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

Fix a metric space M and let Lip 0 (M) be the Banach space of complex-valued Lipschitz functions defined on M. A weighted composition operator on Lip 0 (M) is an operator of the kind wC f : 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 w f 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