Abstract
We study the compactness of the composition operator on de Branges–Rovnyak spaces. Inspired by a paper by Lyubarskii–Malinnikova on model spaces, we give some necessary and some sufficient conditions for compactness. In the paper of Lyubarskii-Malinnikova, the key point is some Bernstein inequality on model spaces due to Cohn (and based on a deep inequality of Axler–Chang–Sarason involving the Hardy–Littlewood maximal function). We generalize the result of Cohn to some subspace of a de Branges–Rovnyak space (in many cases dense) and then get a sufficient condition (analogue to Lyubarskii–Malinnikova’s condition) for compactness of the composition operator on that subspace.
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We would like to thank the anonymous referee for his/her remarks especially concerning the presentation of the paper.
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Emmanuel Fricain was supported by Labex CEMPI (ANR-11-LABX-0007-01) and the Grant ANR-17-CE40-0021 of the French National Research Agency ANR (Project Front). Javad Mashreghi was supported by Grants from NSERC (Canada)
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Fricain, E., Karaki, M. & Mashreghi, J. Composition Operators on de Branges–Rovnyak Spaces. Results Math 74, 61 (2019). https://doi.org/10.1007/s00025-019-0985-z
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DOI: https://doi.org/10.1007/s00025-019-0985-z