Fix a metric space M and let Lip0(M) be the Banach space of
complex-valued Lipschitz functions defined on M. A weighted composition
operator on Lip0(M) is an operator of the kind wCf:g↦w⋅g∘f, where w:M→C and f:M→M are any map.
When such an operator is bounded, it is actually the adjoint operator of a
so-called weighted Lipschitz operator wf acting on the
Lipschitz-free space 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≡1 case: the
spectrum is finite and made of roots of unity