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

  • Arafat Abbar,
  • Clément Coine,
  • Colin Petitjean

摘要

We consider weighted composition operators, that is operators of the type \(g \mapsto w \cdot g \circ f\) g w · g 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.