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

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 maps. When such an operator is bounded, it is actually the adjoint operator of a so-called weighted Lipschitz operator wˆf acting on the Lipschitz-free space 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≡1 case: the spectrum is finite and made of roots of unity.