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

Abstract

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 $w{C}_f:g\mapsto w\cdot g\circ f$, where $w:M\to \mathbb{C}$ and $f:M\to M$ are any map. When such an operator is bounded, it is actually the adjoint operator of a so-called weighted Lipschitz operator $w\hat{f}$ acting on the Lipschitz-free space $\mathcal{F}(M)$. In this note, we study the spectrum of such operators, with a special emphasize 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.