [3] and [22], where the result is already proved by different methods.
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The authors are partially supported by Plan Nacional I+D grant no. MTM2013-42105-P.
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Chalendar, I., Gallardo-Gutiérrez, E.A. & Partington, J.R. Weighted composition operators on the Dirichlet space: boundedness and spectral properties. Math. Ann. 363, 1265–1279 (2015). https://doi.org/10.1007/s00208-015-1195-y
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DOI: https://doi.org/10.1007/s00208-015-1195-y