Journal of Operator Theory

Volume 94, Issue 1, Summer 2025 pp. 175-193.

A note on the spectrum of Lipschitz operators and composition operators on Lipschitz spaces

Authors: Arafat Abbar (1), Clement Coine (2), Colin Petitjean (3)
Author institution: (1) Universite Gustave Eiffel, Universite Paris Est Creteil, CNRS, F-77447, Marne-la-Vallee, France
(2) Normandie Universite, UNICAEN, CNRS, LMNO, 14000 Caen, France
(3) LAMA, Universite Gustave Eiffel, Universite Paris Est Creteil, CNRS, F-77447, Marne-la-Vallee, France

Summary: Fix a metric space $M$ and let $\mathrm{Lip}_0(M)$ be the Banach space of complex-valued Lipschitz functions defined on $M$. A weighted composition operator on $\mathrm{Lip}_0(M)$ is an operator of the kind $wC_f : g \mapsto w \cdot g \circ f$, where $w : M \to \mathbb{C}$ and $f: M \to M$ are any maps. When such an operator is bounded, it is actually the adjoint operator of a so-called weighted Lipschitz operator $w\widehat{f}$ acting on the Lipschitz-free space $\mathcal{F}(M)$. In this note, we study the spectrum of such operators, with a special emphasis when they are compact. Notably, we obtain a precise description in the non-weighted $w \equiv 1$ case: the spectrum is finite and made of roots of unity.

DOI: http://dx.doi.org/10.7900/jot.2023oct13.2467
Keywords: spectrum, point spectrum, compact operator, composition operator, Lipschitz-free space

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