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A Pre-adjoint Approach on Weighted Composition Operators Between Spaces of Lipschitz Functions

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Abstract

We consider weighted composition operators, that is operators of the type \(g \mapsto w \cdot g \circ f\), acting on spaces of Lipschitz functions. Bounded weighted composition operators, as well as some compact weighted composition operators, have been characterized quite recently. In this paper, we provide a different approach involving their pre-adjoint operators, namely the weighted Lipschitz operators acting on Lipschitz free spaces. This angle allows us to retrieve and sometimes improve some results from the literature. Notably, we obtain a distinct characterization of boundedness with a precise estimate of the norm. We also characterise injectivity, surjectivity, compactness and weak compactness in full generality.

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Funding

The third author was supported by the French ANR Project No. ANR-20-CE40-0006.

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Author notes

  1. Arafat Abbar and Colin Petitjean have contributed equally to this work.

Authors and Affiliations

  1. LAMA, CNRS, Univ Gustave Eiffel, Univ Paris Est Creteil, 77447, Marne-la-Vallée, France

    Arafat Abbar & Colin Petitjean

  2. UNICAEN, CNRS, LMNO, Normandie Univ, 14000, Caen, France

    Clément Coine

Authors
  1. Arafat Abbar
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  2. Clément Coine
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  3. Colin Petitjean
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Correspondence to Clément Coine.

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Abbar, A., Coine, C. & Petitjean, C. A Pre-adjoint Approach on Weighted Composition Operators Between Spaces of Lipschitz Functions. Results Math 79, 85 (2024). https://doi.org/10.1007/s00025-023-02115-x

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